Consider the following profit function that has been obtained from a technology that uses a single input π(P.W) = P^2W^BWhere P is the output price and W is the input price and ßis a parameter value i. For which values of ẞ the profit function is a real profit function with all of the appropriate properties? (9 marks) ii. Calculate the supply function of the product and the demand for inputs
i. For the profit function to be a real profit function with all the appropriate properties, it needs to satisfy certain conditions. In this case, the profit function is given by π(P,W) = P^2 * W^ß.
For the profit function to be real, both P and W must be greater than or equal to 0, as negative prices or quantities do not make sense in this context. Therefore, P ≥ 0 and W ≥ 0.
The profit function should also be concave in both P and W to ensure that the profit is maximized at the optimal level of output and input. This requires the second-order partial derivatives to be negative:
∂^2π/∂P^2 = 2(ß-1)P^2W^ß < 0
∂^2π/∂W^2 = ß(ß-1)P^2W^(ß-2) < 0
Solving these inequalities, we get:
ß > 1 for concavity in P
ß > 1 for concavity in W
Therefore, for the profit function to be a real profit function with all the appropriate properties, the parameter ß must be greater than 1.
ii. To find the supply function, we need to determine the optimal level of output that maximizes profit. This involves setting the marginal revenue equal to the marginal cost:
MR = ∂π/∂P = 2PW^ß = MC
Assuming that the marginal cost is equal to the input price W, we have:
2PW^ß = W
2P = 1
P = 1/2
Therefore, the optimal output level is P = 1/2. This represents the supply function of the product.
To find the demand for inputs, we can maximize profit by setting the marginal cost of inputs equal to the input price:
MC = ∂π/∂W = P^2ßW^(ß-1) = W
P^2ßW^(ß-1) = W
P^2ßW^(ß-1) = W
P^2ß = 1
(1/2)^2ß = 1
2ß = 4
ß = 2
Therefore, the demand for inputs can be calculated by setting the parameter ß to 2, which ensures profit maximization.