Consumer TheoryJonathan Levin and Paul MilgromOctober 20041 The Consumer ProblemConsumer theory is concerned with how a rational consumer would mak e consump-tion decisions. What makes this problem worthy of separate study, apart from thegeneral problem of choice theory, is its particular structure that allows us to de-rive economically meaningful results. The structure arises because the consumer’schoice sets sets are assumed to be defined by certain prices and the consumer’sincome or wealth. With this in mind, w e define the consumer problem (CP) as:maxx∈Rn+u(x)s.t. p · x ≤ wThe idea is that the consumer chooses a vector of goods x =(x1, ..., xn)tomaximizeherutilitysubjecttoabudget constraint that says she cannot spend more thanher total w ealth.What exactly is a “good”? The answer lies in the eye of the modeler. Dependingon the problem to be analyzed, goods might be very specific, like tickets to differentworld series games, or very aggregated like food and shelter, or consumption andleisure. The components of x might refer to quantities of different goods, as ifall consumption takes place at a moment in time, or they might refer to averagerates of consumption of eac h good over time. If we w ant to emphasize the roles ofquality, time and place, the description of a good could be something like “Number12 grade Red Win ter Wheat in Chicago.” Of course, the way we specify goods canaffect the kinds of assumptions that make sense in a model. Some assumptionsimplicit in this formulation will be discussed below.Given prices p and wealth w,wecanwritetheagent’schoiceset(whichwasXin the general choice model) as the budget set:B(p, w)=©x ∈ Rn+: p · x ≤ wªThe consumer’s problem is to choose the element x ∈ B(p, w) that is most preferredor, equivalently, that has the greatest utility. If we restrict ourselves to just twogoods, the budget set has a nice graphical representation, as is shown in Figure 1.6-AAAAAAAAAAAAAA@@@@@@@@@@@@@@x1x2B(ˆp, w) B(p, w)ˆp1>p1ˆp2= p2Figure1:Thebudgetsetatdifferent prices.Let’s make a few observations about the model:1. The assumption of perfect information is built deeply into the formulationof this choice problem, just as it is in the underlying c hoice theory. Som ealternative models treat the consumer as rational but uncertain about theproducts, for example how a particular food will taste or a how well a clean-ing product will perform. Some goods may be experience goods which theconsumer can best learn about by trying (“experiencing”) the good. In thatcase, the consumer might want to buy some now and decide later whetherto buy more. That situation would need a different formulation. Similarly,2if the agent thinks that high price goods are more likely to perform in asatisfactory way, that, too, would suggest quite a different formulation.2. Agents are price-takers. The agent takes prices p as known, fixed and ex-ogenous. This assumption excludes things like searching for better prices orbargaining for a discount.3. Prices are linear. Every unit of a particular good k comes at the same pricepk. So, for instance, there are no quantity discounts (though these could beaccommodated with relatively minor changes in the formulation).4. Goods are divisible. Formally, this is expressed by the condition x ∈ Rn+,which means that the agent may purch ase good k in any amount she canafford (e.g. 7.5 units or π units). Notice that this divisibility assumption, byitself, does not prevent us from applying the model to situations with discrete,indivisible goods. For example, if the commodit y space includes automobile ofwhich consumers may buy only an integer number, we can accommodate thatby specifying that the consumer’s utility depends only on the integer part ofthe number of automobiles purchased. In these notes, with the exception ofthe theorems that assume con vex preferences, all of the results remain trueeven when some of the goods may be indivisible.2 Marshallian DemandIn this section and the next, we derive some key properties of the consumer prob-lem.Proposition 1 (Budget Sets) For al l λ>0, B(λp, λw)=B(p, w).Moreover,if p À 0,thenB(p, w) is compact.Proof. For λ>0, B(λp, λw)={x ∈ Rn+|λp · x ≤ λw} = {x ∈ Rn+|p · x ≤ w} =B(p, w). Also, if p À 0, then B(p, w) is a closed and bounded subset of Rn+.Hence, it is compact. Q.E.D.3Proposition 2 (Existence) If u is continuous, and p À 0, then (CP) has asolution.Proof. ...because a continuous function on a compact set achieves its maximum.Q.E.D.We call the solution to the consumer problem, x(p, w), the Marshallian (orWalrasian or uncompensated) demand. In general, x(p, w) is a set, rather than asingle point. Thus x : Rn+× R+⇒ Rn+is a correspondence.Itmapspricesp ∈ Rn+and wealth w ∈ R+into a set of possible consumption bundles. One needs moreassumptions (we’re getting there) to ensure that x(p, w) is single-valued, so thatx(·, ·)isafunction.Proposition 3 (Homogeneity) Marshallian demand is homogeneous of degreezero: for all p, w and λ>0, x(λp, λw)=x(p, w).Proof. This one’s easy. Since B(λp, λw)=B(p, w), x(λp, λw)andx(p, w)aresolutions to the same problem! Q.E.D.The upshot of this result is that if prices go up by a factor λ, but so does wealth,the purchasing pattern of an economic agent will not ch ange. Similarly, it doesnot matter whether prices andincomesareexpressedindollars,rupees,eurosoryuan: demand is still the same.Proposition 4 (Walras’ Law) If preferences are locally non-satiated, then forany (p, w) and x ∈ x(p, w), p · x = w.Proof. By contradiction. Suppose x ∈ x(p, w)withp · x<w.Thenthereissome ε>0 such that for all y with ||x − y||<ε, p · y<w.Butthenbylocalnon-satiation, there m ust be some bundle y for which p · y<wand y  x.Hencex/∈ x(p, w) – a contradiction. Q.E.D.Walras’ Law says that a consumer with locally non-satiated preferences willconsume her entire budget. In particular, this allows us to re-express the consumerproblem as:4maxx∈Rn+u(x)s.t. p · x = wwhere the budget inequality is replaced with an equality.The next result speaks to our earlier observation that there might be manysolutions to the consumer problem.Proposition 5 (Conv exity/Uniqueness) If preferences are convex, then x(p, w)is convex-valued. If preferences are strictly convex, then the consumer optimum isalways unique, that is, x(p, w) is a singleton.Proof. Suppose preferences are convex and x, x0∈ x(p, w). For any t