Asset Prices in an Exchange Economy

Lucas sets out to give a theoretical account of how equilibrium asset prices behave in a “one-good, pure exchange economy with identical consumers,” in which a single perishable good is produced costlessly on a fixed number of productive units whose output fluctuates stochastically, and to understand “the relationship between these exogenously determined productivity changes and market determined movements in asset prices” (Introduction, p. 1429). The economy has a single representative consumer standing in for many identical consumers, maximizing expected discounted utility from consumption of the one good; ownership of each productive unit is represented by one perfectly divisible equity share, traded each period after dividends are paid at a competitively determined price (Section 2, pp. 1429-1431). Since all output must be consumed and all shares must be held in equilibrium, “the main analytical issue… will be the determination of equilibrium price behavior” (p. 1431).

The paper’s central analytical device is a functional equation for the vector of equilibrium asset prices as a function of the current state of the economy (Introduction, p. 1429). Formally, Lucas shows that if consumers value end-of-period portfolios using the correct (equilibrium) value function, and prices are set to clear the market each period, the resulting price function must satisfy a stochastic Euler equation (equation 6, p. 1434): the marginal utility of current consumption times today’s price of an asset equals the expected discounted value of next period’s marginal utility times next period’s payoff (dividend plus resale price). This “equat[es] the marginal rate of substitution of current for future consumption to the market rate of transformation” implicit in each security’s return (p. 1434). Lucas proves (Propositions 1-3, pp. 1431-1435) that this equation, reduced to a system of n independent functional equations, has exactly one continuous, bounded solution, obtainable as the limit of a contraction-mapping (successive-approximation) iteration.

An equilibrium consists of a continuous price function p(y) and a continuous, bounded value function v(z, y) such that the value function solves the consumer’s dynamic optimization problem given prices p(y), and, at the solution, consumers choose to hold exactly one share of each asset and consume exactly aggregate output (Section 3, Definition, p. 1432). Condition (i) requires that the consumer allocate resources optimally between current consumption and end-of-period share holdings given prices; condition (ii) requires that this optimal choice be market-clearing (aggregate consumption equals aggregate output, and share holdings equal one) for every possible state y (p. 1432). Lucas notes this is, in substance if not by explicit proof, a standard Arrow-Debreu equilibrium in which the commodity space is the space of possible realizations of aggregate output (p. 1432, fn. 4).

Section 5 develops a second, dynamic-programming characterization of the identical equilibrium: a functional equation (equation 10) in which an agent’s end-of-period portfolio is valued via a function r(z,y) satisfying its own Bellman-type recursion, and Lucas shows (Propositions 6-8) that the price function which attains the right side of this dynamic program is exactly the equilibrium price function derived in Section 4. This second construction “is slightly more general” than the first because it “does not require differentiability of U,” and, crucially, “it is also suggestive for stability theory” (p. 1436) – it is this dual, dynamic-programming form that Section 6’s stability argument is built on.

Lucas argues that if agents value their asset holdings using some arbitrary (not necessarily correct) continuous, concave, increasing valuation function u instead of the true equilibrium value function v, the market-clearing price this generates gives agents a realized utility experience described by the operator (Mu)(z,y); if agents then replace u with Mu, and Mu with M²u, and so on, the sequence of prices converges to the true equilibrium price function, because M is a contraction mapping (Section 6, pp. 1437-1438). This convergence does not require that agents “be familiar with the theory of Markov processes or of dynamic programming,” nor that they be skilled at articulating their expectations; it requires only that they have consistent preferences and revise their valuation of asset holdings in the direction of the consumption utility those holdings actually yielded (p. 1438). Lucas is careful to frame this as a theoretical argument for expecting rational expectations to be “a good approximation to behavior,” not as a claim that any specific successive approximation describes actually observed adjustment dynamics (p. 1438).

In the linear-utility case, the price of each asset reduces to the standard present-value formula – the expected, discounted sum of the asset’s future dividend stream conditional on current information (equation 14, Section 7.1, p. 1439). In the one-asset case with more general (concave) utility, Lucas derives (equation 20, Section 7.2, pp. 1440-1441) that the elasticity of the equilibrium price with respect to current income decomposes into two terms: a positive “income effect” equal to the Arrow-Pratt measure of relative risk aversion (agents facing a high-income period try to smooth the windfall forward via securities purchases, which – since storage is impossible – is frustrated by a rise in asset prices), and an “information effect” whose sign follows the informativeness of current output about future output. Lucas concludes explicitly that “the relationship of asset prices to real output is far from simple and possibly not even monotonic,” suggesting it may be “good judgment, not merely timidity,” that has kept aggregate theorists from trying to fully “understand the market” (p. 1441).

Section 7.3 shows that when there are many, sufficiently independent productive units, replacing each unit’s exact marginal-utility term with one based on mean aggregate output yields a price function that approximates the true equilibrium arbitrarily well as the number of independent units grows (pp. 1441-1443). This is established via a general contraction-mapping approximation lemma (Lemma 2) bounding the distance between the fixed points of two contractions by the sup-norm distance between the operators themselves; applied to the asset-pricing setting, the approximation error is shown to shrink toward zero as the variance of each unit’s output conditional on its own history, and its covariance with the outputs of others, both shrink (p. 1443).

Lucas shows that raw equilibrium asset prices in this economy generally do not have the Martingale property; instead, it is the specific marginal-utility-weighted series w_it – built from U’(aggregate consumption) times (dividend plus price) – that is a Martingale, since equation (6) guarantees its conditional expectation is zero (Section 8, equation 22, pp. 1443-1444). Lucas notes this weighted series only closely approximates a Martingale in the raw prices themselves when marginal utility of aggregate consumption barely varies – either because agents are close to risk-neutral or because aggregate risk is small – and even then a further correction for the discount factor is needed; “neither rationale… seems likely to closely approximate reality” (p. 1444). The paper’s stated contribution here is not the general point (already stressed by Fama and other efficient-market theorists) that market equilibrium conditions should be stated in terms of expected returns, but “an explicit framework within which one can judge what this requirement means and whether or not it is satisfied” – within which “the presence of a diminishing marginal rate of substitution of future for current consumption is inconsistent with” the raw-price Martingale property (p. 1444).

Lucas argues that because rigorous, fully rational economic models can be constructed both with and without the Martingale property in raw prices, “the outcomes of tests as to whether actual price series have the Martingale property do not in themselves shed light on the generally posed issue of market ’efficiency’” (Section 9, pp. 1444-1445). He frames the paper as “primarily methodological: an illustration of the use of some methods which may help to bring financial and economic theories closer together” (p. 1444), rather than as an empirical claim about any particular market.

Lucas is explicit that the time-additive preference structure used throughout “is… a nuisance, and it has no rationale beyond tractability,” suggesting it could be replaced with recursive but non-additive (Koopmans-Diamond-Williamson-type) preferences given sufficient “impatience” (Conclusions, p. 1444). He also flags that introducing capital accumulation is a natural next step, but cautions that the marginal, Euler-equation-based analysis of Section 4 “is probably a dead-end” for that extension, since stochastic Euler equations are not generally tractable once capital enters non-trivially; he suggests the dynamic-programming formulation of Section 5 “appears more promising” for such extensions (p. 1444). The model itself assumes production is entirely exogenous (no investment or resource allocation choices affect output), a single consumption good, and a representative-consumer device that requires treating all agents as making the same valuation errors during the out-of-equilibrium adjustment discussed in Section 6 (pp. 1429-1430, 1436-1437).