Question

1. Consider a risk-neutral firm that operates for two periods with a production function

that depends only on the amount of labor hired: f(L) = 100L

1/2

. Assume that the

interest rate (r) is equal to 5%. The wage in the first period is equal to $10 per hour,

but the second period’s wage is either $10 (with probability 0.4) or $20 (with

probability 0.6). The current price for the firm’s output is P

0

=$20. In the second

period, the price is either P

1

=$20 (with probability ½) or P

1

=$30 (with probability ½).

a) Assuming the firm cannot store output over time, what is the expected value of

the presented discounted value of lifetime profits for this firm?

(Hint: Recall that if you receive a cash flow = CF in T years in the future, the

value of this cash flow today (i.e. the present discounted value of CF) is equal to

CF/(1+r)

T

.)

b) Find the expected profit maximizing choice of labor in both periods.

c) Given the levels of labor you found in part b, what are the corresponding levels of

output?

d) Now suppose the firm hires union workers that demand a two period contract. In

other words, the same amount of labor must be hired in both periods (but the

wages may be different in the two periods). How will the firm’s optimal choice of

labor be affected by this labor market rigidity?

e) Now suppose the firm has access to inventory storage that costs h(s) = s

2

in the

second period, where s is the amount stored in inventories. Assuming labor

market rigidity as in part d, how does the introduction of inventory storage change

the firm’s behavior?

f) Explain the intuition behind the firm’s behavior in part e.

Answer 1

Solution - GUENO- 5%. (2). . Interest rate = Production function with two periods that depends only on the art of labor hired. lil. f (L) = 100272 Here, the wage in the 1st period $10 tho. 1 be. Wo= 10. The wage in the 2nd period is either $10 with Probability 0.4 or 4 20 with poobability 0.6 2nd Output price of 1st period Po = $20 Oufput price of period of either P, = $20 with probability / or with $30 with probability / cas Expected value of the presented discounted value of lifetime profits for the firm. TT = (PoQo- Nobo ) + (2010«+ 30x0.5)2, - (10x04 +20x0.6) 4,7 Iru T (20810044 - 1010) + ftp (25 x 1004 – 161] T= 2000.' – lobo + tir (2500 2,4 - 1621) T = 20004%2 – lobo + 1 2 1+r

(6) maximizing The expected profit in both periods. choice of labor auto - 2000 - 10 = » toe L'e > Le = 100 Log = 100 - 10,000 duro ( 2.500 ,Y- 16) = 0 * 12501,4-16 co > X = 12.50 = based >L, Y = 48.125 16 L = 6, 103:52 Henice maximum choice of labor for 1st period = L t = 10,000 and 2 period =6,103.52

(C) Given the levels of labor you found in part ba what are the corresponding levels of output ? 2. = 10002 %* = 100x10.000ks = 10000 Moco Q, = 100 %= 100 x 6103.5e/z R = 100x 780125 Q = 4812.5 (d) Firm hires union workers that demand a tivo period contract. The firm's optimal choice of labor be affected by this labor market rigidity. Under the condition the expected profit condition Lo=4,= L l Giren). ' & the same amount of labours must be - hired in both the period contract. 14. 7 20002 - 10L + I (2500W - 161) 1 +7 EdT -0 1. dL .

Hence, dr =0. * 2000,"a. n) + top 2.500 i9_16)=0 -1002_o) + Tivos ( 12602%..) - (1000 + 1190. 48) '* - (10 + 15.24) = (219.48)L" 25.24 1932: 25.24 7.86.79 + 1190 219.48 LE - 753).82 choice under this Hence, the optimum labor - Condition = 1531.82 storage changes (e) The introduction of inventory the firm's behaviour. Yes, it changes from the we find that L LL above and circumstances t>".

if) Explanation: The inventory storage changes the firm's behaviour from part (e) 1 i.e. t < to * > L, fence, In the 1st period the employment . decrease, but in the 2nd period employ -ment increases as compared to the last case of the firm's behavior. - - Thanking you -pls like y support