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Lecture Notes for Econ 101A
David Card
Dept. of Economics
UC Berkeley
The manuscript was typeset by Daniel Nolan in LATEX. The figures were created in Asymptote, Inkscape, R,and Excel (the marjority in Inkscape). Please address comments/corrections to daniel [email protected], with CardLecture Notes in the subject line.
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Contents
1 Optimization 71.1 Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.2 SOC in Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Consumer Choice 142.1 Budget Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Consumers Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Consumers Optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Special Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Two Applications of Indifference Curve Analysis 233.1 Analysis of a Subsidy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 The Consumer Price Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Indirect Utility and the Expenditure Function 284.1 Indirect Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Expenditure Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Comparative Statics of Consumer Choice 315.1 Change in Demand with Respect to Income, Engel Curves . . . . . . . . . . . . . . . 315.2 Change in Demand with Respect to Price . . . . . . . . . . . . . . . . . . . . . . . . 335.3 Graphical Decomposition of a Change in Demand . . . . . . . . . . . . . . . . . . . . 345.4 Substitution Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.5 Income Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6 Slutskys Equation 38
6.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.2 Slutsky Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7 Using Market Level Demand Curves 427.1 An Increase in Income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.2 Tax Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8 Labor Supply 48
9 Intertemporal Consumption 52
10 Production and Cost I 5510.1 One-Factor Production and Cost Functions . . . . . . . . . . . . . . . . . . . . . . . 55
10.1.1 Production Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.1.2 Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.1.3 Connection between MC and MP . . . . . . . . . . . . . . . . . . . . . . . . 5810.1.4 Geometry of c, AC, and MC . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
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11 Production and Cost II 6211.1 Derivation of the Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.2 Marginal Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12 Cost Functions and IRFs 6812.1 Sheppards Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
13 Supply 7013.1 Supply Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.2 The Law of Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.3 Changes in Input Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
14 Input Demand for a Competitive Firm 75
15 Industry Supply 80
16 Monopoly I 8216.1 Monopolists Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
16.2 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8316.3 Monopoly in Two or More Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
17 Monopoly II 87
18 Consumers Surplus 91
19 Duopoly 9419.1 Monopolization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.2 Duopoly Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9519.3 Price Setting vs. Quantity Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
20 Symmetric Cournot Equilibria 99
20.1 n-Firm Symmetric Cournot Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 9920.2 Alternatives to the Cournot Assumption . . . . . . . . . . . . . . . . . . . . . . . . . 100
21 Game Theory I 102
22 Game Theory II 10622.1 Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10622.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
23 Uncertainty I: Income Lotteries 11023.1 Review of Basic Statistical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 11023.2 Choices Over Uncertain Incomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
24 Uncertainty II: Expected Utility 114
24.1 Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11424.2 The Demand for Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
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25 Uncertainty III: Moral Hazard 11825.1 Solution with No Moral Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11925.2 A Partial Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
26 Uncertainty IV: The State-preference Approach and Adverse Selection 12226.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12226.2 Adverse Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
27 Auctions I: Types of Auctions 12727.1 Basic Types of Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12727.2 Important Results Concerning the Private Values Case . . . . . . . . . . . . . . . . . 12827.3 Bidding in a First-price Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
28 Auctions II: Winners Curse 13128.1 Appendix: Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
28.1.1 Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13428.1.2 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
29 Finance I: Capital Asset Pricing Model 13529.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13529.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13729.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
30 Finance II: Efficient Market Hypothesis 13930.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13930.2 Efficient Market Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
31 Public and Near-public Goods 14331.1 Optimal Provision of Goods with No-rivalry Characteristics . . . . . . . . . . . . . . 143
31.1.1 Case 1: one consumer; x = t1/p. . . . . . . . . . . . . . . . . . . . . . . . . . 14331.1.2 Case 2: two consumers; x = (t1 + t2)/p. . . . . . . . . . . . . . . . . . . . . . 143
31.1.3 Case 3: n consumers; x = /p, where = ni=1 ti. . . . . . . . . . . . . . . . 14531.2 Appendix: Social Optimum with Ordinary Goods . . . . . . . . . . . . . . . . . . . . 146
32 Externalities 14832.1 Consumption Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
32.1.1 Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14832.1.2 Social Optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14932.1.3 Market Equilibrium versus Social Optimum . . . . . . . . . . . . . . . . . . . 15032.1.4 Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
32.2 Production Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
33 Empirical Methods in Microeconomics 15433.1 Experiments and Counterfactuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
33.1.1 The Self Sufficiency Project (SSP) . . . . . . . . . . . . . . . . . . . . . . . . 15533.2 Research Designs Based on Natural Experiments . . . . . . . . . . . . . . . . . . . . 157
33.2.1 The Mariel Boatlift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
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33.3 Natural Experiments with Several Control Groups . . . . . . . . . . . . . . . . . . . 15733.3.1 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 linearalgebra 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) andwith basic statistics (random variables, moments, etc.). The course material will be presented in afairly mathematical way and the problem sets and examinations will require you to apply modelsand derive results. Students who are concerned about their mathematical ability should considerEcon 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 theoreticaltreatment of the same material is Varians Intermediate Microeconomics: A Modern Approach.Another, slightly more application-oriented alternative is Perloffs Microeconomics: Theory andApplications with Calculus. Any of the these is a good supplement to the lectures, but the lectureswill 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 setsare due at the end of the last lecture each week. We will not accept late problem sets. Instead, wedrop your two worst scores. Thus, you can miss up to two problem sets without any penalty. Youare encouraged to work in groups but every student must hand in his or her own version of thesolutions.
Course grades will be determined by a combination of weekly problem sets (20 percent), twomidterm exams (15 percent each), and a final exam (50 percent). The midterm exams will beheld 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, Slutskys Equation
5 Market Level Demand and Supply
6 Labor Supply
7 Intertemporal Consumption & Savings
89 Production & Cost, Sheppards Lemma
1011 Supply Determination
12 Monopoly and Price Discrimination
13 Consumer/Producer Surplus & Applications
1415 Duopoly
1617 Game Theory
1821 Uncertainty and Insurance Markets
2223 Auctions
2425 Finance: CAPM and Efficient Markets
2627 Public Goods, Externalities
28 Empirical Methods in Microeconomics
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1 Optimization
1.1 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) = maxaxb f(x). (Read: f(x0) is the maximum value of f(x) when xis selected from the interval [a, b].)
What can we say generally? Obviously, if x0 is a potential candidate for a maximizer, then it mustbe the case that we cant 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) = maxaxb 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) = 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. Asthe function f crosses x0 from left to right, it goes from positive to negative, i.e. its decreasing.On the other hand, as f crosses x1 from left to right, it goes from negative to positive, i.e. itsincreasing. 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) = maxaxb f(x) even thoughf(b) = 1 = 0. The maximum occurs on the boundary. (b) f(x) = 0 has two solutions, xand x but neither one is a maximizer. f(x) is a local maximum while f(x) is a minimum.
(c) f(x) = x
3
. 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 tonegative. 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 andf(x0) = 0, then x0 is a global maximum. A function f such that f
(x) < 0 for all x is calledconcave.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 tos.t.some constraint on x1 and x2which we denote by g(x1, x2) = g0. The two important examples of this in economics are:
1See Appendix 1.3.
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In the study of consumer behavior, maximize utility u(x1, x2) s.t. the budget constraintp1x1 +p2x2 = 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) = g0
?
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 + x
22; 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) = x1x2; f(x1, x2) = m is the equation ofa hyperbola.
The constrained maximum of the function f occurs where a contour of f is tangent to the contourof g corresponding to g0. 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 thetotal differential of g is zero, i.e.
dg = g1(x1, x2)dx1 + g2(x1, x2)dx2 = 0
(where gi denotes g/xi), which implies
dx2dx1
= g1(x1, x2)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
= f1(x1, x2)dx1 + f2(x1, x2)
dx2
dx1
dx1
=
f1(x1, x2) f2(x1, x2) g1(x1, x2)
g2(x1, x2)
dx1
Now in order for (x01, x02) 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 thismeans the above expression is 0 for all dx1, or in other words
f1(x1, x2)
f2(x1, x2)=
g1(x1, x2)
g2(x1, x2)
But this expression says that at (x01, x02), 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 (x
01, x
02)
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 Frenchmathematician named Lagrange noted that one gets the right answer by setting up an artificial,unconstrained maximization problem with an additional variable, :
L(x1, x2, ) = f(x1, x2) [g(x1, x2) g0]The FONC for L, with respect to x1, x2, and are:
L1 = f1(x1, x2) g1(x1, x2) = 0L2 = f2(x1, x2) g2(x1, x2) = 0L = g(x1, x2) g0 = 0
Dividing the first of these by the second gives
f1(x1, x2)f2(x1, x2)
= g1(x1, x2)g2(x1, x2)
while the third simply restates the constraint! Thus by writing down the Lagrangian L and settingits first derivatives equal to zero we get the necessary conditions for a constrained maximum.
We also get a new variable, , called the Lagrange multiplier. How do we interpret ? It turns outthat the value of tells us how much the maximum value off changes if we relax the constraint by asmall amount. Specifically, suppose we are to maximize f(x1, x2) s.t. the constraint g(x1, x2) = g0.Call the solution (x01, x
02). Now suppose we relax the constraint and instead maximize f(x1, x2) s.t.
g(x1, x2) = g0 + dg0. How do we change our optimal choices of x1 and x2? Suppose we decide touse 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 newconstraint is
dx1 =1
g1(x1, x2)dg0
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The increase in f that accompanies this increase in x1 is
df = f1(x1, x2)dx1 =f1(x1, x2)
g1(x1, x2)=
You are encouraged to check for yourself that if you were to use up the added constraint on x2, dfwould again be . This suggests another interpretation of the tangency condition: at a maximum,if we had a bit more constraint, then we would be indifferent as to whether to use it on x1 or x2.
As with unconstrained optimization, there are also second order conditions. These can be expressedalgebraically; however, they amount to the condition that the objective function has contours thatare more convex than the constraint.2
(a) (b)
Figure 1.7: (a) Contours of f are more convex than g(x1, x2) = g0: SOC satisfied. (b) Contours of f arelinear, less convex than g(x1, x2) = g0: SOC not satisfied.
1.3 Appendix
1.3.1 Convexity
A set S R2 is convex if, for every pair of points u = (u1, u2) and v = (v1, v2) in S, [0, 1] = u + (1 )v S
i.e. the line segment joining u and v lies entirely in S. A set that is not convex is called concave.
A function f : [a, b] R is called convex if, for every x1 and x2 in [a, b],
[0, 1] =
f(x1 + (1
)x2)
f(x1) + (1
)f(x2)
2See Appendix 1.3.
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Or, equivalently, f : [a, b] R is convex if the set S = {(x, y) [a, b] R : y f(x)} is convex. Afunction g : [a, b] R is called concave if g is convex. Let f be twice differentiable. Then
f is convex f(x) > 0 for all xf is concave f
(x) < 0 for all x
Throughout these notes, if f(x) >[[ 0 for all i, then z0 satisfies the SOC for a local minimum. On
the other hand, if sgn(|Hi(f)|) = (1)i for all i, then z0 satisfies the SOC for a local maximum.
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2 Consumer Choice
In this section we apply the methods of optimization of Section 1 to the analysis of consumer choicesubject to a budget constraint. The problem has three elements:
1. Describe the budget constraint.
2. Describe the consumers objective, i.e. his or her utility.
3. Set up and solve the constrained optimization.
2.1 Budget Constraint
We assume that a consumer must choose among bundles (x1, . . . , xn) of commodities 1 through nthat fall within his or her budget. In the case of just two goods x1 and x2 let their prices be p1and p2, respectively. Let the consumer have income I. Then the bundle (x1, x2) is affordable iff
p1x1 +p2x2 I.
Figure 2.1: Graphically, the set of affordable bundles (the budget set) is the triangular region boundedby the coordinate axes and the line x2 = (p1/p2)x1 + I/p2.
Note the following:
if all income is spent on x1, the total amount available is I/p1 (and likewise for x2) we are implicitly assuming that you cannot buy negative amounts of x1 or x2 the slope of the budget line (the outer boundary of the budget set) is p1/p2
2.2 Consumers Objective
We seek a simple way of summarizing how the consumer evaluates alternative bundles, say ( x01, x02)
and (x1, x2).
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Figure 2.2: If we give up one unit of x1, we save p1, which can be used to purchase p1/p2 units of x2.The market trades x1 for x2 at the rate p1/p2. This ratio represents the relative price of x1and x2.
Graphically, the device we use is the indifference curve: a curve connecting bundles that are equallygood. Consider the indifference curve through (x01, x
02), i.e. the set of bundles that are as good
as (x01, x02).
Now take a look at Figure 2.4. If both x1 and x2 are desirable, then bundles with more x1 andmore x2 must be preferred to (x
01, x
02). By the same token, (x1, x2) must be preferred to bundles
with less x1 and less x2. This means that indifference curves must have negative slope.
In more advanced treatments of economic theory, indifference curves are derived from a set ofassumptions about how consumers evaluate alternative bundles. Some types of preferences cannotbe represented by indifference curves. The classic example is lexicographic preferences: theconsumer evaluates a bundle (x1, x2) first by the amount of x1, then by the amount of x2. If
x0
1 > x
1, then (x0
1, x0
2) is strictly preferred to (x
1, x
2) regardless of x0
2 and x
2. However, if x0
1 = x
1,then the consumer compares x02 and x2. (This is the same way alphabetical order works.) As an
exercise, try to graph the indifference curves of a consumer with lexicographic preferences.
Analytically, we represent preferences by a utility function u(x1, x2) with domain equal to the setof possible consumption bundles. We construct u such that higher values are preferred.
Examples:
u(x1, x2) = x1x2 u(x1, x2) = x1 + x2 u(x1, x2) = min {x1, x2}
Facts:
The contours of u are the indifference curves. The bundles (x01, x02) and (x1, x2) lie on the same indifference curve iff u(x01, x02) = u(x1, x2).
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Figure 2.3: How does a consumer decide between (x01, x02) and (x
1, x
2)?
Figure 2.4: If both x1 and x2 are desirable, then it follows that indifference curves are downward-sloping.
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Let h > 0. If more of x1 is always preferred, then u(x1 + h, x2) > u(x1, x2), which impliesu1(x1, x2) > 0 for every bundle (x1, x2). (Likewise for x2.) You are encouraged to verify thisfor each of the above examples.
The slope of the indifference curve through (x1, x2), at (x1, x2), is
u1(x1, x2)/u2(x1, x2).
We call the absolute value of this ratio the marginal rate of substitution (MRS) because itis the amount of x2 the consumer would need to compensate for the loss of one unit of x1,or in other words the amount of x2 needed, per unit of x1 given up, in order to keep utilityconstant.
Figure 2.5: The slope of the indifference curve through (x01, x02) is MRS = u1(x
01, x
02)/u2(x
01, x
02).
Examples:
u(x1, x2) = x1 x2 (Cobb-Douglas)u1(x1, x2) = x
11 x
2
u2(x1, x2) = x
1 x
1
2
M RS =u1(x1, x2)
u2(x1, x2)=
x2
x1
u(x1, x2) = x1 + x2
MRS =u1u2
= 1, a constant for every bundle (x1, x2)
u(x1, x2) = 2log x1 + x2
MRS =u1u2
=2/x1
1=
2
x1, independent of x2
As an exercise, graph the indifference curves for these three examples.
Note: If your utility function is u(x1, x2) and mine is v(x1, x2) = au(x1, x2) + b, where a > 0, thenwe have the same preferences. Why? It can be shown that we have the same indifference curves,
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only with different labels. The result holds for v = f(u), where f is a monatonically increasingfunction.
You may be familiar with the concept of diminishing marginal rate of substitution (DMRS). Unlessstated otherwise, we shall assume DMRS in most of the examples throughout these notes.
(a) (b) (c)
Figure 2.6: (a) DMRS (b) constant MRS (c) increasing MRS
Along an indifference curve, (holding utility constant), the MRS decreases with x1. As one obtainsmore x1, the less one values an additional unit of x1 in terms of x2. DMRS implies that consumersalways prefer averages. Suppose we have two bundles ( x01, x
02) and (x
1, x
2), on the same indifference
curve. Then a bundle that is a weighted average of (x01, x02) and (x
1, x
2), e.g. (x
01, x
02) + (1
)(x1, x2), where 0 < < 1, is strictly preferred to either of the original bundles.
Figure 2.7: The dashed line represents the set of all weighted averages of x0 and x, that is, the setS = {x0 + (1 )x : 0 < < 1}. Clearly these are strictly preferred to both x0 and x.Equivalently, the set S = {x R2 : u(x) > u(x0)} is convex. (One can see this by noting theshape of the region above the indifference curve.)
It is important to understand that DMRS is not the same as diminishing marginal utility, nor arethe two even related. Given a utility function u, the marginal utility of x1 is u1. We say that uexhibits diminishing marginal utility if u11 = (u1)1 < 0. However, the sign of u11 says nothing
about the MRS, as the following examples show:
u(x1, x2) = (x21 + x22)1/4u1(x1, x2) = (1/2)(x
21 + x
22)3/4
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u11(x1, x2) = (3/4)(x21 + x22)7/4 < 0 = decreasing marginal utility but the indifferencecurves are circles, which exhibit increasing MRS.
u(x1, x2) = x31x32u1(x1, x2) = 3x
21x
32
u11(x1, x2) = 6x1x32 > 0 = increasing marginal utility but the indifference curves arehyperbolas, which exhibit DMRS.
2.3 Consumers Optimum
Analytically, the consumers problem is to solve
maxx1,x2
u(x1, x2) s.t. p1x1 +p2x2 = I
Have a look at Figure 2.8. Clearly, a bundle (x01, x02) is optimal if two things are true:
Figure 2.8: The consumer chooses the bundle that lands her on the highest indifference curve while stilllying on the budget line.
1. p1x01 +p2x
02 = I,
2. MRS(x01, x02) = p1/p2.
Condition (2), the tangency condition, expresses the simple fact that if (x01, x02) is optimal, then
there are no gains to be made by trading in the market any further. If MRS > p1/p2, then theconsumer values x1 more than the market does, in terms of x2, so it would benefit the consumerto sell x2 and buy more x1 as you can see in Figure 2.9.
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Figure 2.9: MRS > p1/p2. On the margin, the consumer values x1 more than the market does, in termsof x2, and there is room for a profitable trade! What happens if MRS < p1/p2?
To proceed analytically, lets use the Lagrangian method:
L(x1, x2, ) = u(x1, x2) (p1x1 +p2x2 I)L1 = u1(x1, x2) p1 = 0 (2.1)L2 = u2(x1, x2) p2 = 0 (2.2)L = p1x1 p2x2 + I = 0 (2.3)
Dividing (2.1) by (2.2) gives the tangency condition
u1(x1, x2)
u2(x1, x2)=
p1p2
Also,
=u1(x1, x2)
p1
=u2(x1, x2)
p2With an extra dollar to spend one could either
(a) buy 1/p1 units of x1 and increase utility by u1(x1, x2)/p1 = , or
(b) buy 1/p2 units of x1 and increase utility by u2(x1, x2)/p2 = .
For this reason, is sometimes called the marginal utility of income.
For example, if u(x1, x2) = x1x2, then L = x1x2 (p1x1 +p2x2 I), and the FONC are:L1 = x2 p1 = 0L2 = x1 p2 = 0L = p1x1 p2x2 + I = 0
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Therefore, x1 = p2 and x2 = p1. Plugging these results back into (2.3):
p1(p2) +p2(p1) = I
= 2p1p2 = I= = I
2p1p2
=
x1 = x1(p1, p2, I) = I/2p1,x2 = x2(p1, p2, I) = I/2p2
The functions x1(p1, p2, I) and x2(p1, p2, I) are called the demand functions. Notice that p1x1 =p2x2 = I/2, so the consumer spends half his or her income on each good! As an exercise, re-do the
analysis for U(x1, x2) = x1 x
2 with different values of and .
2.4 Special Problems
Preferences do not satisfy DMRS (Figure 2.10). Often, we restrict preferences by requiring theindifference curves to be convex to the origin. (Functions with this property are called quasi-concave. A function u : R2 R is quasi-concave if the upper contour sets Sk = {(x1, x2) R
2 : u(x1, x2) k} are convex for all k.) Even with quasi-concave preferences, i.e. with convex indifference curves, we still can run into
problems (Figure 2.11). Most consumers consume zero units of most goods, so the endpointproblem is potentially one that economists must deal with. The problem is much worse themore narrowly goods are defined, (e.g. Coke versus Pepsi), and becomes less serious themore broadly they are defined (e.g. beverages in general). A considerable amount of appliedresearch regarding consumer demand involves the so-called discrete choice approach, focusingon whether consumers buy some or none of a given commodity. Daniel McFadden won theNobel Prize for his research showing how to link the buy, dont buy decision to underlyingutility functions.
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(a) (b)
Figure 2.10: (a) Indifference curves exhibit CMRS, and there is no bundle with MRS = p1/p2. (b)MRS = p1/p2 but this point is not a maximumwhats wrong?
(a) (b)
Figure 2.11: Endpoint optima: (a) MRS < p1/p2, (x1, x2) = (0,I/p2) (b) MRS > p1/p2, (x1, x2) =
(I/p1, 0).
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3 Two Applications of Indifference Curve Analysis
We have seen that the consumers optimum is represented by a tangency between an indifferencecurve and the budget constraint. This condition expresses the simple economic idea that the
consumer, on the margin, cannot adjust her consumption bundle to spend the same amount ofmoney and simultaneously achieve higher utility. Recall that the tangency condition is only truewhen the indifference curves exhibit DMRS, and we dont have an endpoint optimum.
3.1 Analysis of a Subsidy
In many economies, certain commodities are subsidized by the government. A subsidy is a negativetax that is usually introduced to aid low income consumers. Economists generally argue thatsubsidies are inefficient. Why?
Let there by two commodities: food f and other stuff x. The price of other stuff is px, andthe price of food is pf. A typical consumer has income I and normal preferences, (quasi-concave
indifference curves with DMRS). The budget constraint is pxx +pff = I. See Figure 3.1.
Figure 3.1: Budget constraints with and without food subsidy. (x, f) denotes the optimal choice underthe subsidy arrangement.
Suppose now that a subsidy of $s per unit is introduced on food. The budget constraint becomespxx + (pf s)f = I. If the consumer chooses the bundle (x, f), then the cost of the subsidy tothe government (for this consumer alone) is $sf. Most economists would argue that you shouldinstead give the consumer $sf directly and leave the price of food alone. To see this, suppose thelump sum is given to the consumer directly, but she is forced to pay the market, unsubsidized pricefor food. In this case her budget constraint is
pxx +pff = I+ sf (3.1)
Notice that the bundle (x, f) satisfies the budget constraint, since originally
pxx + (pf s)f = I
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In other words, if I give the consumer $sf she still can afford (x, f). But she can do even better,as shown in Figure 3.2.
Figure 3.2: The unsubsidized budget constraint corresponding to I + sf cuts the original indifferencecurve and therefore enables the consumer to achieve higher utility.
The reason is that the budget line (3.1), with the lump sum, is flatter than the budget line withthe subsidy. They both pass through (x, f), so the budget line (3.1) cuts through an indifferencecurve and therefore enables the consumer to choose a bundle with higher utility.
Figure 3.3 illustrates the same point.
3.2 The Consumer Price Index
The CPI is a measure of how much it costs today (in todays dollars) to buy a fixed bundle ofcommodities. We currently use 1982-84 as our reference period, which means the CPI is calculatedby finding the cost of the bundle relative to its cost in 1982-84, $100.
Suppose the CPI is 177.5, (which it was in July 2001). That means it now costs 1.775 times asmuch to purchase the standard bundle as it did on average in 1982-84. If someone earns 1.78times as much as he did in the early 80s, then he is at least as well off as he was then.
Does your nominal income necessarily have to rise in proportion with the CPI? Suppose that in1983 you purchased (x0, y0) at prices (p0x, p
0y). Your income was I
0, and
x0p0x + y0p0y = I
0
Now suppose that in 2001 prices are (p0x(1 + ), p0y(1 + )). In this case both prices increased at the
rate of . How much would your income have to increase in order to offset the increase in prices?See Figure 3.4.
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Figure 3.3: Note that = sf/px, or the subsidy at initial optimum, in terms of x.
On the other hand, suppose px rises by 3/2 and py rises by /2, i.e.
px = p0x
1 +
3
2
,
py = p0y
1 +
1
2
.
The increase in the cost of living is represented by the increase in the cost of the reference bundle
(x0, y0):
p0x
1 +
3
2
+p0y
1 +
1
2
p0xx0 p0yy0 =
3
2p0xx
0 +1
2p0yy
0.
If you initially spent half your income on each of x and y, then p0xx0 = p0yy
0 = I0/2, and theincrease in the cost of living is
3
2 I
0
2+
2 I
0
2= I0,
a proportional increase of . But, if your income increases by , you are better off!
The reasoning is as follows: If your income increases by enough to allow you to buy (x0, y0) yourbudget is represented by the dashed line. But with that budget, you will not consume (x0, y0); youwill consume a bundle with more y, less x, and higher utility. You respond to the change in relativeprices by altering your consumption. See Figure 3.5.
The CPI is really a weighted average of prices for a fixed set of purchases. See Table 1 for anexample of some of the major categories and their weights. Note the slow growth of apparel prices(usually attributed to the rapid rise in cheap imports) and the very rapid growth in medical prices.
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Figure 3.4: If all prices rise by the same factor, the consumer is in fact worse off.
Figure 3.5: If some prices rise more than others, the new budget line, (assuming income rises in proportion
to CPI), cuts the original indifference curve.
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Table 1: Major Purchase Categories in CPI and Corresponding Weights
Category Weight Price Index (Dec. 2000)All 100.0 174.1
Food & Beverage 16.3 169.5Housing 39.6 171.6Apparel 4.7 131.8Transportation 17.5 155.2Medical 5.8 264.1Recreation 6.0 103.7
Education 2.7 115.4
Communication 2.7 92.3
Other Items 4.7 276.2* Reference period is Dec. 1997, not 1982-84.
The difference between the rate of increase in the average price of the reference bundle and the
minimum increase in income necessary in order to maintain the original level of utility is called thesubstitution bias in the CPI. Note that it depends on two things: how disproportionately pricesfor different goods are rising, and how convex ones indifference curves are. The more convex theindifference curves, and the more dispersion in relative price increases, the bigger the substitutionbias. The Boskin Commission estimates that on average substitution bias was about 0.5% per yearin the U.S. over the past couple decades.
There are lots of other, bigger sources of bias in the CPI. One that is hard to measure is quality bias:consumer goods change over time, which makes it hard to hold the reference bundle constant. Somenew inventions since the early 80s: CD/DVD players, airbags and anti-lock breaks, the internet,laser printers, portable PCs, cell phones, The X-Files. Roughly speaking, quality changes arehandled in the CPI by attempting to subtract the part of any price change that is due to quality,measured at the time the higher quality product is introduced. So, for example, when airbags firstbecame available manufacturers charged about $500 extra for them. Thus, when we compare theprice of a new car in 2001 that is equipped with airbags, to a similar model in 1990 without airbags,we subtract $500 from the 2001 price before computing the price ratio.
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4 Indirect Utility and the Expenditure Function
4.1 Indirect Utility
We characterized the solution to the problem
maxx1,x2
u(x1, x2) s.t. p1x1 +p2x2 = I
as an optimal pair (x01, x02) that satisfies the first order conditions (tangency, budjet constraint).
Note that (x01, x02) varies with (p1, p2, I). We call the optimal choices at a given level of prices and
income the demand functions and write:
x1 = x01(p1, p2, I)
x2 = x02(p1, p2, I)
Note that p1x01(p1, p2, I)+p2x
02(p1, p2, I) = I, so the demand functions satisfy the budget constraint
by definition, even as prices vary. This gives rise to restrictions on the demand functions.
The highest level of utility that can be achieved under (p1, p2, I) is u(x01(p1, p2, I), x
02(p1, p2, I)),
which is the utility of the optimal choices under the budget parameters. We define the indirectutility function to be
v(p1, p2, I) = maxx1,x2
u(x1, x2) s.t. p1x1 +p2x2 = I
= u(x01(p1, p2, I), x02(p1, p2, I))
It should be clear to the reader that v is decreasing in p1 and p2, and increasing in I.
Example: u(x1, x2) = x1 x2 , where + = 1. We saw in Section 2.3 that x
01(p1, p2, I) = I/p1
and x02(p1, p2, I) = I/p2. Note that x01 does not depend on p2, and x
02 does not depend on p1. The
indirect utility function is given by
v(p1, p2, I) = p1 p2 I
4.2 Expenditure Function
Instead of maximizing utility subject to a budget constraint, one could minimize spending, subjectto a utility constraint:
minx1,x2
p1x1 +p2x2 s.t. u(x1, x2) = u0
The Lagrangian isL(x1, x2, ) = p1x1 +p2x2 [u(x1, x2) u0]
The FONC are:
p1 u1(x1, x2) = 0p2 u2(x1, x2) = 0
u(x1, x2) = u0
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Note that the first two conditions are equivalent to the tangency condition p1/p2 = u1/u2. Take alook at Figure 4.1. The parallel lines represent iso-cost lines: combinations such that p1x1 +p2x2is constant. These can be thought of as the contours of the objective function. Their slope isp1/p2. (Why?)
Figure 4.1: How does the consumer reach u0 with as little income as possible?
The utility maximization (u-max) and expenditure minimization (e-min) problems are called dualproblems, since they reverse the objective and the constraint.
What are the solutions to the e-min problem? The choices (x1, x2) that minimize spending subjectto a utility constraint are like demand functions, with the exception that they take utility, ratherthan income, as given. We call these compensated demand functions, and denote them as follows:
x1 = xc1(p1, p2, u
0)
x2 = x
c
2(p1, p2, u
0
)Sometimes these are called Hicksian demand functions, after John Hicks, the English economistwho discovered them (and won the second Nobel prize in economics).
Under (p1, p2, I), and having chosen xc1, xc2, one spends a total of
p1xc1(p1, p2, I) +p2x
c2(p1, p2, I)
We define the expenditure function, (analagous to the indirect utility function for it gives theamount spent assuming one has solved the e-min problem), to be
e(p1, p2, u0) = min
x1,x2p1x1 +p2x2 s.t. u(x1, x2) = u
0
= p1xc1(p1, p2, u
0) +p2xc2(p1, p2, u
0)
Note that e(p1, p2, u0) tells you the minimum amount of money necessary to achieve utility u0 underprices (p1, p2).
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Example: u(x1 , x2 ) = x
1 x
2 , where + = 1. The Lagrangian is
L = p1x1 +p2x2 (x1 x2 u0)FONC:
L1 = p1 x11 x2 = 0L2 = p1 x1 x12 = 0
= p1
p2=
x2
x1= x2 =
p1
p2x1
Substituting this into the budget constraint,
x1
p1
p2x1
= u0
which implies
x1 = u0
p2p1
x2 = u0 p1p2
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5 Comparative Statics of Consumer Choice
In this section we characterize the changes in consumer demands that occur as income and pricesvary. Our goal is to describe the consumers demand functions. Analytically, the demand functions
for the goods x and y are a pair of functions
x = x(px, py, I)
y = y(px, py, I)
that describe the consumers optimal choices of x and y, given prices and income. As you canimagine, the nature of these functions is important in a wide variety of applications.
5.1 Change in Demand with Respect to Income, Engel Curves
As income changes, the budget constrint shifts in a parallel fashion: inward if I decreases, outwardif I increases.
In commodity space, (xy-space, or in our case the plane), the tangencies of the budget constraintswith higher and higher indifference curves trace out the income expansion path shown in Figure 5.1.For a good x, if the quantity of x demanded increases with income, then x is said to be a normalgood. For some goods, the quantity demanded falls with incomesuch goods are called inferior.Analytically, x/I > 0 = x normal, while x/I < 0 = x inferior.
Figure 5.1: Fix prices. Then x(px, py, I) = x(I), and y(px, py, I) = y(I). The income expansion path is{(x(I), y(I)) : I 0}.
A couple interesting implications of the budget constraint for changes in x and y with respect toincome:
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(a) (b) (c)
Figure 5.2: (a) x, y normal (b) x normal, y borderline inferior (c) x inferior, y normal
Using the fact that income is always exhausted,I = pxx +pyy
= dI = pxdx +pydy= 1 = px dx
dI+py
dy
dI
so clearly both goods cannot be inferior for in that case the RHS would be negative.
Starting from the previous equation,xpx
I I
x
dx
dI+
ypyI
Iy
dy
dI= 1
which is equivalent tosxex + syey = 1
where sx and sy are the expenditure shares, (the fraction of income spent on each good),and ex and ey are the income elasticies, (the percent change in demand x/x divided by thepercent change in income I/I, or, in the limit as I 0, (dx/x)/(dI/I)). This equationcan be summarized as follows: the expenditure-weighted sum of income elasticies is unity.
The relation between x and I, holding prices constant, is called the Engel curve, and is shown inFigure 5.3.
The data in Table 2 confirm Engels Law, that as income increases, the expenditure share of fooddecreases. The implication is that income elasticity of food is less than unity. Why? Let x be food.Then sx = xpx/I is the expenditure share of food, and
dsxdI
=px
dxdI
I 1
I2xpx =
xpxI
IxdxdI
I 1
I
xpxI
=sxI
(ex 1)or
I
sx
dsxdI
= ex 1
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Figure 5.3: The Engel curve starts from the origin if x = 0 when I = 0, (which is a reasonable assumption).The Engel curve has positive slope if x is a normal good.
(a) (b) (c)
Figure 5.4: (a) Linear Engel curves: dx/dI = x/I = ex = 1. (b) Convex Engel curves: dx/dI >x/I = ex > 1. (c) Concave Engel curves: dx/dI < x/I = ex < 1.
So, if ex < 1, then food share is declining with income. An alternative proof employs a favoritetrick of economists, taking natural logs:
log sx = log x + logpx log Id log sxd log I
=d log x
d log I 1
orI
sx
dsxdI
= ex 1In some contexts, the food share is used as an indicator of welfare. It has been proposed thatfamilies in different countries with the same food share are equally well off.
5.2 Change in Demand with Respect to Price
A change in one of the prices causes the budget line to rotate; as it does so, the tangencies withhigher and higher indifference curves trace out the price consumption path.
You should be familiar with the demand curve, which is the graph of the demand function x(px) =x(px, p
0y, I
0), where p0y and I0 are fixed. See Figure 5.6.
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Table 2: Food Share of Std. Budget in Various Years
Year Food Share in Std. Budget
1935-39 35.4
1952 32.21963 25.21992 19.62000 16.3* Budget used in calculation of CPI.
Figure 5.5: A rise in px is accompanied by a reduction in x.
Note that we traditionally plot demand, (the dependent variable), on the horizontal axis and theprice, (the independent variable), on the vertical axis.3 The negative slope of the demand curve
reflects the idea that consumption of a commodity falls as its price increases. However, demandcurves are not necessarily downward sloping! We turn now to a decomposition of the change indemand due to a change in price. We show that there are two factors:
1. the curvature of the indifference curves
2. the nature of the income effect on demand
5.3 Graphical Decomposition of a Change in Demand
Suppose px increases from p0x to p
1x; demand changes from (x
0, y0) to (x1, y1). We can deocmposethe change from x0 to x1 as follows:
1. First, think of the change in x that arises purely due to the fact that x now costs more.Draw a budget line with slope p1x/py that still allows the consumer to reach the indifference
3We owe this convention to Alfred Marshall. As a result of this, steep demand curves are inelastic, whereas flatdemand curves are elastic.
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Figure 5.6: The reader is presumed to be famililar with the demand curve.
Figure 5.7: The movement from x0, y0) to (x, y) takes place along the indifference curve.
curve through (x0, y0) (call this indifference curve u0). Note that, since its steeper than theold budget line, it has a tangency with u0 to the left of (x0, y0).4 This artificial budgetconstraint is represented by the dashed line in Figure 5.7.
2. Second, move from this intermediate point to the final optimum. Observe that this movementis a movement along an income expansion path, since the intermediate optimum occurs whereu0 has a tangency with a budget line with slope p1x/py.
Analytically,x = x1 x0 = (x1 x) + (x x0)
where x
denotes the aforementioned intermediate optimum. We refer to the first change (x1
x
),holding utility constant, as the substitution effect. We refer to the second change (x x0), as the
4Assuming DMRS.
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(a) (b)
Figure 5.8: (a) Step 1: move to new tangency on old indifference curve. (b) Step 2: Move along IEP tonew optimum.
income effect. Thus we write
x = xS + xI
5.4 Substitution Effect
The substitution effect represents movement along an indifference curve. It tells you how far tomove in order for the indifference curve to be parallel to the new budget line, i.e. in order for theMRS to equal the new price ratio. Obviously, then, if the indifference curves are relatively flat,you have to go a long way before the MRS equals the new price ratio, and the substitution effect issubstantial. If the indifference curves are highly convex, the MRS changes rapidly and you do notneed to go far: the substitution effect is small. See Figure 5.9.
(a) (b)
Figure 5.9: (a) u0
flat = more substantial substitution effect (b) u0
highly curved = lesser substi-tution effect
Note that if px > 0, the substitution effect is negative. (Why?) What about the substitution
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effect of px on y?
5.5 Income Effect
Intuitively, one might think the income effect is larger the greater x0, i.e. the greater x was inthe first place. If, initially, you consumed very little x, the income effect would be relatively small.Take a look at Figure 5.10:
Notice that the intermediate budget constraint almost passes through (x0, y0). (It alwayscuts below, if not by much.)
So, the income effect is approximately proportional to the change in income from the budgetline through (x0, y0) to the final budget line.
Figure 5.10: The income effect is approximately proportional to the perpendicular distance between thebudget lines.
What is the change in income? The final budget constraint limits the consumer to I, just as theinitial constraint does. Therefore I = p0xx
0 +pyy0. In order to be able to afford (x0, y0) under the
new prices, you would need p1xx0 + pyy
0, or I = pxx0 more than before. For a small change
in px, the intermediate optimum is close to the initial one, so the difference in income from theintermediate constraint to the final one is approximately pxx
0. (The approximation is exact inthe limit px 0.)This confirms our intuition: the movement along the income expansion path from the intermediateoptimum to the final optimumthe income effectwill be larger, the larger was x0, our initial levelof consumption of x.
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6 Slutskys Equation
6.1 Review
Expenditure function:
e(p1, p2, u0) = min
x1,x2p1x1 +p2x2 s.t. u(x1, x2) = u
0
= p1xc1(p1, p2, u
0) +p2xc2(p1, p2, u
0)
where xc1 and xc2 are the compensated demands, the cheapest choices that enable one to achieve
utility level u0 at prices (p1, p2).
The Lagrangian for the e-min problem is
L(x1, x2, ) = p1x1 +p2x2 [u(x1, x2) u0]The FONC are:
p1 u1(x1, x2) = 0p2 u2(x1, x2) = 0
u(x1, x2) = u0
As for the derivatives of the expenditure function with respect to prices,
e(p1, p2, u0)
p1= xc1(p1, p2, u
0) +p1xc1(p1, p2, u
0)
p1+p2
xc2(p1, p2, u0)
p1. (6.1)
The reader is presumed to be familiar with the Envelope Theorem, which says the second and thirdterms on the RHS cancel.
Proof: Recall that u(xc1(p1, p2, u0), xc2(p1, p2, u
0)) = u0. Differentiate both sides with respect to p1:
u1xc1p1
+ u2xc2p1
= 0
But u1 = p1/ and u2 = p2/ by the FONC. It follows by substitution that
p1
xc1
p1+
p2
xc2
p1= 0
which means
p1xc1p1
+p2xc2p1
= 0
Thus we havee(p1, p2, u
0)
p1 = x
c
1(p1, p2, u
0
)
There is a story we tell to go along with this. If you initially are minimizing expenditure, and theprice of good 1 rises, what do you do? Your first order response is simply to continue buying the
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old bundlethis increases your spending by xc1 p1. That is the first term on the RHS of (6.1).But then you would like to adjust your choices of goods 1 and 2 to reflect the new prices. Theadjustments are the second and third terms on the RHS of (6.1). But because your initial choiceswere optimalthey satisfied the FONCwhen you attempt to adjust x1 and x2 you dont save any
more.
6.2 Slutsky Decomposition
Now we are ready to analyze what happens to the uncompensated, or regular demand functionswhen prices rise/fall. Suppose we start with prices (p01, p
02) and income I
0. Initially the optimalchoices are x01 = x1(p
01, p
02, I
0) and x02 = x2(p01, p
02, I
0), where x1() and x2() are the regular demandfunctions.
We decompose the effect of a change in price p1 = p11 p01 as follows:(a) Starting from (x01, x
02), imagine the adjustment you would make if you could remain on the
old indifference curve. This would lead you to a new bundle ( x1, x2). Since prices have risen
this bundle costs more than you were spending before. This move is called the substitutioneffect of the price increase.
(b) Then, from (x1, x2), imagine the adjustment you would make to get back to the original
income level. This would be a move inward along an income expansion path (IEP), andwould lead you to (x11, x
12). This move is called the income effect of a price increase.
Figure 6.1: A decomposition of the change in demand into its constituent parts: movement along theindifference curve followed by movement inward along an IEP.
Note that the total change in x1 is
x1 = x11 x01 = (x11 x1) + (x1 x01) = xI1 + xS1
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What are the relative magnitudes of the constituent parts? To begin, observe that (x01, x02) and
(x1, x2) are on u
0. Now,x01 = x1(p
01, p
02, I
0) = xc1(p1, p2, u0) (6.2)
Also,
x1 = xc1(p11, p02, u0)
so
xS1 = x1 x01 = xc1(p11, p02, u0) xc1(p01, p02, u0)
xc1(p01, p
02, u
0)
p1 p1
The substitution effect depends on the rate at which compensated demands change: this is purely a
function of the curvature of the indifference curves.
How about the income effect?xI1 = x
11 x1
First note that x11 = x1(p11, p
02, I
0): it is the regular demand given (p1,1 , p02, I
0). But what is x1? Itis the choice one would make with enough income remain on u0 even at the new prices. How muchmoney would it take? The answer is e(p11, p
02, u
0)! So,
x1 = x1(p11, p
02, e(p
11, p
02, u
0))
Thus
xI1 = x1(p11, p
02, I
0) x1(p11, p02, e(p11, p02, u0))
x1(p01, p
02, I
0)
I(I0 e(p11, p02, u0))
So the income effect depends on the income derivative of demand times the change in incomeI = I0 e(p11, p02, u0). Note that I < 0 since one would need more than I0 to achieve U = u0at prices (p11, p
02).
But how big is I? We need one last trick. We know that I0 = e(p01, p02, u
0), so we can write
I = I0 e(p11, p02, u0)= e(p01, p
02, u
0) e(p11, p02, u0)
e(p01, p
02, u
0)
p1(p01 p11)
=e(p01, p
02, u
0)
p1 (p1)
= e(p01, p
02, u
0)
p1 p1
(which is negative for an increase in p1). Finally we have
e(p01, p02, u0)p1
= xc1(p01, p
02, u
0) by (6.1)
= x01 by (6.2)
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and combining the last few results,I x01p1
Note that the size of the income effect depends on the original level of consumption of x1.
Putting it all together,
xI1 =x1(p
01, p
02, I
0)
I I = x1(p
01, p
02, I
0)
I x01p1
Thus
x1 = xI1 + x
S1
= x1(p01, p
02, I
0)
I x01p1 +
xc1(p01, p
02, u
0)
p1 p1
orx1p1
= x01x1(p01, p
02, I
0)
I+
xc1(p01, p
02, u
0)
p1
Now in the limit p1 0 the ratio x1/p1 equals the derivative of the regular demand functionwith respect to p1. We have established:
x1(p01, p
02, u
0)
p1= x01
x1(p01, p
02, I
0)
I+
xc1(p01, p
02, u
0)
p1
This is called Slutskys equation, after the Russian economist who proved it over 100 years ago.Slutskys equation says the derivative of the regular demand function with respect to p1 is a com-bination of the income and substitution effects. The income effect depends on the derivative ofdemand with respect to income, times the original level of consumption of x1. The substitutioneffect depends on the derivative of the compensated demand function.
A useful feature of Slutskys equation is that it provides a way to recover information about indif-
ference curves from the derivatives of the demand functions with respect to prices and incomes. Inprinciple, we can observe x1/p1 and x1/I, which would enable us to infer
xc1(p01, p
02, u
0)
p1=
x1(p01, p
02, I
0)
p1+ x01
x1(p01, p
02, I
0)
I
Suppose we get an estimate of xc1/p1 that is nearly zero. The indifference curves must thereforebe almost Leontief (right angles).
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7 Using Market Level Demand Curves
Since the demand curve graphs x = f(px, py, I), if py or I changes, the demand curve shifts. Forexample, if income were to increase by dI > 0, then at a given price, demand would increase by
dx = (x/I)dI. For a normal good x/I > 0, so the demand curve would shift to the right as inFigure 7.1.
Figure 7.1: A shift in the demand curve to to an increase in I, assuming x is a normal good.
If the elasticities of demand are approximately constant, then
d(log x) =dx
x=
x
I I
x
dI
I= ex
dI
I= exd(log I)
where ex is the income elasticity of demand for x.5 Similarly, ifpy changes, the demand curve shifts
unless x/py = 0 (as in the case of Cobb-Douglas preferences). If x/py < 0, and increase in theprice of y causes the demand curve to shift to the right.
For the purposes of evaluating the effect of relatively small changes in prices and income, we oftenassume the demand function has constant elasticities:
x
px px
x=
log x
logpx= xx (constant)
x
py py
x=
log x
logpy= xy (constant)
x
I px
x=
log x
log I= ex (constant)
This is equivalent to assuming that the demand function is log-linear:
log x = xx logpx + xy logpy + ex log I+ c5You should be familiar with the concept of elasticity from Econ 1. In particular, you should be able to verify
that elasticity is a unitless quantity.
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where c is a constant. Note that homogeneity implies xx + xy + ex = 0. Put differently, if pricesand income all rise by one percent, then x remains constant.6
As you recall from introductory economics, the market is constructed by introducing a supply curveof the form x = S(px). (See Figure 7.2.) It is usually assumed that supply is upward sloping. (We
defer the derivation of market supply curves until later.) For now, we shall assume that elasticityof supply is constant:
dS(px)
dpx px
S(px)= x
where x denotes elasticity of supply. We now can combine supply and demand curves to analyzethe effects of exogenous shocks to income or other prices. We have
x = S(px) = f(px, py, I)
a system of two equations in two unknowns, px and x (unit price ofx and quantity ofx, respectively),given income and other prices. This is pictured in Figure 7.3.
Figure 7.2: The reader is presumed to be familiar with the upward sloping supply curve.
7.1 An Increase in Income
Obviously, both x and px increase with I. But by how much? Take a look at Figure 7.4. Startingat equilibrium, with x = x0 and px = p
0x, the changes in demand and supply are:
x
x= xx
pxpx
+ exI
I(demand)
x
x= x
pxpx
(supply)
6A proof would involve recognizing that ifx remains constant, then so does log x, and therefore setting the totaldifferential of log x equal to zero. The details are left to the reader.
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Figure 7.3: The market is in equilibruim when the price is such that supply and demand are balanced.
Figure 7.4: How much does px increase due to an outward shift in the demand curve?
The proportional changes in supply and demand have to be the same in order to restore equilibrium.Therefore
xxpxpx
+ exI
I= x
pxpx
which impliespxpx
=
ex
x xx
I
I
Note that x > 0 and xx < 0, so x xx is strictly positive. Furthermore,
xx = x pxpx
= xexx xx II
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For example, suppose the following:
x = 0.60 (short run)
xx = 1.40ex = 0.40
If I/I = 0.10 (10% increase), then
pxpx
= (0.40)(0.10) 0.02x
x 0.012
As an exercise, calculate the effect of a 10% drop in the price of a substitute good (good y) on themarket for x. Use an estimate for the cros-price elasticity between x and y of 0.67 (xy = 0.67).
7.2 Tax Incidence
If a tax oft dollars per unit is imposed on x, it creates a gap between the price that consumers payand the price that producers receive, of t dollars per unit. You are presumed to be familiar withthe diagram shown in Figure 7.5.
Starting from an equilibrium at (p0x, x0), price received by producers falls to p1x, the price paid
by consumers rises to p1x + t, and the quantity falls to x1. Consider the two marekts shown in
Figure 7.6, each with the same tax. Obviously, the effect of the tax on the prices paid/received bythe two sides depends on the relative elasticities of supply and demand. To see this more formally,we proceed based on the assumption that elasticities are roughly constant. Letting px denote theprice received by producers, the change in supply is
x
x= x
pxpx
The change in prices for consumers is px + t. Therefore, the change in quantity demanded is
x
x= xx
px + t
px
Market equilibrium requires that change in demand equals change in supply:
xx
px + t
px
= x
pxpx
Solving for the equilibrium change in prices, we have
xxt
px=
px
px(x
xx)
andpxpx
=
xx
x xx
t
px
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where t/px is the proportional tax rate. Since x > 0 and xx < 0, so x xx is strictly positive,and therefore px < 0. With regard to quantity,
x
x
= xpx
px
= xxx
x xxt
px
< 0
For producers, the change in price is
pxpx
=
xx
x xx
t
px
and for consumers it is
px + t
px=
xx
x xx
t
px+
t
px=
x
x xx
t
px> 0
Notice that the ratio of the changes in prices for producers versus consumers is xx/x. So, if demand is highly inelastic, i.e. |xx| is small (e.g. xx = 0.1), and supply is moderately elastic(e.g. x = 1.0), then producer prices dont fall by much relative to consumer prices. On the otherhand, if demand is highly elastic, i.e. if xx is big (e.g. xx = 3.0), then producer prices are moreaffected.
Last we consider the effect of a per unit subsidy of s on the price of x. (For example, prior tothe recent rise in electricity rates, electricity prices were subsidized throughout most of California.)The change in price received by producers is px, whereas the change in price paid by consumersis px s. The proportional changes in quantity are:
x
x= xx
px s
px
(demand)
x
x= x
pxx
(supply)
Setting the two equal, we havepxpx
=
xxx xx
s
px> 0
which implies that part of the effect of the subsidy is mitigated by a rise in prices. In fact, thechange in price paid by consumers is
px spx
=
xxx xx
s
px s
px=
xx xx
s
px< 0
Note that x/(x xx) is less than one in absolute value.
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Figure 7.5: The new price p1x is such that when consumers pay p1x+ t and suppliers receive p1x, equilibriumis restored.
(a) (b)
Figure 7.6: (a) Demand inelastic, supply elastic. (b) Demand elastic, supply inelastic.
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8 Labor Supply
In this section we consider the choice of how many hours to work by an individual who faces anhourly wage w > 0, and also has non-labor income y. The individual is assumed to value leisure
and consumption of goods x, using a utility function u(x, ). We assume there is an upper boundT on leisure, and that the sum of leisure and hours of work h is T:
+ h = T, or h = T The graph looks a little unusual since preferences are only defined up to the point where = T asthe reader can see in Figure 8.1.
Figure 8.1: The budget constraint for an agent who works for w/h and consumes a numeraire good x.
The budget constraint is px = wh + y but we shall assume p = 1. The consumers objective is
maxx,
u(x, ) s.t. x = w(T ) + y, or x + w = y + wT
Note that if you think of the consumption bundle as (x, ), then the budget constraint says thetotal cost of the bundle has to be y + wT for this is all the income you would have if you boughtno leisure. This full income depends on w, and therein lies the key difference between laborsupply and other consumer choice problems: as the price of one good (leisure) rises, the consumeris actually richer. Intuitively this is because a worker is a net seller of leisure: he or she starts atan endowment point (x, ) = (y, T). From there he or she can trade with the market by givingup leisure in return for cash, which is then used to purchase goods.
We proceed by the method of Lagrange:
L(x,,) = u(x, ) (x + w y + wT)Lx = ux(x, ) = 0L = u(x, ) w = 0L = x w + y wT = 0
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The first two FONC imply the usual tangency condition: u(x, )/ux(x, ) = w. The solutions are:
x = x(w, y)
= (w, y)
h(w, y) = T (w, y)Now consider the rise in w (from w0 to w1) shown in Figure 8.2. As you can see, the substitution
Figure 8.2: For this individual the income and substitution effects have opposite signs.
effect causes a drop in , or equivalently a rise in h. But the income effect works in the oppositedirection: as a net seller of leisure the agent is better off and uses some of her extra income to buymore leisure.
To formally analyze the income and substitution effects we rely on the expenditure function for the
labor supply case: this is the amount of non-labor income needed to achieve utility u0
, given w:e(w, u0) = min
x,x w(T ) s.t. u(x, ) = u0
L(x,,) = x w(T ) [u(x, ) u0]Lx = 1 ux(x, ) = 0L = w u(x, ) = 0L = u(x, ) + u0 = 0
The first two FONC imply the tangency condition: u(x, )/ux(x, ) = w. The solutions are:
x = xc(w, u0)
= c(w, u0)
hc
(w, u0
) = T c
(w, u0
)The expenditure function is thus
e(w, u0) = xc(w, u0) w[T c(w, u0)] = xc(w, u0) whc(w, u0)
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ande
w=
xc
w w h
c
w
0hc = hc
To see that xc
/w whc
/w = 0, we use the same trick as we did in Section 6 when dealingwith the usual expenditure function. So, recalling that ( xc(w, u0), c(w, u0)) yields utility u0,
u(xc(w, u0), c(w, u0)) = u0
and therefore differentiating both sides,
ux(xc(w, u0), c(w, u0))
xc
w+ u(x
c(w, u0), c(w, u0))c
w= 0
But wux = u by the tangency condition, and hc/w = c/w, hence the desired result.
(Again, this is an example of the Envelope Theorem.)
To summarize, we have shown that e/w = hc(w, u0). To understand this, think of your momwhen she finds out you got a raise at your summer job: she reduces your allowance by an amount
proportional to how much you were working.
Now lets see how leisure choice depends on wages. Assume we start with (w0, y0), and that w risesfrom w0 to w1. The rise in w causes a substitution effect and an income effect:
= S + I
As usual, we can write
S =c
ww
representing the compensated adjustment to the higher cost of leisure on the indifference curvecorresponding to level u0. Also,
I = (w1, y0) (w1, y1)where y0 = original non-labor income, and y1 = e(w1, u0). We use our standard trick of takingfirst order approximations, based on the expenditure function. First, we can approximate
(w1, y0) (w1, y1) (w1, y1)
y (y0 y1)
and recognizing that y0 = e(w0, u0),
y0 y1 = e(w0, u0) e(w1, u0)
e(w0, u0)
w(w)
= hc(w0, u0)(w)= h0w
So,
I (w1, y1)
y h0w
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The income effect is proportional to h0w: if you had been working more, there would be a biggerpositive income effect. Finally, then, we have
= S + I =c(w0, u0)
w
w +(w1, y1)
y h0w
Dividing both sides w, and taking the limit w 0,
w= lim
w0
w=
c(w0, u0)
w+ h0
(w0, y0)
y
This is Slutskys equation for leisure demand. In terms of hours, recall that h = T , soh
w=
wand
h
y=
y
and thereforeh
w=
hc(w0, u0)
w+ h0
h(w0, y0)
y
When the wage rises there is a positive substitution effect and a negative income effect on laborsupply. Note in particular that when a person gets a raise, he wont necessarily work more.
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9 Intertemporal Consumption
The two-period consumption model concerns a consumer whose lifetime spans two periods. Inperiod one the consumer has income y1 and spends c1; in period two the consumer has income y2
and spends c2. The consumer can borrow or lend at a rate of interest equal to r.
We express the consumers budget constraint in terms of period-two dollars. The choice is arbitrary,but this way it ends up simplifying the algebra for then we basically have two goods with prices 1+ rand 1, respectively (rather than 1 and 1/(1 + r), which would be the case in period-one dollars).Having 1 + r in the numerator, not the denominator, is a big help. Total consumption is limitedby total income, so the budget constraint is given by
(1 + r)c1 + c2 = (1 + r)y1 + y2
The consumers objective is to solve
max u(c1, c2) s.t. (1 + r)c1 + c2 = (1 + r)y1 + y2
The Lagrangean is
L(c1, c2, ) = u(c1, c2) [(1 + r)c1 + c2 (1 + r)y 1 y2]and the FONC are
L1 = u1(c1, c2) (1 + r) = 0L2 = u2(c1, c2) = 0L = (1 + r)c1 c2 + (1 + r)y1 + y2 = 0
These give a rise to the tangency condition u1/u2 = 1 + r and the budget constraint, as usual. Thesolutions are functions of r, y1, and y2:
c1 = c1(r, y1, y2)c2 = c2(r, y1, y2)
These demand functions are a little unusual because they specify not just total available resources,or wealth w = (1 + r)y1 + y2, but also the composition of w. To clarify the effects of a changein r on c1 it is helpful to define two other consumption functions, that depend on the interest rateand total wealth (measured in period-two dollars):
c1 = cw1 (r, w)
c2 = cw2 (r, w)
These optimal choice functions are related by:
c1(r, y1, y2) = cw1 (r, (1 + r)y1 + y2)
c2(r, y1, y2) = cw2 (r, (1 + r)y1 + y2)
You can see that as we change r, the effect on c1(r, y1, y2) depends on both c1/r and c1/w.
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Now lets define the expenditure function as the minimum cost to reach a given level of utility(again, measured in period-two dollars). Specifically, define e as follows:
e(r, u0) = min(1 + r)c1 + c2 s.t. u(c1, c2) = u0
The Lagrangian isL(c1, c2, ) = (1 + r)c1 + c2 [u(c1, c2) u0]
and the FONC are
L1 = 1 + r u1(c1, c2) = 0L2 = 1 u2(c1, c2) = 0L = u(c1, c2) + u0 = 0
The solutions are the compensated demand functions cc1(r, u0) and cc2(r, u
0). As usual
e(r, u0) = (1 + r)cc1(r, u0) + cc2(r, u
0)
Differentiating,e(r, u0)
r= cc1(r, u
0) + (1 + r)cc1r
+cc2r
and (as usual) it is easy to show that (1 + r)cc1/r + cc2/r = 0, so
e(r, u0)
r= cc1(r, u
0)
Thus we have three optimal consumption functions for first period consumption:
c1(r, y1, y2), which depends on y1 and y2 cw1 (r, w), which depends only on w
cc
1(r, u0), which depends on utility
We also have two relations connecting the three:
c1(r, y1, y2) = cw1 (r, (1 + r)y1 + y2) (9.1)
cc1(r, u0) = cw1 (r, e(r, u
0)) (9.2)
Now it may seem clear why we defined cw1 : its the function that links the compensated demand andthe demand we ultimately are interested in, c1(r, y1, y2). We can differentiate these two equationswith respect to r. Starting with (9.1),
c1(r, y1, y2)
r=
cw1 (r, (1 + r)y1 + y2)
r+ y1
cw1 (r, (1 + r)y1 + y2)
w(9.3)
This means that when you change r, the response of the demand for c1 as a function of (r, y1, y2)has an income effect, reflecting the fact that as r rises, so does the value of wealth.
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From (9.2) we get an expression like weve seen before:
cc1(r, u0)
r=
cw1 (r, e(r, u0))
r+
cw1 (r, e(r, u0))
w e(r, u
0)
r
= cw
1 (r, e(r, u0
))r
+ cw
1 (r, e(r, u0
))w
cc1(r, u0)
Rearranging, we get a Slutsky equation for cw1 :
cw1 (r, e(r, u0))
r=
cc1(r, u0)
r c
w1 (r, u
0)
wcc1(r, u
0)
=cc1(r, u
0)
r c1(r, y1, y2) (9.4)
assuming u0 is the level of utility one can achieve with income ( y1, y2) and interest rate r.
Finally, plugging (9.4) into (9.3),
c1(r, y1, y2)
r =cw
1(r, (1 + r)y1 + y2)
r + y1cw
1(r, (1 + r)y1 + y2)
w
=cc1(r, u
0)
r+
cw1 (r, e(r, u0))
w[y1 c1(r, y1, y2)]
=cc1(r, u
0)
r+
cw1 (r, e(r, u0))
ws1(r, y1, y2)
where s1(r, y1, y2) = y1 c1(r, y1, y2) is the optimal level of period-one savings.The income effect of a rise in r on optimal consumption c1(r, y1, y2) is positive or negative, dependingwhether s1 is positive or negative. For a saver, s1 > 0 and a rise in r has a positive income effect(because the consumer is a net supplier of funds to the market, as in the case of labor supply). Butfor a borrower, s1 < 0 and a rise in r has a negative income effect (because the consumer is a netdemander of funds, as in the case of basic commodity demand).
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10 Production and Cost I
The technology available to a given firm is is summarized by its production function. This functiongives the quantities of output produced by various combinations of inputs. For example, an airline
uses labor inputs, fuel, and machinery (airplanes, loading equipment, etc.) to produce the outputpassenger seats. We write y = f(a, b) to signify that with inputs a and b, it is possible to producey units of output.
Examples:
One Input
y = a
y =
0 a < a1 a > a
Two Inputs
y = ab (Cobb-Douglas)
y = min{a, b} (Leontief, CRS) y = a + b (Additive, CRS)
For two or more inputs, production functions are a lot like utility functions. The important dif-ference is that output is measurable and has natural units (e.g. passenger seats). Its as if theindifference curves have numbers attached to them that matter.
A second, less obvious, way to summarize technology is to compute the cost associated with pro-ducing a given output level y, at fixed prices for the inputs. In principle, if you know the productionfunction, it is easy to find the cost function in two steps:
1. enumerate all possible ways of producing y
2. determine the cheapest one, and evaluate its cost
Most of the economic behavior of firms is studied via the cost function. In the next few sections,we demonstrate how to derive the cost function and illustrate the connection between its propertiesand those of the production function.
10.1 One-Factor Production and Cost Functions
10.1.1 Production Functions
Suppose there is only one input (apart from, perhaps a set-up cost). Then we have a picturealong the lines of Figure 10.1. Note that f(0) = 0 by convention.
Definitions and Facts:
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Figure 10.1: A representative production function. Note the S shape.
The marginal product of factor a is the increase in y that accompanies a unit increase in a:
MPa =f(a)
a= f(a)
Factor a is said to be useful if f(a) > 0.
The average product of factor a is the ratio of total output to total input of a:
APa =f(a)
a
If the M P of factor a is increasing, then f(a) > 0 and we say that there are increasingmarginal returns: as the scale of output is expanded, each additional unit of input contributesmore. If the MP is decreasing, then f(a) < 0 and we say there are diminishing marginalreturns. See Figure 10.2.
(a) (b)
Figure 10.2: (a) Increasing marginal returns. (b) Decreasing marginal returns.
If MPa > APa, then APa is increasing; if M Pa < APa, then APa is decreasing.Think baseball, with AP = career batting average and MP = season batting average. Ahitter who has a better-than-average season raises his career average. See Figure 10.3. In
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general,dAPa
da=
af(a) f(a)a2
=1
a
f(a) f(a)
a
=
1
a(M Pa APa)
Figure 10.3: At a = a1, AP = f(a1)/a < f(a) = MP, AP is increasing. At a = a2, the opposite is true.
Examples:
f(a) = ka, where k > 0 (linear). APa = MPa = k. f(a) = a , where 0 < < 1 (concave). See Figure 10.4.
Figure 10.4: The greater , the less concave the production function, up to = 1.
f(a) = 9a2 a3, a < 6. See Figure 10.5. For this function we have the following:f(a) = 18a 3a2 = [f(a) 0 a 6]
f(a) = 18
6a =
f(a) > 0 a < 3f
(a) < 0 a > 3
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Figure 10.5: The production function of the example on page 57.
10.1.2 Cost Functions
What is the cost function for a one-factor production function? Let w dentoe the price per unit offactor a. Then
c(y, w) = min wa s.t. y = f(a)But y = f(a) implies a = f1(y).7 Therefore c(y, w) = wf1(y). See Figure 10.6 for an illustrationof this process. If w is fixed, then we often write the cost function as a function of y only: c(y).Define marginal cost MC(y) = c(y), and average cost AC(y) = c(y)/y.
Examples:
y = f(a) = ka (linear) = a = y/k (linear input requirement function)
c(y, w) = wy
2
=
1
2wy (linear in both y and w)
y = f(a) = a = a = y2 (convex input requirement function)
c(y, w) = wy2 (linear in w but convex in ysee Figure 10.7)
10.1.3 Connection between M C and MP
Marginal cost is the amount it would cost, at the current level of output, to produce an additonalunit. By definition of MPa, one unit of input adds MPa = f(a) units of output. It follows that
1/MPa = 1/f(a) units of a are needed to produce one unit of y the marginal cost of an additional unit is MC(y) = w/f(a), when the production function
is given by y = f(a)
7Assume, for the moment, that f is one-to-one.
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(a)
(b)
Figure 10.6: The graph in (b) is obtained by rotating quadrant II in (a) 90 degrees clockwise.
Alternatively, c(y) = wf1(y), using as input requirement function a = f1(y). Thus8
C(y) = w df1
(y)dy
= wf(a)
10.1.4 Geometry ofc, AC, and MC
Take a look at Figure 10.8a. Note the following:
when MC < AC, AC is falling when MC > AC, AC is rising when AC is at a minimum, AC = MC8Recall that iff(x0) = 0, then
df1(y)dy
y=f(x0)
=
1
f(x0) .
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(a) (b)
Figure 10.7: The production function y =
a and the corresponding cost function c = wy2, where w isthe per-unit cost of a.
We sometimes add a set up cost F, (also called a fixed cost). The total cost is then
c(y) = fixed cost + variable cost = F + V C(y)
The implications of this model are illustrated in Figure 10.8b.
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(a) (b)
Figure 10.8: Compare (b) to (a) and note the following: 1. min AC occurs to the right of min AV C.Why? 2. MC intersects both AC and AV C at their respective minimumns. Why?
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11 Production and Cost II
The analysis of production and cost is more interesting when it involves combinations of two ormore inputs to produce y. The production function is y = f(a, b). As in consumer theory, we begin
by thinking about combinations of inputs that produce the same level of output. In the firm casethese are called isoquants.
We define the marginal rate of technical substitution(MRTS) as the slope of an isoquant. It indicateshow many units of b one would need to add, per unit of a given up, to keep output constant. SeeFigure 11.1.
Figure 11.1: The marginal rate of technical substitution is analogous to the consumers MRS. This bearscomparison to Figure 2.5.
Formally, suppose y = f(a0, b0), and consider varying a and b in such a way that output remainsfixed at y0:
dy = fada + fbdb = 0
which implies db
da
y0
= fa(a0, b0)
fb(a0, b0= MPa
MPb
The MRTS is analogous to the marginal rate of substitution (MRS) in consumer theory. Whenthere are two or more inputs, the production function is characterized by both the degree of sub-stitutability between inputs (curvature of isoquants) and the extent to which output expands asinputs are expanded proportionately. The latter gives rise to the idea of returns to scale. Recallthat for a production function y = f(a, b), we say f has constant returns to scale (CRS) if
f(a,b) = f(a, b), > 0
We say that f has decreasing returns to scale (DRS) if
f(a,b) < f(a, b), > 1
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With DRS, if you double both inputs, you get less than twice the output. On the other hand, thesame inequality implies that if you reduce inputs by some proportion, your output falls by a smallerproportion. So DRS suggests that smaller firms are necessarily more efficient. Conversely we saythat f has increasing returns to scale (IRS) if
f(a,b) > f(a, b), > 1
(a) (b)
Figure 11.2: (a) CRS and (b) DRS. This can be seen by noting the shape of the intersection of the surfacewith the plane a = b for example.
