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Adaptive economizing in disequilibrium: Essays on economic dynamics

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Content ADAPTIVE ECONOMIZING IN DISEQUILIBRIUM: ESSAYS ON ECONOMIC DYNAMICS by Kenneth Allen Hanson A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Economics) May 1986 UMI Number: DP23338 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Dissertation Publishing UMI DP23338 Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CAUFORNIA 90089 This dissertation, written by Kenneth Allen Hanson ( ^ under the direction of h.J.$. Dissertation Committee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of re quirements for the degree of Ph.D. H ZSt 3 / &$£/. qg DO CTO R O F PH ILO SO PHY Deantaf Graduate Studies Date . . . . . . 1 . 7 . j., 1985 DISSERTATION COMMITTEE Chairperson A great Scientist without this high gift <of imagina tion) is impossible. I do not mean an imagination that goes into the vague and imagines things that do not exist; I mean one that does not abandon the actual soil of the earth, and steps to supposed and conjectured things by the standard of the real and the known. Then it may prove whether this or that supposition be possible, and whether it is not in contradiction with known laws. Goethe, J. Conversations with Eckermann Trs. John Oxenford, p. 280. ii ACKNOWLEDGEMENT To explore the unknown is a somewhat haphazard ad venture. The support of experienced guides is invalu able, for the youthful persistence to wander needs con trol, while at the same time freedom to err and discover. Somehow, I think, a balance has occurred with Richard Day as dissertation chairman. Others at the University of Southern California have contributed to the development of this work. Robert Kalaba has shared his experience in dynamic pro gramming. Michael Magill has been a guide through the rationality-equilibrium literature, allowing bounded rationality to be seen from the perspective of what it is not. Timur Kuran has in the final stages offered to participate in the procedure of dissertation examina tion and approval. To Gerry Nadler of the industrial engineering department, as outside member of the thesis committee, I thank, for what must be an arduous task. For the last year and a half, the Industriens Ut- redningsinstitut of Stockholm, under the supervision of Gunnar Eliasson has provided the opportunity to com plete the dissertation work within an environment where the bounded rationality-disequilibrium perspective is iii accepted tradition inherent in the Stockholm School of dynamics. Finally, there are the influences outside the aca demic world which provide the basis for all that is done. To these friends and family I thank as well. iv TABLE OF CONTENTS 1 INTRODUCTION: ADAPTIVE ECONOMIZING IN DISEQUILIBRIUM I ECONOMIC DYNAMICS OF BOUNDED RATIONALITY II ADAPTIVE ECONOMIZING IN A SEQUENCE OF TEMPORARY EQUILIBRIA i) Basic Model of Naive Temporal Optimizing a) Decision Operator b) Solution Structure and Dynamics ii) Stock Valuation III MARKET DISEQUILIBRIUM i) Firm Behavior with Multisector Interactions ii) Policy Analysis 2 COMPLICATED DYNAMICS IN A GENERALIZED COBWEB MODEL I INTRODUCTION II GENERALIZED COBWEB MODEL i) Model Structure ii) Solution Dynamics Page 1 8 23 33 35 III SOME ILLUSTRATIONS 45 i) Phase Zones in State Space: An Example of Phase Switching ii) Qualitative Modes of Dynamics: An Example of Chaos IV SUMMARY 59 3 SUBOPTIMIZING STRATEGIES IN A ROBINSON CRUSOE ECONOMY I INTRODUCTION 60 II BOUNDED RATIONALITY: CASE OF NAIVE TEMPORAL OPTIMIZING 63 i) A Renewable Resource in a Robinson Crusoe Economy ii) Comparative Dynamics: Biological Growth with Hunting III INTERTEMPORAL OPTIMIZATION 78 IV BOUNDED RATIONALITY: CASE OF PROXIMATE DYNAMIC PROGRAMMING 89 i) The Model ii) Imputing a Value to Stocks V SUMMARY 107 vi 4 A PROTOTYPICAL FIRM: A MICROFOUNDATION TO MACRODYNAMICS I INTRODUCTION 109 II FIRM ORGANIZATION AND BEHAVIOR 113 III MODEL i) ii) COMPONENTS Targeting of Top Management Pricing 120 iii) Investments and Capacity Expansion iv) Finance and Cash Flow Budgeting v) Production and Factor Inventory Control vi) Sales and Product Inventory and Order Control IV ALTERNATIVE COMPUTABLE FIRM MODELS 150 i) Price Taking in Temporary Equilibrium ii) Price Setting in Market Disequilibrium iii) Comparative Simulations with Multisector Interactions V SUMMARY 172 BIBLIOGRAPHY 174 vii LIST OF FIGURES Figure 1.1: Adaptive economizing Page 15 Figure 2.1: Data zones 46 Figure 2.2: Phase zones in state space 49 Figure 2.3: Chaotic cobweb supply dynamics 58 Figure 3.1: Temporal optimizing <M- = 2 . 2 5, f3=.3) 75 Figure 3.2: Temporal optimizing (m. = 2.25, P=.5) 76 Figure 3.3: Temporal optimizing (pl = 2.75, (3=.5) 77 Figure 3.4: Intertemporal optimizing <ix = 2.75, p=.5) 88 Figure 3.5: Proximate dynamic programming Figure 3.6: <|x = 2.75, p=.5) with pricing rule 42a and expectations 40 Proximate dynamic programming 102 Figure 3.7: (m- = 2.75, p=.5) with pricing rule 42b and expectations 40 Proximate dynamic programming <pi = 2.75, 103 Figure 3.8: p=.5) with pricing rule 42c and expectations 40 Proximate dynamic programming <m-=.75. 104 p=.5) with pricing rule 42a and expectations 41 105 viii LIST OF FIGURES (continued) Figure 3.9: Proximate dynamic programming (m-=.75, P=.5) with pricing rule 42b and expectations 41 106 Figure 4.1: A firm market feedback structure 112 Figure 4.2: Firm organizational structure 117 Figure 4.3: Sequential planning 119 Figure 4.4: Resource sector activities 168 Figure 4.5: Capital sector activities 169 Figure 4.6: Manufacturing sector activities 170 Figure 4.7: Aggregate economic activity 171 LIST OF TABLES Table 2.1: Phase Structures 43 Table 2.2: Phase Conditions 43 Table 2.3: Phase Activities 44 Table 4.1: Organizational Structure with planning Tasks 118 ix ABSTRACT The economizing behavior of agents, such as house holds and firms, and the market process of coordinating plans and mediating transactions are the microfounda tions of macroeconomic theory. It is generally agreed that a better understanding of these foundations is needed for more effective policy analysis. As Koopmans observed until we succeed in specifying fruitful as sumptions for behavior in an uncertain and changing economic environment, we shall continue to be groping for the proper tools of reasoning. In this work I explore some of these foundations from the perspective of an economy of boundedly ra tional agents. This requires an appropriate method of dynamic analysis, one which focusses on the long run implications of agent disequilibrium adjustment behav ior. In the introductory chapter adaptive economizing is presented as such a method. The basic modeling framework is presented and the illustrations and exten sions carried out in the chapters to follow are brought into context. In chapter two I use a version of adaptive econo mizing, involving constrained temporal optimizing and naive expectations to illustrate the potential for com x plicated dynamics in a sequence of temporary equilib ria . In chapter three I extend this approach while mak ing a comparison among intertemporal resource alloca tions in a Robinson Crusoe economy, using different be havioral rules for imputing a value to end of period stocks. A comparison is also made using temporal op timizing and dynamic programming. In chapter four I explore firm behavior in a dis equilibrium market environment. This involves specify ing organizational structure and planning process. With boundedly rational economizing all activities are not planned simultaneously, rather different departments plan different tasks coordinated through behavioral rules of decomposition. According to economic and tech nological conditions, firms in different sectors rely upon different planning routines. Some produce to order while others produce for stock. Some are price takers while others set price. I develop a prototypical firm model which incorporates these distinguishing charac teristics and which is flexible enough to represent these different types of firms. A set of firm-sector models are linked into a disequilibrium, multisector model of interindustry activity. xi CHAPTER 1 INTRODUCTION: ADAPTIVE ECONOMIZING IN DISEQUILIBRIUM I ECONOMIC DYNAMICS OF BOUNDED RATIONALITY Rational behavior is action designed for adapting to an anticipated future situation, as inferred from existing knowledge. How is a rational decision process to be represented? The answer depends in an essential way on a preconceived notion of the complexity of change occurring in the economic environment and the cognitive capacity of agents. Early psychological research directed towards critical analysis of the rationality assumption in eco nomics suggests that agents are boundedly rational (Simon, 1959; Katona, 1951). This is to claim they have limited foresight and computational ability, that er rors in expectations are the rule and not the excep tion, and that a sequential process of short term plan ning with behavioral rules of adjustment to 1 market-environment feedback is characteristic of plan ning procedures. Subsequent research in cognitive psychology and surveys on firm planning supports this hypothesis (Kahneman, Slovic and Tversky, 1982; Scott and Johnson, 1982). Both reveal that there is an essential impact of uncertainty on agent behavior and that few agents prac tice probabilistic methods of planning. Using the distinction that Knight (1921) made be tween risk and uncertainty, it is to be understood that boundedly rational agents plan in context of uncer tainty with imperfect information and simplified plan ning procedures, reflecting caution and flexibility. Knight as did Keynes recognized that 'The hypothesis of a calculable future leads to a wrong interpretation of the principles of behavior which the need for action compels us to take' (Keynes, 1937, p. 222). Viewing economic development as a process where the emergence of novelty is the main spring of economic growth and where the complexity of change is a reason for cautious behavior, it seems appropriate to assume that agents are boundedly rational, and allow 'the peo ple in one's models to not know what is going to hap pen, and know that they do not know just what is going to happen. As in History!' (Hicks, 1977, preface). 2 Since the founding of behavioral economics in the work of Hart <1940), Simon (1959), Cyert and March (1963) Modigliani and Cohen <1963) and others, consid erable effort has been put into developing models of boundedly rational economizing. Far less has been done in analyzing the dynamics of economic activity generic to such models, which is a central aim in this work. Some issues of dynamics have been addressed from the perspective of convergence to a stable equilibrium. For instance, there is the issue, anticipated by Alchian (1950), concerning the convergence of satisfic ing to marginalism, illustrated by Baumol and Quandt <1964), Day (1967), and Winter (1971). Similarly, there is the attempt to assess the extent to which myopic behavior approximates and under what conditions is equivalent to dynamic optimization (Day and Fan, 1976; Tesfatsion, 1980). In the current development of tempo rary equilibrium models analysis is focussed on the ex istence of a stable equilibrium with respect to the ex pectations function (Fuchs and Laroque, 1976). As noted by Grandmont (1977) there is a need to investigate fur ther the dynamic properties to a sequence of temporary equilibria. This is what I do here. To pursue the analysis of dynamics derived from economic models based upon principles of bounded ra- 3 tionality, an acceptable modeling framework is neces sary. This work illustrates adaptive economizing as such a framework. Historically the approach is based on the sequential process analysis of the Stockholm school, codified in Hicks (1939) method of temporary equilibrium and modified by Day (1963) with the behav ioral suboptimization of boundly rational agents. This later modification distinguishes the models developed here from the current practice in temporary equilibrium modeling as surveyed by Grandmont (1977). Consideration for boundedly rational economizing introduces a richer perspective to dynamic analysis as will be brought out below. See Eliasson (1976), Forrester (1961) and Day and Cigno (1978) for models reflecting this perspec tive . In developing a theory of dynamic analysis the Stockholm school recognized that it must be based on an investigation into the character of the planning of the economic subjects, founded upon assumptions regarding technical, institutional and psychological conditions (Lindahl, 1939). The mathematical formulation of a method of dynam ics was not carried out by members of the Stockholm school. It was Hicks (1939) who formalized a method by assuming markets in each period arrive at an equilib 4 rium prior to transactions, and errors in expectations with market feedback induce temporal disequilibrium in the sequence of temporary equilibria. The model has subsequently been modified in several ways. One direc tion of research has followed the suggestion of Clower (1965) developing into the nonwalrasian, quantity con strained temporary equilibrium model (Grandmont, 1977; Benassy, 1982). An alternative development is the re cursive programming model, which incorporates the be havioral suboptimization of boundedly rational agents (Day, 1971). It is from the perspective of complicated nonlinear dynamics derived from these later models that this work proceeds. Before going on to an introduction of the chapters to follow, a distinction needs to be made between two forms of disequilibrium. The distinction is methodo logical, simplifying analysis, for they are inseparable outcomes of bounded rationality. The traditional form of disequilibrium analysis starts with market clearing and considers temporal disequilibrium, the stability and convergence properties to a sequence of temporary market equilibria. Market disequilibrium may also arise in which cur rent period plans need not be realized. Consequently, disequilibrium buffer stock mechanisms and transaction 5 processes are integral parts of an economic model, which add complexity to model structure and analysis. Restricting analysis to temporal disequilibrium is a simplification allowing mathematical tools of dynamic analysis to be used. The further assumption of linearity allows closed form solutions to be derived. With nonlinearities and discontinuities the dynamics even in a temporary equilibrium framework become diffi cult to analyze. Keeping to simple models makes it pos sible to combine analysis with computer simulation, illustrating the complicated dynamics which can arise. This can prepare one for the counterintuitive results of models involving market disequilibrium where ana lytical tools are more limited. In the next section adaptive economizing is de scribed as a general modeling framework where agents are boundedly rational. Examples in the following two chapters illustrate this method in a sequence of tempo rary equilibria. First, using a generalized cobweb model where producer supply is influenced by financial feedback and a flexibility constraint on the use of working capital. Second, using a renewable resource management problem in a boundedly rational Robinson Crusoe economy where an imputed value to stocks influ ences their use. 6 In section three of this introduction the need for removing the temporary equilibrium condition of market clearing is discussed. In chapter four a firm model is developed to capture the essential ingredients of be havior in market disequilibrium. This requires speci fication of a prototypical firm's organizational struc ture and planning routine. Finally the prototypical firm model is distin guished into different types of firms which are united into a computable dynamic multisector model of economic activity where boundedly rational agents transact on markets which may be in disequilibrium. This is discussed in section three and illustrated in chapter four. The final section of the introduction suggests some policy implications of a dynamic disequilibrium modeling framework. 7 II ADAPTIVE ECONOMIZING IN A SEQUENCE OF TEMPORARY EQUILIBRIA The economizing behavior of a boundedly rational agent in this work involves temporal suboptimizing with behavioral rules of adjustment to market-environment feedback. Principles of behavior are embedded into an exogenous market environment. Through a sequence of temporary equilibria the agent and environment interact in a dynamic feedback process. Agent behavior is re ferred to as adaptive economizing. The dynamic agent-environment feedback process is referred to as a recursive programming model. With the later, simulation experiments of a modeled representation of a real time process can be performed, and the dynamics of temporal disequilibrium analyzed. Two versions of adaptive economizing in a sequence of temporary equilibria are described below and illus trated in chapters two and three. They can be distin guished in how stocks carried over from one period to another are treated. In the first, basic model, there is no imputation of current value to the future use of stocks, rather behavioral flexibility constraints are imposed. 8 For instance, firms maintain cash on hand as the outcome of a working capital constraint, not because they impute a value to money. In chapter two, a gener alized cobweb model of producer supply is used to il lustrate the complicated dynamics which may arise from a cash flexibility constraint and financial feedback. In chapter three, behavioral rules for imputing a current value to the future use of stocks in a tempo rary equilibrium framework are studied. This is done with a behavioral approximation to the principle of dy namic programming in a simple consumption-savings deci sion situation. i) Basic Model of Naive Temporal Optimizing Adaptive economizing in a sequence of temporary equilibria consists of three elementary processes, car ried out by a decision operator, data processor, and feedback generator. Each component can be treated sepa rately though they are interdependent in a dynamic agent-environment feedback process. The decision operator represents how an agent se lects a plan based upon partial information about the problem situation. Various strucures of the decision operator may be used, such as behavioral rules of thumb 9 or constrained optimizing with a planning horizon ex tending from one to n periods. The data processor serves as an information proc essor for the decision operator. Observations of the market-environment are taken and processed into antici pations data and parameters of the decision operator. The feedback generator serves as the modeler's representation of the market-environment, where the modeled agents' activities are realized and the suc ceeding state of the system is generated from the past state, current activities, and any exogenous factors involved. By introducing this feedback generator ex ogenous to the modeled agents' decision operator, it is possible to perform computer simulation experiments of a boundedly rational agent interacting with other agents in a market process. With temporal optimizing, in which there is a one period planning horizon, the modeled agent needs no representation of the exogenous feedback process, but only anticipations data on the new state. With a multiperiod decision operator, there needs to be a feedback model endogenous to the decision operator, which is the agents representation of the exogenous market-environment. With perfect foresight both the endogenous and exogenous feedback operators 10 generate the same data, implying the agent correctly anticipates conditions in the market-environment. The imperfect foresight of boundedly rational agents suggests that there will generally be a differ ence between the data generated by the endogenous and exogenous feedback operators, and, hence errors in planning. The problem of errors in current period plans is one of market disequilibrium and will not be consid ered until chapter four. Here temporary equilibrium is assumed and the temporal disequilibrium of short term planning with feedback is investigated. For temporal optimizing with naive expectations, the adaptive economizing planning procedure can be sum marized as follows, see Figure 1.1: Data processor: & : + ^ = 6(v^., e^+l* Decision operator: <p : (x,y)^+^ = <p(d^. + 2 ) Feedback generator: oj : = w((x,y)t+^, dt+^, vt, et + l * where v is a state variable; d is anticipations data and estimates for model parameters; x and y are deci sion imputation pairs; and e is an exogenous event. 11 a) Decision Operator The boundedly rational principles of planning and adjustment which comprise the decision process in a sequence of rolling plans can take a variety of repre sentations. One way to distinguish the different ap proaches is by the local search process which selects the current plan from those considered feasible.-*- Be havioral economics pioneered by Simon (1959) emphasizes a local non-optimizing search in which a current plan is selected by a movement away from past plans until a satisficing criterion is met. The direction and step size of the search movement, as well as the parameters of the satisficing criterion, adjust over time as a behavioral feedback response to the outcome of past activity (Cyert and March, 1963; Forrester, 1961; and Eliasson, 1976). Target planning within a region of perceived fea sibility is a local search procedure which may involve optimizing calculations. In this approach goals are specified along with a distance measure which is mini mized in deriving a feasible solution closest to the Perceived feasibility is relative to past activi ties and is limited by the foresight of agents to a subset of what may actually be feasible. 12 target set (Day and Singh, 1977). The various formula tions of goal programming and multiobjective program ming exemplify this approach <Zelany, 1982). An alternative formulation of the local search process involving optimizing calculations is cautious suboptimizing within a zone of flexible response (Day, 1979). Here optimizing calculations are used to specify the current plan within a bounded region centered on past behavior. There are not only constraints of tech nical feasibility, but also constraints of behavioral flexibility and caution. Such constraints arise out of a boundedly rational agent's need to simplify the complexity of the actual situation and apply rules of thumb rather than globally optimal decision rules. Ac cording to rules of adjustment to feedback, these flexibility constraints, as well as coefficients of the decision criterion, adjust over time. The choice of algorithm for specifying the plan within the zone of flexible response can take on a va riety of representations. The recursive programming models in Day and Cigno (1978) involve a single period horizon and a linear programming solution algorithm. Consideration for risk has involved chance-constraints along the lines of Charnes and Cooper, as well as a linear variant of a modified expected return-variance 13 criterion (Day, Aigner and Smith, 1971; Kingma and Kerridge, 1977). Consideration for a multiple period planning horizon has also been taken (Day and Cigno, 1978; ch 10) . In this work the planning procedure of cautious suboptimizing is used with a decision operator of con strained temporal optimization, which consists of a feasibility operator; r<b,c) = (xj^(x,b) < c, x>0) and a data dependent optimizor: Tt(a,b,c> = max <p(x,a) S.T. x e T(b,c) which combine into the optimizing decision operator: $ ( a , b, c ) = ( x I < p < x , a ) > Tc(a,b,c))Ar<b,c) In explanation, the feasibility operator consists of a set of equality and inequality constraints, y, on the activity vector, x(t), with constraint coeffi cients, b(t), and limitation coefficients, c(t), both of which may vary over time as the result of market- 14 environment feedback. The optimizer specifies those values of x which maximize the decisioncriterion, 9 , with time varying criterion coefficients, a(t). The combined decision operator determines those feasible values of x(t) which maximizes the criterion. « § Datum Vector Generate Data mm Optimize o • c • .2 °?5 * > 1 ■ ■ ^ Feedback Figure 1.1: Adaptive economizing (Day and Cigno, 1978, p. 10) 15 b> Solution Structures and Dynamics In a generalized version of temporal optimizing with market-environment feedback and naive expecta tions, Day and Kennedy <1970) show that the model pos sesses a stationary state and compact orbits. This means that under appropriate conditions the model is globally stable with bounded trajectories, but these trajectories need not be asymptotically stable, con verging to a stationary equilibrium. In fact, it is shown in chapter two that they need not even display an orderly pattern of fluctuation. The dynamics can be characterized by a nonlinear difference equation for the state variable, v^. . The order of the difference equation depends on the lag structure of expectations in the decision operator. Assuming a one period lag, as in the cobweb model and given by naive adaptive expectations, a first order nonlinear difference equation can be derived by substi tuting for d^ + j ^ with the data operator into the feed back and decision operators: vt+1 = o<5(vt, et+1),cp(B<vt, et+1>>, vt, efc + 1) 16 Leading to the composite mapping: vt+l = 0(vt' et+l> Where given initial condition Vq and a sequence of exogenous events, et, the dynamics of vt is determined. A complexity in the dynamics with constrained optimiz ing and inequality constraints, is that a discontinu ity, or at least nondifferentiability, occurs in the difference equation at points in the state space for which equated constraints change. This is termed a change in regime or phase structure of activity (Day and Cigno, 1978). Each phase structure is characterized by the binding constraints and nonzero activities. From period to period there may be endogenous phase switch ing since the constraints adjust in response to raarket- environment feedback. The dynamics of adaptive econo mizing is not only characterized by the qualitative mode of dynamics for a particular phase of activity, but by multi-phase switching as well. From the phase theory of recursive programming, an application of Kuhn-Tucker theory, primal and dual phase switching conditions can be derived, along with the equation of dynamics in each phase. From these con ditions and assumed initial state, it is possible to 17 ! express a complete characterization of the model’s be havior over time. With endogenous phase switching and with the dy namics in each phase characterized by a different equa tion of dynamics, the complex dynamics which results is not amendable to conventional comparative dynamics. Even though it is possible to specify conditions for phase switching in terms of the state variables and parameters of the model, it is not easy to specify when the switching will occur. In general, computer simula tion is the primary method of dynamic analysis. It is possible in the two-dimensional case to explore the dynamics in computer graphics, using the phase zones in state space which are based on the conditions for phase switching. This technique is illustrated in chapter two. Only by imposing conditions on parameters which restrict activities to a single phase, i.e. phase sta bility, is it possible to specify analytically the qualitative mode of dynamics for the entire trajectory. This is also done in chapter two, illustrating non- convergent but bounded trajectories with irregular or chaotic fluctuations. 18 ii) Stock Valuation The limited foresight and computational ability of boundedly rational agents can result in agents to incorrectly perceive the future impact of current ac tivities. In planning the current use and accumulation of resource stocks and capital assets, there is only an approximate valuation of these stocks in their future use, and, hence current plans may be suboptimal in the long run. Such suboptimizing strategies are a charac teristic of bounded rationality, in that, agents make short term plans with feedback adjustment, rather than intertemporal optimizations with perfect foresight (Hart, 1940; Leontief, 1958; Day, 1969; Tesfatsion, 1980) . So far, adaptive economizing has captured such be havioral suboptimizing with flexibility constraints on the use and accumulation of stocks. It is also of in terest to account for the intertemporal dependencies in the management of stocks with a behavioral rule for imputing a current value to the future returns from the stocks. The problem of imputing a value to stocks in a temporary equilibrium framework is usually resolved by considering a two period horizon in which an asset 19 transferred between periods indirectly enters the cur rent period utility function, and thus receives an im puted value (Grandmont, 1977). In this work a similar approach is developed, but one which reveals the infor mation and imputation problem a boundedly rational agent actually has. Using a behavioral approximation to the principle of optimality in dynamic programming, an intertemporal optimization problem is decomposed into a sequence of temporal optimizations with a feedback pricing rule for imputing a value to stocks. This extension of naive temporal optimizing is called proximate dynamic pro gramming . The imputed price is designed to capture the value of future returns from the stocks. In a decentralized economy, markets exist in which agents impute a value to stocks. The efficiency of these markets, how they influence the stability of real economic activity, and the manner in which the agents involved impute values are issues in need of assessment outside the efficient market assumptions (Keynes, 1937; Schiller,1984) . The efficiency of commodity and asset markets has received attention from others as well, since conditions in the 1970's have led to unexpected changes in commodity prices and asset valuations (Cooper and Lawrence, 20 1975; Brainard et al., 1980). Nevertheless, the general approach to modeling such markets is based on the ex tensive theory of efficient futures and capital asset markets. Though suggestive as to what might be expected in a stable economy of agents able to anticipate sto chastic characteristics of future returns to capital, the approach relies on a basic premise that agents 'Have a knowledge of the future of a kind quite differ ent from that which we actually have' (Keynes, 1937, p. 222) . To produce a theory of asset prices or commodity price speculation in a market of boundedly rational agents is a challenging issue. For a stationary economy the rule used in the temporary equilibrium literature can be shown to be valid, allowing an efficient trade off between current and future consumption. In economies which are changing over time and where agents are boundedly rational, the valuation of stocks involves a set of, essentially arbitrary rules, and has long been a subject of controversy, but one usually set aside by the stationarity and rationality conditions assumed in economic models (Hicks, 1983). It is not the intent to resolve this complicated issue of objectively specifying the value of stocks, because it is inherent to the nature of real economies. What can 21 be done here is to; direct attention to the issue that agents subjectively impute a value to resource stocks and captial assets; suggest a few simple rules for rep resenting such behavior in a temporary equilibrium framework; and derive some implications for the dynam ics of resource allocations from the application of these rules. In chapter three, the approach is developed and illustrated. A boundedly rational Crusoe makes a choice between hunting goats (a renewable resource) and farm ing corn. The single agent model eliminates complicat ing market considerations and the concern for only a single resource stock eliminates the portfolio problem, leaving the simplest of consumption-savings decisions. Eventually these other factors will have to be intro duced for a theory of asset markets with boundedly ra tional agents, but still the impact of speculation on the dynamics of resource allocation can be explored in a temporary equilibrium framework. 22 Ill MARKET DISEQUILIBRIUM Current period plans need not be fulfilled where agents make errors in expectations, and hence realized activities are mediated through disequilibrium transac tion processes. In such a setting of market disequilib rium, the inconsistent plans of agents lead to excess demands which are not arbitraged away prior to transac tions. Instead, they are in a sense, held in limbo with stock-flow disequilibrium buffer mechanisms. Agents then in real time, have a feedback response to the discrepancy between plans and realizations. In fact, agents recognizing the potential for the nonfulfillment of plans will maintain flexibility with such disequilibrium mechanisms as inventories and order backlogs on the product and commodity markets, cash liquidity on the financial market, and even special contractual arrangements with built in measures for flexibility, such as inflation indexing of wages on the labor market. With unrealized plans and maladjustment of stocks and flows, firms will initiate feedback adjustment processes in the planning of activities. Flexprice and fixprice adjustment are two extreme representations of this response, which occur as a tatonnement prior to 23 transactions. An alternative adjustment process which occurs as a real time feedback response, involves a "hybrid", disequilibrium firm in a sequence of rolling plans. According to various factors, such as industrial structure, market competition, demand elasticity, in ventory cost, and production technology such a firm will use some combination of price and production ad justments, the use of which will also depend on the state of the macroeconomy. The extent to which market disequilibrium persists is unclear, for there is no agreed upon measure to characterize it and little in the way of conclusive analysis about how an economy functions when current plans are not realized. Data on both plans and realized activities which would reveal the existence of disequi librium is scarce (Nerlove, 1983). Though, common expe rience, another form of data, is filled with indicators of disequilibrium. For instance: Delays and abandon ments of capital projects, fluctuations in inventory and order backlog of manufactured goods, financial com plications leading to loan default and bankruptcy, im balance with foreign exchange balance of payments, mass unemployment for extended periods of time, and commod ity price speculation. 24 There is reason enough to move away from the mar ket equilibrium assumption, with its implication of realized plans. The short-run adjustment behavior of firms and households, and what occurs on markets during the 'Hicksian week', may well have a bearing on medium to long-run dynamics. This is what needs to be inves tigated, and, I suggest, what Hicks realizes when he states, that where he went wrong in the dynamics was to represent the markets of the week in equilibrium. Rather, the 'working of the economy within the week, should be a matter of the structure of markets', how plans are mediated into transactions and how spillover effects from one market influence activities on other markets. In preparation for the next week, agents for mulate new plans as behavioral feedback adjustment to the outcome of past transactions and in anticipation of future conditions (Hicks, 1977, preface). What is required for such a disequilibrium analy sis is a representation of the economic system, dis tinct from the market equilibrium models based on neoclassical firm theory and assumed stability of mar ket processes. The method of dynamic analysis needs to refer back to how firms behave and how markets func tion. 'What we need is a theory capable of describing system behavior as a temporal process, in or out of 25 equilibrium, which requires a prior account of how trade is organized in the system and of how business and household units behave when the system is not in equilibrium and is predicated on how trade is organ ized' (Clower and Leijonhufvud, 1975, p.183). That the interaction of firms in different sectors of an economy who use different principles of behavior in planning activities under conditions of disequilib rium, may have implications for the dynamics of a macroeconomy and the impact of government policy has been suggested by several economists, such as Hicks (1974), Kaldor (1976) and Okun (1981). They discuss how the coordination of transactions between flexprice pri mary good producers and fixprice industrial manufac tures is a source of adverse effects on the macro econ omy, inparticular inflation and imbalanced growth among sectors. This work attempts to move towards the development of such a micro to macro dynamic framework by develop ing a prototypical firm model in chapter four, based upon principles of adaptive economizing in disequilib rium. A dynamic disequilibrium multisector model will be completed when various firm-sector submodels as devel oped in chapter four are linked together in a network 26 of intersector flows. Such a model of the disequilib rium "circular flow of economic activity" can then be used to study how important macroeconomic problems are generated by microeconomic forces in a way called for by Clower and Leijonhufvud (1975), Hicks (1974), Kaldor (1976), Okun (1981), Malinvaud (1984), Perry (1984), and others. x) Firm Behavior with Multisector Interactions Market equilibrium in one form or another is at the core of most micro models of the firm. Either Walrasian or non-Walrasian quantity constrained equi librium is assumed. Such equilibrium assumptions are powerful in economic analysis, making realized activi ties equivalent to plans, at least in the current pe riod, which allows considerable simplification in the representation of economic phenomena. For instance, the multiple decisions of a firm can be made consistently in a single optimizing computation in neoclassical the ory. Therefore, it is not necessary to investigate the internal organization of the firm or its actual plan ning routines. With the assumed institutional structure of mar kets and rationality of agents in neoclassical theory 27 the stability and convergence of markets to a long run equilibrium may well be the case. As is suggested, economies at one time may even be adequately repre sented by this model structure (Hicks, 1977). But, con ditions have changed. The political economic structure of societies have become more complex and information intensive. One dimension of the historical development of in dustrialized economies is the changing structure and role of firms and markets in the allocation of re sources. Firms and markets are alternative informa- tion-planning-control systems, which combine to form the institutional structure of a decentralized market economy. As a self-organizing system, there is a mar ket-firm feedback process which endogenously modifies the institutional structure of resource allocation. The changing technology and new energy supplies of the early 1900's permitted high volume production, mass marketing, and standardization of price and quality. With boundedly rational management, organizational in novation was necessary to govern the growing modern corporation. (Williamson, 1981, Hicks, 1977). An implication of the change in firm-market struc ture for economic theory is that the Walrasian market with neoclassical firm behavior is no longer adequate 28 for representing industrialized economies. The quantity constrained non-Walrasian market with fixprice monopo listic firm behavior has been proposed as an alterna tive (Negishi, 1979; Benassy, 1982). Under existing institutional structure of markets and the bounded rationality of agents, there is no a priori grounds for representing the economy in a Walrasian or non-Walrasian equilibrium. The study of disequilibrium dynamics, 'where the processes of ad justment are determined endogenously by the behavior of agents and the way markets are organized, is of para mount importance' (Fisher, 1983, p.11). It is of interest to introduce into a disequilib rium firm model, principles of behavior which have a 'family resemblance' to what we find firms using. To do so requires a closer look at the organizational and planning structure of a firm. For economic and techni cal reasons firms in different sectors apply different principles of behavior. Some produce to order while others produce for stock. Some are price takers while others are price setters (Scherer, 1980). In chapter four a prototypical firm model which is flexible enough to capture these distinguishing characteristics is de veloped. Computable models for three types of firms are united into a multisector model of interindustry activ 29 ity. Product market transactions are explicitly mod eled, whereas, labor and financial market interactions remain for future development. xx) Policy Analysis Policy analysis in a micro-to-macro dynamic dis equilibrium framework is to be contrasted with fiscal and monetary policy analysis in an equilibrium frame work. The equilibrium based models diminish the use of economic structure, how agents behave and markets func tion. In exchange for these simplifications, models are developed which can be statistically fitted to aggre gate historical data and used for extrapolative fore casting. Discretionary fiscal and monetary policy are then analyzed, at least for the short run. In a dynamic disequilibrium framework analysis is shifted to comparative dynamics, refocusing attention upon the long run implications of short run disequilib rium adjustment behavior, a dimension of which past policy has been criticized (Feldstein, 1982). Further more, the disequilibrium micro-to-macro approach is not suited for aggregated statistical estimation nor for extrapolative forecasting. It is not precise short-run predictions that are aimed for, but rather, dynamic 30 trends which are derived from hypothesized principles of behavior and market interactions. It is then possi ble to ask how change in the institutional structure of markets arise out of the need to cope with disequilib rium. As with Schumpeter (1934) interest is on how economic change creates and destroys legal-institution al structures. Postwar economics has shown little interest in the analysis of reforms for the institutional structure of markets (Lucas, 1981, p. 251). Recognition of a struc tural change approach to policy analysis is most simi lar to the school of historical and institutional eco nomics, where the legal foundations of economies is under investigation (North, 1978). A few examples of such a view toward policy analysis will be cited. Under the political philosophy of Laissez faire, Henry Simons (1948) argues that state responsibility is to maintain a legal and institutional framework within which competition and the free movement of relative prices can function effectively as agency of control. Minsky (1969) states that it is the task of monetary analysis to design a financial system that would make serious financial disturbances impossible. Friedman (1985) suggests the need to overhaul the framework of monetary policy. Malinvaud (1964) and others have rec 31 ognized the need to assess the institutional structure of markets to understand unemployment problems. Other areas of structure receiving attention are the use of foreign trade restrictions, regulation of financial institutions, and revision of tax policy (Salvatore, 1985; Friend, 1979; Sinai and Eckstein, 1983). It is policy related to changes in institutional structure for which a micro-to-macro dynamic disequi librium model of economic activity is designed. Struc tural analysis with a disequilibrium model is not in conflict with equilibrium analysis, it is used for a different purpose. Disequilibrium analysis is an at tempt to apply mathematical analysis and computer simu lation to institutional economics where the behavior of agents and the legal-institutional structure of markets is essential. 32 CHAPTER 2 COMPLICATED DYNAMICS IN A GENERALIZED COBWEB MODEL I INTRODUCTION The adaptive economizing of boundedly rational agents involves behavioral suboptimization with feed back. The structure of economizing with feedback is illustrated here, while at the same time illustrating the complicated dynamics which may arise in a general ized cobweb model. The maintenance of financial working capital as a dimension of producer budgeting illustrates how a be havioral flexibility constraint operates in naive adap tive economizing. Through such a constraint the tradi tional cobweb model is generalized into one where a richer variety of dynamics may occur. This includes endogenous switching among multiple regimes of activity as shown by Day and Tinney (1969), as well as nonlinear dynamics within a particular phase of activity. For several examples of such a model, I give an exact derivation, and using computer graphics, an il 33 lustration of the phase zones in state space, and dis cuss how they may be used to explore endogenous phase switching. I also present sufficient conditions for chaotic two-commodity cobweb fluctuations. These are details worked out for illustrative purposes, but they bear more than passing interest. The examples demon strate how instabilities and structural change can arise endogenously from basic principles underlying agent behavior. More complex, information intensive behavior can be incorporated into the adaptive economizing frame work, but guidance into complicated dynamics by ana lytical tools will be reduced, so for initial explora tions a simple setting is used. 34 II GENERALIZED COBWEB MODEL The cobweb model describes the dynamics of pro ducer supply as a response to expected price. Assuming temporary equilibrium and naive expectations, the price expected is the previous periods market clearing price. In a stationary state expectations will be fulfilled, otherwise, the familiar cobweb dynamics arise. Given demand and supply as linear equations of price there is linear dynamics (Leontief, 1934; Samuelson, 1948). Day and Tinney (1969) generalized the cobweb model by introducing behavioral suboptimization on the part of firms. Each firm determines the supply of m commodi ties with a constrained temporal optimizing calcula tion. Given constant average cost the criterion is to maximize expected total revenue, which is equivalent to maximizing end of period capital. The constraints are linear and consist of a technical land constraint and a financial working capital constraint. The former constraint can be viewed as a linear constant coefficient production technology which de pends on the factor of land, which is available in lim ited supply. The working capital constraint is a behav ioral bound on firm activity due to a revenue lag and demand for cash on hand. The importance of this budget- 35 ing consideration in producer behavior was first empha sized by Sune Carlson (1939) and later by Williams (1967). With no borrowing in the model, the working capital maintained in any one period is a fixed frac tion of the difference between the revenue received from the previous period sales and overhead costs. This available working capital must be greater than or equal to the operating costs for the period. Market feedback influences the plans of firms through price expecta tions and through the available working capital, which depends on revenue received, and, hence market price. This version of a generalized cobweb model is an introductory example to adaptive economizing with a constrained temporal optimizing decision operator and financial feedback. Specification of the model is as follows. i) Model: Single Firm Two Commodity Example Producer behavior is represented by a constrained temporal optimizing decision operator: 36 max afcxt s.t. (btxt < ct; xt > 0) <xt > where X-j.: vector of m activities a^.: vector of m criterion coefficients C£: vector of n limitation coefficients bt: n,m matrix of constraint coefficients Through the data processor, the parameters of the model dj. = (a^-,b^_ ,c^_) may be updated over time with behavioral rules of adjustment to market-environment feedback. From the assumptions of two commodities, a land and working capital constraint, and the technical relation of the constraint coefficients to activities, the model takes the following form: (1) (2) (3) (4) max ( X-i ,x5 l P x It It ne 2tx2t s.t. x It + X2t < c± blxlt + b2x2t < c2t = kt 2t > 0 37 The firm has the choice of producing none, one, or both of the commodities, using two inputs, land, L and working capital, K^_. Each product is measured in units equal to the yields per unit of land, and assume that one unit of land yields one unit of each of the two commodities. Given these assumptions on the units and productivity of land, the activity levels, x^t and x-2t' are the amount of land devoted to and the output of each commodity in period t, and the land constraint, (2), has constant unitary coefficients. The working capital constraint has the current operating cost, with average cost per unit of land for commodity one and two given by b-^ and b2 , respectively, less than or equal to the available working capital, K^_, which is influenced by the market price through the revenue received in the previous period. Kt = s(Plt-lxlt-l + P2t-lx2t-l ~ h> (5) where S(>0) is a precautionary working capital factor * and h(>0) is overhead cost. The criterion of maximizing expected total revenue in a competitive market implies the criterion coeffi cients, a-j^ and &2t are expected market prices. With naive expectations the previous periods market equilib 38 rium prices, as given by inverse demand functions, pro vide the expected price: lt-1 °1 (xlt-lr X2t-1) (6a) 2t-l °2 <xlt-l' x2t-l (6b) In the analysis to follow, linear inverse demand functions are used: The generalized cobweb model consists of the con strained temporal optimizing decision operator, equa tions 1-4, and the feedback-adjustment equations 5-7. To follow is an analysis of the dynamics to a sequence of temporary equilibria. First, a general discussion of solution structures, and then two specific examples. Plt-1 g10 gllxl't-l + g12X2t-l (7a) P2t-1 g20 + g21Xlt-l g22X2t-l (7b) where i = 1,2; j = 0, 1, 2 39 ii) Solution Structures and Dynamics Under appropriate conditions of demand and cost, the generalized cobweb model is globally stable with bounded, non-negative trajectories for individual ac tivity levels, outputs and prices. These trajectories, however, need not be asymptotically stable, nor, in deed, need they approach an orderly pattern of behav ior. The qualitative mode of dynamics will be investi gated below. First, however, we must consider that the firm will exhaust the supply of particular resources, leav ing others in excess. It may, or may not, use all of the available working capital. Likewise, the firm may produce some of a given subset of commodities while not producing some of the goods. Each combination of active nonzero activities and tight or binding constraints resulting from the firm's economizing choice can be represented by a set of equated constraints. This set, together with the feedback-data functions (5)-(7), give a phase structure that characterizes behavior for a given period of time. From time to time these struc tures may switch as the firm responds to feedback from the market through prices and the changing supply of working capital. 40 The phase theory of recursive programming is used to derive the phase switching conditions and the equa- tion of dynamics for each phase (Day and Cigno, 1978). The phase theory is an application of the Kuhn-Tucker conditions to an inequality constrained maximization problem. Introduce the dual variables y^ and y^,, associated with the land and working capital constraints, form the langrangian maximization problem, and derive the fol lowing first-order Kuhn-Tucker conditions for maximization: dXl X1 “ (Plt " yLt * blyKt)xlt = 0 (8a) dx2 x2 = <P2t ' yLt ' b2yKt,x2t = = 0 ( 8b) dy1 yL “ 11 ‘ xlt x2t,yLt " 0 <8c) dyR yK = (Kt - blxlt - b2x2t,yKt = 0 (8d) Let i = 1,2; j=L,k if x.. > 0 then = 0 xt dx . l < 9a) if x.. = 0 then §r~ < 0 xt dx . X <9b) if y.. > 0 then = 0 it dy_. (9c) 41 then - > 0 dy • dy . J < 9d) From these Kuhn-Tucker conditions, Eqns. (8) and (9), it is possible to specify a primal and dual condi tion for each phase along with the equation of dynamics for x-^. and associated with these phases. They are derived by taking the set of possible nonzero activi ties and binding constraints, assume the appropriate equality-inequality relations in Eqns. (9), and then substitute and rearrange terms in Eqns. <8). The solution structure consists of six possible combinations of nonzero activities and binding con straints, which are summarized in Table 2.1, phase structures. The phase conditions and associated equa tion of dynamics are summarized in the phase condition Table 2.2, and phase activity Table 2.3. Upon substitu tion for the available working capital and the price expectations, the phase conditions and activities can be expressed in terms of the demand and cost parameters and activities in the preceding period. Further analysis of the potential dynamics to a sequence of temporary equilibria will be given below in context of two examples, one illustrating endogenous phase switching, and the other illustrating bounded but nonconvergent nonlinear dynamics in phase 12KL. 42 Table 2.1: Phase structures Case Activity Constraint 0 none none 1L X1 land 2L x2 working capital IK X1 land 2K x2 working capital 12KL x±, x2 both Table 2.2: Phase conditions Condition Phase Primal: k = Kt/L Dual: P21 = p2t/plt 0 k = 0 p 21 0 1L k > bl P21 < 1 2L k > b2 P21 > 1 IK k < bl P21 < b2/bl 2K k < b2 P21 > b2 /b-^ 12KL b2 < k < b-^ b2/bl < P21 < 1 43 Table 2.3: Phase activities Activity xlt x2t yRt Phase 0 0 0 0 0 1L L 0 0 plt 2L 0 L 0 P2t IK Kfc/bi 0 Plt/b2 0 2K 0 Kt/b2 p2t/b2 0 12KL Kt~b2L Plt"P2t Pltblyi bl"b2 L_Xlt bl"b2 44 Ill SOME ILLUSTRATIONS i) Phase Zones in State Space: An Example of Phase Switching With multiple regimes and endogenous phase switch ing it is necessary to characterize system dynamics as a set of switching rules with associated equation of dynamics, and provide simulation experiments for spe cific model parameters and initial conditions. A graphical presentation of phase zones in state space allows phase switching dynamics to be illustrated in a compact form. It is possible to see how, for a given set of parameters, the phase zones are shaped, and given initial conditions to trace out the dynamics. Furthermore, the impact of parameter changes can be illustrated. In the phase conditions, see Table 2.2, it is the relation of the relative expected prices and the capi- tal-land ratio to the cost coefficients, b^ and b2 ^ which is crucial to phase switching. This relationship is brought out in the data zones of Figure 2.1. The shape of the data zones depend on the cost coeffi cients. The phase in which current activities occur 45 depends on the expected relative prices and capital to land ratio. Since these are a function of the activity levels from the previous period, x^t-l and x2t-l' ; * - t : ' - s Pos” ! I ! ! a . 8 2K oj £ a . ^ o 12KL o o o 0.0 2.0 4.0 6.0 8.0 10.0 12.0 KITJ/L Figure 2.1: Data zones 46 sible to transform the phase conditions in data space into phase conditions in state space. From the equation of supply response and the location of phase zones in state space, it is possible to see which phase the new activities fall in. If it is in the same phase then the new supply response is the same. If the phase zone changes then the structure of supply response switches and the cobweb dynamics enters a new regime of activ ity. In this way a sequence of regimes, or phase struc tures, is generated. Each regime is characterized by a distinct type of activity mix and a distinct situation of factor supply (capital scarcity, land scarcity, or capital and land scarcity). Which sequence of phases and regime switchings occur depends on the form and parameters of the demand function, the cost parameters, the supply of land and initial working capital. The construction of the phase zones is straight forward, though somewhat tedious. First set the primal and dual phase conditions equal to the phase switching value, given in the table of phase conditions. This includes three equations for both the primal and dual, since a condition for the null phase is to be included. From these equations solve for *2t. as a function of x-^.. For the primal conditions the result is one of the 47 hyperbolic functions (ellipse, circle, parabole) de- 1 pending on the nonzero demand parameters/ The dual conditions generate a linear relation between x-^ and X2t. In addition to these six phase switching condi tions the land constraint and nonnegativity constraints on the activities are included, to complete the graph of phase zones in state space. Specification of which phase occurs between which curves involves an associa tion of the areas in data space to state space, as re lated by the curves for P-|/Pf and K/L. Using the phase zones in state space and the ini tial state the dynamics of endogenous phase switching may be characterized. The phase zone of the existing state determines the supply response for the upcoming period. The location of the next state of the system in the phase zones will tell which phase of activity will prevail in the succeeding state, etc. See Figure 2.2, for an illustration of phase switching dynamics, traced out with the connected boxes. 1 For instance, with no cross partial terms in the demand equations < g^ 2~*321 = ® ^ ’ then circles result. With the cross partial terms nonzero, ellipses occur. 2 Parameters for this example are: g10 gll g 12 g2 0 g 21 g22 bl b 2 L h 16.25 .00525 .001 10. .001 .0025 6 . 3. 2000. 4400. S = 1. XjtO) = 500. x2 (0) = 900. 48 I o ! = P i o Q O X. o o • o o - 1000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 Figure 2.2: Phase zones in state space 49 ii) Qualitative Mode of Dynamics: An Example of Chaos The complicated dynamics generic to the general ized cobweb model includes, besides endogenous phase switching, nonlinear dynamics in phase 12KL, where both activities occur and both constraints binding. The pos sible qualitative modes of dynamics include the four conventional modes, common to linear and nonlinear equations, imparticular, convergence and divergence which is either monotonic or cyclical. In addition, there is also bounded but nonconvergent trajetories, some of which are limit cycles, while others involve irregular fluctuations called chaos. In this example it is of interest to characterize the nonlinear dynam ics to a sequence of temporary equilibria in phase 12KL and derive demand and supply conditions which lead to chaotic dynamics. To determine the conditions for chaos, or any other qualitative mode of activity for phase 12KL sev eral steps are followed. First, transform the non linear equation of dynamics for one activity or the other into a difference equation whose qualitative be havior is well known. From the dynamics of the trans formed equation, conditions on cost and demand parame ters can be derived which are sufficient for chaos. 50 Second, derive a set of parameter restrictions on cost, demand, and last periods production for phase stabil ity . The supply response for commodity one in phase 12KL is: xlt = <Kt ~ b2L)/(b1 -b2) Substituting for K^_ , ^it-l' B2t-1 an<^ x2t ^rom Eqs. (2), (5), <7a) and (7b), the supply dynamics be comes a nonlinear (quadratic) first order difference equation. (Let x ^ = x^.) : X . J . = f(x^._2 ) = C + (10) where A, B and C are functions of the demand and cost parameters: C = ((g2 Q—b2~g22L)L — h) / (b ^ —b2) (11a) B = *9l0 ~ g 20 + ^g22L + (gi2 + ^2 1 >L)7 <bl“b2 ) (lib) A = (g-^ • L+g-^2 +g2 ^+g22 ) / (b^-b2 ) (11c) The analysis of nonlinear difference equations is not always possible by analytical techniques. In this case it is possible to transform the quadratic differ ence equation into one whose qualitative behavior can 51 be characterized by a single parameter, m, which is a function of the parameters, A, B, C, given above. Using the transformation zfc = D + Ext D > 0, E > 0 (12) The difference Equation (10) becomes-1 zt = mzt_1(l - zt_1) 0 < zt < 1 (13) where m is: m = 1 + (1+f) 1/2 m > 0 f = B2 - 2B + 4AC (14a) (14b) An additional restriction is on the range of x^. since z^. is confined by the range (0,1). With x^. con strained by the available land, the upper bound is: Substitute (12) into (11) and rearrange to get: xt = - (D/E)(m<1-D)-1) + m(l-2D)xt_2 - mEx^_q Make the association of A, B and C from equation (1 0 ) with the parameters of equation (1 2 ) and solve for m in terms of A, B and C to get: A = (D/E)(m (1-D) - 1); B = m(l-2D>; C = mE. To solve for m in terms of A, B and C substitute for E from the third equation into the first and sub stitute for D from the second into the first. 52 x^_ e<0 , xmax) xniax = <m+B)/2A < L < 15a) < 15b > The dynamics of zt can be characterized with re spect to the stationary state and bifurcation points for the changing qualitative mode of dynamics, which can be expressed in terms of the parameter m. The two stationary states, z, are given by the solutions to: z = m(1-z)z which are z = {0 , (m-1 )/m) The bifurcation points for changing qualitative modes of dynamics are derived from a stability analysis of the difference equation around the stationary states. Taking the derivative of with respect to zt results in the condition for local stability, or convergence of the difference equation to its station ary states: d2t + l 1 < -- d2t m (l-2z) < 1 zfc = z 53 The qualitative modes of dynamics for the general- ized cobweb model in phase 12KL are: 1 ) Monotonic convergence to the stationary state z = 0 , if 0 < m < 1 2 ) Stable convergence to the stationary state z = (m-l)/m, if 1 < m < 3 where if 1 < m < 2 then monotonic convergence if 2 < m < 3 then cyclical convergence 3) Unstable fluctuations about z = (m-l)/m if 3 < m < 4 where a) if 3 < m < 3.57 even cycles of increasing order b) if 3.57 < m < 3.83 odd cycles of decreasing order c) if 3.83 < m < 4 irregular or chaotic fluctuations 4) Unstable divergence if m > 4. The convergence property depends on the value of m. With nonlinearities values of m between three and four, rather than being divergent, as one might suspect from standard linear stability analysis, leads to 54 bounded but nonconvergent trajectories, which are ei ther limit cycles or irregular fluctuations. For chaos, parameters are specified in accordance to the transformation conditions of Eqs. (14) and (15), and the conditions for phase stability. Primal and dual conditions for phase stability can be expressed as re strictions on the maximum and minimum values of supply response, x-^. These values, xmax and xmin, can be de rived from Eq. (10), and may also be used in expressing a sufficient condition for chaos.^ The maximum value of x ^ occurs when x^-i equals xstar which can be found by maximizing f(xlt_^). Taking the derivative xstar is: xstar = B/2A (16) Substitute xstar into f(*) to find xmax and then substitute xmax into f<*) to find xmin. xmax = C + B2/4A (17) xmin = C(1+B-AC) - <B2/16A)<8AC+B(B-4>)3 (18) A sufficient condition for chaos involves the fol lowing sequential order among four consecutive iter ations of Eq. (10): 0 < xmin < xc < xstar < xmax < L where xc is the smallest root of xstar = f(xc). 55 For the cobweb model to maintain phase stability a primal and dual phase condition must be satisfied. For phase 12KL, these conditions are as follows, see Table 2.2: 1) Primal phase condition b2 < Kt/L < bjL which upon substitution and rearranging; 0 < xlt < L (19) assuming h± > b2 (2 0 ) The primal phase condition restricts the amount of production, where production is in terms of the area of land cultivated. Substituting the maximum and minimum values of x^t, xmax and xmin from Eqs. (17) and (18), Eq. (19) becomes a constraint on A, B and C. 56 2) Dual phase condition b 2 2 2t b < < 1 1 z It The dual phase condition is a restriction on the ratio of per unit gross profit for the two commodities, see Table 2.2. The per unit profit is the expected price. With the assumption that price for the second commodity is constant the only variable factor is the expected price of the first commodity. Given naive price expectations, this dual phase condition can be expressed as a restriction on Xxt-1* Substituting for zt with P^t-l an<* p2t-l an<* uPon considerable rearrang ing the dual phase condition reduces to Just as xmax and xmin are the maximum and minimum values for x-^. they are for xit-l' an<^ hence the dual condition is a set of restrictions on A, B and C, and the structural parameters 920 an(^ ^2 * ) < X-(t-1) < ^ 1 A <21) assuming Eq. (29) and ^20 “ b2 > 0 <22) 57 In summary, the conditions necessary for a stable phase 12KL and chaos in the dynamics of supply are Eqs. (14)-(15), (19)-(22) and the information in Eqs. (11) and (17)-(18). As one can see there are many combina tions of cost and demand conditions which may result in chaos. One step in specifying a set of parameters is to assume a value for m(= 3.9) and consistently specify the structural parameters using the restrictions men- 1 tioned above. For an example see Figure 2.3. / LJ v —' o Z ™ o 0.0 10.0 20.0 30.0 SO.O TIME Figure 2.3: Chaotic cobweb supply dynamics The parameters for this chaos example are: 910 911 912 920 921 922 bl b 2 L h 00 • 00 • 0 . 5.0 0 . 0 . 2 . 1 . 5. 20. x±( 0 ) x2 (0 ) A B C xmin xc xstar xmax S .5 4.5 CO r - > • 3.9 0 . .475 .755 2.5 4.875 1. 58 IV) SUMMARY The purpose of this chapter is to illustrate adap tive economizing and to demonstrate that complicated dynamics may arise endogenously from basic principles underlying agent behavior. An important point to realize is that with bounded but nonconvergent dynamics, analysis must be focused on the long-run implications of short-run disequilibrium adjustment. Stability cannot be relied upon as the ba sis for long-run equilibrium analysis. The role of stock flexibility constraints is to be contrasted with the approach of imputing values in the following chapter. 59 CHAPTER 3 SOBOPTIMIZING STRATEGIES IN A ROBINSON CRDSOE ECONOMY I INTRODUCTION In planning the current use and accumulation of resource stocks and capital assets, a valuation of their future use is made, generally with error, and hence current plans may be suboptimal over the long run. How to approximate an assets value of future use on the basis of existing information in a temporary equilibrium setting is the issue. The solution proposed here involves an approach called proximate dynamic pro gramming which decomposes an intertemporal optimization problem into a sequence of temporal optimizations using a behavioral stock pricing rule derived from the prin ciple of optimality in dynamic programming. The ap proach is proposed as an alternative to that taken in the temporary equilibrium literature (Grandmont, 1977). The management of a renewable resource serves as an example for exploring the impact that alternative behavioral stock pricing rules has upon the utility 60 derived from resource using activities. Rather than hypothesizing a market demand function for the renew able resource and assuming market equilibrium, the re source allocation problem is treated in context of a Robinson Crusoe economy, where an opportunity cost to allocating labor to exploiting the resource (goats) is made explicit with an alternative of farming corn. The technique of intertemporal optimization has been applied to this example by Smith (1975), who, us ing a qualitative analysis restricts the dynamics to saddle point stability and monotonic convergence. Day (1980) has reformulated the model as one of naive tem poral optimizing in which complicated dynamics occurs. Here proximate dynamic programming extends naive tempo ral optimizing with stock valuation rules while main taining a temporary equilibrium framework. To experiment with the impact of behavioral asset pricing on the utility of Crusoe over a finite planning horizon, the solution to the resource management prob lem is compared using three approaches, namely naive temporal optimizing, proximate dynamic programming and n-period dynamic programming. Each is characterized by a particular approach to stock valuation. In the first section to follow a Robinson Crusoe economy is outlined and the dynamics from naive tempo 61 ral optimizing with feedback is illustrated. A compara tive dynamic analysis is performed with respect to the efficiency of technology in the use of the resource and the natural growth rate of the resource stock. In the third section a discrete time dynamic program is ana lyzed for parameters consistent to those considered in the previous section. In the fourth section proximate dynamic programming is formulated, a set of behavioral approximations to the imputed value of end of period resource stocks is derived, and the resource alloca tions generated are compared with the previous two cases. In the final section several concluding comments are made on intertemporal efficiency in context of bounded rationality, imputed prices, asset and commod ity markets, and the use of institutional constraints or quantity controls. 62 II BOUNDED RATIONALITY: CASE OF NAIVE TEMPORAL OP TIMIZING With naive temporal optimizing, a boundedly ra tional Crusoe does not account for the feedback effect of current activities on the productivity of future resource using activities. Even though Crusoe is a cal culating individual, balancing the value of marginal product from current activities, there is no valuation for the resource stock in its future use. The extent to which the behavior of Crusoe may be suboptimal in the long run due to the limited horizon over which the im pact of current activities are considered is discussed in the following sections. In this section the Robinson Crusoe economy is outlined and the dynamic properties to a sequence of temporary equilibria derived from na ive temporal optimizing is explored. i) A Renewable Resource in a Robinson Crusoe Economy Consider a Robinson Crusoe economy where a single agent is both producer and consumer. At the beginning of each period Crusoe observes the resource stock and allocates a fixed labor supply between exploiting a 63 renewable resource, hunting goats, and participating in an alternative activity, farming corn. With temporal optimizing the utility from consum ing the current end of period harvest of corn and goat meat is maximized. Through a biological growth process the goat herd is replenished, if not hunted into ex tinction, and the next period labor allocation decision is made. The effect of hunting on the dynamics of the resource stock depends on the preference function for corn and meat, as represented by the utility function in Equation (1>, the production technologies for corn and goat meat in Equations (2) and (3), the labor sup ply constraint of Equation (4), and the available re source stock which is updated with Equation (5). With the addilog utility function having positive constants, c and g, it is possible for Crusoe to main tain a positive utility while consuming only meat or only corn. The efficiency of labor in the production of corn and goat meat is represented by the coefficients y and PrOf., respectively. The biological growth in the goat herd population is a given percentage adjustment, p., around a sustainable yield, in. The actual growth is then adjusted according to the hunting activity of Crusoe. With the efficiency of labor in hunting depend ing on the size of the resource stock, , there is a 64 feedback effect from past decisions. The future impli cations of this intertemporal link is not accounted for in making the current decision with naive temporal op timizing. It is the dynamic implication of this limited foresight which is of interest to investigate. The variables and equations are defined as fol lows : ct: corn production per capita in period t; g^.: goat meat production per capita in period t; f.j.: farming effort in period t; h^.: hunting effort in period t; m^.: resource stock available at beginning of period t; T: total labor supply available. Crusoe's utility function for corn and goat meat is: u(ct,gt) = «ln(c+ct) + (l-<x) In ( g + gt) (1 ) The production function for corn is: ct = -yft { 2 ) 65 The production function for goat meat is: gt = (3m,. ht ( 3 ) The labor supply constraint is: ft + ht < T (4) The percentage change in resource stock is bio logical growth minus hunting: (m^ +2 -m^ ) /m,_ = Mm-m^.} - g^./m^. Substituting for g^_ the equation of dynamics for the resource stock is: m^ +2 = m,_ + p. ) m^_ - Pm^h^. (5) To analyze the dynamics it is first necessary to derive the period ' t' allocation of labor to hunting, hj. . The labor allocation decision can be formulated as a lagrangian maximization problem. In each period the activities are derived and the resource stock is up dated with the equation of dynamics corresponding to the phase of activity, i.e. all farming, all hunting, 66 or a mix of hunting and farming. Consequently, it is also necessary to derive conditions which specify the phase of activity. Both the phase conditions and the corresponding equation of dynamics are derived from the phase theory of recursive programming, which is an ap plication of the Kuhn-Tucker conditions (Day and Cigno, 1978; and Chapter 1 above). Express the lagrangian as a function of the deci sion variables, f^. and h^., by substituting the produc tion technologies into the utility function and intro ducing the lagrange multiplier, y^., for the constraint on the available labor. L = max (cxln (c+yf ^ ) + (1-oc) In ( g +(3m^.h^_) - y^. < f ^.+h^.-T ) ) (6 ) The Kuhn-Tucker conditions are: dL , <xy , v ai ft = < ^ “T ' Yt)ft = 0 171 c v f t . (l-oc)pm. h. = ( - y. )h. = 0 ( 8 > d h t t t g+Pmtht y = (T-f ~h ) y = 0 (9) dy 2 1 t t J t Assume the available labor, T, is always fully al located. If ft or hj. is zero then an inequality occurs 67 in Equation (7) or (8 ), respectively, from which the phase switching conditions are derived in terms of the existing resource stock and the parameters of the model. The two phase conditions in terms of the exist ing resource stock will be labeled phf for the switch from farming to the mixed phase and phh for the switch from the mixed phase to all hunting. I farm phase: 0 < m^. < phf; f^. = T, h^. = 0 II mix phase: phf < m^. < phh; ft + ht = T, ft,ht > 0 III hunt phase: phh < i t i j . ; f^. = 0, h^. = T From Equations (7) and (8 ) the phase conditions with respect to the beginning of period resource stock, m^., are: phf = -------- <1 0 ) | 3 < 1 —ex ) {c+yT ) phh = °n/g ^------------ <11) P( <l-cx) (c+yT) - yT) For the mixed hunting-farming phase the allocation of labor to hunting is derived from Equations <7) and (8) : 68 ht = < l-oc) {T+c/y > ~ gcx/Pm^. (12) The equation of dynamics for the resource stock in each phase can now be expressed by substituting for ht. Introduce parameters A, B, and C, such that Equation (5) has the general form: mt+^ = C + Bmt - Am^ (13) where A, B, and C depend on the phase of activity as follows: A = M- for all phases (14) i) 1 + urn ii) B = 1 + um - ( 3 < l-cx) (c+yT > /y (15) iii) 1 + urn - PT i) 0 ii) C = <xg iii) 0 (16) with i) for the no hunting phase; ii) for the mixed hunting and farming phase; and iii) for the hunting phase only. 69 ii) Comparative Dynamics: Biological Growth with Hunt ing In comparative dynamics it is of interest to de termine the impact of hunting efficiency on the sta tionary state, the qualitative mode of dynamics, and endogenous switching in the regime of activity. To ac complish this, the nonlinear difference Equation (13) is transformed such that the dynamics of the no hunting phase can be analytically characterized. The impact of hunting is then assessed through a combination of mathematical analysis and computer simulation. The chaos transformation described in the previous chapter, results in Equation (17). The dynamics of z(t) within the range of zero to one is characterized by the parameter of dynamics, 'k', which can be expressed in terms of the parameters A, B, and C given in Equations (14), (15), and (16): = k(l-zt)zt (17) where; k = 1 + (l+f>1/2 (18) 70 where; f = B2 - 2B + 4AC (19) and the maximum level of the resource stock is: maxm = m + 1/m. (20) Since maxm can be greater than the stationary population, m, it is possible for overpopulation. Whether and how the dynamics without hunting will con verge to the stationary population will depend on the adjustment parameter, K, which is: k = 1 + M-m and the stationary equilibriums are: m = (0 ,m) Let the maximum sustainable resource stock, m, be normalized to one, allowing the dynamics in the no hunting phase to be given by the adjustment parameter for the biological growth process, M-. For example, with M=1.75 there is a cyclical convergence to the station 71 ary equilibrium m=l; with m. = 2.25 there is a cyclical convergence to a two period cycle around the same equi librium; and with m- = 2.83 there is chaotic or irregular fluctuations. Hunting introduced at various levels of effi ciency, p, can have a variety of impacts. The outcome is rather complicated with the potential for phase switching and changing qualitative mode of dynamics. It is not possible to analytically perform a comparative dynamics, and, hence computer simulation is used for specific examples. Consider a given resource stock. There will be a persistence of no hunting for low values of hunting efficiency, p. Not unless the efficiency of hunting, all else fixed, reaches a critical level, given by the phase switching condition, phf, will the activity switch into the mix hunting-farming phase. Or, given a level of hunting efficiency, as the resource stock grows during a no hunting phase, a critical amount of resource stock will be reached at which the phase of activity switches. In the new phase where hunting and farming occurs, the dynamics of the resource stock depends not only on the adjustment parameter, p., but also on the sensitiv ity of the biological growth process to hunting. 72 The parameter of dynamics, K, will be decreased slightly by hunting. Simulation experiments suggest that only when p i is slightly above a bifurcation point will hunting cause a change in qualitative mode of dy namics. Though in general, the impact will lower the stationary state, which the dynamics converges or fluc tuates about.^ During a cyclical trough in the state of the resource stock there can be a reswitching back into the all farming phase. If the efficiency of hunting is increased even further the hunting activity of Crusoe will dominate farming and eventually lead to one of several results: movement into the all hunting phase; an overshoot re sulting in such a small remaining resource stock that a phase switch into all farming occurs; or overhunting resulting in extinction of the resource stock. Take a specific example where the parameters tx and ( 3 are free for adjustment, <x = l/3, all other parameters are set at one, and the initial resource stock is at the stationary population, m(l)=l. With the resource stock adjusting at a percentage rate of m-=2.25 and with the hunting efficiency at p=.3, there is cyclical hunt- 1 The stationary states in the mix hunting-farming phase are the solutions to (m0 >: 0 = <xg + (nm - ( P/t ) (l-« > ( c+yT ) )m0—irm^ 73 ing - no hunting phase switching corresponding to the cyclical growth of the resource stock, see Figure 3.1. A more efficient hunting technology, P=.5, induces Crusoe to hunt in each period, though at a rate which fluctuates with the resource stock, see Figure 3.2. As might be expected the discounted present value of util ity derived over a twenty period horizon is increased. With the more efficient hunting technology, P=.5, consider a faster adjusting resource stock, m-=2.75. In this setting Crusoe overhunts in some periods leading to phases of no hunting interspersed among longer peri ods of mixed hunting and farming, see Figure 3.3. This has the effect of lowering utility over the horizon. Whether this adverse impact can be corrected for, with intertemporal planning is of interest to investigate. 74 A) DYNAMICS OF RESOURCE STOCK o O o I— ^ LO ZD ^ o o to UJ o o 7.5 10.0 12.5 15.0 17.5 20. TIME / 5.0 0.0 2.5 B) DYNAMICS OF HUNTING ACTIVITY O o CD 2 t — 2 3 X o o 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20 TIME Figure 3.1: Temporal optimizing (n = 2.25, p = .3) 75 A) DYNAMICS OF RESOURCE STOCK in / o : uj § 2 co UJ a : i i 7.5 10.0 12.5 15.0 17.5 TIME 5.0 0.0 2.5 B) DYNAMICS OF HUNTING ACTIVITY O CD i o 7.5 10.0 12.5 15.0 17.5 0.0 2.5 5.0 TIME Figure 3.2: Temporal optimizing = 2.25, p = .5) 76 A) DYNAMICS OF RESOURCE STOCK in LD O o ZD in O d to UJ C t L o « o 50. 30.0 20.0 0.0 TIME B) DYNAMICS OF HUNTING ACTIVITY o t— Z ID X o o o • o 40.0 30.0 20.0 10.0 0.0 TIME Figure 3.3: Temporal optimizing (m- = 2.75, P = .5) 77 Ill INTERTEMPORAL OPTIMIZATION A renewable resource allocation problem is a capi tal theoretic issue involving intertemporal tradeoffs. In the previous section the temporal interdependence in the use of the resource stock was decomposed using na ive temporal optimizing such that no future value to the end of period resource stock is accounted for in the current period plans. With intertemporal optimiza tion an imputed value which depends on the future use of the resource stock enters as a new factor in current period planning. In theory, intertemporal optimization leads to an intertemporal efficient valuation and allocation of a resource stock. Unfortunately, resolving the intertem poral dependence among the sequence of hunting activi ties and stock valuations over the planning horizon involves considerable computation. For all but the sim plest of problems, it is necessary to approximate the solution using computer techniques which involve limit ing the planning horizon to n-periods and assuming an end of horizon value of resources. In the approximation technique, the initial period resource allocation decision will be influenced by the length of the planning horizon and the assumed end of 78 horizon value of resources. The impact of extending the planning horizon beyond temporal optimizing should be considered. Computational cost increases as the horizon is extended. There is also a need for anticipations data whose accuracy decreases over the planning hori zon. On the other hand, incorporating the intertemporal dependencies may improve the utility derived from an approximately optimal control sequence. With Crusoe using intertemporal optimization, it is of interest to illustrate the impact of a forward looking imputed value and the impact of extending the planning horizon. It can be expected that a positive value will decrease current use of the resource which will reduce current period utility, but will increase the discounted present value of utility over the plan ning horizon. The renewable resource allocation problem in a Robinson Crusoe economy is formulated as a discrete time dynamic program. Of the various approaches to de riving a solution one based upon Bellman's principle of optimality and iterative search is used. The choice is dictated by the nonlinearity in the model and the exis tence of a bounded set for both the control variable, hunting, and state variable, resource stock. Given an initial resource stock a sequence of approximately op- 79 timal controls is derived which maximizes the dis counted present value of utility over the planning ho rizon plus the assumed end of horizon value of remaining resources. Using the notation in the previous section, the only new notation to be introduced for the intertempo ral optimization problem is: Costate variable to the state transition equation, also referred to as the im puted value to a change in the resource stock; Qfrr^) = Piji+2 : Encl of planning horizon value of the re source stock, assumed zero in computa tions; and R . j - = 1/d + r^.): Time discount factor, assumed constant over time The utility function and state transition equa tion, in terms of the control variable, ht, and state variable, m^., are (see Equations < 1) — (5 > of the previ ous section and assume ht = T-f^.) : 80 uthj-firij.) = <xln(c+i/(T-ht) ) + < 1-cx ) In ( g + Pm^h^. > (21) f(ht ,mt ) = mt+2 - n»t = |Ji<m-mt )mt - |3mth^. (22) where the bounds on ht and m^. are: 0 < ht < T (23) 0 < < maxm = m + 1/ti (24) The discrete intertemporal optimization problem is to determine the sequence of optimal controls, h^., for periods t = 0,1,...,T-l, which maximizes the return function: T-l J(m ) = T . (R^u(h.,m,) + P, - (m. , --m.-'f <h. ,m. ) ) ) (25a) o g t t t+1 t + 1 t t t + Q(n»T ) given mQ J (mQ ) = ... (m^. + ^ -ra^-f ( • ) ) Qtm^,) = PT+1 = 0 < 25b) 0 < h|_ < T (25c) 0 < < m + 1/m- (25d) Let the equation of dynamic programming J(m^.) be the return for the process beginning in state mt at 81 time t = 0 and using an optimal control policy ht for 0<t<T-l. By the principle of optimality the optimiza tion problem is: J(m.) = max (R^u(h. ,m. ) + Rt + ^J(m. - )) (26) t , t t t + 1 ht where mt+l = mt + f<mt'ht) <27) From Bellman's principle of optimality the inter temporal optimization problem is transformed into maxi mizing the utility from current period consumption plus the discounted present value of the utility which is anticipated from the future use of the end of period resource stock. There is a problem of maximizing con flicting objectives, involving the current and future use of the resource stock. A balance is found where conditions of optimality are met (Dorfman, 1969; In- triligator, 1971). Following Dorfman (1969) the two optimality condi tions, one relating to the optimal path of hunting and the other to the optimal path of resource stock dynam ics, can be interpreted as follows. First, Crusoe 82 should choose hunting activity in every period so that the marginal current period utility just equals the value of the resource stock, imputed price, multiplied by the effect of hunting on the accumulation of the resource stock. Ih; + pt*i aKt = 0 {281 The second optimality condition has the decreased value of a unit of resource stock, the temporal de crease in the imputed price, equal to the sum of its marginal utility in the current period and its contri bution to enhancing the value of the resource stock at the end of the period. _ gtjl = du + df_ 29 dt dm, t+1 dm, t t As one can see the current period optimal hunting activity depends on the future hunting activity and dynamics of the resource stock. To capture the inter temporal dependency in an explicit solution is computationally complex. It is necessary to apply com puter techniques in which these optimality conditions are approximated. 83 In this work an iterative search and comparison is made in a dynamic programming solution algorithm. As suming an end of planning horizon value of resource stocks, J(mT) = P - J - + 1' it is possible to compute back ward in time the optimal control of hunting effort for each level of resource stock in each period. Given such a table of values it is possible to specify, for ward in time, the sequence of optimal controls and state variables, given the initial resource stock (Bellman and Dreyfus, 1962; Bellman, 1961). The imputed value for the end of period resource stocks can also be approximated in each period with the change in the fu ture period return function from an incremental change, A, in the end of period resource stock: Pt + 1 = {J< <ra+<d) t+1* " J<mt+1))/d (30) A computer algorithm is used to derive the se quence of optimal hunting activities, the derived util ity and the dynamics for the resource stock. Different planning horizons are assumed to experiment with the Both the level of resource stock and quantity of hunting effort are considered at discrete intervals with arbitrary small increments. 84 impact of forward looking behavior. But in each case the end of horizon value of resources was assumed zero. First compare the outcome from a sequence of tem poral optimizing, which is intertemporal optimizing with a one period horizon, with the solution from in tertemporal optimizing with a twenty period planning horizon. Both hunting and utility in the first period will be lower, but the discounted present value of utility over a twenty period simulation horizon does not decrease. The improvement in utility over the hori zon depends on parameter values. It is possible, due to the relative preference of corn to goat meat, <x, the relative efficiency of hunting to farming, P/y, and the biological growth rate, , that Crusoe's temporal op timizing hunting and derived utility may be equivalent to the intertemporal optimizing solution. For the comparison made in the previous section, where (3=.5 and M- is changed from 2.25 to 2.75, the utility derived is greater in both cases with intertem poral optimizing, and in contrast to the outcome with temporal optimizing there is an increase in utility corresponding with a faster resource growth rate. The impact of an approximately optimal imputed value of resources improves the balance of Crusoe's activity with the dynamics of the resource stock, and hence 85 phase switching is reduced and greater utility is de rived. Compare Figure 3.4 with Figure 3.3. It is of interest to consider the impact of chang ing the horizon on the initial hunting decision. In going from a one to two period horizon the magnitude of hunting is decreased. This is reasonable for a positive imputed value reduces the incentive to hunt. Going from a two to five period horizon there is some increase in hunting, but it does not surpass the level of temporal optimizing. As the horizon is extended beyond five pe riods hunting activity does not change significantly. With different planning horizons it is less mean ingful to compare utility unless a recursive solution over the twenty period simulation horizon is used. This is expensive to pursue for a recursive solution in volves a repetitive solution, one for each period of the simulation horizon, of a multiple period dynamic program. For the parameters used above, in a two period planning horizon with recursive solution there is some, but little, improvement in utility. With a five period horizon there is some improvement over temporal op timizing and the two period solutions. But, neither re cursive solution improves utility from the one time twenty period optimization. 86 Part of the poor performance of recursive inter temporal optimizing with a short (2-5) planning horizon is that the end of horizon value of resources is set at zero. With a fluctuating resource dynamics this may be out of phase with the size of the resource stock, lead ing to inefficient intertemporal tradeoffs. 87 A) DYNAMICS OF RESOURCE STOCK / i o O o I i n LJ t x. 10.0 12.5 15.0 17.5 20. TIME 5.0 0.0 2.5 B) DYNAMICS OF HUNTING ACTIVITY C J D \ 7.S 10.0 12.5 15.0 17.5 TIME 0.0 2.5 5.0 C) DYNAMICS OF IMPUTED PRICE o o DC O C l. o o 20. 10.0 12.5 15.0 TIME 7.5 0.0 2.5 5.0 I Figure 3.4: Intertemporal optimizing (i j l = 2.75, P = .5) 88 IV BOUNDED RATIONALITY: CASE OF PROXIMATE DYNAMIC PROGRAMMING To resolve the intertemporal dependencies among a sequence of resource using activities, a boundedly ra tional agent uses anticipations of future conditions in making current period plans rather than trying to de rive a sequence of intertemporal optimal plans. A rep resentation of behavior can be derived from the theory of dynamic programming with a behavioral approximation to the principle of optimality. An approach taken here, that of proximate dynamic programming, is different from that taken in the temporary equilibrium models. In the later, the backward optimization of dynamic pro- gremming is applied to a two period horizon in which an asset transferred between periods indirectly enters the current period utility function, and, hence receiving an imputed value (Grandmont, 1977). With proximate dynamic programming an imputed value to end of period resource stocks is specified with a behavioral pricing rule, decomposing an inter temporal optimization problem into a sequence of tempo ral optimizations with feedback. Two issues are how to specify the imputed value at the beginning of each pe riod on the basis of existing data, and what impact 89 does such a valuation procedure have on the activity of Crusoe, the utility derived over the planning horizon, and the dynamics of the resource stock. To explore these issues the results from simulation experiments, using several different pricing rules, are compared among themselves and with the results from naive tem poral optimizing and n-period dynamic programming. First, the model of proximate dynamic programming will be presented. Then rules for imputing the present value to the end of period resource stock are derived. i) The Model Derivation of proximate dynamic programming from an intertgmporal optimization problem can be made in two steps. The first follows a standard dynamic pro gramming argument, applying Bellman's principle of op timality and taking a linear approximation to the fu ture return function (Bellman, 1957; Dorfman, 1969). The second step, following the work of Cigno (1970), introduces a behavioral pricing rule for specifying the end of period value of resources, decomposing the in tertemporal dependencies among the sequence of activi ties and simplifying the problem into one of temporal optimizing with feedback. 90 Consider the intertemporal problem introduced in the previous section, and apply the principle of opti mality to get: J(m^) = max(u(h^,m^) + RJtn^)) (31) hi s.t. m2 = m^ + ffm-^h^) m^ = given beginning of period resource stock With a linear approximation to the future return, this optimization problem can be further simplified. First consider an increment of time A, for which the return function can be rewritten by substituting for m 2 from the equation of dynamics for the resource stock: J(m^) = max (u( h^ ,m^ )A + (1-r-^ ) J (m^ +Af (m^ , h^ ) ) ) (32) hi The linear approximation to the future return re sults in: J(m-) = max(u(h1,m-)A + (1-rA)(J(m-) + AJ f(m1,h.)>) (33) 1 , 1 1 1 m i l Approximate the discount factor as follows: R - * - = 1 - rA; where R = l/(l + r). 91 Subtract J(m^) from both sides, divide by A and let A go to 0: 0 = max<u<h1,m1) - rJ(m1) + J f(m1,h-)) (34) , 11 1 m i l hi Finally, since rJ(m-^) is by definition evaluated prior to determination of h-^, the term can be removed from the maximizing problem, the result is: rJ(m-) = max{u(m-,h-) + J f(h.,m.)) (35) 1 , 11 m i l hi The optimization problem is now one of choosing the current period hunting activity, h-^, which maxi mizes the sum of the current period utility plus an imputed value to the change in the resource stock. The imputed value, Jm , the equivalent to the costate vari able in optimal control, provides the intertemporal link between the current and future activities. Its calculation should reflect the discounted present value to the future use of the resource stock. In proximate dynamic programming a behavioral ap proximation based upon data existing prior to the cur rent period decision, is used to specify the imputed value. With such an approach, a sequence of temporary equilibria can be derived as with naive temporal op 92 timizing. One significant difference is that the phase switching conditions and the optimal level of hunting in the mixed hunting-farming phase depend on the im puted value which, as will be discussed below, depends on the past level of resource stocks and hunting ac- 1 tivities. Consequently, the form and order of the non linear difference equation describing the dynamics of the resource stock is no longer subject to analytical tools, as with the naive temporal optimizing and, hence computer simulation is the necessary form of dynamic analysis. First, it is necessary to explore the speci fication of the imputed value. Phase conditions and corresponding Equation of Dynamics are: i) all farming phase: ht = 0; ft = T phf : 0 < i r i j . - (on/m) / < (c+l/T) P (l-oc-mPt+^ ) > ii) all hunting phase: ht = T; ft = 0 phh: 0 < (ocy/c) - Prn^ ( (l-« ) / (m+ PTm.^) ) - P^ + l iii) mix hunting farming phase: ht = H(mt,Pt+^) a quadratic formula. iv) In all three phases the equation of dynamics is as follows, given the appropriate ht value. mt + l " mt + ^ * ™~mt ^ mt “ ^mt^t 93 xi> Imputing Future Values The imputed value to a current change in the re source stock is the discounted present value of the utility which could be derived from the future use of the resource stock. But, it is this dependency on fu ture decisions which is of interest to eliminate with a behavioral approximation. The problem is to derive a pricing rule which can be specified on the basis of data existing at the time of planning current period activities. This is what agents do in assessing the value of capital assets and resource stocks. According to the assumptions on the dynamics for an economy and the rationality of agents, different rules for imputing values to the future use of stocks can be suggested. In general economic theory suggests that the imputed value (price) per unit of an asset is the marginal utility of the stock from its future use, discounted by the rate of time preference and accumu lated over the life of the asset (see Mussa, 1977). For instance, with the economy in a stationary state and the resource stock producing a constant stream of re turns over an infinite life, the imputed price per unit 94 of stock simplifies to the present value of a perpetual 1 flow. Pt+1 = < V r where P^. + ^: Imputed value to a unit of stock Um: Marginal utility derived from the stock r: Rate of interest This is the formula used in the temporary equilib rium literature except with a two period horizon the discount factor is omitted. In other words, the imputed value is the current period marginal utility. For a dynamic process with agents having perfect foresight the conditions of the future can be capitalized into a current value. With stationary sto chastic future returns and agents having rational ex- pectations imputed values are still calculable. But as If the stock is depreciating over time at an ex ponential rate, d, then divide marginal utility by r+d (see Mussa, 1977). 2 The subject of valuing uncertain future income streams in a stationary stochastic environment is a current research issue which will not be addressed here, but note its relationship to the problem on hand. 95 more and more complexity in the economy is allowed for and the bounded rationality of agents accounted for, the means for specifying imputed values becomes subjec tive, relying on some form of behavioral rule. This di lemma of subjectivity has been more noticable during the 1970s and 80s with inflation and speculative booms in commodity prices (Modigliani and Cohen, 1979; Brainard, et al., 1980; Cooper and Lawrence, 1975). The decomposition technique proposed here is designed as a behavioral representation of a boundedly rational agent imputing future values under such dynamic conditions. For deriving the pricing rules consider Crusoe who attempts to maximize the present value of utility from using a renewable resource, i x i j - , over a time horizon of ' T' periods. Given an initial resource stock and assum ing a discounted present value of final resource stocks, Q(m,j,), a sequence of hunting decisions, ht, is made from which a time stream of utility is derived: T-l J(mt,ht> = max £ u(m^_,ht,t) + Q(mT> (36) ht t=1 where the resource stock equation of dynamics is: ™ t+1 = f(mt,ht, t ) (37) 96 From the principle of optimality and a linear ap proximation to the future return, the criterion is (see previous section): rJ(m.) = max(u(h.,m,) + J f(h,,m,)) (38) t - i t u m t t ht Taking the derivative with respect to mfc, letting the imputed value Jm be + ^ and h^- equal the optimal hunting activity, h£, then: rI>t + l ^ut + + ^t+l^m* (39) To specify the imputed value from this formula it is necessary to know the optimal level of hunting. This simultaneity problem is to be resolved by using an ex pected optimal hunting activity, E(h£), for an antici pated level of resource stock E(m|_). Various expecta tions formulas can be used. In the simulation experiments to follow, two formulations are proposed. First naive expectations: E(m^_) = m^__ 2 <40a) E(h£) = h^_x (40b) 97 Second, assume the agent observes the beginning of period resource stock, i i i j . , and then anticipates the optimal hunting activity as follows: E<m^.) = m^ (41a) E(h£) = ht_^ + p (mt-mt_-^)/mt_-L (41b) where p = parameter; 0 < p < 1 Using these anticipated conditions the specifica tion of the imputed price is: u + J f p = —13 mm_ (42a) Pt+1 r - f m Some simplifications can add insight. First con sider Jmm = 0, which has the implication that the im puted price is invariant to the level of the resource stock. In general this is not true, for with a larger resource stock the imputed value can be expected to decrease, < 0.^ As a first approximation to the An approximation is: Jmm = J(m+^J) - 2J(m) + 98 pricing formula this assumption will be used along with fm = 0, to get the formula for the present value to a perpetual flow: um/r < 42b) A second simplification is from assuming a sta tionary state where f<mt) = 0, fm = -|i.E<mt>, and, hence Pt = u^j/(r + M-E(mt>) (42c) This formula is similar to the one for a present value to a perpetual flow. A difference is that fm = -(j.E<m|.> is a rate of appreciation for the resource stock, and is subtracted from the rate of interest. In other words a rate of depreciation is added to the in terest rate. Outside a stationary state, the imputed value will change with the resource stock and hunting activity in a dynamic feedback process, given by Equa tion < 42a). From a behavioral approximation to the intertempo ral optimization problem, a variety of rules for imput ing values to end of period resource stocks are de rived, namely Equations (42a, b, c). These are only three of many possible rules, which do not incorporate 99 speculative behavior where the potential for short-run arbitrage gains are behind the valuations. Both opti mism and pessimism, not captured in current data can sway the valuation of resource stocks. The impact of such speculative behavior can be assessed relative to the above formulas, which attempt to approximate the intertemporal optimal imputed value. Questions of interest for investigation with a comparative simulation analysis pertain to the relative impact among these different pricing rules and with a comparison to the outcome from temporal optimizing. In no case did proximate dynamic programming result in greater utility relative to intertemporal optimization. As discussed several pricing rules are tried and the outcome, discounted present value of utility over a twenty period horizon, varies. Both (42a) and (42c) improve the outcome relative to temporal optimizing, whereas the simpler rule, (42b), has a tendency to re duce utility, see Figure 3.5 in comparison with Figures 3.3 and 3.4. Even in cases where the resource stock with hunting converges to a steady state with the more sophisticated pricing rules, the simpler rule can be destabilizing, leading to fluctuations about the steady state. See Figures 3.8 and 3.9 where pricing rule 42a and b are compared for m.= .75 and p=.5. Outside a sta 100 tionary state the asset pricing rule used in the tempo rary equilibrium models can be improved upon. Using the parameters as previously specified and comparing the outcome between pricing rule (42a) and (42c) there is a tendency for (42a) to be better, see Figures 3.5-3.7. This can be explained in part by the fact that Jnmj is fixed exogenously. When the resource stock is fluctuating, the rate at which the price changes with respect to the size of the resource stock can be important and may bias the result when poorly specified. As for the two expectation formulas the first seems to provide a better outcome. This result may be sensitive to the parameter, p, which has been assumed but not experimented with. A final observation on the simulation results is that the initial imputed price must be prespecified and different values influence the outcome. There is a dynamic feedback effect from past price. Using the initial price from intertemporal op timizing improves the outcome. 101 A) DYNAMICS OF RESOURCE STOCK \ 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20. TIME - X B) DYNAMICS OF HUNTING ACTIVITY C) DYNAMICS OF IMPUTED PRICE a o LJ O o_ o o o o i I 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 TIME Figure 3.5: Proximate dynamic programming (u = 2.75, P = .5) with pricing rule 42a and expecta tions 40 102 A) DYNAMICS OF RESOURCE STOCK i n o O o o 10.0 12.5 15.0 17.5 20. TIME y 0.0 2.5 5.0 7.5 B) DYNAMICS OF HUNTING ACTIVITY o / CJD | Z ° z> O 7.5 10.0 12.5 15.0 17.5 20.1 TIME J 5.0 0.0 2.5 C) DYNAMICS OF IMPUTED PRICE / O t o fs . o. d UJ Q O g CE d CL t o OJ o„ o o o o 10.0 12.5 15.0 17.5 20. TIME 7.5 5.0 0.0 2.5 Figure 3.6: Proximate dynamic programming = 2.75, P = .5) with pricing rule 42b and expecta tions 40 103 A) DYNAMICS OF RESOURCE STOCK t / > LJ O cn ID «> I ! o " o 20.' 10.0 12.5 15.0 TIME 7.5 0.0 2.5 5.0 B) DYNAMICS OF HUNTING ACTIVITY CD o 7.5 10.0 12.5 15.0 17.5 20.. TIME 0.0 2.5 5.0 C) DYNAMICS OF IMPUTED PRICE i d o LJ CD ce Q_ o 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20. TIME Figure 3.7: Proximate dynamic programming (m- = 2.75, P = .5) with pricing rule 42c and expecta tions 40 104 A) DYNAMICS OF RESOURCE STOCK I I o o o S" i o o c . ZD O o ud LJ Q C i I i o o 10.0 12.5 15.0 17.5 20. 7.5 0.0 2.5 5.0 TIME B) DYNAMICS OF HUNTING ACTIVITY / cs 2 ° o o 10.0 12.5 15.0 17.5 20./ 0.0 2.5 5.0 7.5 TIME C) DYNAMICS OF IMPUTED PRICE in i o 7 .5 10.0 12.5 15.0 17.5 0.0 2.S 5.0 TIME Figure 3.8: Proximate dynamic programming (m- = .75, P = .5) with pricing rule 42a and expecta tions 41 105 A) DYNAMICS OF RESOURCE STOCK o O o LU < _ > o c O o to Lj o n I I o o 10.0 12.5 15.0 17.5 20. 7.5 5.0 0.0 2.5 TIME B) DYNAMICS OF HUNTING ACTIVITY < 5 i I o i z ° r> o o 10.0 12.5 15.0 17.5 5.0 0.0 2.5 TIME C) DYNAMICS OF IMPUTED PRICE / o 7.5 10.0 12.5 15.0 17.5 TIME 0.0 2.5 5.0 Figure 3.9: Proximate dynamic programming (m- = .75, p = .5) with pricing rule 42b and expecta tions 41 106 V SUMMARY Crusoe's problem of managing the use of a renew able resource is one involving intertemporal tradeoffs. There may well be an intertemporal efficient use of resources, but it is questionable whether boundedly rational behavior will achieve such efficiency. The computational problem of imputing a present value to the future use of resource stocks and capital assets has implications for economic analysis which have not been fully explored here. Smith (1975) discusses how earlier societies have relied upon constraints and social values, imposed through religious and social custom, to control the use of resources. A decentralized market economy attempts to rely upon pricing through market transactions. The development capital asset and commodity future markets are examples. The efficiency of pricing in these mar kets is in need of investigation, particularly outside the efficient market framework. Some economists such as Kaldor and Keynes have suggested the use of interna tional commodity controls as a supplement to market pricing. Hopefully, the behavioral approach to deriving pricing rules from the principle of optimality in dy- 107 namic programming is a step in this direction. I leave for future work the development of these alternative pricing rules in context of a consumption-savings port folio problem in a temporary equilibrium. 108 CHAPTER 4 A PROTOTYPICAL FIRM: A MICROFODNDATION TO MACRO DYNAMICS I INTRODUCTION Consider the firm as a governance structure for planning, monitoring, and adapting the use of produc tive and financial resources. Its internal organization consists of a hierarchical structure involving top management, product divisions, and departmental func tions. Representation of firm behavior as the outcome of a coalition of these subunits which may have con flicting objectives and be influenced differently by market feedback, is a complex and unresolved issue. Simplification is necessary to reduce actual complexity into a theoretical structure and computable model, ade quate for the purpose of analysis and consistent with assumed rationality of agents and assumed market-envi- ronment. The micro detail, behavior and organization of the firm, considered in this work is dictated by principles of boundedly rational economizing and the aim to inte 109 grate a set of firm models into a computable dynamic multisector model of industrial development where in terindustry transactions can occur in or out of market equilibrium. There is no general agreement as to the appropri ate representation of a prototypical firm in market disequilibrium. For some time a trial and error search process has and will continue to occur, during which model structure is somewhat unique to the modeler. Some examples can be given. To start, there is the classic behavioral theory of the firm by Cyert and March (1963). Principles of satisficing behavior are used in simulating price, production and sales activity adjust ments, coordinated by a profit goal set by top manage ment. Using similar principles of behavior, Forrester (1961) has emphasized the feedback effect of delivery delays, whereas Kornai (1980) has emphasized the impli cations of shortages and the role of queues. Eliasson (1976) uses an additive targeting formula to decompose production and investment decisions and introduces a bidding-negotiating-bargaining process to determine price and mediate plans into transactions. Using constrained optimizing computations, Benassy (1982) and others developing the nonwalrasian equilib rium firm model have emphasized the role of perceived 110 quantity constraints and rationing in the market. The recursive programming models of industrial development by Day and others emphasize the impact of cautious sub- optimizing on firm production and investment planning in a sequence of temporary equilibria, with and without financial feedback. See Day and Cigno (1978) for the later and Day, Morely and Smith (1974) for the former. In this work I use these recursive programming models as the basis for developing a prototypical firm model, within an adaptive economizing framework. The planning and control structure of the firm is embedded into a market-environment feedback process, see Figure 4.1. The market-environment is specified so that the current period's plans are fulfilled or errors in ex pectations lead to the nonfulfillment of plans. The former results in a sequence of temporary equilibria, the later in market disequilibrium. In both, analysis is directed towards the dynamic and long-run implica tions of hypothesized firm planning and adjustment be havior . Firm planning involves production, investment and finance activities. These are derived in conjunction with pricing, cash flow budgeting and adjustment of in ventory and order backlog. According to economic and technological conditions, firms in different sectors 111 rely upon different principles of planning. Some pro duce to order while others produce for stock. Some are price takers while others set price. It is of interest to develop the firm model to incorporate these distin guishing characteristics, and be flexible enough to represent these different types of firms. In this chapter I discuss the organization and behavior of a prototypical firm and develop several variants as computable simulation models. The firm-sector models are then linked into a multisector model of economic activity in or out of equilibrium. Data Product market: Demand and Cost Labor Market: Supply and Wage Decisions: Investment,Finance, and Production Financial Market: Supply of Funds Interest Rate Planning Control Figure 4.1: A firm-market feedback structure 112 II FIRM ORGANIZATION AND BEHAVIOR The prototypical firm's planning process is con sistently specified with its hierarchical structure of internal organization. The multidivision firm coordi nated by top management is common, where each product division has a set of functional departments which plan and carry out activities. Issues pertaining to the planning process of hierarchical organizations focus around rules of decomposition and coordination among subunits and with top management, as well as the deci sion rules used in each subunit. In this work I only consider a single division. This leaves out some important changes which have oc curred in industrialized economies, where vertical and horizontal integration have replaced market coordina tion functions. Still in these multidivision firms, functional decomposition and coordination is a part of firm structure and behavior in need of representation, particularly outside the rationality-equilibrium frame work. Motivation for the structural change which has occurred may eventually be explored with such a model. Perhaps, as suggested by Williamson <1975), it may be shown that due to bounded rationality, adaptation to uncertainty may be better accomplished by administra- 113 tive processes in a sequential fashion, rather than by markets. The single division firm involves a hierarchical ordering of functional decisions. A firm's production, investment and finance plans are not derived simultane ously from any single criterion such as value maximiz ing. Different departments carry out different planning tasks in a recursive sequential process, coordinated through behavioral rules of decomposition. The prototypical firm model has the organizational structure of that in Figure 4.2. Top management sets targets with which the different departments carry out their planning tasks. How will be discussed more fully in the next section, see Table 4.1 for a brief outline. In the remainder of this section the sequential process of planning is outlined as in Figure 4.3: To start demand expectations are formed and for a price setting firm, a cost markup price is specified. For those firms which are price takers, an expected market price is formed. The planning of investment and finance activities is in pursuit of anticipated growth in demand. Finan cial targets, cash flow budget, and anticipated demand serve as constraints in an expected revenue maximizing computation. The investment strategy involves a choice 114 between capacity expansion or investment in external market opportunities, represented by banking. The ex pansion of capacity involves a choice of technologies, each characterized by cost and productivity coeffi cients . Financing involves, first, internal funds retained from the previous period and external funds in the form of debt and equity. There is a constraint on the new issue of equity, a maximum acceptable debt to equity ratio, and a cost differential between the two. The scheduling of production with the available capital is accomplished through revenue maximizing com putations. There are capacity constraints, factor sup ply constraints, and a demand constraint. The form of the latter will vary according to whether the firm pro duces to order or produces for stock. Orders for factors of production are derived from production plans, but ones which extend into the future since an order-delivery lag exists. The sales department for a price setting firm han dles shipments, fulfilling orders. Rules for rationing available supply to a backlog of orders and using in ventory stocks are specified. For a price taking firm sales are equated to demand and a temporary equilibrium price is determined. 115 The budgeting department completes the activities of the period. The revenue received from sales are dis tributed. First, to pay obligated costs for materials, labor, interest on debt, and taxes. The remaining net revenue is divided between dividends and earnings to be retained as working capital and internal funds for in vestment . Omitted from analysis are several dimensions of firm behavior which need to be recognized. First, are activities relating to multifirm competition in a sin gle sector, which involves sales and pricing policy effort to gain market share. These are not considered here, for the aim is to first develop a multisector model comprised of single firm sectors. Second, is re search and development and endogenous technological change. Third, feedback from labor and financial mar kets which are not explicitly represented. 116 Pricing Budgeting Capacity Expansion External Finance Production Sales Investments Engineering Accounting Top Management Finance Figure 4.2: Firm organizational structure 117 Table 4.1: Organizational structure with planning task I) Top management and targeting i) Leverage, dividend payout, and working capital ii) Price markup and payoff period for capital II) Engineering and accounting planning data i) Capital and production cost coefficients ii) Production activity coefficients iii) Anticipations data III) Production department i) Schedule output and operation of capital ii) Control factor inventory and order backlog IV) Sales department i) Shipments ii) Control product inventory and order backlog V) Investment and finance departments i) Investments a) Capacity expansion and banking b) Acquisition of external financing ii) Pricing a) Cost markup and adjustment for b) Demand elasticity and competition c) Inventory and order backlog disequilibrium iii) Budgeting a) Collect revenues and pay obligated costs b) Distribute dividends c) Maintain working capital 11£ Stop Top management targeting Initialize stocks and data Generate end of period report Pricing department Cost markup plus adjustment Investment-finance department Capacity expansion and external finance Engineering and accounting planning data for production, investment and finance Production department Schedule output and control factor inventory Budget department Distribute revenues, maintain working capital Sales department Shipments and control inventory and order backlog Figure 4.3: Sequential planning 119 Ill MODEL COMPONENTS In this section I discuss the process by which de partments plan and carry out activities. The central features are capacity expansion, production and pric ing. The first two are derived from constrained linear programming computations. Their technical specification follows closely a class of recursive programming models previously developed for describing and projecting in vestment and production planning of individual economic sectors in a temporary equilibrium framework (Day and Cigno, 1978, Ch. 4; Day and Nelson, 1973). These models have been applied to the U.S. petroleum, steel, and coal industry illustrating a capability to simulate in dustry response to a variety of market-environment situations. Finance and budgeting considerations are also taken into account along the lines of the model by Day, Morley and Smith <1974). The approach taken is similar to the programming approach to corporate financial man agement which goes back to Weingartner (1963). Building on these earlier computable firm models it is of interest to be more explicit with the impact of construction lead times on capacity expansion and cash flows, the control mechanisms for inventory and 120 order backlog for final products and factors of produc tion, as well as price setting behavior. For reasons of expediency in presentation the specification of plan ning data by the engineering and accounting planning staff is incorporated into the discussion of the rele vant functional department. This involves the cost co efficients in capacity expansion, factor productivity coefficients in production, and anticipations data con cerning future demand, factor prices, and interest rates. i) Targeting of Top Management With bounded rationality the concept of a corpo rate criterion for deriving a consistent set of optimal production, investment and finance activities is re placed with a system of targets from top management. What optimizing computations used by the departments are heuristic conveniences for deriving a unique set of activities within a constrained set. The constraints, based on the targets, play the significant role in de termining the activities of the firm. Two difficulties in setting targets are internal consistency and anticipating market response. Targets serve to decompose and coordinate the activities of the 121 departments. Due to inconsistencies in their specifica tion the internal conflict may be resolved in a suboptimal fashion. Implications on firm performance is discussed further in Donaldson (1985). A second set of issues relate to anticipating mar ket response to activities planned consistently with targets. For instance, there is a sales response to production and marketing, and there is financial market valuation of the firm in the form of equity prices and bond ratings. Targets decompose the determination of activities from such market considerations. Though, in response to the market outcome top management will ad just targets. A causal relation between planned activi ties and market valuations is not used to determine an optimal set of activities. Targets are characterized with a priority ordering and a value. Both evolve out of past behavior. Economic man, being boundedly rational, is habit forming and relies on rules of thumb which have worked in the past. Some old rules, formed during different stages of in dustrial development and states of economic prosperity, may not work well under existing conditions. For exam ple, Ellsworth (1983) suggests that long accepted fi nancial standards prevent firms from taking profitable investment opportunities.On the other hand, Drucker 122 (1980) finds that during prosperous times firms lower liquidity standards only to find themselves in a cash flow crisis when turbulent times reappear. Top management has the role of specifying and ad justing targets. In the models developed here, feedback adjustments are omitted. Not until market processes are explicitly modeled will the feedback adjustment mechanisms appear. Specification of targets for the various departments will be discussed in the appropri ate model component. Here only a few comments will be made in regards to anticipations data. The driving force of the firm is anticipated sales growth. Since the model does not entail market competition, targets refer to anticipated growth in demand. There is a short-run forecast for production planning and long-run forecasts for investments. There will also be forecast of factor prices and interest rates. In the exploratory models of this work simple moving averages are used (Wheelwright and Makridakis, 1977). Errors in these expectations, which are common in actual business practice, is the basis for introduc ing market disequilibrium behavior (Feldstein and Auerbach, 1976; Nerlove, 1983). 123 ii) Pricing Firms set price according to costs, conditions of market demand and competition, and financial market feedback. There is a sensitivity of sales and revenue to price which depends on highly uncertain market con ditions. The price which maximizes revenue is not com putable where bounded rationality prevails. What occurs is the application of a behavioral rule which serves as a link or principle of decomposition between the real (sales) and financial (budgeting) activities. Production, sales, and cash flow budgeting need not be optimally nor consistently coordinated with price, since errors in demand and cost expectations prevail. The discrepancies between demand and sales and between cash inflows and outflows are allowed for through the existence of inventory and order backlog buffer stocks and cash liquidity. As a feedback re sponse to disequilibrium there will be adjustment around a cost markup price. Cost markup is the basis for a pricing rule which has its roots in observed behavior (Hall and Hitch, 1939; Silberston, 1970). Within limits imposed by mar ket conditions, price is set to generate revenue con sistent with costs, financial rate of return, and pay 124 off targets. Since early pricing studies, controversy remains over which costs and product volumes to use, how to specify the markup, and the factors of impor tance in price adjustment <Eckstein and Fromm, 1968; Silberston, 1970; Godley and Nordhaus, 1972; Nordhaus, 1974; Okun,1981). To illustrate some specification problems consider a simple stationary equilibrium setting where average cost is constant both over time and with the level of production. The costs per unit of capacity include variable cost, c, and capital cost, p, depreciated into a per period cost, &p. The pricing rule with markup, d, is: P = (1+d)(c+&p) With c and p given, it remains to specify the tar get payoff period used in specifying depreciated capi tal cost, 1/&, and the markup, d. The payoff period is treated as a top management target. Firms rely upon payoff periods shorter than the productive life of capital, as a hedge against uncer tainty (Gordon, 1955; Smith, 1961; Weingartner, 1969) The markup is specified to achieve the target rate of return to the equity owners of the firm. Original 125 equity is the only source of external funds in this stationary equilibrium. With all equity financing, both operating and capital expenditures are from the income of equity owners, and, hence should earn the markup return. The tradeoff between current and future consump tion by the equity owners is the basis for the markup. How can be illustrated by looking at the owners con- sumption-savings decision in a stationary equilibrium. Building on the price rule above, let production x be equivalent to capacity, y, and the firm's equity be the expenditures, m, then: m = <c+&p)x Suppose dividend policy is a fraction, d, of eq uity, which is of interest to derive from the owners time preference for consumption. D = dm The firms budget constraint incorporates a revenue lag, so that revenue received, Px, is either paid out as dividends, D, or retained as next periods equity for capital and operating expenditures. With the economy in 126 a stationary equilibrium time subscripts can be dropped, and the budget constraint is: Px - D = m Upon substituting for D and m into the budget con straint, the price rule, as given above, is derived with markup d, the dividend payout ratio. The return to equity in the following period is one plus the firms rate of return, r, times equity: (1+r)m = (1 + r)(Px-D) Now introduce the owner of the firm who consumes dividend income in each period. Current consumption, CQ, equals D. Consumption in each period following is d times the return to equity: C^_ = d(l + r)(Px-D) t = 1, 2, ..., n Let C' be the perpetual flow of all future con sumption discounted to a present value: C' = d(1+r)(Px-D)/r 127 From the owners tradeoff for current versus future consumption a simple rule for the dividend payout and markup, d, can be derived. Using an addilog utility *| function the constrained maximizing problem is: max <xlnC0 + (l-oc)lnC' D s.t. C0 = D C' = d{1+r)(Px-D)/r The derived dividend policy is: with dividend payout and markup: d = - z — — &d/£><x > 0 1 - <x Given a fixed revenue requirement for working capital (stationary equilibrium) the markup varies with the demand for dividends which increases with the rate of time preference of current consumption to future consumption. A higher price is required to maintain the ^ There are simplifying implications from this utility function allowing d to be independent of r. 128 stationary equilibrium the greater is the demand for current dividends. It is the responsibility of top man agement to specify the markup on the basis of owner time preference for consumption and the revenue re quirement for working capital. This is no easy task. Out of the stationary equilibrium where demand is changing and average cost is not necessarily the same for all levels of output nor over time, other specifi cation issues exist. The extent price varies with changing demand is one way to distinguish cost markup pricing rules. Whether forecasted demand and costs are used, or whether a standard volume of demand and normal cost, using averages over the past, are used, is a basic dis tinction. In the former, price responds to changing demand while maintaining a target profit rate. In the latter, short-run fluctuations in demand does not imme diately stimulate price adjustment. With either pricing rule, inflationary trends in variable costs are generally passed on through price increases, an aspect of interindustry contracting (Okun, 1981; Wachter and Williamson, 1978). How best to include inflation in capital costs and capture the dif ference between historical cost and replacement cost remains an unresolved issue. The economic suggestion is 129 to use replacement cost, but this involves unavailable information and a change in accounting practice (van Dam, 1978; Hicks, 1976; Lee, 1980; Drucker, 1980). In this work historical capital cost is used. Cost markup pricing relies upon a systematic cost account. The costs for each department can be divided into variable, capital and overhead. Since the models of this work do not have decentralized budgeting, costs will be expressed for the firm as a whole. Labor and material costs which can be computed relative to pro duction using technical activity coefficients, are re ferred to as variable costs per unit of production. Capital costs are those for construction and mainte nance which can be expressed per unit of capacity. A purchasing price per unit of capacity times a rate of depreciation, as specified by the target payoff period, provides the capital cost coefficent in pricing. Costs which cannot be expressed in a technical relation to production or capacity are referred to as overhead costs. These will be included as an average cost per period given an anticipated product volume. There is a choice of cost component to which the markup is applied. What firms actually do is uncertain and the versions estimated with aggregate data do not provide clear evidence for a dominant pricing formula 130 (Eckstein and Fromm, 1968; Nordhaus, 1974). In this work the variable and capital cost components are used as the cost basis for markup. In addition to these costs there is a return to financial sources and a demand for working capital which are incorporated into the markup. Consideration for debt-equity structure and rates of return, dividend payout and retained earnings policy are, at least im plicitly, the basis to the targeted markup. The discrepancy between financial targets and the revenue generated from sales given price, is the basis for financial feedback on price. An imbalance of cash flows with financial targets requires the firm to learn about market conditions. Just as a monopolist seeks rents through adjusting price and learning about de mand, a boundedly rational firm in an uncertain mar- ket-environment will experiment with price adjustment seeking the target return to equity. Such a price ad justment and learning process is given in Day (1967) which can be, but currently is not incorporated into the price adjustment formula. A discrepancy in supply and demand is captured in the deviation of inventory and order backlog from de sired levels. In response there is a disequilibrium adjustment. To what extent price adjusts rather than 131 production is not clear. It is suggested that only when disequilibrium signals point strongly to changed demand conditions is a decision to revise the price structure seriously entertained (Scherer, 1970, p. 149-157). The division of response from industry to industry will vary according to technical and economic conditions specific to the sector. In the prototypical firm model adjustment parameters can be set accordingly. iii) Investments and Capacity Expansion The primary function of the investments department in the model developed here is expansion of production capacity. When demand is slack, or production not pro fitable, the internal funds available may be invested into market opportunities, represented by banking with a rate of return less than the borrowing rate. Cer tainly, a larger array of investment opportunities ex ist, such as the acquisition of other firms, but these will not be considered. Analysis is focused on expan sion of capacity and choice of technology. Financial and cash flow budgeting considerations will be taken up in the next section. At this point it only needs to be clear that they are constraints in an investment pro gram . 132 Investment in capacity is motivated by two consid erations, forecasted demand growth and replacement of depreciated stocks. Net revenue maximizing linear programming computations are made in each period for initializing capacity expansion projects. Once started a time stream of construction activity occurs over a prespecified lead time. In the models of this work there is no choice for project delay or abandonment, but due to market conditions a construction delay may occur. The extent of expansion is constrained by a behavioral bound on adjustment, that reflects hedging against uncertainty in the demand forecast. The choice of technology depends on relative capital and operating costs and may be constrained by behavioral bounds on the adoption of new technologies. Revenues and costs of capital projects being spread over time cannot be compared directly, rather they are converted into annual levelized revenue and cost coefficients for use in a temporal optimizing choice of technology. This takes the place of present value calculations and comparison. Consider a situation where the revenue from a capital project, once it is in operation (initial period), remains constant over time. The annual 133 levelized revenue flow is given by the initial period cost mark up price and anticipated demand. If demand and/or product price are expected to fall, while the annual levelized revenue flow remains approximated as above, the firm may not generate revenues over the target payoff period to cover costs. Under changing de mand and/or price conditions it is in the firms interest to consider some forward looking measure of project worth. This can be accomplished as with present value computations by using annual levelized revenue flow and cost coefficients in temporal optimizing computations. Another way to consider the acceptance of a capital project with a forward looking measure is the payoff period criterion. Can the firm set price to generate revenues adequate to cover costs within a target payoff period? If demand, price and cost conditions are anticipated to remain constant over the payoff period, then a temporal optimizing investment criterion is appropriate. If conditions are anticipated to change some allowance should be made in specifying the revenue and cost coefficients in the criterion. In the models developed here, the annual levelized revenue flow is given by the initial period cost mark up price and anticipated demand. The annual levelized cost coefficients are specified to account for changing conditions as will be discussed next. Specifying annual levelized capital and operating cost coefficients is a technical detail performed by the engineering planning staff. How coefficients are specified in the electric utility industry provides the systematic approach to be used in the models developed here (Marsh, 1980). The levelized annual cost of capital is computed by multiplying the price of capital per unit of effective capacity with the annual capital charge rate. The latter rate is a percentage which includes the capital recovery factor plus a percentage allowance for insurance and taxes. The capital recovery factor is the sum of interest and depreciation in terms of a sinking fund. To compute the capital recovery factor an economic life of capital and a rate of time discount must be specified. The discount rate will be treated as a weighted average of the interest rate on debt and the rate of return on equity. The economic life of capital is the payoff period which is allowed for recovering the investment costs. Using the weighted average cost of capital as the discount rate, the financial cost per unit of capacity is incorporated into the capital cost coefficient. 135 The levelized annual operating cost is also expressed as a cost per unit of effective capacity. To avoid computing operating schedules of capital over the long run, utilization rates are assumed which are consistent with anticipated demand. In this way the levelized annual capital cost and operating cost can be added and treated as one cost coefficient per unit of effective capacity. Detailed short-run production plans are made separate from capacity expansion planning. The cost coefficients for the different factors of production, namely labor, fuel, and intermediate goods purchased from other firms can be computed separately and then added together. This way the different rates of price escalation anticipated for these factors can be accounted for in the cost coefficients for the different technologies. The choice of technology then becomes dependent on changing factor prices as anticipated into the future. Switching and even re switching in the choice of technology may arise out of anticipated commodity price speculation for - raw materials and labor demand for wage increases. The factor cost coefficients are computed by multiplying the current price with a production activity coefficient, a capacity availability factor, and the utilization rate. The result is then levelized 136 by multiplying with the capital recovery factor, as discussed above, and a factor which accounts for any inflationary increase in the price which is expected to prevail over the life of capital. Innovations in technology can be accomodated by introducing new vectors of activities into the technology matrix of alternative production processes. It is also possible to allow the activity coefficients to change over time as well as the cost coefficients. The rate and direction of technological change is treated as exogenous since no research and development program is considered. When new technologies are intro duced it will be possible to trace the interindustry effects by using derived factor demands in a multisector model. iv) Finance and Budgeting The budgeting department works in conjunction with the investment and pricing departments, handling the flow of cash and acquiring external funds. Top manage ment provides key financial targets pertaining to liquidity, leverage and dividend payout. These targets are based upon measures of a firm’s financial soundness 137 which may be influenced by such market valuations as equity prices and bond ratings. Long ago behavioral rules of finance have evolved into practice as controlling factors in firm investment. The acceptance or rejection of capital projects is based not so much on present value and rate of return calculations, but more on such financial considerations as payoff period and availability of funds. This is a behavioral response to the extreme uncertainty of capital projects (Cyert, DeGroot and Holt, 1979). The availability of funds serves as a constraint on the revenue maximizing choice of investments. Past conditions accumulate into current constraints. Since existing rules on the availability of funds came into practice financial innovations have broadened the opportunities in investments strategy and firms have taken on a greater diversity of holdings. These changes bring into question the strict adherence to long accepted financial standards (Ellsworth, 1983). Still basic rules of thumb pertaining to new issue of debt, dividend payout, and liquidity persist and will be the ones considered in the prototypical firm model. In addition to rules of thumb pertaining to financial strategy there are others related to 138 budgeting. In particular are depreciation, taxation, bond repayment, and payment schedule for materials of production. Simplifications are made in specifying these cash flows as will be discussed below. Consideration for budgeting as an explicit factor in the economic theory of the firm goes back at least to Sune Carlson (1938). More recent programming formulations in a static context are typified by Baumol and Quandt (1965), Weingartner (1963), and Vickers (1968). This work has blossomed into a constrained optimizing approach to modeling corporate financial management (Myers and Pogue, 1974; Carleton, Dick, and Downes, 1973). In a dynamic context with a lag in the use of revenues, the availability of funds is a channel through which financial feedback can modify firm investment and production activity (Williams, 1967; Day, Morley, and Smith, 1974). The balance in source and use of funds, with a minimum liquidity and dividend requirement, is a constraint on firm behavior. The bud get constraint as such is incorporated into the proto typical firm model. In doing so, the issue of firm financial strategy must be confronted and simplified to its basic ingredients so as to be included as well. 139 Modern theories of firm finance assume a rationality where agents have probabilistic information for planning in a sto-chastic environment, and assume markets are in equilibrium. The outcome is a debt-equity-dividend strategy which maximizes expected market value. What the strategy is depends on the market imperfections and costs which are considered. To a large extent these theories fail to explain actual financing behavior (Myers, 1984, 575). Early empirical studies, see Anderson (1964) and Donaldson (1961), suggest an 'Old fashioned pecking order' as one behavioral explanation of actual financing strategy. Myers (1984) provides a more recent account of and support for the approach. Firms decompose financial planning into a sequence of tradeoffs involving internal versus external financing, debt versus equity, dividends versus retained earnings, and liquidity versus investment of cash into fixed assets. In this work there is a sequential application of rules of thumb to these tradeoffs. To illustrate the flow of cash from gross revenue and acquisition of funds will be traced. Consider a firm at the end of a period receiving gross revenues from sales. From this, there is a payment for production costs, a payment for debt cost and a 140 depreciation allowance. Operating costs are specified per unit of production and supplemented by costs for inventories of both factors and final goods. With debt either one period bonds require repayment plus interest or a perpetuity requires an annual interest payment. Financial depreciation is a bookkeeping operation which is subtracted from pretax revenue and added back into internal cash flow. Straight line depreciation of book value of capital is used. Taxes, as a fixed fraction, are taken out and the net earnings (net profit) which remain is divided into retained earnings and dividends. The tradeoff is made with a target adjustment rule. The dividend payment depends on net profits, last periods payment, and a target dividend rate. If revenue permits the dividend payment will not be set below last periods and will be a partial adjustment from that level towards the target fraction of earnings. The dividend formula is: Divt = Divt_-^ + Divf (DivTrg - Divj__-^) DivTrg = Divr Revnet^ 141 where the dividend target (DivTrg) is the target dividend fraction (Divr) of net earnings (Revnett). The actual total dividend payment (Div^.) is last periods payment plus a fraction of the difference between the target and last year's payment. The actual dividend payment may differ from the target, but with an adequate flow of net earnings over a sufficient time, convergence will occur. It is also possible that the payment will differ from the return to equity implicit in the pricing equation. Here, top management can make adjustments to the dividend and pricing targets. The net earnings retained are combined with the depreciation allowance into a fund of internal cash available for the next period. The cash can be designated for several purposes. First, top management may impose a liquidity requirement. Second, under conditions of revenue shortage the depreciation allowance can be used to help cover dividend, interest and/or operating cost payments. What internal funds which remain is internal cash for capital investment. The investment program involves the acquisition of external funds. The decision is heuristically repre sented by a constrained linear program. If investment in capital is not warranted then the available internal funds earn a minimum rate of return from banking. With 142 capital investment requiring funds which exceed available internal funds a new issue of debt is made. This will only occur if the internal rate of return on investment, as reflected in the dual to the financial constraint, is greater than the borrowing rate. A lev erage ratio constrains the amount on the basis of the firms equity and existing debt to equity ratio. A gene ral rule is no more than fifty-fifty. If new equity is allowed, a limit is imposed as a fraction of existing equity, otherwise dilution occurs and the return on existing equity falls. The rules of thumb and behavioral parameters attempt to reflect acceptable practice. Over time the parameters may be adjusted by top management in response to financial market feedback, in particular, with respect to equity valuation and bond rating. These feedback features will not be considered until a fi nancial market is incorporated into a micro to macro model. v) Production and Factor Inventory Control A central feature in representing the production department is the activity analysis approach to specifying a firms production technology. The structure 143 of production is given by a set of alternative production processes each based around a specific capital structure. Each production process involves a sequence of activities using factors of production in conjunction with the capital stock, specified by a vector of constant activity coefficients (Koopmans, 1951; Manne and Markowitz, 1963; Day and Cigno, 1978, Ch. 4). Through constrained linear programming computations, the cost minimizing utilization of available capital stocks is determined, from which factor demands can be derived. As expected relative costs to these factors change it is possible to sim ulate substitution in factor demand. The constraints involve capacity available, factor supply, and production flexibility. The department is not considered to have a budget constraint. Available capacity is updated from period to period according to construction lead times in capacity expansion. The flexibility constraint is a behavioral bound on the extent to which production is allowed to deviate from past activities. This constraint will also depend on anticipated change in demand and whether the firm produces for stock or to orders. 144 Factor inventories accumulate through the temporal relationship among orders, arrivals, and use of materials. Current period production activities are constrained by existing factor supplies. Factors ordered in the previous period arrive for current period production given no unexpected delays. These orders are made using anticipated demand and factor prices. Desired levels of factor inventory will evolve into practice. They will be maintained to avoid shortages and as a speculative demand. Firms desire to maintain flexibility with production scheduling, adjusting rates of output to maladjustment of inventory and order backlog of final goods. This requires inventory stocks of materials. Desired levels of in ventories are expressed as a fraction of current production an average of past levels. The magnitude of the fraction will depend on demand variability and other factors (Feldstein and Auerbach, 1976). The demand for factors of production will depend not only on the rate of anticipated production, but also on the choice of technology in production. This choice will depend on anticipated factor prices. A speculative demand can be motivated by anticipated increase in prices. In the models developed here such 145 speculative demand will not be considered though the potential for inclusion exists. vi) Sales and Product Inventory and Order Backlog In a temporary equilibrium setting this department function is trivial. With disequilibrium, inventory and order backlog stocks accumulate, and the temporal rela tion among orders, production and shipments is mediated through the functioning of the sales department. Priority rules for fulfilling orders must be specified when a backlog accumulates. A similar priority rule must be specified for using inventories. Issues of desired inventory and order backlog, and the rates at which these stocks decay also arise. Marketing and sales effort directed towards gaining market share are omitted in order to emphasize multisector interactions of single firm-sectors. As for the functions of the sales department mentioned, they will be resolved as follows. Orders are fulfilled either starting with the most recent and working back in time or vice versa. The former priority rule allows the firm to derive maximum revenue from sales, for price will be rising with an order backlog and the sales price is that which exists at the time of 146 ordering. Without market competition to compete away the order backlog it is assumed that a fixed rate of withdrawal after the first year of delay occurs. As inventories accumulate the oldest are used first in fulfilling orders and existing stocks decay at a fixed rate. Inventory valuation occurs at the sales price at the time in which the goods enter the inventory stock. Desired level of these stocks evolve into practice, at levels which depend on technical and econ omic conditions. Firms in some sectors will allow order backlog to accumulate towards a desired level. Other types of firms will aim to maintain a desired in ventory. Production technology and product storability are significant factors in who does what. For instance, order backlogs are used by the con struction industry due to the nature of production technology. Most manufacturing industries will produce for stock, allowing inventories to accumulate, though order backlogs do arise as well. The desired level of inventory and order backlog stocks will be related to production smoothing, demand variability, inventory cost, and customer loss from delivery delay. The complicated relationship is proxied by letting desired stocks be a fraction of past sales, 147 using either sales in the last period or an average over the past ' n* periods (Lovell, 1962). As economic and technical conditions change, which occurs over the business cycle, there will be a feedback response, in the adjustment of these desired levels. For instance, if a recessionary phase is entered, firms may cut back on production allowing desired inventory stocks to diminish. Early inventory studies allow desired stocks to maintain a constant proportion to anticipated sales, implying a rapid adjustment of desired stocks to changing conditions. A more recent study by Feldstein and Auerbach (1976) criti cizes these models and suggest that desired stocks adjust slowly to changing conditions and that actual stocks adjust rapidly towards desired levels. An outcome of errors in sales expectations is that accumulated stocks deviate from desired levels. In response, firms initiate production and price adjustments. So long as the errors in expectations is considered a short-term deviation from a correctly perceived long-term trend, adjustment is made with production (Scherer, 1970; Feldstein and Auerbach, 1976). Only after sales expectations are persistently biased in one direction or another will prices be adjusted. Of course, in some industries this will not 148 be the case. For instance, with primary resources price adjustment occurs as rapidly as production, making for a complicated business environment. In some agricultural sectors, inventory and order backlogs are of little significance, due to market competition and product perishability, consequently, price adjustment occurs. In these sectors temporary equilibrium prices are assumed to clear markets. 149 IV ALTERNATIVE COMPUTABLE FIRM MODELS The prototypical firm is formulated into a set of alternative computable simulation models. Traditional two, three and multisector models take on an equilibrium structure and characterize firms with identical neoclassical principles of behavior (Johansen, 1974). More recent reflections on micro to macro modeling suggests that different types of product markets exist and the behavior of firms supplying these markets should be distinguished (Hicks, 1974; Kaldor, 1976; Okun, 1981). Hicks makes the distinction between flexprice and fixprice markets which is similar to the distinction made by Okun between auction and customer markets. Kaldor refers to the first as primary pro ducers and the later as manufacturers. In this work three types of firms will be distinguished on the basis of planning and adjustment behavior. They are to typify resource producers, capital goods industries, and manufacturers. These firms are distinguished as suggested above, but with the fixprice firms either producing to order or for stock. 150 Firms in the flexprice-auction markets provide the primary resources for industrial activity. Prices adjust to equilibrate supply and demand. Firms in the f ixprice-customer markets use cost markup pricing and produce either to order or for stock. For technological reasons the capital goods industry is characterized to produce as orders arrive. Due primarily to uncertainty of market demand and competition and the cost of stockouts in the loss of sales, a prototypical manufacturer of final goods is characterized as a fixprice firm which produces for stock in anticipation of orders. In this section a computable simulation model is specified for each firm type, and a multisector simulation is made. In doing so the temporal relation among orders, arrivals and use of goods by the three firm sectors is an important consideration. The flow of goods which will be considered involve the resource sector using capital and supplying goods to both the other sectors and final demand. The capital goods in dustry will use resources and capital and provide capital for both other sectors as well as itself. The manufacturing sector will use capital and resources and provide goods for final demand only. 151 Relative to the discussion of a prototypical firm there will be a number of simplifying assumptions which are common to the different models. These are as follows: 1. Assume an initial stock of equity but allow no new issue of equity; 2. Borrowed funds are repaid at end of period; 3. One capital type with construction lead time of one period; 4. Interest rates and cost coefficients are constant and no overhead costs; 5. Demand expectations are from exponential smoothing; 6. Bankruptcy occurs if money stock goes negative. i) Price Taking in Temporary Equilibrium The resource sector is represented as a price taking firm in temporary equilibrium. Investment, finance and production activities are planned on the basis of price expectations, in constrained expected net revenue maximizing computations. Output is sold at the market clearing price given an inverse demand 152 function. Either other firms purchase the resource or it is consumed in final demand. No inventory nor order backlog is maintained, though in a more complete rep resentation of the resource sector, speculators will exist maintaining inventories. The firm model follows directly from the one in Day, Morley and Smith (1974). What modifications arise from introducing a capital order-delivery lag of LT periods, set at one, the expectations mechanism and the dividend rule. Planning begins with price expectations for LT periods ahead. These are given by an inverse demand function with expected demand formed by exponential smoothing of the past. The functional form will be assumed identical to the actual inverse demand function which appears when the market clearing price is determined. The investment-finance activities are planned separately from production planning. Once derived, the demand for capital and finance as well as the supply of output are taken to the markets. What happens in the finance market will not be considered here, by assuming demand is fulfilled at a fixed interest rate. On the product market, for this type of firm, supply is equated with actual demand and a market clearing price 153 is determined. The revenue generated is processed through the budgeting department. So long as the working capital retained at the end of the period is nonnegative the firm moves into the planning routine for the next period, evolving through a sequence of rolling plans. The production, investment-finance, and budget planning routines will now be described. The firm during period t produces a resource in amount, Xt, at constant unit cost, C, using capital stock, ZKj.. So long as expected price, PE^r exceeds unit cost full capacity output will be supplied, otherwise zero production will occur. A further consideration is that the supply of production factors, YFS^., derived from orders in the previous period may limit production below full capacity. Also, a flexibility constraint which will be different for the three types of firms may limit production, xmaxp. For the flexprice firm fluctuation in price expectations may lead to unacceptable change in production from period to period. An upper and even lower bound may be imposed on the extent current production, if it is to occur, is allowed to deviate from past production. Though no budget constraint limits production, the ex 154 pected cost does influence the working capital avail able for investment. The production linear program tableau is as follows: max DX DK DF min 1J PE where Primal Activity Xt: Production of resource Dual Variables DX: Marginal value of production flexibility DK: Marginal value of capital stock DF: Marginal value of factor supply Limitation Coefficients xmap^. = (1 +(3P ) Xt_^ : Production flexibility ZKt : (I): Stock of capital YFS^.: Factor supply Criterion Coefficient PE.J.-C: Expected net revenue per unit of pro duction 155 Constraint Coefficient a^j: Factor utilization per unit of production. Investment into capacity, Yt, is in anticipation of demand LT periods ahead, DEt+LT, and accounts for the capital stock which will be inherited from the previous period, ZKt+LT-l * I-6 > ' w^ere & i-s the rate of depreciation. Other investment-finance activities are to either borrow, b^., or save, S^.. These decisions are related through a source equals use cash flow constraint. Other constraints on these activities include a borrowing constraint and a flexibility con straint. The later limits capacity expansion in response to uncertain long-run expectations. The former is a financial constraint on capital expenditures. The cash flow constraint involves the budgeting of working capital, 2Mfc, derived from last periods revenue. This financial stock is allocated among current expected production cost, CBP^., investment in capacity, PYt, at price, p, and savings, St, with rate of return, ris. The internal funds available can be supplemented with borrowing up to percent of working capital, which must be repaid in full at the end of the period with interest, at a rate, rib, which is greater than the savings rate, ris. 156 The investment-finance linear program tableau is as follows: max Yt St Bt < rain DK -1 0 0 xmaxlj. - ZKlag^. DM P 1 -1 ZMt- CBPt DB 0 0 1 © ZMt > PCt rib where Primal Activities Yt. Investment in capacity during period t, to be available in period t+LT S.J.: Savings Bt: Borrowing Dual Variables DK: Marginal value of capacity flexibility DM: Marginal value of working capital DB: Marginal value of borrowing capacity Limitation Coefficients ZKlag^. - xmaxlj.: Maximum demand for new capacity ZMj. - CBP|_: Working capital available for investment ©ZMt: Borrowing limit as a fraction of working capital 157 Criterion Coefficients: PCj. = PEt+LT - C - 5p: Expected price minus per unit production and capital cost ris: Interest rate on savings rib: Interest rate on borrowing Constraint Coefficient: p: price of capital per unit of capacity Definitions: ZKlag^. = ZKt + LT-l: CaPacity inherited in period t+LT xmaxlj. = (1-pf ) DE^. + Lr j , : maximum investment in capacity ZMj.: Working capital in period I (see below) CBPt = C DEfc: Production budget It remains to show how the budget department derives the working capital to be used in the next period. Given production and an actual demand function a market clearing price, P^., is derived and revenue is received, REV^. . DAt = F(Pt) = dconst - dslope P^. Therefore Pt - F^IX,.) 158 and REVfc = Pt Xfc From the revenue a dividend payment, DIVfc+^, is computed as discussed in the model component. Savings with interest is returned, borrowing with interest is paid and production cost overruns computed. With production planning performed separately, but simul taneously with investment-finance, there is a budgeting conflict. In computing the working capital available for investment there is subtracted an expected cost of production. No budget constraint is imposed on the production department, allowing cost overruns. At the end of the period when actual production costs are known, what discrepancies must be accounted for in computing working capital available in the next period. The cost overrun is: CXOR = CBPt - C Xt The working capital available for the next period is: ZMt+1 = REVfc + <l+ris)St - <l+rib)Bt - DIVt+1 - CXOR 159 where the firm is declared bankrupt if: ZM fc+i < 0 ii) Price Setting in Market Disequilibrium With fixprice-custoraer markets new elements of firm behavior must be introduced relative to a price taking firm in temporary equilibrium. Of particular importance are cost markup pricing and the use of inventory and order backlog buffer stocks. To make these additions to firm behavior two functional departments are introduced, namely a pricing department and sales department. The planning routines in the investment-finance, production and budgeting departments are similar among the three types of firms allowing the core of model structure to be shared. One exception is that the production flexibility constraint, xmaxp, will differ, as will be discussed below. A second distinction is that actual price rather than expected price is used in production planning. For the price setting firms the production flexi bility constraint, xmaxp, is defined with respect to a demand measure, D, desired inventory and order 160 backlog stocks, (YPinvd, YPord), and the existing state of these stocks, (ZPinvt, ZPor^.). The formula is: xmaxp^. = D + YPinvd - ZPinv^. - YPord + ZPor^. Specification of the demand measure and desired level of stocks distinguishes the price setting firms. The manufacturer who produces for stock in anticipation of orders will use expectations as the demand measure, D = DE^.. The capital goods producer will receive actual orders before planning production, therefore D = DAt . The desired level of inventory and order backlog will also differ. Firms which produce to order will have a zero desired inventory, but may have a positive desired order backlog. Those who produce for stock will have a zero desired order backlog and positive desired inventory. Due to market circumstances, either actual order backlog or inventories may arise for the manufacturer. Whereas, for the capital goods industry only order backlogs may technically exist. For symmetry of presentation, note that the production flexibility constraint for the price taking firm can be expressed with the demand measure depending on last periods production and a behavioral flexibility 161 parameter, D = (1 + PP)X.j -. Desired and existing inventory and order backlog stocks are zero. A second consideration is specification of price. The basic formula is a cost markup, which under the assumption of constant unit costs and no overhead costs is: PP = (1+Pmup)(c+&p) Where the cost parameters are as defined above and the markup is a top management target, pmup, accounting for financial considerations. From this cost basis, price will be adjusted in response to product market disequilibrium as captured in the deviation of inventory and order backlog stocks from desired levels. Pt = PP(1+PAd) The disequilibrium price adjustment, PAd, will be positive for an order backlog and negative for inventories when they exceed desired stocks: PAd = PAdor - PAdin 162 To maintain a maximum price change, let price flexibility parameters, (admor, admin), restrict adjustment: PAdor = miniadmor, padorl) PAdin = min(admin, padinl) It remains to define whether price adjusts to last periods conditions or to a weighted average of conditions in the past. For the purpose of simplifying this exposition, let the former occur as follows: PAdorl = <xor{ZPOR^_ - YPord)/SaleS|._^ PAdinl = txin(ZPinvt - YPinvd)/Salest_^ where the adjustment parameters (ocor, ocin) are defined relative to conditions specific to firm-sectors. Other measures for disequilibrium adjustment of price may be used. There is no definitive solution to how prices adjust to market disequilibrium. Eckstein and Fromm (1968) provide a discussion and empirical test of some options. A third consideration involves the activities of the sales department. Several things arise here. One relates to the accumulation and dating of order 163 backlog, and that revenue generated from sales depend on the ordering time and the price which prevails then. Second is determination of desired inventory and order backlog. These are specified as a fraction of last period sales: YPinvd = Rinvd Sales^._^ YPord = Rord Salest_-^ where the parameters (Rinvd, Rord) depend on conditions of the firm-sector. xxi) A Simulation Comparison with Multisector Inter actions Uniting the three firm-sectors into a multisector model involves interactions on product, labor and financial markets. In this work I only consider product market transactions. The labor and financial market processes and spillover effects on production and investment activity are certainly important, but before considering them an adequate representation of interindustry transactions is aimed for. Each firm-sector has a corresponding product market where the flow of goods occur through a transaction process. There are also cash flows 164 circulating among the firms, which influences product price in the following period through cost markup pricing. I describe briefly in this section the coinci dental process of physical and cash flows among firm sectors and with final demand. The firm manufacturing goods for final demand produces in anticipation of orders. Final demand is an exogenously given function of price, unknown to the firm. Total demand is current final demand plus order backlog. Total supply is current output plus inventories where production is scheduled on the basis of demand expectations adjusted for inventory and order backlog. The sales department, as described above, in vokes a rationing scheme whenever demand is different from supply. Due to errors in expectations either inventories or order backlogs accumulate. From such product market disequilibrium there is a feedback response in price, production and investment. These disequilibrium adjustments have repercussions throughout the multisectors of the economy and may have long run implications for industrial development. That is what I am interested in investigating. The capital goods producer initializes capacity construction activity upon arrival of orders from the 165 three firm-sectors. There is a specified construction schedule for each project after which the ordering firm receives completed capacity. If delays occur, due to excess demand, construction is shifted forward in time postponing completion. The sales department of the capital goods industry in cooperation with a capital updating department in the ordering firm, keeps track of project construction and when new capacity is avail able. The order backlog for the capital goods industry is associated with delayed projects. Previous orders on schedule are treated as current demand. In this way the order backlog serves as a signal of product market disequilibrium, stimulating price, production and investment response. As with the manufacturing industry, the long run implications of short run disequilibrium adjustment is of interest to investi gate . The resource market process results in a temporary equilibrium. Firm production based on price expectations is taken to the market where resource demand is a function of price. Demand is derived from firm orders and from orders by speculators. Firm demand is in anticipation of future production activities. So long as there is potential for factor substitution in 166 production, resource demand will depend on price relative to the other factor prices. With a single technology having fixed coefficients no factor substitution occurs and current resource demand by firms is independent of price. In this case firm demand does not adjust to equal supply at the market clearing price. If supply is greater than demand speculators purchase the excess, inducing a market clearing price. With a supply shortage, speculators sell their inventories and if that is not adequate firms are rationed. From simulation experiments a bottleneck in multisector activities often arises in the capital goods industry. When demand exceeds capacity in this industry, first, it must expand its own capacity and then produce the capital for the other sectors. During the delay orders accumulate. After fulfillment the economy takes off into a period of growth, see Figures 4.4-4.7. During the later periods of growth a downturn begins. One probable cause is the borrowing constraint for the financing of expansion. Working capital is re duced during the growth phase which leads to bankrupt cies in the next five years which extend beyond the years portrayed. 167 f A) INVESTMENT Z o UJ - n i I 7.5 10.0 12.5 15.0 17.5 20. TIME 0.0 2.5 5.0 B> PRODUCTION I t — o o I o o 10.0 12.5 15.0 17.5 20. 0.0 2.5 5.0 7.5 TIME C) PRICING < / > UJ o o 10.0 12.5 15.0 17.5 20. 0.0 2.5 5.0 7,5 TIME Figure 4.4: Resource sector activities 168 A) INVESTMENT o o o 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20. TIME B) PRODUCTION o 8 z 2 0 8 O o o * a ' C L O o 0.0 2.S 10.0 12.5 15.0 17.5 20. 5.0 7.5 TIME C) PRICING LJ O QC 0- o o o 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20. TIME Figure 4.5: Capital sector activities A) INVESTMENT o t — Z LJ n H ! o o 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20 TIME B) PRODUCTION i z ° O o < _ > i O I o o 10.0 12.5 15.0 17.5 7.5 0.0 2.5 5.0 TIME C) PRICING UJ C _ ) o 10.0 12.5 15.0 17.5 20. 0.0 2.5 5.0 7.5 TIME Figure 4.6: Manufacturing sector activities 170 o <M O O o o 7 .5 10.0 12.5 15.0 17.5 TIME 0 .0 2 .5 5 .0 Figure 4.7: Aggregate economic activity (gross natio nal product) 171 V SUMMARY In this chapter a computable prototypical firm model has been presented, which can function outside of market equilibrium. Using some suggestions from the industrial organization literature the planning and adjustment behavior of firms in different sectors of the economy are distinguished. Three firm-sector models are united into a multisector model of interindustry activity where both physical and cash flows occur. Demand expectations is the driving force for firm ac tivities and errors in these expectations is the stimulus for disequilibrium adjustment behavior. The computable dynamic multisector model is still in its infant stage of development. Currently, only product market interactions are explicitly modeled, leaving labor and financial market processes to be represented in future work. It is not until the multimarket interactions among firms and households is modeled that important spillover effects can be captured. Then it may be possible to study the impact of money and the activities of financial intermediaries on real economic activity and the demand for labor. An inquiry that has stimulated this work from the beginning. 172 Unexpected changes in the economy during the 1970s have brought out a new wave of alternative macro models which emphasize the medium to long run. 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Creator Hanson, Kenneth Allen (author)

Core Title Adaptive economizing in disequilibrium: Essays on economic dynamics

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Degree Program Economics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Economics, General,OAI-PMH Harvest

Language English

Advisor Day, Richard H. (committee chair), Kuran, Timur (committee member), Nadler, Gerald (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c20-269650

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