Math Camp Questions - Section 2 The questions below are designed to illustrate three important concepts in constrained optimization: the Kuhn-Tucker algorithm (Question 1), the Envelope Theorem (Question 2), and the Implicit Function Theorem (Question 3). They are written in two “languages”: math and economics. If you find that you do not understand some of the economics yet, that is fine; the problems should still be solvable purely as math problems. 1 The Kuhn-Tucker Algorithm In economics: A consumer is choosing between two goods x and y, and has utility function u(x, y) = x + log(y). She has w dollars to spend, and the prices of the two goods are p = (px , py ). Solve for the consumer’s demand as a function of her wealth and prices. In math: Solve the following optimization problem: max x + log(y) x,y subject to px x + py y w and x, y 0. That is, find the solution functions x⇤ (p, w) and y ⇤ (p, w) (be careful of corner solutions). 2 The Envelope Theorem (This is question 5.C.13 from MWG.) In economics: A price-taking firm produces output q from inputs z1 and z2 according to a diﬀerentiable, concave production function f (z1 , z2 ). The price of the output is p > 0 and the price of the inputs are (w1 , w2 ) > 0. However, there are two unusual things about this firm. First, rather than maximizing profits, the firm maximizes revenue. Second, the firm is cash constrained. In particular, it only has C dollars on hand before production and, as a result, its total expenditures cannot exceed C. Suppose one of your econometrician friends tells you she has used repeated observations of the firm’s revenues under various output prices, input prices, 1 and levels of financial constraint and has determined that the firm’s revenue level R can be expressed as the following function of the variables (p, w1 , w2 , C): R(p, w1 , w2 , C) = p( + log C ↵ log w1 (1 ↵) log w2 ). and ↵ are scalars that she tells you. What is the firm’s use of output z1 when prices are (p, w1 , w2 ) and it has C dollars of cash on hand? In math: The following is an alternate expression for R: R(p, w1 , w2 , C) = max pf (z1 , z2 ) s.t. w1 z1 + w2 z2 C z1 ,z2 0 where f is unknown. Apply the envelope theorem to find z1⇤ (p, w1 , w2 , C), the revenue-maximizing choice of input z1 . 3 The Implicit Function Theorem In economics: Let f (x, y) be a concave production function with two inputs, x and y, with prices px and py . Let the price of the output good be p. Suppose @f that @f @x , @y > 0 everywhere. Use the first order conditions and implicit function theorem to show:1 1. An increase in the output price always increases the profit-maximizing level of output. 2. An increase in the output price increases the demand for some input. 3. An increase in the price of an input leads to a reduction in the demand for that input. In math: Consider the following optimization problem: max pf (x, y) x,y 0 px x py y Use the first order conditions and implicit function theorem to show: 1. @f (x⇤ (p,px ,py ),y ⇤ (p,px ,py )) @p 2. @x⇤ (p,px ,py ) @p > 0 or 3. @x⇤ (p,px ,py ) @px < 0 and 1 You >0 @y ⇤ (p,px ,py ) @p > 0 (or both) @y ⇤ (p,px ,py ) @py <0 may assume the solutions are interior and ignore the nonnegativity constraints. 2

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