Question

Consider a farmer that grows hazelnuts using the production q = AL^1/3, where q is the amount of hazelnuts produced in a year (in tonnes), L represents the number of labor hours employed on the farm during the year, and A is the size of orchard which is fixed. The annual winter pruning of the orchard cost $1200, and is already paid by the farmer. The hourly wage rate for labor is $12.

a) Derive the equation for the marginal product of labor.

b) Does the production function exhibit diminishing returns to labor? Explain.

c) Derive variable cost (VC), fixed cost (FC), and total cost (C) functions.

d) Derive the average cost, average variable cost, and average fixed cost functions.

e) Derive the marginal cost function

f) Write the farmer's profit maximization problem.

g) Derive the farmer's short run supply equation using your answer for f.

h) Suppose the rainfall in the region where the farmer operates in was far lower than what was expected at harvest. This increased productivity of labor by 40%. Derive the new marginal cost function assuming the wage rate remains unchanged.

Answer 1

a) Marginal product of labor, MPL =

b) q = AL^1/3

Let L = tL where t > 0
So, q' = A(tL)^1/3
So, q' = t^1/3AL^1/3
So, q' = t^1/3q
Thus, power of t is 1/3 which is less than 1. This means that there are decreasing returns to scale.

c) Wage rate (w) = $12
Variable cost (VC) = wL = 12L
And, q = AL^1/3
So, L^1/3 = q/A
So, L = (q/A)³

VC = 12(q/A)³
FC =  annual winter pruning of the orchard cost = $1200
So, FC = $1,200

Total cost (C) = FC + VC = 1200 + 12(q/A)³
So, C = 1200 + 12(q/A)³

d) Average Cost = C/q = (1200/q) + [(12/q)((q/A)³]
So, Average Cost =

Average Variable cost = VC/q = [12(q/A)³]/q
So, Average Variable Cost =

Average Fixed Cost = FC/q = 1200/q

(Note: Post four subparts at a time.)