©February 21, 2019, Christopher D. Carroll Envelope

The Envelope Theorem and the Euler Equation

This handout shows how the Envelope theorem is used to derive the consumption Euler equation in a multiperiod optimization problem with geometric discounting and intertemporally separable utility.

The consumer’s goal from the perspective of date t is to

| (1) |

subject to the dynamic budget constraint

The problem can be written in Bellman equation form as

The first order condition for (3) can be written as

and we can define a function cNow define a function

The Chain Rule of differentiation tells us that

Here’s the key insight: The assumption that consumers are optimizing
means that we will always be evaluating the value function and its
derivatives at a c_{t} that satisfies the first-order optimality condition
(6).^{1}
Thus we have from (8) that

Now notice that the RHS’s of (5) and (14) are identical, so we can equate the left hand sides,

The general principle can be condensed into a rule of thumb by realizing that
the Envelope theorem will always imply that the total derivative of a value
function with respect to any choice variable must be equal to zero for optimizing
consumers (because the first order condition holds). Thus we could have
obtained the result immediately by treating c_{t} as though it were a constant (that
is, treating the problem as though c_{t}^{′}(m_{t}) = 0) and taking the derivative of
Bellman’s equation with respect to m_{t} directly. This leads immediately to the
key result: