12 May DOES THIS PRODUCTION FUNCTION SATISFY THE LAW OF DIMINISHING RETURNS?
Homework #12(due April 11) Except where noted you should solve problems analytically, not using a calculator. You should also simplifyyour answers where possible. 1. A firm uses two inputs, X and Y and its production function is Q = (xy)b, where x and y represent thequantities of the two inputs.(a)(b)(c)(d) Calculate the marginal products of X and Y.Does this production function have increasing, decreasing, or constant returns to scale?Does this production function satisfy the Law of Diminishing Returns?What is the firmâs marginal rate of technical substitution of X for Y, at the point where it choosesx=5 and y=25? For the rest of the problem, assume that px=8 and py=3.(e)(f) Calculate the firmâs FOC for cost-minimization (the tangency condition).If the firm wants to produce Q=576, then calculate its cost-minimizing combination of inputs (x,y),and the minimized cost of production. (The FOC for this problem is the FOC from part (e).) For the rest of the problem, assume that the demand for the firmâs product is Q(P) = 90 ! 6P. The firmâsobjective is to choose the input combination (x,y) that maximizes profit.(g)(h)(i)(j)(k)(l) Calculate the FOCâs for the firmâs profit-maximation problem.Find an integer solution to the FOCâs. (This is difficult analytically, so you can use any method.)You can ignore any other solution to the FOCâs.What are the boundary points of the profit-maximization problem? Under what conditions could theoptimal point occur on the boundary?Calculate the firmâs price, quantity, and profit, if it maximizes profit.With input X on the horizontal axis and good Y on the vertical axis, draw the isocost line that passesthrough the optimal point. Show the exact coordinates of its intercepts.On the same diagram, draw the isoquant that passes through the optimal point. Show the exactcoordinates of at least six integer points on the curve, such that none of the coordinates exceeds 50.Based on these points, is the isoquant convex? 2. Consider the invasion game in the practice questions, except that the German payoffs are reversed: thepayoffs in the first column become the payoffs in the second column, and the payoffs in the second columnbecome the payoffs in the first column. Given this adjusted game, repeat questions (a)-(d) from the practiceproblem.
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