Question

In: Economics

The production function of a firm is Y = x11/2x21/2 . Suppose in the short run...

  1. The production function of a firm is Y = x11/2x21/2 . Suppose in the short run factor 2 is fixed at 400 units. The cost of factor 1 is $2 and the cost of factor 2 is $1. The price of the output is $6 each. What's the maximum profit that the firm can get in the short run?

    1400

    3600

    1800

    2000

suppose the firm is in the long run and can now choose both inputs freely. What's the optimal level of input 1 in terms of y?

x1 = Y2/x2

x1= Y/1.41

x1= Y1/2/2

x1= Y/2.45

Solutions

Expert Solution

Y = x11/2x21/2

x2 = 400

Y= 20x11/2

Squaring both sides:

x1 = (Y/20)2

Total cost(TC)= 2(x1 )+x2

TC= 2 (Y2 /400) +400

TC= (Y2 /200) +400

MC= Differentiation of TC wrt Y= Y/100

Total revenue(TR)= P x Y= 6Y

Marginal revenue= Differentiation of TR wrt Y= 6

Optimal condition for maximum profit:

MC=MR

Y/100= 6

Y= 600

TC= = (Y2 /200) +400= 1800+400= 2200

TR= 6 (600)= 3600

Maximum profit= TR-TC= 3600-2200= 1400

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In the long run:

Optimal condition :

Marginal product of x1 /Marginal product of x2 = cost of factor 1/cost of factor 2

Marginal product of x1 = Differentiation of Y wrt x1 =  (1/2)x1-1/2x21/2

Marginal product of x2 = Differentiation of Y wrt x2 =  (1/2)x11/2x2-1/2

Marginal product of x1 /Marginal product of x2 = x2 / x1

x2 / x1 = 2/1

x2 = 2x1

Use this relation in production function:

Y = x11/2 (2x1)1/2

Y= 21/2 x1

x1 = Y/21/2

x1 = Y/1.41


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