ArrowDebreu Pricing, Part I
Chapter Outline
9.1 Introduction 247
9.2 Setting: An ArrowDebreu Economy 248
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9.3 Competitive Equilibrium and Pareto Optimality Illustrated 250
9.4 Pareto Optimality and Risk Sharing 257
9.5 Implementing Pareto Optimal Allocations: On the Possibility of Market Failure 260 9.6 Risk-Neutral Valuations 263
9.7 Conclusions 266
References 267
9.1 Introduction
As interesting and popular as it is, the CAPM is a very limited theory of equilibrium pricing. We will devote the next chapters to reviewing alternative theories, each of which goes beyond the CAPM in one direction or another. The ArrowDebreu pricing theory discussed in this chapter is a full general equilibrium theory as opposed to the partial equilibrium static view of the CAPM. Although also static in nature, it is applicable to a multiperiod setup
and can be generalized to a broad set of situations. In particular, it is free of any preference restrictions and any distributional assumptions on returns. The Consumption CAPM considered subsequently (Chapter 10) is a fully dynamic construct. It is also an equilibrium theory, though of a somewhat specialized nature. With the Risk-Neutral Valuation Model and the Arbitrage Pricing Theory (APT), taken up in Chapters 1214, we will move into the domain of arbitrage-based theories, after observing, however, that the ArrowDebreu pricing theory itself may also be interpreted from the arbitrage perspective (Chapter 11).
The ArrowDebreu model takes a more standard equilibrium view than the CAPM: it is explicit in stating that equilibrium means supply equals demand in every market. It is a very general theory accommodating production and, as already stated, very broad hypotheses on preferences. Moreover, no restriction on the distribution of returns is necessary. We will not, however, fully exploit the generality of the theory: In keeping with the objective of this text, we shall often limit ourselves to illustrating the theory with examples.
Intermediate Financial Theory.
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248 Chapter 9
ArrowDebreu modeling will be especially useful for equilibrium security pricing, especially the pricing of complex securities that pay returns in many different time periods and states of nature, such as common stocks or 30-year government coupon bonds. The theory will also enrich our understanding of project valuation because of the formal equivalence, underlined in Chapter 2, between a project and an asset. In so doing we will move beyond a pure equilibrium analysis and start using the concept of arbitrage. It is in the light of a set of no-arbitrage relationships that the ArrowDebreu pricing takes its full force. As noted earlier, the arbitrage perspective on the ArrowDebreu theory will be developed in Chapter 11.
9.2 Setting: An ArrowDebreu Economy
In the basic setting that we shall use, the following apply:
1. There are two dates: 0, 1. This setup, however, is fully generalizable to multiple periods; see discussion that follows.
2. There are N possible states of nature at date 1, which we index by θ 5 1,2,. . ., N with probabilities πθ.1
3. There is one perishable (nonstorable) consumption good.
4. There are K agents, indexed by k 5 1,. . ., K, with preferences:
U0kðck0Þ 1 δk
5. Agent k’s endowment is described by the vector fek0;ðekθÞθ51;2;…;Ng:
In this description, ckθ denotes agent k’s consumption of the sole consumption good in state θ, U is the real-valued utility representation of agent k’s period preferences, and δk is the agent’s time discount factor. In fact, the theory allows for more general preferences than the time-additive expected utility form. Specifically, we could adopt the following representation of preferences:
U k ð c k0 ; c kθ 1 ; c kθ 2 ; . . . ; c kθ N Þ :
This formulation allows not only for a different way of discounting the future (implicit in the relative taste for present consumption relative to all future consumption), but it also permits heterogeneous, subjective views on the state probabilities (again implicit in the
1 In the present chapter and in most of this text going forward, we assume that all agents hold the same (objective) probability beliefs. Such an assumption is most appropriate to a context where the economy’s probabilistic structure over endowments and states does not change, allowing agents to learn the true structure, revising their own beliefs accordingly. In Chapter 18, the topic of heterogeneous beliefs is considered.
πθUkðckθÞ;
ðPÞ max U0kðck0Þ 1 δk ð c k0 ; c k1 ; . . . ; c kN Þ
qθckθ # ek0 1 ck0;ck1;…;ckN $0
The first inequality constraint will typically hold with equality in a world of nonsatiation. That is, the total value of goods and security purchases made by the agent (the left-hand side of the inequality) will exhaust the total value of his endowments (the right-hand side).
Equilibrium for this economy is a set of contingent claim prices (q1, q2,. . ., qN) such that 1. at those prices ðck0 ; . . .; ckN Þ solve problem (P), for all k, and
2. PKk51 ck0 5 PKk51 ek0; PKk51 ckθ 5 PKk51 ekθ; for every θ:
Note that here the agents are solving for desired future and present consumption holdings rather than holdings of ArrowDebreu securities. This is justified because, as just noted, there is a one-to-one relationship between the amount consumed by an individual in a given state θ and his holdings of the ArrowDebreu security corresponding to that particular state θ, the latter being a promise to deliver one unit of the consumption good if that state occurs.
Note also that nothing in this formulation inherently restricts matters to two periods, if we define our notion of a state somewhat more richly, as a date-state pair. Consider three
2 So named after the originators of modern equilibrium theory: see Arrow (1951) and Debreu (1959).
ArrowDebreu Pricing, Part I 249
representation of relative preference for, say, ckθ2 vs ckθ3 ). In addition, it assumes neither time-additivity nor an expected utility representation. Since our main objective is not generality, we choose to work with the more restrictive but easier to manipulate time- additive expected utility form.
In this economy, the only traded securities are of the following type: One unit of security θ, with price qθ, pays one unit of consumption if state θ occurs and nothing otherwise.
Its payout can thus be summarized by a vector with all entries equal to zero except for column θ where the entry is 1: (0,. . .,0, 1, 0,. . .0). These primitive securities are called ArrowDebreu securities,2 or state-contingent claims or simply state claims. Of course, the consumption of any individual k if state θ occurs equals the number of units of security θ that he holds. This follows from the fact that buying the relevant contingent claim is the only way for a consumer to secure purchasing power at a future date-state (recall that the good is perishable). An agent’s decision problem can then be characterized by:
250 Chapter 9
t = 1 t= 2 θ12
The structure of an economy with two dates and (1 1 NJ) states.
periods, for example. There are N possible states in date 1 and J possible states in date 2, regardless of the state achieved in date 1. Define θ^ new states to be of the form θ^s 5 ð k; θkj Þ; where k denotes the state in date 1 and θjk denotes the state j in date 2, conditional that state k was observed in date 1 (refer to Figure 9.1). So ð1; θ51Þ would be a state and ð2; θ32Þ another state. Under this interpretation, the number of states expands to 1 1 NJ, with:
1: the date 0 state
N: the number of date 1 states J: the number of date 2 states
With minor modifications, we can thus accommodate many periods and states. In this sense, our model is fully general and can represent as complex an environment as we might desire. In this model, the real productive side of the economy is in the background. We are, in effect, viewing that part of the economy as invariant to securities trading. The unusual and unrealistic aspect of this economy is that all trades occur at t 5 0.3 We will relax this assumption in Chapter 10.
9.3 Competitive Equilibrium and Pareto Optimality Illustrated
Let us now develop an example. The essentials are found in Table 9.1.
There are two dates and, at the future date, two possible states of nature with probabilities 13
and 23: It is an exchange economy, and the issue is to share the existing endowments 3 Interestingly, this is less of a restriction for project valuation than for asset pricing.
Figure 9.1
ArrowDebreu Pricing, Part I 251 Table 9.1: Endowments and preferences in our reference example
Preferences
Agent 1 10 1 2 12 c01 10:913 lnc1123lnc21 Agent 2 5 4 6 12 c02 10:913 lnc12123lnc2
between two individuals. Their (identical) preferences are linear in date 0 consumption, with constant marginal utility equal to 12: This choice is made for ease of computation,
but great care must be exercised in interpreting the results obtained in such a simplified framework. Date 1 preferences are concave and identical. The discount factor is 0.9.
Let q1 be the price of a unit of consumption in date 1 state 1, and q2 the price of one unit of the consumption good in date 1 state 2. We will solve for optimal consumption directly, knowing that this will define the equilibrium holdings of the securities. The prices of these consumption goods coincide with the prices of the corresponding state-contingent claims; period 0 consumption is taken as the numeraire, and its price is normalized to 1. This means that all prices are expressed in units of period 0 consumption: q1, q2 are prices for the consumption good at date 1, in states 1 and 2, respectively, measured in units of date 0 consumption. They can thus be used to add up or compare units of consumption at different dates and in different states, making it possible to add different date cash flows, with the qi being the appropriate weights. This, in turn, permits computing an individual’s wealth. Thus, in the previous problem, agent 1’s wealth, which equals the present value of his current and future endowments, is 10 1 1q1 1 2q2, while agent 2’s wealth is 5 1 4q1 1 6q2.
Endowments
The respective agent problems are: Agent 1.
max 12 ð10 1 1q1 1 2q2 2 c1q1 2 c12q2Þ 1 0:9 13 lnðc1Þ 1 23 lnðc12Þ s:t: c1q1 1c12q2 #101q1 12q2; and c1;c12 $0
max12ð514q1 16q2 2c21q1 2c2q2Þ10:9 13lnðc21Þ1 23lnðc2Þ s:t: c21q1 1c2q2 #514q1 16q2 and c21;c2 $0
Note that in this formation, we have substituted out for the date 0 consumption; in other words, the first term in the max expression stands for 12 (c0), where we have substituted
252 Chapter 9
for c0 its value obtained from the constraint: c0 1 c1q1 1 c12q2 5 10 1 1q1 1 2q2: With this trick, the only constraints remaining are the nonnegativity constraints requiring consumption to be nonnegative in all date-states.
The FOCs state that the intertemporal rate of substitution between future (in either state) and present consumption (i.e., the ratio of the relevant marginal utilities) should equal the price ratio. The latter is effectively measured by the price of the ArrowDebreu security, the date 0 price of consumption being the numeraire. These first order conditions (FOCs) are (assuming interior solutions)
>c2:q1 50:9 1 1 < 1 2 3 c21
of the form ðqθ=1Þ 5 ðð0:9Þπθð1=ckθÞÞ=ð1=2Þ; k; θ 5 1; 2; or δπ @Uk
qθ5 @U0k ;k;θ51;2:
Together with the market-clearing conditions, Eq. (9.1) reveals the determinants of the
>c1:q1 50:9 1 1 < 1 2 3 c1
Agent 2:> ! >c2:q2 50:9 2 1
Agent 1:> ! >c1:q2 50:9 2 1
: 2 2 3 c2
: 2 2 3 c12
while the market-clearing conditions read: c1 1 c21 5 5 and c12 1 c2 5 8: Each of the FOCs is
equilibrium ArrowDebreu security prices. It is of the form:
Price of the good if state θ is realized MUθk
Price of the good today 5 MU0k :
In other words, the ratio of the price of the ArrowDebreu security to the price of the date 0 consumption good must equal (at an interior solution; see Box 9.1) the ratio of the marginal utility of consumption tomorrow if state θ is realized to the marginal utility of today’s consumption (the latter being constant at 12). This is the marginal rate of substitution (MRS) between the contingent consumption in state θ and today’s consumption. From this system of equations, one clearly obtains c1 5 c21 5 2:5 and c12 5 c2 5 4 from which one, in turn, derives:
q151ð0:9Þ 1 1 52ð0:9Þ 1 1 5ð0:9Þ 1 4 50:24 12 3 c1 3 2:5 3 5
q251ð0:9Þ 2 1 52ð0:9Þ 2 1 5ð0:9Þ 2 4 50:3 123c12 3438
ArrowDebreu Pricing, Part I 253
BOX 9.1 Interior Versus Corner Solutions
We have described the interior solution to the maximization problem. By that restriction we generally mean the following: the problem under maximization is constrained by the condition that consumption at all dates should be nonnegative. There is no interpretation given to a negative level of consumption, and, generally, even a zero consumption level is precluded. Indeed, when we make the assumption of a log utility function, the marginal utility at zero is infinity, meaning that by construction the agent will do all that is in his power to avoid that situation. Effectively, an equation such as Eq. (9.1) will never be satisfied for finite and nonzero prices with log utility and period one consumption level equal to zero; that is, it will never be optimal to select a zero consumption level. Such is not the case with the linear utility function assumed to prevail at date 0. Here it is conceivable that, no matter what, the marginal utility in either state at date 1 (the numerator in the RHS of Eq. (9.1)) will be larger than 12 times the ArrowDebreu price (the denominator of the RHS in Eq. (9.1) multiplied
by the state price). Intuitively, this would be a situation where the agent derives more utility from the good tomorrow than from consuming today, even when his consumption level today is zero. Fundamentally, the interior optimum is one where he would like to consume less than zero today to increase even further consumption tomorrow, something that is impossible. Thus the only solution is at a corner, that is, at the boundary of the feasible set, with c0k 5 0 and the condition in Eq. (9.1) taking the form of an inequality.
In the present case, we can argue that corner solutions cannot occur with regard to future consumption (because of the log utility assumption). The full and complete description of the FOCs for problem (P) spelled out in Section 9.2 is then
@Uk0 @Uk k
qθ @c0k #δπθ @cθk ; and if c0 .0; and k;θ51;2: (9.2)
In line with our goal of being as transparent as possible, we will often, in the sequel, satisfy ourselves with a description of interior solutions to optimizing problems, taking care to ascertain, ex-post, that the solutions do indeed occur at the interior of the choice set. This can be done in the present case by verifying that the optimal c0k is strictly positive for both agents at the interior solutions, so that Eq. (9.1) must indeed apply.
Notice how the ArrowDebreu state-contingent prices reflect probabilities, on the one hand, and marginal rates of substitution (taking the time discount factor into account and computed at consumption levels compatible with market clearing) and thus relative scarcities, on the other. The prices computed above differ in that they take account both of the different state probabilities 13 for state 1, 23 for state 2 and the differing marginal utilities as a result of the differing total quantities of the consumption good available in state 1 (5 units) and in state 2 (8 units). In our particular formulation, the total amount of goods available at date 0 is made irrelevant by the fact that date 0 marginal utility is constant. Note that if the date 1 marginal utilities were constant, as would be the case with
254 Chapter 9
Agent 1 Agent 2
9.04 2.5 4 5.96 2.5 4
15.00 5.0 8
Table 9.2: Post-trade equilibrium consumptions
linear (risk-neutral) utility functions, the goods endowments would not influence the ArrowDebreu prices, which would then be exactly proportional to the state probabilities.
The date 0 consumptions, at the equilibrium prices, are given by
c10 5 10 1 1ð0:24Þ 1 2ð0:3Þ 2 2:5ð0:24Þ 2 4ð0:3Þ 5 9:04
c20 5 5 1 4ð0:24Þ 1 6ð0:3Þ 2 2:5ð0:24Þ 2 4ð0:3Þ 5 5:96 The post-trade equilibrium consumptions are found in Table 9.2.
This allocation is the best each agent can achieve at the equilibrium prices q1 5 0.24 and q2 5 0.3. Furthermore, at those prices, supply equals demand in each market, in every state and time period. These are the characteristics of a (general) competitive equilibrium.
In light of this example, it is interesting to return to some of the concepts discussed in our introductory chapter. In particular, let us confirm the (Pareto) optimality of the allocation emerging from the competitive equilibrium. Indeed, we have assumed as many markets as there are states of nature, so assumption H1 is satisfied. We have
de facto assumed competitive behavior on the part of our two consumers (they have taken prices as given when solving their optimization problems), so H2 is satisfied. (Of course, in reality such behavior would not be privately optimal if indeed there were only two agents. Our example would not have changed materially had we assumed a large number of agents, but the notation would have become much more cumbersome.)
In order to guarantee the existence of an equilibrium, we need hypotheses H3 and H4 as well. H3 is satisfied in a weak form (no curvature in date 0 utility). Finally, ours is an exchange economy where H4 does not apply (or, if one prefers, it is trivially satisfied). Once the equilibrium is known to exist, as is the case here, H1 and H2 are sufficient to guarantee the optimality of the resulting allocation of resources. Thus, we expect to find that the above competitive allocation is Pareto optimal (PO); that is, it is impossible to
ArrowDebreu Pricing, Part I 255 rearrange the allocation of consumptions so that the utility of one agent is higher without
diminishing the utility of the other agent.
One way to verify the optimality of the competitive allocation is to establish the precise conditions that must be satisfied for an allocation to be PO in the exchange economy context of our example. It is intuitively clear that the above Pareto superior real allocations will be impossible if the initial allocation maximizes the weighted sum of the two agents’ utilities. That is, an allocation is optimal in our example if, for some weight λ it solves the following maximization problem.4
max U1c1; c1; c1 1 λU2c2; c2; c2 012 012
c1; c1; c1 012
c10 1c20 515; c1 1c21 55; c12 1c2 58; c10; c1; c12; c20; c21; c2 $ 0
This problem can be interpreted as the problem of a benevolent central planner constrained by an economy’s total endowment (15, 5, 8) and weighting the two agents utilities according to a parameter λ, possibly equal to 1. The decision variables at his disposal are the consumption levels of the two agents in the two dates and the two states. With Uik denoting the derivative of agent k’s utility function with respect to cki ði 5 1; 2; 3Þ; the FOCs for an interior solution to the above problem are found in Eq. (9.3).
U01 U1 U21
U02 5U12 5U2 5λ (9.3)
This condition states that, in a PO allocation, the ratio of the two agents’ marginal utilities with respect to the three goods (i.e., the consumption good at date 0, the consumption good at date 1 if state 1, and the consumption good at date 1 if state 2) should be identical.5 In an exchange economy this condition, properly extended to take account of the possibility of a corner solution, together with the condition that the agents’ consumption adds up to the endowment in each date-state, is necessary and
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