Contracts and Markets Citation Maretto, Guido Tulio Andrea (2010) Contracts and Markets. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/FBS0-F288. https://resolver.caltech.edu/CaltechTHESIS:05282010-090118586 Abstract I merge the standard Principal Agent model with a CAPM-type financial market, to study the interactions of contracts and financial markets. I prove existence of equilibrium in two models, a more general economy allowing for hidden type and action under generic mean variance preferences and a hidden action economy with Markowitz mean-variance preferences. I study economies for which markets have an insurance effect on compensation contracts. I show sufficient conditions for lower variance to obtain in large economies, even with asymmetric information. In this context I show the effect of markets' size on efficiency. I also study moral hazard economies, for which I prove existence of a unique pure strategy equilibrium, and I show that financial markets negatively affect the equilibrium returns of firms. In the final chapter I study the efficiency of securities issued under symmetric information. I find that small markets and low correlation of firms' returns generate inefficiency. I also show that the assumption of symmetry or independence is crucial to obtaining the insurance results in the previous Chapters. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Economics, Finance Degree Grantor: California Institute of Technology Division: Humanities and Social Sciences Major Option: Social Science Thesis Availability: Public (worldwide access) Research Advisor(s): Ledyard, John O. Thesis Committee: Ledyard, John O. (chair) Echenique, Federico Cvitanić, Jakša Defense Date: September 2009 Record Number: CaltechTHESIS:05282010-090118586 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05282010-090118586 DOI: 10.7907/FBS0-F288 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5876 Collection: CaltechTHESIS Deposited By: Guido Maretto Deposited On: 04 Jun 2010 16:35 Last Modified: 28 Oct 2021 19:18 Thesis Files Preview PDF (Dissertation) - Final Version See Usage Policy. 506kB Repository Staff Only: item control page

Contracts and Markets Thesis by Guido Tullio Andrea Maretto In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2010 (Submitted May 28, 2010) ii c 2010 Guido Tullio Andrea Maretto All Rights Reserved iii To L, K, Z & T iv Acknowledgements I want to thank John Ledyard, Federico Echenique, Jaksa Cvitanic for academic guidance and more. v Abstract I merge the standard Principal Agent model with a CAPM-type financial market, to study the interactions of contracts and financial markets. I prove existence of equilibrium in two models, a more general economy allowing for hidden type and action under generic mean variance preferences and a hidden action economy with Markowitz mean-variance preferences. I study economies for which markets have an insurance effect on compensation contracts. I show sufficient conditions for lower variance to obtain in large economies, even with asymmetric information. In this context I show the effect of markets’ size on efficiency. I also study moral hazard economies, for which I prove existence of a unique pure strategy equilibrium, and I show that financial markets negatively affect the equilibrium returns of firms. In the final chapter I study the efficiency of securities issued under symmetric information. I find that small markets and low correlation of firms’ returns generate inefficiency. I also show that the assumption of symmetry or independence is crucial to obtaining the insurance results in the previous Chapters. 1 Chapter 1 Contracts and Aftermarkets Hidden Type 1.1 Introduction Labor compensation is arguably the most relevant expense for a corporation. In the US, for example, more than 60% of the payment to factors group in the 2008 GDP was in fact to labor. Since the stock of a company is a claim to its profits, the firm’s decisions on workers’ compensation affect the returns of its stock. In the aggreagate this affects financial markets. By the same token, diversification opportunities offered by markets should influence the design of compensation packages. In view of this consideration it is rather surprising that the economic and financial literature has devoted relatively little attention to these interactions. Here I address two questions: • How does the existence of asset markets affect the design of compensation packages? • How does asymmetric information inside firms affect aggregate risk? The answer to the former question depends on preferences of individuals and on the distribution of returns of companies. The market provides provides insurance and diversification opportunties to risk averse principals, allowing them to reach a higher utility and eventually changing equilibrium compensation. I show that in large markets with symmetric information and independent returns, the compensation packages offered are always less risky. A weaker version of this result applies to the asymmetric information case. The answer to the latter question is that, under the same 2 assumptions as above, the constraints created by asymmetric information will induce securities that are at least as risky as those issued under complete information. Since this will turn out to be the case for every firm in the economy, it will be the case that this type of asymmetric information implies excessive risk also at the aggregate level. I reach these conclusions constructing a model of firms in financial markets. Each firm is formed by an owner and a worker. The skill of workers are initially unobservable. This is the only source of information asymmetry in the model. I address the effect of contracting inside firms on financial markets, and the effect of markets on firms’ efficiency. I do not study the effects of asymmetric information in markets, but rather the effects of asymmetric information inside firms on markets. This marks the first difference from the General Equilibrium works on insurance markets, starting from the seminal paper of Rothschild and Stiglitz . Another important difference is that, in those papers, the fact that some individuals are risk neutral and act as firms is usually an assumption with the notable exception of Dubey and Geneakoplos, where individuals endogenously form pools to share risk. In this work there are many risk averse investors, who access financial markets to trade away part of the risk they are exposed to. Traditionally, the assumption of risk neutrality of a principal is motivated by the existence of diversification opportunities. The present work also enquires when the usual motivation, the opportunity to trade risks on a financial market, actually provides a justification for the risk neutrality assumption and its implications. A strand of the finance literature looks at asset pricing in the presence of delegated portfolio management (for a survey, see Stracca, 2003). An example of the approach typical in these papers can be seen in Ou Yang’s paper. These studies look at the effects on prices and returns of the classical informational asymmetries phenomena. Moral hazard and adverse selection are largely studied in a CAPM or APT setting, in which a representative principal delegates his investing decisions to an agent. In this literature inefficiencies take the form of deviations from the non-delegated case equilibrium. These deviations can take the form of changes in asset prices and optimal portfolio composition. Besides the different object of interest, the perspective in these works is in a sense opposite of the one taken here. There we have informed parties trading, whereas in the present work it is the uninformed parties accessing markets. A branch of the general equilibrium literature focuses on the organization of firms and the employment choices of infividuals. Rahman (2005a, 2005b), Ellickson, Grodahl, Scotchmer and Zame (1999, 2001, 2005) and Zame (2008) separately study economies where agents form team or 3 clubs, which can have different functions such as production or consumption. While none of these models includes financial markets, some of them allow for asymmetric information. Their approach is very general, firms and contracts are both endogenously determined. However, this generality makes it hard to derive any predictions on the shape of contracts. Finally, the works technically closest to mine are those by Magill and Quinzii (2005) and Parlour and Walden (2009) who use models which bear some similarities to this one. As in the present chapter, they take firm formation as an exogenous process, abstracting from labor market considerations, and they allow for contracts inside firms and financial markets across firms. However they use their model to study economies with hidden action. I address the problem of moral hazard in a separate chapter. The paper is constructed as follows. In Section 2 I introduce the problem and an example. In Section 3 I present the model. In Section 4 I define the notion of equilibrium. In Section 5 I prove existence of equilibrium. In Section 6 I provide sufficient conditions for compensation to be closer to a wage when principals access markets. 1.2 A Simple Example In this example, I show how markets can affect contracts in a very simple setting. Markets provide diversification for principals. This diversification opportunity makes Principals insure agents more than in a standard P-A model.
2 There are four individuals, with identical preferences over random variables, U (X)F µX , σX =
b 1 2 2 µX − 2 µX + σX with b = 4 . Two of them, the Principals, own an identical technology. Two of them, the Agents, have the skills to operate the technology. Their skills are private information at the contracting stage. Both agents have the same reservation utility of ū = 1. The skills of agents are identified as average returns. The performance of one agent is stochastically independent from the performance of the other. The mean returns for an agent is given by µl = 2, µh = 3 and the variance is σl2 = σh2 = 1. Each principal designs a menu of two linear contracts to offer to the agent he is going to be randomly matched with. A linear contract is a function of the form y = α+βx, and is characterized by a pair (α, β) Principals will receive yp = −α + (1 − β)x, the agent will receive ya = α + βx. 4 1.2.1 Standard Principal-Agent Model: No Market The problem of principals is going to be: max 1 αH ,βH ,αL ,βL 2
1
2 2 2 F αH + βH µH , βH σH + F αL + βL µL , βL2 σL 2 subject to IRH , ICH , IRL , ICL The solution to a standard P-A model with linear contracts is to offer a menu of these two contracts: yL = 1.17 yH = −.65 + (.64) x 1.2.2 Principal-Agent with Financial Markets Now suppose that the Principals can trade their claims on the asset market. The objective function of principals is now different, because it includes the outcome of markets. With mean-variance preferences the asset market equilibrium is determined by a few simple equations, which can be substituted in the objective function. The outcome of the CAPM market: • The market portfolio will be characterized by mean and variance - µM KT (αH , βH , αL , βL ) = αL + αH + βL tL + βH tH 2 2 2 - σM KT (αH , βH , αL , βL ) = βH tH (1 − tH ) + βL tL (1 − tL ) • Equilibrium shares will be - Market Portfolio: θH = θL = 12 - Riskless asset: 2 θiP = (−αi + (1 − βi )ti − a2 (1 − βH )2 σH − 5 2 2 2 2 a −αH +(1−βH )tH − a 2 (1−βH ) σH −αL +(1−βL )tL − 2 (1−βL ) σL 2 The Principals’ problem when Markets are available will be:   
1 2
1 qH + qL 1 max F qH − + µM KT αH , βH , αL , β L , σM KT αH , βH , αL , β L αH ,βH ,αL ,βL 2 2 2 4   
1 2
1 qH + qL 1 F qL − + µM KT αL , βL , αH , β H , σM KT αL , βL , αH , β H 2 2 2 4 subject to IRH , ICH , IRL , ICL The optimal contracts in this setting will be M KT yL = 1.17 M KT yH = −.43 + (.54) x 1.2.3 Insurance Effect An important feature exhibited by this example is that optimal contracts are different when asset markets are present: they are more similar to wages. In this example the returns are independent, so that the market offers diversification opportunitie. In this case these are sufficient for principals to offer safer contracts to agents and achieve a higher expected utility. However, this is not always going to be the case. Consider an economy identical to the above, but in which agents reservation utility is equal do 21 . These are the resulting contracts. • The equilibrium contracts without markets: yh = −1.42 + (.7x) yl = .53 6 • The equilibrium contracts with markets: yh = −1.4 + (.65)x yl = .42 + (.06)x In this case type L gets a riskier contract when markets are present. Why is this the case? In the rest of the paper I will give sufficient conditions for the insurance effect to obtain. 1.3 The Model 1.3.1 Primitives There are 2N individuals. All individuals have identical preferences. They all have quadratic utility on outcomes, in the form u (x) = x − 2b x2 . Hence their preferences over random variables can be represented by the function U (X) = E [u (X)]
b E (X) − E X 2 2 b b 2 E (X) − E (X) − V ar (X) 2 2 N individuals are Principals, and N are Agents . There are N firms. Firm n is formed by Principal Pn and Agent An . Principal n owns a productive technology Xn , Agent n is a skilled worker, whose skill tn is drawn from from a distribution F with finite support {1, ..., T }. The profits of firm n are described by a random variable Xn,t . In other words profits depend on technology and skill. 1.3.2 Timeline Initially firms are identical in terms of internal organization. The Principal owns the firm. Her technology and the Agent’s work are necessary for creating profits. Their position differs because 7 while the technology is public, the Agent’s skill is his private information. 1. Each principal n will design a mechanism Mn = (An , Cn ), in which the agent can chooose an action an out of set An . Depending on the agent’s actions in the mechanism, he will be compensated with a contract C ∈ Cn . A contract comprises of shares of the company (1 − βn ) and a transfer −αn . A contract Cn is hence represented by a pair (αn , βn ) ∈ R × [0, 1] and it will be a function of the action taken by the agent Cn (an ). If he does not work he will get a reservation utility of u ≥ 0 and there will be no profits. From now on, I will slightly abuse notation and indentify an agent’s choice of action in a mechanism, with the resulting contract. 2. Once the shares/wage decisions have been taken, production starts, and agents’ skills will become public. 3. Finally principals can trade their claims to profits αn + βn Xn,tn on a market, in which a riskless asset L is also available in zero net supply. 4. After trading takes place, all uncertainty is realized. Compensation and securities pay off. When Who What Knowing What 0 Pn Mn CMN 1 An Cn ∈ An CMN, t (n) , Mn 2 Pn θn CMN, t Table 1.1: Timing 1.3.3 Strategies and Payoffs Let T be the set of possible realizations of workers’ skill values and let t = (t1 , ...tN ) be its generic element, distributed according to F = (F1 , ..., FN ) Let C be the vector of contracts across the economy. N N C = {Cn }n=1 = {αn , βn }n=1 8 Let θ = (θR |θL ) be the portfolio held by an agent. θR is a N -dimensional vector of positive holdings of the N risky assets, whereas θL is the position an investor holds in the riskless asset. The ex-ante utility from a portfolio θ fixing the matching t and the contracts CN , is given by Up3n (θ, C, t) = U θ · (~ µ (t, C) |1) , θR Ω (t, C) θR
Demand θ will depend on available assets (and hence con contracts C and their prices q (but prices are also a function of contracts). Agents payoffs depend on the action chosen and on the technology of their principal and on their type t: UA2 n = U (Cn , tn ) A Principal’s expected utility when mechanisms are M, the resulting contracts are C (M) and he holds portfolio θ is UP1n (M, C (·) , θ) = 1.4 Equilibrium 1.4.1 Description Z T UP3n (θ, C (M) , t) F (t) Because individuals take their decisions at each stage looking at the final payoffs, Equilibrium is more easily described starting from the final stage of the game. Asset Market At this stage the realized skills are observed. There is no asymmetric information. Principals hold one unit of a security equal to their share of returns in their firm, and a riskless asset is available in net zero supply. The solution concept used here is that of ArrowDebreu Equilibrium. Since securities payoffs are determined by contracts C and by agents’ skills t, the equilibrium portfolios and prices will be function (θ, q) (C, t), Contracting, Agents’ turn Each agent An observes the mechanism he is offered, Mn , and he knows his own type and the technology of the principal. This is all the payoff relevant information, so every agent is facing a choice between lotteries, and he is not interacting with any other individual, 9 except of course Pn . They pick an action maximizing UA2 n (·). Their strategies will be functions an (Mn , tn ). Contracting, Principals’ turn Each principal offers designs a mechanism, without knowing the skills of any agent. However they correctly forecast the strategies of each agent, and the outcome of asset markets, for any possible mechanism. In other words, they can forecast the equilibrium path for all profiles of mechanisms M and types t . Principals at this stage play a game against each other. A mixed strategy for Pn is a lottery on possible mechanisms, M̃n ∈ ∆ (Mn ). The flow of decisions is described schematically below, and the information available at each stage is summarized by the argument of the strategies. M̃ C → C̃(M, t)) → t → [(θ, q)] (C, t) t Based on this timeline we can write the utility in the first stage in this form:   h      i Vn M̃n |M̃n = EM̃n UP1n M̃n |M̃−n , C M̃n |M̃−n , t̃ , θ C M̃n |M̃−n , t̃ Note that t̃ is a random vector and UP1n defined above includes an expectation with respect to its distribution F . 1.4.2 Definition An Equilibrium consists of • A trading strategy θn∗ for each Principal n and prices q ∗ ∈ RN +1 such that [θ∗ , q ∗ ] (C, t) is an Arrow-Debreu Equilibrium for the symmetric information asset market taking place after contracting. Each principal is endowed with one unit of one asset so that the endowment of principal n is wn = [0, 0, ..., 1, ..., 0, 0] with 1 being in the nth position. 10 θn∗ (C∗ , t) ∈ arg max UP3n (θ, C∗ , t) +1 θ∈RN + s.t. q∗ (C∗ , t) · θ (C∗ , t) ≤ qn∗ (C∗ , t) X θn∗ = [1N |0] n∈N • For each agent An a strategy Cn∗ (Mn∗ , t (n)) such that Cn∗ (Mn∗ , t (n)) ∈ arg max UA2 n (C, tn ) C∈Cn • For each principal Pn ,a lottery M̃n∗ of deterministic mechanisms such that h i supp M̃n∗ ⊆ arg max Vn∗ (Mn |M∗−n ) Mn ∈M 1.4.3 Result To prove existence of equilibrium I need the following result, the mechanism design classic, “customized” for this setting in which principals compete with each other.. Let an instance of the previously described economy be G, and the set of its equilibria E(G). Consider now an economy GT which is identical in all respects, but where principals are restricted to offer to agents menus of contracts of size |T |, instead of designing general game forms. I am
going to show that E (G) = E GT . In this way the strategy space of the Principals at the first |T | stage will be finite dimensional.1 The space of Pn ’s strategies is going to be Mn . Proposition 1. The unrestricted menus economy G and the restricted menu economy GT , have
the same equilibria: E (G) = E GT   Proof. E(G) ⊆ E(GT ) Consider an equilibrium  = M̃, C, θ, q now construct a strategy profile T
and show it is an equilibrium for GT : Let a menu MnT of |T | contracts offered in  GT be made 1 In this model, T is finite. 11 of contracts {(yn )}t∈T : y (t) = Cn (Mn , t) For agents, it is an optimal action to choose the contract y (t). Suppose not. Than it would be the case that for some t0 U (y (t0 ) , t) > U (y (t) , t) = U (Cn (Mn , t) , t) 0 By construction Cn (Mn , t0 ) played in Mp yields the same payoff as y (t0 ), which contradicts Cn being an equilibrium strategy. The above described menus are optimal for every principal. Suppose not, then for some principal p there is a |T | sized menu M̃ 0 such that       Vn M 0 |M̃T−n > Vn M̃T = Vn∗ M̃ The equality follows by construction, and since n could have offered M̃ 0 in the unrestricted economy, this would contradict  being an equilibrium. E(GT ) ⊆ E(G) Consider an equilibrium eT : the agents equilibrium strategies CT are simply the equilibrium strategies of the real economy, restricted to the domain of |T | sized menus. Moreover, M̃T , are equilibrium strategies also in the unrestricted game. Suppose it was not the case, then for some n there is an unrestricted mechanisms lottery M̃ such that Vn∗ (M̃ |M̃T−n ) > Vn∗ (M̃T ) 0 Note that Vn (M̃ |M̃T−n ) can be attained by n by offering a lotteries of restricted menus M̃ T made of the following contracts y 0 : yn0 (t) = Cn (Mn , t) This implies ∗ T Vn∗ (M̃ 0 |M̃T −n ) > Vn (M̃ ) which contradicts M̃T being an equilibrium for GT 12 The utility function of a Principal is continuous in the contracts chosen by each type of Agent so it is continuous in C T . To make it continuous in the contracts offered by Principals’ it has to be that these contracts form Incentive Compatible menus. In other words, menus must come from as subset of C T such that C 1 is always chosen by agent 1, C 2 is always chosen by agent 2, and so on. To achieve this the menu in a contracts must satisfy incentive compatibility constraints. We can be sure that we can restrict attention to these menus because all and only the equilibria of the original unrestricted game are obtained in a game where Principals are allowed to offer only an incentive compatible menu of T contracts. I will call this economy, GIC . Corollary 2. The restricted menus economy G and the IC economy GT , have the same equilibria E(GT ) = E(GIC ) Proof. E(GT ) ⊆ E(GIC ) The contracts in the direct revelation menus in the proof of Theorem 1 are Incentive Compatible by construction. E(GIC ) ⊆ E(GT ) This part of the proof goes as the second part of the proof in Theorem 1: any outcome that a principal can achieve by menus of size T can be achieved by incentive compatibles menu of size T . 1.5 Existence of Equilibrium 1.5.1 Assumptions 1.5.1.1 Monotonic Preferences It is well known that Mean-Variance Preferences are not monotonic. This of course can cause problems for the existence of an equilibrium. In a standard CAPM setting, monotonicity of preferences is solved by imposing a bound on the variance aversion of every individual. I will require agents to not be satiated if they owned every return in the economy, regardless of state. I will then show that this implies they will not be satiated in the asset market of the later stage. It is possible to show that if preferences are monotonic for given returns (µ, Ω), they will be monotonic for the returns induced by any contracts in that economy, (α + βµ, βΩβ 0 ). Definition 1. Let X be a generic Random Variable on the state space S = (s1 , ..., sS ) taking values ∂U (x1 , ..., xS ). We say that U (X) is monotonic if ∂x > 0, ∀i. i 13 Lemma 3. Consider the preferences induced by the utility function U (X) = E(X) −  b 2 E (X) + V ar (X) 2 They are monotonic on a set of variables X defined on a finite state space S, if b < min 1 X,s X(s) Proof. The proof amounts to checking (by differentiating) under which conditions on a the utility function is increasing. Definition 2. Preferences are monotonic for the entire economy if b < P max1s Xn (s) n This definition amounts to stating that if an agent owned the entire returns available to the economy, he would still not be satiated. 2 Lemma 4. If preferences are monotonic for the entire economy with assets characterized by returns (µ, Ω), then they will be monotonic on all feasible portfolios for any contracts (αn , βn )n∈N . Proof. For preferences to be monotonic for all feasible portfolios it has to be that 1 θ n∈N n Xn (s) b < min P θn ,Xn ,s Where the min is taken across portfolios θ such that θn ∈ (0, 1) and outcomes Xn (s). So that the lemma can be restated as max s X θn (αn + βn Xn (s)) < n∈N X θn max Xn (s) s n∈N Note that the solution to the maximization on both sides is going to be reached at the aggregate market portfolio ( θn = 1 for all n) so that the previous is equivalent to Claim 1 max s X (αn + βn Xn (s)) < n∈N 2 The definition is equivalent to the less stringent b < |ρmn | 6= 1, ∀m, n X max Xn (s) n∈N maxs P1 n Xn (s) s as long as no firms are perfectly correlated 14 To prove this, note first how Claim 2 max αn + βn Xn (s) < max Xn (s), ∀n, αn , βn s s Suppose by means of contradiction ∃n, αn , βn s.t. max −αn + (1 − βn ) Xn (s) < 0 s Which in turn implies 0 > −αn + (1 − βn ) E (Xn ) = E (−αn + (1 − βn ) Xn ) > EU (−αn + (1 − βn ) Xn ) However EU (−αn + (1 − βn )) has to be greater than zero in equilibrium, so that 0 > EU (−αn + (1 − βn ) Xn ) > 0 which is impossible. This proves Claim 2 It follows from Claim 2 that X αn + βn max Xn (s) < s n X n max Xn (s) s Finally X n∈N max Xn (s) > s X n∈N  X αn + βn max Xn (s) > max (αn + βn Xn (s)) s s n∈N 15 1.5.2 The existence result Theorem 5. If preferences are monotonic for the entire economy then there exists an equilibrium in the CAPM contracting economy. Proof. I am going to use a well known fixed point result by Glicksberg (1952) to show that there is an equilibrium in the first stage of the game, given that the asset market develops as predicted by the CAPM model. I need to show that 1. The strategy space ∆(MM V ) is a convex, compact subset of a locally convex Hausdorff space. 2. The best response correspondence of all principals is upper hemi-continuous, convex valued, and nonempty. For the first part note that the space of Incentive Compatible menus MM V is a subset of a Euclidean space. It is closed because it is defined by a finite number of weak inequalities, and it is bounded because the larger set of feasible contracts are bounded. Hence it is compact. The space of lotteries (identified with Borel probability measures) over these Menus is of course convex. It is also compact with respect to the weak* topology. This space of probabilities is a subset of the space of continuous functions C(MM V ), which is locally convex (and Hausdorff) with respect to the weak* topology.3 For the second part, convexity of the best response correspondence follows from preferences on random variables being represented by expected utility. I will use Berge’s Maximum theorem to show that it is non empty, compact-valued and upper hemi-continous. To apply the maximum theorem to individuals’ best response, it has to be that constraints vary continuously with other principals’ strategies, and that the payoff function is continuous in one’s own actions. First note how the constraints correspondence is constant with respect to other principals strategies, and is therefore continuous. Also note how the constraints correspondence maps to the space of Borel probability measures on menus, which is a Hausdorff space as noted above. We also need to make sure that the payoff function of a principal is continuous in menus. To do this we need to show that 3 For a treatment of these and other results on weak topologies, and also to see the theorems of Berge and Glicksberg, see Aliprantis, Border (2005) 16 1. Payoffs at the market stage are a continuous function of the contracts chosen by agents. 2. The contracts chosen by agents are a continuous function of the menus offered. Claim 1 By Lemma 4, if the preferences are monotonic for (µ, Ω), they are going to be monotonic for the asset markets resulting from all possible contracts C. Under the present assumptions a CAPM equilibrium exists once contracts are chosen.4 Because in such equilibrium the price of
a security can be expressed as qn = αn + βn µn − Nb βn2 σn2 + Σm6=n ρmn βm βn σm σn The indirect utility from a contract profile in the CAPM function is continuous in contracts. Claim 2 Recall that we can restrict attention to Incentive Compatible menus. If a principal makes a small change to the menu he offers while remaining in this set, every type of agent t , will still find it optimal to pick the contract intended for him, Ct . Hence any small change, will correspond to a small change in the contract picked by each type of agent. We can conclude that the indirect utility for a principal facing type t is a continuous function of the menus offered. Taking expectation with respect to F over these indirect utilities yields a continuous functional on the domain of lotteries on IC menus. By the maximum theorem the best response correspondence of each player is now UHC and compact valued, which implies that the game best response is as well. By Glicksberg’s theorem there is a fix point, which is an equilibrium by construction. 1.6 The Insurance Effect of Markets Throughout this section I will consider the case of firms whose returns are independent. 1.6.1 Utility from Markets The point of this section is incorporating the outcome of markets in the principals’ utility functions. First consider again what the final holdings are in equilibrium. Every individual will hold the same risky portfolio, an equal fraction N1 of the aggregate endowment, and will spend the rest on the 4 In the literature briefly reviewed by Nielsen (1990), one can find many sufficient conditions for the existence of CAPM equilibrium, most of them deal with the possibility of satiation of preferences. Things are particularly simple when returns are bounded (which includes this model): monotonicity and local non satiation are guaranteed by a low enough risk aversion 17 riskless asset (or short it if their remaining endowment is negative). With this in mind the mean and variance of the portfolio held by the agent is readily computed as a function of contracts. For a general principal i we have that • The holding of riskless asset is qi − N1 • The mean of the risky portfolio is N1 P j∈P qj P j∈N (αj + βj µj ) • The variance of the risky portfolio is N12 2 2 j∈N βj σj P Since qi = αi + βi µi − Nb βi2 σi2 , we have that the mean of the portfolio simplifies to αi + βi µi − b X 2 2 b 2 2 βi σ i + 2 βj σ j N N j∈N P and the variance is of course the variance of the risky part N12 j∈N βj2 σj2
So that if U (αi , βi ) = F αi + βi µi , βi2 σi2 is the utility a principal obtains from contract α, β when no markets are available, markets will change this into U M (αi , βi ) =
F M αi + βi µi , βi2 σi2 =   X X b 1 b βj2 σj2 , 2 βj2 σj2  F αi + βi µi − βi2 σi2 + 2 N N N j∈N j∈N Lemma 6. If the principals are allowed to trade their claims, they will act as if their utility functions were Et U M (αi , βi ) =
2 Et F M αi + βi µi,ti , βi2 σi,t = i   b b X 2 2 1 X 2 2  2 Et F αi + βi µi,ti − βi2 σi,t + 2 βj σj,tj , 2 βj σj,tj i N N N j∈N The partial derivatives of these functions are given by j∈N 18   P 2 X σ
∂F b ∂F M b j j∈N µi − σi2 +  µi , σi2 = σj2 , ∂µi ∂µ N N2 N2 j∈N   P   2 X σ
∂F M b b b ∂F b j j∈N 2 2 2   µi , σi = µi − σi + 2 − σj , + ∂σi2 ∂µ N N N2 N2 N j∈N   P 2 b 2 ∂F  b X 2 j∈N σj  1 µi − σi + 2 σj , ∂σ 2 N N N2 N2 j∈N and so that   P 2 X σ b b j∈N j  UαMi (αi , βi ) = Fµ µi − σi2 + 2 σj2 , N N N2 j∈N   P 2 X σ b b j j∈N + σj2 , UβMi (αi , βi ) = µi Fµ µi − σi2 + 2 N N N2 j∈N    P   2 X σ b b b b j j∈N 2 2 2   2βi σi µi Fµ µi − σi + 2 σj , + − N N N2 N2 N j∈N    P 2 X σ b 1 b j j∈N   σj2 , Fσ2 µi − σi2 + 2 N N N2 N2 j∈N whereas the partial derivatives without markets are Uαi (αi , βi ) =Fµ αi + βi µi , βi2 σi2


Uβi (αi , βi ) =µi Fµ αi + βi µi , βi2 σi2 + 2βi σi2 Fσ2 αi + βi µi , βi2 σi2 19 1.6.2 First Best Theorem 7. When information is complete and symmetric, the variance of optimal contracts is smaller in a large market than without markets. There is a number of principals N such that βt∗ < βt∗ (N ) , ∀N ≥ N Proof. Consider the problem of a principal. max (αt ,βt )T t=1 T X U (αt , βt |t) t=1 s.t. U (−αt , 1 − βt |t) ≥ u, ∀t = 1, ...T Which can be broken into T separate problems max U (αt , βt |t) (αt ,βt ) s.t. U (−αt , 1 − βt |t) ≥ u Note that
U (αt , βt |t) = F αt + βt µt , βt2 σt2   2 U (−αt , 1 − βt |t) = F −αt + (1 − βt ) µt , (1 − βt ) σt2 Dropping the type subscripts for convenience, we can proceed to solve an individual problem max F α + βµ, β 2 σ 2
(α,β)   2 s.t. F −α + (1 − β) µ, (1 − β) σ 2 ≥ u 20 The first order conditions for this problem amount to  
∂L 2 = Fµ α + βµ, β 2 σ 2 − λFµ −α + (1 − β) µ, (1 − β) σ 2 = 0 ∂α

∂L = µFµ α + βµ, β 2 σ 2 + 2βσ 2 Fσ2 α + βµ, β 2 σ 2 + ∂β i   h  2 2 2 −λ µFµ −α + (1 − β) µ, (1 − β) σ + 2 (1 − β) σ 2 Fσ2 −α + (1 − β) µ, (1 − β) σ 2 = 0 Solving for the optimal contract we have that βM ∗ = NUM DEN where  
2 N U M = Fσ2 −α + (1 − β) µ, (1 − β) σ 2 Fµ α + βµ, β 2 σ 2  
2 DEN = Fσ2 −α + (1 − β) µ, (1 − β) σ 2 Fµ α + βµ, β 2 σ 2 +  
2 Fσ2 α + βµ, β 2 σ 2 Fµ −α + (1 − β) µ, (1 − β) σ 2 Similarly we can “break” the problem with markets in smaller optimizations like the following. max F M α + βµ, β 2 σ 2
(α,β)   2 s.t. F −α + (1 − β) µ, (1 − β) σ 2 ≥ u The first order conditions for this problem amount to  
∂LM 2 = FµM α + βµ, β 2 σ 2 − λFµM −α + (1 − β) µ, (1 − β) σ 2 = 0 ∂α

∂LM µFµM α + βµ, β 2 σ 2 + 2βσ 2 FσM2 α + βµ, β 2 σ 2 + ∂β h    i 2 2 −λ µFµM −α + (1 − β) µ, (1 − β) σ 2 + 2 (1 − β) σ 2 FσM2 −α + (1 − β) µ, (1 − β) σ 2 = 0 Solving for the optimal contract we have that β∗ = N U MM DENM 21 where  
2 N U MM = Fσ2 −α + (1 − β) µ, (1 − β) σ 2 FµM α + βµ, β 2 σ 2  
2 DENM = Fσ2 −α + (1 − β) µ, (1 − β) σ 2 FµM α + βµ, β 2 σ 2 +  
2 FσM2 α + βµ, β 2 σ 2 Fµ −α + (1 − β) µ, (1 − β) σ 2 By the considerations from Section 1.6.1, we note that, since FσM2 gets arbitrarily small as N gets large ( Nb − Nb2 and N12 are arbitrarily small for large enough N , so is FσM2 ) . β ∗ is arbitrarily close to 1 in a large economy. Which concludes the proof. 1.6.3 Ordering Types To solve a risk sharing principal agent problem with multiple types and asymmetric information, I will need to adapt methods use in the contract theory literature.5 I will require that, for any firm, the type space is ’ordered’ in the sense of Single Crossing Property (ie: any two indifference curves from any two types will cross only once). The following cumbersome notation means exactly this. Given a firm, agents can be ranked (by the slopes of their indifference curves). The ranking need not be the same in every firm. Let SCP (n) = (1(n), 2(n), ..., T (n)) be a function from N to T N such that i (n) 6= j (n) , ∀i (n) , j (n) ∈ 1, ...T, ∀n . Definition 3. An economy satisfies Single Crossing Property if and only if there is a function SCP,


as defined above such that types µ1(n) ), σ1(n) , ..., µT (n) , σT (n) satisfy Single Crossing Property for all n . That is, if U2 (−α, 1 − β|t(n)) U2 (−α, 1 − β|t(n) + 1) > U1 (−α, 1 − β|t(n)) U1 (−α, 1 − β|t(n) + 1) ∀t, p, α, β In the case of quadratic utility, SCP amounts to 5 Similar techniques are used in the literature on second order price discrimination. See for example Maskin and Riley, 1984 and Stiglitz, 1980 22 µt − 2 bt+1 σt+1 (1 − β) bt σt2 (1 − β) > µt+1 − 1 − bt (−α) − bt µt (1 − β) 1 − bt+1 (−α) − bt+1 µt+1 (1 − β) Proposition 8. The following cases6 imply SCP : • Agents have the same mean and different variance • Agents have different mean and the same variance • Agents generate the same outcomes, but the probabilities of success are different. The proof amounts to the algebra necessary to verify the definition. The following two facts will come handy for the proving Theorem 11 Fact 9. ∀t, (α, β) , U (−α, 1 − β|t) > U (−α, 1 − β|t + 1) Fact 10. ∀t < s, ∀ (α, β) , (α0 , β 0 ) : β ≤ β 0 U (−α, (1 − β)| t) − U (−α0 , (1 − β 0 )| t) > U (−α, (1 − β)| s) − U (−α0 , (1 − β 0 )| s) 1.6.4 Second Best Theorem 11. If types satisfy Single Crossing Property and markets are large enough, the variance of the average contract is smaller when markets are present. Proof. The strategy of the proof is to show that -at an optimum- all types shares’ ( (1 − β) ) are bounded by the share of the “best” type, t = 1. And that the share of this type become small for large enough market. A generic principal’s problem is given by max (αt ,βt )T t=1 T X U (αt , βt |t) f (t) t=1 s.t. U (−αt , 1 − βt |t) ≥ u, ∀t ∈ {1, ...T } U (−αt , 1 − βt |t) ≥ U (−αt0 , 1 − βt0 |t) , ∀t, t0 ∈ {1, ..., T } (IR) (IC t t’) 6 It is to be noted that, although it is not formalized in the general model, SCP holds also if agents are distinguished by their risk aversion, and the following results would still apply 23 SCP and IC constraints imply by usual arguments that yield that β ∗ (s) ≥ β ∗ (t) , ∀s < t ∈ {1, ...T }. This imply that β1 ≤ βt , ∀t Now we have to solve for the contract of type 1. To do this I show that the IC12 constraint will always be binding, and that this is enough to attain the result. The first thing to do is to reduce the set of relevant constraints. Fact 9 implies that U (−αt , 1 − βt |t) ≥ U (−αt , 1 − βt |T ) This together with IR holding for type T and IC holding for type t with respect to the contracts of type T, implies that IR holds for type t. In other words, if U (−αT , 1 − βT |T ) ≥ u U (−αt , 1 − βt |t) ≥ U (−αT , 1 − βT |t) we will also have that U (−αt , 1 − βt |t) ≥ u We can hence solve the problem without worrying about any of the IR constraints except that of type T . We can also infer that ICt−1,t will be binding at an optimum for any t. Suppose that it were not binding, U (−αt−1 , 1 − βt−1 |t − 1) > U (−αt , 1 − βt |t − 1) Since we are at an optimum it has to be that the IC constraints are satisfied 24 U (−αt , 1 − βt |t) ≥ U (−αk , 1 − βk |t) , ∀k By Fact 10 it has to be that U (−αt , 1 − βt |s) > U (−αk , 1 − βk |s) , ∀k ≥ t, ∀s < t Consider an alternative incentive scheme, which gives less transfer −αs to all types s lower t. Because their IC constraints for contracts Ck , with k > t ( ICs,k ) are not binding, we are increasing the maximand while remaining in the admissible set of contracts, which contradicts the original scheme being an optimum. This and SCP imply that constraints ICt,t−1 will not be binding. It also implies that no other IC constraint will bind at the optimum. Fact 9, ICt−1,t , and ICt,t+1 imply that ICt−1,t+1 is satisfied with a strict inequality. This means that the only relevant constraints for determining the optimal β1 are in the form U (−α1 , 1 − β1 |1) = U (−α2 , 1 − β2 |1) I now have to solve a much simpler problem max (αt ,βt )T t=1 T X U (αt , βt |t) f (t) t=1 s.t. U (−αT , 1 − βT |t) = u U (−αt , 1 − βt |t) = U (−αt+1 , 1 − βt+1 |t) , ∀t ∈ {1, ..., T − 1} The only first order conditions involving α1 and β1 are given by (IR) (IC t t +1) 25  
2 q1 Fµ α + βµ, β 2 σ 2 − λFµ −α + (1 − β) µ, (1 − β) σ 2 = 0


q1 µFµ α + βµ, β 2 σ 2 + 2βσ 2 Fσ2 α + βµ, β 2 σ 2 + i   h  2 2 −λ µFµ −α + (1 − β) µ, (1 − β) σ 2 + 2 (1 − β) σ 2 Fσ2 −α + (1 − β) µ, (1 − β) σ 2 = 0 Which yield an expression for 1 identical to the first best case. β1∗ = N U MM DENM  
2 N U MM = Fσ2 −α + (1 − β) µ, (1 − β) σ 2 FµM α + βµ, β 2 σ 2  
2 DENM = Fσ2 −α + (1 − β) µ, (1 − β) σ 2 FµM α + βµ, β 2 σ 2 +  
2 FσM2 α + βµ, β 2 σ 2 Fµ −α + (1 − β) µ, (1 − β) σ 2 Solving for β1 we obtain a formula similar to the first best case. Just as in the first best case we can observe that β ∗ is arbitrarily close to 1 in a large economy, since FσM2 can again be made arbitrarily small by choosing an appropriately large N . This bound allows us to reach the result P P t t Claim: t,n f (t) βn ≥ t,n f (t) βn (1) Let t be the bijection satisfying SCP for all matches. The claim is equivalent to XX n t f (t) βnt ≥ X βn1(n) (n) n Note that ∀n, t, ∃N s.t. βn1(n) (N ) , ∀N ≥ N 26 and naturally ∀n, ∃N s.t. βn1(n) (N ) , ∀N ≥ N , ∀tn This implies βn1(n) > X f (tn ) βnt , ∀N ≥ N t and X n βn1(n) > XX n f (t) βnt , ∀N ≥ N t which concludes the proof. 1.6.5 Effects on Welfare and on Asset Markets As it is usually the case, asymmetric information entails a loss of efficiency. Corollary 12. Under the assumptions of Theorem 11, the equilibrium is inefficient compared to the complete information case. Proof. It follows from the fact that IRT and ICT −1,T are both binding that at least the contract of type T is Pareto dominated by the complete information solution of Theorem 7. Let ∆t (N ) = β t∗ (N ) − β t1 (N ), where β t1 (N ) is the first best solution. I will use this as a measure of inefficiency. Corollary 13. For any principal-agent pair, there is a number N , such that any market with N ≥ N traders induces more efficient contracts, ∆t (N ) < ∆t (1) 27 Proof. The proof again leverages on the fact that both first and second best contracts are arbitrarily close to 1 for a large enough markets. The claim is trivially satisfied for β 1∗ (N ) (the second best solution, with markets), since it is a Pareto Efficient Contract by construction. For β t∗ (N ) , t > 1 consider ∆t (N ) = β t∗ (N ) − β t1 (N ), where β t1 (N ) is the first best solution with markets. Since β t∗ (N ) and β t1 (N ) both go to one as N gets large, there must be a N such that ∆t (N ) < ∆t (1) for all N > N . 1.7 Conclusion This chapter integrates a model of principal-agent interaction with asset markets. Principals and Agents are randomly matched. Each pair produces random returns, whose distribution is known only to the agent at the contracting stage. Every Principal offers a menu of contracts to the Agent he is matched with, and the Agents make their pick. What marks the difference from the standard contracting model is that Principals have access to an asset market on which they trade their shares of returns. I present a general framework and define a notion of equilibrium. I prove a revelation principle result, to simplify the study of firms’ structure and prove existence of equilibrium. Under standard assumptions of contract theory, I study the interactions of financial markets on contracts. The existence of markets, induces less risky compensation for agents. In a large market diversification opportunities multiply, and contracts become less and less risky. Contracting inside firms induces inefficient aggregate risk in an economy, however the size of this inefficiency is reduced by a large enough market. As noted this is a limiting result and it leaves the open question of the behavior in small markets. Two extensions of this model seem particularly interesting. The first one is studying the effect of correlation of firms returns (systemic risk) on the screening problem and the loss of welfare. The second one is studying the effect of financial markets on the labour market, in particular on the matching of workers’ with firms. 28 Chapter 2 Contracts and Aftermarkets Hidden Action 2.1 Introduction It is well known that financial markets can favor the efficient allocation of resources to production and the sharing of risk. Less is known about their effects on incentives, for those who have limited access to markets. In this chapter I propose a stylized model of firms and financial markets, to capture these effects. The concern here is how different access to markets can affect the incentives to production. This chapter relates to two branches of economic theory. The literature on Walrasian economies in presence of Moral Hazard issues, and the literature on endogenous securities. I will discuss the fit of my work in these literatures, but first I am going to relate my work to two recent papers, which are particularly relevant to the discussion. Parlour and Walden (2008) construct a model based on CAPM where workers are also the dispersed owners of the firms in the economy. Their assumptions and their main object of interest are opposite of mine. By assuming workers are the only traders, and cleverly constructing a simple shock that cannot be traded away on markets, they mostly analyze the effects of moral hazard on financial markets. I am on the other hand more interested in the effect of financial markets on contracts. For this reason I cannot take firms as abstract risk neutral principals at the contracting stage. In fact Risk Neutrality corresponds only to a special case (firms with independent returns). Magill and Quinzii (2005) ask a different although related question. What set of securities is 29 needed, for risk sharing and incentives to coexist? They study several cases, distinguished by the amount of information available for contracting, and for each case characterize the set of securities which allow (constrained) efficiency. In this chapter securities are determined in equilibrium, but they are chosen out of a very simple set. In the future it would be worth describing the endogenous choice of complex securities, to study the efficiency of financial engineering, in terms of incentives and risk sharing, but this is out of the scope of this chapter. There is by now a vast literature on moral hazard in general equilibrium settings. Helpman and Laffont (1975) and Prescott and Townsend (1984) are among the first to tackle the topic. Their works are concerned with efficiency in exchange and production economies, in which individuals can exert a costly, unobservable payoff relevant action. These papers are the first in a long, but not large, series of works, which extend the study of efficiency to more general economies. My work is different in that I include financial markets, and I trade off some generality for more precise comparative statics results. A strand of the asset pricing literature looks at asset pricing in the presence of delegated portfolio management (for a survey, see Stracca, 2003). An example of the approach typical in these papers can be seen in Ou Yang’s paper. These studies look at the effects on prices and returns of the classical informational asymmetries phenomena. Moral hazard and adverse selection are largely studied in a CAPM or APT setting, in which a representative principal delegates his investing decisions to an agent. In this literature inefficiencies take the form of deviations from the non-delegated case equilibrium. These deviations can take the form of changes in asset prices and optimal portfolio composition. Besides the different object of interest, the perspective in these works is in a sense opposite of the one taken here. There we have informed parties trading, whereas in the present work it is the uninformed parties accessing markets. In Section 2 I present the model. In Section 3 I define the notion of equilibrium. In section 4 I show existence and uniqueness. In section 5 I analyze the effect of markets on production. Section 6 concludes. The appendix discusses issues concerning Markowitz preferences, when used in a dynamic contest. 30 2.2 The Model This stylized model is meant to capture certain features of interactions within firms and across firms in financial markets. It is helpful to introduce first the model of a single firm, and then proceed to define how they interact. 2.2.1 Primitives A generic firm i is constituted by a principal-agent pair pi , ai . Each of these firms i generates random returns. All individuals have identical Mean Variance Preferences over random variables in the familiar form 1 b U (X) = µ (X) − σ 2 (X) 2 Agents have a reservation utility, u. Agents choose a costly action (which can be interpreted as the effort put into production) ei from the interval E = [e, e]. The cost of effort is a continuous, strictly increasing real valued function c (e). The returns of a firm X̃i (ei ) depend stochastically on the effort level of the agent employed. Because of the assumptions on preferences, I can restrict my attention to the mean and the variance of the firm’s returns, and express them as a function of effort levels. µi (ei ) σ 2 (ei ) A contract Ci is a contingent agreement on how to split returns between the Principal and the Agent forming firm i. I impose the restriction that these sharing rules be affine. If X is the random variable describing the profits of the firm, an admissible rule describing the principal’s and agent’s 1 Note that all the analysis in this paper would apply to the CARA/normal framework, except that conditions for existence would be less demanding. 31 share must be of the form: Xai = −α + (1 − β)X Xpi = α + βX with α ∈ R and β ∈ [0, 1]. A contract Ci amounts to a pair αi , βi . In a binary setting this imposes the restriction that both shares be increasing in firm’s returns. Now consider that there are N firms. Let e = [ei (Ci )]1≤i≤N be the vector of efforts, and C = [Ci ]1≤i≤N be the vector of contracts in each firm . The joint distribution of the vector of firms’ profits X (e) = [Xi (ei )]1≤i≤N firms’ profits will be uniquely determined by the effort levels across firms.    µ1 (e1 )        .. µ (e) =   .       µN (eN ) Ωjk (e) = ρjk (ej , ek ) σj (ej ) σk (ek ) I will denote this vector and matrix as µ (e) and Ω (e). Principals will have access to a financial market where they can trade their claims to profits and a riskless asset available in zero net supply. The risky assets available can be described by a random vector, Xp (e, C) = [Xpi (ei , Ci )]1≤i≤N = [αi + βXi (ei )]1≤i≤N Because of our assumption on preferences, from now on I will simply identify the principals’ shares with the vector of means µ (C, e) = α + βµ (e) and the variance-covariance matrix Ω (C, e) = β 0 Ω (e) β . 2.2.2 Timeline I will consider here the case of “hidden action”. In other words, the two parties cannot write contracts on the actual effort level. The conditions for existence trivially imply the conditions for 32 existence when effort is observable. CMN will denote the primitives of the game that are common knowledge at every stage, which are: • The vector of means and the variance-covariance matrix of payoffs as a function of effort levels µ (·) Ω (··) • The preferences of individuals (the variance aversion parameter b). • The reservation utility of agents u. The economy reaches its equilibrium in 3 stages. 1. Each Principal pi designs a contract Ci = (αi , βi ). Ci is the set of all possible contracts for firm i. At this stage every principal knows CMN. The profile of offered contracts is C = {C1 , ..., CN } ∈ C = ΠN i=1 Ci 2. Each agent ai chooses his effort level, based on the contract Ci . A strategy of an agent in firm i is a function of contracts mapping to possible effort levels. ei : Ci → E ei : Ci 7→ ei (Ci ) The strategy profile of all agents can be written as a vector of functions e = [ei (Ci )]1≤i≤N of contracts offered. 3. Before uncertainty is realized, principals trade their claims to returns on an asset market, where a riskless asset L is available in zero supply. At this stage principals can observe all contracts and make conjectures on the level of effort and hence the distribution of returns of all firms. Every principal j will form some beliefs γij on the effort level ei . γij (ei |Ci ) is the 33 cumulative distribution function induced by the beliefs on firm i under contract Ci . Principal j holds θji shares of firm i and the price of the stock of firm i is qi This table summarizes the choices each individual faces at a given time, and the information available to them. When Who What Knowing What 0 pi Ci CMN 1 ai ei CMN, Ci 2 pi θi CMN, C Table 2.1: Timing 2.2.3 Payoffs Let θ be the portfolio held by an agent. Let θ = (θR |θL ) Where θR is an N -dimensional vector of positive holdings of the N risky assets, whereas θL is the position an investor holds in the riskless asset. The ex-ante utility from a portfolio θ fixing the contracts C and effort choices e, is given by b 0 θ · (µ (C, e) |1) − θR Ω (C, e) θR 2 However principals don’t observe the efforts e so they evaluate utility of portfolios based on contracts C and the beliefs they induce γ (·|C) Z θ· µ (C, e|1) dγ (e|C) − E b 2 Z Upi (C, θ, γ) =  2 ! Z 0 θR Ω (C, e) θR + µ (C, e) − µ (C, e) dγ (e|C) E E Note how the squared term is always positive. This will pose a problem, but in equilibrium beliefs will be degenerate and the term will be equal to zero. 34 Demand θ will depend on available assets and their prices (but prices are also a function of contracts). Agents payoffs depend on the effort chosen and the contract in place. b Uai (Ci , ei ) = µi (Ci , ei ) − σi2 (Ci , ei ) − c (ei ) 2 2.3 Equilibrium 2.3.1 Description Because individuals take their decisions at each stage looking at the final payoffs, Equilibrium is more easily described starting from the final stage of the game. Asset Market Principals hold one unit of a security equal to their share of returns in their firm, and they all have the same information. Because all individuals have the same beliefs the solution concept used here is that of Arrow-Debreu Equilibrium. The equilibrium portfolios and prices will be based on expected payoffs induced by C. They will be a function (θ, q) (C, γ (e|C)). Contracting, the Agents’ turn Each agent ai observes the contract he is offered, Ci , and he knows his own type and the technology of the principal. This is all the payoff relevant information, so every agent is facing a choice between lotteries, and he is not playing against other players. They simply choose an effort level maximizing Uai (·). As noted their strategies will be functions ei (Ci ). Contracting, the Principals’ turn Each principal designs a contract. They correctly conjecture the action of each agent, and the outcome of asset markets, given contracts. They can forecast the equilibrium path for all possible strategy profiles. Hence, this stage can be seen as a game principals play against each other. I will focus on equilibria in pure strategy equilibria (and show their existence). The flow of decisions is described schematically below, and the information available at each stage is summarized by the argument of the strategies. C → e (C) → θ (C, q (C) , γ (e|C)) The utility in the first stage can be written as: 35 Vpi (C) = Upi (C, θ (γ (e|C)) , γ (e|C)) 2.3.2 Definition An Equilibrium consists of • A trading strategy θi∗ for each Principal pi and prices q ∗ ∈ RN such that [θ∗ , q ∗ ] (C, γ) is an Arrow-Debreu Equilibrium for the asset market when contracts are C . Each principal is endowed with one unit of one asset so that the endowment of principal pi is wi = [0, 0, ..., 1, ..., 0, 0] with 1 being in the i-th position. θi∗ (C, γ (e|C)) ∈ arg max Upi (C, θi , γ (e|C)) +1 θi ∈RN + such that q ∗ (C, γ (e|C)) · θi (C, γ (e|C)) ≤ q ∗ (C, γ (e|C)) · wi X θi∗ = [1N |0] i∈N • Beliefs γ ∗ (e|C) such that supp (γi∗ ) ⊆ arg max Uai (Ci , ẽi ) ẽi ∈E • For each agent ai a strategy e∗i (Ci ) such that ei (Ci ) ∈ arg max Uai (Ci , ei ) ei ∈E • For each principal pi , a contract Ci∗ such that ∗ Ci∗ ∈ arg max Vi∗ Ci , C−i
Ci ∈Ci = U pi
∗
∗ ∗ ∗ Ci , C−i , γ e|Ci , C−i ,θ
∗


∗ ∗ Ci , C−i , γ e|Ci , C−i 36 2.4 Existence of Equilibrium 2.4.1 Assumptions 2.4.1.1 Monotonic Preferences It is well known that Mean-Variance Preferences need not be monotonic. This could pose problems for the existence of equilibrium. In a standard CAPM setting, monotonicity of preferences is solved by imposing a bound on the variance aversion of every individual. Because I restrict attention to linear contracts, it is possible to show that, if preferences are monotonic for given returns, they will be monotonic for any prevailing contracts. Definition 4. Let X be a generic Random Variable on the state space S = (s1 , ..., sM ) taking values ∂U (x1 , ..., xM ). U (X) is monotonic if ∂x > 0, ∀i. i Lemma 14. Consider the preferences induced by the utility function b U (X) = E(X) − V ar(X) 2 They are monotonic on a set of variables X defined on a finite state space S, if b < min 1 X,s |xs − µX | Proof. The proof amounts to checking (by differentiating) under which conditions on b the utility function is increasing. Lemma 15. If preferences are monotonic for all feasible portfolios in an economy with assets characterized by returns (µ, Ω), then they will be monotonic on all feasible portfolios for any contracts (αi , βi )i∈N . Proof. For preferences to be monotonic for all feasible portfolios it has to be that 1 P θ x i∈N i i − i∈N θi µi | 1 P . = | max i∈N θi (xi − µi ) | b< | max P 37 Where the max is taken across portfolios θ such that θi ∈ (0, 1) and outcomes xi ∈ supp (Xi ). Note that X θi [(−αi + (1 − βi )] xi − [−αi + (1 − βi ) µi )] i∈N = X θi ((1 − βi ) xi − (1 − βi ) µi ) i∈N = X θi (1 − βi ) (xi − µi ) i∈N I claim that max | X θi (1 − βi ) (xi − µi ) | ≤ max | i∈N X θi (xi − µi ) | i∈N Note that the solution to the maximization on both sides is going to be reached at the aggregate market portfolio so that the previous is equivalent to max | X (1 − βi ) (xi − µi ) | ≤ max | i∈N X (xi − µi ) | i∈N Since it will also be the case that at the maximum all the xi ’s chosen will be greater (or smaller) than the µi ’s so that max X (1 − βi ) | (xi − µi ) | ≤ max i∈N X | (xi − µi ) | i∈N Observing that βi ∈ [0, 1] concludes the proof Assumption 1. For a given set of assets, for every individual i, their risk tolerance parameter bi
1 lies on the interval 0, b , where b = minX,s |xs −µ X| This ensures that everyone’s preferences will be monotonic. 2.4.1.2 Cost and Productivity of Effort The purpose of the following assumptions is making sure that the only randomness in the economy is due to the uncertainty of firms’ returns. Markowitz preferences are dynamically inconsistent and 38 introducing further randomizations (such as some individual playing a mixed strategy) is undesirable in many ways. 2 Moreover, if only one effort choice is optimal for an agent, then principals can correctly infer the equilibrium effort by observing contracts. To achieve this I need each agent’s objective function to be concave in effort for any possible contract (α, β). Assumption 2. 1. The cost function of an agent c(e) is strictly increasing and strictly convex. ∂c ∂2c > 0, 2 > 0 ∂e ∂e 2. The effect of effort on the mean distribution of returns is such that utility is strictly increasing
and concave in effort, the effect on the variance is strictly decreasing and convex for all 0, b ∂µ ∂2µ > 0, 2 ≤ 0 ∂e ∂e ∂σ 2 ∂ 2 σ2 < 0, >0 ∂e ∂e2 3. Moreover, I require the effect of effort on the variance to be bounded relative to the effect on mean returns. µe > |σe2 | 2 |µee | > |σee | Note how the first set of conditions implies that 2 ∂µX(e) b ∂σX(e) − >0 ∂e 2 ∂e 2 2 ∂ 2 µX(e) b ∂ σX(e) − <0 ∂e2 2 ∂e2 Lemma 16. Under Assumptions 1 and 2, the solution to the agent’s problem, e∗ (β) is • unique • continuous 2 See Appendix 39 • decreasing • concave Proof. Note how e∗ (α, β) solves max −α + (1 − β) µ(e) − e b 2 (1 − β) σ 2 (e) − c(e) 2 The First Order Conditions of these problems amount to (1 − β) µe (e) − b 2 (1 − β) σe2 (e) − ce (e) = 0 2 Uniqueness. It follows from Assumption 2 that this derivative is a strictly decreasing function on E = [e, e]. To see this note that by, Assumption 2, concavity is guaranteed for any risk tolerance
parameter in the interval 0, b . This implies that the assumptions for derivatives will also be true for µe − b (1 − β) σe2 2 for β between 0 and 1. This and the fact that cost is convex imply uniquness of the solution. Continuity. for β in [0, 1) follows from the implicit function theorem. When β = 1 the optimal effort is e . Monotonicity and Concavity. Applying the implicit function theorem to the FOCs we obtain that ∂e∗ µe − b (1 − β) σe2 = <0 2 2 ∂β (1 − β) µee − b (1 − β) σee − cee 2 ∗ ∂ e = ∂β 2 2 2 2 2 b2 2 2 2 (1 − β) σe σee + µee µe − b (1 − β) σee µe − bcee σe <0  2 2 2 b (1 − β) µee − 2 (1 − β) σee − cee 40 Corollary 17. In equilibrium, principals correctly conjecture the equilibrium effort of agents. γi∗ (ei |Ci ) = 1, ei ≥ e∗i (Ci ) = 0, ei < e∗i (Ci ) This follows immediately from the definition of equilibrium and Lemma 16. By doing this, I am removing one potential layer of randomization. The objective function of a principal at the first stage will be. b 0 Upi (C, θ, γ ∗ ) = θ · (µ (C, e∗ (C)) |1) − θR Ω (C, e∗ (C)) θR 2 Lemma 18. The best response of a principal is single valued. Proof. The proof amounts to showing that every principal is maximizing a strictly concave function on a convex set. Once the reaction function of the agent in incorporated in his individual rationality constraint, the resulting set need not be convex. However, by substituting the IR constraint in the objective function, I define a simple maximization problem on β in [0, 1] . By Assumption 2.3, the resulting maximand is a strictly concave function. Let’s start by studying the sign of the second derivative with respect to β of the objective function of a generic principal i. To make the proof more readable I abuse notation and suppress all the i subscripts.   2eβ µe − 2  e2β βµee −   eββ βµe − b b − N 2N 2 b b − N 2N 2   b b − N 2N 2  βσe2 − 2 2 β 2 σee −   β 2 σe2 −  b b − 2 N N  j6=i    X  ρij βj σj  βσee  + b b − 2 N N    b b − 2 N N    X  ρij βj σj  σe  + j6=i    X  ρij βj σj  βσe  + j6=i  −2 b b − N 2N 2  σ2 41 If P j6=i ρij βj σj  is positive, it follows from Assumptions 2 and 1 that the addend on each line 
P is negative. If the coefficient is negative we can observe that Nb − Nb2 ρ β σ is smaller ij j j j6=i than 1, by Assumption 1. This together with Assumption 2.3 implies that all the addends are negative as desired. Let’s now study the second derivative of the IR constraint reducing it to   b 2 2 (1 − β) µee − (1 − β) σee − cee + 2   b 2 2 eββ (1 − β) µe − (1 − β) σe − ce + 2   −2eβ µe − b (1 − β) σe2 e2β By Lemma 16 eβ = µe − b (1 − β) σe2 2 2 −c (1 − β) µee − 2b (1 − β) σee ee   Substituting the first term with eβ µe − b (1 − β) σe2   b 2 2 eββ (1 − β) µe − (1 − β) σe − ce + 2   −eβ µe − b (1 − β) σe2 The last term is problematic because it is positive. Substituting for α in the objective function, we have V (β) defined on [0, 1] The derivatives of this function will be given by ∂kU ∂ k IR ∂kV = + ∂β k ∂β k ∂β k 2 2 Adding the last term of ∂∂βIR to the next to last term of ∂∂βU2 2 42       X    b b b b βσe2 − 2 eβ 2 µe − 2 − − 2  ρij βj σj  σe  − µe − b (1 − β) σe2  = N 2N 2 N N j6=i         X  1 1 b b eβ µe − 4 β + (1 − β) bσe2 − 2 − − 2  ρij βj σj  σe  N 2N 2 N N    j6=i The term multiplying eβ needs to be positive. Inspecting the expression, it is clear that the P  “worse” possible scenario is when β = 0 and ρ β σ takes the lowest possible value (the ij j j j6=i P 
largest negative value). As noted above Nb − Nb2 j6=i ρij βj σj is smaller than 1. A sufficient condition for this expression to be always positive is µe + bσe2 + bσe > 0 But this is implied by Assumption 2.3 . The second derivative of V is hence negative, which concludes the proof. 2.4.2 Existence Theorem 19. If the mean-variance preferences are monotonic for an asset market economy characterized by the mean vector µ and variance-covariance matrix Ω then there exists an equilibrium in the CAPM contracting economy. Proof. Because the Markowitz preferences are not expected utility -and they also fail to satisfy the ’betweenness axiom’ (see Dekel, Safra, Segal (1991) -the best response correspondences of individuals would not be convexified by allowing for lotteries. This makes using the fixed point theorems by Glicksberg or Kakuthani impossible. I will instead show that the game can be reduced to a one-shot game in which principals have a continuous best response function function and apply Brouwer’s fixed point theorem. Principals’ Turn: Asset Market. Principal utility will be given by the CAPM analytical solution. 43 P i∈N qi Upi = qi − N P + i∈N µi N − b 1’Ω1 2 N2 As noted earlier prices and returns q, µ, Ω are all continuous functions of contracts C and efforts e. Agent’s Turn: Effort Choice. By Lemma 16 effort levels e are continuous functions of contracts C, hence the final utility of a principal can be described as a continuous function of contracts C . Principals’ Turn: Contract Design. By Lemma 18 the Principals’ problem is equivalent to a strictly concave optimization so that their best response function is always unique. By the maximum theorem it is also continuous. The strategy space C is a rectangle, which is a convex, compact subset of R2N . This satisfies the hypotheses of Brower fixed point theorem. There exist a fix point C ∗ which determines uniquely equilibrium efforts, beliefs, prices and portfolios. C ∗ , e∗ , γ ∗ , θ∗ , q ∗ form an equilibrium by construction. 2.5 The Insurance Effect of Markets - Moral Hazard Consider the following special case. Individuals are identical in the sense that they all have Markowitz type of Mean-Variance preferences, and they all have the same risk tolerance coefficient. Technologies have different variances (denoted by σi2 and different correlation coefficients ρij ). The mean returns of firms are determined by the effort of agents ei , specifically µi (ei ) = ei . Cost is quadratic c(ei ) = 2c e2i 2.5.1 First Best Equilibrium For the purpose of having a benchmark for optimal risk sharing and optimal effort, let’s have a look at the first best (observable action) case. 2.5.1.1 No Markets A Principal and an Agent agree on an action and a (random) payment. 44 b max α + βe − β 2 σ 2 α,β,e 2 such that − α + (1 − β) e − b c 2 (1 − β) σ 2 − e2 ≥ u 2 2 (IR) The optimal solution, action and contract is e∗ = 1 c 1 b (α∗ , β ∗ ) = − σ 2 + u, 8 2  Proof. Substitute for α in the objective function to obtain max U (β, e) = (1 − β) e − β,e b c b 2 (1 − β) σ 2 − e2i − u + βe − β 2 σ 2 2 2 2 which yields the first order conditions ∂U = a (1 − β) σ 2 − aβσ 2 = 0 ∂β ∂U = 1 − ce = 0 ∂e Which give the solution. 2.5.1.2 Financial Markets Now I consider the case of many firms. To do this I solve the optimization problem of an arbitrary principal, who has now access to a financial market. The form of the utility function follows from similar considerations as in the hidden type case seen in Chapter 1.. Lemma 20. Principal i behaves as if his utility function were 45 αi + βi ei +  b b − + βi2 σi2 + 2N 2 N   b X b − + ρij βi βj σi σj N2 N  j6=i Proof. Using the CAPM pricing formula, the riskless share of principal from firm i   b  2 2 X ρil βi βl σi σl  + βi σ i + αi + βi µi − N l6=i   P b X 2 2 X j∈N αj + βj µj − + 2 βj σ j + ρjk βj βk σj σk  N N j∈N k6=j Each principal holds a fraction of the aggregate portfolio, in particular he holds a random variable from which he gets utility  P j∈N αj + βj µj N − b  N2  X j∈N βj2 σj2 + 2 X ρjk βj βk σj σk  k6=j Because utility is linear in mean, and the riskless asset has variance zero, adding the two and simplifying obtains the claim. 46 max UiM KT (αi , βi ) = α i + βi e i +   b b − βi2 σi2 + + 2N 2 N   b X b − + ρij βi βj σi σj N2 N j6=i s.t. − αi + (1 − βi ) ei − c b 2 (1 − βi ) σ 2 − e2i ≥ u 2 2 (IR) The optimal solution, action and contract is KT eM = i βiM KT = 1 c σi2 − N −1 N2 σi2 +
P j6=i ρij βj σi σj
2 2N −1 σi N2 Proof. Substitute for αi in the objective function to obtain c b 2 max UiM KT (1 − βi ) ei − (1 − βi ) σ 2 − e2i − u + βi ei + βi ,ei 2 2   b b + − βi2 σi2 + 2N 2 N   b b X + − ρij βj σi σj N2 N j6=i which yields the first order conditions ∂UiM KT = a (1 − β) σi2 − ∂β  b a2 − 2 N N  βi σi2 −  b b − 2 N N X ρij βi βj σj = 0 j6=i ∂UiM KT = 1 − ce = 0 ∂e Which give the solution. The key observation is that the optimal action stays the same, but the first best contract changes. 47 As noted in the case of hidden type economies, the effect of markets is that principal acts as if they were less risk averse. This changes the optimal risk sharing. To better understand the effects of markets, let’s focus on the special case in which all technologies are identical and so are their correlation coefficients. σi = σj = σ, ∀i, j ρij = ρ, ∀i, j The symmetric solution in this case is 2 1 − (NN−1) ρβ 2 β= 2N −1 1 + N2 So that in equilibrium β M KT = N2 N 2 + 2N − 1 + ρ (N 2 − 2N + 1) Note how, unless there is perfect correlation, in equilibrium, a principal always take more risk than
in the no the market case βi = 21 . In fact the equilibrium contracts will be identical to the no market case only if technologies are perfectly correlated (ρ = 1) . ∀ρ < 1, 2 β M KT < N 1 = N 2 + 2N − 1 + N 2 − 2N + 1 2 2.5.2 Second Best Equilibrium 2.5.2.1 No Markets Let’s now turn to the more interesting case of unobservable actions. How do markets affect the equilibrium actions and returns? Here the decisions on risk sharing and effort are interdependent. Again let’s first look at a firm “in isolation” 48 b max α + βe − β 2 σ 2 α,β,e 2 b c such that − α + (1 − β)e − (1 − β)2 σ 2 − e2 ≥ u 2 2 b c e ∈ arg max −α + (1 − β)ẽ − (1 − β)2 σ 2 − ẽ2 ẽ 2 2 (IR) (IC) To simplify the problem let’s start by solving the problem of an agent facing a given contract (α, β) . b c max −α + (1 − β)ẽ − (1 − β)2 σ 2 − ẽ2 ẽ 2 2 b c max −α + (1 − β)e − (1 − β)2 σ 2 − e2 e 2 2 Because Individual Rationality can be optimally attained with the transfer α, the optimal action e can be obtained from the first order condition of the agent problem. (1 − β) − ce = o back into the Principal’s objective function and the agent’s IR constraint Plugging e∗ = 1−β c yields the following problem  max α + β α,β 1−β c  2 such that − α + The solution to this problem is b − β 2 σ2 2 2 (1 − β) b (1 − β) − (1 − β)2 σ 2 − ≥u c 2 2c (IR) 49 β= bσ 2 2bσ 2 + 1c e= bcσ 2 + 1c 2bσ 2 + 1 Proof. Again substituting α results in  max β α,β 1−β c  2 2 b (1 − β) b (1 − β) − β 2 σ2 + − (1 − β)2 σ 2 − −u 2 c 2 2c Differentiating with respect to β gives the first order condition −2bβσ 2 + bσ 2 − β =0 c The solution follows immediately from this and the fact that e = 1−β c Note how β < 21 . Risk sharing is distorted to give the proper incentive to the agent. Note that β > 0, so that e < 1c . Because there is a trade-off between incentivizing the agent and optimally sharing risk between two risk averse individuals, the first best cannot be achieved. I am now going to analyze how markets affect this tradeoff. 2.5.2.2 Financial Markets Consider the market with many principals from the previous section, except now the action in each firm is unobservable. 50 max UiM KT (αi , βi ) = αi + βi ei +   b b − βi2 σi2 + + 2N 2 N   b X b − + ρij βj σi σj N2 N j6=i c b 2 (1 − βi ) σ 2 − e2i ≥ u 2 2 b c ei ∈ arg max −α + (1 − β)ẽ − + (1 − β)2 σ 2 − ẽ2 ẽ 2 2 s.t. − αi + (1 − βi ) ei − (IR) (IC) The solution is now βiM KT =  b σi2 − N −1 N2
P j6=i ρij βj σi σj

2N −1 2 b σi + N 2 σi2 + 1c  In the symmetric case β M KT = bσ 2 N 2  2 bσ 2 N 2 + 2N − 1 + ρ (N − 1)  2 + Nc 2 The solution differs from the first best case because of the Nc term added to the denominator. So the equilibrium securities will be less risky than in the first best case. Moreover as observed for the non market case, the equilibrium effort is lower than at the optimum. Because β M KT is again increasing in N , the equilibrium effort and hence returns are decreasing N . In a fully symmetric case this result holds for every firm. This will not be the case if we drop the assumption of symmetry. However, the result will still hold in the aggregate in an economy where firms marginal distributions are identical, but not their conditionals. In other words I allow for different ρij coefficients in the variance covariance matrix. This means that while every firm is identical when there are no markets, they are ex-ante different in terms of diversification opportunities they face (and offer). This can be interpreted as the existence of different “sectors”. 51 Proposition 21. Consider an economy with identical marginal distributions characterized by mean ei , variance σ 2 , cost of effort 2c e2 . The aggregate output with markets is lower than the aggregate output without markets. ∗ N M ΣN i=1 ei ≥ Σi=1 ei with the inequality holding strictly unless ρij = 1, ∀i, j Proof. Claim 1. ∗ N M ∗ N M ΣN ⇐⇒ ΣN i=1 ei ≥ Σi=1 ei i=1 βi ≤ Σi=1 βi To see this note that ∗ N M ΣN ⇐⇒ i=1 ei ≥ Σi=1 ei
1 − βiM (1 − βi∗ ) ΣN ≥ ΣN ⇐⇒ i=1 i=1 c c
∗ N M ΣN ⇐⇒ i=1 (1 − βi ) ≥ Σi=1 1 − βi ∗ N M ΣN i=1 βi ≤ Σi=1 βi After estabilishing the above inequality. Observe that under the assumptions βi∗ = βiM = 1 1 2 + cbσ 2 M 1 − NN−1 2 Σj6=i ρij βj 1 2n−1 1 + cbσ 2 + N2 I need to prove that ΣN i=1 Claim 2. 1 1 2 + cbσ 2 ≤ ΣN i=1 M 1 − NN−1 2 Σj6=i ρij βj 1 2N −1 1 + cbσ 2 + N2 52 ∗ N M ΣN ⇐⇒ i=1 βi ≤ Σi=1 βi ∗ N ΣN i=1 βi ≥ Σi=1 Σj6=i 1 ρij βjM N −1 2 (N −1) 1 Note that the denominator of βiM can be rewritten as 2 + cbσ 2 − N 2 . The first inequality amounts to N −1 N M N 2 Σi=1 Σj6=i ρij βj
1 1 N −1 2 2 + cbσ 2 − cbσ 2 − N N N −1 N M Σ Σ ρ β N (N − 1) 2 j6 = i ij i=1 j N 
≥0

− 1 N −1 2 1 N −1 2 1 2 + − 2 + cbσ − 2 + 2 cbσ 2 N N cbσ 2 N N ≤ 1 2 + cbσ 2 2+
− N −1 2 ⇐⇒ ⇐⇒ Simplifying the last inequality  N (N − 1) − 2 + 1 cbσ 2  M ΣN i=1 Σj6=i ρij βj Claim 2 immediately follows. To conclude the proof, suppose by means of contradiction that the proposition did not hold, by 1 M ∗ N M N ∗ N Claim 1, ΣN i=1 βi > Σi=1 βi . By claim 2 this implies that Σi=1 βi < Σi=1 Σj6=i N −1 ρij βj Then it should be that ΣN i=1 Σj6=i 1 M ρij βjM > ΣN i=1 βi N −1 This can be rewritten as ΣN i=1   1 1 − Σj6=i ρij βiM < 0 N −1 53 which is impossible since ρij < 1, ∀i, j Financial Markets change the terms of risk sharing inside firms. Unlike the first best case, this has an effect on the equilibrium action, because it is exactly risk providing incentives for the agent to exert some positive effort. As the relative terms of risk sharing change, we move further away from the case in which the agent is risk neutral and optimal effort is obtained. In this case, it seems markets reward firms for variance against returns, but that is not exactly the case. Markets reward firms for providing opportunities for hedging and diversification in specific states of the world. To get a better intuition for this result, suppose there is a state space S = {SU N, RAIN } Consider a principal owning the an ice cream factory. Her firm returns can take two values, x + e > x + e respectively, in state SU N and in state RAIN . The security she is selling to the market will return α + βx + βe and α + βx + βe. When designing an incentive contract she is playing with 3 variables α, β, e. When markets are not available, the underlying state representation does not matter, the solution α∗ , β ∗ , e∗ will be some tradeoff between risk sharing and incentives (increasing β decreases e and viceversa . However consider now what happens if there is another firm, whose securities return α0 + β 0 x + βe0 when the state is RAIN and α0 + β 0 x + βe0 when the state is SU N . Because both principals are risk averse and there is now more return available in state RAIN , the owner of the ice cream factory will want to design a security generating more return in state SU N . Let’s consider what happens to returns in each state when she increases the 54 3 components of her contract ∆α RAIN : ∆α SU N : ∆α ∆β RAIN : ∆βx + ∆βe SU N : ∆βx + ∆βe ∆e RAIN : ∆eβ SU N : ∆eβ Note how β is the only component of the contract which has a different effect in different states, and specifically it provides more exactly where needed: in state SU N . On the other hand, the agent, who does not access markets, values equally returns in equally likely states (risk aversion would make returns in RAIN more valuable, if anything). Because markets change the relative value of returns in difference states for principals, the solution under markets will exhibit different risk sharing β M KT > β ∗ and consequently lower equilibrium effort eM KT < e∗ . Understanding this effect allows one to construct an example in which contracts and firms’ output are affected in the opposite way as described above. Consider an economy with two types of firms (50 of each kind). The low variance firms have variance .25, the high variance firms have variance 1. They are perfectly positively correlated ρij = 1, ∀i, j. Risk aversion is .2, cost of effort is .5 . The average output without markets is given by 106 56 . Numerical evaluation yields an average output 2.11 . In chapter 3 I discuss extensively how this particular distortion of risk sharing can occur even in a simple first best setting. The key intuition is that the low variance firm has a comparative advantage in providing returns in the low state, where the other firms are obtaining an even lower performance. 55 2.6 Conclusion This chapter integrates a model of principal-agent interaction with asset markets. Each pair/firm produces random returns, whose distribution depends on agents’ efforts. Every Principal offers a contract to the Agent he is matched with, and the Agents choose their costly action. What marks the difference from the standard contracting model is that Principals have access to an asset market on which they trade their shares of returns. This assumption is meant to capture the limited degree of access to financial markets, available to the average worker, who cannot entirely insure his labor risk. I present a unified framework for which I define a notion of equilibrium and prove its existence and uniqueness. Under standard assumptions of contract theory, I study the interactions of financial markets on contracts. On one hand, Moral Hazard inside firms induces suboptimal aggregate risk in an economy. On the other hand, introducing markets for principals, has ambiguous effects. If the marginal distributions of returns of firms are identical, markets will induce lower production levels. I construct an example, where asymmetry of marginal distributions and high degree of correlations induces riskier compensation packages in certain sectors and a positive effect on aggregate output. Interesting directions for the future include making welfare comparisons with the case where workers can (at least partially) access financial markets, and making the decision of entering markets endogenous in a non trivial way (making access costly). 2.7 Appendix 2.7.1 Preferences Markowitz preferences cannot be represented by expected utility. That is they cannot be represented by a linear functional in the space of mixtures. As a result they do not satisfy the independence axiom or -equivalently, but more significantly in this context- they are not “dynamically consistent”. Furthermore they pose a threat to equilibrium existence. 56 Consider two random variables X1 and X2 . We will have that b U (X1 ) = E(X1 ) − V ar(X1 ) 2 b U (X2 ) = E(X2 ) − V ar(X2 ) 2 Now consider a mixture with probability p of X1 and X2 . which we will call Y If individuals’ preferences were “linear”, we would have that. UEU (Y ) = pU (X1 ) + (1 − p)U (X2 ) Note that if a “linear” decision maker is indifferent between two random variables, he will also be indifferent between any mixture of them. Now compare with a Markowitz Individual a U (Y ) > E(Y ) − V ar(Y ) = 2  a 2 2 pE (X1 − E(Y )) + (1 − p) (X1 − E(Y )) pU (X1 ) + (1 − p)U (X2 ) − 2 He would choose any of the two RV he is indifferent to above any mixture of the two. This implies that his best response correspondence is not convex in the space of mixtures, unless it is a singleton, which makes it impossible to use Kakutani’s fixed point Theorem.3 To address this issue, I make sure that the model of Chapter 2 and 3 admits an equilibrium in pure strategies. As noted in the literature, it is possible to construct sequences of choices in between which some uncertainty is resolved, such that the agent makes plans he would not stick to. In this paper, this would lead to contracts which are not efficient as soon as they are signed even in a first best setting with symmetric information. Here is an example of this issue. Consider a standard principal agent problem with two types of agents. The type of the agent is known as soon as principals and agents are matched If Mean-Variance Preferences had a linear representation, the principal’s problem would look like this 3 This also implies that these preferences fail to satisfy Between-ness as in Dekel (1986). This renders also other methods for proving existence ineffective 57 h i a ρ α1 + β1 µ1 − β12 σ12 + 2 (α1 ,β1 ,α2 ,β2 ) i h a (1 − ρ) α2 + β2 µ2 − β22 σ22 2 a s.t. IRt : −αt + (1 − βt )µt − (1 − βt )2 σt2 ≥ u 2 max However the actual problem will look different because the variance of a mixture is not the convex combination of variances: h i a 2 2 2 ρ α1 + β1 µ1 − β1 σ1 + (α1 + β1 µ1 ) + 2 (α1 ,β1 ,α2 ,β2 ) i h a 2 2 2 β2 σ2 + (α2 + β2 µ2 ) + (1 − ρ) α2 + β2 µ2 − 2 a 2 [ρ (α1 + β1 µ1 ) + (1 − ρ) (α2 + β2 µ2 )] 2 a s.t. IRt : −αt + (1 − βt )µt − (1 − βt )2 σt2 ≥ u 2 max To see the dynamic inconsistency, let’s turn this into a numeric example. µ1 = 3,σ12 = 1 µ2 = 2,σ22 = ρ= 1 2 1 2 The random portion of optimal contracts in the first case are given by β1 = 12 , β2 = 12 , which are exactly the same that the principal would offer to each type if he knew which type is the agent he was dealing with. The Markowitz principal’s solution is instead β1 = .467, β2 = .539, to which he would prefer 12 , 21 as soon as uncertainty is resolved. Note that this Pareto improving renegotiation is not induced by voluntary information revelation from the agent, but simply by uncertainty resolving. Similarly, but more complicated, examples could be constructed for the model in this paper. 4 4 To read more on the topic of dynamic non-expected utility preferences, Machina (1989) 58 Chapter 3 A Few Observations on Security Design with Symmetric Information 3.1 Introduction In this chapter I study symmetric information economies. I am going to discuss several issues arising from the different risk sharing possibilities introduced by financial markets. I will study economies without any informational frictions, in which inefficiencies will arise from strategic behavior. In Section 3.2 I study the extent to which the results obtained under complete information in Chapter 1 and 2 can be generalized. It turns out that some of the assumptions made in those chapters are crucial to obtain an insurance effect. In particular the role of uncorrelated returns and the symmetry of firms will become clearer. In Section 3.3 I will analyze the behavior of principals in designing securities, when they do not act fully competitively. I will study how the number of traders affects efficiency of securities issued. In Section 3.4 I look at a large market, I consider N identical productive units, but I depart from the assumptions I made so far, by allowing one firm to be a merger of several productive units. I will describe the effects of market power on both large and small firms. 59 3.2 Markets Make Contracts Riskier In Chapters 1 and 2 first best risk sharing follows a simple pattern. More diversification opportunities for principals imply less risky contracts for agents who do not access markets. However I will show here that need not be the case under more general assumptions. The insurance results of Chapter 1 were possible because of the independence assumption. In Chapter 2 they were possible because of the symmetry assumption (every firm had the same mean and the same variance). The following examples show that enough positive correlation among firms returns, and enough difference in the variance of returns will induce some firms to pay workers with more risky contracts than they would without markets. Surprisingly this can be the case even in a large economy. 3.2.1 A Small Market Consider a model similar to that of Chapter 2. A firm constitutes of a principal and an agent, but no action is needed for production. Principals will only need to compensate the agent to the point in which their participation constraint is satisfied. This will be achieved with a contract which induces optimal risk sharing plus some transfer. The problem of a Principal who does not have access to markets is given by b max α + βµ − β 2 σ 2 , α,β 2 b such that − α + (1 − β)µ − (1 − β)2 σ 2 ≥ u. 2 (IR) The optimal contract will induce symmetric risk sharing in equal parts. The only notable difference from the First Best section of Chapter 2 is the lack of action e and its cost. This does not affect the optimal β as the first order conditions are the same as in Chapter 2. Let us first see what happens in a small market. Consider two firms. Both firms will have mean returns µ. One firm has variance σ 2 and the other has variance σ 2 , with σ 2 > σ 2 . Proposition 22. There are values of ρ, σ, σ , with ρ ∈ (0, 1] , σ > σ defining two firms economies for which the low risk firm offers riskier contracts when markets are introduced. 60 Proof. The problem of the Principal in the low risk firm. max U M KT α, β
1 3 =α + βµ − bβ 2 σ 2 − bρββσσ, 8 4

2 b such that − α + 1 − β µ − 1 − β σ 2 2 ≥ u. 2 (IR) And the problem of the principal in the high risk firm. max U M KT α, β
3 2 1 =α + βµ − bβ σ 2 − bρββσσ, 8 4

2 b such that − α + 1 − β µ − 1 − β σ 2 2 ≥ u. 2 (IR) From Chapter 2 we can derive the optimal contracts given the other firm contracts. 4σ − ρβσ , 7σ 4σ − ρβσ . β= 7σ β= And solve for equilibrium contracts. 28σ − 4ρσ , 49σ + ρ2 σ 28σ − 4ρσ ∗ β = . 49σ + ρ2 σ β∗ = 2ρ Note how β ∗ < 12 if σ < 7−ρ 2 σ. Similarly to Chapter 1 we have a case in which asset markets induce a riskier contract. One could be tempted to conclude that Markowitz preferences suffer from the same issues as quadratic expected utility does, as noted in Chapter 1 in markets with a small number of agents. It will turn out it is not the case, and in fact large markets can actually exacerbate this phenomenon. 61 3.2.2 A Large Market Consider an economy with N firms. All firms have the same mean returns µ. A fraction γ of firms have variance σ 2 and 1 − γ have variance σ 2 , with σ 2 > σ 2 .1 In this simple setup it is easy to relax one or both the assumptions from the other chapters. ρ controls the degree of correlation across firms. The difference in the variances controls how similar are the marginal distribution of returns for each firm. Proposition 23. There are values of γ, ρ, σ, σ with γ ∈ (0, 1) , ρ ∈ (0, 1] , σ > σ such that in a large economy low risk firms offer riskier contracts when markets are introduced. Proof. A generic Principal’s problem max UiM KT (αi , βi ) =αi + βi µ   b b + − βi2 σi2 2N 2 N   b b X + − ρij βi βj σi σj , N2 N j6=i such that − αi + (1 − βi ) µ − b 2 (1 − βi ) σ 2 ≥ u. 2 (IR) From which we get the following condition for optimal contracts σi2 − βiM KT = N −1 N2 σi2 +
P j6=i ρij βj σi σj
2 , ∀i. 2N −1 σi N2 Looking for a symmetric equilibrium we can write the optimal contracts as functions of other contracts for both types of firms. 1 Of course γ should be rational and γN should be always an integer. This can be easily obtained by defining an initial economy by two integers g and h with g firms with high variance and h firms with low variance, and then replicating them. However, in this section I am interested in the behavior of large economies in which the two approaches yield the same results. 62 β=   (N − 1) (1 − γ) ρβσ 2 + (N − 1) γρβσσ σ 2 − NN−1 2 β=   σ 2 − NN−1 (N − 1) γρβσ 2 + (N − 1) (1 − γ) ρβσσ 2 −1 2 σ 2 + 2N N2 σ , −1 2 σ 2 + 2N N2 σ I am considering large economies so I will look at the contracts as N → ∞, 2 σ − (1 − γ) ρβσ − γρβσ , σ σ − γρβσ − (1 − γ) ρβσ β= . σ β= Solving the system we first obtain contracts for one type as a function of the other type. σ − γρβσ , σ + (1 − γ) ρσ σ − (1 − γ) ρβσ β= . σ + γρσ β= And finally the equilibrium contracts σ − γρ (σ − σ) , σ (1 + ρ) σ + (1 − γ) ρ (σ − σ) β= . σ (1 + ρ) β= We can now observe that β< 1 2γρ ⇐⇒ σ < σ. 2 2γ + (1 − ρ) 2 Because the problem is strictly convex, and the assumptions of the Maximum Theorem are satisfied, best responses are continuos functions of others strategies and of the asset market equilibrium (which is also a continuous function of strategies). Hence taking limits on N and then solving for equilibrium corresponds to taking the limit of the sequence of equilbria. 63 Note how this is never satisfied if ρ = 0 and always satisfied when ρ = 1 (since we assumed all along that σ > σ). An interesting fact is that in the limit, every worker gets exactly the same compensation package (ie: the same random variable).

Corollary 24. −α + 1 − β X = −α + 1 − β X Proof.
ρσ + γ (σ − σ) ρσ + γρ (σ − σ) 1−β σ = = , 1+ρ 1+ρ
ρσ − (1 − γ) (σ − σ) ρσ + γρ (σ − σ) 1−β σ = = . 1+ρ 1+ρ We can conclude that agents obtain contract with the same standard deviation, and hence variance. Because agents are pushed to their IR constraint, if contracts have the same variance, they will also have the same mean. Because we are working with binary random variables it is enough to show that the compensation packages have the same mean and variance, to show that they are identical. We have seen in Chapter 1 that small markets do not necessarily induce less risky contracts, what is striking is that this can be the case also for large markets. By setting γ = 12 we have a replica economy of the initial two firms example. We can see that for certain parameters the contracts in the large economy will be even riskier than in the two firms case. Proposition 25. ∀ρ > 1, ∃σ, σ such that β (2) > lim β (N ) . N →∞ Proof. The claim amounts to showing that there are variances such that σ − 12 ρ (σ − σ) 28σ − 4ρσ > . 49σ + ρ2 σ σ (1 + ρ) 64 Some algebra shows that this boils down to  σ<  1 2 41 2 ρ − 4ρ + 2 ρσ. 21 + 12 ρ3 + ρ2 − 72 ρ The fraction on the right side is strictly positive, which allows us to conclude that we can find parameters satisfying our claim if and only if ρ > 0. How do markets (small and large) induce principals to behave as if they were more “risk averse”? A good intuition for this comes from thinking of two perfectly correlated technologies (ρ = 1) with different variances. In this case we can think of two states of the world L and H, with the returns of both types of firms being higher in H. The low variance firms have higher returns in L and lower in H than high variance firms do. Because of this they have an advantage in providing returns in state L. Markets make sure that this advantage is exploited by giving incentives to these firms to issue lower variance securities, compared to the share of profits they would keep for themselves when markets are not present. When correlation is not perfect, this effect still persists but to a lesser degree. 3.3 Number of Traders and Efficiency It is a relevant question, which I did not tackle so far, weather the securities issued are efficient in the case of symmetric information. When Principals design securities their utility functions take into account the effect they have on the value of other securities, which will ultimately affect their holding of the riskless asset. I will study this problem within a model of identical firms in which I will let the number of firms vary. In the symmetric examples from Chapter 2, all principals are identical ex-ante and after the contracting stage equilibrium. They end up with identical securities, and hence with 0 holdings of risky assets. I am going to show that outcome is non efficient. 65 max UiM KT (αi , βi ) = αi + βi µi   b b − βi2 σi2 + 2N 2 N   b X b − + ρij βi βj σi σj N2 N j6=i such that − αi + (1 − βi ) µi − b 2 (1 − βi ) σ 2 ≥ u. 2 (IR) The optimal contract is σi2 − βiM KT = N −1 N2 σi2 +
P j6=i ρij βj σi σj
2 . 2N −1 σi N2 In the symmetric case σi = σj = σ, ρij = ρ, ∀i, j The first order conditions for a symmetric equilibrium is in this case 2 1 − (NN−1) ρβ 2 . β= 2N −1 1 + N2 From which the equilibrium contracts β M KT = N2 N 2 + 2N − 1 + ρ (N 2 − 2N + 1) . To see this, consider a different game, in which each principal designs a piece of the aggregate endowment to be equally split, and no riskless asset transfers take place.3 3 A straightforward way to generalize this to the asymmetric case, would be to let individuals design securities and then give them unconditionally the transfers, and the fractions of aggreagte portfolio, they would have had received had they traded on the market 66 max Ui (αi , βi ) N 1 X N i=1  (αi + βi µi ) − such that − αi + (1 − βi ) µi − b 2N 2    N X X  βi2 σi2 + 2 ρij βi βj σi σj  , i=1 j6=i b 2 (1 − βi ) σ 2 ≥ u. 2 (IR) β E is the solution to this problem. βiE = N σi − P j6=i ρij βj σj (N + 1) σi . Note how, once we fix other players strategies, we can already observe how the best response induces riskier share for a principal than in the market case. This is because in the market setting a principal takes into account the effect of her actions on market prices. She takes the pricing mechanism as given, but because she can determine the characteristics of her security, she can influence prices of all securities, and hence other traders’ endowments. Again let’s focus on the symmetric case, in which case the equilibrium will be βE = N . (N + 1) + ρ (N − 1) Lemma 26. The contracts induced by βE give every individual a higher utility than those induced by βM Proof. Without loss of generality assume that u = 0 Claim b N (1 − ρ) + ρN 2 2 2 βE σ ≥ 2 N2 b N (1 − ρ) + ρN 2 2 2 αM + βM µ − βM σ . 2 N2 αE + βE µ − 67 We can substitute the α ’s from the reservation constraints to obtain b b N (1 − ρ) + ρN 2 2 2 2 βE σ ≥ (1 − βE ) σ 2 + βE µ − 2 2 N2 b b N (1 − ρ) + ρN 2 2 2 2 (1 − βM ) µ − (1 − βM ) σ 2 + βM µ − βM σ . 2 2 N2 (1 − βE ) µ − Rearranging and expanding we obtain
b 2 b N (1 − ρ) + ρN 2 βE σ 2 ≥ βE − 2βE σ 2 − 2 2 N2
b 2 b N (1 − ρ) + ρN 2 βM σ 2 . − βM − 2βM σ 2 − 2 2 N2 −  
(1 − ρ) N + ρN 2 + N 2 2 2 β − β ≥ 0. bσ 2 (βE − βM ) − E M 2N 2 We can divide by βE − βM and simplify N in the fraction above to obtain   (1 − ρ) + (1 + ρ) N bσ 2 1 − (βE + βM ) ≥ 0. 2N which boils down to 2N − [(1 − ρ) + (N + 1 + ρ (N − 1))] (βE + βM ) ≥ 0. Some tedious algebra shows that (βE + βM ) is equal to 2 (1 + ρ) N 2 + 3 (1 − ρ) N − (1 + ρ) . ((N + 1) + ρ (N − 1)) (N 2 + 2N − 1 + ρ (N 2 − 2N + 1)) So that we have 68   2 (1 + ρ) N 2 + 3 (1 − ρ) N − (1 + ρ) bσ 2N − ≥ 0. N 2 + 2N − 1 + ρ (N 2 − 2N + 1) 2 Multiplying and observing that the denominator is positive, we have left to show that the following polynomial is positive 2 (1 + ρ) N 3 − 6ρN 2 − 5 (1 − ρ) + (1 + ρ) . This has to be the case because it is positive for N = 2, and it is an increasing function of N ( its derivative is in fact 6 (1 + ρ) N 2 − 12ρN − 5 (1 − ρ) ). 1 . As N gets large the Good news come from the fact that limN →∞ βE = limN →∞ βM = 1+ρ difference between the efficient allocation and the market outcome gets smaller. Unfortunately, It is also straightforward to see that βE > βM ⇐⇒ ρ < 1, ∀N < ∞. This means that markets do better when we have little use for them, when firms returns are perfectly correlated. 3.4 Large firms In this section I study the equilibrium property of economies in which one firm is large relative to the market. I consider N identical productive units, each with returns characterized by mean µ, variance σ 2 , and pairwise correlation coefficient ρ. Consider a large number of firms N . One principal owns γN productive units, and the remaining (1 − γ) N constitute separate firms.4 The first tedious step is to write the quantities relevant for the principal utility functions. Firms are going to be of two types: one large firm and many small firms, I will look for a symmetric equilibrium, in which all small firms issue the same contracts/securities. The large firm will issue a security characterized by α, β and the small firms’ by αi , βi (or αj , βj when I need to distinguish among them). The mean and variance of the aggregate endowment 4 Again, it would be proper to have K units owned by a firm and the remaining N − K being individual firms. This however would lead to a more complicated notation whose effects become negligible as N and K get large. 69 α + βµ + Σαi + βi µi = γN (α + βµ) + (1 − γ) N (αi + βi µ) , 0 2 [β|β] Ω [β|β] = γN β 2 σ 2 + (1 − γ) N βi σ 2 + 2γ 2 N 2 ρβ 2 σ 2 + 2 (1 − γ) N 2 ρβi2 σ 2 + 2γ (1 − γ) N 2 ρββi σ 2 . The price of the security issued by the large firm is qγ = γN (α + βµ) −   b γN β 2 + 2γ 2 N 2 ρβ 2 + γ (1 − γ) N 2 ρββi σ 2 . (1 − γ) N The price of a security issued by a small firm i is qi = αi + βi µi −  2  b βi + (1 − γ) N ρβi2 + γN ρββi σ 2 . (1 − γ) N Note that the covariance of the productive units within the large firm are counted twice. This is because they are part of the variance of the security issued, and this is what marks the difference with the smaller firms. The large firm partially internalizes the effects of contracting inside the single unit on asset prices. The average price needed to compute the riskless asset holding is 1 qγ + Σqi = [γN (α + βµ) + (1 − γ) µ] N (1 − γ) N 

 1 − γ N β 2 + 2γN 2 ρβ 2 + (1 − γ) N 2 ρββi + (1 − γ) N βi2 + (1 − γ) N 2 ρβi2 + γN 2 ρββi . 2 (1 − γ) N 2 With this in mind we can construct the utility function of the principal of the large firm and drop the parts where β does not appear as they are not relevant to the solution. 70 − b b − 2 (1 − γ) N 2 (1 − γ) N 2 − Uγ (α, β) = γN (α + βµ) !
γN β 2 + 2γ 2 N 2 ρβ 2 σ 2 b b − 2 (1 − γ) N (1 − γ) N 2 !
γ (1 − γ) N 2 ρββi σ 2 . For optimization purposes, we can divide this function by γN and use this objective function. U (α, β) (α + βµ) !
b b − β 2 + 2γN ρβ 2 σ 2 2 (1 − γ) N 2 (1 − γ) N 2 ! b b − ((1 − γ) N ρββi ) σ 2 . − 2 (1 − γ) N (1 − γ) N 2 − This is the utility function for a principal in a generic small firm i, also trimmed of parts irrelevant to the optimization problem. − − Ui (αi , βi ) = αi + βi µ ! b b − βi2 σ 2 2 (1 − γ) N 2 (1 − γ) N 2 ! b b − 2 (1 − γ) N (1 − γ) N 2 (γN ρββi + (1 − γ) N ρβi βj ) . From the usual procedure of substituting α and αi from the agents’ IR constraints, we obtain the optimal contracts as a function of others’.   1 ρβi 1 − 1 − (1−γ)N   , β∗ = 1 1 1 + 2 (1−γ)N − 2(1−γ) (1 + γN ρ) 2 N2 βi∗ = N 2 − (N − 1) [γN ρβ + (1 − γ) N ρβi ] . N 2 + 2N − 1 71 As N gets large we have (1 − γ) (1 − ρβi ) , (1 − γ) + 2γρ 1 − γρβ βi = . 1 + (1 − γ) ρ β= Solve for β yields β∗ = (1 − γρ) . 2γ 1 + (1 − γ) ρ + γρ2 + 1−γ ρ We can now draw some conclusion on the contracts. Proposition 27. The large firms offers riskier contracts than in the symmetric case. Small firms offer less risky contracts. β < β symm < βi . 1 Proof. We know from Chapter 2 or Section 3.3 that β symm is equal to 1+ρ . 1 Part 1. β < 1+ρ . We want to show that (1 − γρ) 1 . < 2γ 2 1+ρ 1 + (1 − γ) ρ + γρ + 1−γ ρ 2γ expanding this yields 1−γ > 0, which is true ∀γ ∈ (0, 1). 1 Part 2. βi > 1+ρ . 1−γρβ 1 and β < 1+ρ it has to be the case that Since βi = 1+(1−γ)ρ βi > γρ 1 − 1+ρ 1 + (1 − γ) ρ = 1 + ρ − γρ 1 + (1 − γ) ρ 1 = = . (1 + ρ) (1 + (1 − γ) ρ) (1 + ρ) (1 + (1 − γ) ρ) 1+ρ This concludes the proof. I conjecture that similarly to the case of a small number of trader- this equilibrium is inefficient 72 1 and Pareto-dominated by an allocation with the securities induced by β = 1+ρ and the same transfers as the riskless asset holdings of this equilibrium. 3.5 Conclusions In this chapter, I study some implications on the design of securities and contracts. In Section 1 I look at what seems an“anomaly”. Even when markets are large, diversification opportunities do not necessarily translate to those who do not access financial markets. In Section 2 I look at inefficiencies induced by a small market on security design. A reassuring feature of this inefficiency is that its size becomes negligibly small with market size. A less reassuring feature is that it is larger when more diversification opportunities are available, which is exactly when we need financial markets. In Section 3 I consider the case of a firm that is large relative to the market, and the properties of equilibrium securities. I finalize the study of efficiency in this setting. This chapter tackles important questions on financial markets. More work needs to be done in terms of generalizing the phenomena exhibited here, to better understand how relevant they are to the real economy. Another important question, also in light of recent economic event, is weather firms (or financial institutions) choose an efficient amount of diversification in equilibrium. This would amount to making the ρi ’s coefficients endogenous. This is better suited to a model in which there is competition on the “agents” side, a direction I intend to pursue. 73 Bibliography [1] Aliprantis, Charalambos D., Kim C. Border,”Infinite Dimensional Analysis: A Hitchhiker’s Guide”, Springer, 2006. [2] Bolton, Patrick, Mathias Dewatripont, “Contract Theory”, MIT press, 2005. [3] Dekel, Eddie, “An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom”, Journal of Economic Theory, 40, 1986, 304-318 [4] Ellickson, Bryan, Birgit Grodal, Suzanne Scotchmer and William R. Zame, “The Organization of Production, Consumption and Learning”, Papers in Honor of Birgit Grodal, New York: Springer (forthcoming). 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