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Labor contracts under general equilibrium: Three essays on the comparative statics of employment

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Labor contracts under general equilibrium: Three essays on the comparative statics of employment

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. U M I films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bJeedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UM I a complete manuscript and there are missing pages, these w ill be noted. Also, if unauthorized copyright material had to be removed, a note w ill indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, M l 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LABOR CONTRACTS UNDER GENERAL EQUILIBRIUM : THREE ESSAYS ON THE COMPARATIVE STATICS OF EMPLOYMENT by Sunanda Roy 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) December 2001 Copyright 2001 Sunanda Roy Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. U M I Number 3065843 Copyright 2001 by Roy, Sunanda A ll rights reserved. U M I* U M I Microform 3065843 Copyright 2002 by ProQuest Information and Learning Company. A ll rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, M l 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE (RtADUATE SCHOOL UNTVEKSITY PARK LOS ANGELES, CALIFORNIA 90007 This dissertation, written by &>Nf\rvn>A R .0V under the direction of h&X. 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 DOCTOR OF PHILOSOPHY Dean of Graduate Studies Date .....^SOTiter.i7J..gO Q ;...... DISSERTATION COMMITTEE 1m. Cd ^ J i M . .j L j _____ 1 ^^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. SUNANDA ROY Co Chairs: MARHNE QUINZII MICHAEL MAGILL ABSTRACT LABOR CONTRACTS UNDER GENERAL EQUILIBRIUM: THREE ESSAYS ON THE COMPARATIVE STATICS OF EMPLOYMENT Labor contracts are a way of sharing production risks between entrepreneurs and workers. This dissertation examines how these are influenced in equilibrium by risk related factors, such as, risk aversion of agents and risk sharing opportunities in financial markets. The first essay develops a production based general equilibrium model with many sectors and incomplete asset markets and proves that employment levels in these sectors are inversely related to the risk aversion of agents under certain conditions. Traditional tools of comparative static analysis in multivariate systems are difficult to apply in General Equilibrium (GE) models such as die one in Essay 1, because these tools are based on the Tarski theorem which requires the mapping (the fixed point of which is the equilibrium) to be montone increasing - an assumption rarely satisfied in GE models, on account of negative feedbacks. The result of Essay 1 uses a comparative statics theorem, which does not assume that the mapping is isotone. Essay 2 develops the required mathematical tools and proves this general theorem, which can take care of some negative feedbacks and is thus more readily applicable to GE models than the traditional theorems. A practical implication of the model in Essay 1 is that a low paying, productively inefficient outside option to working for private firms may be desirable to households as an insurance instrument. An example of such an option of widespread current interest is state firms in a transition economy. Essay 3 uses numerical simulations of die model to explain how employment levels in state and private firms vary under different labor market structures (competitive and monopsorust), risk characteristics of sectors, and number and types of assets traded. It provides a partial explanation for the persistence of state enterprises in transition economies, on grounds of risk sharing. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS First and foremost. I would like to thank Michael Magill for teaching a course on the Eco nomic Theory of Financial Markets, a seminal experience in my intellectual development which eventually inspired my choice of a dissertation area 1 would like to thank my advisors Michael Magill and Martine Quinzii for their help and suggestions at every stage of the work. I owe a special debt of gratitude to Martine Quinzii for taking immense pains to check the details of the writing and the proofs. I owe a similar debt of gratitude to my committee member Herbert Dawid. All remaining errors are of course mine. I would also like to thank my other committee members - Caroline Betts. Fernando Zapatero and Pablo Andres Neumeyer for their valuable comments and suggestions at various stages of the work. Finally, only my husband Tirthankar and son Kausteya, know what I owe them. My hus band's personal love for research and their patience, understanding and sacrifice, saw me through the difficult years of writing a dissertation. I dedicate this work to them. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CONTENTS ACKNOWLEDGEMENTS u LIST OF TABLES v 1 Introduction I 2 Risk Sharing through Labor Contracts - Risk Aversion and Employment 8 2.1 Introduction................................................................................................. 8 2.2 The M odel................................................................................................... 1 3 22.1 Stock Market Equilibrium.................................................................. 20 2.3 Existence of the Stock Market Equilibrium.................................................... 21 2.3.1 No Arbitrage Equilibrium (N A E )..................................................... 23 2.32 Existence of a normalized N A E ........................................................ 26 2-3.3 Equivalence of normalized NAE and SME........................................... 33 2.4 Comparative Statics of Employment.............................................................. 35 2.4.1 Employment and Relative Risk Aversion .......................................... 36 2.42 Risk Aversion and Market Incompleteness.......................................... 44 2.5 Conclusion................................................................................................... 45 3 Comparative Statics of Fixed Points for Non Isotone Mappings in Product Spaces 46 3.1 Introduction................................................................................................ 46 32 Comparative Statics Theorems.................................................................. 50 33 Conclusion................................................................................................... 59 4 Government-Private Ownership Equilibrium with Incomplete Markets 60 4.1 Introduction................................................................................................ 60 4.1.1 Summary of comparative static results................................................ 63 4.2 The M odel................................................................................................... 65 42.1 Competitive price perceptions in financial and labor markets ............. 68 4.22 Non competitive labor markets, competitive price perceptions in asset markets ........................................................................................... 72 4.2.3 Stock Market Equilibrium.................................................................. 73 42.4 A benchmark economy........................................................................ 76 43 Comparative Statics of Employment.............................................................. 79 43.1 Competitive labor markets................................................................. 80 4 32 Monopsonistic wage setting............................................................... 87 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4 Conclusion....................................................................................................... 8 9 Reference List 90 Appendix A .............................................................................................................................. 92 Appendix B ............................................................................................................................. 93 Appendix C .............................................................................................................................. 94 Appendix D .............................................................................................................................. 96 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES 2.1 SME under incomplete asset markets .......................................................... 45 4.1 SME and CME for competitive labor markets, r = -0.75.0 7 /i =0.91.0.75 . . 84 43 SME and CME for competitive labor markets, r = -032. a-/ /x = 0.61,0.26 . . 85 43 SME and CME for competitive labor markets, r = 033,0 / / 1 = 0.61,0.26 . . . 86 4.4 SME and CME under monopsony, r = -0.22,0 //1 = 0.61.036............ 88 4.5 SME and CME under monopsony, r = 0.33, a/n = 0.61,036 ....................... 88 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction Labor contracts are a way of sharing idiosyncratic production risks between entrepreneurs and workers, especially when such risks are too complex for a complete set of contingent contracts to be written on them and traded in organized markets. A question of potentially important policy implication is therefore how such contracts are influenced by risk related factors, such as the risk characteristics of preferences and sectors of production and the risk sharing opportunities in the economy. This dissertation looks at the relationship between equilibrium employment, risk aversion of agents and availability of risk sharing opportunities in financial markets. The idea that a relationship exists between the risk aversion of households and employment (or unemployment) goes back to the Implicit Contract theories in macroeconomics. A central theme in these models - two classic expositionsof which are the papers by Azariadis (1975) and Baily (1974) - is that risk averse households are willing to accept a variable employment status in return for a non stochastic or at least a less variable real wage stream. This wage rate would typically be higher than the full employment wage rate, the difference between the two being the unemployment premium. This story has widely been cited as an explanation of the downward rigidity of real wages in times of unemployment. 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A key lemma of Azariadis’ s paper states that given any state contingent real wage contract which yields the worker/household a certain level of utility, it is always possible to find a fixed real wage contract which Pareto dominates it • i.e. the latter provides the worker greater or equal utility and the firms greater or equal expected profits. As he notes, this result is not robust with respect to a number of assumptions in the model, in particular to that of the indivisibilty of leisure available to the household. Given these however, the lemma implies that it is profitable to both the worker and the firm to agree to a contract which specifies a non stochastic real wage rate and a stochastic employment profile rather than to one in which both are stochastic. The next important (and central to the literature) theorem shows that under certain condi tions, a fixed wage full employment labor contract may be Pareto dominated by a fixed wage underemployment one. A rise in the relative risk aversion of households makes this condition less likely to be satisfied - leading one to conclude that such a rise makes (involuntary) unem ployment less likely in equilibrium (with further policy implications). The intuition behind this seemingly surprising result is clear - the higherthe risk aversion, the higher is the unemployment premium desired by households and therefore the higher is the non stochastic underemployment equilibrium wage rate required to be. The full employment labor contract is likely to be more profitable to firms under the circumstances. Note that the above intuition can also work in a completely opposite way. If one were to start off with a labor contract in which wages are allowed to be stochastic but not employment, a rise in the relative risk aversion of households would require the expected wages to go up and hence the employment level to fall rather than rise in an equilibrium. What this discussion suggests is that in equilibrium, wage variability can substitute for employment variability. A rise in relative risk aversion can positively or negatively affect employment ievel(s) depending on what we specify 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the stochastic and non stochastic components of the labor contract to be. There is a lack of consensus among economists on this issue. Baily’ s paper (1974) lends theoretical support to Azariadis’s specification, as his model demonstrates that (under restrictive assump rions) firms prefer employment variation to wage variation in an equilibrium contract. The empirical evidence as to which of the two specifications discussed above is more common in practice, is mixed. There are economies in which for legal and institutional reasons, it is less easy to lay off workers, easier to negotiate bonuses, overtime rates and hours and other benefits. In his 1982 paper, Gordon notes for example, that wage and hours variation is large but employment variation small in Japan. In the US it is the other way round and UK is somewhere in between the two countries in this respect. Along a different dimension, a problem with these models is that they are not full scale general equilibrium set ups. Azariadis’ s model for example assumes exogenously given out put prices and lacks asset markets. It is not obvious that his conclusions about risk aversion and employment are robust with respect to endogenizing prices or introducing asset markets, even if one were to assume non stochastic wages. This is because of the various kinds of feedbacks, neg ative and positive on employment and wages from the other variables in response to a change in risk aversion. Thus from the point of view of macroeconomic analysis it is important to address this old issue in a general equilibrium set up. A major difficulty in attempting comparative static analysis in a general equilibrium model is the lack of appropriate tools. Traditionally, the literature on comparative statics has developed around the Tarski theorem and supermodular functions. As we explain below, direct applica tions of these therems are not possible in a GE set up because of negative feedbacks. The first two essays (Chapters 2 and 3) of this dissertation fills a theoretical and methodolical gap in the 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Macro/GE literature by (i) developing a production based general equilibrium model with la bor and asset markets to study the relationship between risk aversion and employment and (ii) developing the necessary tools of comparative statics for this. The first essay develops the model, addresses the question of existence of equilibrium - which is a little problematic when asset markets are incomplete - and proves the main comparative statics result about relative risk aversion and equilibrium employment when asset markets are complete. The difficulties that crop up in the proof when markets are incomplete are also dis cussed. Wage contracts in this model are specified as a stochastic wage stream and a non stochas tic employment level. This is a first cut which keeps the analysis tractable. The model is thus appropriate for those economies in which labor laws and institutions make layoffs difficult for private firms but they can nevertheless share some of the production risks through the wages. W e claim that the model and the tools can be easily adapted to address the same question under the other kind of contract specification, namely, stochastic employment and fixed wages. House holds have an outside option to working for private firms in the model and this prevents open unemployment The important question of whether unemployment (if this option did not exist) would be voluntary or involuntary and subsequent policy implications are also left for future research. The second essay (Chapter 2) develops the comparative statics tools which are used to prove the main result in the first essay. Two main approaches exist to comparative static analysis or the problem of determining how the solution to a system of equations behave with respect to a change in the parameter. (i) The supermodularity approach (or more generally, the quasisupermodularity approach) is useful when the system of equations are the first order conditions of an optimization problem. 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Suppose X = xJLjJf, where each .V , is a chain (i.e. all pairs of elements in A ', are ordered), and f(x) : X -+ X , then supermodularity is equivalent to f(x) having increasing differences on X. If /(x ) is twice differentiable, this in turn is equivalent to having d2f(x)/dxtdx} > 0 for all distinct i and j and all x. The fundamental theorem in this approach states that if the objective function(s) of an optimization program have certain supermodularity properties the optimal solutions are increasing in the parameter (for a detailed and comprehensive treatment of supermodular functions and their optimization, see Topkis, 1998). The results are widely applied in game theoretic models. (ii) Thefixed point approach is more useful for general equilibrium models in which the sys tem of equations are derived either from (a) agents’ objective functions which are not supermod- ular or (b) assumptions about the way institutions function e.g. market pricing equations under competitive assumptions. The Tarski theorem, which is the fundamental analytical tool in this approach (see Milgrom and Roberts, 1994) states that if the multivariate mapping (whose fixed points are the equilibria) is isotone in the variables and parameter, with respect to a partial order, the extreme fixed points will be monotone in the parameter. Although several useful extensions of the main theorem have been made and applied to a variety of economic problems, a direct application to production based general equilibrium models continues to be difficult because of the isotonicity assumption. A partial order with respect to which the mapping is isotone, is often hard to define in such models because of negative feedbacks from some variables. The second essay develops two comparative statics theorems when there are negative asym metric feedbacks from certain variables in the model. It is first of all shown that the Tarski theo rem can be directly applied when these negative feedbacks are symmetric. However, a problem arises when such negative feedbacks are asymmetric. Asymmetric negative feedbacks are said 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to be present if / is increasing in *, decreasing in y and e is increasing in x, also decreasing in y in the mapping, (f(x.y,t).e(x,y.t )) : X x Y x T X x Y . From a general equilibrium point of view, the comparative static results derived in this es say are more useful when extended to correspondences rather than restricted to mappings. We however leave such extensions of the above theorems for the future. As pointed out earlier, households/workers have an outside option to working for a private firm in our model. This alternative pays less than the private firms to the workers and reduces productive efficiency for the economy as a whole. A practical implication of the first essay is that this option can be attractive as an insurance instrument for households. The third essay (Chap ter 4) develops this theme. One can think of a number of examples of such outside options in real economies, e.g. household production, self cultivation of land with family labor etc. W e identify this option with state firms engaged in similar productive activities as private firms but less efficient and less paying to the household. A sustantive motivation is to provide a partial ex planation for the persistence of the public sector in several economies including erstwhile com mand economies which are trying to privatize rapidly. The main conclusion of this paper is that asset market incompleteness restricts the size of the private sector in an economy. Transition economies are particularly good examples of incomplete asset markets. The main conclusion is drawn on the basis of numerical computation and comparision (as analytical tools fail) of the equilibrium under market incompleteness with the equilibrium of an idealized model in which a complete set of Arrow securities are traded. Non competitive ele ments are introduced into the labor markets to estimate whether the effect of market 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. incompleteness can be modified in any direction by competition. A natural question at this point is • how well does the market incompleteness hypothesis ex plain the actual data on privatization in transition economies? It is difficult to answer this ques tion given the limited empirical data available to date and is therefore left for the future. The dissertation focuses on the comparative statics of employment levels only, with respect to some of the fundamentals. Important related issues such as the extent of risk sharing among households or entrepreneurs, welfare losses from market incompleteness, fiscal and monetary implications are left for future research. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Risk Sharing through Labor Contracts - Risk Aversion and Employment 2.1 Introduction Economists agree that labor contracts are a way of sharing idiosyncratic production risks be tween entrepreneurs and workers. From a macroeconomic point of view, it is therefore impor tant to understand how equilibrium employment and wages are influenced by risk related factors, such as, risk aversion of entrepreneurs and workers and lack of complete risk sharing opportuni ties in the economy. The paper addresses the first of these questions. While this has been asked before in the literature, the novelty of the paper lies in attempting to answer it within a general equilibrium set up. The Implicit Contract theories (see Rosen (1998) for a survey) were amongst the early at tempts in the literature to explore the relationship between employment and risk aversion of agents and its potential policy implications. Azariadis’s classic paper (1975) states that under certain conditions, a labor contract which specifies a fixed real wage rate and stochastic employ ment profile is better for both households (risk averse) and firms (risk neutral) rather than one in 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which both components are stochastic. Further, a fixed wage Ml employment labor contract may be Pareto dominated by a fixed wage underemployment labor contract under certain conditions, the difference between the two wage rates to be interpreted as an unemployment premium. A rise in relative risk aversion raises the level of the unemployment premium in equilibrium, making the full employment wage contract more profitable for firms and hence more likely. A limitation of this set up is that it is not a general equilibrium one. Product prices are not endogenous and there are no asset markets to allow agents to diversify their risks. The question remains therefore whether the above conclusion is robust with respect to feedbacks from goods and asset markets. Contractual labor resembles a financial security in that it yields a risky income stream at a future date against a current investment (disutility of labor, which may be zero). In a model with multiple risky assets, it is non obvious how a rise in risk aversion will affect the equilibrium use of a specific asset (labor) on account of substitution effects. Similarly, within Azariadis's set up. if output prices are allowed to adjust, it is not clear that the Ml employment contract is likely to be more profitable as risk aversion goes up. A major difficulty in attempting to address the issue of the relationship between risk aversion and employment within a general equilibrium set up is the lack of appropriate tools for compar ative static analysis. Later in this section, we explain that the traditional tools, such as the Tarski Theorem, fail under a GE set up on account of negative feedbacks from some of the variables. Roy (2000b) derives some comparative static results which hold even in the presence of such feedbacks. The present paper develops a general equilibrium set up and utilizes the results of Roy (2000b) to explore the above relationship. The model has several sectors of production which are subject to idiosyncratic productivity shocks, two inputs - labor and capital - and security markets which help diversify sectoral risks 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. although not completely. We prove the existence of equilibrium for this general model. Assum ing CRRA utility functions and a complete set of contingent markets, we go on to prove that the equilibrium employment levels in the private sector vary inversely with the coefficient of rela tive risk aversion of agents. Lack of closed form solutions of demand functions when markets are incomplete, make it difficult to prove the same result under the latter. However, numerical simulations of the incomplete markets model show that the inverse relationship extends to this situation also. Our result which is contrary to Azariadis (1975), is driven by the model's assumption that labor contracts specify a non stochastic employment and stochastic wage profile, as opposed to a fixed wage/stochastic employment specification. A rise in relative risk aversion raises expected real wages in equilibrium and lowers employment. What this analysis essentially suggests there fore is that if households are risk averse, wage variability and employment variability are substi tutes, in equilibrium. The relationship between risk aversion and employment may be negative or positive depending on how the contracts are specified. The empirical evidence on which spec ification is more common in practice is mixed.1 Our specification suits well economies in which for institutional and legal reasons it is difficult to lay off workers, but relatively easy to negotiate earnings through bonuses, overtime rates and hours. However, the techniques discussed in the paper are easily extended to the other specification. There are several productive activities producing the same good (income) which we call sec tors. Sectors differ from each other in their risk profiles only. Production can be organized ei ther by private firms operating on the profit motive, or by state firms which are extensions of a benevolent government The households distribute their given supplies of labor between these 1 see Gordon (1982), for example 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. two types of firms. It is costly to work for a private firm (requires more effort/training/skills etc.) and costless to work for state firms. Workers thus have an outside alternative (working in state firms) to working for private firms. This option is productively less efficient from the macro point of view and pays less to the households. However; as the relative risk aversion goes up so does the use of this option. The model thus has a practical implication, namely, a productively inefficient, low paying alternative to working for private firms may be desirable as an insurance instrument (for further discussion on this theme, see Roy (2000c)). One can think of other ex amples of such outside options in real economics e.g. household production (cottage industry), self cultivation of land with the help of family labor as an alternative to wage labor in the man ufacturing sector, government financed unemployment doles etc. Workers and private entrepreneurs have sector specific skills which expose them to sectoral shocks. The extent to which these risks can be shared through the wage contract depends on the labor market structure. Two extreme scenarios may be potentially considered - (i) Competitive under which there are innumerable workers and private entrepreneurs in each sector. Competi tion among firms and workers ensure that wages are equal to marginal product in each state of Nature in equilibrium (see Section 2). Both parties are exposed to sectoral risks in equal mea sure under this scenario, (ii) MonopsonisUc under which there is one private firm in each sector who acts as the wage leader by taking the worker’s optimal labor supply response into account to decide on the optimal wage contract Workers are fully insured against sectoral risks under this. In this paper as a first cut and as mathematically the more tractable case, the competitive structure is assumed. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Sectoral shocks can also be diversified by trading in the financial securities. The model be low is that of a stock market economy with equities as the only assets. The number of equities (sectors) is less than the number of states, which make asset markets incomplete. State firms by contrast to private firms always pay their employees an average output (aver aged across sectors) per worker in each state. State employees are thus protected against sectoral but not aggregate shocks. From a pure modelling point of view the paper shares the feature of having contractual la bor with the RBC models of Boldrin and Horvath (1995. Gomme and Greenwood (1995). The financial market structure is however richer because it does not assume representative agent and market completeness both of which simplify many of the problems associated with risk sharing. The model in this paper is close in spirit to Dreze’s (1991) CAPM model with labor contracts. The CAPM assumption is dropped however and laborers are assumed to have sector specific skills (unlike in Dreze) which make them suitable for employment in only one sector at a time. W e consider this feature to be important and realistic as it partly explains why labor income may be subject to idiosyncratic risks in the first place. Section 2 lays down the details of the model Section 3 proves the existence of an equilib rium. Proving existence of an equilibrium in a production model with incomp iete markets is difficult in general because the market subspace may be influenced by the action of the agents. In this, we are helped by the competitive assumption for labor markets. The concept of a no- arbitrage equilibrium (NAE) common in exchange based finance models2 is extended for a pro duction economy to rewrite the Stock Market Equilibrium as a constrained Arrow-Debreu Equi librium. In Section 4.1 we prove the main comparative static result, namely, the equilibrium 2 See Magill and Qumzii (19%) for a comprehensive discussion. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. employment levels in the private sector vary inversely with the coefficient of relative risk aver sion of agents. Section 42 reports the numerical simulations results for the case of incomplete markets. Traditional comparative static analysis uses extensions of the Tarski theorem and the litera ture on this is well developed following the seminal work of Milgrom and Roberts (1994), Villas Boas (1997) and others. There are two difficulties however in using these established results for general equilibrium models. Firstly, the multivariate mapping whose fixed points are the equi libria may be non isotone (non monotone increasing) with respect to the partial order defined, because of negative feedbacks from some variables. These make it impossible to use the Tarski theorem which holds only for isotone mappings. Secondly, the required montonicity proper ties of the mapping with respect to the variables and the parameters may not hold globally but only locally for most of these models. The paper uses a comparative statics result proved in Roy (2000b) for non isotone multivariate mappings, and the implicit function theorem to prove the main comparative static proposition of Section 4. 2.2 The Model There are two periods 0 and 1, and J sectors of production indexed by j - 1 ... J. Produc tion in each sector is organized by state and privately owned firms. Production decisions (i.e. employment and investment decisions) are made at date 0. The actual production takes place at date 1. At date 1, Nature subjects each sector j to a total productivity shock 7 7* with prob ability p3 . All sectors produce the same good (income) and differ only in their risk profiles t )j — Shocks are multiplicative. The production function of private firms in sector j is given byy^ = {y^(s)} = {Vj F ^j)} where/and it stand for labor and capital. 1 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Private and government firms in any sector are subject to the same productivity shocks but have different state independent production functions. In particular government firms operate with an exogenous and historically given stock of capital.3 The production function of the state firm in sector j is = {y?(s)} = { ^ ( / p } - All functions are assumed to be continuous and differentiable. The production functions sat isfy. Assumption 1 /. P(l3. 0) = p( 0. k}) = 0. Both inputs are essential. 2. p . is strictly concave and f f > 0,/jJ > 0 3. f J(lj.k3), is linear homogeneous 4. //(0,k3) = oc. ff(lj.O) = 0,Pk(lj.0) = oc.fp0.kj) = 0. (Inadaconditions) 5. ^(0) = 0,gJ > > 0 Note that no concavity/convexity assumption is made about g} at this stage. Such assump tions will be made in Section 4 when they become necessary to prove the comparative static results. The model is interesting only when for k} above a critical minimum, the level of output in private firms is sufficiently higher than that in state firms given the same employment levels in both such that private firms are able to pay their workers more than the state firms in equilibrium. Two reasons suggested for the lower productivity of workers in state firms and assumed in the model are - firstly, the state firms operate with a fixed and outdated capital stock, and secondly, workers in state firms lack incentives to put in quality effort because of a free rider problem in volved in the government wage contract So, JThe state film’s investment decision is not modelled here. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Assumption 2 There exists k^ such thatfor k3 > kf^.g1 (lj) < f 1 k3) The state independent utility function u(c) is assumed to be identical for workers and en trepreneurs with. Assumption 3 u'{c) > 0. u"(c) < 0. u'(0) — * • x Private firms are initially (at date 0) created and owned by the entrepreneurs. Labor and en trepreneurship are sector specific, which means that each household has the skills to work and each entrepreneur the leadership to organize production in one sector only. There are however numerous identical entrepreneurs and households in each sector j . With regard to labor and stock markets, this implies that entrepreneurs and households perceive their private actions as not in fluencing the market wage rates or the security pay-off structures. In other words they make private decisions taking the market wage contracts and the market subspace as given. The house holds and entrepreneurs are said to be having competitive price perceptions in both the labor and stock markets when this is the case (see Magill and Quinzii, 1996). Entrepreneurs maximize expected utility as consumers and total dividends (output minus costs) from production as producers. As initial owners of firms, entrepreneurs make capital in vestment and employment decisions, k} and respectively, for their firms at date 0. Capital investment can be financed by selling ownership shares of firm j to households and other en trepreneurs. Trading equities is also a way of sharing sectoral production risks. These are as sumed to be the only assets in the economy. Introducing a bond into the model does not change any of the results qualitatively. We shall however need to discuss this issue again in Section 42. Labor is hired at date 0 for date 1 and paid a contract W J = { W*}f=1. Employment levels are thus not state contingent but wages are. 1 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Agent’s actions lj and k} influence the dividend payments of the jth representative firm. However, agents do not perceive this as causing the market subspace to change and hence their date 1 income streams from securities (in particular from the share ofthejth representative firm) to be affected. One way to explain this is to think that a household or an entrepreneur in buying a share of the 7 th representative firm is actually investing the amount on the income stream offered by the industry. He can always change his portfolio for the numerous firms within the industry to take care of any changes in the dividend stream offered by a single firm without changing his portfolio for the industry. Let V represent the dividend streams of the sectors, VJ the sth row, 7 th column element, and V , the *th row of the matrix. Under a competitive market structure in sector j the representative firm has zero retained earnings in equilibrium. This means that output net of wage costs are paid out as dividends to the shareholders by way of return on capital invested in the firm. Another way to think about this is that the firm’ s budget constraint (revenue > wage costs + dividends) must bind in equilibrium. It must not be left with a surplus. Thus in equilibrium, (2. 1) Let us define k — {kj}/=1,1 = {M/=i> and W = {W1 }j=l the vector of private wage contracts. Then in equilibrium, V — V (fc, I, W) i.e. the dividend streams depend on the ac tions chosen by the agents. Since production functions are constant returns to scale, they can be written as yj = l)/;. Hence V(.) can also be written as V = V(j, I, W) in equi librium, where j = { }^=l. We shall choose either representation according as whichever is more convenient. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We are now ready to discuss the optimal programs of the representative household and en trepreneur. For convenience of notation we shall regard date 0 also as another state, namely, the state 0 and s € {0,1.. .5}. Given an optimal choice of kj (which he makes as a producer), the jth representative en trepreneur is restricted to the following budget set as a consumer * 0 - *4 + (i ~ )Qj ~ i tfQi kj BHQ.V) = ■X3 = (Xq, Xi^Xg) 6 Ri+l < el + Z U W Vs 6 5} where xJ represents consumption, eJ = (e^}f_ 0 initial endowment, 53 = ownership shares of representative firms in other sectors, Q = {Qj}j=l the price of full ownership of firm J- Define II' (xJ) = (Tl^ (xJ), HlJ e (x1)) as the >th entrepreneur’s vector of personal valua tions of income streams - i.e. his present value vector. Assuming the wage contract W Jt the security prices Q. and the market subspace or < V > to be given, the entrepreneur j, 1. chooses kj and /*, to maximize n ^ a r'K y ? - WU'j) - nk(*>)Ao as a producer. Since entrepreneurs are initial owners of their firms, production projects are evaluated using their personal valuation vectors (the Grossman - Hart (1985) criterion). 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 . given his choice of k3 above, chooses x 3 and S3 to maximize s = u(x£) + Y . *=1 subject to Equation 22, as a consumer. Since labor is sector specific, the index j can also be used for the representative worker or household working in sector j. The jth representative household has 1 unit of labour which it distributes between the private and the government firms. Working for a government firm is costless and working for a private firm is costly for the laborer, either because they have to invest to acquire more skills or because they have to put in harder efforts. The cost of supplying labour to the private firm is ^(/j) where c^(0) = 0, c^(/j) > Ofor/, > Oandc^' > 0. We also assume that c^(0) = 0 which ensures that certain relevant sets are bounded, in Section 3 where we discuss the existence of equilibrium. Since it is costless for households to work for state firms, it is optimal for them to supply any residual labour to the government Thus lj+lj = 1- Henceforth, we shall denote the private sector employment by, I,, and employment in state firms by, 1 - l}. The j th household's budget set is given by. M 3(W 3.G.Q, V) = < ”* o ^ -'o ~ m J = {mJ 0, m^.m3) € R^+l VI S + Wih + (i - h)G. + Vs 6 (1 ,.. .5} (23) 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where m-' is consumption of household, initial endowment, G = {G.jjL, the state wage contract, and 0 1 = {& ■ ownership shares of private firms. Household> chooses mJ, l}, and 61 assuming the private and state wage contracts, W J and G, security prices Q and the < V > to be given, to maximize, s L'(mJ.lj) = u(mJ 0) + ^ Psu{TnJ s) - Cj(lj) «=1 subject to Equation 2.3. We need to discuss the characteristics of the competitive equilibrium labor contracts at this point There is no enforcement mechanism for the private wage contracts in the model. This means that there is no penalty for the households or firms to renege on a wage contract at date 1 . Since there are many firms in each sector, it is therefore possible for a worker in sector j to join one firm at date 0 and leave it to work for another at date 1 for a higher wage. Similarly a firm can attract workers from a competitor by paying higher wages at date 1 . It is clear that incentives to renege on a contract drawn at date 0 will generally exist if at date 1 Nature moves before the agents and announces a state. W e want however to look at an equilibrium in which at date 1 workers and entrepreneurs do not have any incentive to switch parties with whom they have drawn contracts at date 0. Under this equilibrium all entrepreneurs in any individual sector j must therefore offer the same wage contract to any worker in this sector. Of the set of profit maximizing wage contracts, under the competitive assumption, there is only one which is robust with respect to parties reneging on their date 0 contracts in this sense. W e argue in Section 3.2 that this is also the equilibrium wage contract which is robust with respect to opening of spot markets although in our model there are no spot markets for labor. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. There are no enforcement problems with the state wage contracts. Wages paid to the em ployees are the only expenses of the government and the revenue from the state firms its only income. In equilibrium the two must balance. We assume that the Government is interested in maximizing the welfare of its employees all of whom are given the same weights in its objective function. So the government firms distribute the total revenue that is generated from all their activities equally among all the employees by way of wages. Thus the wage profile in the state firms is. G{1) = (2'4) 2.2.1 Stock Market Equilibrium Letx = {xJ }j~i, rn = represent the consumption allocations, and 0 — 6 = {8: }j=1 the portfolio allocations. Then, Definition 1 A Stock Market Equilibrium (SME), ((x, (m. /)). (k.C*). (0.6). (W. G. V .Q)), is a 4-tuple ofconsumption plans ofentrepreneurs, consumption and labor supply plans o f house holds. production plans o f entrepreneurs, portfolio plans ofhouseholds and entrepreneurs, pri vate and state wage contracts, dividends and security prices such that, I. For each representative entrepreneur j , (x], 83) = argmaxlUix3} and (x }, 63,k j) € BJ(W 3.Q, V) (k},l*) = argmaxn1 3e(xi)(yp J - W 3l}) - n ^ (x J)^ , 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2. For each representative household j. (ro>,0>J]) = argmax{U{m3,/,)} and e M J{W 3.G .Q .V ) 3. At date I. workers and entrepreneurs have no incentives to switch parties with whom they have drawn contracts at date 0. 4. Firms in each sector j have zero retained earnings. 5. Labor markets clear ld = i G = G(l) 6. Equity markets clear. 1=1 i=i 2 J Existence of the Stock Market Equilibrium To prove the existence of a SME we proceed along the following steps. We first define in Section 3.1 the concept of a normalized No-Arbitrage Equilibrium (NAE) which is a constrained Arrow-Debreu Equilibrium. By this is meant that under this equilibrium the agents are allowed to trade in a complete set of contingent goods as in an Arrow-Debreu set up. But all of them excepting one are allowed to trade only those commodity bundles which lie on a subspace of the whole commodity space. This concept, which has been used before 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to prove existence of equilibrium in exchange based financial models (see Magill and Quinzii (1996)) needs some explanation. When there are no arbitrage opportunities in financial markets, security prices in equilibrium are equal to the present value oftheir income streams. When mar kets are incomplete the present value vectors of agents generally differ in equilibrium. However they are unanimous in their evaluation of the income stream of a marketed security. Thus the present value vector of any agent at his equilibrium consumption can be used to evaluate the income stream of a marketed security. W e can then use these no-arbitrage pricing equations to eliminate the demand functions for securities 0, S and security prices Q, from the descrip tion of a Stock Market Equilibrium and replace these by demand functions for goods x . m and state prices which can be chosen to be the present value vector of any agent in equilibrium. The Stock Market Equilibrium looks very much like a Contingent Market Equilibrium with these substitutions. However, as we pointed out, the present value vector of any agent can be chosen to represent the state price vector under this equilibrium. And so equilibrium state prices are not uniquely defined. The usual convention in the literature is to normalize state prices in equi librium by choosing agent 1 ’s present value vector to be the equilibrium state price vector. His budget set then becomes an unconstrained Arrow-Debreu budget set. Section 3.1 adapts the concept of the normalized NAE to a production model with labor. Section 3 2 proves the existence of a NAE for our model. Then in Section 33 we show that a SME is equivalent to a normalized NAE thus defined. Hence as a normalized NAE exists, so does a SME. 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 J .l No Arbitrage Equilibrium (NAE) Let 7r = {77,};% denote the state price vector and define it1 = Let 11^(mJ) = )* n 1^(m-')} denote the present value vector of household j. Also let us denote by m iJ and x i J the date 1 consumption vectors. When there are no arbitrage opportunities in the financial markets, there exists a jt 6 A^ +1 such that, *oQ = * l V (2.5) W e can use these equations to eliminate the security prices from the the budget constraints of the jth household and write these as, s -o ^ ) - l j 3 - W3l} - (1 - 1 })G3) = 0 5=1 W3l} - (1 - l3)G, = V,V. Vs e {1... .5} The date 0 budget constraint in the above expression is the Arrow-Debreu contingent market budget set. Since the j th household is free to choose any portfolio (short sales are allowed) the date 1 constraints merely imply that the “net trade” vector (demand minus endowments minus earnings from production) must lie in the market subspace < V >. Thus the date 1 budget con straints of the households can be written without any explicit reference to the portfolio variables 01. Following similar steps the date 1 budget constraints ofthe entrepreneurs can also be written without any explicit reference to the portfolio variables S3. Thus the No Arbitrage budget sets for household j and entrepreneur j, can symbolically be written as some kind of constrained Arrow-Debreu budget sets with constraints on date 1 trade. 23 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. The budget sets for households and entrepreneurs are respectively. M J na(v. W J.G. V) = m J € R i+l *o(” *o ~ * ^0) + E L i - * 4 - W ? / , - ( l -/,)< ?,) = 0 {mi - 4 - W’lj - (1 - 0)GS } € < V > e Ri * 1 € < V > It is to be noted at this point that in exchange based financial models < V > is a fixed subspace of R%+ 1 because security pay-offs are exogenous. In production models the market subspace is endogenous. When markets are incomplete i.e dim(V') < 5, the no-arbitrage price equations 2.5 cannot uniquely solve for the state price vector given the equilibrium security prices. There are thus in general many state price vectors associated with a no-arbitrage equilibrium allocation.4 To uniquely define the state prices in equilibrium we shall follow the usual convention and choose the present-value vector of the first household in equilibrium to represent them. This assumption converts the first household's budget set into a non-constrained Arrow-Debreu budget set The definition of this normalized No-Arbitrage Equilibrium for our production model is, Definition 2 A normalized NAEfora stock market economy with statefirms is a 3-tuple ((z , m. I). (fc. ld), (W . it. G, V)) ofconsumptionand labor supply plans ofentrepreneurs and households, production plans o f entrepreneurs, and contracts and state prices, such that. 4 see Magill and Quinzii, 1996 24 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 1 . For household 1 . (to1. l\) £ argmax{Lr (ml, lx) | (m M i) satisfies s *o(mi - 4 ) + Y . - *Wi - (1 - fi)G.) = 0 )} »=1 2. For all other households j £ {2....J} (m>,lj) £ argmax{L'(mJ.l}) | (mS.lj) £ Af£a(ir. W J.G. V 3 . For entrepreneurs j £ { 1 J ) (xi) £ argmaxii^r1) | (z-f, k}) £ W 1. V) (kj.l*) = argmaxi* 1 (jfi - W ]lj) - x0kj} 4. At date I. workers and entrepreneurs have no incentives to switch parties with whom they have drawn contracts at date 0. 5. Firms in each sector j have zero retained earnings. 6. Labor markets clear ld = I G = G(Z) 25 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 7 . Markets clear at dates 0 and I. T .U i(mo + *£ + * > -- > 6 -4 ) = o £/= i(n»iJ + * i J - « 1*-ex'- * £ ( £ > , = 0 It should be noted that in Part 3 of Definition 2 we use state prices rather than the personal present value vector of entrepreneur j to define the profit function of his firm. W e show in the subsection below that this is valid because the expression (y^ - W ]l3) belongs to the market subspace in equilibrium. 2 J .2 Existence o f a normalized NAE We now derive the set of equations that describe a NAE for a stock market economy. Let V — { jt } denote the space of state price vectors. Since utility functions are strictly concave and the NAE budget set of household 1 is convex, the household I’s demand and labor supply as functions of state prices and contracts are given by. ( m V . W ^ . G M i l i r . H ^ . G ) ) = argmaxC/fm1' lt )\ S 0 = xo(m£ - ->o) + £ 7 r *(mi - - W ,1 /, - (1 - li)G,) *=i Since M*s(.) is continuous in (*\ W l,G) the functions m 1^) and/[(.) which are the set of maximal elements in M*a (.) are continuous. 26 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. The budget sets for all other households j 6 {2,... J } are constrained Arrow-Debreu and is the intersection of the unconstrained budget set and < V > which is given to the individual agents under competitive price perceptions. The constrained budget set is thus an intersection of the unconstrained Arrow-Debreu budget set and a given subspace of /t^ +1 and hence is convex. Thus the demand and labor supply of household j, j ^ 1 are functions defined by, (mJ{ic. W J.G. V).lj(ir. W J.G. V)) = argmax{r(mJ./,) (2.6) I *o(™ o “ U 7 o) s + - n v , - (1 - MG.) = 0 5= 1 {mi - * > - W’l, - (1 - Ij)G3} €< V >} The household’s demand and supply functions are continuous because the budget sets are continuous in rr. W J. G and the dividend streams V 1. Given his optimal choice of k}, the entrepreneur j's utility maximization program yields his demand functions. W J. V) = argmax{tr(xJ) | x0{io + k} - e^) + £ - ej) - w1 (fc, I, " 0 = o J=I { x i- e i} € < V > ) (2.7) 27 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. These functions are continuous because the budget sets are continuous in t . W 1 and V J. The profit maximization program yields the optimal capital/labor ratio and the first order condi tion that the optimal wage contract must satisfy. »=i j & ) - * ! ) = 0 5= 1 J (2.9) (2.8) As defined, the NAE wage contract must be such that parties to it must not have the incentive to switch at date 1 . W e now claim that the only contract which maximizes net earnings for any entrepreneur in sector j , and which is robust with respect to such incentives is. It is easy to see that if all firms in sector j pay this there will be no incentives either for the workers or for the firms to withdraw from an existing contract To understand why this is the only contract with this feature, note firstly that all firms in sector j must pay the same contract so that the incentives to withdraw do not exist Combine this with the feature of competitive markets which allow for free entry and exit of firms and households and our claim is true. Equation 2.8 yields the optimal capital/labor ratio employed as a function of state prices, j (2. 10) (2.11) 28 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. Combining this with the labor market clearing condition = ld, i.e. assum ing that the en trepreneur decides to employ all the labor that is offered at given state prices, Equation 2.11 alternatively yields the optimal capital stock k} of firm j as a function of state prices and labor supply. kj = < 2 l2 > Equation 2.10 yields the optimal wage contract as a continuous function of the capital/labor ratio employed by the firm j or after substitution of Equation 2.11 into Equation 2.10, as a func tion of jr. W } = W J{k -±) = V V j (k(it)) = WJ{*) (2.13) v REMARK 1: Since production functions are constant returns to scale, = r?J /* (7^) J where f J k ( ) is a scalar, under competitive assumptions and optimal behavior of entrepreneurs. This implies that < V > is fixed in equilibrium and equal to <r\ >. Thus all agents agree on the equilibrium valuation of wages W and dividends (y^ - W Jl}). This means that ir1 (y^ - W Jlj) - = n 1J r(*J)(t^ - W Jlj) - n J e 0 {xi)k}. Hence to define the profit function ofthe representative firm in sector j, (in Definition 2), using state prices or using the personal present value vector of the entrepreneur j are equivalent. The aggregate excess demand functions are defined by, j £ m J(jr, W },G, V) + x>(k, W>, V) - !£(*„/;) - j f t l - l j ) - * ’ -e* i = 1 29 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. = £ W .G . V) + x>{n. W>, V) - $ & ,{ * , /,(*-. W ’.G. V )),/,(.)) j = i = Z{*,W,G,V) (2.14) whereZ(rr. W.G. V) = {Zs(ir, W.G, K ) } ^ . The aggregate excess demand functions are continuous. Define k(it)1 = I(tt. W.G.V) = {lj(v. W.G. V)}/=I, and W ( k) = Then a No-Arbitrage Equilibrium is a 6-tuple (ir, 1. fc, IV, G. V ") such that Z{it. W.G, V) < 0 (2.15) 1 — l(it. W.G, V) = 0 (2.16) k - k(x )1 = 0 (2.17) W -W {n) = 0 (2.18) G -G (Z ) — 0 (2.19) V - V{k, I, W) 0 (2 .2 0 ) The excess demand functions defined above satisfies Walras’ Law. There are in effect four types of “agents” in the stock market economy whose budget constraints must be satisfied given their optimal choices, even when the economy is not in equilibrium. These are the workers, the entrepreneurs, the government and the firms. Thus so long as the government has a balanced 30 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. budget Equation 2.4 must be true for any I. Similarly the firm’ s output must be at least as large as the sum of wages and dividends for any W , I and k. Thus Walras’ Law in this model implies that for any (*\ W, k, t), s j £ r. W . G(Z), V(k, I, W) + * „ (£ k}) < 0 s=0 j=l S = > Y , w , G (/), V{k,l,W)< o j= 0 since capital and state prices are always assumed to be non-negative in our model. To prove the existence of a NAE, we shall work with a reduced form of the system described by Equations 2.15- 220. Equations 2.17,2.18 and 2.20 can be used to eliminate W , V and k from the set. The reduced form set of equations which determine the NAE (7r. /. G) are given by, Z(H.lG) < 0 (2.21) l- f ( x ,G ) = 0 (222) G - G{1) = 0 (223) Since the budget sets of the agents are invariant with respect to a scalar multiple of ir, the excess demand functions are homogeneous of degree zero in state prices. Thus we can choose an appropriate normalization for i t . 31 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. We choose a normalisation, such that the set of prices is compact and convex. s = 1 } i s O As total available labor in each sector has been normalized to 1, Z € £ = [0, l ]*7 which is compact and convex. Since markets have to clear at date 0 , there is an upper bound of^^= t + tJ 0) for each k}. Thus production functions for each sector is bounded above which places a lower bound on excess demand functions. Next note that G(Z) has an upper bound given by Z = {0,0...0} so that Q — {Q} is a closed cube of length Gmar = G (0 ) in R f. Proposition 1 There exists a normalized NAE for the stock market economy. Proof: We use a standard technique in the literature (see Varian, 1986). Define = C x V x £ which is compact and convex since component sets are. Define the function, , ( » . t . Q = - i t ln“ (l> ' ? f c < -C)) ,2.24) l+ E L o tn a x ( 0 ,Z,(ir,Z,G)) /i(x, Z , G) maps V into itself and is continuous. Define the function, = (Z , ir,G)) = (Z(x,G),/r(ir, Z,G),G(Z)) from Q to itself, v is continuous. Since £ 2 is compact and convex, all conditions of Brower’s theorem are satisfied. Hence t- has a fixed point, J" = (Z - , jr“, G*). Following Varian this can be shown to be an equilibrium and interior in rr if consumption in every state is desirable. A 32 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 2 J J Equivalence of normalized NAE and SME We need to prove that, Proposition 2 (i) If((x. m. I). (k. 1*), (0.5). (W.G.Q.V)) is a SME. and ifU \ is agent Is present value vector under this equilibrium, then ((x, m , /), (ik.f*). (W.G, IlJ) is a normalized NAE. < ii) lf((x. m. /). (k. C *). (W. G. tt. V)) is a normalized NAE then there exist portfolios (6 ,8 ) and security prices Q = n lV(k. I. W) such that ((x. m . I), (k. ld). (0,6 ). (W. G. Q)) is a SME. Proof: (i) Note first that under a SME (from the first order conditions of the entrepreneurs) the equi librium private wage contract in sector j is also equal to the marginal product of labor at each state, i.e. W J = tj3 f f (^ ) (as under a NAE). Since lf£ and V are in equilibrium we have 11^ = Il£(m l./i) and V(k.l. W) = V. Sincem1-u /1-lV 'l/I- ( l - f 1)G = V 9l and from the first order conditions ofthe 1 st household ri£ [-Q , V]T = 0 then )+ £ * _ , n L (mJ “ -J - Will ~ (1 - h)G.) = 0 . Therefore n il € W l,G). Since H i (m 1 ./- ,) = n i , the first order conditions for maximizing U(m 1,/j) over Af^a( n ‘. W l. G) are satisfied at (m l. /i). At private and state wage contracts (W, G), we have M 3 na(Il£, W J,G, V) = M J(WJ, G, Q, V) for households j E {2 So for the households (m> ,l}) are opti mal in M J na. Since the profit function for entrepreneur j, We(x^)(tf} ] - W 3l}) is identical to nj(y^ - W }lj) by no-arbitrage and the fact that (y* - W 31 }) is a marketed security for W 3, the pair (kjJd) maximizes profits for entrepreneur j under state prices II For kj — k}, W J, V ) = B 3(WJ,Q , V) for entrepreneurs j 6 {1,.. - J}. So for the entrepreneurs 33 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. (x3) are optimal in B 3 na. Since (x. to) are clearly feasible ((x, m, /), (k, t 1 ), (W , II*, G)) is a NAE. (ii) ((x.rh./). (k.P), (W .x.G )) is a normalized NAE. Define V(fc. /. W) = V . Then the equilibrium present value vector ofthe 1st household is given by n * = n *(m 1./i) = k. Define Q = and 03 as solutions of (mj - u >\3 — W 3l} - (1 - 1 })G) = VO3 for households j £ {2. ...J } and S3 as solutions of (x{ - eiJ) = VS3 for entrepreneurs j £ {1 J }. Define 0l = 1 - Y.j=2 & ~ 5Zj=i & ■ Then the market clearing conditions for date 1. -I- x iJ -u>1 J - eiJ - - y^(l - l}) = 0 implies that (to1./!) satisfy the 1 st household's date 1 budget constraints, i.e. to i 1 - u j i 1 - W llx - (1 - l\)G = V0l. Since II* = I l ^ m 1. lx), (ml.l x.0l) satisfies the FOC’s of the 1st household and is utility maximizing over M 1(W'1.G .Q . V). For all other households, the NA budget sets are identical to the SM budget sets with the variables defined as above. So that (nW./,) are utility maximizing for the respective households. Since - W 3l,) = H3(x3)(yp J - W 3l}) by REMARK 1 , the pair (kj.fj) maximizes the profit function H3(x3)(yj - W 3l3). Given k} = k} the SM budget sets of the entrepreneurs are identical to the NA budget sets, so that x3) are optimal for the entrepreneurs. Since (nv'.x-O are feasible as well, ((x.to.Z). (ife . f*). (9.S). (W . G.Q)) is a SME. A. We are now ready to combine Propositions 1 and 2 to prove the existence of a SME. Proposition 3 A Stock Market Equilibrium (Definition 1) exists. Proof: Follows Propositions 1 and 2. A 34 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 2.4 Comparative Statics of Employment When the utility function is CRRA, the solutions of Equations ( 2 2 1) - ( 2.23) are defined for a given value of 3, the coefficient of relative risk aversion. The m ain question that we ask in this section is • Are private employment levels in equilibrium decreasing in 3. assuming that the labor supply function /(x. G. 3) is decreasing in 31s . The answer to this question is non- obvious because of positive and negative feedbacks on l} from x and G, in equilibrium. The Tarski theorem with its extensions (see Milgrom and Roberts, (1994), Villas Boas, (1997)) is the usual tool for comparative statics analysis of fixed points in equilibrium models. A direct application of these existing theorems is however difficult in general equilibrium models for two main reasons. Firstly, all these theorems require the multivariate mapping, whose fixed points are the equilibria, to be isotone (monotone increasing) with respect to a defined partial order. Then, if the mapping is also monotone increasing with respect to the parameter, under this or der, the theorems predict that the fixed point will also be monotone increasing in the parameter. It is difficult to define such a partial order in many general equilibrium models because of negative feedbacks from some variables. With respect to #(/, x,G ) = (i(x.G. 3).^{k.I,G, J).G{1)) these negative feedbacks come from I and G. The second problem is that any montonicity prop erty (isotone or non isotone) of the mapping with respect to the variables and/or the parameter may be at most local not global. Roy (2000b) proves a comparative statics result for a multivariate mapping with negative feedbacks from several variables. This theorem6 can be applied on a transformation o f the map ping ip. A direct application however continues to be difficult because the required sFor conditions under which 1(*. G, 3) is decreasing in 3 see Appendix 5.1 6 Proposition 4 reports the main result without the proof 35 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. monotonicity properties can be checked only locally for t (or the transformation that we use). W e cannot therefore make assertions about the extreme fixed points of this mapping. However, assuming differentiability of c (or its transformation) and regularity of a fixed point for a given 3. we can use this theorem to predict changes in the fixed point locally as 3 changes. That is, assuming differentiability and regularity around a fixed point (/*, x*) for 3 = 3lt by the im plicit function theorem, there is a locally differentiable function (r(3). *m {3)) which can pre dict changes in the fixed point as 3 changes locally. Our objective is to deduce the monotonicity properties of I" (3) from the monotonicity properties of r . 2.4.1 Employment and Relative Risk Aversion A problem of a different nature than the ones discussed above occurs when comparative statics analysis is attempted within a GEI set up. In terms of the present model, such an analysis entails studying the monotonicity properties of the mappings f(x, G, 3 ), (*(*. /. G) and G(l). To do this for I we need to substitute the closed form expressions for and mJ 3 in the first order con ditions of households. For /i, these expressions have to be added across agents to arrive at the excess demand functions. For CRRA utility functions, closed form expressions for and mJ 3 are possible only for the household with the unconstrained budget set (see Appendix S3). For the constrained agents this is difficult because individual demand functions for securities are not defined. To proceed with the analysis we thus effectively need to assume that all agents are unconstrained, i.e. markets are complete. We shall make this assumption in this section. 36 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. To apply Proposition 4 to our present model we need to (i) eliminate G from t and (ii) re place absolute state prices {“5)5=0 by relative state prices We proceed along the following steps. 1 . G(l) is increasing in each /, if, and decreasing in each l} if. The first condition is usually satisfied if g}’s are concave and is thus the typical case. It implies that state wages increase (or stay constant) as the number of state employees decrease because there are fewer people to share the pie with. The second inequality which is likely to be satisfied for sufficiently convex g}’ s is also not an implausible scenario. Since state firms operate with historically fixed capital stocks which had absorbed the entire labor force till date 0 , these firms are likely to have increasing returns to labor over the range of available labor at date 1 . The second inequality requires that the marginal product of labor in each sector be sufficiently high (compared to the average of the average products) so that as labor is transferred from state firms to private firms the per capita wages in the former do not increase. For this paper we assume that each g3 is concave and therefore G(l) is increasing in each h- 37 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 2. Under a NAE, the first order condition ofthe household with respect to /, (which underlies the function J)) depends on whether it has a constrained or unconstrained budget set. For household j. j ^ I (constrained household) the foe is. £ - G.) + - -V (m ji))(l*7 - G.) - <'(!,) = 0 (225) s = t *«» % I *0 For j = 1, this condition is s Y . - G.) - c'(Zi) = 0 (2.26) 7?\ T « > Note however that since W J = under competition and profit maximization by firms, the market subspace < V > is equal to < tj >7 Since G is a linear combination of {'Hj}. we have (W 1 - G) €< V >. Since (W] - G) is marketed, it follows from the first order condition of households that SZfLj f£u,(mo)(W» 1 ~ Gs) = 0. Equation ( 2.25) therefore reduces to ( 226). Thus the first order conditions of all households with respect to labor have the same form ( 226), whether they have constrained or unconstrained budget sets. The left hand side of equation ( 226) is the net marginal value of the extra income that house hold j makes from working for private rather than state firms (net of costs of labor). W e denote this as NVMG (net value of marginal gains) for short For an interior equilibrium this must be zero. We next substitute for W1 , , and G in this expression. ‘ This equality is also true outside the equilibrium, so long as firms are optimizing and labor markets are competitive. 38 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. W e use equations ( 2.13) and (2.4) to replace and G?(Z). Since Wl is increasing in each of the relative prices the expression f^(W/(.) - G,(l)) is increasing in each fj. Once we substitute for (see Appendix 5 J), equation ( 2.26) expresses l} as an implicit function lj (.) of 7r and l-j , in particular of relative prices — rather than absolute prices and . For the rest of the paper the function!, (w. G. d) will be redefined as l-j- J) whose mono- tonicity properties we discuss now. By the implicit function theorem Jpfc - — ( ) rn w n * u'(mq) is diminishing in tm q and is increasing in l} if (W'/ - Gs) is positive and suf- ficiendy large (see Appendix). The expression (W} - G,) is diminishing in /; , and c'(lj) is increasing in l3. Hence the expression is negative if - G, ) is positive and suffi ciently high. The signs of and ^ therefore depend on the signs of i > N . ^ G and dS^ G. »0 ' * *0 * ,A \lU Ci for i £ j is negative because mJ Q is increasing in G, hence in /, and (W3 - G,) is decreasing for the same reason. Therefore |jj- is negative. The sign of is more difficult to specify. A rise in ^ for any s has three effects on mJ Q *0 0 as given by the expression in Appendix, (i) Current consumption becomes cheaper relative to future consumption causing m \ to be substituted by m3 Q . This is the usual positive substitution effect, (ii) Value of income in state s is higher relative to value of income at date 0. (iii) Real income at state s rises because of rise in capital-output ratio. The last two causes m{ to move up relatively to mJ Q . These are negative income effects. For < 1 the substitution effect is stronger with m jj positively and u’(mJ Q ) negatively related to each When i3 > 1, however, income effects may be stronger with diminishing and u'(m£) increasing in fj. 39 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. (A) When income effects are stronger, decreases and u'(mg) increases with each f*. NVMG therefore unambiguously rises with each Thus is monotone increas ing in each (B) When substitution effects are stronger, mJ Q increases and decreases with each There are conflicting effects on NVMG. Differentiating the NVMG expression with respect to and simplifying, > or < according as ? “'K ) £ - G‘>& - * 0 .) + A " i ) W i -G ,)> o or < 0 . (227) * S i 0 where t is the elasticity of mJ u with respect to jjj and si the elasticity of the marginal prod uct of labor, rfijf (« _ , (w)) with respect to Since c J 0t > 0, the sign of expression ( 227) de pends on the sign and magnitude of (si - 3tJ 0a). The sign of this is more likely to be positive as Appendix 5.3 explains. We shall therefore look only at cases in which . J) is increas- 3. W e discuss the pricing equations (224) next Note that when utility functions are CRRA and households and entrepreneurs have demand functions given by the closed form expression, aggregate excess demand functions depend only on relative prices ^ -.f and J, not on G. Define M , _ M .j W j) «0 M o(tT.O ) ^ ih _ 7 ra + max(0. ZJ(w, I, $)) 1 4- max(0, Z,(x~, f, G)) V -o 1 -(- m ax(0,Z,(ir,/,G))’ x0 + max(0, Z0(ir, (, ^)) _ irs + m ax(0,Za(x-, (,/?)) ir0 + maa(0, Z0(jt, /, /?)) (228) 40 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. £ + m « ( 0 ,± Z .(» ,i.* )) 1 + m ax(0. ^ Z o ( w ,/,;? )) = £ + ma*(°-U + XlLi ^ (1 + y ^ ^ ) = l l + ma*(0 .(l + £SL1 £ Z b ( £ , «,<*)) ’ . t t ^ o ffo J_ = H -E L ,m ax(O .Z .(y.l.G )) Ho *o + m ax (0 .Z o( w ./.d ) ) 1 + E L i k + £»=° max(°’ (1 + £ ) * .( £ • '• fl) 1 + m ax(0. ( l + 2 ^ i £ Z 0( £ . ^ ) ) We can define the mapping, H o H o *o as one from £ x V -► £ x V. Then these sets being compact and convex, and the functions being continuous the mapping has a fixed point W e however choose to proceed in a different way. We define (£(.)./) instead as a mapping from £ x V -> £ x V where V is the set ■“* * 0 # 0- Thus we write (/(£ £ . d)). V is not com pact. This however is not a problem since we know that the fixed point of v in (/, jt) is a fixed point of this mapping in (1, and vice versa, for a given d. W e are now ready to make use of the following comparative statics proposition. Proposition 4 Let X C R n and Y C jRro be compact and convex and let T be an ordered set. Let the mappings f(x,y.t) : X x Y xT — > X ande(x,y. t) : X x Y x T — ► Y be continuous in z, y and t. /, (x,, x_,. y, t) is weakly increasing in se_, and t and weakly decreasing in y. tj(x.y.t) is weakly increasing in x, increasing in t and decreasing in y. Then there exist lowest and highestfixed points ofthe product mapping (f(x,y,t),e(x,y,t)): X x Y x T X x Y , 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. denoted by (x’(t), ym (t)) and (x"(f), y"“{t)) respectively, such that the components y"(t) and ym m (t) are non decreasing in t. Alternately, if[f, c) are decreasing in t, y’(t) and y“ {t) are non increasing in t. Proof: see Roy (2000b). The mapping (/(.). in particular £(.)), does not satisfy the monotonicity properties required by Proposition 4 above, globally. Hence we cannot make comparative statics assertions about its extreme fixed points. However as noted earlier local comparisions at certain points are possible. First we need to check for the local monotonicity of £(.)) with respect to the variables and parameter. We require that ^ be (i) non decreasing in ^ where t £ s and (ii) non increasing in / ,. Whether these properties hold locally or not depends on the relative percentage changes of Z, and Zu with respect to these variables. ^ has positive substitution and income effects on ag gregate demand in both states. A necessary condition for (i) is thus, consumption at state t is no more a gross substitute for consumption at date 0 than it is for consumption at state s. In addition positively affects the capitalrlabor ratio and hence production in states s = 1.. .5. Thus it has an additional negative effect on Z ,. We thus require the positive demand effect on Zs to be actually larger than on Z0, locally, (need to push this point further?) I] on the other hand has positive income effects on consumption at states 0 and s. Addition ally it has a negative effect on Z , through production. It is therefore quite likely that ^ will be non increasing in lj globally. The required monotonicity properties (local or global) of j-j-, namely, (i) non decreasing in for all 6- = 1 ... 5 and (ii) non increasing in l} follow for similar reasons. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Finally, ^,l-j./3) is globally decreasing in 3 because NVMG is. W e need ^ to be decreasing in 3 which can happen if demand at state 5 falls relative to demand at state 0, locally. (For conditions under which this is true, see Appendix 52.) The mapping (/(/. ^ . 2.. 3),p(l. ^ .J ) ) has a fixed point (/".*■•) (a zero of the map- ping (/(/. 2-. J) - l . p ( * £ .1.3)- ( f ^ .^ ))) corresponding to 3 = di since by Propo sition 1. the mapping r has a fixed point We are now ready to make the following claim. Proposition 5 If(1(1. f^-. 2-. 3).p(^-. 2., /, 3)) satisfies the monotonicity properties o f Propo sition 4 and is differentiable at (/’. x") and if (P. x*) is a regular point then a local decrease o f 3 from Jj increases 1 “ fo r some j. Proof: Since (/(/. 2^. 3).p(^. 2^.1. 3)) is differentiable at (f\ x*), and if (/*. x ') is a regu lar zero of (/(/. ^ .2-. 3) - _ (f~> j")),by the implicit function theorem, we can define neighborhoods.Vc(/*.x*)and:V;(/^1 )andatnapping(/(d).x(^)): .Y { -» • .V ?, such that for all 3 € .Yj(Jj), (Z ( 3). x(j)) is a zero of (/(/,., 3) - l.p(..l,3) - (fj-, ^ )). Since the mapping (/(/... 3).fi(..l. J)) has the monotonicity properties stated in Proposition 4, locally at (/*. x*), the fixed point function 1(3) has the stated monotonicity properties. S.. REMARKS 1 . Proposition 5 (also 4) can be proved in its weaker form, namely, employment in at least some sectors will decrease as 3 increases. The reason for this weaker assertion (as opposed to the stronger one that employment in all sectors will decrease) is that state wages increase as in any one j falls and this causes marginal gains from private employment to diminish. This is the negative feedback of on lj. The strength of this negative feedback however depends on the absolute size of the marginal gains. If the marginal productivity of labor in private firms is 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. considerably higher than the average product of state firms and the marginal product is highly sensitive to changes in relative prices (Le. the ratio kj/lj is highly sensitive to changes in relative state prices) the lower will be the strength of this negative feedback and the larger the number of sectors that will be adversely affected by a rise in 3 . 2. The essence of Proposition 5 is that it is a necessary but not sufficient condition for the comparative statics result that the NVMG of the households be decreasing in 3- The monotonic ity of NVMG in 3 follow from the separability of utility in income and labor. 3 adversely affects the marginal utility of income but not the marginal disutility or costs of labor. Separability how ever is not necessary for this property. What we need for the monotonicity property to hold, is that the coefficient of relative risk aversion affect marginal utility of income more than the marginal disutility of labor. 2.4 J. Risk Aversion and Market Incompleteness Proposition 5 is difficult to prove when markets are incomplete. Numerical simulations of the stock market model however reveal that an inverse relationship exists between 3 and private employment levels. Table 1 gives an idea of the sensitivity of employment levels with respect to 3. The model is parameterized by assuming 5 = 5, J = 2, and » ? i = {1.6.6,1,1} and T fi — {4.1.1,2.2}. The figures show that employment levels are very sensitive to the coefficient of relative risk aversion. The other interesting point to note is that no such inverse relationship exist between 3 and capital stock k} which like labor is also a risky asset As discussed at the Introduction of the paper, the non obviousness of the inverse relationship between 3 and in a model with multiple risky assets is reconfirmed. 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 2.1: SME under incomplete asset m arkets A h h *2 0.9 0.78 44 0.60 32 1.0 0.50 40 0.41 32 1 .1 032 37 028 32 1J 0.15 33 0.14 34 1.6 0.06 32 0.06 38 2.0 0.02 34 0.02 47 2.5 Conclusion The paper develops a general equilibrium model with incomplete asset markets to discuss the influence of two risk related factors on labor contracts - namely risk aversion and market incom pleteness. We have proved the existence of equilibrium for this general model assuming compe tition in the labor market. The comparative statics of risk aversion has been proved analytically for CRRA utility and Cobb-Douglas production functions. As we pointed out, the method shown can be extended relatively easily for many other utility functions of the HARA class for which closed form solutions for demand functions are available. The model and main results of the pa per has a practical implication. Low paying, productively inefficient outside options for workers may be attractive from the risk sharing point of view. The policy implications of this observa tion has not been explored in this paper and is a subject for future research. The paper also does not address the economically more interesting but mathematically less tractable casefs) of non competitive labor markets. An extreme example (and therefore a first cut) of a non-competitive situation is discussed in another paper (Roy, 1999). 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Comparative Statics of Fixed Points for Non Isotone Mappings in Product Spaces 3.1 Introduction The literature on monotone comparative statics of fixed points of multivariate mappings - i.e. the problem of determining how the solution of a system of equations behave with respect to a change in the parameter - is well developed for cases in which the mappings are isotone or mono tone increasing under some partial order. Milgrom and Roberts (1990,1994), Topkis (1998), Vil las Boas (1997) etc. are some of the major contributions in this area. One of the main analytical tools in this literature is the Tarski theorem which says that an isotone mapping from a partially ordered set to itself has a fixed point, in particular a highest and a lowest fixed point. In their seminal paper (1994) Milgrom and Roberts adds to this theorem the result that the lowest and highest fixed points of the mapping are monotone increasing in a parameter, if the mapping itself is also increasing in the same parameter (Theorem 3 of the paper). Topkis (1998) extends these results for correspondences and Villas Boas (1997) shows that a mapping I \ which is weakly in creasing and higher than another mapping Tj, on an ordered set, always has a higher fixed point 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. corresponding to any fixed point of Ti. In a situation of multiple fixed points, we cannot com pare an interior fixed point of one mapping with an interior one of another mapping. Therefore, although this theorem does not directly allow us to compare interim- equilibria in face of multi plicity. it is useful for local comparative statics. The major assumption in all of these theorems is that mappings are isotone. In the 1994 paper, Milgrom and Roberts use the Tarski approach a second time to prove the monotonicity result for a more general case (Theorem 4) in which each component function of the overall multivariate mapping is allowed to be non isotone with respect to only one endoge nous variable at a time. In symbols, let /(x , t) = {/,(x,, x _ ,,£)}"_, : X x T -* X where X C R n and T is a partially ordered set representing the parameter space. Then so long as each /, is monotone increasing with respect to x_, and t but only continuous (increasing or de creasing) but for upward jumps with respect to x ,, the monotonicity results for the extreme fixed points of f(x.t) hold. This last theorem is especially useful, as they point out, for economic ap plications. In particular, pure exchange based general equilibrium models are well described by such a system because the individual excess demand functions are decreasing with respect to own pnces but increasing with respect to prices of other goods (on account of gross substitutability). The last theorem (Theorem 4, to be referred as the Tarski, Milgrom and Roberts (TMR) theo rem, henceforth) is however difficult to apply directly to models in which the negative feedbacks among the variables are more complicated, such as in production based general equilibrium mod els. In such models, there are typically several types of variables, for example price variables and non price variables such as employment or investment and negative feedbacks between the types exist. For example prices may be adversely affected by an increase in production because of a rise in employment, etc. A component function of the model like /, in the above example, will 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. typically be decreasing in more than one variable and increasing in others in such a situation. The Tarski theorem may apply only in special cases in such multivariate models because only in a few cases it may be possible to define a partial order on the variable space with respect to which the mapping (whose fixed point is the equilibrium of the model that we are looking for) is monotone increasing. We consider below two typical examples to try to understand the cases in which the theorem directly applies and those in which it does not 1. Let X C f2” and Y c R ”1 and J be an ordered set Let /( x , y. t) : X x Y x J' — ► X and e(x.y.t) : X x Y x T -+ Y. Let f(x, y, t) be increasing in x and de creasing in y and t. Let e(x. y, t) be increasing in y and t and decreasing in x. We want to know how the highest and lowest fixed points of the product mapping (/(x ,y ,f),e (x .y . t)) : X x Y x T X x Y (which are equilibria of the model) behave with respect to t. So we define the following partial order on the product space X x Y . (x1. y 1) > (x1, y 1) iff x 1 < x 2 and y 1 > y 2 in the componentwise order. It is easy to check that under this partial order ( /( x ^ y '.tj.e f x '.y 1. t) v (/( x 2.y 2.t),e(x2,y2.f)), i.e. the product mapping is iso tone. Moreover since (/(x .y .f).e (x ,y ,ti)) > (/(x ,y ,f),e (x ,y ,t2)) for £1 > we can conclude that the highest and lowest fixed points are monotone increasing in the same order. A slightly different way (which is useful at this point to look at) to get some but not all of the TMR results is to consider the following transformation of the above model. Define z = -y , f( x ,z j ) = f{x.~z,t) = f{x,y,t) and e(x,z,t) = - e (x ,-z ,f) = -e (x ,y .f). It is easy to check that / and e are increasing in x, z and decreasing in t. Hence the TMR theorem applies on the transformed mapping (/. e) and its extreme fixed points are monotone decreasing in t. It is possible to show that the extreme fixed points of the transformed mapping are fixed points of the primitive mapping subject to the transformation y{t) = -z(t). A fixed point of the primitive 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. mapping exists therefore. The monotonicity results however do not extend from the transformed to the primitive mapping because the extreme fixed points of the former are not extreme fixed points of the latter. We come back to this point later on in the paper (see Comments at the end of Theorem 2). 2. We now consider an example in which the TMR theorem does not apply directly. Let X C i f and Y C Rm and T be an ordered set. Let /(* . y,t) : X x Y x T -> X and e(x.y.t) : X x Y x T Y. Let f(x . y,t) be increasing in z and decreasing in y in the componentwise order. Let e(x.y.t) also be increasing in x and decreasing in y. Finally assume both to be increasing in the parameter t. We cannot define a partial order with respect to which the product mapping (f{x.y,t),e(x,y,t)) : X x Y x T -¥ X x Y is isotone. Suppose we transform the variables and functions in the following way. Define z = -y , ~f{x. z. t) = f(x.-z.t) = f(x,y.t) ande(z.z.f) = -e(x,-z,t) - -e(x.y.t). The TMR theorem will not apply to the transformed mapping , (f.e) : X x Z x T -> X x Z because although / is increasing in (z. z), e is decreasing these. The theorem applies directly in the first case because the component mappings are symmetric with respect to the negative feedbacks from the variables. / which maps into z is positively affected by z and negatively by y. e which maps into y is positively affected by y and negatively by z. A transformation of any one of the variables restores the isotone property of the product mapping. In the second case the negative feedbacks are not symmetric. / is negatively affected by y whereas e is positively affected by z. A transformation of a variable does not make the product mapping isotone. This is a problem from the applications point of view because a large number of models particularly of the general equilibrium type have negative feedbacks amongst variables which are asymmetric. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This paper describes a method of doing comparative static analysis in two cases of asym metric negative feedbacks. The basic model is the product mapping (/(z , y. t).e(x.y.t)) : X x Y x T — > X x Y . In the first theorem both / and e are increasing in x and f, but decreas ing in y. In the second one/is increasing in x,y and t, whereas e is decreasing in all three. The essence of the method is in reducing the product mapping to a mapping on the component space Y only. A general property for decreasing maps (discussed in Villas Boas (1997))is then used to get the comparative statics result The method allows us to make comparative static predictions about only y, not z and we see why in the proofs. So the theorems are useful when the variable y happens to be of interest from the economics point of view. 3.2 Comparative Statics Theorems At the cost of being repetitive, we review the following definitions for partially ordered sets, as some are less frequently used than others in the literature. Let X c Rn and x l,x 2 € X . We define the order relation > on X as follows, z 1 > x 2, if xj > x2 for all i = 1 ... n. This relationship can also be written as x 2 < x l. We read this as ~z‘ is as big as x 2” (alternately, “x2 is as small as x2"). If x- > x2 for all i = 1.. .n then z 1 is strictly bigger than z 2. The relation x l £ z 2 is read as “x 1 is no bigger than x 2 ” implies that there is at least one t for which x- < xf. Thus x 1 £ x2 implies that either z 2 > z 1 or z 1 and x2 are non comparable under the componentwise order. This relationship can alternately be written as x 2 jC **. The mapping g(x) : X - * ■ X is weakly increasing in z if for all z 1 > z 2 with x l ^ x 2, ^(x 1) > g{x2) which thus subsumes the possibility ^(x1 ) = g(x2). It is strictly increasing in x ifff(xl) > g{x2) for all x 1 > x 2 withx1 ^ x 2. Similarly, it is weakly or strictly decreasing 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. according as g(x*) < g{x?) (which subsumes the possibility ^ (z 1) = g ^ O o r f f f i 1) < g(x2) for all x 1 > x2 with x 1 x 2. W e define g{x) to be never decreasing in x, if for all x 1 > x2 w ithx1 ^ x 2, g(xl) £ ^(x2), Le. if at least one component of g(xl ) is bigger than the corresponding component of g(x2). The may is defined to be never increasing if for all x 1 > x2 withx1 £ x 2, g(x‘) £ g(x2), i.e. at least one component of g(x') is smaller than the corresponding component of g(x2). The above definitions imply that a constant mapping is both weakly increasing and weakly decreasing. A never decreasing mapping can be either weakly or strictly increasing or elseg(x‘) and g(x2) may be non comparable in the componentwise order. Similarly a never increasing mapping is either weakly or strictly decreasing, or the values may be non comparable. Let X' C X be partially ordered under the componentwise order, x 6 X is an upper bound (lower bound) for X ' if x > x ' (x < x') for all x ' 6 X '. If additionally, x € X ' then x is the greatest (least) element of X '. If x € X ' and there does not exist any x f € X with x < x' (x > x'), then x is a maximal (minimal) element of X '. The infimum and supremum of X ' (denoted inf(X') and sup(X') respectively) are as usual defined as the least upper and greatest lower bounds. It is clear from the definitions that a greatest (least) element is a maximal (minimal) element but not the reverse. The greatest (least) element and the supremum (infimum) of X ' are unique if they exist. There can however be any number of maximal (minimal) elements of X '. In a slight abuse of notation, we shall denote by max(X') and min (X'), a maximal and minimal point of X '. The inf (X') and sup(X') are the unique minimal and maximal points of X ', if they belong to the set We are now ready to discuss the following set of lemmas about fixed points of ordered sets in JT. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L em m a 1 Let g(x. t) : X x T -+ X be never increasing in x. lfxm (t) is a lowest fixed point ofg(x.t), then x‘[t) 6 {m in{x | g (x .f) < x}}. Proof: Since it is a fixed point, x’(t) 6 {* I ff(*.f) < *}• Let*' < x ’(t) withx' £ x’{t). Since g(x.t) is never increasing in x, 3j such that g^x'.t) > £,(x*(f),f) = xj(f) > r '. Hence x ' g {x | g{x.t) < x } which proves our Lemma. The proof obviously holds for decreasing functions as well. Hence we have the corollary, Corollary 1 Ifg(x,t): X x T -* X is decreasing in x and x ’(t) is a lowest fixed point, then x*(f) € {min{x | g(x.t) < x}}. Proof : As above. The counterparts of Lemma 1 and Corollary 1 for a highest fixed point are as follows. Lemma 2 Let g(x. t) : X x T -* X be never increasing in x. lfxm m {t) is a highest fixed point, thenx"{t) 6 {max{x | g(x.t) > x}}. Proof : Since x**(f) is a fixed point, x” {t) 6 {x | g(x,t) > x}. Let x' > x*’ (f) with *' x " (t). Since p(.) is never increasing in x, 3j such that ^ (x '.f) < flr,(x**(f).f) = *“ (/) < x'y Hence x' $ {x | p(x.f) > x} which proves our claim. The corollary for decreasing functions is obvious. Corollary 2 lfg(x.t) : X x T — > X is decreasing in x and x ” (t) is a highest fixed point, then x**(f) € {max{x | g{x,t) > x}}. Proof: As above. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comment 1. As discussed above, the sets {min{z | g(x,t) < x}} and {max{x | g{x.t) > z}} may in general contain more than one element The above lemmas state that the lowest and highest fixed points of a never increasing mapping are members of these sets. Comment 2. The Tarski theorem states that if g(x, t) : X x T -* X be weakly increasing in x then the highest fixed point is x*’{t) = sup{z | g{x,t) > x} and the lowest fixed point is x ‘(t) — inf {x | g(x. t) < x}. Thus for weakly increasing functions, the lowest and highest fixed points are identical to the sets {min{z | g(x.t) < z}}and{max{x | g(x.t) > z}}) respectively which are singletons. The uniqueness of the extreme fixed points is thus also es tablished. For never increasing and decreasing functions a lowest and a highest fixed point must belong to the above sets. It is thus quite possible that such a mapping may have multiple lowest and highest fixed points which are not comparable amongst themselves. Theorem 1 Let X C IV 1 and Y C R m be compact and convex and let T be an ordered set. Let the mappings f(x,y,t) : X x Y x T -> X ande(x,y,t) : X x Y x T — > Y be continuous in x , y and t. f, (z,. x_,. y. t ) is weakly increasing in x_t and t and weakly decreasing in y. tj(x.yJ) is weakly increasing in x, increasing in t and decreasing in y. Then there exist lowest and highestfixed points ofthe product mapping(f{x,y.t) ,e(x,y.t)) : X x Y x T -¥ X xY. denoted by (x'(t). y'(t)) and (x**(t), y“ {t)) respectively, such that the components ym {t) and y"(t) are never decreasing in t. Alternately, if (f,e) are decreasing int. y ’(t) and y"{t) are never increasing in t. PROOF: Define/,(x_„y,t) = inf{z, | f,{xi,x-i,y,t) < z ja n d /(x ,» ,t) = {/,(*_,,y,t)}"=1. Then /(x . y, t) is (i) increasing in x and t, decreasing in y, (ii) continuous in all its arguments, since /,(.) is continuous. Property (i), discussed in Milgrom and Roberts (1990,1994) follows 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. from the fact that the set {x* | /,(z,, x_,, y, t) < x,} becomes smaller as x and/or t increase and larger as y increases. Define f[y.t) = inf{x | f{z.y,t) < x}. Since f{x,y,t) is increasing in x, f(y,t) is (i) the lowest fixed point of f(x, y. t) given y and t (ii) is continuous because /(.) is so. and (iii) increasing in t and decreasing in y since f(x, y, t) is so, following the same arguments as those used for f(x. y. t) above. Property (i) is the Tarski theorem. Define the composite mapping e(y,t) = e (/(y ,f),y ,t). The following properties hold for e(y. /). (i) Since / ( y , t) and e(x. y. t) are continuous, e(y. t) is continuous in y, has a fixed point, in particular a lowest fixed point, (ii) Since e(x, y. t) is increasing in x and f ( y . t) is decreasing in y, e (y ./) is decreasing in y. (iii) Since e(x, y, t) and /(y , t) are increasing in t, e(y. /) is increasing in t. Let y* (/) be a lowest fixed point of e(y ,/). Then, Claim 1: y'(t) is never decreasing (alternately, never increasing) in t if ( /. e) are increasing (alternately, decreasing) in t. Proof: Let tx > t 2 and y*(ti), ym (t2) be two lowest fixed points corresponding to ti and t2 respectively. Supposey’(ti) < y’(t2). Sincee(y,t) is decreasing in y, we have, y’ (/i) - e(y’ (*iMi) > e(y-(/2)./,j > e(y’{t 2),t 2),siacee(y,t) is increasing int. Bute(y*(t2)./ 2) = y'[t2), which is therefore a contradiction. Following the same logic we can show that y “(i) is never increasing if ( /, e) are decreasing in t. Claim 1 follows from a general property for decreasing functions which is discussed in Villas Boas (1997). Claim 2: Denote x*(f) = f(y‘(t),t). Then (x*(i), y*(f)) is a fixed point of the primitive mapping (f(z,y.t),e(x,y,t)). 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Proof: y*(0 = e(y*(f)-0 = e{f{y'(t).t).ym {t),t). Hence, (x*(*),y’(f)) = = (/(*•(«),»*(0.0,e(*-(t),ir-(0.0) = (/( * “(*). t), e(x*(t). y*(f). t)). which proves the above claim. Claim 3: (x"(f). y* (0) is a lowest fixed point of the primitive mapping ( /( z , y, f), e(x, y, t)). Proof Suppose not. Then there exists a fixed point (xt(t),yL(t)) of the primitive such that (xt(<). yL(/)) < (x'(t).y-(f)). Since (xL(t).yL(t)) is a fixed point, f{xL{t),yL{t),t) = xL{t) ande(xL(t).yL(t).t) = yL{t). Therefore G {x, | f,{x,.x-tL{t),yL{t),t) < x,} for alii. Since /, is de creasing in x„ it also follows that, x./,(f) = inf{x, | /,(x,, x_,*,(£). yL(f).f) < x,} for all i. Hence from the definition of / it follows that xi{t) = f(xi,(t),yL(t)J). Similarly, € {y | e(x/.(f).y.f) < y}. Since e is decreasing in y, it follows that for any yf < yi(t), e(xL(t).y/.t) > e(xL(t).yL(t),t) = y L(t) > y'. Hence y7 g {y | e(xL(t),y,t) < y}. Therefore yL{t) 6 {min{y | e(x£,(<)*y,0 < y}}- Since yt (t) is a fixed point of e(xL(t), y, t), and the above property holds, y^(t) is a lowest fixed point of e (x t (f), y, t). Thus {xl (t), yL (t)) is a lowest fixed point of (/(x, y, t), e(x, y, t)). 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since y’(t) is a low est fixed point o f e (y , t), by Lemma 1, y'(t) € {min{y| e(y,t) <y}} =► (*‘(0 -y‘(0 ) € (min{(x,y) | f(y,t) = x ,e(x,y,f) < y}} = {min{(x.y) | f(x.y.t) < x.e(x .y .f) < y}} Hence (xm (t),ym (f)) is also a lowest fixed point of (/(* . y. t).e(x. t)). The last statement implies that (xi(t).yL(t)) < (x'(t).y'(t)) is a contradiction. It remains to show that the claims are hue for a highest fixed point of the primitive mapping as well. We redefine the functions f and / as follows. /,(x_,, y, t) = sup{x, | /,(x ,, x_,, y. t) > x,} and f(x.y.t) = {/,(*_*.y.f)},n =i- /(*»y.f) is increasing in x and f, decreasing in y. f(y.t) = sup{x | f(x.y.t) > x} is a highest fixed point of / ( x ,y ./), increasing in t and decreasing in y. W e define e(y. t) = e (/(y . t). y, t), which has the same properties as before. The steps to show that y"(t) is never decreasing in t and that (x” (/). y**(f)) is a fixed point of the primitive mapping are identical to the previous ones. To prove that (x**(t). y** (f)) is a highest fixed point of the primitive map, we use Lemma 2 and Corollary 2 and use symmetric arguments. The case of (/. e) decreasing in t is straightforward. Comment: The method above does not allow us to make any monotonicity statement about the components x'(t) and x ’“{t). Although /(y , t) is decreasing in y, since the most that we 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. can say about y~(t) andy**(t) is that these are never decreasing, we cannot make any conclusive statements about xm (t) and x “ (t). Theorem 2 Let X C R n and Y C /T1 be compact and convex and let T bean ordered set. Let the mappings f(x,y, t): X x Y x T X ande(x,y,t): X x Y x T - * ■ Y be continu ous in x ,y and t. fi(xi, x_,t y, t) is weakly increasing in z_„ y and t. e(x, y. t) is weakly decreasing in x and y and decreasing in t. Then there exist lowest and highest fixed points of{f(x,y.t).e{x,y,t)) denoted by (x"(f),y*(f)) and (x*’ (f),y*'(t)) respectively, such that the components y'(t) andy"(t) are never increasing int. Alternately if f is weakly decreasing in t ande is increasing in t, y’(t) and ym ’(t) are never decreasing in t. PROOF: The functions f(x,y,t), f(y, t) and e(y,t) are defined as before. Most of the properties of these functions carry over with the following exceptions. (i) Since f{x, y, t) is increasing in *, y and t, the function f(y , t) is increasing in y and t. (ii) Since e(z, y , t)) is decreasing in x, and f(y, t) is increasing in y and t, the composite function e(y, t) is decreasing in y and t. We then follow the same steps as in Theorem 1. The case of / weakly decreasing in t and e increasing in t is straightforward. Comment 1 . Consider the mapping (/(z , y, t), e(x, y, t)) of Theorem 1. Define the trans formation z - — y, /(x , z,t) = f{x,-z,t) = f{x,y,t) ande(z,z,t) = -e(x,-z,t) - -e (x , y, f)- Then /( z , z, t) is increasing in z, z and t. e(z, z, t) is decreasing in z, y and t. The model of Theorem 2 is thus a transformation of the model of Theorem 1 and vice versa. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We can show that the extreme fixed points of (/(x , z, t),e{x. z. f)) are fixed points of the mapping,(f(x.y.t).e(x,y.t)). Let (xi,(f),zj,(f)) and (x#(f), z//(f)) be respectively alow- est and highest fixed point of the first mapping. Hence, = /(x*,(t).zL(f),t) = f{xL{t),-zL{t),t) = /(**.(*)• y t( 0 - 0 - Similarly, 2 l ( 0 = -V l^) = e(*Z.(0 -2 t ( * ) - 0 = -e(xL{t),-zL[t)J) = -e(xL{t).yL(t).t) or yL(t) = e(xL(t).yL{t).t). Hence, (xt,(0-yz,(0) is a fixed point of the mapping (/, e) (hence also of the primitive mapping). The proof for (x//(t), yH (t)) is analogous. It is not true however that (xi(t). y^t)) is a lowest fixed point of (/, e). To see this note thatz^(t) is a lowest fixed point of e(xi{t),z.t) i.e. o fe(x .z.t) given that x = xt(t). In otherwords, - y ^ t) = zi(t) isalowestfixed point of -e (x t(f),y ,t). Hencey/^f) is&high- est fixed point of e(x/,(f).y,t), i.e. a highest fixed point of e given x = x*.(f)- Similarly, t///(0 can be shown to be the lowest fixed point of e given x = x//(f). We cannot conclude from these observations however that y ^ (f) and y H (t) are even components o f the highest and lowest fixed points o f e(x. y. t), in the global sense. A direct proof of Theorem 2 is therefore necessary even though one model is a transformation of the other. Comment 2. The method above once again does not allow us to make any monotonicity statement about the components x'(f) and x**(f) for the same reasons as in the previous case. 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 3 Conclusion Under a general equilibrium set up, the building blocks are very often correspondences rather than mappings. Topkis (1998) has extended the Tarski theorem to mappings. Extensions ofThe- orems 1 and 2 to cases when either/both of the components of ( /. g) are correspondences rather than mappings is thus an useful and important exercise which is left for the future. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Government-Private Ownership Equilibrium with Incomplete Markets 4.1 Introduction Economists agree that labor contracts are a way of sharing idiosyncratic production risks be tween entrepreneurs and workers. From a macroeconomic point of view, it is therefore important to understand how equilibrium employment and wages are influenced by risk related factors, in particular, risk characteristics of productive activities and availability of risk sharing opportu nities in the economy. The paper develops a computable general equilibrium model to address these questions. The model has several sectors of production which are subject to idiosyncratic productivity shocks, two inputs - labor and capital - and stock markets which help diversify sectoral risks although not completely. The question of existence of equilibrium in this model, which is prob lematic since the market subspace is endogenous, is discussed elsewhere (see Roy 2000a). In this paper, we numerically solve for the equilibrium levels of employment and wage profiles and study how these are influenced by the risk profiles of the sectors - namely, by the variability 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the sectoral outputs and the correlation between them - and by the extent of risk sharing in the financial markets. For the latter study, we compare the equilibrium of the stock market econ omy under which there is incomplete risk sharing with that of a benchmark economy in which a complete set of Arrow securities are traded and hence risk sharing is complete. There are several productive activities producing the same good (income) which we call sec tors. Sectors differ from each other in their risk profiles only. Production can be organized ei ther by private firms operating on the profit motive, or by less productive state firms which are extensions of a benevolent government, interested in maTimiring the welfare of its employees. The households distribute their given supplies of labor between these two types of firms. It is costly to work for a private firm (requires more effort/training/skills etc.) and costless to work for state firms. Workers thus have a best outside option (working in state firms) to working for private firms. This option is productively less efficient from the macro point of view and pays less (expected wages) to the households. Our study shows that the households supply less labor to private and more labor to state firms when financial markets are incomplete as compared to the situation of complete markets. The difference in the employment levels between these two cases is influenced by the variability of the sectoral outputs and the correlation between them. The practical implication of the study is therefore this. A productively inefficient (from the social point of view) and less paying outside option for households can be useful as an insur ance instrument. State firms displace private firms as employers of households leading to loss of productive efficiency. There are however offsetting gains in risk sharing which lead to wel fare improvements for households. An argument somewhat similar in spirit is that of Amott and Stiglitz (1991), who show that non-market institutions can improve welfare sometimes by pro viding insurance in the presence of moral Hazard. One can think of a number of other examples 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of such low paying, productively inefficient outside options in real economies, e.g. - in a devel oping economy, household production or self cultivation of land with the help of family labor as an alternative to wage labor in the manufacturing sector. In more industrialized economies, government financed unemployment doles is an example of a costless but low paying outside option. Our choice to identify the outside option with state firms is partly motivated by the cur rent interest and concern in the profession about the slow pace of privatization in several erst while command or regulated economies. Several existing studies, such as Ramamurti and Ver non (1991). Cook and Kirkpatrick (1988), and Commander and Coricelli (1995) suggest that the public - private balance has not been dramatically altered in favor of the private in most of these economies, particularly in large scale manufacturing sectors. The paper offers insurance markets incompleteness as an explanation for higher employment levels in state firms. Since transition economies are good examples of incomplete insurance markets, the paper offers an explanation for the persistence of state enterprises in these economies. Several other alternative explanations have been attempted in the literature such as, job search costs (Atkeson and Ke- hoe, 1996), training costs (Arabadjiev, 1999), political compromises (Dewatripont and Roland, (1992), Fernandez and Rodrik (1991) amongst others. The explanation that we provide which is based on risk sharing supplements but does not supplant the others, as available empirical ev idence is insufficient to build a strong case for any one of these theories. Although the role of the public sector in transition economies is the focus, the present paper does not address the issue of transition dynamics. Dynamizing the present model to study the evolution of the public sector over time would be a move in this direction and is left as a future exercise. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As pointed out above, household production is another example of a productively inefficient and low paying outside option. There is a big and related literature on household production models within the RBC framework (see Greenwood, Rogerson & Wright (1995)) whose focus is the explanation of the stylized facts about business cycles. Our study focuses more on the risk sharing aspects of these outside options. The main conclusion is that there are substantial employment gains from market completion. The calculation of associated welfare gains is left as a future exercise. Workers and private entrepreneurs have sector specific skills which expose them to sectoral shocks. The extent to which these risks can be shared through the wage contract depends on the labor market structure. Two extreme scenarios are considered as first cuts, (i) Competitive, un der which there are many workers and entrepreneurs in each sector. Wages ate equal to marginal products in each state in equilibrium (Section 2 explains why). Workers and entrepreneurs are exposed to risks in equal measure when this is the case, (ii) Monopsonistic, under which there is one firm in each sector acting as a wage leader. The firm takes the worker’s optimal labor supply response to a given wage contract into account and decides on the optimal wage contract. The optimal wage contract thus provides complete insurance to the workers against sectoral risks. The paper studies how the public-private employment division is affected across these two sce narios. 4.1.1 Summary of comparative static results Employment levels in private firms may be higher when markets are complete than when these are not because of the following reason. Complete markets allow entrepreneurs and workers 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (in particular workers) to diversify idiosyncratic production risks more. Thus, given a risky in come/wage stream, it allows households (and entrepreneurs) to smoothen its consumption and thereby increase utility. This in turn induces households to accept lower wages for a given labor supply or equivalently increase labor supply for given wage contract leading to lower private em ployment levels in equilibrium. This implies that as the best outside option for households, state firms employ larger numbers when markets are incomplete relative to when these are not The difference between the equilibrium employment levels under complete and incomplete markets depend on a complex of factors. We summarize some of the main findings below. 1 . Private employment levels are inversely to the coefficient of relative risk aversion, both under complete and incomplete markets and under both wage settings. The difference between the two levels is also in general larger for lower values of risk aversion. A strict monotonicity relationship however cannot be established on the basis of the numerical analysis. 2. The difference is strictly positive under competitive labor markets and zero under monop- sonistic conditions if firms, as distinct from their initial owners the entrepreneurs, are assumed to maximize expected profits. If firms are assumed to be risk averse and maximize expected utility, the difference is higher under competition. 3. Employment gains from market completion are larger when sectoral shocks are negatively correlated compared to when these are positively correlated. Similarly gains are larger when sectoral variabilities are higher. Section 2 develops the general model of a stock market economy under the two wage settings and defines the complete and incomplete markets equilibrium The comparative statics results about the government-private equilibrium are presented in section 3. 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 2 The Model There are two periods 0 and 1, and J sectors of production indexed by j = Produc tion in each sector is organized by state and privately owned firms. Production decisions (i.e. employment and investment decisions) are made at date 0. The actual production takes place at date 1. At date 1, Nature subjects each sector j to a total productivity shock t j * with prob ability pa. All sectors produce the same good (income) and differ only in their risk profiles t )j = {t j*}. Shocks are multiplicative. The production function of private firms in sector j is given by = {yj’(s)} = {jfjP{Pr k,)} where/and k stand for labor and capital. Private and government firms in any sector are subject to the same productivity shocks but have different state independent production functions. In particular government firms operate with an exogenous and historically given stock of capital. 1 The production function of the state firm in sector j is = {y?(s)} = {rfig 3 {I*)} ■ All functions are assumed to be continuous and differentiable. The production functions sat isfy. Assumption 4 I. f 3 (l}. 0) = f 3 (0, k ,) = 0. Both inputs are essential. 2. f 3 is strictly concave and f \ > 0, f l > 0. 3. f 3 (Ij. kj) is linear homogeneous. 4. ff{0.kj) = 3c.//(/j,0) = 0,/£(/,,0) = oc,/j((0, kj) = 0. (Inada conditions) 5 . yJ(0) = O.g3 ' > Q.g3 " < 0 1 The sate firm’s investment decision is not modelled here. 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The model is interesting only when for k} above a critical minimum, the level of output in private firms is sufficiently higher than that in state firms given the same employment levels in both such that private firms are able to pay their workers more than the state firms in equilibrium. Two reasons suggested for the lower productivity of workers in state firms and assumed in the model are - firstly, the state firms operate with a fixed and outdated capital stock, and secondly, workers in state firms lack incentives to put in quality effort because of a free rider problem involved in the government wage contract So, Assumptions There exists kc } such that for k} > kc y ^(Ij) < P(lr kj) Private firms are initially (at date 0) created and owned by the entrepreneurs. Labor and en trepreneurship are sector specific, which means that each household has the skills to work and each entrepreneur the leadership to organize production in one sector only. Entrepreneurs max imize expected utility as consumers and total dividends (output minus costs) from production in their capacity as producers. Workers/households maximize expected utility only. We need to discuss the competitive structures in the labor and asset markets at this point. Two different types of competitive structures in the labor market will be considered in the paper. Under scenario 1, numerous identical entrepreneurs and households are assumed to exist in each sector j. This implies that entrepreneurs and households perceive their private actions as not influencing the market wage rates. In particular, they make private decisions taking the market wage contract as given. The representative firm in sector j only decides therefore how much labor to employ and the representative household only decides how much to supply, at a given wage contract. Under scenario 2, we assume that there is one firm per sector which therefore has some monopsony power in the labor market The monopsony power is translated into the equilibrium 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. labor contract in the following way. The household accepts a wage contract as given and decides its optimal labor supply. The entrepreneur is assumed to know the preferences of the households and hence its labor supply curve. He chooses the optimal wage contract which maximizes his expected profits. Thus the entrepreneur acts as a wage leader and in equilibrium ends up by par tially/completely (depending on the firm’s objective function) risk insuring the worker against wage risks. Under both scenarios, there are innumerable traders in the asset markets. This is obviously true under scenario 1 because each sector has numerous firms. In scenario 2, the number of sec tors and households has to be large enough to ensure this. This condition implies that under both scenarios, private agents (workers and entrepreneurs) perceive the security pay-off structures in the asset markets to remain unaffected by their actions. In a stock market economy, agent’s (worker’s or entrepreneur’s) actions, /_ , and k, influence the dividend payments of the 7 th rep resentative firm in equilibrium and hence security payoffs. However our assumption implies that agents do not perceive this as causing the market subspace to change and hence their date 1 income streams from securities to be affected. One way to explain this under the competitive scenario is to think that a household or an entrepreneur in buying a share of the j th representative firm is actually investing the amount on the income stream offered by the industry. He can al ways change his portfolio for the numerous firms within the industry to take care of any changes in the dividend stream offered by a single firm without changing his portfolio for the industry. In other words, private agents make their portfolio decisions assuming the market subspace to be given. The households and entrepreneurs are said to be having competitive price perceptions in the stock market when this is the case (see Magill and Quinzii, 1996). 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 .1 Competitive price perceptions in financial and labor markets As initial owners of firms, entrepreneurs make capital investment and employment decisions, kj and I* respectively, for their firms at date 0. Capital investment can be financed by selling ownership shares of firm j to households and other entrepreneurs. Trading equities is thus also a way of sharing sectoral production risks. Equities are assumed to be the only assets in the economy. Labor is hired at date 0 for date 1 and paid a contract W } — . Employment levels are not state contingent although wages are. Let V represent the dividend streams of the equities, V > the sth row, jth column element, and V , the sth row of the matrix. Under a competitive asset market structure in sector j the rep resentative firm have zero retained earnings in equilibrium. This means that output net of wage costs are paid out as dividends to the shareholders by way of return on capital invested in the firm. Another way to think about this is that the firm’s budget constraint (revenue > wage costs + dividends) must bind in equilibrium. It must not be left with a surplus. Thus in equilibrium, v = - w *h}U < 41> Let us define k = {k]}'j^l, I = {/7}j=1, and W = the vector of private wage contracts. Then in equilibrium, V = V{k, I, W) i.e. the dividend streams depend on the ac tions chosen by the agents. Since production functions are constant returns to scale, they can be writtenasj^ = Hence V(.) can also be written as V = W ) in equi- • i t i librium, where j = { }j=1. We shall choose either representation according as whichever is more convenient. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We are now ready to discuss the optimal programs of the representative household and en trepreneur. For convenience of notation we shall regard date 0 also as another state, namely, the state0 ands € {0.1. ..5}. The state independent utility function u(c) is assumed to be identical for workers and en trepreneurs with. Assumption 6 u'(c) > 0. u"(c) < 0. u'(0) -► oo Given an optimal choice of k} (which he makes as a producer), the jth representative en trepreneur is restricted to the following budget set as a consumer zi < *0 + (1 - SJ)Qj - SiQ, - k} BJ(Q.V) = <xJ = (x1 0.x\.jci s) e < ei + E ^,^V ;* Vse {1....5} (4.2) where xJ represents consumption, eJ = {e ^ } ^ _ 0 initial endowment, SJ = { < ^}/=1 ownership shares of representative firms in other sectors, Q = {Q}}j=l the price of Ml ownership of firm j- Define II^x^) = II1 ^ ^ ) ) as the jth entrepreneur’s vector of personal valua tions of income streams - i.e. his present value vector. Assuming the wage contract W }, the security prices Q, and the market subspace or < V > to be given, the entrepreneur j, 1. chooses kj and ld, to maximize n - w t y - n i ' W k j 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. as a producer. Since entrepreneurs are initial owners of their firms, production projects are evaluated using their personal valuation vectors. 2 . given his choice of from above, chooses x3 and & to maximize s i'ix3) = u(x£) + 5 Zpstt(ri) subject to Equation ( 42), as a consumer. Since labor is sector specific, the index j can also be used for the representative worker/household working in sector j. The jth representative household has 1 unit of labour which it distributes between the private and the government firms. Working for a government firm is costless and working for a private firm is costly for the laborer, either because they have to invest to acquire more skills or because they have to put in harder efforts. The cost of supplying labour to the private firm is Cj(l} ), </,(/,) > 0 for lj > 0 and dj > 0. Since it is costless for households to work for state firms, it is optimal for them to supply any residual labour to the government Thus, lp } - r - lj — 1. Henceforth, private employment will be denoted as /, and state employment as 1 - lj. The j th household’s budget set is given by, M J(WJ.G,Q,V) = VI I f m J = (tjIq, € f?f+ l mi — ^ + Wilj + (1 - /,)& ’. + a L i W {1....S} (4-3) 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where m J is consumption of household, ui3 = {u^}~Lo initial endowment, G = {Gs }*=1 the state wage contract, and 03 = }^.lt ownership shares of private firms. Household j chooses m J, l}, and & as«niming the private and state wage contracts, W 1 and G, security prices Q and the < V > to be given, to maximize, s = u(m £) + £ p s it(m ') “ M M j=i subject to Equation ( 4.3). W e need to discuss the characteristics of the competitive equilibrium labor contracts at this point. There is no enforcement mechanism for the private wage contracts in the model. This means that there is no penalty for the households or firms to renege on a wage contract at date 1 . Since there are many firms in each sector, it is therefore possible for a worker in sector j to join one firm at date 0 and leave it to work for another at date 1 for a higher wage. Similarly a firm can attract workers from a competitor by paying higher wages at date I. It is clear that incentives to renege on a contract drawn at date 0 will generally exist if at date 1 Nature moves before the agents and announces a state. W e want however to look at an equilibrium in which at date 1 workers and entrepreneurs do not have any incentive to switch parties with whom they have drawn contracts at date 0. Under this equilibrium all entrepreneurs in any individual sector j must therefore offer the same wage contract to any worker in this sector. Of the set of profit maximizing wage contracts, under the competitive assumption, there is only one which is robust with respect to parties reneging on their date 0 contracts in this sense. It is clear that this is also the equilibrium wage contract which is robust with respect to opening of spot markets although in our model there are no spot markets for labor. 7 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. There are no enforcement problems with the state wage contracts. Wages paid to the em ployees are the only expenses of the government and the revenue from the state firms its only income. In equilibrium the two must balance. W e assume that the Government is interested in maximizing the welfare of its employees all of whom are given the same weights in its objective function. So the government firms distribute the total revenue that is generated from all their activities equally among all the employees by way of wages. Thus the wage profile in the state firms is. G{1) = ~ /j) (4 .4 ) Z U i - h ) 4.2.2 Non competitive labor markets, competitive price perceptions in asset markets When labor markets are monopsonistic, households are wage takers. The first order condition of the household with respect to l3 defines the optimal labor supply as an implicit function of the private and state wage contracts, security prices and market subspace. W e denote this by, l]{W3.G.Q.V). The entrepreneur j assumes Q, and < V > to be given and 1 . chooses kj and W 3, to maximize n^(a^)(j^ - w > ij) - n^ix3 )^ as a producer. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 . given his choice of k: from above, chooses x* and 6 * to maximize s £/(**) = «(*S) + 5 > « (x J ) 9 = 1 subject to Equation ( 42), as a consumer. The first order condition of the firm with respect to Wf assuming differentiability and the partial derivatives to exist, is given by. v , = ‘ ' s - i4- s > Equation (4.5) thus implicitly defines the optimal wage contract as a function of all the other variables. W e now define the Stock Market Equilibrium for an economy with competitive and non com petitive labor markets respectively. 42 3 Stock M arket Equilibrium Let x = {xJ } -f=1, m = represent the consumption allocations, and 0 = {0J}^_1 , S = { < 5 J }j=l the portfolio allocations. Then, Definition 3 A Stock Market Equilibrium (SME) with competitive labor markets is a 4-tuple ((x. (m, /)). (k, ld). (0.6), {W , G. V, Q)) of consumption allocations to entrepreneurs and households, labor supplyplans o f households, production plans o f entrepreneurs.portfolioplans 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ofhouseholds and entrepreneurs, private and state wage contracts, dividends and security prices such that, 1 . For each representative entrepreneur j, (;&.V)=argmax{U{x>)} and (x'.&.kj) e B J(W J.Q.V) (M j) = argmaxn1i(xJ)(yP J ~ W ’l,) - 2. For each representative household j, [m f.V jj) = argmax{L(mJ,l])} and {rk},9>) G M J(W 3.G .Q .V) 3. At date I. workers and entrepreneurs have no incentives to switch parties with whom they have drawn contracts at date 0. 4. Firms in each sector j have zero retained earnings. v = WtiJ,)-wib}Ui 5. Labor markets clear fi = i G = G(l) 6. Equity markets clear, i l k + = i=i 1=1 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Definition 4 A Stock Market Equilibrium (SME) with non competitive labor markets is a 4- tuple ((x. m). (/. k), (0.6), [W. G, V. Q)) o f consumption allocations to entrepreneurs and households, employment, investment by entrepreneurs, portfolio plans o f households and en trepreneurs, private and state wage contracts, dividends and security prices such that, 1. For each representative entrepreneur j, (U .SJ) = argmax{l'(x3)} and {xi.&.k,) e B 3{Q. V) (kj, W 3) = argmaxn^xJHy^ - W 3i}(W3.G,Q. V) - 2. For each representative household j. [m’.O’Jj) = argmax{U(m},lj)} and (n V > ) e M 3{W3.G.Q. V) where I, = 1 }(W J.G.Q. V '). 3. Firms in each sector j have zero retained earnings. v = tyikjJ,) - w ’ftU 4. Labor markets clear. G = G(l) 5. Equity markets clear, 1=1 i=i 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Stock markets are a way of sharing idiosyncratic production risks when such risks are too complex for agents to write and exchange contingent contracts on these. The objective of this paper is to compare the equilibria of such an economy with that of an idealized one in which a complete set of such contracts are traded. In the next section we describe this economy and define its equilibrium. 42.4 A benchmark economy In an economy in which a complete set of Arrow securities are traded, entrepreneurs have no incentives to sell ownership shares of their firms and hence are sole proprietors. With competitive labor markets, the budget set of the jth entrepreneur is: r 0 — ^0 B ^ P . W 3) ^ - x3 € < ei + K /(a)-W 7 /J + « * where P = {p,}f=1 represents the prices of Arrow securities, represents quantities of Arrow securities purchased. The budget set of the jth household is: G) = m 3 € Rf+l VI r t^ o — X **=i p*Q VI Wi + Wjlj + i l - l J G . + C l 4 (4.7) where Q 3 = {CiJiLi represents quantities of Arrow securities purchased. Let £ = and C = {C J}/=i Then, 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Definition 5 A Complete Markets Equilibrium (CME) with competitive labor markets is a 4- tuple ((z. m . I). (fe, ld), ({, Q). (W ,G, P) o f consumption plans o f entrepreneurs and con sumption and labor supply plans o f households, production plans o f entrepreneurs, portfolios ofhouseholds and entrepreneurs, private and government wage contracts and state prices, such that: 1 . For each entrepreneur j . (* '.£ ) = argmax{U{x>) | ( z ^ ) 6 B ^ P .W * ) } (kj.ld) = argmaxIl1 J e(xJ)(yP - WH,) - 2. For each household j, (mJ,CJ.fJ)=argmax{U(mflJ) | ( r t , ? ) 6 M ^ ( P . W>.G)} 3. At date I, workers and entrepreneurs have no incentives to switch parties with whom they have drawn contracts at date 0. 4. Labor markets clear i d = i G = G{1) 5. Markets fo r Arrow Securities clear, £ K J +CJ) = o ,V s e { i,...s } j=i 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Definition 6 A Complete Markets Equilibrium (CME) with non competitive labor markets is a 4-tuple ((x.m), (/. k). (£.£). (iV.G, P) o f consumption plans o f entrepreneurs and house holds. employment and investment by entrepreneurs, portfolios o f households and entrepreneurs, private and government wage contracts and state prices, such that: 1 . For each entrepreneur j, (& .?) = argmax{U(x>) | € B *m (P)) (kj. W J) = argmaxn1i(xJ)(yp J - W>l](W> G. P)) - ll^ x J )^ , 2. For each household j, .1]) =argmax{U(mJ. I j) | (nv>,CJ) € M ^ P . W 1 .G)} where = fj(W J.G.P). 3 . Labor markets clear. G = G(i) 4 . Markets for Arrow Securities clear, J £ K J +Cj ) = 0,VS6 { 1 ,...5 } 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 J Comparative Statics of Employment In this section we present the results from the numerical simulations of the Stock Market and Benchmark economies under competitive and non competitive conditions in the labor markets. The models are not calibrated to fit actual data for the present Such an exercise is left for future research. Results for this stylized model(s) should therefore be taken as qualitative rather than quantitative. Equilibrium values of I, k ,W ,0 and S are solutions of the following first order conditions of houesholds and firms. j £ P . «'(-•* + W’l, + (1 - l,)G. + £ W ( s ) - iv ;/,))(H V - G.) 3 = 1 1 = 1 -c '(/j) = O.Vj. (4.8) A ’ J + W’ .l, + (1 - h)G. + £ W ( a ) - WUi))(Y,, (») - » ’ .*/.) »=i i = i j - B V 0 - E W = O .V ,i (4.9) i = i , W,p(») K; = — — .Vs, ./when markets are competitive (4.10) dlj rV( A J # V P(s) j=i 1=1 J -«V0 + (i - q n j - £<s*q, - kj) = o ,v/ ( 4 .1 1 ) 1 3 I X > u V , + £ W ( s ) - wiii)){Y?{s) - IV*/,) 3 = 1 |= 1 J - « '( e j + (1 - tyQ j - £ * * < ? , ~ k,)Q> = ‘ (4-12) ■ si 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E ^ + E ^ 1-^' (4I3) 1=1 1=1 when labor markets are competitive. When labor markets ate monopsonistic these are given by solutions of the above with equation 4.10 replaced by y > su V ,+ y v ,(V 'p(s) - a(/j) - w y - ^ - ) S 0(W7) 'd(WY) + £ f y W ) - W 'tl,))i, = 0. Vf = 1 .. .5. i=i The model is now parameterized as follows. 5 is assumed to be equal to 5, and are equiprob- able. J is set equal to 4. There are thus effectively 4 “assets” for each household - 2 equities, labor for private firms and labor for state firms. The “degree” of market incompleteness (no. of assets relative to no. of states of Nature) is thus not very high. The utility function is assumed to be CRRA. The production function is Cobb-Douglas, . l/j' fc1 " J . Both sectors have identical linear cost function c.lj. The state firm’s production function is (1 — lj)T. The parameters .4. a. c.andr are all chosen so that (i) interior solutions in employment levels exist in all the cases and (ii) government wages are less than private wages in every state. 4 J .l Competitive labor markets The linearity of the cost function of labor keeps the model simple for the purpose of getting nu merical solutions. This leads to another problem of a different nature, however. Our model al lows for short sales in equities. As wages and dividend streams for any firm are collinear in equi librium (both are scaler multiples of the risk profile of that sector), with a linear cost function of labor the two assets for household j and0j - end up having very similar payoff structures. If 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. we allow for short sales in the numerical examples below, each of the households end up short- selling over 30% of the firms that they work for, in equilibrium. This boosts the equilibrium employment levels in the private firms considerably. What happens is that household j short- sells at date 0 , sells more labor to firm j and pays dividends to other shareholders at date 1 from its labor earnings. Short sales of this magnitude is clearly not realistic. To keep the model simple for numerical solutions and at the same time meaningful, we therefore impose a no short sales restriction in equilibrium. Since die whole point of this section is to form some idea about the effect of market incompleteness on employment based on a stylized model such a step is not unjustified. W e first consider a model with high sectoral risks (measured by the ratio of standard devia tion to mean) and negatively correlated sectoral shocks. The productivity shocks are, m = {1.6.6.1.1}, n = {4,1,1.2.2} Qualitatively, there are 4 levels of shocks - 6 represents very high, 4 high, 2 medium and 1 very low. Thus sector 1 has higher variability and mean in productivity compared to sector 2. The sectoral shocks are negatively correlated - around -0.75. Sectoral variabilities (measured by standard deviation/mean) are high also at around 0.91 and 0.75. Equities in the Stock Market model and the Arrow securities in the Benchmark model perform well in hedging sectoral risks in this model because of high negative correlation. The risk aversion coefficients are chosen keeping in mind the usual conventions in the RBC literature, i.e. between 0.8 and 2.0. The results are reported in Table 1. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the second model the sectoral variabilitcs arc lowered to around 0.61 and 0.26. The cor relation between the sectors is around - 0.22. The shock configurations are, T il = {‘ 2.6, 6.2,2} IJ2 = {4.4,2.3,3} The results for this example is reported in Table 2. For the third model, we alter the shock pattern of sector 2 above slightly, so that the sectoral risks are positively correlated, around033. Equities and Arrow securities perform less well with regard to hedging sectoral risks. The sectoral variabilities are the same as before. The configu rations are, T fi = {2.6 . 6 .2.2} m = {4,4.3.2,3} The results for this example are reported in Table 3. Observations and Explanations The following conclusions can be drawn from the results reported in these three tables. 1. Private employment levels are inversely related to the degree of relative risk aversion. Within the set up of competitive markets this result has been established analytically and dis cussed elsewhere (see Roy 2000a). However this relationship is difficult to establish analytically when wage settings are monopsonistic. 2. There are situations under which CME employment levels are higher than SME employ ment levels under competitive wage settings. The difference between these depends on a com plex of factors, in particular (i) the degree of relative risk aversion (ii) the risk characteristics 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the sectors, namely, the variability (measured by normalized standard deviation) of the sec toral shocks, and the correlation between these, (a) In general, the difference goes down as the degree of risk aversion goes up as the table below shows, (b) The difference is larger when sec toral shocks are negatively rather than posiitvely correlated, (c) The difference is larger when sectoral variabilities are higher. Complete markets allow entrepreneurs and workers (in particular workers) to diversify id iosyncratic production risks more than the stock market economy does. Thus, given an income stream, it allows households (and entrepreneurs) to smoothen its consumption and thereby in crease utility. This in turn induces households to accept lower wages for a given labor supply or equivalently increase labor supply for given wage contract. Expected wages are thus found to be lower under a CME than under a SME. In particular, expected wage differences between the private and state firms (mean W 1 - G) are lower under a CME than under a SME. For the same reason the variability of this difference, measured by the ratio of the standard deviation and mean of the expression, is slightly higher under a CME than under a SME. The higher the degree of negative correlation between sectoral shocks, the greater is the extent of consumption smoothening through trade in contingent contracts. Hence the larger is the difference between CME and SME employment levels. Differences are larger, for negative rather than positive correlations for the same reason. Similarly, the higher the sectoral variabil- ties, the larger is the utility gains from consumption smoothening and hence the difference. In general, differences are larger for lower values of the risk aversion coefficient But there does not seem to be a strictly monotonic relationship in this respect 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.1: SM E and CME for com petitive labor markets, r = -0.75, a/p = 0.91,0.75 Sector 1 3 l\ k\ *i !L {W l - G) liY \) 0.9 0.78 0.85 44 40 0.94 0.95 0.91 0.91 1 .0 0.50 0.56 40 36 0.93 0.93 0.91 0.91 1.1 032 0J7 37 32 0.92 0.92 0.91 0.91 13 0.15 0.18 33 29 0.92 0.92 0.91 0.91 1.6 0.06 0.07 32 28 0.92 0.92 0.91 0.91 2 0 0 .0 2 0 .0 2 34 30 0.91 0.91 0.91 0.91 Sector 2 J h h k* k2 °-{W* - G) i i W 2 - G ) K YI) z (Y’ 2) 0.9 0.60 0.72 32 31 0.67 0.70 0.61 0.61 1.0 0.41 0.51 32 30 0.64 0.65 0.61 0.61 1.1 0.28 0.36 32 30 0.63 0.64 0.61 0.61 1.3 0.14 0.18 34 31 0.62 0.63 0.61 0.61 1.6 0.06 0.07 38 35 0.62 0.62 0.61 0.61 2 .0 0 .0 2 0 .0 2 47 43 0.61 0.62 0.61 0.61 3 0.9 1.0 1.1 13 1.6 2 .0 f{Y) 0.48 0.45 0.43 0.41 0.38 0.36 f(Y) 0.45 0.43 0.41 039 036 0.34 I j = employment under a SME, = employment under a CME, k} = physical investment under a SME, k3 = physical investment under a CME, ^ (W 3 - G) = standard deviation/mean of wage differentials under SME, ^(W J - G) = standard deviation of wage differentials under CME, (Y *) = standard deviation of sectoral output under SME, %(Yj) = standard deviation of sec toral output under CME, ^{Y) = standard deviation of aggregate output under SME, ^{Y) = standard deviation of aggregate output under CME. Figures for £( W 3 - G) and - G) have been rounded offat the second place after decimal. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4 2 : SM E and CME for competitive labor markets, r = -0 2 2 ,< r/^ = 0 .6 1 ,0 2 6 Sector 1 2 *i ki Z W ' - G ) 0.85 0.90 0.93 41 39 0.67 0 .6 8 0.9 0.79 0.81 38 36 0.63 0.64 1.0 0.51 0.53 34 32 0.62 0.62 1.1 023 025 31 29 0.62 0.62 1.3 0.15 0.16 28 26 0.61 0.61 1.6 0.06 0.06 27 25 0.61 0.61 2 .0 0 .0 2 0 .0 2 29 27 0.61 0.61 Sector 2 J h h *2 *2 i( W * - G ) Z(W2-G ) 0.85 0 .8 8 0.91 39 38 0 2 1 0.33 0.9 0.77 0.81 37 37 0.28 029 1.0 0.51 0.55 35 34 021 021 1.1 0.34 027 34 32 021 021 1.3 0.16 0.17 32 31 026 026 1.6 0.06 0.06 34 32 026 026 2 .0 0 .0 2 0 .0 2 39 37 026 026 lj - employment under a SME, I, = employment under a CME, k} = physical investment un der a SME, A r ; = physical investment under a CME, ^{W} - G) = standard deviation of wage differentials under SME, WJ - G) = standard deviation of wage differentials under CME. Figures for W J - G) and - (?) have been rounded off at the second place after decimal. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.3: SME and CM E for competitive labor markets, r = 0 2 3 , a/p = 0 .6 1 ,0 2 6 Sector 1 3 h *t *i i ( W l - G ) Z ( W '- G ) 0.85 0.95 0.96 41 41 0.661 0.669 0.9 0.81 0.83 38 38 0.630 0.633 1.0 0.52 0.54 34 33 0.617 0.618 1.1 034 035 31 30 0.614 0.614 1.3 0.16 0.16 27 27 0.612 0.612 1.6 0.06 0.06 27 26 0.610 0.610 1.7 0.04 0.04 27 26 0.610 0.610 Sector 2 J h h ki *2 Z(W2- G ) %{W2-G ) 0.85 0.87 0 .8 8 37 37 0276 0278 0.9 0.77 0.79 36 35 0266 0267 1.0 0.52 034 34 33 0263 0263 1.1 035 036 32 3 1 0262 0262 1.3 0.16 0.17 31 30 0262 0262 1.6 0.06 0.07 33 32 0262 0262 1.7 0.05 0.05 34 33 0262 0262 l: = employment under a SME, l2 = employment under a CME, k} = physical investment un der a SME, kj = physical investment under a CME, ^(VVJ - G) = standard deviation of wage differentials under SME, - G) = standard deviation of wage differentials under CME. Figures for lj and lj, have been rounded off at the second place after decimal. Figures for ^{WJ - G) and - G) have been rounded off at the third place after decimal. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 J J Monopsonistic wage setting We now compare the competitive wage setting with the monopsonist one. Table 4 presents the results. The shock configurations are the same as in Tables 2 and 3. Observations and Explanations Since firms, as different from entrepreneurs, maximize present value of profits, they are risk neutral relative to households which are risk averse. Under this wage setting they therefore pro vide complete insurance to the households. This does not imply that the wages are constant across states ofNature because there are aggregate risks in the economy. Wages are variable but much less than under the competitive case. Wage contracts are identical under CME and SME. In particular, employment levels are identical under the two as are ^(W J-G ) also. ^(W J-G) is lower under this model than under the competitive model. Wage contracts are also less variable across sectors. Under competition, the more variable sector paid the more variable wage contract (because wages were equal to marginal product). The difference in the variability of wages across sectors is much less in this model for obvious reasons. If we alter the assumptions of the monopsony model and make the firm risk averse rather than risk neutral, i.e. if firms maximize the expected utility of its initial owner, differences in the employment levels under CME and SME surface again. The reason is that firms provide partial insurance to the workers in this kind of a set up, as opposed to complete insurance in the present one. Compared to the competitive situation the differences however are much less, given everything else. The two models discussed in the paper (the competitive one and the monopsonistic one with profit m axim izing firms) thus represent two extremes with regard to risk exposure ofhouseholds. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.4: SM E and CM E under monopsony, r = -0 .2 2 , a/n = 0 .6 1 ,0 .2 6 Sector 1 J h ki ° z [W l -G ) 1 { W '- G ) 0.85 0.71 0.71 39 39 032 032 0.9 0.55 0.55 36 36 032 032 1.0 034 0.34* 32 32 031 031 1.1 022 022 30 30 031 031 Sector 2 3 h I'i ki k2 1 s Z(W2 -G ) 0.85 0.70 0.70 38 38 0.32 032 0.9 0.55 0.55 36 36 032 0.32 1 .0 0.34 0.34 34 34 0.31 0.31 1.1 0 .2 2 0 .2 2 32 32 0.31 0.31 Table 4.5: SME and CME under monopsony, r = 0.33, a/n =0.61,0.26 Sector 1 3 *t ki °-[Wl -G ) 0.85 0.71 0.71 40 40 039 039 0.9 0.55 035 37 37 038 0.38 1.0 0.34 034 33 33 038 038 1.1 0 .2 2 0 .2 2 30 30 038 038 Sector 2 3 h h k2 ki i(W * -G ) ° Z(W * -G ) 0.85 0.69 0.69 37 37 039 0.39 0.9 0.54 0.54 35 35 038 0.38 1 .0 0.34 034 33 33 038 0.38 1.1 0 .2 2 0 2 2 32 32 038 0.38 lj = employment under a SME, = employment under a CME, k, = physical investment un der a SME, k} = physical investment under a CME, ^(W 1 -G )= standard deviation of wage differentials under SME, J (W 1 —G) = standard deviation of wage differentials under CME. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4 Conclusion The paper develops a computable general equilibrium model to study the effect of asset market incompleteness on labor contracts. Specifically we study how the level of employment varies as we complete markets. Two wage settings are chosen to study the effects of market comple tion. These represent two extreme types of competitive structures in the labor market. The paper shows that employment gains for private firms from market completion can be significant under certain conditions. Employment gains are larger for competitive labor markets everything else remaining the same. A practical implication of the model is that a low paying and productively inefficient (from the social point of view) outside option for households may be attractive for risk sharing reasons. This provides a rationale for state enterprises in economies in which opportunities for diversify ing idiosyncratic risks in the financial merkets are limited. The results reported in the paper are based on numerical simulations of stylized models as closed form solutions of the equilibria are hard to find. These models are at this point not calibrated to actual data. This exercise is left for future research. The analysis presented here should be regarded as a first cut 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1] B. B. Arabadjiev. Essavs in Transition. PhD thesis. University of Southern California, 1999. [2] R. Arnott and J. Stiglitz. Moral hazard and nonmarket institutions: Dysfunctional crowd ing out or peer monitoring. American Economic Review, 81:179-90,1991. [3] K. Arrow. Essays in the Theory o f Risk Bearing. Markham Publishing Company, 1971. [4] K. Arrow and G. Debreu. Existence of an equilibrium for a competitive economy. Econo- metrica, 1954. [5] A. AtkesonandP. Kehoe. Social insurance and transition. International Economic Review', 37:377-401,1996. [6] C. Azariadis. Implicit contracts and underemployment equilibria. Journal o f Political Economy. 83:1183-1202. 1975. [7] Martin N. Baily. Wages and employment under uncertain demand. Review o f Economic Studies, 1974. [8] Miguel J. Villas Boas. Comparative statics of fixed points. Journal o f Economic Theory, 1997. [9] M. Boldrin and M. Horvath. Labour contracts and business cycles. Journal o f Political Economy, 103:972-1004,1995. [10] S. Commander and F. Coricelli. Unemployment, Restructuring, and the Labor Market in Eastern Europe and Russia. EDI, World Bank, 1995. [11] P . Cook and C. Kirkpatrick. Privatization in Less Developed Countries. Wheatsheaf Books, 1998. [12] T. F . Cooley. Frontiers o f Business Cycle Research. Princeton University Press, 1995. [13] G. Debreu. Theory o f Value. 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Cooley, editor. Frontiers o f Business Cycle Research. Princeton University Press, 1995. [21] S. Grossman and O. Hart The costs and benefits of ownership: A theory of vertical and lateral integration. In O. E. Williamson, editor. Industrial Organization. Aldershot 1990. [22] O. Hart and B. Holmstrom. The theory of contracts. In T. Bewley, editor. Advances in Economic Theory: Fifth World Congress. Cambridge University Press., 1987. [23] W . Hildenbrand and A. P . Kirman. Equilibrium Analysis. North Holland, 1988. [24] M. J. P . Magill and M. Quinzii. Theory o f Incomplete Markets. MIT Press, 19%. [25] M. J. P . Magill and M. Quinzii. Incentives and risk sharing in a stock market equilibrium. Working Paper, 1997. [26] P . Milgrom and J. Roberts. Comparing equilibria. American Economic Review, 199. [27] P Milgrom and C. Shannon. Monotone comparative statics. Econometrica, 1994. [28] R. Ramamurti and R. Vernon. Privatization and Control ofState-owned Enterprises. EDI, World Bank, 1991. [29] S. Rosen. Implicit Contract Theory. Aldershot 1994. [30] S. Roy. Risk sharing through labor contracts - risk aversion and employment Working paper, 2 0 0 0 a. [31] S. Roy. Comparative statics of fixed points of non isotone mappings in product spaces. Working paper, 2000b. [32] S. Roy. Government-private ownership equilibrium under incomplete markets. Working paper, 2 0 0 0 c. [33] D. Topkis. Supermodularity and Complementarity. Princeton University Press, 1998. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A The foe of the j th household is given by, £ - g .) - «/(/,) = o irr *o (A.1) where W3 is a function of { }f=1 and is function of { and 3. d{u(mJ 0 J , . ) j a mQ (3..) —--------= (mJ Q (J,.)) a{-lnm{3{3..) - 3—— — ) dJ mJ 0(3..) where mJ 0 (J..) represents the partial derivative with respect to 3. (A.2) which is clearly satisfied when mJ Q (3, ■ ) > 0. When t t i q (3, •) < 0, the inequality requires, d In m } 0(3,.) 83 We shall assume this inequality to be strictly satisfied because then /, (jr. G, 3) is strictly de creasing in 3 and this is the interesting case for this paper. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B The sign of mJ 0 (J..) depends on the behavior of the individual terms (£iS.)1 /J . Since mJ s = (see Appendix 5.2 below), we have, j ) > 0 iff ( ^ 2 .)‘/fl < 1, < 0 otherwise n il m* jr. 93 mQ Whether m„ and ( ) increases with J or not (the signs of these are crucial for the comparative m o statics propositions) in the neighborhood of the fixed point (/*, x ’) depends on the magnitudes °n -;k = o - 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix C For CRRA utility functions, from the foe’s of household 1 , the demand functions can be shown to be. <+ tL i few? + wih + ( i - w .) ( p j ( — )*mJ 0 *» when J £ 1. When J = 1, these are given by, < = + n v , + a - w ) «^i ° j j W o = p,mQ — w * Since production functions are Cobb-Douglas IV/f, = ctYf{s.lr k3). If G, is approxi mately constant with respect to l3 and wage differentials between private and public firms (WI - Gs) is always positive, mJ 0 is positively related to l}. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. When J = 1, clearly mJ Q is increasing in When (3 < 1, a rise in ^ reduces the denom inator and increases the numerator. So increases. When S > 1, both die numerator and the denominator move in the same direction, so the difference in the rate of increase of the two will determine the direction. Thus the rate of growth of the numerator which is equal to state income in s relative to total income, will determine whether increases or not 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D - ^ i ( - ) E W} + + (1 - /JG ,)} 0 t=l ^0 - ( i - j ) ( p , ) l/fl( - ) (l' I/fl) (D-n T 0 We shall look at cases when the middle term dominates, so that the expression is negative and expression 2.27 is negative. Then 3) is monotone decreasing in each 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Creator Roy, Sunanda (author)

Core Title Labor contracts under general equilibrium: Three essays on the comparative statics of employment

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

Contributor Digitized by ProQuest (provenance)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-203687

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