The Envelope Theorem and the Euler Equation

$©$ February 14, 2012, Christopher D. Carroll Envelope

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 is to

∑T max βs - tu (cs) s=t

(1)

subject to the dynamic budget constraint

The problem can be written in Bellman equation form as

vt(mt ) = m{acx} {u (ct) + βvt+1 ((mt - ct)R + yt+1) }. (3)
t

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

=- R from (2)
◜ ---◞◟----◝
( dm )
0 = u ′(ct) + ---t+1- βv ′ (mt+1 ) (4)
dct t+1
′ ′
u (ct) = R βv t+1(mt+1 ), (5)

and we can define a function $ct(mt )$ that returns the $ct$ that solves the max problem for any given $mt$ . That is, for $ct = ct(mt )$ the first order condition (5) will hold so that

′ ′
u (ct(mt )) - R βv t+1((mt - ct(mt ))R + yt+1 ) = 0. (6)

Now define a function

v-(mt, ct) = u (ct) + βvt+1 ((mt - ct)R + yt+1 ) (7)
t

with partial derivatives

( )
∂vt- ≡ v c(m ,c ) = u′(c ) - R βvm ((m - c )R + y ) (8)
∂ct --t t t t t+1 t t t+1
( )
∂vt-- m m
≡ v-t (mt, ct) = R βv t+1 (mt+1 ) (9)
∂mt

and note that by definition

The Chain Rule of differentiation tells us that

( ) ( )
dv ∂c (m )
v′t(m ) ≡ vmt (mt ) ≡ ---t- = vmt (mt, ct(mt )) + --t----t- vct(mt, ct(mt )).
dmt ∂mt

(11 )

Now 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 $ct$ that satisfies the first-order optimality condition (6).¹ Thus we have from (8) that

c ′ ′
v-t(mt, ct(mt )) = u (ct(mt )) - R βv t+1((mt - ct(mt ))R + yt+1 ) (12)

= 0. (13)

This means that the second term in (11) is always equal to zero, so from (9) we obtain

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

and since a corresponding equation will hold in period $t + 1$ we can rewrite (14) as

The general principle can be condensed by realizing that the Envelope theorem will always imply that the derivative of a value function with respect to any choice variable must be equal to zero for optimizing consumers. 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 $′ ct(mt ) = 0$ ) and taking the derivative of Bellman’s equation with respect to $m t$ directly. This leads immediately to the key result:

v (m ) = u(c (m )) + βv ((m - c (m ))R + y ) (17)
t t t t+1 t t t t+1
v ′t(mt ) = βRv ′t+1 (mt+1 ). (18)