Answer to Question #220351 in Microeconomics for tim

Question

The production function of a firm is given as

f(x)=20x-x²

 The price of output is equal to 1. Let w be the price of the x-input. We must have x > 0.

 a) What is the first-order condition for profit maximization if x > 0?

b) For what values of w will the optimal x be zero?

 c) For what values of w will the optimal x be 10?

d) What is the factor demand function?

 e) What is the profit function?

 f) What is the derivative of the profit function with respect to w?

Answer 1

a. f(x)=20x-x²

Price of output=1

Price of input=w

Max_xP(20x-x²)-WX

But P=1

First Order Condition with respect to X

= 20-2x-w=0

b. From FOC W= 20-2X

BUT x=0

W= 20-2(0)

W=20

c. From FOC W= 20-2X

BUT x =10

W=20-2(10)

w=0

d. factor demand function

FOC is 20-2x-w=0

Making X the subject by rearranging the FOC function

X^* = 10- (w/2)

e. Profit function

П (w,p)=P(20-10+ w/2)(10-w/2) - w(10-w/2)

=(20p- w/2)(10-w/2)

but P=1

Hence the profit function is =(20- w/2)(10-w/2)

f. the derivative of the profit function with respect to w

П (w,p)=(20- w/2)(10-w/2)

ɚП (w,p)/ɚw= -15-w/2