We derive two forms of Roy’s identity for a dynamic consumption model. The results are potentially useful in theoretical and empirical studies.
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.
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
subject to the budget constraints
where
The first-order conditions are
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 utility function
In the second stage, the consumer makes decisions on how much to buy in
subject to
The indirect utility function in each period
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
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
Similarly,
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
A Roy’s identity similar to the static form can be obtained by considering the “cake-eating” problem by assuming that income
If we define
then the intertemporal utility maximization problem becomes
Notice that
Roy’s identity, in this case, is
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.