b) A short run production function is given as Q = x1/2 where Q is output and X is the input (i). Is the production function concave? Show your workings (3 marks) (ii). Let P represent the output price and W represent the input price. Derive the firm's profit function (9 marks) (iii). If a legitimate profit function is convex and positively linearly homogeneous. Is the profit function above legitimate? Show your workings
(i) To determine if the production function is concave, we need to check the second derivative of the function.
Q = x^(1/2)
The first derivative is: dQ/dx = (1/2)x^(-1/2)
The second derivative is: d^2Q/dx^2 = (-1/4)x^(-3/2)
Since the second derivative is negative for all values of x, the production function is concave.
(ii) The firm's profit function can be derived by taking the difference between total revenue and total cost.
Total Revenue = P * Q
Total Cost = W * X
Profit = Total Revenue - Total Cost
Profit = P * Q - W * X
Profit = P * x^(1/2) - W * X
(iii) A legitimate profit function must satisfy the following properties:
1. Convexity: The profit function should be concave.
2. Positively linearly homogeneous: This means that if all input prices and output prices are multiplied by a positive constant, the profit function should also be multiplied by the same constant.
From part (i), we know that the profit function is concave. Now, to check if the profit function is positively linearly homogeneous:
Let's multiply both output price (P) and input price (W) by a positive constant (k):
New Profit = k * P * x^(1/2) - k * W * X
New Profit = k * (P * x^(1/2) - W * X)
We can see that the profit function is multiplied by the constant k. Therefore, the profit function above is legitimate as it satisfies both conditions.