Let Q = FIK, L) = KL74 denoted a production function with K as capital input and L as labour input. The price per unit of output is p, the price per unit of capital is r, and the price per unit of labor is w.(a) Write down the profit function .(b) Write down the first-order conditions for maximum profit.(c) Find the choices K =K* (p, r, w) of K and L =L* (p,r, w) of L that maximize .(d) Let * (p, , w) = R(K, L.P.r. w) be the optimal profit as a function of (p, r.w), Show that dã* / dp = F(K.L*) = Q• (Special case of Hotelling's lemma.)(a)Write down the profit functionK L p. r, w) =(b)Write down the first-order conditions for maximum profit.1/4-r= 0aK- w = 0(c)Find the choices K =K* (p, r, w) of K and L =L (p. r, w) of L that maximize 1.-1/216PL. =-1/2(4)Let * (p, r, w) = E(K•,L•. p. r, w) be the optimal profit as a function of (p, r, w). Show that d / ớp = F(K, L.) = Q.. (Special caseHotelling's lemma.)Begin by finding Q. = (K•)'/L• }'/« = !pr wNext compute*•,L•.P,r, w) = p(K•)(L•)'4 - rk• - wL =The condusions ar" I op = F(K, L) = Q• follows immediately. So, increasing the price of output by 1 unit increases the optimal profit by Q*, the optimal number of produced units.

We are going to answer this question using two variable maximization technique. Note: As per…

Let Q = F(K, L) = KL denoted a production function with K as capital input and L as labour input. The price per unit of output is p, the price per unit of capital is r, and the price per unit of labor is w. (a) Write down the profit function (b) Write down the first-order conditions for maximum profit. (c) Find the choices K =K* (p, r, w) of K and L = L* (p,r, w) of L that maximize . (d) Let * (p, , w) = R(K, L.P.r. w) be the optimal profit as a function of (p. r.w), Show that dã* / dp = F(K.L*) = Q• (Special case of Hotelling's lemma.) (a) Write down the profit function K L p, r, w) = Write down the first-order conditions for maximum profit. 1/4 -rE 0 aK - w = 0 (c) Find the choices K =K* (p, r, w) of K and L =L (p, r, w) of L that maximize 1. -1/2 16P -1/2 (4) Let * (p, r, w) = E(K•,L•.p.r, w) be the optimal profit as a function of (p, r, w). Show that dz / ớp = F(K, L) = Q.. (Special case Hotelling's lemma.) Begin by finding a - (K•'(L•)/4 = Next compute K•,L*. p.r, w) = p(K (L)4 - rk - wL = The condusions ar I op = F(K, L) = Q• follows immediately. So, increasing the price of output by 1 unit increases the optimal profit by Q, the optimal number of produced units.

Let Q = F(K, L) = KL denoted a production function with K as capital input and L as labour input. The price per unit of output is p, the price per unit of capital is r, and the price per unit of labor is w. (a) Write down the profit function (b) Write down the first-order conditions for maximum profit. (c) Find the choices K =K* (p, r, w) of K and L = L* (p,r, w) of L that maximize . (d) Let * (p, , w) = R(K, L.P.r. w) be the optimal profit as a function of (p. r.w), Show that dã* / dp = F(K.L) = Q• (Special case of Hotelling's lemma.) (a) Write down the profit function K L p, r, w) = Write down the first-order conditions for maximum profit. 1/4 -rE 0 aK - w = 0 (c) Find the choices K =K (p, r, w) of K and L =L (p, r, w) of L that maximize 1. -1/2 16P -1/2 (4) Let * (p, r, w) = E(K•,L•.p.r, w) be the optimal profit as a function of (p, r, w). Show that dz / ớp = F(K, L) = Q.. (Special case Hotelling's lemma.) Begin by finding a - (K•'(L•)/4 = Next compute K•,L*. p.r, w) = p(K (L)4 - rk - wL = The condusions ar I op = F(K, L) = Q• follows immediately. So, increasing the price of output by 1 unit increases the optimal profit by Q, the optimal number of produced units.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter11: Profit Maximization

Section: Chapter Questions

Problem 11.4P

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Question

Let Q = FIK, L) = KL74 denoted a production function with K as capital input and L as labour input. The price per unit of output is p, the price per unit of capital is r, and the price per unit of labor is w.
(a) Write down the profit function .
(b) Write down the first-order conditions for maximum profit.
(c) Find the choices K =K* (p, r, w) of K and L =L* (p,r, w) of L that maximize .
(d) Let * (p, , w) = R(K, L.P.r. w) be the optimal profit as a function of (p, r.w), Show that dã* / dp = F(K.L*) = Q• (Special case of Hotelling's lemma.)
(a)
Write down the profit function
K L p. r, w) =
(b)
Write down the first-order conditions for maximum profit.
1/4
-r= 0
aK
- w = 0
(c)
Find the choices K =K* (p, r, w) of K and L =L (p. r, w) of L that maximize 1.
-1/2
16P
L. =
-1/2
(4)
Let * (p, r, w) = E(K•,L•. p. r, w) be the optimal profit as a function of (p, r, w). Show that d / ớp = F(K, L.) = Q.. (Special case
Hotelling's lemma.)
Begin by finding Q. = (K•)'/L• }'/« = !pr w
Next compute
*•,L•.P,r, w) = p(K•)(L•)'4 - rk• - wL =
The condusions ar" I op = F(K, L) = Q• follows immediately. So, increasing the price of output by 1 unit increases the optimal profit by Q*, the optimal number of produced units.

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