Question

1. A competitive, profit-maximizing firm uses two inputs a and b. Its production function is F(a, b) = a^1/2+b^1/2. Its output sells for $4 per unit. The price of input a is $1 per unit. The price of input b is $3 per unit.

What is the profit maximizing amount for factor a?

**(SOLVED, DO NOT ANSWER)**

Answer 1

Profit maximizing amount for factor a is given by the equality of value of marginal product of input a (VMPa) with price of input a (Pa).

And, VMPa = Price of output*(Marginal product of input a, MPa) = 4*MPa

MPa =

So, VMPa = Pa gives
4*(1/2)a^-1/2 = 1
So, 2 a^-1/2 = 1
So, 2(1/a)^1/2 = 1
So, (1/a)^1/2 = 1/2
So, 1/a = (1/2)²
So, 1/a = 1/4
So, a = 4

The profit maximizing amount for factor a is 4 units.