A producer produces good y using a single input x according to the production fu
ID: 1181015 • Letter: A
Question
A producer produces good y using a single input x according to the production function y=x^a where 0<a<1. The producer can sell as much as he wants at a unit price p, but must pay wage w for every unit of x used.
1) What are the first and second-order condition for the profit maximization problem?
2) Derive the expression for the profit maximizing level of output y*(p,w,a), the profit maximizing level of input x*(p,w,a), and the optimal profit pi*(p,w,a) = py*(p,w,a)
A producer produces good y using a single input x according to the production function y=x^a where 0Explanation / Answer
Given: Output, y = x^a
=> Revenue = p*y = p*x^a
=> Cost = w*x
=> Profit = Revenue - Cost = p*x^a - wx ----- (i)
To maximize profit, d(Profit)/dx = 0 and the second order derivative is negative.
Hence, pax^(a-1) -w =0 and the second order derivative = pa(a-1)x^(a-2) <0. Here, p>0, a>0, a-1 <0. So, the second order differential is negative anyway.
1) So, the first and second order condition for the profit maximization problem are
pax^(a-1) -w =0, and
pa(a-1)x^(a-2) <0
2) From the condition, x* = (w/pa)^[1/(a-1)]
Substituting this in equation for y, we get y* = (w/pa)^[a/(a-1)]
From (i), Profit = p * (w/pa)^[(a/(a-1)] - w * (w/pa)^[1/(a-1)]. You can simplify this.
3) When a = 1, y=x. Profit function = py - wx = px - wx = (p-w) x. The first order differential d(profit)/dx = p-w. This is not a function in x. So, the profit is a linear function and the profit is maximum with the maximum input used and maximum output produced.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.