Question

5. Suppose a firm has the production function Q = K^1/2 L^1/4 M^1/4 The wage rate w = 16 , rental rate r = 2 , and the price of the materials m = 1.

1) Suppose in the short run, K is fixed at K *. What’s the solution to the firm’s short run cost-minimization problem?

2) What is the solution to the firm’s long run cost minimization problem given that the firm wants to produce Q units of output?

3) Suppose that Q = 10, K* = 20. Compare the long run and short run demands.

Answer 1

Suppose a firm has the production function. Q: К% L4 м У . Now инв, in the short ren kis fined at k* да 41 24 - к+ + 2494 про щаяј0 +4 5 Давот (1) MPL = 8. к2-4 . Mosajimg + 4 + лаш маві (ч) :- MP м е н з якњ.24 m - 3/4 - Now . wage rate w= 16 and rental rate, 8=2 Price Now, optimal in put mir condition is i- of material M=1 - Да К. Ка7 2. К%-4 м 4 як «незу, а на 14 дм = 162 “MP ји : . : CS Scanned with CamScanner

Now, substituting M=16L in the Production function we get's att Q = k*%2/4 MY4 : a) Q = k* 1/2 2/4 (164)/4 =) Q **-1/2 = 22 =) Q**EL and as, M = 164 = 16 Q k* - * * Q! 2 28Q K*- Ance short hay demands will be LE QK*-Ź : 2 . .: K= k* . and im= 8QK*- - Now in the long run both k, Land the M .: are all variable. cs Scanned with CamScanner

4 | Nov @ = к-чи, Now MPL = do 4 K/22 -3/4 m/4 MP = 20 = Ź K-²2 14 M | bet : , чи 4 - -) MPя 20 : +к 42 - 2 Now optimal input mix condition will be MPK Ž K ² Lanta - , як - 2 — сії) 16 - 4 -ii) 1 / K / LY/4 M-1/4 men = set equation (2) = ук-1 L4 м, Equation (1) к. 2 «н-4 - - 2 - 1 . k- 4 2 Уч мм К% 244 м-° м = 1 +7 k= 1 -- tіv) 2) ма К Scanned with CamScanner

Now set equation (ill equation (1) - к+-4 м 4 Тка. **и - — 5 кк, 6 - К% -/4 м _4 м -34 K42 ** m-34 =) ~ - 16 КИ. [-4 мм - 16 , , , . К% 4м-V4 м= 16L няк и = 16 =) а) м. L=4 what? Huce substituting kam and in the Production fraction we get, - К., 4 м 4 -) Q3 м * (и 4 м 4 2) QО.sм ә) 2.0 = м o k2 м = 24 on L= = 1% Scanned with CamScanner

Hnce ren demands are i the long K=2Q L = 20 M=20 3 suppose that k*=20, Q=10 Shord ren demands will be Ź 10 (20)/2 2 <= QK* 2 L = 2 = 2.12 - k*= 20 and M = 80 k* -% = 8 (10) (20) 12 = 17.89 Huce short te ren demands are ! L = 1.12 : K*= 20 and M = 17189 why Q = 10 long run demands are: K= 2Q = (2x10) = 20 . L=2Q - 12/10 - 1:25 mis 20 = (2x 10) = 20 Hunce long run demands are:- k=20 L = 1.25 and M.=20 16 cs Scanned with CamScanner