Question

2) Earl’s production function is f(x1, x2) = x1^(1/2) * x2^(1/3), where x1 is the number of pounds of lemons he uses and x2 is the number of hours he spends squeezing them. His cost function is c(w1, w2, y) = 2w1^(1/2) * w2^(1/2) * y^(3/2), where w1 is the cost per pound, w2 is the wage rate, and y is the number of units of lemonade produced.

a) If lemons cost $1 per pound, the wage rate is $1 per hour, and the price of lemonade is p, find Earl’s marginal cost function and his supply function. If lemons cost $4 per pound and the wage rate is $9 per hour, what will be his supply function be?

b) In general, Earl’s marginal cost depends on the price of lemons and the wage rate. At prices w1 for lemons and w2 for labour, what is his marginal cost when he is producing y units of lemonade? The amount that Earl will supply depends on the three variables, p, w1, w2. As a function of these three variables, what is Earl’s supply?


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my answers:

2)

a)
MC(y) = 3y^(1/2)
S(p) = p^2/3
S(p) = p^2/18

b)
MC(w1, w2, y) = 3w1^(1/2) * w2^(1/2) * y^(1/2)
S (p, w1, w2) = p^2 / (3 w1 * w2)