1 MUST-HAVE MATH TOOLS FOR GRADUATE STUDY IN ECONOMICS William Neilson Department of Economics University of Tennessee Knoxville September by William Neilson web.utk.edu/~wneilson/mathbook.pdf
2 Acknowledgments Valentina Kozlova, Kelly Padden, and John Tilstra provided valuable proofreading assistance on the first version of this book, and I am grateful. Other mistakes were found by the students in my class. Of course, if they missed anything it is still my fault. Valentina and Bruno Wichmann have both suggested additions to the book, including the sections on stability of dynamic systems and order statistics. The cover picture was provided by my son, Henry, who also proofread parts of the book. I have always liked this picture, and I thank him for letting me use it.
3 CONTENTS Econ and math. Some important graphs Math, micro, and metrics I Optimization (Multivariate calculus) 6 2 Single variable optimization 7 2. A graphical approach Derivatives Uses of derivatives Maximum or minimum? Logarithms and the exponential function Problems Optimization with several variables 2 3. A more complicated pro t function Vectors and Euclidean space Partial derivatives Multidimensional optimization i
4 3.5 Comparative statics analysis An alternative approach (that I don t like) Problems Constrained optimization A graphical approach Lagrangians A 2-dimensional example Interpreting the Lagrange multiplier A useful example - Cobb-Douglas Problems Inequality constraints Lame example - capacity constraints A binding constraint A nonbinding constraint A new approach Multiple inequality constraints A linear programming example Kuhn-Tucker conditions Problems ii II Solving systems of equations (Linear algebra) 7 6 Matrices Matrix algebra Uses of matrices Determinants Cramer s rule Inverses of matrices Problems Systems of equations Identifying the number of solutions The inverse approach Row-echelon decomposition Graphing in (x,y) space
5 7..4 Graphing in column space Summary of results Problems Using linear algebra in economics IS-LM analysis Econometrics Least squares analysis A lame example Graphing in column space Interpreting some matrices Stability of dynamic systems Stability with a single variable Stability with two variables Eigenvalues and eigenvectors Back to the dynamic system Problems Second-order conditions 4 9. Taylor approximations for R! R Second order conditions for R! R Taylor approximations for R m! R Second order conditions for R m! R Negative semide nite matrices Application to second-order conditions Examples Concave and convex functions Quasiconcave and quasiconvex functions Problems iii III Econometrics (Probability and statistics) 3 Probability 3. Some de nitions De ning probability abstractly De ning probabilities concretely Conditional probability
6 .5 Bayes rule Monty Hall problem Statistical independence Problems Random variables 43. Random variables Distribution functions Density functions Useful distributions Binomial (or Bernoulli) distribution Uniform distribution Normal (or Gaussian) distribution Exponential distribution Lognormal distribution Logistic distribution Integration Interpreting integrals Integration by parts Application: Choice between lotteries Di erentiating integrals Application: Second-price auctions Problems Moments Mathematical expectation The mean Uniform distribution Normal distribution Variance Uniform distribution Normal distribution Application: Order statistics Problems iv
7 4 Multivariate distributions Bivariate distributions Marginal and conditional densities Expectations Conditional expectations Using conditional expectations - calculating the bene t of search The Law of Iterated Expectations Problems Statistics Some de nitions Sample mean Sample variance Convergence of random variables Law of Large Numbers Central Limit Theorem Problems Sampling distributions Chi-square distribution Sampling from the normal distribution t and F distributions Sampling from the binomial distribution Hypothesis testing 2 7. Structure of hypothesis tests One-tailed and two-tailed tests Examples Example Example Example Problems Solutions to end-of-chapter problems 2 8. Solutions for Chapter Index 267 v
8 CHAPTER Econ and math Every academic discipline has its own standards by which it judges the merits of what researchers claim to be true. In the physical sciences this typically requires experimental veri cation. In history it requires links to the original sources. In sociology one can often get by with anecdotal evidence, that is, with giving examples. In economics there are two primary ways one can justify an assertion, either using empirical evidence (econometrics or experimental work) or mathematical arguments. Both of these techniques require some math, and one purpose of this course is to provide you with the mathematical tools needed to make and understand economic arguments. A second goal, though, is to teach you to speak mathematics as a second language, that is, to make you comfortable talking about economics using the shorthand of mathematics. In undergraduate courses economic arguments are often made using graphs. In graduate courses we tend to use equations. But equations often have graphical counterparts and vice versa. Part of getting comfortable about using math to do economics is knowing how to go from graphs to the underlying equations, and part is going from equations to the appropriate graphs.
9 CHAPTER. ECON AND MATH 2 Figure.: A constrained choice problem. Some important graphs One of the fundamental graphs is shown in Figure.. The axes and curves are not labeled, but that just ampli es its importance. If the axes are commodities, the line is a budget line, and the curve is an indi erence curve, the graph depicts the fundamental consumer choice problem. If the axes are inputs, the curve is an isoquant, and the line is an iso-cost line, the graph illustrates the rm s cost-minimization problem. Figure. raises several issues. How do we write the equations for the line and the curve? The line and curve seem to be tangent. How do we characterize tangency? At an even more basic level, how do we nd slopes of curves? How do we write conditions for the curve to be curved the way it is? And how do we do all of this with equations instead of a picture? Figure.2 depicts a di erent situation. If the upward-sloping line is a supply curve and the downward-sloping one is a demand curve, the graph shows how the market price is determined. If the upward-sloping line is marginal cost and the downward-sloping line is marginal bene t, the gure shows how an individual or rm chooses an amount of some activity. The questions for Figure.2 are: How do we nd the point where the two lines intersect? How do we nd the change from one intersection point to another? And how do we know that two curves will intersect in the rst place? Figure.3 is completely di erent. It shows a collection of points with a
10 CHAPTER. ECON AND MATH 3 Figure.2: Solving simultaneous equations line tting through them. How do we t the best line through these points? This is the key to doing empirical work. For example, if the horizontal axis measures the quantity of a good and the vertical axis measures its price, the points could be observations of a demand curve. How do we nd the demand curve that best ts the data? These three graphs are fundamental to economics. There are more as well. All of them, though, require that we restrict attention to two dimensions. For the rst graph that means consumer choice with only two commodities, but we might want to talk about more. For the second graph it means supply and demand for one commodity, but we might want to consider several markets simultaneously. The third graph allows quantity demanded to depend on price, but not on income, prices of other goods, or any other factors. So, an important question, and a primary reason for using equations instead of graphs, is how do we handle more than two dimensions? Math does more for us than just allow us to expand the number of dimensions. It provides rigor; that is, it allows us to make sure that our statements are true. All of our assertions will be logical conclusions from our initial assumptions, and so we know that our arguments are correct and we can then devote attention to the quality of the assumptions underlying them.
11 CHAPTER. ECON AND MATH 4 Figure.3: Fitting a line to data points.2 Math, micro, and metrics The theory of microeconomics is based on two primary concepts: optimization and equilibrium. Finding how much a rm produces to maximize pro t is an example of an optimization problem, as is nding what a consumer purchases to maximize utility. Optimization problems usually require nding maxima or minima, and calculus is the mathematical tool used to do this. The rst section of the book is devoted to the theory of optimization, and it begins with basic calculus. It moves beyond basic calculus in two ways, though. First, economic problems often have agents simultaneously choosing the values of more than one variable. For example, consumers choose commodity bundles, not the amount of a single commodity. To analyze problems with several choice variables, we need multivariate calculus. Second, as illustrated in Figure., the problem is not just a simple maximization problem. Instead, consumers maximize utility subject to a budget constraint. We must gure out how to perform constrained optimization. Finding the market-clearing price is an equilibrium problem. An equilibrium is simply a state in which there is no pressure for anything to change, and the market-clearing price is the one at which suppliers have no incentive to raise or lower their prices and consumers have no incentive to raise or lower their o ers. Solutions to games are also based on the concept of equilibrium. Graphically, equilibrium analysis requires nding the intersection of two curves, as in Figure.2. Mathematically, it involves the solution of
12 CHAPTER. ECON AND MATH 5 several equations in several unknowns. The branch of mathematics used for this is linear (or matrix) algebra, and so we must learn to manipulate matrices and use them to solve systems of equations. Economic exercises often involve comparative statics analysis, which involves nding how the optimum or equilibrium changes when one of the underlying parameters changes. For example, how does a consumer s optimal bundle change when the underlying commodity prices change? How does a rm s optimal output change when an input or an output price changes? How does the market-clearing price change when an input price changes? All of these questions are answered using comparative statics analysis. Mathematically, comparative statics analysis involves multivariable calculus, often in combination with matrix algebra. This makes it sound hard. It isn t really. But getting you to the point where you can perform comparative statics analysis means going through these two parts of mathematics. Comparative statics analysis is also at the heart of empirical work, that is, econometrics. A typical empirical project involves estimating an equation that relates a dependent variable to a set of independent variables. The estimated equation then tells how the dependent variable changes, on average, when one of the independent variables changes. So, for example, if one estimates a demand equation in which quantity demanded is the dependent variable and the good s price, some substitute good prices, some complement good prices, and income are independent variables, the resulting equation tells how much quantity demanded changes when income rises, for example. But this is a comparative statics question. A good empirical project uses some math to derive the comparative statics results rst, and then uses data to estimate the comparative statics results second. Consequently, econometrics and comparative statics analysis go hand-in-hand. Econometrics itself is the task of tting the best line to a set of data points, as in Figure.3. There is some math behind that task. Much of it is linear algebra, because matrices turn out to provide an easy way to present the relevant equations. A little bit of the math is calculus, because "best" implies "optimal," and we use calculus to nd optima. Econometrics also requires a knowledge of probability and statistics, which is the third branch of mathematics we will study.
13 PART I OPTIMIZATION (multivariate calculus)
14 CHAPTER 2 Single variable optimization One feature that separates economics from the other social sciences is the premise that individual actors, whether they are consumers, rms, workers, or government agencies, act rationally to make themselves as well o as possible. In other words, in economics everybody maximizes something. So, doing mathematical economics requires an ability to nd maxima and minima of functions. This chapter takes a rst step using the simplest possible case, the one in which the agent must choose the value of only a single variable. In later chapters we explore optimization problems in which the agent chooses the values of several variables simultaneously. Remember that one purpose of this course is to introduce you to the mathematical tools and techniques needed to do economics at the graduate level, and that the other is to teach you to frame economic questions, and their answers, mathematically. In light of the second goal, we will begin with a graphical analysis of optimization and then nd the math that underlies the graph. Many of you have already taken calculus, and this chapter concerns singlevariable, di erential calculus. One di erence between teaching calculus in 7
15 CHAPTER 2. SINGLE VARIABLE OPTIMIZATION 8 $ π * π(q) q * q Figure 2.: A pro t function with a maximum an economics course and teaching it in a math course is that economists almost never use trigonometric functions. The economy has cycles, but none regular enough to model using sines and cosines. So, we will skip trigonometric functions. We will, however, need logarithms and exponential functions, and they are introduced in this chapter. 2. A graphical approach Consider the case of a competitive rm choosing how much output to produce. When the rm produces and sells q units it earns revenue R(q) and incurs costs of C(q). The pro t function is (q) = R(q) C(q): The rst term on the right-hand side is the rm s revenue, and the second term is its cost. Pro t, as always, is revenue minus cost. More importantly for this chapter, Figure 2. shows the rm s pro t function. The maximum level of pro t is, which is achieved when output is q. Graphically this is very easy. The question is, how do we do it with equations instead? Two features of Figure 2. stand out. First, at the maximum the slope of the pro t function is zero. Increasing q beyond q reduces pro t, and
16 CHAPTER 2. SINGLE VARIABLE OPTIMIZATION 9 $ π(q) q Figure 2.2: A pro t function with a minimum decreasing q below q also reduces pro t. Second, the pro t function rises up to q and then falls. To see why this is important, compare it to Figure 2.2, where the pro t function has a minimum. In Figure 2.2 the pro t function falls to the minimum then rises, while in Figure 2. it rises to the maximum then falls. To make sure we have a maximum, we have to make sure that the pro t function is rising then falling. This leaves us with several tasks. () We must nd the slope of the pro t function. (2) We must nd q by nding where the slope is zero. (3) We must make sure that pro t really is maximized at q, and not minimized. (4) We must relate our ndings back to economics. 2.2 Derivatives The derivative of a function provides its slope at a point. It can be denoted in two ways: f (x) or df(x)=dx. The derivative of the function f at x is de ned as df(x) dx = lim f(x + h) f(x) : (2.) h! h The idea is as follows, with the help of Figure 2.3. Suppose we start at x and consider a change to x + h. Then f changes from f(x) to f(x + h). The ratio of the change in f to the change in x is a measure of the slope:
17 CHAPTER 2. SINGLE VARIABLE OPTIMIZATION f(x) f(x+h) f(x) f(x) x x+h x Figure 2.3: Approximating the slope of a function [f(x + h) f(x)]=[(x + h) x]. Make the change in x smaller and smaller to get a more precise measure of the slope, and, in the limit, you end up with the derivative. Finding the derivative comes from applying the formula in equation (2.). And it helps to have a few simple rules in hand. We present these rules as a series of theorems. Theorem Suppose f(x) = a. Then f (x) =. Proof. f f(x + h) f(x) (x) = lim h! h a a = lim h! h = : Graphically, a constant function, that is, one that yields the same value for every possible x, is just a horizontal line, and horizontal lines have slopes of zero. The theorem says that the derivative of a constant function is zero. Theorem 2 Suppose f(x) = x. Then f (x) =.
18 CHAPTER 2. SINGLE VARIABLE OPTIMIZATION Proof. f (x) = f(x + h) f(x) lim h! h = (x + h) x lim h! h = lim h = : h! h Graphically, the function f(x) = x is just a 45-degree line, and the slope of the 45-degree line is one. The theorem con rms that the derivative of this function is one. Theorem 3 Suppose f(x) = au(x). Proof. Then f (x) = au (x). f (x) = f(x + h) f(x) lim h! h = au(x + h) au(x) lim h! h = u(x + h) u(x) a lim h! h = au (x): This theorem provides a useful rule. When you multiply a function by a scalar (or constant), you also multiply the derivative by the same scalar. Graphically, multiplying by a scalar rotates the curve. Theorem 4 Suppose f(x) = u(x) + v(x). Proof. Then f (x) = u (x) + v (x). f f(x + h) f(x) (x) = lim h! h [u(x + h) + v(x + h)] [u(x) + v(x)] = lim h! h u(x + h) u(x) v(x + h) u(x) = lim + h! h h = u (x) + v (x):
19 CHAPTER 2. SINGLE VARIABLE OPTIMIZATION 2 This rule says that the derivative of a sum is the sum of the derivatives. The next theorem is the product rule, which tells how to take the derivative of the product of two functions. Theorem 5 Suppose f(x) = u(x)v(x). Then f (x) = u (x)v(x)+u(x)v (x). Proof. f f(x + h) f(x) (x) = lim h! h [u(x + h)v(x + h)] [u(x)v(x)] = lim h! h [u(x + h) u(x)]v(x) u(x + h)[v(x + h) v(x)] = lim + h! h h where the move from line 2 to line 3 entails adding then subtracting lim h! u(x+ h)v(x)=h. Remembering that the limit of a product is the product of the limits, the above expression reduces to f (x) = [u(x + h) u(x)] [v(x + h) v(x)] lim v(x) + lim u(x + h) h! h h! h = u (x)v(x) + u(x)v (x): We need a rule for functions of the form f(x) = =u(x), and it is provided in the next theorem. Theorem 6 Suppose f(x) = =u(x). Then f (x) = u (x)=[u(x)] 2. Proof. f f(x + h) f(x) (x) = lim h! h = lim h! u(x+h) u(x) h u(x) u(x + h) = lim h! h[u(x + h)u(x)] [u(x + h) u(x)] = lim h! h = u (x) [u(x)] 2 : lim h! u(x + h)u(x)
20 CHAPTER 2. SINGLE VARIABLE OPTIMIZATION 3 Our nal rule concerns composite functions, that is, functions of functions. This rule is called the chain rule. Theorem 7 Suppose f(x) = u(v(x)). Then f (x) = u (v(x)) v (x). Proof. First suppose that there is some sequence h ; h 2 ; ::: with lim i! h i = and v(x + h i ) v(x) 6= for all i. Then v(x) f (x) = f(x + h) f(x) lim h! h u(v(x + h)) u(v(x)) = lim h! h u(v(x + h)) u(v(x)) v(x + h) = lim h! v(x + h) v(x) h = u(v(x) + k)) u(v(x)) v(x + h) v(x) lim lim k! k h! h = u (v(x)) v (x): Now suppose that there is no sequence as de ned above. Then there exists a sequence h ; h 2 ; ::: with lim i! h i = and v(x + h i ) v(x) = for all i. Let b = v(x) for all x,and f (x) = f(x + h) f(x) lim h! h = u(v(x + h)) u(v(x)) lim h! h = u(b) u(b) lim h! h = : But u (v(x)) v (x) = since v (x) =, and we are done. Combining these rules leads to the following really helpful rule: d dx a[f(x)]n = an[f(x)] n f (x): (2.2)
21 CHAPTER 2. SINGLE VARIABLE OPTIMIZATION 4 This holds even if n is negative, and even if n is not an integer. So, for example, the derivative of x n is nx n, and the derivative of (2x + ) 5 is (2x + ) 4. The derivative of (4x 2 ) :4 is :4(4x 2 ) :4 (8x). Combining the rules also gives us the familiar division rule: d u(x) = u (x)v(x) v (x)u(x) dx v(x) [v(x)] 2 : (2.3) To get it, rewrite u(x)=v(x) as u(x) [v(x)] rule and expression (2.2) to get. We can then use the product d u(x)v (x) = u (x)v (x) + ( )u(x)v 2 (x)v (x) dx = u (x) v(x) v (x)u(x) : v 2 (x) Multiplying both the numerator and denominator of the rst term by v(x) yields (2.3). Getting more ridiculously complicated, consider f(x) = (x3 + 2x)(4x ) x 3 : To di erentiate this thing, split f into three component functions, f (x) = x 3 + 2x, f 2 (x) = 4x, and f 3 (x) = x 3. Then f(x) = f (x) f 2 (x)=f 3 (x), and f (x) = f (x)f 2 (x) + f (x)f2(x) f (x)f 2 (x)f3(x) : f 3 (x) f 3 (x) [f 3 (x)] 2 We can di erentiate the component functions to get f (x) = 3x 2 +2, f 2(x) = 4, and f 3(x) = 3x 2. Plugging this all into the formula above gives us f (x) = (3x2 + 2)(4x ) x 3 + 4(x3 + 2x) x 3 3(x 3 + 2x)(4x )x 2 x 6 : 2.3 Uses of derivatives In economics there are three major uses of derivatives. The rst use comes from the economics idea of "marginal this" and "marginal that." In principles of economics courses, for example, marginal cost is
22 CHAPTER 2. SINGLE VARIABLE OPTIMIZATION 5 de ned as the additional cost a rm incurs when it produces one more unit of output. If the cost function is C(q), where q is quantity, marginal cost is C(q + ) C(q). We could divide output up into smaller units, though, by measuring in grams instead of kilograms, for example. Continually dividing output into smaller and smaller units of size h leads to the de nition of marginal cost as c(q + h) c(q) MC(q) = lim : h! h Marginal cost is simply the derivative of the cost function. Similarly, marginal revenue is the derivative of the revenue function, and so on. The second use of derivatives comes from looking at their signs (the astrology of derivatives). Consider the function y = f(x). We might ask whether an increase in x leads to an increase in y or a decrease in y. The derivative f (x) measures the change in y when x changes, and so if f (x) we know that y increases when x increases, and if f (x) we know that y decreases when x increases. So, for example, if the marginal cost function MC(q) or, equivalently, C (q) is positive we know that an increase in output leads to an increase in cost. The third use of derivatives is for nding maxima and minima of functions. This is where we started the chapter, with a competitive rm choosing output to maximize pro t. The pro t function is (q) = R(q) C(q). As we saw in Figure 2., pro t is maximized when the slope of the pro t function is zero, or d dq = : This condition is called a rst-order condition, often abbreviated as FOC. Using our rules for di erentiation, we can rewrite the FOC as d dq = R (q ) C (q ) = ; (2.4) which reduces to the familiar rule that a rm maximizes pro t by producing where marginal revenue equals marginal cost. Notice what we have done here. We have not used numbers or speci c functions and, aside from homework exercises, we rarely will. Using general functions leads to expressions involving general functions, and we want to interpret these. We know that R (q) is marginal revenue and C (q) is marginal cost. We end up in the same place we do using graphs, which is a good
23 CHAPTER 2. SINGLE VARIABLE OPTIMIZATION 6 thing. The power of the mathematical approach is that it allows us to apply the same techniques in situations where graphs will not work. 2.4 Maximum or minimum? Figure 2. shows a pro t function with a maximum, but Figure 2.2 shows one with a minimum. Both of them generate the same rst-order condition: d=dq =. So what property of the function tells us that we are getting a maximum and not a minimum? In Figure 2. the slope of the curve decreases as q increases, while in Figure 2.2 the slope of the curve increases as q increases. Since slopes are just derivatives of the function, we can express these conditions mathematically by taking derivatives of derivatives, or second derivatives. The second derivative of the function f(x) is denoted f (x) or d 2 f=dx 2. For the function to have a maximum, like in Figure 2., the derivative should be decreasing, which means that the second derivative should be negative. For the function to have a minimum, like in Figure 2.2, the derivative should be increasing, which means that the second derivative should be positive. Each of these is called a second-order condition or SOC. The second-order condition for a maximum is f (x), and the second-order condition for a minimum is f (x). We can guarantee that pro t is maximized, at least locally, if (q ). We can guarantee that pro t is maximized globally if (q) for all possible values of q. Let s look at the condition a little more closely. The rst derivative of the pro t function is (q) = R (q) C (q) and the second derivative is (q) = R (q) C (q). The second-order condition for a maximum is (q), which holds if R (q) and C (q). So, we can guarantee that pro t is maximized if the second derivative of the revenue function is nonpositive and the second derivative of the cost function is nonnegative. Remembering that C (q) is marginal cost, the condition C (q) means that marginal cost is increasing, and this has an economic interpretation: each additional unit of output adds more to total cost than any unit preceding it. The condition R (q) means that marginal revenue is decreasing, which means that the rm earns less from each additional unit it sells. One special case that receives considerable attention in economics is the one in which R(q) = pq, where p is the price of the good. This is the
24 CHAPTER 2. SINGLE VARIABLE OPTIMIZATION 7 revenue function for a price-taking rm in a perfectly competitive industry. Then R (q) = p and R (q) =, and the rst-order condition for pro t maximization is p C (q) =, which is the familiar condition that price equals marginal cost. The second-order condition reduces to C (q), which says that marginal cost must be nondecreasing. 2.5 Logarithms and the exponential function The functions ln x and e x turn out to play an important role in economics. The rst is the natural logarithm, and the second is the exponential function. They are related: ln e x = e ln x = x. The number e 2:78. Without going into why these functions are special for economics, let me show you why they are special for math. We know that d x n = x n. dx n We can get the function x 2 by di erentiating x 3 =3, the function x by di erentiating x 2 =2, the function x 2 by di erentiating x, the function x 3 by di erentiating x 2 =2, and so on. But how can we get the function x? We cannot get it by di erentiating x =, because that expression does not exist. We cannot get it by di erentiating x, because dx =dx =. So how do we get x as a derivative? The answer is the natural logarithm: d dx ln x = x. Logarithms have two additional useful properties: ln xy = ln x + ln y: and Combining these yields ln(x a ) = a ln x: ln(x a y b ) = a ln x + b ln y: (2.5)
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