CITY AND REGIONAL PLANNING 7230 Consumer Behavior Philip A. Viton March 4, 2015 Contents 1 Introduction 2 2 Foundations 2 2.1 Consumption bundles........................ 2 2.2 Preference relations......................... 2 2.3 Rationality.............................. 4 2.4 Utility functions........................... 5 2.5 Counter-example.......................... 6 3 Utility maximization 7 3.1 Setup................................ 7 3.2 Maximizing utility......................... 8 3.3 Marshallian demands........................ 9 3.4 Indirect utility function....................... 9 3.5 Properties of the indirect utility function.............. 9 3.6 Properties of the Marshallian demands............... 10 3.7 Elasticities and their properties................... 11 3.8 Comparative statics......................... 12 4 Expenditure Minimization 13 4.1 Minimizing expenditure....................... 13 4.2 The Hicksian (compensated) demands............... 14 4.3 The expenditure function...................... 15 4.4 Properties of the expenditure function............... 15 4.5 Properties of the Hicksian demands................ 15 1
5 Comparative Statics of Utility Maximization 16 5.1 The Slutsky equation........................ 18 1 Introduction In this note we study the behavior of an individual consumer. To economize on notation, we will not index the individual concerned: however, when two or more individuals are being discussed, almost everything here would need a subscript denoting which individual is in question. (We will need to do this later when we discuss general equilibrium and taxation theory). 2 Foundations 2.1 Consumption bundles We assume that our individual wishes to select quantities of n available goods to consume. If the quantity of good i is x i then a consumption bundle is just a vector of quantities of all goods: x = (x 1, x 2,..., x n ). The (feasible) consumption set X is assumed to be all bundles with non-negative quantities (components), so for feasibility we require x R n +. Note that we allow for fractional (real-valued) consumption quantities. Notation: x is an n-vector (consumption bundle): the quantity of the i-th good in bundle x is x i. y is another bundle: the quantity of the i-th good in bundle y is y i. When we need to refer to more than two bundles, we will sometimes use superscripts to identify them: x j is a third bundle of n goods: the quantity of good i in x j is x j i. 2.2 Preference relations We assume that our individual has preferences over different bundles, and that our task is to describe those preferences. (In other words, we do not attempt to say where these preferences come from: that is the job of psychology and/or sociology). It is important to note at the outset that we distinguish between prefer- 2
ences and opportunities to satisfy those preferences: for example, we can think of a homeless person s preferring a bundle consisting of 1 Maserati and 2 Rolls- Royce s to a bundle consisting of a ticket to the Space Station and a condominium on Fifth Avenue in Manhattan, though neither bundle is (or is likely to be) actually available to him or her. (This is implicit in our description of the consumption set as R n + ). In this section we are concerned with preferences, not opportunities. We will describe our individual s preferences in terms of a binary preference relation such that for two consumption bundles x and y the statement x y says that bundle x is at least as good as bundle y. 1,2 Using our basic preference relation we can define two more (derived) preference relations: Preferred to: we write x y and say that x is (strictly) preferred to y if x y and not-(y x). Indifferent to: we write x y and say that x is indifferent to y if both x y and y x. The set of all bundles x that are indifferent to a given bundle y generates the y-indifference-curve for this individual. Here are some properties we may wish to attribute to our preference relations: Completeness of : for all x, y X : either x y or y x (or both). Completeness says that the relation is capable of dealing with all feasible bundles. Equivalently, any two bundles can always be compared using. Reflexivity of : for all x X : x x. Any bundle x is at least as good as itself. This should be uncontroversial. Transitivity of : for all x, y, z X : x y and y z implies x z. Transitivity is a bit more controversial, but is needed in order for there to be a best choice of a consumption bundle. 1 This applies to the individual we are describing: remember that different individuals will have different preference relations, and when we are comparing different individuals we would have to write j and k for individuals j and k. 2 In class we will probably use xry for x y because there is less possibility of confusing with when written on a blackboard. For the derived relations and we will probably use P and I. 3
To see why, suppose, intransitively, that x y and y z but z x. Suppose we start with bundle z. Then since y z we would prefer to switch to bundle y. But given x y, we would prefer to switch from y to x. But since z x we d prefer to switch from x to z, and we re back where we started, going in circles. With intransitive preferences there s no way we can settle on a most-preferred bundle. Continuity of : for all y X, the sets {x : x y} and {x : x y} are closed. The two sets in question are the set of all bundles that are at least as good as a given bundle y, and those that are no better than y. Suppose we have a sequence of bundles x 1, x 2,... such that x 1 y, x 2 y,... etc. Suppose the sequence converges to a bundle x. Under continuity, x y. (Similarly for x y). Intuitively, this says that small changes in bundles do not lead to big changes in the way those bundles are evaluated. Strong monotonicity: if x y and x y then x y. In this, x y means that x i y i for all components i. Strong monotonicity is a way of saying more is better. Note that we may need to redefine the goods in order to make this plausible: for example, clearly, more air pollution is not better than less; but we can retain the monotonicity assumption if we redefine the good in question to be clean air. Local nonsatiation: for any x X and any vector ε > 0, there is some bundle y with x y < ε such that y x. This says that given any bundle x, we can always find a better one. It should be clear that strong monotonicity implies local nonsatiation, but not viceversa. Strict convexity: For all x, y, z X such that x y, if x y and y z then [αx + (1 α)y] z. Strict convexity says that our individual prefers averages to extremes. 2.3 Rationality The minimal assumptions needed for a useful preference relation are that is complete and transitive, and in that case we say that is rational. It can be shown that if is rational then for the derived relations and : 4
is irreflexive (not-(x x), ie x is not preferred to itself) and transitive. is reflexive, transitive and symmetric (x y implies y x). 2.4 Utility functions We could do all our work solely in terms of the preference relation and its derived relations and. But this will be cumbersome. Can we find a better way? It would be nice to be able to summarize a bundle by a single (real) number, in such a way that comparisons between bundles come down to comparisons between numbers. It would also be nice if the summary measures were continuous in the bundle components, so that we could use the tools of calculus. Of course, for this to be useful, any summary function must be related to the individual s preferences. We shall therefore say that a (continuous) function u(x) = u(x 1, x 2,..., x n ), which is a function from R n to R 1, represents an individual s preferences if u(x) u(y) if and only if x y ; and u(x) > u(y) if and only if x y. Such a representing function is called a utility function. What it does is reduce the problem of comparing bundles via preference orderings to the potentially simpler problem of plugging in the bundles into the utility function and then just comparing single numbers. But can we find a utility function that represents an individual s preferences? (This is the existence question). The following fundamental result is due to Gerard Debreu, but will not be proved here: Theorem: suppose preferences are complete, transitive (ie, rational), reflexive, and strongly monotonic. Then there exists a utility function that represents those preferences. If in addition preferences are continuous, then the utility function is, too. (Note that the hypotheses here do not involve convexity). It is worth noting that this representation is not unique. If u represents someone s preferences, and if g is a positive monotonic transformation, then g(u(x)) also represents the same preferences. So the actual values of the utility function have no special meaning. We say that utility is ordinal, not cardinal: all that matters is whether one value is greater than, less than, or equal to another value. At the 5
level of a specific individual this causes no difficulties, but if we want to compare different individuals, we need to remember that the utility numbers (values) are in a certain sense arbitrary. 3 By way of a weaker result we prove: Theorem: The relation can be represented by a real-valued utility function u only if it is rational. Proof: Suppose u represents. Then we have to show that is rational, ie complete and transitive. Completeness: since u is real-valued, then for any x and y we have either u(x) u(y) or u(y) u(x). (This is just the order property of the real numbers). But since u represents, the first case says x y and the second says y x. So is complete. Transitivity: Suppose x y and y z. We have to show that x z. But since u represents, x y implies u(x) u(y), and y z implies u(y) u(z). By the transitivity of, u(x) u(z), and since u represents, we have x z. Thus is transitive. 2.5 Counter-example It is important to realize that not all preferences can be represented by a continuous utility function. The standard counter-example is given by the idea of lexicographic preferences. Suppose that we are in a two-good world, so that bundles are 2-vectors. And suppose that preferences are as follows: if x 1 > y 1 then x y if x 1 = y 1 then x y if x 2 > y 2 What s going on here is that we first rank bundles based only on their quantities of the first good. Only if the quantities of the first good are the same do we look at the quantity of the second good. The name lexicographic comes from the analogy of the order of words in a dictionary: words beginning with a come before those beginning with b, and if two words both begin with a then we order then by their next letter (aa comes before bc and aa comes before ab), etc. 3 This leads to the vexed problem of interpersonal comparisons of utility, and is one motivation for basing such comparisons on willingness-to-pay. 6
Lexicographic preferences are inconsistent with continuity. To see this, take y = (2, 1) and consider a sequence of bundles x i, (i = 1, 2,... ) where x i = (2 1, 0). (The first few bundles in the sequence are (2.1, 0), (2.01, 0), (2.001, 0)). 10 i Clearly under lexicographic preferences we have x 1 y, x 2 y, and in fact for any i we have x i y. But the sequence of bundles x i converges to x = (2, 0), and if we compare x and y then with lexicographic preferences we have y x. This violates the characterization of continuity given above on p. 4. 3 Utility maximization 3.1 Setup We focus on an individual in an n-good world whose preferences are rational, reflexive continuous and strongly monotonic, and therefore can be represented by a continuous utility function u(x) = u(x 1, x 2,..., x n ). We assume that the individual has available income M to spend on the goods in question, and write p i for the market price of good i. The individual is assumed to be a price-taker in the usual sense that he/she regards the p i as fixed (parametric). The set of affordable bundles is the set of x s such that p x = i p i x i M. We shall write, as usual, px for p x. The individual s problem is to choose a best (most-preferred) consumption bundle. Since we assume that preferences are strongly monotonic, hence locally non-satiated, we know that any bundle that does not exhaust available income can be improved upon: thus we know that a best bundle must satisfy the individual s budget constraint: px = p 1 x 1 + p 2 x 2 + + p n x n = M (note the equality). We could make this a bit more intuitively plausible by defining one of the goods to be, eg, savings, so that it is not necessary to spend all one s income on current consumption. It is also possible to extend the approach in a natural way to an intertemporal choice setting, where the problem for the individual is to select quantities of goods indexed by the time at which they will be consumed; but we do not do that here. (The interesting part of intertemporal choice is the issue of whether time-denominated prices exist). 7
3.2 Maximizing utility The problem of the consumer is to choose x = (x 1, x 2,..., x n ) to: max u(x) s.t. px = M This is a 1-equality-constraint maximization problem. The Lagrangian is: L = u(x) + λ(m px) leading to the FOCs: L u = 0 : λp i = 0, x i x i L = 0 : M px = 0 x i i The quantity u/ x i u i is the marginal utility of good i. We can also express the FOCs in the x s as: u i = p i u j p j On the left we have the slope of an indifference curve, called the marginal rate of substitution; and on the right we have the slope of the budget constraint. In equilibrium, a level curve of the utility function must be tangent to the budget constraint. The SOC involves the bordered Hessian: u 11 u 11... u 1n p 1 u 21 u 22... u 2n p 2 H =... u n1 u n2... u nn p n p 1 p 2... p n 0 and since this is a 1-equlity-constraint maximization problem we must have the (leading) border-preserving principal minors of the Hessian alternating in sign, beginning with the order-2 minor, which is the determinant of a 3 3 matrix whose sign is ( 1) 2 ie positive: H 2 = u 11 u 11 p 1 u 21 u 22 p 2 p 1 p 2 0 8 > 0
Note that if we assume that the problem generates a unique interior solution, we are in effect assuming that the underlying utility function is convex. 3.3 Marshallian demands Assuming that the conditions of the Implicit Function Theorem hold, we can solve the FOCs for the optimal choices of the x s: these will depend on the problem parameters, and we write them as: x (p, M) = (x1 (p, M), x 2 (p, M),..., x n (p, M)) These are the Marshallian (or ordinary) demand functions. They tell us the best quantity of x i to consume when faced with parametric prices p and with available income M. We also obtain the optimal value of the Lagrange multiplier: λ (p, M). 3.4 Indirect utility function The indirect objective function is obtained by evaluating the (direct) objective function at the optimal choices. The result in this case is the indirect utility function, usually written as V (p, M), and we have V (p, M) u(x (p, M)) The indirect utility function tells us the greatest value of the utility index that we can obtain, given prices p and income M. 3.5 Properties of the indirect utility function We have two important and useful results: Shephard s Lemma: We have V (p, M) = λ (p, M) xi (p, M) V (p, M) M = λ (p, M) 9
Proof: Envelope Theorem. The second result gives us an interpretation of the Lagrange multiplier: it is the marginal utility of income. Intuitively, it converts income to utility values, so that in the maximization process we re not trying to maximize apples (the utility function) subject to oranges (the budget constraint). Roy s Identity: V (p, M)/ V (p, M)/ M = x i (p, M) Proof: Follows from Shephard s Lemma by direct calculation. Other properties (without proofs) are: V (p, M) is homogeneous of degree 0 in (p, M). V ( p, M) is non-decreasing in M and non-increasing in p. V ( p, M) is quasi-convex When u(x) is continuous, V (p, M) is continuous too. (The proofs of the first two largely parallel the corresponding properties of the profit function). 3.6 Properties of the Marshallian demands We can also see that the Marshallian demands are homogeneous of degree zero in prices and income: xi (α p, αm) = x i (p, M). This is because scaling up all prices and income by the same proportion doesn t change the budget constraint (in the 2-good case, the intercepts of the budget constraint are x i = M/p i and this will be unchanged at αm/α p i ) and hence won t change the solution, since the utility function doesn t depend on prices and income. 10
3.7 Elasticities and their properties We define the price- j-elasticity of demand for good i as η i j = x i p j p j x i Intuitively, this is the percentage change in the demand for good i, per percentage change in the price of good j. The case j = i gives rise to the own-price elasticity of demand; otherwise we have a cross-elasticity. Similarly, we define the income elasticity of demand for good i as ηi M = x i M M xi (Intuitively: the percentage change in the demand for good i, per percent change in income). The advantage of elasticities over slopes (derivatives) is that they are dimensionless quantities, whereas if we are told that the own-price slope of demand is, say, 2, we need to know the units in which both demand and price are measured in order to make sense of it. First, since the Marshallian demands are homogeneous of degree zero in prices and income, Euler s (adding-up) Theorem implies: j x i p j p j + x i M M = 0 Now divide through by x i and obtain: j x i p j p j x i + x i M M xi = 0 Thus we see that the elasticities satisfy: ηi j + η i M = 0 i Next, recall that the Marshallian demands satisfy the FOCs as identities. This means (for the FOC corresponding to the Lagrange multiplier) that p x (p, M) M 11
Now differentiate both sides with respect to M : i x i M p i = 1 and multiply and divide by terms which cancel out: x ( ) i M M p x i i = 1 x i i M xi M ( pi x ) i = 1 M x i i M ηi M κ i = 1 i that is, if we define a weight κ i as good i s share in total expenditures, then the weighted sum of the income elasticities of demand equals 1. Similarly, differentiating with respect to p i : and multiplying and dividing: j xi + j j j x j p i = 0 x j p i = x i x j p i p j x j p j = p i xi Mx j M x j p i p j x j x j M = p i xi M η ji κ j = κ i j 3.8 Comparative statics There are no straightforward comparative statics results for the uility-maximization problem. To see this, focus on the 2-good setting. Then the basic comparative 12
statics equation in p 1 is H x 1 /p 1 x 2 /p 1 λ /p 1 = λ 0 x 1 ie: u 11 u 11 p 1 u 21 u 22 p 2 p 1 p 2 0 x 1 /p 1 x 2 /p 1 λ /p 1 = so if we solve for the own-price slope of the Marshallian demand we get (using Cramer s Rule): λ u 11 p 1 0 u 22 p 2 x1 x1 p 2 0 = p 1 H The numerator isn t a border-preserving principal minor, and trying our next trick, doing the actual computation, yields λ 0 x 1 λ p 2 2 u 11x 1 p 2 + p 1 u 22 x 1 which is hardly helpful. As it turns out, there is a useful result here, but in order to develop it, we first need to consider a second problem facing our individual. 4 Expenditure Minimization 4.1 Minimizing expenditure We consider the problem of minimizing the cost (expenditure) necessary to realize a specific level of utility, which we will write as ū. Expenditure on a consumption bundle is px, so the problem is to choose x to: min s.t. px u(x) = ū The Lagrangian is: L = px + µ(ū u(x)) 13
leading to the FOCs: L x i = 0 : p i µu i = 0, i L µ = 0 : ū u(x) = 0 Note that, like the FOCs for the utility maximization problem, these also have the interpretation: at an optimum, the marginal rate of substitution (along the ū isoquant) equals the price ratio. 4.2 The Hicksian (compensated) demands Assuming that the Implicit Function Theorem applies, we can solve the FOCs obtain the optimal choice functions: x i (p, ū) the so-called Hicksian, or compensated demands. These tell us the least-cost quantities of the x s needed to attain the utility index ū. As the notation makes clear, they are different from the Marshallian demands which solve the utility maximization problem. It is also important to note that the Hicksian demands depend on the fixed utility index (ū) : they are therefore unobservable with data, unlike the Marshallian demands, which depend on observables, namely prices and income. We also get the optimal value of the Lagrange multiplier: µ (p, ū). The SOC involves the border-preserving principal minors of the bordered Hessian: µ u 11 µ u 12... µ u 1n u 1 µ u 21 µ u 22... µ u 2n u 2 H =... µ u n1 µ u n2... µ u nn u n u 1 u 2... u n 0 (evaluated at a stationary point, so µ u 11 etc). 14
4.3 The expenditure function The indirect objective function for the expenditure-minimization problem is the Expenditure Function: E(p, ū) = px (p, ū) which tells us the minimum outlay (expenditure) you need at prices p in order to attain the utility level (index) ū. 4.4 Properties of the expenditure function The expenditure-minimization problem studied here is exactly analogous to the firm s cost minimization problem, so the properties, and their proofs, are the same, too. Shephard s Lemma: The expenditure function satisfies: E(p, ū) = xi (p, ū) E(p, ū) = µ (p, ū) ū we say that the Lagrange multiplier µ is the marginal cost of utility. E(p, ū) is homogeneous of degree 1 in p : E(α p, ū) = αe(p, ū) E(p, ū) is nondecreasing in p. E(p, ū) is concave in p. If u(x) is continuous, then E(p, ū) is continuous. 4.5 Properties of the Hicksian demands The Hicksian demands are the analogues of the cost-minimizing factor demands, hence we have a determinate comparative-statics result: x i (p, ū) < 0 15
Next, since the Hicksian demands are the derivatives of the expenditure function, they are homogeneous of degree 0 in p : xi (α p, ū) = xi (p, ū). If all prices scale up by α, the same demands result (though of course it takes α times as much expenditure to purchase them). We can also define compensated elasticities for the Hicksian demands and use Euler s adding-up theorem to derive a set of elasticity relations, just as we did with the Marshallian demands. 5 Comparative Statics of Utility Maximization How do all these concepts fit together? First, consider the utility-maximization problem: max u(x) s.t. px = M whose solution is x (p, M). Here we take the budget constraint (ie p and M) as given and find the highest possible attainable indifference curve, which is V ( p, M). Now take this indifference curve as given, and try to minimize the cost of getting onto it at the same prices p. A bit of thought, plus an examination of the pictures below, should convince you that the solution to the second problem has the same consumption bundle as the first, and moreover, the minimum expenditure you will need is exactly E(p, V (p, M). x.2 x.2 data: px = M x * 2(p,M) V(p,M) x ** 2(p,V(p,M)) data: V(p,M) E(p,V(p,M)) = px ** x * 1(p,M) x.1 x ** 1(p,V(p,M)) x.1 16
It works the other way, too. Suppose you solve the expenditure-minimization problem: min s.t. px u(x) = ū whose solution is x (p, ū). Here we take the indifference curve labelled ū as given and minimize the cost of attaining it at prices p. That minimum cost is E(p, ū). Now give our individual income E(p, ū) so that the budget constraint is px = E(p, ū), and find the highest indifference curve you can reach. A bit of thought, together with the picture below, should convince you that the utility-max problem will have the same quantities of the x s as the expenditure-min problem, and moreover the label of the highest indifference curve will be V (p, M). x.2 x.2 data: px = E(p,u) x ** 2(p,u) data: u E(p,u) x * 2(p,M) V(p,M) x ** 1(p,u) x.1 x * 1(p,M) x.1 These observations lead to four useful identities: 1. E(p, V (p, M)) = M, p, M 2. V (p, E(p, ū)) = ū, p, ū 3. x i (p, E(p, ū)) = x i (p, ū), p, ū 4. xi (p, V (p, M)) = xi (p, M), p, M 17
5.1 The Slutsky equation Now we can finally derive our comparative statics result. Let x maximize utility at (p, M) and write ū = v(p, M). Now look at the third identity in the previous section, which we write as: xi (p, ū) = xi (p, E(p, ū)) Since it is an identity in p and ū, we can differentiate it on both sides with respect to p j. The result is: xi = x i + x i E p j p j M p j But we know that E(p, ū)/ p j = x j (p, ū) = x j (p, V (p, M)) = x j (p, M). The first equality is Shephard s Lemma for expenditure functions, the second is the definition of ū, and the third is identity (4) in the previous section. So we have: x i p j = x i p j + x i M x j the Slutsky Equation. An important special case is when i = j. The Slutsky Equation says: x i = x i + x i M x i and because the Hicksian demand slope downwards in own-price, we conclude that: xi + x i M x i < 0 This is the major comparative statics result for the utility-maximization problem. Note that this does not rule out the possibility that xi / > 0, the case of a Giffen (upward-sloping) demand function. But it does rule out something. Suppose we define an superior good (sometimes called normal in the literature) as one for which xi / M > 0. A superior good is one which, as your income goes up, you 18
consume more. 4,5 Then for a superior good whose consumption is non-zero, we can conclude that its demand function must slope downwards in own-price (ie a superior good cannot be Giffen). By assumption, xi > 0 and xi / M > 0, so ( xi / M) xi > 0. But by the Slutsky equation the sum of ( xi / M) xi and xi / must be negative, hence xi / must also be negative. Even though (inferior) Giffen goods are possible, this doesn t mean that they actually occur. In empirical practice, a positive own-price coefficient is usually taken to indicate that something has gone wrong. Finally, we can also derive a reciprocity relation. By Young s Theorem we know that: 2 E = 2 E p j p j so we get: ( ) E p j x j = = p j x i p j ( ) E (using Shephard s Lemma). Hence, the Slutsky Equation tells us that S ji = x j + x j M x i = x i p j + x i M x j = S i j the matrix S of Slutsky terms is symmetric. 4 The opposite case is that of an inferior good for which xi / M < 0 : as your income goes up, you tend to consume less of it. Possible examples for me at least are that a good French bordeaux wine is superior, while Gallo s cabernet wine is inferior. 5 Note that in this context superior and inferior arenot intended to express a moral judgement: they just refer to how demand reacts to income. Still, the terminology is probably a bit unfortunate. As far as I know, the terms superior and inferior date from the days of Alfred Marshall late nineteenth century when, at least at Cambridge, UK, economics was known as Moral Science and was thought of as a practical branch of ethics. 19