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Lecture Notes for Econ 101A David Card∗ Dept. of Economics UC Berkeley

∗ The manuscript was typeset by Daniel Nolan in LA TEX. The figures were created in Asymptote, Inkscape, R, and Excel (the marjority in Inkscape). Please address comments/corrections to daniel [email protected], with “Card Lecture Notes” in the subject line.

Contents 1 Optimization 7 1.1 Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.1 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.2 SOC in Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Consumer Choice 2.1 Budget Constraint . . 2.2 Consumer’s Objective 2.3 Consumer’s Optimum 2.4 Special Problems . . .

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14 . 14 . 14 . 19 . 21

3 Two Applications of Indifference Curve Analysis 23 3.1 Analysis of a Subsidy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 The Consumer Price Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4 Indirect Utility and the Expenditure Function 28 4.1 Indirect Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Expenditure Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5 Comparative Statics of Consumer Choice 5.1 Change in Demand with Respect to Income, Engel Curves . . . . . . . . . . . . . . . 5.2 Change in Demand with Respect to Price . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Graphical Decomposition of a Change in Demand . . . . . . . . . . . . . . . . . . . . 5.4 Substitution Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Income Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 34 36 37

6 Slutsky’s Equation 38 6.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.2 Slutsky Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7 Using Market Level Demand Curves 42 7.1 An Increase in Income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7.2 Tax Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 8 Labor Supply

48

9 Intertemporal Consumption

52

10 Production and Cost I 10.1 One-Factor Production and Cost Functions . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Production Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Connection between MC and MP . . . . . . . . . . . . . . . . . . . . . . . . 10.1.4 Geometry of c, AC, and M C . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

55 55 55 58 58 59

11 Production and Cost II 62 11.1 Derivation of the Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 11.2 Marginal Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 12 Cost Functions and IRFs 12.1 Sheppard’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68 68

13 Supply 70 13.1 Supply Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 13.2 The Law of Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 13.3 Changes in Input Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 14 Input Demand for a Competitive Firm

75

15 Industry Supply

80

16 Monopoly I 82 16.1 Monopolist’s Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 16.2 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 16.3 Monopoly in Two or More Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 17 Monopoly II

87

18 Consumer’s Surplus

91

19 Duopoly 94 19.1 Monopolization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 19.2 Duopoly Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 19.3 Price Setting vs. Quantity Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 20 Symmetric Cournot Equilibria 99 20.1 n-Firm Symmetric Cournot Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 99 20.2 Alternatives to the Cournot Assumption . . . . . . . . . . . . . . . . . . . . . . . . . 100 21 Game Theory I

102

22 Game Theory II 106 22.1 Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 22.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 23 Uncertainty I: Income Lotteries 110 23.1 Review of Basic Statistical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 23.2 Choices Over Uncertain Incomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 24 Uncertainty II: Expected Utility 114 24.1 Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 24.2 The Demand for Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

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25 Uncertainty III: Moral Hazard 118 25.1 Solution with No Moral Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 25.2 A Partial Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 26 Uncertainty IV: The State-preference Approach and Adverse Selection 122 26.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 26.2 Adverse Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 27 Auctions I: Types of Auctions 127 27.1 Basic Types of Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 27.2 Important Results Concerning the Private Values Case . . . . . . . . . . . . . . . . . 128 27.3 Bidding in a First-price Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 28 Auctions II: Winner’s Curse 131 28.1 Appendix: Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 28.1.1 Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 28.1.2 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 29 Finance I: Capital Asset Pricing Model 135 29.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 29.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 29.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 30 Finance II: Efficient Market Hypothesis 139 30.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 30.2 Efficient Market Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 31 Public and Near-public Goods 143 31.1 Optimal Provision of Goods with No-rivalry Characteristics . . . . . . . . . . . . . . 143 31.1.1 Case 1: one consumer; x = t1 /p. . . . . . . . . . . . . . . . . . . . . . . . . . 143 31.1.2 Case 2: two consumers; x = (t1 + t2 )/p. . P . . . . . . . . . . . . . . . . . . . . 143 n 31.1.3 Case 3: n consumers; x = τ /p, where τ = i=1 ti . . . . . . . . . . . . . . . . 145 31.2 Appendix: Social Optimum with Ordinary Goods . . . . . . . . . . . . . . . . . . . . 146 32 Externalities 148 32.1 Consumption Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 32.1.1 Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 32.1.2 Social Optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 32.1.3 Market Equilibrium versus Social Optimum . . . . . . . . . . . . . . . . . . . 150 32.1.4 Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 32.2 Production Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 33 Empirical Methods in Microeconomics 33.1 Experiments and Counterfactuals . . . . . . . . . 33.1.1 The Self Sufficiency Project (SSP) . . . . 33.2 Research Designs Based on Natural Experiments 33.2.1 The Mariel Boatlift . . . . . . . . . . . . .

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154 . . . . . . . . . . . . . . . . . . . . 154 . . . . . . . . . . . . . . . . . . . . 155 . . . . . . . . . . . . . . . . . . . . 157 . . . . . . . . . . . . . . . . . . . . 157

33.3 Natural Experiments with Several Control Groups . . . . . . . . . . . . . . . . . . . 157 33.3.1 The New Jersey Minimum Wage . . . . . . . . . . . . . . . . . . . . . . . . . 158 33.4 The Discontinuity Research Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

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Course Description This is a course in intermediate microeconomics, emphasizing the applications of calculus and linear algebra to the problems of consumer choice, firm behavior, and market interactions. Students are presumed to be familiar with multivariate calculus (including e.g. limits, derivatives, integrals) and with basic statistics (random variables, moments, etc.). The course material will be presented in a fairly mathematical way and the problem sets and examinations will require you to apply models and derive results. Students who are concerned about their mathematical ability should consider Econ 100A. The basic text is Microeconomic Theory: Basic Principles and Extensions, by Nicholson & Snyder, which should be available at the campus book store. An alternative, slightly more theoretical treatment of the same material is Varian’s Intermediate Microeconomics: A Modern Approach. Another, slightly more application-oriented alternative is Perloffs Microeconomics: Theory and Applications with Calculus. Any of the these is a good supplement to the lectures, but the lectures will be at a somewhat higher level, and will not follow the texts closely. Problem sets and practice exams will be made available on the course website. The GSIs will present some additional material in section (for which all students will be responsible) and also will review the solutions to problem sets, practice exams, and problems from the lectures, etc. Weekly problem sets will be assigned most weeks throughout the course. Completed problem sets are due at the end of the last lecture each week. We will not accept late problem sets. Instead, we drop your two worst scores. Thus, you can miss up to two problem sets without any penalty. You are encouraged to work in groups but every student must hand in his or her own version of the solutions. Course grades will be determined by a combination of weekly problem sets (20 percent), two midterm exams (15 percent each), and a final exam (50 percent). The midterm exams will be held in class.

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Lecture

Topics

1

Methods of Optimization

2

Consumer Choice

3

Applications of Indifference Curve Analysis, Expenditure Function

4

Comparative Statics, Slutsky’s Equation

5

Market Level Demand and Supply

6

Labor Supply

7

Intertemporal Consumption & Savings

8–9

Production & Cost, Sheppard’s Lemma

10–11

Supply Determination

12

Monopoly and Price Discrimination

13

Consumer/Producer Surplus & Applications

14–15

Duopoly

16–17

Game Theory

18–21

Uncertainty and Insurance Markets

22–23

Auctions

24–25

Finance: CAPM and Efficient Markets

26–27

Public Goods, Externalities

28

Empirical Methods in Microeconomics

6

1 1.1

Optimization Unconstrained Optimization

Consider a smooth function y = f (x). How do we go about finding a point x0 such that y0 = f (x0 ) ≥ f (x) for any x in [a, b]?

Figure 1.1: In this picture f (x0 ) = maxa≤x≤b f (x). (Read: “f (x0 ) is the maximum value of f (x) when x is selected from the interval [a, b].”)

What can we say generally? Obviously, if x0 is a potential candidate for a maximizer, then it must be the case that we can’t move around x0 and reach a higher value of f . But this means f ′ (x0 ) = 0. Why? Let 0 < h ≪ 1. If f ′ (x) > 0, then f (x + h) ≈ f (x) + hf ′ (x) > f (x).

If f ′ (x) < 0, then f (x − h) ≈ f (x) − hf ′ (x) > f (x). This leads us to Rule 1: If f (x0 ) = maxa≤x≤b f (x), then f ′ (x0 ) = 0. This is called the first order necessary condition (FONC) for an interior maximum. Does f ′ (x0 ) = 0 always mean that x0 is a maximizer? Are there maximizers with f ′ (x0 ) 6= 0? Consider the examples illustrated in Figure 1.3. How can we be certain that we have located a maximum (not a minimum, nor an inflection point)? We examine the properties of f ′ (x), which is itself a function of x. Take a look at Figure 1.4. As the function f ′ crosses x0 from left to right, it goes from positive to negative, i.e. it’s decreasing. On the other hand, as f ′ crosses x1 from left to right, it goes from negative to positive, i.e. it’s increasing. In general, at a local maximum f ′ (x) has negative slope, or in other words f ′′(x) < 0, while at a local minimum f ′ (x) has positive slope, that is f ′′ (x) > 0. These considerations lead us to Rule 2: If f ′ (x0 ) = 0 and f ′′ (x0 ) < 0, then f (x0 ) is a local maximum. If f ′ (x0 ) = 0 and f ′′ (x) > 0, then f (x0 ) is a local minimum.

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Figure 1.2: Notice that Rule 1 also holds for a function of several variables.

(a)

(b)

(c)

Figure 1.3: Exceptions to the converse of Rule 1: (a) f (x) = x. Thus f (b) = maxa≤x≤b f (x) even though f ′ (b) = 1 6= 0. The maximum occurs on the boundary. (b) f ′ (x) = 0 has two solutions, x′ and x′′ but neither one is a maximizer. f (x′ ) is a local maximum while f (x′′ ) is a minimum. (c) f (x) = x3 . Solving f ′ (x) = 0 gives x = 0, which is an inflection point.

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Figure 1.4: Properties of f ′ (x): at a local max f ′ is decreasing since the tangent lines go from positive to negative. The reverse is true for a local min.

This generalizes to two or more dimensions. How do we determine whether a local maximum is a global maximum? If f ′′ (x) < 0 for all x and f ′ (x0 ) = 0, then x0 is a global maximum. A function f such that f ′′(x) < 0 for all x is called concave.1

Figure 1.5: A concave function always lies below any line tangent to its graph.

1.2

Constrained Optimization

Now we consider maximizing a function f (x1 , x2 ) subject to—“s.t.”—some constraint on x1 and x2 which we denote by g(x1 , x2 ) = g 0 . The two important examples of this in economics are: 1 See

Appendix 1.3.

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• In the study of consumer behavior, maximize utility u(x1 , x2 ) s.t. the budget constraint p1 x1 + p2 x2 = I. • In the study of firm behavior, maximize profit py − wx s.t. the production function y = f (x).

How do we go about a graphical analysis of the problem of maximizing f (x1 , x2 ) s.t g(x1 , x2 ) = g 0 ?

Figure 1.6: Illustration of two-step approach described on p. 10.

A two-step approach: 1. Plot the contours of √ the function g. E.g. g(x1 , x2 ) = x21 + x22 ; g(x1 , x2 ) = k is the equation of a circle with radius k and center O = (0, 0). 2. Plot the contours of the function f . E.g. f (x1 , x2 ) = x1 x2 ; f (x1 , x2 ) = m is the equation of a hyperbola. The constrained maximum of the function f occurs where a contour of f is tangent to the contour of g corresponding to g 0 . Why? Suppose we add a small amount dx1 to x1 in such a way as to keep g(x1 , x2 ) constant. If so, then we must have a corresponding reduction in x2 such that the total differential of g is zero, i.e. dg = g1 (x1 , x2 )dx1 + g2 (x1 , x2 )dx2 = 0 (where gi denotes ∂g/∂xi ), which implies dx2 g1 (x1 , x2 ) =− dx1 g2 (x1 , x2 ) If we increase x1 by one unit, we must increase x2 by −g1 (x1 , x2 )/g2 (x1 , x2 )—or, equivalently, decrease x2 by g1 (x1 , x2 )/g2 (x1 , x2 )—in order to keep the value of g constant. The net effect of

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such a change in x1 on the value of f is df = f1 (x1 , x2 )dx1 + f2 (x1 , x2 )dx2 dx2 = f1 (x1 , x2 )dx1 + f2 (x1 , x2 ) × dx1 dx1   g1 (x1 , x2 ) = f1 (x1 , x2 ) − f2 (x1 , x2 ) × dx1 g2 (x1 , x2 ) Now in order for (x01 , x20) to be a constrained maximum, it must be the case that we cannot increase f by adding or subtracting a small amount to x1 while keeping the value of g constant. But this means the above expression is 0 for all dx1 , or in other words g1 (x1 , x2 ) f1 (x1 , x2 ) = g2 (x1 , x2 ) f2 (x1 , x2 ) But this expression says that at (x01 , x20), the contours of f and g are tangent, i.e. have the same slope. Note that this argument applies only if (x01 , x02 ) lies in the interior of the domain for if (x01 , x02 ) lies on the boundary then we cannot increase or decrease one of x1 or x2 . How do we convert a constrained maximization problem into an unconstrained one? A French mathematician named Lagrange noted that one gets the right answer by setting up an artificial, unconstrained maximization problem with an additional variable, λ: L(x1 , x2 , λ) = f (x1 , x2 ) − λ[g(x1 , x2 ) − g 0 ] The FONC for L, with respect to x1 , x2 , and λ are: L1 = f1 (x1 , x2 ) − λg1 (x1 , x2 ) = 0

L2 = f2 (x1 , x2 ) − λg2 (x1 , x2 ) = 0 Lλ = g (x1 , x2 ) − g 0 = 0

Dividing the first of these by the second gives g1 (x1 , x2 ) f1 (x1 , x2 ) = g2 (x1 , x2 ) f2 (x1 , x2 ) while the third simply restates the constraint! Thus by writing down the Lagrangian L and setting its first derivatives equal to zero we get the necessary conditions for a constrained maximum. We also get a new variable, λ, called the Lagrange multiplier. How do we interpret λ? It turns out that the value of λ tells us how much the maximum value of f changes if we relax the constraint by a small amount. Specifically, suppose we are to maximize f (x1 , x2 ) s.t. the constraint g(x1 , x2 ) = g 0 . Call the solution (x10 , x02 ). Now suppose we relax the constraint and instead maximize f (x1 , x2 ) s.t. g(x1 , x2 ) = g 0 + dg 0 . How do we change our optimal choices of x1 and x2 ? Suppose we decide to use more x1 , enough to use up the added constraint. Since the total differential of g is dg = g1 (x1 , x2 )dx1 + g2 (x1 , x2 )dx2 if we change only x1 , (that is, if dx2 = 0), the amount we can change x1 while satisfying the new constraint is 1 dx1 = dg 0 g1 (x1 , x2 ) 11

The increase in f that accompanies this increase in x1 is df = f1 (x1 , x2 )dx1 =

f1 (x1 , x2 ) =λ g1 (x1 , x2 )

You are encouraged to check for yourself that if you were to us...