∗
                                 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).
                                                     x∈D(θ)
                      Here f is the objective function (e.g. pro¯ts, 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 a®ect 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 θde¯nes a speci¯c
                      problem (e.g. how much beer to buy given that I have $20 and beer costs
                      $4a bottle). If we let £ denote the set of all possible parameter values, then
                      £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),
                                                             x∈D(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.
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                       that gives the solution(s) for any parameter 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)
                                                   x∈D(θ)
                       that gives the value of the function at the solution for any parameter
                                                     ∗
                       θ ∈ £ (V(θ) = f(y,θ) for any y ∈ 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)
                                     x∈D(θ)                   x∈D(θ)
                   and
                             V(θ) = min f(x,θ) ⇐⇒ V(θ) = − max −f(x,θ).
                                    x∈D(θ)                 x∈D(θ)
                   2   Notions of Continuity
                   Before starting on optimization, we ¯rst 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.
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                   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
                         n          m
                   X⊆R andY ⊆R .             k
                      Recall that for any x,y ∈ R ,
                                                s X           2
                                       kx−yk=          (xi − yi)
                                                  i=1,...,k
                   denotestheEuclideandistancebetweenxandy. Usingthisnotionofdistance
                   we can formally de¯ne continuity, using either of following two equivalent
                   de¯nitions:
                   De¯nition 1 A function h : X → Y is continuous at x if for every ε > 0
                   there exists δ > 0 such that kx − yk < δ and y ∈ X ⇒ kh(x)−h(y)k < ε.
                   De¯nition 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 de¯nitions 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.
                   De¯nition 3 A function h : X → Y is continuous if it is continuous at
                   every x ∈ 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” ε. If ε is small, each point in the interval will be less than A. To
                   satisfy continuity, we must ¯nd a distance δ such that, as long as we stay
                   within a distance δ of x, the function stays within ε 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.
                      Thebottom¯gure illustrates the second de¯nition of continuity. To meet
                   this requirement at the point x, it must be the case that for every sequence
                   x converging to x, the sequence h(x ) converges to h(x). But consider
                    n                               n
                                                  3
                                                                                                        ✻
                                                                                                A q                                             ❛
                                                                                          h(x) q 2ε                                             q
                                                                                                                                              2δ
                                                                                                                                                q                                                      ✲
                                                                                                                                               x
                                                                                                        ✻
                                                                                                        q h(z )                                           q
                                                                                                        q ❄ n                                   ❛  ✏✮
                                                                                                  A
                                                                                           h(x)q                                        ✟✯ q
                                                                                         h(yn)q✻                                      q
                                                                                                                                      q ✲ q ✛ q                                                        ✲
                                                                                                                                   y           x z
                                                                                                                                      n                    n
                                                                                                                Figure 1: Testing for Continuity.
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