Financial Economics: Risk Sharing and Asset Pricing in General Equilibrium c

Transcription

1 1 / 170 Contents Financial Economics: Risk Sharing and Asset Pricing in General Equilibrium c Lutz Arnold University of Regensburg

2 Contents 1. Introduction 2. Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy 5. The fundamental equations of asset pricing 6. Applications 7. Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing 9. Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model 12. Time and uncertainty 13. Equilibrium and the fundamental equations of asset pricing 14. Fundamental value Appendix. Incomplete markets 2 / 170

3 3 / Introduction 1. Introduction

4 4 / Introduction This course in financial economics addresses two main questions. Efficiency: does trade in financial assets lead to an efficient allocation of the economy s risks? Asset pricing: what determines asset asset prices? Does price reflect fundamental value? To answer these questions, we approach the field of financial economics from the perspective of general equilibrium theory.

5 5 / Introduction Results in this field come at two strikingly different levels of mathematical difficulty: Almost all results require only very basic math. Examples include the first welfare theorem with financial markets the fundamental equations of asset pricing in two-period and multi-period setups the shareholder unanimity theorem the Modigliani-Miller theorem the CAPM the efficient markets hypothesis.

6 6 / Introduction Some results, which assert the existence of certain entities, are very hard to prove and require the use of separating hyperplane or fixed point theorems. Examples include existence of equilibrium the 2nd welfare theorem (for any Pareto-optimal allocation, there exist endowments and prices such that the allocation and the prices are an equilibrium) the fundamental theorem of asset pricing (in the absence of arbitrage opportunities, there exist state prices that can be used to price assets). These slides focus on the former set of results. A second set of slides Financial Economics: Risk Sharing and Asset Pricing in General Equilibrium II c summarizes the latter set of results and the necessary math.

7 7 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy EFFICIENT RISK SHARING

8 8 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy 2. Two-period two-state model

9 9 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Endowment economy with I consumers i = 1, 2,..., I, one (physical) good, two periods, t and t + 1, two states s = 1, 2 with probabilities π s (π s > 0, 2 s=1 π s = 1).

10 10 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Endowments: y i t, y i t+1,1, y i t+1,2. Preferences: utility functions U i (c i t, ci t+1,1, ci t+1,2 ) (Ui : R 3 + R). The only assumption we place on the U i s is that they are strictly increasing.

11 11 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy When it comes to asset pricing, we assume a time-separable expected utility function: with U i (c i t, c i t+1,1, ci t+1,2 ) = ui (c i t) + β i positive discounting (0 < β i < 1), u i : R + R twice differentiable, 2 π s u i (ct+1,s i ). s=1 (u i ) (c) > 0 > (u i ) (c) for all c > 0, and (u i ) (0) =. But that s inessential for much of the equilibrium analysis.

12 12 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy The model is simple to the point of triviality. That doesn t mean that it s easy to analyze, however! It offers deep insights into the nature of general equilibrium, its welfare properties, and efficient risk bearing.

13 13 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Literature: Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory, Oxford University Press (1995), Chapters 16, 19. Magill, Michael, and Martine Quinzii, Theory of Incomplete Markets, MIT (2002), Section 6.

14 14 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy 3 Efficient risk sharing: contingent-commodity markets

15 15 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy To begin with, ignore financial markets. Suppose there are contingent-commodity markets (CCMs). In these markets, claims to the delivery of one unit of the consumption good contingent upon the realization of state s are traded at price q t+1,s. If s does not materialize, nothing is delivered. spot market q t t t +1, 1 t +1, 2 q t+1,1 q t+1,2 = delivery of one unit of the commodity

16 2. Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy q = (q t, q t+1,1, q t+1,2 ): price vector y i = (yt i, y t+1,1 i, y t+1,2 i ): i s endowment vector c i = (ct i, ci t+1,1, ci t+1,2 ): i s consumption vector. Given that the U i s are strictly increasing, we can restrict attention to q R Without further comment, we let ci R 3 + (and analogously when we consider the model with more than two states). i s budget constraint is: q(c i y i ) 0. Convention: the product of two vectors is the scalar product! In equilibrium, due to strictly increasing utility, the budget equations will hold as equalities. 16 / 170

17 17 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Definition: An allocation (c i ) I i=1 is feasible if I c i i=1 I y i. i=1 An allocation is Pareto-optimal if it is feasible and there is no other feasible allocation (c i ) I i=1 such that U i (c i ) U i (c i ) for all i with strict inequality for at least one i.

18 18 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Definition: An allocation (c i ) I i=1 and a price vector q are an equilibrium with CCMs (ECCM) if c i maximizes U i subject to the individual s budget constraint for all consumers i and markets clear: I c i = i=1 I y i. i=1

19 19 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Theorem (1st Welfare Theorem with CCMs): Let ((c i ) I i=1, q) be an ECCM. Then (ci ) I i=1 is Pareto-optimal. Proof: Suppose not. Then there is (c i ) I i=1 which is feasible and Pareto-superior. Pareto-superiority implies qc i qc i for all i with strict inequality for at least one i. This is because otherwise i could have afforded a better consumption bundle than c i (use is made of strictly increasing utility).

20 20 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy So I I qc i > qc i. i=1 i=1 Feasibility of the allocation (c i ) I i=1 requires I i=1 ci I hence I I qc i qy i. i=1 i=1 i=1 yi,

21 21 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy The fact that ((c i ) I i=1, q) is an equilibrium (i.e., I i=1 ci = I i=1 yi ) implies I qc i = i=1 I qy i. i=1 Taking stock, we get a contradiction: I qy i i=1 I qc i > i=1 I qc i = i=1 I qy i. i=1 Q.E.D.

22 22 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy As usual in non-monetary micro models, we can arbitrarily normalize one price (e.g., q t = 1) without affecting the equilibrium allocation or the equilibrium relative prices. Theorem (irrelevance of price normalization): If ((c i ) I i=1, q) is an ECCM, then ((ci ) I i=1, λq) is an ECCM for any λ > 0. Proof: Markets clear by construction. (λq)(c i y i ) 0 if, and only if, q(c i y i ) 0. That is, the budget set remains the same after multiplying all the prices by a constant, so the optimal c i also remains the same. Q.E.D.

23 23 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Two other important theorems are: Existence of equilibrium: An ECCM exists. 2nd Welfare Theorem: For any Pareto-optimal allocation (c i ) I i=1, there exists endowments (yi ) I i=1 and a price vector q such that ((c i ) I i=1, q) is an ECCM The proofs of these theorems require additional (convexity, continuity, and boundary) assumptions and the use of a fixed point theorem and a separating hyperplane theorem, respectively, and can be found in the slides Financial Economics: Risk Sharing and Asset Pricing in General Equilibrium II c.

24 24 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy The analysis is unchanged if one interprets the three commodities as three goods traded in spot markets at a given point in time. Hence, the 1st Welfare Theorem above is in essence a reinterpretation of the standard theorem for non-contingent commodity markets. It shows that the standard analysis of trade at a point in time without uncertainty carries over to intertemporal trade in the presence of risk. Two drawbacks of the model are are: In reality, futures markets exist for very few commodities (energy, oil, wheat, metals, etc.), and even in those markets delivery is most often not state-contingent. If we allowed for different consumption goods, the number of CCMs (S times the number of different physical goods) would be large. In what follows, we replace CCMs with financial markets.

25 25 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Literature: Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory, Oxford University Press (1995), Chapter 19. Magill, Michael, and Martine Quinzii, Theory of Incomplete Markets, MIT (2002), Section 7.

26 26 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy 4 Efficient risk sharing: finance economy

27 27 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy The CCM economy achieves Pareto-optimal risk sharing as CCMs allow state-contingent consumption plans. We now assume that the only goods markets are (usual) spot markets: the price is paid on delivery. The spot price is normalized to unity. There are no CCMs. Financial markets (FMs) serve the same purpose as CCMs (indirectly): they allow the state-contingent transfer of purchasing power which can then be used to buy goods.

28 28 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy There are a risky and a safe asset, also called asset 1 and asset 2, respectively. The risky asset trades at prices p t at t and pays a dividend a t+1,1 = 1 in state 1 and a t+1,2 = 0 in state 2 at date t + 1. The safe asset trades at price 1/(1 + r t+1 ) and pays one unit of income in either state at date t + 1. By buying 1/(1 + r t+1 ) units of the safe asset and (short-) selling one unit of the risky asset, one gets nothing in state 1 and one unit of income in state 2. The financial market is complete, in that it is possible to buy purchasing power for the different states. t t +1, 1 t +1, 2 spot market 1 p t spot market 1 spot market 1 1+r t+1 p t 1

29 29 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Let z1 i denote consumer i s investment in the risky asset and zi 2 his purchases of the safe asset and z i = (z1 i, zi 2 ). His budget constraints are: c i t y i t p t z i r t+1 z i 2 c i t+1,1 y i t+1,1 + zi 1 + zi 2 c i t+1,2 y i t+1,2 + zi 2.

30 30 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Definition: An allocation (c i ) I i=1, asset holdings (zi ) I i=1, and a price vector (p t, r t+1 ) are an equilibrium with FMs (EFM) if c i and z i maximize U i subject to the individual s budget constraints for all consumers i and markets clear: and I c i = i=1 I i=1 I z i = 0. i=1 y i

31 31 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Consider an ECCM. Without loss of generality (as the absolute price level is indeterminate), set q t = 1. By buying one unit of the risky asset, one gets the purchasing power needed to buy one unit of the good in state 1. If p t = q t+1,1, this costs the same amount of money as one unit of contingent commodity 1 in the ECCM. By buying one unit of the safe asset and short-selling the risky asset, one gets the purchasing power needed to buy one unit of the good in state 2. If r t+1 = q t+1,1 + q t+1,2, this costs the same amount of money as one unit of contingent commodity 2 in the ECCM.

32 32 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Theorem (ECCM and EFM): Let (c i ) I i=1 and q with q t = 1 be an ECCM. Let p t = q t+1,1 Let r t+1 = q t+1,1 + q t+1,2. ( ) ( ) z1 i = ct+1,1 i y t+1,1 i ct+1,2 i y t+1,2 i z i 2 = ci t+1,2 y i t+1,2. Then ((c i, z i ) I i=1, (p t, r t+1 )) is an EFM.

33 2. Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Proof: To prove the theorem we have to show that c i and z i maximize i s utility and markets clear. Let B i denote the set of affordable consumption vectors c i for consumer i with CCMs given equilibrium CCM prices q, and let B i denote the set of affordable consumption vectors c i in the finance economy given the asset prices in the theorem. To prove that c i maximizes utility in the finance economy, it suffices to show that c i B i B i. That is, c i is affordable, and there is no better consumption vector in B i, because it would have been chosen in the CCM economy. c i B i B i 33 / 170

34 34 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy We first prove c i B i. The choice of the portfolio z i in the theorem implies that i can afford the same date-t + 1 consumption levels as in the ECCM. The question is whether he can also afford ct i. Since c i satisfies the CCM budget constraint with equality, we have q(c i y i ) = 0. Using the asset prices and the equations for i s portfolio in the theorem, this can be rewritten as ( ) t yt i ) + p t (z1 i + zi 2 ) + 1 p t z2 i 1 + r = 0. t+1 (c i Rearranging terms shows that the date-t budget constraint is satisfied. This proves c i B i.

35 35 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy To prove B i B i, suppose c i B i, i.e., c i and some z i satisfy the finance economy budget constraints. Using the pricing formulas in the theorem, the date-t budget constraint becomes c i t y i t q t+1,1 (z i 1 + zi 2 ) q t+1,2z i 2. From the date-t + 1 budget constraints, z i 1 + zi 2 ci t+1,1 y i t+1,1 and z i 2 ci t+1,2 y i t+1,2. So c i t y i t q t+1,1 (c i t+1,1 y i t+1,1 ) q t+1,2(c i t+1,2 y i t+1,2 ), i.e., q(c i y i ) 0. Thus, c i B i implies c i B i. That is, B i B i.

36 36 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy It remains for us to show that markets clear. For the goods markets, this holds true by construction. For the asset markets, this follows immediately upon summing the individuals asset demands in the theorem: I i=1 zi = 0. Q.E.D. The equilibrium allocation (c i ) I i=1 is Pareto optimal, as it is identical to the allocation in the CCM economy. So a 1st Welfare Theorem follows immediately from the theorem that relates an EFM to an ECCM: if an EFM replicates an ECCM, then the equilibrium allocation is Pareto optimal. The theorem does not rule out that there are other EFMs, which do no replicate an ECCM and with a Pareto-nonoptimal allocation. In what follows, without further mention, we consider only EFMs which replicate an ECCM.

37 37 / Two-period two-state model 3. Efficient risk sharing 1: contingent commodity markets 4. Efficient risk sharing: finance economy Literature: Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory, Oxford University Press (1995), Chapter 19. Magill, Michael, and Martine Quinzii, Theory of Incomplete Markets, MIT (2002), Section 8.

38 38 / The fundamental equations of asset pricing 6. Applications CONSUMPTION-BASED ASSET PRICING

39 39 / The fundamental equations of asset pricing 6. Applications 5 The fundamental equations of asset pricing

40 40 / The fundamental equations of asset pricing 6. Applications The central idea in consumption-based asset pricing is to exploit the necessary optimality conditions which characterize consumer choice in a market equilibrium. So we consider an EFM and derive its implications for asset prices. To do so, we now consider the time-separable expected utility function: U i (c i t, c i t+1,1, ci t+1,2 ) = ui (c i t) + β i 2 π s u i (ct+1,s i ) s=1 with (u i ) (c) > 0 > (u i ) (c) for all c > 0 and (u i ) (0) =. Let a t+1 denote the payoff of the risky asset (i.e., a t+1,1 = 1 or a t+1,2 = 0).

41 41 / The fundamental equations of asset pricing 6. Applications Consider individual i s utility maximization problem. The budget constraints hold with equality. Substituting for the consumption levels ct i, ci t+1,1, and ci t+1,2 yields the following problem: ( max : u i y i z i t p t z1 i 1 [ z2 )+β ( )] i i E t u i yt+1 i 1 + r + a t+1z1 i + zi 2. t+1 The optimality conditions can be written as: [ p t = E t β i (ui ) (ct+1 i ) ] (u i ) (ct i) a t r t+1 = E t [ β i (ui ) (c i t+1 ) (u i ) (c i t ) These equations relate the asset prices to individual i s marginal utility function and his consumption levels. As such, these are not remarkable asset pricing formulas. ].

42 42 / The fundamental equations of asset pricing 6. Applications The reason why they do yield interesting implications is that they can be used to express asset pricing without reference to individual-specific variables. To see this, note that individual i s MRS between consumption at t and consumption in state s at t + 1 is determined by That is: du i = (u i ) (c i t)dc i t + π s β i (u i ) (c i t+1,s )dci t+1,s = 0. MRS i t,t+1,s = dci t dc i t+1,s = π s β i (ui ) (c i t+1,s ) (u i ) (c i t ).

43 43 / The fundamental equations of asset pricing 6. Applications Theorem (Equality of MRSs when the allocation is Pareto-optimal): Let (c i ) I i=1 be a Pareto-optimal allocation. Then for each s, for all i and i. MRS i t,t+1,s = MRSi t,t+1,s Proof: Suppose for s = 1 or s = 2 the MRSs of individuals i and i are different. Without loss of generality, let MRSt,t+1,s i > MRSi t,t+1,s. That is, i has a stronger preference for consumption in s at t + 1 than i.

44 44 / The fundamental equations of asset pricing 6. Applications Let 0 < dc i t+1,s = dci t+1,s dct i = MRSi t,t+1,s + MRSi t,t+1,s dct+1,s i 2 = dci t. The resulting allocation is feasible. Then du i (u i ) (ct i) = MRSi t,t+1,s MRSi 2 t,t+1,s dc i t+1,s > 0 du i (u i ) (ct i ) = MRSi t,t+1,s MRSi 2 This contradicts Pareto-optimality. t,t+1,s dc i t+1,s > 0. Q.E.D.

45 45 / The fundamental equations of asset pricing 6. Applications Theorem (The fundamental equations of asset pricing): Let the utility function be given by U i (c i t, c i t+1,1, ci t+1,2 ) = ui (c i t) + β i 2 π s u i (ct+1,s i ), s=1 where u i is twice differentiable, with (u i ) (c) > 0 > (u i ) (c) for all c > 0 and (u i ) (0) =. Then, at an EFM with a Pareto optimal allocation there exists a stochastic discount factor (SDF) M t,t+1,s such that p t = E t ( Mt,t+1 a t+1 ) r t+1 = E t (M t,t+1 ).

46 46 / The fundamental equations of asset pricing 6. Applications Proof: Since the MRSs MRS i t,t+1,s = π sβ i (ui ) (c i t+1,s ) (u i ) (c i t ) are uniform across individuals, so is M t,t+1,s MRSi t,t+1,s π s = β i (ui ) (c i t+1,s ) (u i ) (c i t ) The assertion then follows immediately from the necessary optimality conditions for the individuals utility maximization problems. Q.E.D.

47 47 / The fundamental equations of asset pricing 6. Applications Future returns are discounted weakly (M t,t+1,s is high) if consumption is low in state s. Future returns are discounted heavily (M t,t+1,s is low) if consumption is high in s. That is, the SDF takes care that returns are valued according to whether they provide a hedge against low consumption. Notice that what makes this an asset pricing theory is the underlying general equilibrium model, which ensures that the MRSs are uniform, so that a single SDF exists which can be used to discount payoffs.

48 48 / The fundamental equations of asset pricing 6. Applications Notice the difference between the stochastic discount factor p t = E t (SDF a t+1 ) and a common, non-stochastic, discount factor DF where DF = p t = DF E t (a t+1 ), r t+1 + risk premium. The next section brings the two ways of discounting closer together.

49 49 / The fundamental equations of asset pricing 6. Applications Literature: Cochrane, John F., Asset Pricing, 2nd Edition, Princeton University Press (2005), Chapter 1.

50 50 / The fundamental equations of asset pricing 6. Applications 6 Applications

51 51 / The fundamental equations of asset pricing 6. Applications The fundamental equations of asset pricing look quite different from standard asset pricing formulas. This section shows that they can be rearranged such that they become much more similar. For now, the asset pricing formulas we derive apply only to the risky asset with payoff 1 in state 1 and 0 in state 2. Later on, we will see that they generalize to assets with arbitrary payoff vectors and to a multi-period model. In that multi-period model, the risky asset can be resold at date t + 1, so the date-t + 1 value of the asset is the sum of the resale price p t+1 and the dividend a t+1. We replace a t+1 with p t+1 + a t+1 here, so that the asset pricing formulas apply later too, where for now p t+1 = 0.

52 52 / The fundamental equations of asset pricing 6. Applications Application 1: Covariance Define σ M,p+a as the covariance between the SDF and the risky asset s return: σ M,p+a E t [M t,t+1 (p t+1 + a t+1 )] E t (M t,t+1 )E t (p t+1 + a t+1 ). Then p t = E t (M t,t+1 )E t (p t+1 + a t+1 ) + σ M,p+a = E t(p t+1 + a t+1 ) 1 + r t+1 + σ M,p+a. That is, the asset price is the expected payoff discounted at the riskless rate plus a risk adjustment, and the risk adjustment is simply the covariance of its return with the SDF. The risky asset is expensive if the SDF is high (i.e., consumption is low) in state 1, in which it pays off.

53 53 / The fundamental equations of asset pricing 6. Applications Application 2: Systematic risk Let R t+1 = p t+1 + a t+1 p t 1 denote the rate of return on the risky asset. Then, using σ M,p+a = p t σ M,R, E t (R t+1 ) r t+1 = (1 + r t+1 )σ M,R. The risky asset s risk premium is proportional to the covariance between its returns and the SDF (its systematic risk).

54 54 / The fundamental equations of asset pricing 6. Applications Application 3: Idiosyncratic risk If the asset s payoff is uncorrelated with the SDF, then the asset does not pay a risk premium, irrespective of how volatile its returns are: E t (R t+1 ) r t+1 = 0 if σ M,R = 0.

55 55 / The fundamental equations of asset pricing 6. Applications Application 4: beta Another way to write the fundamental pricing equation for the risky asset is where σ 2 M E t (R t+1 ) r t+1 = β E t (M t,t+1 ), β = σ M,R σm 2. That is, (analogously to the CAPM) the asset s risk premium is the product of a non-asset specific market term and its beta.

56 56 / The fundamental equations of asset pricing 6. Applications Application 5: Random walk Contrary to what has been assumed so far, assume the agents are risk-neutral: u i (c i ) = c i. Consider a short time interval, in which discounting doesn t play a role (β i = 1) and the asset doesn t pay a dividend (a t+1 = 0). The fundamental equation for the risky asset becomes: E t (p t+1 p t ) = 0. The asset price is a random walk.

57 57 / The fundamental equations of asset pricing 6. Applications Literature: Cochrane, John F., Asset Pricing, 2nd Edition, Princeton University Press (2005), Chapter 1.

58 58 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing COMPLETE MARKETS

59 59 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing 7 Complete markets: Efficient risk sharing

60 7. Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing The restriction to S = 2 states is immaterial for most of the analysis. Let S 2. State s occurs with probability π s (π s > 0, S s=1 π s = 1). Define q = (q t, q t+1,1,..., q t+1,s ): price vector y i = (yt i, y t+1,1 i,..., y t+1,s i ): i s endowment vector c i = (ct i, ci t+1,1,..., ci t+1,s ): i s consumption vector. U i is defined over c i (U i : R S+1 + R). The whole analysis of the CCM economy goes through without any modification. 60 / 170

61 61 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing The financial market in the two-state economy in Section 4 is complete in that it is possible to transfer income to a specific state s (= 1, 2) by forming a suitable portfolio. Assume that there are financial products, called Arrow securities (ASs), which do so directly: for each state s (= 1,..., S), there is an AS that costs p s at t and entitles the owner to the payment of one unit of income in s at t + 1 (and nothing in any other state).

62 62 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing As in Section 4, the goods prices are set equal to unity in both periods. Consumer i s budget constraints can be written as: c i t y i t S p s z s i s=1 c i t+1,s y i t+1,s zi s, s = 1,..., S. Let p = ( p 1,..., p S ) denote the vector of AS prices and z i = ( z 1 i,..., zi S ) i s AS holdings. In our one-good model, AS markets are almost the same as CCMs: they allow the purchase of the purchasing power required to purchase the good in one single state.

63 63 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Definition: An allocation (c i ) I i=1, AS holdings ( zi ) I i=1, and a vector of AS prices p are an equilibrium with a complete set of ASs (ECAS) if (c i, z i ) maximizes U i subject to the individual s budget constraints for all consumers i and markets clear: I c i = i=1 I i=1 I z i = 0. i=1 y i

64 64 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Consider an ECCM with q t = 1 (which entails no loss of generality because of the irrelevance of price normalization). Theorem (ECAS and ECCM): Let ((c i ) I i=1, q) be an ECCM, p s = q t+1,s, s = 1,..., S, and z s i = ct+1,s i y t+1,s i, s = 1,..., S. Then ((c i, z i ) I i=1, p) is an ECAS. The former condition says that the cost of buying the purchasing power needed to buy one unit of the good in the spot market if state s occurs is the same as the cost of buying the delivery in the CCM. The second condition says that the portfolio finances the ECCM consumption vector.

65 65 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Proof: The proof works exactly like the proof of the 1st Welfare Theorem for the two-state finance economy. We have to show that c i maximizes i s utility and markets clear. To prove that c i maximizes utility given ECAS prices, it suffices to show that c i B i B i, where B i is the set of affordable consumption vectors c i for consumer i with CCMs given equilibrium CCM prices q and B i is the set of affordable consumption vectors c i with a complete set of ASs given the ECAS prices. That is, c i is affordable, and there is no better consumption vector in B i, because it would have been chosen in the CCM economy. c i B i B i

66 66 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing To prove c i B i, notice that the fact that i chooses c i at the ECCM implies q(c i y i ) = 0. Substituting for q from the former condition of the theorem (and q t = 1) yields c i t y i t + S s=1 p s (c i t+1,s y i t+1,s ) = 0. As stipulated by the theorem, let consumer i choose the portfolio z s i = ct+1,s i y t+1,s i, s = 1,..., S. From the preceding equation, c i t y i t = S p s z s. i s=1 The latter two equations show that c i B i.

67 67 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing To prove B i B i, suppose c i B i, i.e., c i and some portfolio z i satisfy the S + 1 budget constraints. Substituting for z i s from the latter S constraints into the period-t budget constraint yields c i i y i t S p s (ct+1,s i y t+1,s i ). s=1 The former condition of the theorem and q t = 1 yield q(c i y i ) 0. So c i B i. The fact that c i B i c i B i proves B i B i.

68 7. Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Goods market clearing in the CCM economy implies goods market clearing in the economy with ASs: I c i = i=1 I y i. i=1 From the second condition of the theorem, I z s i = i=1 I i=1 (c i t+1,s y i t+1,s ) } {{ } =0 So the AS markets also clear: I z i = 0. i=1 = 0, s = 1,..., S. Q.E.D. 68 / 170

69 69 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Next, we consider a less peculiar asset structure. There are K assets k = 1,..., K with prices p k in t and payoff a sk in state s at t + 1. The payoffs are summarized in the payoff matrix A = a a 1K..... a S1... a SK. The k-th column gives asset k s payoffs in the S states, the s-th row gives the K assets payoffs in state s. The set of assets is exogenously given there is no financial engineering.

70 70 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Let z = (z 1,..., z K ) denote a portfolio. Az = a a 1K..... a S1... a SK z 1. z K = K k=1 a 1kz k. K k=1 a Skz k gives the payoffs generated the portfolio in the S states. Definition: The financial market is complete if A contains S linearly independent payoff vectors. Market completeness implies that for all x R S, there is z such that Az = x. Let the assets be ordered such that the first S payoff vectors (k = 1,..., S) are independent. Then for all x R S, there is z = (z 1,..., z S, 0,..., 0) such that Az = x.

71 71 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing In theory, the condition for market completeness is not too demanding: it can be shown that it is fulfilled if there is one asset with a different payoff in each state of nature and it is possible to write put or call options on this asset. Giving up the matrix notation, the condition can be written as K a sk z k = x s, s = 1,..., S. k=1

72 72 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Consumer i s budget constraints become: c i t y i t c i t+1,s y i t+1,s K k=1 K p k zk i k=1 a sk zk i, s = 1,..., S.

73 73 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Let p = (p 1,..., p K ) denote the asset price vector. Definition: An allocation (c i ) I i=1, asset holdings (zi ) I i=1, and a vector of asset prices p are an equilibrium with complete financial markets (ECFM) if the financial market is complete, (c i, z i ) maximizes U i subject to the individual s budget constraints for all consumers i and markets clear: I c i = i=1 I i=1 I z i = 0. i=1 y i

74 74 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Theorem (ECFM and ECAS): Let ((c i, z i ) I i=1, p) be an ECAS. Let p k = S p s a sk, k = 1,..., K. s=1 Suppose the financial market is complete, and for all i = 1,..., I, let z i = (z i 1,..., z i S, 0,..., 0) denote a solution to S k=1 a sk z i k Then ((c i, z i ) I i=1, p) is an ECFM. = zi s, s = 1,..., S.

75 75 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Proof: Let B i denote the set of affordable consumption vectors c i for i given ECAS prices p, and let B i denote the set of affordable consumption vectors c i for i with complete financial markets. We show that c i B i B i. That is, i can afford c i, and there cannot be a consumption vector he likes better, because he would have chosen it in the ECAS.

76 76 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing The fact that i chooses (c i, z i ) in the ECAS implies that it satisfies his budget constraints for the AS economy with equality: c i t y i t = S s=1 p s z i s c i t+1,s y i t+1,s = zi s, s = 1,..., S. The existence of the portfolio z i in the theorem is implied by market completeness. This portfolio reproduces the payoffs of the ASs portfolio. Substituting the formula for z s i in the theorem into these constraints and using the pricing formula in the theorem and zk i = 0 for k = S + 1,..., K in the first constraint proves that the budget equations for the economy with CFMs are satisfied. This proves c i B i.

77 77 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Suppose c i B i. That is, there is a portfolio z i (not necessarily with zero holdings of assets k = S + 1,..., K ) such that (c i, z i ) satisfies c i t y i t c i t+1,s y i t+1,s K k=1 K p k zk i k=1 a sk zk i, s = 1,..., S. Let z s i = K k=1 a skzk i. Using this and the pricing formula in the theorem, it follows that (c i, z i ) satisfies the budget equations for the economy with ASs. That is, c i B i. This proves B i B i.

78 7. Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Obviously, the goods markets clear. As for the asset markets, let zk = I i=1 zi k. Asset market clearing in the economy with ASs implies So 0 = I i=1 z i s = I S i=1 k=1 a a 1S z a S1... a SS a sk z i k z S = = S k=1 a sk zk, s = 1,..., S. S k=1 a 1kz k. S k=1 a Skz k = From linear independence of the S payoff vectors in the matrix, it follows that the only solution to this system of equations is (z1,..., z S ) = (0,..., 0). zi k = 0 for k = S + 1,..., K implies that the other asset markets also clear. Q.E.D. 78 / 170

79 79 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing 8 Complete markets: Asset pricing

80 80 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Using the utility function U i = u i (c i t) + β i S s=1 π s u i (c i t+1,s ), we can also generalize the fundamental asset pricing equations to the S-states case.

81 81 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing To do so, start with the economy with ASs: ) S U i = u (y S i t i p s z s i + β i π s u i (yt+1,s i + zi s). s=1 s=1 Necessary optimality conditions: (u i ) (ct) p i s + β i π s (u i ) (ct+1,s i ) = 0 or p s = π s β i (ui ) (c i t+1,s ) ( u i ) (c i t ) = π s M t,t+1,s where the SDF M t,t+1 is defined as before. One need not make use of Pareto optimality in oder to prove that M t,t+1 is uniform across individuals i.

82 82 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Let ã s denote the random payoff of AS s, i.e., 1 in state s and 0 in all other states. Then p s = s s π s M t,t+1,s 0 + π s M t,t+1,s 1 or simply p s = E t (M t,t+1 ã s ). This generalizes the fundamental pricing formula for a risky asset.

83 83 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Now consider the economy with a complete set of FMs. From the pricing rule p k = S s=1 p s a sk, we get p k = S π s M t,t+1,s a sk s=1 or, letting a k denote the random payoff of asset k:

84 84 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Theorem (The fundamental equations of asset pricing): Let the utility function be given by U i = u i (c i t) + β i S s=1 π s u i (c i t+1,s ), where u i is twice differentiable, with (u i ) (c) > 0 > (u i ) (c) for all c > 0 and (u i ) (0) =. Then at an ECFM prices obey p k = E t ( Mt,t+1 a k ) and, in particular, p k = E t (M t,t+1 ) for a safe asset with a sk = 1 for all s = 1,..., S.

85 85 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing It follows that the applications in Section 6 go through without any modification in the economy with more than two states and assets with arbitrary payoff vectors. Define q s (1 + r t+1 )M t,t+1,s π s. Then the pricing equation can be rewritten as p k = S s=1 q sa sk 1 + r t+1. That is, the q s s can be interpreted as risk-neutral probabilities: the price of the asset is its discounted expected value, where the q s s are used to calculate the expectation.

86 86 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing An important property of the ECAS and the ECFM is that there are no arbitrage opportunities, i.e., it is not possible to form a portfolio that has non-positive cost and has a non-negative payoff in all states with one inequality strict. This is obvious for the economy with ASs. A portfolio that has a positive payoff in some state costs something. A portfolio with negative cost has a short position in some AS and, hence, a negative payoff in some state.

87 87 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Turning to the economy with CFMs, we formalize the definition above: Definition: Asset prices p are arbitrage-free if there is no portfolio z R K such that pz 0, Az 0, and either Az 0 or pz < 0. Theorem (Arbitrage-freeness of the ECFM): ECFM prices are arbitrage-free.

88 88 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Proof: From the first two conditions and the pricing formula for p k, K a sk z k 0, s = 1,..., S, and k=1 S p s s=1 K k=1 a sk z k 0. If Az 0, then one of the former equalities is strict, which contradicts the latter inequality. If pz < 0, then K k=1 a skz k must be negative for some s, which contradicts the former set of inequalities. Q.E.D.

89 89 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Another important theorem is: The fundamental theorem of asset pricing: If asset prices p > 0 are arbitrage-free, then there exist non-negative state prices p = ( p 1,..., p S ) such that p k = S p s a sk, k = 1,..., K. s=1 The proof requires the use of a separating hyperplane theorem and can be found in the slides Financial Economics: Risk Sharing and Asset Pricing in General Equilibrium II c.

90 90 / Complete markets: Efficient risk sharing 8. Complete markets: Asset pricing Literature: Magill, Michael, and Martine Quinzii, Theory of Incomplete Markets, MIT (2002), Section 10.

91 91 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model FIRMS

92 92 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model 9 Stock market economy

93 93 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model The aim of this section is to show that the results for the exchange economy carry over to a model with firms. This also allows addressing questions about shareholder aims and capital structure. For simplicity, we don t consider production decisions, but take firms outputs as exogenously given.

94 94 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model There are J firms j = 1,..., J with exogenous outputs ỹ j t+1,s in state s in period t + 1. The firms are 100 percent owned by the consumers i and (for now) don t issue debt. x% of firm j s shares entitle the holder to x% of the firm s revenue ỹ j t+1,s in any period-t + 1 state s. Consumer i is endowed with a share θ ij in firm j, where θ ij 0 and I θ i=1 ij = 1 for all j. In period t, consumers have endowments yt i production., and there s no

95 95 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model There are markets for the shares and for a complete set of ASs. The value of firm j as of time t is denoted v j t and the vector of firm values as v = (vt 1,..., v t J ). i s post trade share holdings are denoted θ ij. We also write θ i = (θ i1,..., θ ij ), θ i = ( θ i1,..., θ ij ), and y = y t y t+1,1. y t+1,s I i=1 y i t J = j=1 ỹ j t+1,1. J j=1 ỹ j t+1,s. We call this a stock market economy (SME) and analyze it by comparing it to the exchange economy with ASs.

96 96 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model The SME budget constraints read: c i t y i t c i t+1,s zi s + S p s z s i s=1 J j=1 J j=1 (θ ij θ ij )v j t θ ij ỹ j t+1,s, s = 1,..., S.

97 9. Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model Definition: An allocation (c i ) I i=1, AS holdings ( zi ) I i=1, shareholdings (θ i ) I i=1, a vector of AS prices p, and a vector of firm values v are an equilibrium of the stock market economy (ESME) if (c i, z i, θ i ) maximizes U i subject to the individual s budget constraints for all consumers i and markets clear: I c i = y i=1 I z i = 0 i=1 I θ i = 1. i=1 97 / 170

98 98 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model Theorem (ESME and ECAS): Let (c i, z i ) I i=1 and p be an ECAS of the exchange economy with endowments and let y i t+1,s = J j=1 v j t = θ ij ỹ j t+1,s, s = 1,..., S, i = 1,..., I, S s=1 p s ỹ j t+1,s, j = 1,..., J. Then (c i, z i, θ i ) I i=1 and ( p, v) are an ESME.

99 99 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model That is, given arbitrage-free pricing of the payoffs generated by the firms, the SME behaves like the exchange economy in which individuals are endowed with what their initial shareholdings are worth. Proof: Let B i be the set of consumption vectors attainable in the SME. As usual by now, we show c i B i B i.

100 100 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model The fact that i chooses (c i, z i ) in the ECAS implies that it satisfies the budget constraints of the exchange economy with a complete set of ASs with equality: c i t y i t = S s=1 p s z i s c i t+1,s y i t+1,s = zi s. Using the definition of yt+1,s i in the theorem, it follows that (c i, z i, θ i ) satisfies the SME budget constraints, i.e., c i B i.

101 101 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model Let c i B i. That is, there are z i and θ i such that the SME budget constraints hold. Let z i be given by z i s = z i s + Then (c i, z i ) B i. J (θ ij θ ij )ỹ j t+1,s, s = 1,..., S. j=1

102 102 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model The fact that the period-t budget constraint is satisfied follows from the period-t budget constraint for the SME and the definition of v j t in the theorem: c i t y i t = = = S p s z s i s=1 S p s z s i s=1 S s=1 J j=1 (θ ij θ ij )v j t J (θ ij θ ij ) j=1 p s z i s + S p s z s i. s=1 S p s ỹ j t+1,s s=1 J (θ ij θ ij )ỹ j t+1,s j=1

103 103 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model The period-t + 1 budget constraints also hold: c i t+1,s z i s + = z i s + c i t+1,s y i t+1,s z i s. J θ ij ỹ j t+1,s j=1 J θ ij ỹ j t+1,s j=1 = z i s + y i t+1,s So c i B i and B i B i. The validity of the market clearing conditions in the SME is obvious. Q.E.D.

104 104 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model So the SME behaves essentially like the exchange economy considered before. So if an ESME replicates an ECAS which in turn replicates a ECCM, then the ESME allocation c is Pareto-optimal, since it coincides with the EECM allocation, which is Pareto-optimal due to the 1st welfare theorem. Households do not trade shares at the ESME in the theorem. There are other ESMEs, with trade in shares, however:

105 105 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model Theorem (ESME with trade in shares): Let ((c i, z i, θ i ) I i=1, p, v) be an ESME. Suppose household i changes his stock demands by dθ i = (dθ i1,..., dθ ij ) (not necessarily small) and his AS demands by d z i s = J ỹ j t+1,s dθij, j=1 where I i=1 dθij = 0 for all j. Then ((c i, z i + dz i, θ i + dθ i ) I i=1, p, v) is also an ESME. Proof: From the budget constraints, these asset reallocations do not affect consumption. So if (c i, z i, θ i ) maximizes utility, so does (c i, z i + dz i, θ i + dθ i ). Markets clear. Q.E.D.

106 106 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model Shareholders in the SME are unanimous with regard to what the firms they own shares in should do: maximize value. (Caveat: firms do nothing in the model considered here, as production plans and outputs are exogenous. So we formulate the assertion accordingly.) Theorem (Shareholder Unanimity): An increase in v j t expands the budget set of each individual i with θ ij > 0. Proof: Substituting for z i s from the period-t + 1 budget constraints into the period-t budget constraint yields

107 107 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model c i t y i t = S J p s ct+1,s i θ ij ỹ j t+1,s s=1 S J p s ct+1,s i + s=1 j=1 j=1 θ ij S s=1 J j=1 p s ỹ j t+1,s }{{} (θ ij θ ij )v j t J j=1 (θ ij θ ij )v j t = S J p s ct+1,s i + s=1 j=1 θ ij v j t. =v j t So if v j t rises, the budget sets of the initial shareholders (individuals i with θ ij > 0 expand). Q.E.D.

108 108 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model 10 The Modigliani-Miller theorem

109 109 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model The Modigliani-Miller (MM) theorem states that the division of a firm s cash flow between different kinds of financial claims (i.e., its capital structure) is irrelevant for its value. This section proves a much more general version of the MM theorem (due to Stiglitz, 1969), which proves that firms capital structure is irrelevant for real economic activity in general finance is a veil. In doing so, we neglect the issue of investment finance, i.e., how the choice of different financial instruments affects investment activity (as is well known, capital structure is not irrelevant in this regard in the presence of information asymmetries).

110 110 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model Without uncertainty, the value of a firm s payments to stockholders (ỹ j t+1 bj t ) and creditors (bj t ) at t + 1 is ỹ j t+1 /(1 + r), where r is the safe interest rate. This is the same as in the case of no debt because the present value debt is zero.

111 111 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model With uncertainty, let a single firm j take on debt b j. The value of the firm s payments to stockholders (ỹ j t+1,s bj t ) and creditors (b j t ) at t + 1 is S p s (ỹ j t+1,s bj t ) + p bb j t. s=1 Since the price of debt is this coincides with v j t = p b = S p s, s=1 S p s ỹ j t+1,s, s=1 the value of the unlevered firm.

112 112 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model Now the general case. Let b j t denote the debt issued by firm j. b j t is exogenous, so we can discuss the effects of exogenous changes in the firms capital structure. Let bt i denote the debt held by individual i. Shares θ ij in j entitle the holder to a fraction θ ij of the proceeds of j s debt issue p b b j t, in addition to the same share in its profits (in a model with investment this would translate into a reduction in the contributions to the investment outlays). We rule out bankruptcy: ỹ j t+1,s bj t, s = 1,..., S, j = 1,..., J. This means that any two firms debt obligations are perfect substitutes for the consumers. We call this the stock and debt economy (SDE).

113 113 / Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model i s budget constraints become c i t y i t S p s z s i s=1 c i t+1,s zi s + J j=1 (θ ij θ ij )v j t p bb i t + J θ ij (ỹ j t+1,s bj t ) + bi t, j=1 J θ ij p b b j t where v j t is the value at which the shares of firm j trade in the stock market. j=1

114 9. Stock market economy 10. The Modigliani-Miller theorem 11. The Capital Asset Pricing Model Definition: An allocation (c i ) I i=1, AS holdings ( zi ) I i=1, shareholdings (θ i ) I i=1, debt holdings (bi t )I i=1, a vector of AS prices p, a vector of firm values v, and a price of debt p b are an equilibrium of the stock and debt economy (ESDE) if (c i, z i, θ i, b i t ) maximizes Ui subject to the individual s budget constraints for all consumers i and markets clear: I c i = y i=1 I z i = 0 i=1 I θ i = 1 i=1 I bt i = i=1 J b j t. j=1 114 / 170