Question

Suppose firm j’s output is given by y_j = n^1-a, where 0 < a < 1 ('a' is a parameter). Suppose the firm must pay a fixed cost b < a if it wants to operate. That is, the firm’s profits are given by: Pi(n_j) = 0, if  n_j = 0 and Pi(n_j) = n_j^1-a - wn_j - b, if n_j > 0. Where w is wage  

a. Under what condition on w (in terms of the parameters a and b) would firm j be willing to hire a positive number of workers (i.e., choose n_j > 0)? Let ¯ w denote the level of w for which this condition is just met, and ¯n the amount of labour the would the firm hire if w = ¯ w and the firm chose to operate. Solve for ¯n in terms of a and b. (HINT:
Assuming the firm does operate, what’s the optimal level of n? When is this better than choosing n = 0?)

b. Explain intuitively the reason for the condition on w you found in part (a). In what direction does ¯ w change if the fixed cost b increases? Explain your answer.

c. Give a mathematical statement of firm j’s labour demand function n_j as a function of the wage w.

d. Give a mathematical statement of average labour demand n (as a function of w) across all firm’s. For any cases where an individual firm would be indifferent between operating and not operating, assume that a fraction of them choose to operate and the remainder don’t, where any such fraction between 0 and 1 (inclusive) is a possible outcome. Draw a diagram showing this (average) labour demand curve.

**Please show all steps and Pi is the pi symbol which means profit**

Answer 1

a) A firm would hire a positive number of workers only when > or 0 i.e (1-wj)nj -(aj+b)> or 0 i.e nj> or (aj+b)/(1-wj)

so,nj>0, aj+b >0 and 1-wj>0 i.e wj<1 and aj+b> or = 1-wj i.e (aj+b-1) > or = -wj , wj>or (1-aj-b)

w- = 1-aj-b (n- =1 which is greater than zero)

b) A firm would hire a positive number of workers only when > or 0 and at n=1 since =0 (firm would just be breaking even ) .If fixed cost b rises then w- falls.

c) A firm would hire a positive number of workers only when > or 0 i.e (1-wj)nj -(aj+b)>or 0 i.e nj> or (aj+b)/(1-wj)

d) In a scenario where firm is indifferent between operating and not operating

p be the fraction of firms who decide to operate and 1-p be the fraction of firms deciding not to operate while breaking even

If a firm decides to operate then he would hire

nj=aj+b/(1-wj) where 1-aj+bj <wj<1

but if the firm decides not to operate then nj=0

So,total labor demand function would be = p*aj+b/(1-wj) where 1-aj-bj <wj<1