Useful Math for Microeconomics∗

Useful Math for Microeconomics∗

Useful Math for Microeconomics¤ Jonathan Levin Antonio Rangel September 2001 1 Introduction Most economic models are based on the solution of optimization problems. These notes outline some of the basic tools needed to solve these problems. It is wort spending some time becoming comfortable with them — you will use them a lot! We will consider parametric constrained optimization problems (PCOP) of the form max f(x; θ): x2D(θ) Here f is the objective function (e.g. profits, utility), x is a choice variable (e.g. how many widgets to produce, how much beer to buy), D(θ) is the set of available choices, and θ is an exogeneous parameter that may affect both the objective function and the choice set (the price of widgets or beer, or the number of dollars in one’s wallet). Each parameter θ defines a specific problem (e.g. how much beer to buy given that I have $20 and beer costs $4 a bottle). If we let Θ denote the set of all possible parameter values, then Θ is associated with a whole class of optimization problems. In studying optimization problems, we typically care about two objects: 1. The solution set x¤(θ) ´ arg max f(x; θ); x2D(θ) ¤These notes are intended for students in Economics 202, Stanford University. They were originally written by Antonio in Fall 2000, and revised by Jon in Fall 2001. Leo Rezende provided tremendous help on the original notes. Section 5 draws on an excellent comparative statics handout prepared by Ilya Segal. 1 that gives the solution(s) for any parameter θ 2 Θ. (If the problem has multiple solutions, then x¤(θ) is a set with multiple elements). 2. The value function V (θ) ´ max f(x; θ) x2D(θ) that gives the value of the function at the solution for any parameter θ 2 Θ(V (θ) = f(y; θ) for any y 2 x¤(θ):) In economic models, several questions typically are of interest: 1. Does a solution to the maximization problem exist for each θ? 2. Do the solution set and the value function change continuously with the parameters? In other words, is it the case that a small change in the parameters of the problem produces only a small change in the solution? 3. How can we compute the solution to the problem? 4. How do the solution set and the value function change with the param- eters? You should keep in mind that any result we derive for a maximization problem also can be used in a minimization problem. This follows from the simple fact that x¤(θ) = arg min f(x; θ) () x¤(θ) = arg max ¡f(x; θ) x2D(θ) x2D(θ) and V (θ) = min f(x; θ) () V (θ) = ¡ max ¡f(x; θ): x2D(θ) x2D(θ) 2 Notions of Continuity Before starting on optimization, we first take a small detour to talk about continuity. The idea of continuity is pretty straightforward: a function h is continuous if “small” changes in x produce “small” changes in h(x). We just need to be careful about (a) what exactly we mean by “small,” and (b) what happens if h is not a function, but a correspondence. 2 2.1 Continuity for functions Consider a function h that maps every element in X to an element in Y , where X is the domain of the function and Y is the range. This is denoted by h : X ! Y . We will limit ourselves to functions that map Rn into Rm, so X ⊆ Rn and Y ⊆ Rm. Recall that for any x; y 2 Rk, s X 2 kx ¡ yk = (xi ¡ yi) i=1;:::;k denotes the Euclidean distance between x and y. Using this notion of distance we can formally define continuity, using either of following two equivalent definitions: Definition 1 A function h : X ! Y is continuous at x if for every " > 0 there exists ± > 0 such that kx ¡ yk < ± and y 2 X ) kh(x) ¡ h(y)k < ". Definition 2 A function h : X ! Y is continuous at x if for every sequence xn in X converging to x, the sequence h(xn) converges to f(x). You can think about these two definitions as tests that one applies to a function to see if it is continuous. A function is continuous if it passes the continuity test at each point in its domain. Definition 3 A function h : X ! Y is continuous if it is continuous at every x 2 X. Figure 1 shows a function that is not continuous. Consider the top pic- ture, and the point x. Take an interval centered around h(x) that has a “radius” ". If " is small, each point in the interval will be less than A. To satisfy continuity, we must find a distance ± such that, as long as we stay within a distance ± of x, the function stays within " of h(x). But we cannot do this. A small movement to the right of x, regardless of how small, takes the function above the point A. Thus, the function fails the continuity test at x and is not continuous. The bottom figure illustrates the second definition of continuity. To meet this requirement at the point x, it must be the case that for every sequence xn converging to x, the sequence h(xn) converges to h(x). But consider 3 6 A q a h(x) q 2" q 2± q - x 6 qh(zn) q ³) A q? a h(x) q q ©* q6 q h(yn) q q q - yn- x zn Figure 1: Testing for Continuity. 4 the sequence zn that converges to x from the right. The sequence h(zn) converges to the point A from above. Since A > h(x), the test fails and h is not continuous. We should emphasize that the test must be satisfied for every sequence. In this example, the test is satisfied for the sequence yn that converges to x from the right. In general, to show that a function is continuous, you need to argue that one of the two continuity tests is satisfied at every point in the domain. If you use the first definition, the typical proof has two steps: ² Step 1: Pick any x in the domain and any " > 0. ² Step 2: Show that there is a ±x(") > 0 such that kh(x) ¡ h(y)k < " whenever kx ¡ yk < ±x("). To show this you have to give a formula for ±x(¢) that guarantees this. The problems at the end should give you some practice at this. 2.2 Continuity for correspondences A correspondence Á maps points x in the domain X ⊆ Rn into sets in the range Y ⊆ Rm. That is, Á(x) ⊆ Y for every x. This is denoted by Á : X ¶ Y . Figure 2 provides a couple of examples. We say that a correspondence is: ² non-empty-valued if Á(x) is non-empty for all x in the domain. ² convex if Á(x) is a convex set for all x in the domain. ² compact if Á(x) is a compact set for all x in the domain. For the rest of these notes we assume, unless otherwise noted, that corre- spondences are non-empty-valued. Intuitively, a correspondence is continuous if small changes in x produce small changes in the set Á(x). Figure 3 shows a continuous correspondence. A small move from x to x0 has a small effect since Á(x) and Á(x0) are approx- imately equal. Not only that, the smaller the change in x, the more similar are Á(x) and Á(x0). Unfortunately, giving a formal definition of continuity for correspondences is not so simple. With functions, it’s pretty clear that to evaluate the effect of moving from x to a nearby x0 we simply need to check the distance be- tween the point h(x) and h(x0). With correspondences, we need to make a 5 6 X = Y = [0; 1). q - 6 ©p p ©p p p p p©p p p p p ©p ©p p p p p p p ©p p p p p p p p p p ©p p p p p p p p p p © ©p p p p p p p p p p © p©p p p p p p p p ©p © ©p ©p p p p p p p p ©p ©p p p p p p p p p p© ©p p p p p p p p p p © - ©p p p p p p p p p p © p©p p p p p p p p ©p © ©p ©p p p p p p p p ©p ©p p p p p p p p p p© p p p p p p p p © p p p p p p © X = Y = (¡1; 1). p p p p © ©p © ©p Figure 2: Examples of Correspondences. 6 6 0 p Á(x ) p ©p p ©p ©p p p p £ ©p p p p p p p Á(x) ©p p p p p p p p p £p©p p p p p p p p ©p p ©p p p p p p p p ©p A p ©p p p££ p p p p ©p p© ©p p©p p p p p p ©p p ©p p p pAUp p p p p©p p©p p p p p p p p ©p p ©p p p p p p p p ©p p ©p p p p p p p p p© p p©p p p p p p ©p p © ©p p ©p p p p p p p©p ©p p p p p p p p ©p p p p p p p ©p p p p p p© p ©p p © ©p q q - x x0 Figure 3: A Continuous Correspondence.

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