Question

Question 2. Firms in the lettuce industry compete in a perfectly competitive industry and produce using the following production technology, q = (K − 8) ^1/ 4 L^ 1/ 4 . The inverse demand function for lettuce is P = 40 − Q. (a) Write down an equation for the firm’s demand for L and K as a function of it’s output q when the wage and rental rate of capital are equal to $1 (b) Write down the firm’s long-run cost function C(q) = f(q). (c) What is the price that the firm’s face in the long-run for lettuce? (d) How many firms are operating in the lettuce industry and how many lettuce does each firm produce?

Answer 1

a)

Given,

q=(K-8)^1/4L^1/4

Marginal Product of labor=MPL=dq/dL=(1/4)*(K-8)^1/4L^-3/4

Marginal Product of capital=MPK=dq/dK=(1/4)*(K-8)^-3/4L^1/4

We know that profit maximization requires

(MPL/MPK)=w/r

[(1/4)*(K-8)^1/4L^-3/4]/[(1/4)*(K-8)^-3/4L^1/4]=1/1

(K-8)/L=1

K-8=L

Set L=K-8 in production function

q=(K-8)^1/4L^1/4=(K-8)^1/4(K-8)^1/4=(K-8)^1/2

or K=q²+8 -------- demand function of K

We know L=K-8 hence,

L=q²+8-8=q² -------- demand function of K

b)

Firm's long run cost function is given by

TC=wL+rK=1*L+1*K

TC=L+K

TC=q²+8+q²=2q²+8

c)

ATC=TC/q=2q+(8/q)

MC=dTC/dq=4q

Set ATC=MC for cost minimization

2q+(8/q)=4q

2q=8/q

q=2

Cost minimizing output is 2 units

Minimum Cost=2q+(8/q)=2*2+(8/2)=$8

Price in long run=Minimum ATC=$8

d)

Let us calculate quantity demanded at P=$8

P=40-Q

Put P=8

8=40-Q

Q=32

Number of firms in long run=Quantity demanded/Output of each firm=32/2=16