In: Economics
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 |
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3600 |
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1800 |
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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 |
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x1= Y/1.41 |
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x1= Y1/2/2 |
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x1= Y/2.45 |
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