# ECO Homework #12 SP17

Subject: Economics / Accounting
Question
Homework #12
(due April 11) Except where noted you should solve problems analytically, not using a calculator. You should also simplify
your 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 the
quantities 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 firm’s marginal rate of technical substitution of X for Y, at the point where it chooses
x=5 and y=25? For the rest of the problem, assume that px=8 and py=3.
(e)
(f) Calculate the firm’s 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 firm’s product is Q(P) = 90 ! 6P. The firm’s
objective is to choose the input combination (x,y) that maximizes profit.
(g)
(h)
(i)
(j)
(k)
(l) Calculate the FOC’s for the firm’s profit-maximation problem.
Find an integer solution to the FOC’s. (This is difficult analytically, so you can use any method.)
You can ignore any other solution to the FOC’s.
What are the boundary points of the profit-maximization problem? Under what conditions could the
optimal point occur on the boundary?
Calculate the firm’s 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 passes
through 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 exact
coordinates 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: the
payoffs in the first column become the payoffs in the second column, and the payoffs in the second column
become the payoffs in the first column. Given this adjusted game, repeat questions (a)-(d) from the practice
problem.