Question

ECNS 301 Homework #: 2 5. A firm's output, Q, depends on the number of workers it hires, L. Its production function is Q = 4L0.5. It can sell its output for $2, and workers are paid $1. It has no costs other (20) than what it pays its workers. (a) What are profits expressed in terms of the number of workers hired? (b) What is the optimal number of workers it should hire? What are profits when the optimal number of workers are hired? (c) The firm unionizes, and the union receives 25% of ownership of the firm. The union does not care about profits, but only cares about the number of workers hired (L). The other 75% of firm owners continue to care only about profits. The new objective function of the firm is as follows. 0.757T (L) +0.25L How many workers are hired with the union? What are the firm's profits?

Answer 1

a).

Consider the given problem here the production function is given by, “Q=4*L^0.5”. Now, the output price is given by “P=$2” and the wage rate is given by “W=$1”. So, the profit function is given by as follows.

=> A(L) = P*Q – W*L= 2*4*L^0.5 - 1*L= 8*L^0.5 – L, => A(L) = 8*L^0.5 - L.

b).

The profit function is given by, “A(L) = 8*L^0.5 - L", => the FOC require “dA(L)/dL = 0”.

=> 8*0.5*L^(-0.5) - 1 = 0, => 4/L^0.5 = 1, => L^0.5 = 4, => L*=16. So, the optimum labor hiring is “L=16”. So, the optimum profit is given by.

=> A(L) = 8*L^0.5 - L= 8*16^0.5 - 16 = 16, => A(L) = 16. So, here the optimum profit is given by “A* = 16”.

c).

Now, the firm’s new objective function is given by.

=> Z(L) = 0.75*A(L) + 0.25*L = 0.75*[8*L^0.5 - L] + 0.25*L = 6*L^0.5 – 0.75*L + 0.25*L= 6*L^0.5 – 0.5*L.

=> Z(L) = 6*L^0.5 – 0.5*L, => FOC require “dZ/dL = 0”.

=> 6*0.5*L^(-0.5) – 0.5 = 0, => 3/L^0.5 = 0.5, => L^0.5 = 3/0.5 = 6, => L = 36.

So, here under the union the optimum numbers of hiring is “L=36”, => the optimum profit is given by “Z(L) = 6*L^0.5 - 0.5*L”, => “Z(L) = 6*36^0.5 - 0.5*36” , => “Z(L) = 6*6 – 18 = 16”. So, here the optimum profit is given by “Z* = 16”.