A firm producing two products x and y where x and y are the

Question

Question -

a. A firm producing two products X and Y where x and y are the quantity of product X and Y produced respectively. If the firm produces on the same isocost and has a fixed cost of $1000.

Given the marginal costs, ∂C/∂x = 14xy + 15x² and ∂C/∂y = 7x² + 12y.

i. Show that (∂C/∂x)dx + (∂C/∂y)dy = 0 is an exact differential equation.

ii. Hence, solve the differential equation to find the total cost TC(x, y).

b. A firm produces two goods, X and Y. If x is the quantity of good X demanded and y is the quantity of good Y demanded and their corresponding prices p_x and p_y.

Given that the demand functions of the firm are: x = 2p_y - 4p_x + 80 and y = -2p_y + 2p_x + 50. Further, that the joint total cost function of the firm is C(x, y) = x² + xy + 3y².

i. Find an expression in terms of x and y for the profit function.

ii. Determine the quantities of x and y that maximizes the profit.