Question

A firm produces and sells two commodities. The demand function for one commodity is x= 24− 1/4 Pₓ. The demand function for the other commodity is y = 42− 1/2 Pᵧ. Note, the quantities are measured in tons. The total cost of producing and selling x tons of the first commodity and y tons of the second is given by C(x,y)=2x²+2xy+y²

. a) Solve the firm's profit maximisation problem. What is the demand elasticity of price for each commodity at the maximum point?
b) Suppose the firm's production causes pollution and the local authorities decide to apply a tax on its production, T=t(x+y), where t is a parameter. Given the tax, the firm maximises its profit and the authorities maximise their tax revenue. What is the maximum profit the firm can obtain? What is the demand elasticity of price for each commodity at the maximum point? .

Answer 1

To solve the firm's profit maximization problem, we need to find the maximum point of the profit function and calculate the demand elasticities of price for each commodity. In the case of a tax on production, we need to adjust the profit function and calculate the maximum profit and the demand elasticities of price.

Step-by-step explanation:

To solve the firm's profit maximization problem, we need to find the maximum point of the profit function. The profit function is given by P(x, y) = TR(x, y) - TC(x, y), where TR(x, y) is the total revenue function and TC(x, y) is the total cost function. To find the maximum point, we need to take the partial derivatives of the profit function with respect to x and y, set them equal to zero, and solve the resulting equations simultaneously.

To find the demand elasticities of price for each commodity at the maximum point, we can use the demand functions and the revenue function. The demand elasticity of price is given by the formula ε = (1/2)(P/Q)(dQ/dP), where P is the price, Q is the quantity, and dQ/dP is the derivative of the quantity with respect to the price.

For part b, to find the maximum profit the firm can obtain with the tax, we need to adjust the profit function by subtracting the tax. The new profit function becomes P(x, y) - T(x + y), where T(x + y) is the tax function. To find the demand elasticities of price for each commodity at the maximum point, we can use the adjusted demand functions and the adjusted revenue function.