10. Envelope Theorem Econ 494 Spring 2013

Agenda Indirect profit function Envelope theorem for general unconstrained optimization problems Theorem Proof Application to profit maximization Graphical illustration Reading Silb §

Recall Profit maximization How would you answer the question: “What are the firm’s profits at the optimal solution?” Substitute the optimal solution, x i *, back into the objective function This will give us the indirect objective function or value function. Note: Until now, we have been substituting x i * into the FONC for comparative statics 3

Example: Profit maximization 4 Substitute solution, x i * (p,w 1,w 2 ), into obj. fctn.  *(p,w 1,w 2 ) is the indirect objective function, or indirect profit function Note difference betw.  *(p,w 1,w 2 ) &  (x 1,x 2 ) The indirect objective function is also referred to as the value function.

A simple example 5

Simple example (cont.) FONC: SOSC: Solve FONC: 6

Simple example (cont.) 7

Two different functions 8

Compare 9

Indirect profit function is HOD(1) 10

Envelope Theorem The envelope theorem is one of the most important theorems in economic theory. It concerns the rate of change of the objective function (rather than the choice functions) when a parameter changes. It is useful for deriving comparative static results 11

Unconstrained optimization 12 Assuming the SOSC hold, the IFT implies that the FONC can be solved simultaneously for the choice functions x i = x i * (  1,…,  n )  is the indirect objective function The FONC are: Consider the general unconstrained maximization problem:

Unconstrained optimization 13

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Envelope Theorem Proof 15

Envelope Theorem Proof 16 Rewriting the above yields:

Envelope Theorem Recap 17

Proving vs. using the envelope theorem 18

ET proof for profit max problem 19

Factor demand functions 20 At the  -max input choice, the rate of change of profit when a factor price changes is the same whether we let the input choices vary optimally or they are held fixed at profit maximizing levels.  * wi = – x i * is what many economists refer to as Hotelling’s Lemma As you can see, this is just an application of the envelope theorem applied to a profit max problem.

Proving vs using the ET revisited 21 To prove that you must go through all of the steps just shown Set up indirect objective function Differentiate wrt w 1 Simplify and use FONC=0 to eliminate terms To use the ET Differentiate objective function wrt w 1 Evaluate at optimal solution:

Supply function (prove with ET) 22

Supply function (use ET) 23

Reciprocity relationships 24 Differentiate with respect to w 2 : By Young’s Theorem  * w 1 w 2   * w 1 w 2 Using symmetry

Reciprocity relationships 25 Note that these reciprocity results are identical to those we derived earlier using Cramer’s rule to solve for the comparative statics

26 See Silb §7.2

27 Note that every variable has a superscript 0, except w 1. Remember that x i 0 = x i * (p 0, w 1 0, w 2 0 ) is optimal given that prices are (p 0, w 1 0, w 2 0 ). If prices are something different, say (p 0, w 1 1, w 2 0 ), then x i 0  x i * (p 0, w 1 1, w 2 0 ) is not optimal.

Constrained profit function 28 w12w12  (w 1 2, w 2 0, p 0 ) w11w11  (w 1 1, w 2 0, p 0 )

Indirect profit function 29 x i * are functions that vary optimally with prices and are not parameters of this function. x i 0 are fixed and cannot vary with prices. They are parameters of this function.

Indirect vs. constrained profit function 30

Shape of indirect profit function 31

Shape of indirect profit function 32

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Graphical illustration in more detail 34 w 1  A B C

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Consider another constrained profit fctn 36

Another constrained profit function 37 We can show that the two constrained profit functions are related to each other as follows: w 1 0 w 1   (w 1, w 2 0, p 0, x ) w 1 1 A:  *(w 1 0, w 2 0, p 0 ) >  (w 1 0, w 2 0, p 0, x ) B:  *(w 1 1, w 2 0, p 0 ) =  (w 1 1, w 2 0, p 0, x )  (w 1, w 2 0, p 0, x )

Show relationship in previous graph 38

Indirect profit function as “envelope” of all constrained profit functions 39