Question

A food manufacturer is trying to maximize profit by selling wheat-based cereal (C) and wheat bread(B) with raw wheat (W). The production functions are: Cereal: C = 40Wc - 1.5Wc Bread: B - 80W8 - 2W82 Constraint: Wc+W8 = 8,000 Profit is $1.00 per box of cereal and $0.50 per pack of wheat bread. There are 8,000 units of raw wheat available. How much wheat should go to the bread? Enter as a value.

Answer 1

W_C is the wheat used in cereal.

W_B is the wheat used in bread.

W_C + W_B = 8,000

W_C = 8,000 - W_B

Production function of cereal is as follows -

C = 40W_C - 1.5W_C²

Derive the marginal production function of cereals -

MP_C = dC/dW_C

MP_C = d(40W_C - 1.5W_C²)/dW_C

MP_C = 40 - 3W_C

Production function of bread is as follows -

B = 80W_B - 2W_B²

Derive the marginal production function of bread -

MP_B = dB/dW_B

MP_B = d(80W_B - 2W_B²)/dW_B

MP_B = 80 - 4W_B

At optimum production,

MP_C = MP_B

40 - 3W_C = 80 - 4W_B

40 - 3(8,000 - W_B) = 80 - 4W_B

40 - 24,000 + 3W_B = 80 - 4W_B

40 - 24,000 - 80 = -4W_B - 3W_B

-24,040 = -7W_B

W_B = 24,040/7

W_B = 3,434.28 units

Thus,

The wheat that should go to bread is 3,434.28 units.

This can be rounded off to 3,434 units, if Stated.