14.451 Lecture Notes Economic Growth

where
• The FOC with respect to c j
t gives
George-Marios Angeletos
H j
t = U(c j
t, 1 − l j
t )+λ j £
t (1 + Rt)a j
t + wtl j
t − a j
t+1 − c j¤
t
The FOC with respect to l j
t gives
∂L j
0
∂c j
t
∂L j
0
∂l j
t
t ∂Hj t
= β
∂c j
t
Uc(c j
t,z j
t )=λ j
t
t ∂Hj t
= β
∂l j
t
=0⇔
=0⇔
Uz(c j
t,z j
t )=λ j
twt
Combining, we get
Uz(c j
t,z j
t )
Uc(c j
t,z j = wt.
t )
That is, households equate their marginal rate of substitution between consumption
and leisure with the (common) wage rate.
• The Kuhn-Tucker condition with respect to a j
t+1 gives
∂L j
0
∂a j
t+1
= β t
"
∂H j
t
∂a j
t+1
+ β ∂Hj
t+1
∂a j
t+1
λ j
t ≥ β [1 + Rt] λ j
t+1,
#
≤ 0 ⇔
with equality whenever a j
t+1 >a t+1. That is, the complementary slackness condition is
£ j
λt − β [1 + Rt] λ j ¤£ j ¤
t+1 at+1 − at+1 =0.
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