Question

Let the input prices be w = (w1, w2) and output price be p. Derive the...

Let the input prices be w = (w1, w2) and output price be p. Derive the cost function c (w; y) and the output supply function y (w, p) for firms with the following production functions:

a] f (x1; x2) = sqrt(x1) + 2sqrt(x2)

b] f (x1; x2) = min [sqrt(x1), 2 sqrt(x2)]

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Answer #1

The given production function is y = x1^2 + 2x2^2 , the price of y is p , the input cost of x1 = w1 and x2 = w2

The profit maximization function = yp - x1w1 - x2w2

= sqrt(x1) + 2sqrt(x2) - x1w1 - x2w2

f(x1, x2) = x1^2 + 2x2^2 ,

(a) The cost function would be

C = ( w1 x1 + w2 x2 ) =  ( w1 x1 + w2 x2 )

Derivating with respect to x1 and x2

sqrt(x1) + 2sqrt(x2) - x1w1 - x2w2

= 2x1 - w1 = 0 so w1 = 2x1

sqrt(x1) + 2sqrt(x2) - x1w1 - x2w2

= 4x1 -w2 , w2= 4x1 ,

so the ratio is 1/2 , 1 unit of x1 and 2 unit of x2.

(b)

The profit maximization function = yp - x1w1 - x2w2

= min [sqrt(x1), 2 sqrt(x2)] - x1w1 - x2w2

f(x1, x2) = min [sqrt(x1), 2 sqrt(x2)],

(a) The cost function would be

C = ( w1 x1 + w2 x2 ) =  ( w1 x1 + w2 x2 )

Derivating with respect to x1 and x2

min [sqrt(x1), 2 sqrt(x2)] - x1w1 - x2w2

= minsqrt(x1) - w1 = 0

sqrt(x1) + 2sqrt(x2) - x1w1 - x2w2

= 2sqrt(x2) -w2 ,

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