Question

1. Consider a farmer that grows hazelnuts using the production q = AL^2/3 , 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 $1000, and is already paid by the farmer. The hourly wage rate paid for labor is $12.

1. Derive the equation for the marginal product of labor.
2. Does the production function exhibit diminishing returns to labor? Explain.
3. Derive variable cost (VC), fixed cost (FC), and total cost (C) functions.
4. Derive the average cost, average variable cost, and average fixed cost functions.

Answer 1

Solution:

Production function is: q = A*L^2/3

a) Marginal product of labor, MPL = = (2/3)*A*L^2/3-1

MPL = (2/3)*A/L^1/3

b) A production function shall exhibit diminishing returns to labor if changing the labor by some factor, say k, where k > 1, changes the total output by a factor less than k. In this sense, if scaling up of a factor of production scales up the total production only slightly, the returns are diminishing.

So, if L increases by factor k, let's find by how much does q change. Calling this new q as q', so

q' = A*(kL)^2/3

q' = k^2/3*(A*L^2/3) = k^2/3*y

Since, the power of k is less than 1 (2/3 < 1), we can say that output increases by a factor lower than k, so the given production function does exhibit diminishing returns to labor.

c) Note that the annual cost of orchard pruning costs a fixed amount (irrespective of labor used) of $1,000, so this is Fixed cost (FC). Since, the wage cost changes as the number of labor employed changes, which further changes output, it becomes the variable cost. So, with L denoting the number of labor hours, variable cost (VC) = 12*L. But this function should be function of output, q. So, now using the production function: q = A*L^2/3, we can modify this to

L = (q/A)^3/2

So, variable cost, VC = 12*(q/A)^3/2

Total cost is simply addition of the fixed cost and variable cost. Hence, TC = FC + VC

TC = 1000 + 12*(q/A)^3/2

d) Average cost = TC/q

AC = (1000 + 12*(q/A)^3/2)/q = 1000/q + (12/A^3/2)*q^1/2

Average variable cost, AVC = VC/q = 12*(q/A)^3/2/q = (12/A^3/2)*q^1/2

Average fixed cost, AFC = FC/q = 1000/q

(NOTE: AC = AFC + AVC)