Question

1.1. What is the profit-maximizing inputs if the profit function of a firm is the following:

π(X, Y) = P ln[X + aY] – wX – wY

where

P - price of output

f(X,Y) = ln[X + 0.5Y] - production function

X - input 1, X>=0

Y - input 2, Y>=0

w - same price of input for inputs 1 and 2

a - parameter between 0 and 1

1.2. What is the profit-maximizing inputs if the profit function of a firm is the following:

π(X, Y) = 20 ln[X + .5Y] – 2X – 2Y

X - input 1, X>=0

Y - input 2, Y>=0

Answer 1

According to the question:

X and Y are substitutes. And prices are the same.

maximize profit = ln [X+0.5Y] -wX - wY

x=0 then Y is

from equation (2)

Y =1/w

profit - ln[0.5/w] -1

y = 0

from equation (1)

x =1/w

profit - ln[1/w] -1

so , here profit is greater from employing only x or y depends on w.

b ) Similarly, in this case:

maximize profit =20 ln [X+0.5Y] -2X - 2Y

x=0 then Y is

from equation (2)

Y =10

profit - ln[5] -20 = 1.609 -20 = -18.3

y = 0

from equation (1)

x = 10

profit - ln[10] -20 = 2.30-20 = -17.7

I think firm will emply x as loss is less than loss incurred by emplying y