2 Consider the following production function a 10 points Sol

2. Consider the following production function: (a) (10 points) Solve the cost minimization problem for the firm given the pro- duction function. (b) (5 points) Show that for a-1/2 the cost function can be written as (c) (5 points) For wi = 1 and tv2 = 4 solve the profit maximization for the producer for the cost function given in part (b). [Hint: firm is choosing y and price of output is denoted by p.] profit? profit? (d) (2 points) If p- 2 how much would the firm supply? What would be it\'s (e) (2 points) If p 4 how much would the firm supply? What would be it\'s

Solution

Cost minimisation problem can be given as below

Z=w1x1+w2x2-L(x1^a*x2^(1-a)-f(x1,x2))

dZ/dx1=w1-L*a(x2/x1)^(1-a)

dZ/dx2=w2-L*(1-a)(x1/x2)^a

w1/w2=(a/(1-a))(x2/x1)

w1=(a/(1-a))w2(x2/x1)

if a=1/2 then a/1-a=1

w1=w2x2/x1

C=2w2x2

f(x1,x2)=(x1x2)^0.5

C=2w2x2=2(w1x1*w2x2)^0.5=2(w1w2)^0.5(x1x2)^0.5=2(w1w2)^0.5*f(x1,x2)=2(w1w2)^0.5*y is proved

Ans c)

Profit=p*y-2(w1w2)^0.5y=(p-4)y

Profit=(p-4)(x1x2)^0.5

Ans d)

if p=2 then firm would not produce anything because Profit<0

if p=4 then firm has zero profit

if p=6 then firm has profit=(6-4)*y=2y