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Microeconomics for Policy, Part I, 2014/2015: Answer KeysWeek 1
0 General Calculus
0.1. We want to maximize a function f ( x1 , x2 , a) with respect to two variables x1 , x2 ,
which also depends on a parameter a. Furthermore, maximization is subject to the
constraint g( x1 , x2 , a) = 0.
Let x1 ( a) and x2 ( a) be the optimal values depending on a and let
f ∗ ( a ) : = f ( x1 ( a ) , x2 ( a ) , a )
be the maximum value function. Then, the envelope theorem for the constrained
optimization problems tells us the derivative of f ∗ :
df∗ ∂f ∂g
= +λ | ,
da ∂a ∂a x1 = x1 ( a), x2 = x2 ( a)
which is shown in the following steps.
(a) The Lagrangian is
L( x1 , x2 , a) = f ( x1 , x2 , a) + λg( x1 , x2 , a).
The first-order conditions are
∂L ∂f ∂g
= +λ = 0,
∂x1 ∂x1 ∂x1
∂L ∂f ∂g
= +λ = 0,
∂x2 ∂x2 ∂x2
∂L
= g( x1 , x2 , a) = 0.
∂λ
(b) Taking sums:
∂f ∂f ∂g ∂g
h( a) : = + +λ + = 0.
∂x1 ∂x2 ∂x1 ∂x2
(c) Taking the derivative:
dh( a) ∂ f dx1 ∂ f dx2 ∂g dx1 ∂g dx2
= + +λ + = 0.
da ∂x1 da ∂x2 da ∂x1 da ∂x2 da
(d)
df∗ ∂ f dx1 ∂ f dx2 ∂f
= + +
da ∂x1 da ∂x2 da ∂a
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