Given production function ,Y=X^a in maximum profit .
a) calculate explicitly of the profit function and verify that its homogeneous and convex (P,W)
To calculate the profit function from the given production function, we first need to determine the cost function. The cost function represents the total cost of producing a given level of output. In this case, we can assume that the cost function is represented by the function C(Y).
To find the explicit profit function, we subtract the cost function from the revenue function. The revenue function is simply the price of the output (P) multiplied by the quantity produced (Y). Therefore, the profit function (Π) can be calculated as follows:
Π(P,W) = PY - C(Y)
Now, to verify that the profit function is homogenous and convex with respect to prices (P) and input prices (W), we need to analyze its properties.
1. Homogeneity: A function is homogeneous of degree k if multiplying all its inputs by a positive constant c, the function's output is also multiplied by c^k. In this case, we need to check if Π(cP,cW) = c^k * Π(P,W).
For the given production function Y = X^a, we can transform it into a function of prices and input costs by introducing price (P) and input costs (W). Let's assume the cost function is represented by C(Y, W).
Now, we can write the profit function as:
Π(P,W) = PY - C(Y, W)
If we multiply both P and W by a constant c, the profit function becomes:
Π(cP, cW) = cPY - C(cY, cW)
Applying the original production function Y = X^a, we get:
Π(cP, cW) = cX^a * cP - C(X^a, cW)
Since both P and W are multiplied by a constant c, we can see that the profit function Π(cP, cW) can be written in the form c^k * Π(P,W), where k = a. Therefore, the profit function is homogeneous of degree a.
2. Convexity: A function is convex if, for any two points (P1,W1) and (P2,W2), and for any value λ between 0 and 1, the function satisfies the following condition:
Π(λP1 + (1-λ)P2, λW1 + (1-λ)W2) ≤ λΠ(P1,W1) + (1-λ)Π(P2,W2)
To verify convexity, we can take the second derivative of the profit function with respect to prices (P) and input costs (W). If the second derivative is positive, then the function is convex.
So, let's calculate the second derivatives of the profit function Π(P,W) with respect to P and W:
∂^2Π/∂P^2 = 0
∂^2Π/∂W^2 = 0
Since both second derivatives are zero, we cannot determine the convexity of the profit function with this information alone. Further analysis would be required.
In summary, to calculate the explicit profit function, we subtract the cost function from the revenue function. To check if it is homogeneous, we verify if it satisfies the condition Π(cP,cW) = c^k * Π(P,W), where k is the degree of homogeneity. To determine convexity, we need to take the second derivatives of the profit function with respect to prices and input costs.