Question

1. 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 paid for labor is $12.

a.Derive the marginal cost function.

b.Write the farmer’s profit maximization problem.

c.Derive the farmer’s short run supply equation using your answer for b.

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

Answer 1

a).

Consider the given problem here the production function is given by, “q=A*L^1/3”, => L^1/3 = q/A,

=> L = (q/A)^3, => the cost function is given by, “C=W*L”.

=> C = W*L = W*(q/A)^3 = (W/A^3)*q^3, => C = (W/A^3)*q^3.

=> MC = dC/dq = (W/A^3)*3q^2 = (3W/A^3)*q^2, => MC = (3W/A^3)*q^2.

b, c).

Now, the profit function is the difference between “TR” and “TC”, => the profit function is given by.

=> π = P*q - C= PA*L^1/3 – W*L. Now the FOC for profit maximization require “dπ/dL=0”.

=> (PA/3)*L^(-2/3) – W = 0, => (PA/3)*L^(-2/3) = W, => (PA/3W) = L^2/3, => L = (PA/3W)^3/2.

Now, the production function is given by, “q = A*L^1/3”.

=> q = A*L^1/3 = A*[(PA/3W)^3/2]^1/3 = A*(PA/3W)^1/2, => q = A*(PA/3W)^1/2, be the SR supply curve shows the positive relationship between “P” and “q”.

d).

Now, let’s assume that the labor productivity increase by “40%”, => the new production function is given by.

=> q = A*(1.4*L)^1/3 = (A*1.4^1/3)*L^1/3 = (A*1.12)*L^1/3, => L^1/3 = [q/(A*1.12)].

=> L^1/3 = [q/(A*1.12)], => L = [q/(A*1.12)]^3.

=> the cost function is given by, “C = W*L”.

=> C = W*[q/(A*1.12)]^3= [W*/(A*1.12)]^3*q^3. So, the MC is given by.

=> MC = dC/dq = [3W*/(A*1.12)]^3*q^2.