Math Guide for Econ 11 These notes are intended as a reference guide for the mathematical tools of calculus that are used in Econ 11. The following pages provide all the mathematical results, along with brief explanations, illustrative examples of their use and additional exercises. You should have seen some, if not most of this material in your calculus course, especially the parts that are related to single-variable calculus. Some of the concepts may be new, but these will be discussed in detail, when they are introduced during the lectures; these notes are intended to serve you as a quick reference as you go along with the course. The first two TA sessions will review the material in this handout in detail. 1 Introduction Consider the following two examples: 1. A firm that has to decide how much to produce of a certain good. The market price 1 2 for the firm’s product is p, and it costs the firm C (q)= 2 q to produce a quantity q of the good. The firm wants to maximize its profits, i.e. sales revenue minus production costs, and can choose the quantity q that it produces. With a sales revenue pq,wecanwritethefirm’s 1 2 profits as F (q)=pq 2 q ,andthefirm’s decision problem consists in finding the quantity q that maximizes F (−q). 2. A student has to prepare for an exam the next day. She has 12 hours remaining before the exam. For each additional hour of studying, she expects to raise her grade by 2 points (out of a 100), but she also realizes that her concentration level during the exam depends on how much she sleeps. If she sleeps 7 hours, she is able to work with full concentration. If she sleeps less than that, her concentration goes down. If she didn’t study any more, and slept exactly y hours before the exam, she expects a grade of 11 + y (14 y) (for y=7, this gives a grade of 60; for y=6, this gives a grade of 59, for y=5, this gives a− grade of 56, and so on). Therefore, if the student sleeps y hours and studies x hours until the exam, her grade (as a function of x and y)isF (x, y)=11+y (14 y)+2x. The student wants to allocate her time optimally between sleeping and studying− so as to maximize her expected grade. But her choices of x and y have to satisfy the additional constraint that 12 x + y,i.e.the amount of time that the student studies plus the amount that she sleeps cannot≥ exceed the 12 hours remaining before the exam. These two examples are representative of the decision problems that are studied in Mi- croeconomics, and that we are concerned with in much of this course. Formally, we consider problems in which a decision-maker (i.e. a consumer, a household, a firm, etc. depending on the exact context), has to maximize an objective function F (x) with respect to some choice variable x.Inthefirst example above, the decision-maker was a firm, its objective 1 were profits, and its choice variable was the quantity. In some of these problems, x may consist of multiple elements, or it may be subject to additional constraints. Thisisthe case in the second example, where the decision-maker (the student) had to choose both x (the amount of time spent studying) and y (the amount of sleep), and had to satisfy the additional constraint that g (x, y)=12 x y 0. − − ≥ The advantage of formulating individual decision problems in this mathematical format is that we can then use the mathematical tools of optimization to analyze them. The firm’s problem of choosing how much to produce is an example of an unconstrained optimization problem, generally represented as max F (x) . x Thestudentexampleisaconstrained optimization problem, which is generally represented as max F (x) ,subjecttog (x) 0. x ≥ In both cases, x can consist of one or multiple variables, and the objective function F (x) can take many different forms; as does the constraint g (x) 0. The remainder of this hand-out introduces you to the mathematical techniques used to≥ solve this kind of problem. 2 Preliminaries 2.1 Derivatives Consider the following graph of a function F (x),infigure 1. You want to maximize this function with respect to the variable x.Inthefigure you see that this occurs at the point where the function is “flat”, i.e. at x∗ the “slope” of this function is zero. The slope of a function at a given point is given by the function’s derivative. Formally, for a function F (x), its derivative with respect to x is defined as dF F (x + h) F (x) =lim − dx h 0 h → dF In figure 1, you see that for a given value of x, dx is determined as the slope of F (x) at x. dF As you can easily convince yourself by changing x, dx varies with x, i.e. is itself a function of x.Wecanthendefine a “second derivative” simply as the derivative of the first derivative d2F (and consequently third derivatives etc.), and denote it dx2 . The second derivative gives us the change in the slope of a function at a certain point. In Nicholson, chapter 2, you find a series of useful rules for finding the derivatives of functions. You should memorize these rules! Example 1a: Intheexampleoftheprofit maximizing firm, the profit function is F (q)= pq 1 q2.ThederivativeofF with respect to q is dF = p q. The second derivative of F − 2 dq − with respect to q is d2F = 1. You can verify this using the derivative rules in Nicholson. dq2 − 2 dF dx x=x dF F(x) 1 = 0 dx x=x* F(x) * x x1 x Figure 1: dF Exercise 1a: using the definition of dx , find the derivatives of the functions (i) F (x)= x2; (ii) F (x)=√x (Note: the second example is somewhat harder than the first!). 1 1+2x Exercise 1b: Find the first and second derivatives of F (x)= 1+x ; F (x)=e ; 1+x2 F (x)= 1+x . 2.2 Functions with multiple variables In many cases, we have functions of multiple variables. For example, the grade of the student depended on x, the amount of studying, and on y, the amount of sleep. We can define the derivatives of multivariate functions in the same way as above. Let (x, y) be two variables, and let F (x, y) be a function of these two variables. The partial derivative of F with respect to x is defined as ∂F F (x + h, y) F (x, y) = lim − ∂x h 0 h → ∂F i.e. we simply take the derivative of F (x, y) with respect to x, but holding y constant. ∂x gives us the change of F (x, y) with respect to a change in x; this measures by how much the function F (e.g. the student’s grade) changes in response to a small change in x (the time spent studying), while holding fixed y (the amount of sleep). Likewise, we can define ∂F ∂F the partial derivative of F with respect to y, ,as = limh 0 (F (x, y + h) F (x, y)) /h, ∂y ∂y → − which gives the change of F (x, y) with respect to a change in y; i.e. in the grade example, ∂F ∂y measures by how much the student’s grade changes in response to a small change in y (the amount of sleep), holding fixed the time spent studying, x. These partial derivatives measure the change of a function with respect to a change in one particular direction (changing only x or only y). In many cases, however, we will want to examine how a function changes, as we change simultaneously both variables. For example, we may want to examine how the student’s grade changes as we simultaneously 3 alter the amount of sleep she gets and the amount of time she studies. If F is continuously ∂F ∂F differentiable (i.e. if ∂x and ∂y are continuous functions of the pair of variables (x, y)), then we can do this simply by adding the changes along each dimension to find the total change in the objective function: Let ∆x be the change in x and ∆y be the change in y (and think of both ∆x and ∆y as small). Then the total change in F that results from this change is ∂F ∂F ∆F = ∂x ∆x + ∂y ∆y. There are exceptions to this rule that one has to be careful about (they arise when F is not continuously differentiable), but these will not be a bigconcernforusinthiscourse. Example 2c below discusses one such example. These partial derivatives are themselves multi-variate functions of the pair of variables ∂2F (x, y). If we take their partial derivatives, we find the second-order partial derivatives, ∂x2 ∂2F or ∂y2 ,aswellasthecross-partial derivatives ∂2F ∂ ∂F ∂2F ∂ ∂F = ∂x and = ∂y ∂x∂y ∂y ∂y∂x ∂x As in the single variable case, the second-order partial derivatives measure the rate of change of the first derivatives, along the x or the y dimension. The cross-partial derivative measures how the partial derivatives with respect to x change with y (and vice versa). Young’s ∂2F ∂2F Theorem (Nicholson, ch. 2) states that ∂y∂x = ∂x∂y, i.e. whether we firsttakethederivative with respect to x and then y,orfirsttakethederivativewithrespecttoy and then x,we will find the same answer.