A Note on Dynamic Roy's Identity

Roy's identity is a useful tool in theoretical and empirical studies of static consumption problems. Most dynamic consumer problems, however, concentrate on obtaining the optimal consumption path derived from the Euler equation. In this note we assumes that the consumer makes decision in a two-stage process. In the dynamic stage, two forms of Roy's identity is derived. The first form relates the asset holding in each period to the marginal utility of interest rate and the marginal utility of income. The second form resemble the classic Roy's identity in the static analysis.


Introduction
Roy's identity is a useful tool in theoretical and empirical studies of static consumption problems. Most dynamic consumer problems, however, concentrate on obtaining the optimal consumption path derived from the Euler equation. In this note we assume that the consumer makes decision in a two-stage process. In the dynamic stage, two forms of Roy's identity are derived. The first form relates the asset holding in each period to the marginal utility of interest rate and the marginal utility of income. The second form resembles the classic Roy's identity in the static analysis.

Dynamic Roy's Identity
The consumer is supposed to be making a two-stage decision. In the first stage, an inter-temporal decision on aggregate consumption and saving is made. Abstract from uncertainty about the future, a simple model of the decision problem can be set up as where t a , t m , and t r are exogenous asset, income, and interest rate respectively in period t , with 0 0 r = . The choice variables are t c and 1 t a + . Utility is modelled by a stationary and separable discounted sum of utility function : The first-order conditions are ( ) (4) and the budget constraint (2). The optimal solution is characterized by the Euler equation and the consumption function (in the case of constant interest rate) which implies that in the steady state consumption in each period is a fixed portion of total wealth. The indirect In the second stage, the consumer makes decisions on how much to buy in n number of goods and services in the market given the market prices ( ) Roy's identity relates the optimal consumption of each good or service to the marginal disutility of price and the marginal utility of income, that is, for every good i in every period t . In other words, consumption of each good depends on the consumer's preference structure and the relative price of the good to other goods and services.
Can we have Roy's identity for the inter-temporal consumption problem in the first stage? Apply the envelop theorem to (3), we have  Substituting the first-order condition (4) into the above gives Combining (8) and (9) gives the Roy's identity for the inter-temporal consumption problem as It says that in any period if the consumer is in debt ( ) 0 t a < , the marginal utility of an interest rate increase is negative.
A Roy's identity similar to the static form can be obtained by considering the "cake-eating" problem by assuming that income t m in the budget constraint (2) in each period is zero. Then the inter-temporal budget constraint is reduced to ( )

Conclusion
We have used the envelop theorem to derive two forms of Roy's identity under an infinite horizontal consumption setting. In the first form, asset holding is related to the marginal utility of interest rate and marginal utility of income. The result can be used as a structural restriction on empirical analysis of inter-temporal consumption. In the second form prices are interpreted as future discount factors. The resulting form resembles the static Roy's identity.