Economic behavior under temporal uncertainty.

I. 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.

II. 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 made in period two. Period-two decisions depend on the observed value of the random variables e. The goal here is to identify the characteristics of the uncompensated demands x* = ([x*.sub.1], [x*.sub.2]) associated with the solution to the primal problem (2).

While there are many intertemporal decision problems that could be used to conduct the analysis, our model is simple yet general enough to illustrate the basic method and results. With the appropriate redefinition of variables, the above framework can be used to analyze a variety of production, consumption, and investment problems.(8) Also, note that the generalization from a two-period framework to a T-period framework is straightforward. To see that, consider redefining the variable [x.sub.2] in model (1) as the vector of all future decisions at time t = 2, 3, . . . T. Furthermore, consider restricting [x.sub.2](e) in the normal form of equation (1) to be a function only of the information available at the time of each decision. For example, the [x.sub.2] decisions made at time t would be a function only of the states observable before time t, but not of the states that become observable after time t. This would allow for sequential learning over time. Given this interpretation along with a process for updating expectations, the results in this paper will hold as well when the agent's planning horizon becomes longer. In other words, our two-period model presents the essence of the behavioral implications of dynamic economic theory under temporal uncertainty.(9) Finally, it includes as special cases both the riskless case and the case of "timeless risk" mentioned in the introduction. The riskless case corresponds to the situation where the distribution of the random variables e is degenerate. And the "timeless risk" case can be obtained under two alternative assumptions: 1/either there is no learning taking place between period one and period two (in which case the random variables e would be observed only after t = 2); or 2/there is no flexibility to respond to new information after the period-one decisions are made (as implicitly assumed in Arrow [1], Sandmo [23], Chavas [3], Dalal [6] and others). These special cases should be kept in mind for the interpretation and significance of the results obtained in this paper since they will indicate how temporal uncertainty can influence the validity of basic tools commonly used in economic and welfare analysis.

A Dual Problem

Assuming that the utility function is strictly increasing in the numeraire good z, define the inverse function C([U.sub.0], [x.sub.1], [x.sub.2],[center dot]) as follows:

C([U.sup.0], [x.sub.1], [x.sub.2],[center dot]) = {C : [U.sup.0] = EU([x.sub.1],[x.sub.2], w - [r[prime].sub.1][x.sub.1] - [r.sub.2](e)[prime][x.sub.2] + C,e)}, (3)

where the inverse function C([U.sup.0], [x.sub.1], [x.sub.2],[centerdot]) is interpreted as the income compensation necessary to obtain an expected utility level of [U.sup.0] given any [x.sub.1] and [x.sub.2].

Given the inverse function defined in (3), the indirect compensation function for the dual problem is defined in normal (or strategic) form as:

[Mathematical Expression Omitted].

The compensated choice functions for the dual problem (4) are denoted by [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. Here [Mathematical Expression Omitted] is the vector of ex ante compensated choices made in period one, and [Mathematical Expression Omitted] is the vector of ex post compensated decisions made in period two based on the observed value of the random variables e. The compensated demands functions thus can be written as [Mathematical Expression Omitted].

The indirect compensation function C* in (4) defines the sure amount of money the agent must receive (or pay if negative) in order to remain at the reference level of expected utility [U.sup.0]. Thus, C*(V(w,[center dot]),[center dot]) = 0, and using (3) and (4) the following relationships exist between the compensated and uncompensated choices functions:(10)

[Mathematical Expression Omitted].

Based on this primal-dual formulation of the decision problem under temporal uncertainty, two main areas of theoretical interest lie in the general properties of the compensation function C* and in the impact of ceteris paribus changes in prices on economic behavior ([x*.sub.1], [x*.sub.2]). The properties of C* are important because the compensation function provides the basis for defining ex ante compensation tests commonly used in applied welfare economics (see section IV below). Also, understanding the response of the uncompensated and compensated demands x* and [x.sup.c] to changes in prices [r.sub.1] or [r.sub.2](e) is crucial in the analysis of the impacts of price policies and government regulations on technology, investment choices or savings behavior.

Note that the indirect compensation function C* depends on the probability distribution of the random vector e. The analytical tractability of the function C* for all distribution functions of e is likely to be difficult under general conditions. In order to obtain insights into questions of irreversibility, learning, and flexibility, it will be useful to develop a more refined characterization of the indirect compensation function under temporal uncertainty. This task is presented in the following section in the context of an explicit characterization of the agent's expectations about the future random price variables [r.sub.2](e).

III. Properties of the Indirect Compensation Function C*([U.sup.0],[center dot])

The following formulation of future price expectations will be used in this section:

[Mathematical Expression Omitted];

where [[Epsilon].sub.i](e) is a random variable with mean zero, [Mathematical Expression Omitted] is the ith mean price (a "location" parameter), and [[Sigma].sub.i] [is greater than] 0 is a mean-preserving spread for the ith price (a "scale" parameter). For notational purposes, let [Mathematical Expression Omitted], [Sigma] = {[[Sigma].sub.i]; i = 1, . . . , [n.sub.2]} and [Epsilon] = {[[Epsilon].sub.i]; i = 1, . . . , [n.sub.2]}.

The above characterization of uncertainty in terms of location parameters [Mathematical Expression Omitted] and scale parameters [[Sigma].sub.i] has been analyzed by Meyer [19]. Meyer [19] has shown that this location-scale characterization does not impose any restriction on the nature of risk preferences. Also, it allows any general joint distribution of the [Epsilon]'s.(11) Finally, it has the advantage of being empirically tractable. As such, it appears to provide a reasonable framework for the analysis of the implications of temporal uncertainty for economic behavior.(12)

Given the above assumption on price expectations, the characteristics of the indirect compensation function in terms of parameters ([r.sub.1], [Mathematical Expression Omitted], [Sigma]) are summarized in the following proposition (see the proof in the Appendix).

PROPOSITION 1. The indirect compensation function C*([U.sup.0],[center dot]) has the following properties:

(i) linear decreasing in w with [Delta]C*/[Delta]w = -1;

(ii) nondecreasing and concave in [r.sub.1], [Mathematical Expression Omitted].

If C* is differentiable, then

(iii) [Mathematical Expression Omitted]

(iv) [Mathematical Expression Omitted]

(v) [Mathematical Expression Omitted],

where [U.sub.w] [is equivalent to] [Delta]U([center dot])[Delta]w denotes the marginal utility of income, [Mathematical Expression Omitted], and COV(.,.) denotes a covariance.

From Proposition 1, characteristic (i) shows that initial wealth is a perfect substitute for compensation. Characteristic (ii) shows that the indirect compensation function is nondecreasing and concave in period-one prices and expected period-two prices. Conditions (iii)-(iv) are intertemporal analogues to Shephard's lemma with respect to [r.sub.1] and [Mathematical Expression Omitted]. From (iii), derivatives of C* with respect to [r.sub.1] give the period-one compensated demands directly, which is a well known result in the absence of temporal uncertainty. From (iv), the derivative of C* with respect to [Mathematical Expression Omitted] involves the period-one expected value of the compensated choice in period two, plus the covariance of marginal utility of income with [Mathematical Expression Omitted] divided by the expected marginal utility of income.

Thus, Proposition 1 shows that, under temporal uncertainty, the derivative of the indirect compensation function C* with respect to an expected future price [Mathematical Expression Omitted] is in general not equal to the expected value of future decisions [Mathematical Expression Omitted]; this derivative involves as well the covariance term COV([U.sub.w],[Mathematical Expression Omitted])/E[U.sub.w]. In other words, the covariance between the marginal utility of income and future decisions influences the effect of the expected price [Mathematical Expression Omitted] on the compensation function C*.(13)

Finally, Proposition 1 indicates how price uncertainty (as measured by the mean preserving spread parameters [Sigma]) influences the compensation function C*. Note that the covariance terms in (iv) and (v) can vanish under certain conditions. For example, all the covariance terms are zero in the absence of learning, in which case the optimal [x.sub.2] decisions do not depend on e. All the covariance terms are also zero in the absence of temporal flexibility. This corresponds to the case of "timeless risk" mentioned in the introduction, when the period-two decisions do not respond to new information. Finally, the covariance term vanishes under risk neutrality, where [U.sub.w] is a constant. This suggests that risk neutrality would greatly simplify the analysis of intertemporal decision marking under temporal uncertainty. However, there is strong empirical evidence of risk averse behavior in the literature. Given that learning typically characterizes any decision making process over time, the covariance terms in Proposition 1.iv and 1.v can thus be expected to be non-zero in general.

From the concavity of the function C* in ([r.sub.1], [Mathematical Expression Omitted]) stated in Proposition 1.ii, it follows that the hessian matrix [Mathematical Expression Omitted] is symmetric and negative semidefinite, where:

[Mathematical Expression Omitted].

In (6), the hessian of C*([U.sup.0],[center dot]) with respect to current prices [r.sub.1] is equal to the partial derivative of the compensated choice functions [Mathematical Expression Omitted], implying that compensated demands [Mathematical Expression Omitted] are downward sloping with respect to period-one prices. This is the standard result obtained in a static context. In contrast. (6) shows that the elements of the Hessian of C*([U.sup.0],[center dot]) with respect to [Mathematical Expression Omitted] is in general not equal to the partial derivative of the expected compensated choice functions [Mathematical Expression Omitted], due to the covariance term COV([Mathematical Expression Omitted]) in Proposition 1.iv. As noted above, this covariance term can be expected to be non-zero under risk aversion and temporal uncertainty. In other words, expected compensated demands [Mathematical Expression Omitted] are not necessarily downward sloping in expected period-two prices. Thus, from Proposition 1 and the hessian matrix in (6), risk aversion and learning have significant implications for the properties of the compensation function C* in terms of the location-scale parameters.

Useful results are obtained based on the relationship between the compensated and uncompensated choice functions given in (5). Differentiating equations (5) with respect to prices ([r.sub.1], [Mathematical Expression Omitted]) and using Proposition 1.iii and 1.iv give the following dynamic Slutsky matrix:

[Mathematical Expression Omitted].

Note that a fundamental implication of temporal uncertainty is that the matrix of compensated slopes in (7) is in general not symmetric nor negative semidefinite. The reason is that, although C* is concave in [[r.sub.1], [Mathematical Expression Omitted]] from Proposition 1.ii, the equivalence of Shephard's lemma does not apply with respect to [Mathematical Expression Omitted] from Proposition 1.iv.

The non-symmetry of the Slutsky matrix (7) does not imply the absence of symmetry restrictions on the uncompensated demands. Such symmetry restrictions just take a different form. To see that, consider the negative semi-definite Hessian of the compensation function C* in (6). Using expression (7), it implies that the matrix

[Mathematical Expression Omitted]

is symmetric and negative semidefinite.

The matrix in (8) shows how temporal uncertainty affects the symmetry conditions of optimal choice functions. While the upper left comer of the matrix in (8) is the traditional symmetry of compensated Slutsky price effects in the first period, the rest of the matrix in (8) takes a more complex form. Note that, under risk neutrality (where [U.sub.w] is a constant), then all the covariance terms vanish and (8) reduces to a Slutsky matrix where all random variables (i.e., [r.sub.2] and [Mathematical Expression Omitted]) are replaced by their expected value. Thus, either risk neutrality or the absence of temporal uncertainty is a sufficient condition to imply the symmetry of the slopes of compensated (expected) choice functions with respect to expected prices. In other words, the classical Slutsky symmetry restrictions (which are well known under certainty) remain valid under risk neutrality. However, the conditions for this validity are rather restrictive. As noted above, under learning and risk aversion, the covariance terms in (8) do not vanish and the classical symmetry restrictions take a different form.

These results are of interest. They indicate how temporal uncertainty and attitudes toward risk affect the behavioral implications of economic theory. Proposition 1 shows that the theoretical implications for [Mathematical Expression Omitted] and [Mathematical Expression Omitted] in general depend on risk preferences (through the covariance terms). Thus, a number of basic tools of economic analysis are not robust under temporal uncertainty and risk aversion, and empirical investigations that neglect temporal uncertainty may in general be inappropriate. This point has some troublesome implications for empirical analyses of dynamic choices under temporal uncertainty. It suggests that the empirical testing of the theory in a location-scale framework requires knowledge of risk preferences and of the nature of temporal uncertainty. Such knowledge would allow the evaluation of the covariance terms in Proposition 1 and thus the empirical testing of the theory.

The Special Case of Risk Neutrality

As argued above, risk neutrality (where [U.sub.w] is a constant) is a sufficient (although restrictive) condition for COV([U.sub.w], [x*.sub.2]) = 0. Under this condition, the above results simplify, as stated in the following corollary.

COROLLARY 1. Under risk neutrality, results (iv) and (v) of Proposition 1 become

(i) [Mathematical Expression Omitted]

(ii) [Mathematical Expression Omitted].

Corollary 1 states that risk neutrality is a sufficient condition for Shephard's lemma to hold with respect to expected prices and expected quantities. It also shows that how price uncertainty (as measured by the mean preserving spread parameters [Sigma]) influences the compensation function C* under risk neutrality.

These results can be easily related to previous comparative static analyses of choice under temporal uncertainty [12; 14; 27]. For example, when a firm maximizes expected profits, all the wealth effects are zero and marginal utility of wealth is constant. Then, equations (6) and (7) along with Corollary 1 imply that the following matrix

[Mathematical Expression Omitted]

is symmetric.(14) The symmetry restrictions in (9) show how optimal choices respond to changing (expected) prices and price variability under risk neutrality. From the symmetry of (9), [Mathematical Expression Omitted]. This is the well known symmetry of cross-price effects as obtained in a riskless framework. This shows that the assumption of risk neutrality preserves this static result (using expected values) in a dynamic risky situation with learning. Also, from the symmetry of (9), [Delta][x*.sub.1]/[Delta][Sigma] = COV([Delta][x*.sub.2]/[Delta][r.sub.1], [Epsilon]). This illustrates that the period-one response to price instability involves the second-order properties of the period-two demands (which are in turn third-order properties of the second period optimal value function). As analyzed in Wright [27], these effects depend on the direction and rotation of shifts in the period-two demands. Thus, in the presence of temporal uncertainty, even risk neutral firms respond to changes in expected price variability. Finally, from the symmetry of (9), [Mathematical Expression Omitted]. This indicates that the impact of price uncertainty on expected period-two choices depends on the covariance between the period-two price slope and the period-two price.

IV. Implications for Welfare Analysis

We have just explored how temporal uncertainty and risk aversion affect economic behavior. In this section, we turn our attention to the role of temporal uncertainty in welfare analysis. The compensation function C* defined in (4) provides the basis for conducting compensation tests. Such tests are the basic tools of applied welfare analysis [20]. We focus here on evaluating the effects of some pricing policy that affects current as well as future prices. Let [Gamma] denote some parameter of the price vector r = ([r.sub.1], [r.sub.2]). In the context of the location-scale characterization proposed in preceding section (where [Mathematical Expression Omitted]), we consider the case where a pricing policy influences the vector [Mathematical Expression Omitted]. Then, investigating the welfare effects of the pricing policy becomes equivalent to examining the effects of a change in [Gamma], say from [[Gamma].sub.0] to [[Gamma].sub.1], on the compensation function C*. More specifically, one wants to evaluate the term M = C*([U.sup.0], [[Gamma].sub.1], [center dot]) - C*([U.sup.0], [[Gamma].sub.0], [center dot]).

If [U.sup.0] is the utility level obtained in the original situation ([[Gamma].sub.0]), then C*([U.sup.0], [[Gamma].sub.0], [center dot]) = 0 and M is the compensating variation. It measures the sure amount of money the agent is willing to receive (or pay if negative) to be indifferent to the change from [[Gamma].sub.0] to [[Gamma].sub.1], using [[Gamma].sub.0] as a reference situation. The compensating variation measure has been commonly used in the formulation of Pareto improvement tests in welfare analysis. For example, it provides the basis for the Kaldor-Hicks criterion [20]. However, compensating variation measures have been criticized because they may not give a transitive ranking when concerned with more than two situations [5].

Alternatively, if [U.sup.0] is the utility level obtained in the subsequent situation ([[Gamma].sub.1]), then C*([U.sup.0], [[Gamma].sub.1], [center dot]) = 0 and -M is the equivalent variation. Using [[Gamma].sub.1] as a reference situation, -M measures the sure amount of money the agent must receive (or pay if negative) to give up the change from [[Gamma].sub.0] to [[Gamma].sub.1] and be as well off. The equivalent variation measure has also been commonly used in welfare analysis. For example, it is the basis for the formulation of the Scitovsky reversal test in benefit-cost analysis [20].

The issue then is the empirical measurement of M = C*([U.sup.0], [[Gamma].sub.1], [center dot]) - C*([U.sup.0], [[Gamma].sub.0], [center dot]). Under differentiability, note that

M = C*([U.sup.0], [[Gamma].sup.1], [center dot]) - C*([U.sup.0], [[Gamma].sup.0], [center dot]) = [integral of] [Delta]C*([U.sup.0], [Gamma], [center dot])/[Delta][Gamma]d[Gamma] between limits [[Gamma].sub.1] and [[Gamma].sub.0]. (10)

Equation (10) is of particular interest given the results of Proposition 1. We consider the welfare effects of changing each element of the vector ([r.sub.1], [Mathematical Expression Omitted], [Sigma]). First, consider the case where [Gamma] = [r.sub.1]. Using Proposition 1.iii, equation (9) becomes

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the compensated choice function defined in the previous section. Expression (11) implies that welfare change due to a shift in current prices is given by the "agent's surplus" or welfare triangle measured between the compensated choice function [Mathematical Expression Omitted] and the two prices [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. This well known result, also obtained in the absence of temporal uncertainty, has provided the basis for much applied welfare analysis [20].

Second, consider the case where [Mathematical Expression Omitted]. Using Proposition 1.iv, equation (10) becomes

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the compensated choice function defined in the previous section, and [Mathematical Expression Omitted].

Expression (12), which gives the welfare change due to a shift in future expected prices, shows that the traditional welfare triangle is not an appropriate welfare measure in the presence of temporal uncertainty and risk aversion. The first term on the right-hand side of (12) is the "agent's surplus" or welfare triangle as traditionally measured between the expected compensated choice function [Mathematical Expression Omitted] and the two expected prices [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. The second term on the right hand side of (12), which involves the covariance COV([U.sub.w], [Mathematical Expression Omitted]), can be interpreted as the bias of the traditional welfare triangle. As discussed in the previous section, this bias vanishes under risk neutrality or in the absence of learning or flexibility. Under such conditions, the traditional welfare triangle between the expected compensated choice function [Mathematical Expression Omitted] and the two expected prices [Mathematical Expression Omitted] and [Mathematical Expression Omitted] provides an exact welfare measure of the effects of a change in an expected future price. However, under learning and risk aversion, tle traditional welfare triangle can be expected to give a biased measure of welfare change because of a non-zero covariance COV([U.sub.w], [Mathematical Expression Omitted]) in (12). The magnitude of this bias appears to be largely an empirical issue. However, if it were found to be relatively large, our results could raise serious questions about the validity of much applied welfare analysis in the literature.

Third, consider the case where [Gamma] = [Sigma]. Using Proposition 1.v, equation (10) becomes

[Mathematical Expression Omitted].

Expression (13) provides a measure of the welfare change due to a shift in the scale parameter [Sigma], for example due to a price stabilization policy. The first term on the right-hand side of (13) reflects the role of learning while the second term reflects the role of risk preferences in the welfare analysis of temporal price uncertainty. Both terms vanish in the absence of learning, while the second term vanishes under risk neutrality (where [U.sub.w], is a constant). Under risk aversion and/or learning, equation (13) suggests that any open-loop certainty-equivalent approach will tend to provide systematic biases in welfare analysis. This strongly suggests a need to evaluate carefully the implications of learning in applied benefit-cost analysis.

These results indicate how temporal uncertainty influences the validity of empirical welfare measures. A well known issue concerns the approximation involved with using uncompensated (rather than compensated) choice functions in the empirical evaluation of welfare triangles [20]. A standard result obtained in the riskless case is that the approximation involves no error in the absence of income effects, and only a small error when the income effects are small [26]. Our analysis indicates that income effects are similar under temporal uncertainty. In particular, equation (7) shows that the absence of income effects ([Delta]x*/[Delta]w = 0) implies that compensated and uncompensated price slopes are equal ([Mathematical Expression Omitted]), i.e., that compensated and uncompensated triangles are identical. However, in contrast with the riskless case, zero income effects are not sufficient to validate using these triangles as welfare measures under temporal uncertainty. As illustrated in equation (12), these welfare triangles provide in general a biased welfare measure under learning and risk aversion. This suggests a need to evaluate empirically the covariance terms in equations (12) and (13). Again, this will require empirical assessments of the influence of risk preferences and temporal uncertainty on economic behavior.

The above discussion has focused exclusively on the welfare effects of price changes. It should be noted that the above results can also be used in the welfare evaluation of non-price changes as well. For example, one may be interested in the cost-benefit analysis of a public project [Pi] under temporal uncertainty. As used in the travel-cost method [18], weak complementarity conditions between the public project and some private goods x can make the above results useful. In this context, it is of interest to find some high (expected) price of the private goods for which the agent would obtain no benefit from the complementary public project [Pi]. In other words, we seek a price [[Gamma].sub.1] satisfying C*([U.sup.0], [[Gamma].sub.0], [Pi], [center dot]) = C*([U.sup.0], [[Gamma].sub.1], 0, [center dot]), where 0 denotes the absence of the project and [[Gamma].sub.0] is the actual price of the private good. Provided that such a price [[Gamma].sub.1] exists, the agent's willingness-to-pay for the project then becomes equivalent to the willingness-to-pay for a price increase in the complementary private goods (as given in equations (11) or (12)) where M = C*([U.sup.0], [[Gamma].sub.1], 0, [center dot]) - C*([U.sup.0], [[Gamma].sub.0], 0, [center dot]). This shows how the above results can be used in the welfare evaluation of public projects.

V. Conclusion

This paper has investigated the influence of temporal uncertainty--specifically learning and the flexibility to respond to new information--on the structure of intertemporal demands. The analysis is based on a two-period model of a competitive agent and a location-scale characterization of price uncertainty. In this context, an indirect compensation function is defined and its basic properties are investigated. An intertemporal analogue of the Shephard-Hotelling lemma is derived. The derivatives of the compensation function with respect to expected period-two prices do not in general equal the expected period-two compensated demands. An intertemporal classical Slutsky matrix is derived which is not in general symmetric, although the symmetry of the indirect compensation function provides symmetry restrictions on the uncompensated demands. These results highlight the need to address explicitly the role of risk preferences and temporal uncertainty in the theoretical and empirical analysis of dynamic behavior. They also suggest how uncertainty and risk aversion can be incorporated in empirical welfare measurements. By deriving a number of empirically tractable results for dynamic behavior under learning, it is hoped that this paper will help stimulate additional research on the economics and welfare of intertemporal resource allocation when agents adjust to new information over time.

Appendix

Proof of Proposition 1.

Inverting the relationship [U.sup.0] = EU defined in (3), which is possible due to the monotonicity assumption of EU in z, C([U.sup.0], [x.sub.1], [x.sub.2], [center dot]) takes the general form [Mathematical Expression Omitted]. This implies (i). The nondecreasing part of (ii) follows from the fact that increasing every price ([r.sub.1], [Mathematical Expression Omitted]) cannot lower C* in the minimization problem (4).

To show that C* is concave in [Mathematical Expression Omitted], consider any two price vectors [r.sup.[Alpha]] and [r.sup.[Beta]]. Let [Mathematical Expression Omitted] and [Mathematical Expression Omitted], where [Mathematical Expression Omitted]. For any scalar [Theta] satisfying 0 [is less than or equal to] [Theta] [is less than or equal to] 1, define [Mathematical Expression Omitted]. By definition of the minimization problem in (4), we have

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted].

Multiplying (A1) by [Theta] and (A2) by (1 - [Theta]), and summing yields

[Theta]C*([U.sup.0], w, [r.sup.[Alpha]]) + (1 - [Theta])C*([U.sup.0], w, [r.sup.[Beta]]) [is less than or equal to] C*([U.sup.0], w, [Theta][r.sup.[Alpha]] + (1 - [Theta])[r.sup.[Beta]]),

which proves that C*([U.sup.0], w, r) is concave in r.

In the case where C* is differentiable, the envelope theorem applied to equation (4) yields

[Delta]C*/[Delta][Alpha] = ([Delta]C/[Delta][Alpha]) [where] x = [x.sup.c], (A3)

where [Mathematical Expression Omitted]. Using the implicit function theorem in equation (3), we have

[Delta]C/[Delta][r.sub.1] = E([U.sub.w][x.sub.1])/E[U.sub.w] = [x.sub.1],

[Mathematical Expression Omitted],

and

[Delta]C/[Delta][Sigma] = E[[U.sub.w][x.sub.2](e)[Epsilon](e)]/E[U.sub.w] = E[[x.sub.2](e)[Epsilon](e)] + COV[[U.sub.w], [x.sub.2](e)[Epsilon](e)]/E[U.sub.w]

= COV[[x.sub.2](e), [Epsilon](e)] + COV[[U.sub.w], [x.sub.2](e)[Epsilon](e)]/E[U.sub.w].

Substituting these results into (A3) proves (iii), (iv) and (v).

1. Throughout the paper, we use the terms "risk" and "uncertainty" interchangeably.

2. Meyer [ 19] has argued that this characterization is attractive for two reasons: (1) it is empirically tractable; and (2) it does not impose any a priori restriction on the nature of risk preferences.

3. This is equivalent to a formulation where the prices [r.sub.1] and [r.sub.2] are relative prices, i.e. nominal prices for [x.sub.1] and [x.sub.2] divided by the nominal price of good z.

4. In economic modeling, such dynamics is often represented in terms of a state equation characterizing the underlying dynamic process.

5. Note that our analysis does not strictly require the expected utility hypothesis. For example, the results presented below could be derived in the more general context of a state-preference approach [7].

6. The prices [r.sub.1] and [r.sub.2] in the intertemporal budget constraint are assumed to be appropriately discounted to reflect the opportunity cost of money from one period to the next.

7. To simplify the notation throughout the rest of the paper, we write [r.sub.2]([e.sub.a]) as [r.sub.2] (e), [T.sub.2]([x.sub.1], [e.sub.b]) as [T.sub.2]([x.sub.1], e) and U(.; [e.sub.c]) as U(.;e). Although this involves no loss of generality, it should be kept in mind that price uncertainty, technological uncertainty and preference uncertainty will typically involve different random variables.

8. For example, consider Stigler and Becker's [24] household production model, where the utility function U([a.sub.1], [a.sub.2], e) represents household preferences over the "basic commodities" [a.sub.1] and [a.sub.2] in period one and two. Denote the household production function for [a.sub.1] as [a.sub.1] = [a.sub.1] ([x.sub.1]), and that for [a.sub.2] as [a.sub.2] = [a.sub.2]([x.sub.2], z, [x.sub.1], e), where the household's technology is uncertain and period-one choices influence production possibilities in period two. After substituting the budget constraint in terms of z into the production function, the household's maximization problem becomes:

[Mathematical Expression Omitted].

9. See Epstein [11] for a three-period model that allows for incomplete learning.

10. The relationship between the compensated and uncompensated demands can also be expressed as: [Mathematical Expression Omitted] and [Mathematical Expression Omitted].

11. For example, it imposes no restriction on the covariances between [[Epsilon].sub.i] and [[Epsilon].sub.j], i [is not equal to] j.

12. The only restriction imposed by the expectation assumption is that changes in the distribution of [r.sub.2] must be limited to a location-scale transformation of the random variables [Epsilon].

13. Chavas, Bishop, and Segerson [4] analyze in more detail the covariance between the marginal utility of income and a future choice.

14. Note from (8) that the matrix [Mathematical Expression Omitted] is symmetric and negative semi-definite under risk neutrality.

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