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A firm uses two inputs, X and Y and its production function is Q = √(xy). Assume that the demand for the firm’s product is QD = 24 - 4P, where P is measured in dollars.
For the rest of the problem, assume that px=$4 and py=$1.
(j) Calculate the solution to the firm’s problem. What are its profit-maximizing price and quantity of output? How much of each input does it use? What is the firm’s maximum attainable profit?
(k) At the firm’s profit-maximizing level of output, calculate its average cost and marginal cost.
(l) Draw the isocost line that shows the combinations of X and Y that have a total cost of $20. You should show the exact coordinates of the intercepts. Repeat this question, on the same diagram, for the isocost lines that imply a total cost of $12 and $32.
(m) On the same diagram, show the firm’s optimal point and the isoquant and isocost line that are tangent at that point. What is the firm’s marginal rate of technical substitution of X for Y, at that point?
(n) Suppose that the firm decides to produce Q=8. Its problem is to choose x and y to produce Q=8 at the lowest possible cost. Use the tangency condition (part of the first-order conditions) to solve this problem. Calculate the optimal choices of x and y and the lowest possible cost of producing Q=8.