env-theorem2

# env-theorem2 - Envelope Theorem Kevin Wainwright 1 Maximum...

This preview shows pages 1–3. Sign up to view the full content.

Envelope Theorem Kevin Wainwright Mar 22, 2004 1 Maximum Value Functions A maximum (or minimum) value function is an objective function where the choice variables have been assigned their optimal values. These optimal values of the choice variables are, in turn, functions of the exogenous variables and parameters of the problem. Once the optimal values of the choice variables have been substituted into the original objective function, the function indirectly becomes a function of the parameters (through the parameters’ in f uence on the optmal values of the choice variables). Thus the maximum value function is also referred to as the indirect objective function. What is the signi F cance of the indirect objective function? Consider that in any optimization problem the direct objective function is maximized (or minimized) for a given set of parameters. The indirect objective function gives all the maximum values of the objective function as these prameters vary. Hence the indirect objective func- tion is an ”envelope” of the set of optimized objective functions generated by varying the parameters of the model. For most students of economics the F rst illustration of this notion of an ”envelope” arises in the comparison of short-run and long-run cost curves. Students are typically taught that the long-run average cost curve is an envelope of all the short-run average cost curves (what parameter is varying along the envelope in this case?). A formal derivation of this concept is one of the exercises we will be considering in the following sections. To illustrate, consider the following maximization problem with two choice vari- ables x and y , and one parameter, α : Maximize U = f ( x,y, α ) (1) The F rst order necessary condition are f x ( x,y, α )= f y ( x,y, α )=0 (2) if second-order conditions are met, these two equations implicitly de F ne the solutions x = x ( α ) y = x ( α ) (3) 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
If we subtitute these solutions into the objective function, we obtain a new function V ( α )= f ( x ( α ) ,y ( α ) , α ) (4) where this function is the value of f whentheva lueso f x and y are those that maximize f ( x,y, α ) . Therefore, V ( α ) is the maximum value function (or indirect objective function). If we di f erentiate V with respect to α V ∂α = f x x ∂α + f y y ∂α + f α (5) However, from the f rst order conditions we know f x = f y =0 . Therefore, the f rst two terms disappear and the result becomes V ∂α = f α (6) This result says that, at the optimum, as α varies, with x and y allowed to adjust optimally gives the same result as if x and y were held constant! Note that α enters maximum value function (equation 4) in three places: one direct and two indirect (through x and y ). Equations 5 and 6 show that, at the optimimum, only the direct e f ect of α
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 09/13/2009 for the course EOCNOMICS Econ 331 taught by Professor Kelvinkwainger during the Spring '09 term at Simon Fraser.

### Page1 / 17

env-theorem2 - Envelope Theorem Kevin Wainwright 1 Maximum...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online