Suppose the production function for coffee (C)is C = min(B,W),where B = beans in pounds and W = water in gallons. Suppose the price of water is $.10 per gallon and the price of beans is $10 per pound.The expansion path is
A)B = 10W
B)B = .1W
C)B = W
D)-10 = B + W

Question

Quizplus · Accepted Answer

The production function for coffee is C = min(B,W), which means that the amount of coffee produced depends on the minimum of the amount of beans and water used in the production process.

Since the price of water is $.10 per gallon and the price of beans is $10 per pound, the cost of producing coffee can be expressed as:

Cost = 10B + 0.1W

To find the optimal combination of beans and water, we need to minimize the cost of production subject to the production function C = min(B,W).

If we assume that the price of water is fixed, then the optimal choice of B and W can be found by simply setting the partial derivative of the cost function with respect to B equal to zero, which gives:

10 = λ    (where λ is the Lagrange multiplier)

Similarly, setting the partial derivative of the cost function with respect to W equal to zero gives:

0.1 = λ

Substituting λ = 10 into the production function C = min(B,W) gives us:

C = min(B,W) = min(B, 10)

Therefore, the optimal choice is to use as many beans as possible, subject to the constraint that we must use 10 times more water.

This gives us the following expansion path:

B = 10W

Suppose the Production Function for Coffee (C)is C = Min(B,W),where