Question

Brandon raises apples. Where L is the number of units of labor he uses and T is the number of units of land he uses, his output is ?(?, ?) = 2√?? bushels of apples

a) Write an equation for the isoquant that gives him an output of 100 bushels?

From now on, assume that the number of units of land is fixed at T= 8 in the short run. Per-unit prices of apple,

labor and land are p = 3, wL = 2 and wT = 1

b) Write a short run production function reflecting a short run restriction on the units of land

c) Write a short run profit function that states Brandon’s profit as a function of the amount of labor

d) Find out profit maximizing unit of labor. How much bushels of apples will he produce? What is his

maximized profit?

Answer 1

a)

Q = f(L,T) = 2(LT)1/2  

Q = 2(LT)1/2  

Q/2 =  (LT)1/2  

LT = (Q/2)2  

T = (1/L )(Q/2)2  

Q = 100

T = (1/L)(100/2)2

= 2500/L

Thus equation of isoquant for output of 100 bushels is T = 2500/L  

b)  

Q = 2(LT)1/2

T = 8  

Q =   2(L8)1/2  

Q = 2(222L)1/2

Q = 22(2L)1/2

Q = 4(2L)1/2  

Short run production function is  

c)

f(L) = Q = 4(2L)1/2  

Profit = PQ - (wL)L - (wT)T

= 3[4(2L)1/2 ] - 2L - 18  

= 12(2L)1/2   - 2L - 8

Short run profit function is = 12(2L)1/2   - 2L - 8

d)    = 12(2L)1/2 - 2L - 8

   ddL = 12(1/2)(2)1/2(L)1/2 -1 - 2 - 0  

= 6(2)1/2(L)- 1/2 - 2  

Put   ddL = 0  

6(2)1/2(L)- 1/2 - 2 = 0

6(2)1/2(L)- 1/2 = 2  

6(2)1/2   = 2(L)1/2

(L)1/2 = 3(2)1/2  

L = 9(2) = 18

Q = 4(2L)1/2

Q = 4(218)1/2  

= 4(36) 1/2  

= 46 = 24

Profit = PQ   - (wL)L - (wT)T

= 324 - 218 - 18  

= 72 - 36 - 8  

= 28