Lecture 7.5
The Envelope Theorem

What is the effect of a change in a parameter on the objective function?

Consider the maximization problem:

max     f(x,y)

s.t.     g(x,y,a) = c.

1. Set up the Lagrangian:

   L = f(x,y) + λ[c - g(x,y,a)].

2. Find the FOCs:

   	f1(x,y)		= λg1(x,y,a)			(1)
   	f2(x,y)		= λg2(x,y,a)			(2)
   	g(x,y,a)	= c				(3)
   

3. Solve equations (1) - (3) to obtain the optimal values of x and y in terms of the parameter a:

   x* = x(a)

   y* = y(a)

4. Substitute x* and y* in the objective function to get the maximum value of the function:

   f* = f(x*,y*).

5. Differentiate f* totally to obtain
   df*/da = f₁(dx*/da) + f₂(dy*/da) = λg₁(dx*/da) + λg₂(dy*/da)           (4)


6. Differentiate (3) totally to obtain

   g₁(dx*/da) + g₂(dy*/da) + g₃ = 0.

7. Substitute in equation (4) to obtain

   df*/da = - λg₃.

Result: A unit increase in the parameter a will cause the objective function to increase by -λg₃.