Economic behavior under temporal uncertainty.

• Author: Chavas, Jean-Paul

1. Introduction

   A considerable amount of research has focused on the effects of uncertainty on economic behavior. In the absence of complete risk markets, private risk bearing is widely believed to influence economic decisions.(1) Arrow [1], Sandmo [23], Chavas [4], Dalal [6] and others have investigated the role of uncertainty in resource allocation. For example, Sandmo [23] has proposed a theory of the firm under risk. He showed how risk and risk aversion can have a negative effect on production decisions.

   While this research has provided useful insights in the economics of risk, it has focused in a large part on "timeless risk," where all decisions are assumed to be made ex ante, i.e., before the resolution of uncertainty. It neglects dynamic situations characterized by "temporal risk," where decisions are made over time and where learning takes place from one time period to the next. Mossin [21] and Dreze and Modigliani [8] have argued that temporal uncertainty is the rule in most situations of choice under uncertainty. For example, optimal inventory decisions depend on future market conditions and are revised over time as new information becomes available. Also, investment decisions are influenced by the uncertain future return from capital (depending on future technological possibilities as well as market conditions). Information about this future uncertainty as well as its temporal resolution affects the optimal path of capital. For instance, the presence of sunk cost tends to reduce the flexibility of the firm to respond to new information, which makes investment decisions somewhat irreversible and provides an incentive to delay capital accumulation [22]. Finally, current consumption decisions are made before future prices are known, which typically influence current and future consumption possibilities through the intertemporal budget constraint. In all these examples, learning occurs with the passage of time, either passively or actively as a direct result of decisions in previous periods. Future prices or rates of return on investment eventually become observable. In situations of choice under temporal uncertainty, the flexibility to adjust to new information over time in general affects future as well as current optimal decisions.

   A number of studies have investigated various aspects of dynamic economic behavior under temporal uncertainty, such as production flexibility and quasi-fixed input choice [10; 12; 14; 25; 27], irreversible choices [2], the demand of information [13], the characterization of dynamic preferences [15; 16; 17], and consumer behavior [10; 11]. However, in the absence of complete contingent claim markets, this literature does not provide a general characterization of economic choices under temporal uncertainty and risk aversion (e.g., the envelope and symmetry results analogous to Shephard's/Hotelling's lemma and the Slutsky matrix of traditional demand theory). Such a characterization is important because it provides the foundations for the empirical investigation of economic behavior and welfare analysis under temporal uncertainty.

   The objective of this paper is to explore further the implications of temporal uncertainty for economic behavior under general risk preferences. In section II, a primal two-period decision problem is defined for a competitive agent under the expected utility hypothesis. Decisions in period one are subject to uncertainty over future prices and other random variables. The random variables are observed at the beginning of period two, before period-two choices are made. The model allows for learning and the flexibility to respond to new information--the key features of temporal uncertainty--under risk aversion. In this context, a compensation function dual to the primal decision problem is defined. In section III, this compensation function and a location-scale representation of price uncertainty(2) are used to derive envelope results and an intertemporal Slutsky matrix that characterizes ex ante compensated and uncompensated choice functions. Implications of the results for welfare analysis are presented in section IV.

2. The Model

   Consider a competitive agent facing a two-period problem, t = 1, 2. Decisions are made each period. The period-one decisions are denoted by (z, [x.sub.1]), where z is a scalar representing the numeraire good, and [x.sub.1] [is greater than or equal to] is a ([n.sub.1] x 1) vector. The period-two decisions are denoted by [x.sub.2], where [x.sub.2] [is greater than or equal to] is a ([n.sub.2] x 1) vector. The goods (z,[x.sub.1] [x.sub.2]) are purchased or sold on competitive markets. Let [r.sub.t] be the ([n.sub.1] x 1) vector of market prices for [x.sub.t], t = 1, 2. The good z being the numeraire has a unit price.(3) Throughout the paper, we will use the convention that prices are positive for purchases and negative for sales made by the agent.

   The period-one decisions are made subject to future uncertainty, as represented by a random vector e. We assume that, at time t = 1, the agent has some given subjective probability distribution about e. Learning takes place over time through the observation of the realized values of the random vector e. These observations are made at the beginning of period two, i.e., before the decisions [x.sub.2] are made. As a result, the period-two decisions are made knowing the actual value taken by the uncertain variables e. The random variables e can be partitioned as e = ([e.sub.a], [e.sub.b], [e.sub.c]) to represent three possible sources of uncertainty: 1/price uncertainty ([e.sub.a]); 2/technological uncertainty facing the agent ([e.sub.b]); and 3/preference uncertainty ([e.sub.c]). Assuming that the prices [r.sub.1] of [x.sub.1] are known in period one, the price uncertainty concerns the prices [r.sub.2] denoted by [r.sub.2]([e.sub.a]). This reflects the situation where the prices [r.sub.2]([e.sub.a]) are uncertain in period one, but become known in period two. Let the technology facing the agent be represented by the feasible sets [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. The technology constraints then take the form [x.sub.1] [is an element of] [T.sub.i] and [x.sub.2] [is an element of] [T.sub.2]([x.sub.1], [e.sub.b]). The feasible set [T.sub.2]([x.sub.1], [e.sub.b]) depends on [x.sub.1] to reflect technological dynamics (e.g., the dynamics of capital accumulation).(4) It also depends on e, indicating that technology [T.sub.2] can be uncertain in period one, but becomes known in period two through learning. Finally, let the intertemporal preference function of the agent be denoted by the von Neumann-Morgenstern utility function U([x.sub.1], [x.sub.2], z, [e.sub.c]). The effect of [e.sub.c] on U([center dot]) reflects the possibility that the agent's tastes and preferences can change in ways that are not predictable ahead of time. It indicates that the period-two preferences may not be known with certainty in period one.

   The Primal Problem

   Given this general characterization of uncertainty, we assume that the agent behaves in a way consistent with the expected utility hypothesis(5) and that the utility function U([center dot]) is quasiconcave and increasing in [x.sub.1], [x.sub.2], and z. Then, the objective of the agent is to make decisions so as to maximize expected utility subject to technological constraints and a budget constraint. Let w denote the agent's initial wealth. The intertemporal budget constraint takes the form: z + [r[prime].sub.1][x.sub.1] + [r.sub.2]([e.sub.c])[prime][x.sub.2] [is less than or equal to] w.(6) Then, the agent's primal problem is::(7)

   [Mathematical Expression Omitted],

   where E is the expectation operator conditional on all information available at the beginning of period one, i.e., based on the subjective joint distribution of e. The intertemporal budget constraint ([r[prime].sub.1][x.sub.1] + [r.sub.2](e)[prime][x.sub.2] + z [is less than or equal to] w) and the technological constraints {[x.sub.1] [is an element of] [T.sub.1]; [x.sub.2] [is an element of] [T.sub.2]([x.sub.1], e)} define the feasible choice set. Through the utility function, the budget constraint and the feasible set [T.sub.2]([x.sub.1], e), the period-two choices are influenced by the realized value of the random variables e and by the period-one decisions [x.sub.1]. This reflects the fact that the period-one decisions [x.sub.1] (e.g., quasi-fixed inputs or consumption levels) can alter future benefits as well as technological and budget possibilities. Also, uncertainty about future price and rate of return makes the budget constraint uncertain in period one. And future technological constraints can be uncertain due to weather effects, random capital depreciation, or new technologies. The linkage between period-one choices and either period-two budget or technological constraints (reflecting possible irreversibilities) represents the dynamics characterizing most economic decisions under temporal uncertainty and learning.

   Assuming non-satiation with respect to z, the budget constraint is necessarily binding. It can be solved for the numeraire good z = w - [r[prime].sub.1][x.sub.1] - [r.sub.2](e)[prime][x.sub.2] and substituted into the utility function to give:

   [Mathematical Expression Omitted].

   V(w,[center dot]) is the agent's indirect objective function, where "[center dot]" represents all other relevant parameters (such as [r.sub.1] and the distribution of e). The first formulation of the problem in (2) is in extensive form (based on a dynamic programming approach), while the second formulation is in normal or strategic form.

   Let [x*.sub.1](w,[center dot]) and [x*.sub.2](w, e,[center dot]) denote the uncompensated choice functions obtained as the solution of the primal problem (2). Here [x*.sub.1](w,[center dot]) is the vector of ex ante choices made in period one, while [x*.sub.2](w, e,[center dot]) is the vector of ex post decisions...