Question #85573

c) Alpha industries sells two products, X and Y, in related markets, with demand functions given by:

Px – 13 + 2X + Y = 0

Py– 13 + X + 2Y = 0

The total cost, in shillings, is given by:

TC = X + Y

Determine the price and the output for each good, which will maximize profits.

Px – 13 + 2X + Y = 0

Py– 13 + X + 2Y = 0

The total cost, in shillings, is given by:

TC = X + Y

Determine the price and the output for each good, which will maximize profits.

Expert's answer

c) Px – 13 + 2X + Y = 0, Px = 13 - 2X - Y,

Py – 13 + X + 2Y = 0, Py = 13 - X - 2Y.

The total cost, in shillings, is given by:

TC = X + Y.

The price and the output maximize profits, if MR = MC = P.

MRx = TR'(x) = (Px*X)' = 13 - 4X - Y,

MCx = TC'(x) = 1,

MRy = TR'(y) = (Py*Y)' = 13 - X - 4Y,

MCx = TC'(x) = 1,

13 - 4X - Y = 1 -> 4X + Y = 12 -> Y = 12 - 4X,

13 - X - 4Y = 1 -> X + 4Y = 12, so:

X + 4*(12 - 4X) = 12,

15X = 36,

X = 2.4,

Y = 12 - 4*2.4 = 2.4,

Px = 13 - 2*2.4 - 2.4 = 5.8,

Py = 13 - 2.4 - 2*2.4 = 5.8.

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