The Envelope Theorem

13J0. A hrm in perfect competition with the production function Q = f(K.L) and a production limit of seeks to maximize profit tt= PQ- rK - wL

Assuming conditions arc satisfied for the implicit function theorem, the Lagrangjan function and the indirect objective function can be written, respectively.

iI(/C. L, Q. A; r. iv, P. Q0) - PQ(r, w, P, Qu) - rK(r. w. P. (?<>) - wL(r. P, Q0)

+ \[Qo-f(K.L)\ Q. r. »v. P. Q0) = PQ(r. iv. P, (?„) - rR(r, iv. P. C?0) - H». P. Q0)

Use the envelope theorem to find and comment on the changes in the indirect objective function signified by (a) dvttr. (b) diHd*. (c) dvtdP.(d) diHdQ^

Differentiating ihe profil function with respect to input prices gives the firm's demand for inputs. Notice here where input prices (r. w) appear in the objective function and not in the constraint, the desired derivatives can be readily found from either function.

Diffcrentiatirg the profit function with respect to output prices gives the firm's supply function. Since output price appears only in the objective function, the derivative can once again easily be found from either function.

Differentiating the profit function with respect to an output constraint gives the marginal value of relaxing that constraint. Lc., the extra profit the firm could earn if it could increase output b> one unit. Notice here where the output limit (Q„) appears only in the constraint, the derivative can be derived more quickly and readily from the Lagrangian function.

13JL A consumer wants to minimize the cost of attaining a specific level of utility:

If the implicit function theorem conditions are fulfilled, the Lagrangian and indirect object functions arc

C(x.y. A:pt.py. Do) = p,x(p„p,, U0) * pvy(p..p„ U0) + A(i/, - u(x.y)] c(x,y\pJtpv C/0) - ptx{pt.p,. U0) + p,y(p„py. U0)

Use the envelope theorem to find 3nd comment on the changes in the indirect objective function signified by (0) del dp t, (fr) ficJrip,. (c) HcIAUq.

dc AC

In both cases, a change in the price of a good has a positive effect on the cost that is weighted by the amount of the good consumed. S.nce prices appear only in the objective function and not the constraint, the desired derivatives can be easilv taken from cither function.

Here A measures the marginal cost of changing the given level of utility. Since the utility limit U0 appears only in the constraint, the derivative is more easily found from the Lagrangian function.

13.32. Assume the model in Example 7 is a function with only a single exogenous variable a. Show that at the optimal solution the total derivative of the Lagrangian function with respect to a is equal to the partial derivative of the same Lagrangian function with respect 10 a.

The new Lagrangian function and first-order conditions arc

Z(x.y.A:fl) = zl*(a).y(<i)-.fl| + A(a)flx(fl).y(a):a) Z. = :,(x.y-.a) + AUx.y-.a) » 0 Z, - :,(x.y.o) + xfr(x.y:a) = 0 Z. - /(.f.y;a) - 0

Taking the total derivative of the original lagrangian function with respeci to a. dZ - dx - d\ dk

But from the first-order conditions.

so the first three terms cancel and the total derivative of the function with respect to the exogenous variable a ends up equal to the partial derivative of the function with respect to the exogenous variable a:

"This suggests we can find the total effect of u change in a single exogenous vanable on the optimal value of the Lagrangian function by simply taking the partial derivative of the Lagrangi&n function with rcspccl to that exogenous variable.

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