Frank Cowell Microeconomics October 2011 Consumption Basics MICROECONOMICS

Frank Cowell: Microeconomics October 2011 Consumption Basics MICROECONOMICS Principles and Analysis Frank Cowell

Overview. . . Frank Cowell: Microeconomics Consumption: Basics The setting The environment for the basic consumer optimisation problem. Budget sets Revealed Preference Axiomatic Approach

A method of analysis Frank Cowell: Microeconomics n n Some treatments of micro-economics handle consumer analysis first But we have gone through theory of the firm first for a good reason: We can learn a lot from the ideas and techniques in theory of the firm… …and reuse them

Reusing results from the firm Frank Cowell: Microeconomics n What could we learn from the way we analysed the firm. . ? n How to set up the description of the environment How to model optimisation problems How solutions may be carried over from one problem to the other. . . and more n n n Begin with notation

Notation Frank Cowell: Microeconomics Quantities a “basket of goods xi x = (x 1, x 2 , . . . , xn) • amount X • consumption Prices • commodity vector set x X denotes feasibility pi p = (p 1 , p 2 , . . . , pn) y of commodity i • price vector • income

Frank Cowell: Microeconomics Things that shape the consumer's problem n n The set X and the number y are both important. But they are associated with two distinct types of constraint. We'll save y for later and handle X now. (And we haven't said anything yet about objectives. . . )

The consumption set Frank Cowell: Microeconomics n n The set X describes the basic entities of the consumption problem. Not a description of the consumer’s opportunities. u n Use it to make clear the type of choice problem we are dealing with; for example: u u n That comes later. Discrete versus continuous choice (refrigerators vs. contents of refrigerators) Is negative consumption ruled out? “x X ” means “x belongs the set of logically feasible baskets. ”

The set X: standard assumptions Frank Cowell: Microeconomics §Axes indicate quantities of the two goods x 1 and x 2 §Usually assume that X consists of the whole nonnegative orthant. §Zero consumptions make good economic sense §But negative consumptions ruled out by definition Consumption goods are (theoretically) divisible… no points here… § x 1 …or here …and indefinitely extendable… § But only in the ++ direction §

Rules out this case. . . Frank Cowell: Microeconomics §Consumption set X consists of a countable number of points x 2 Conventional assumption does not allow for indivisible objects. § x 1 But suitably modified assumptions may be appropriate §

. . . and this Frank Cowell: Microeconomics §Consumption x 2 holes in it x 1 set X has

. . . and this Frank Cowell: Microeconomics §Consumption set X has the restriction x 1 < xˉ x 2 Conventional assumption does not allow for physical upper bounds § ˉx x 1 But there are several economic applications where this is relevant §

Overview. . . Frank Cowell: Microeconomics Consumption: Basics The setting Budget constraints: prices, incomes and resources Budget sets Revealed Preference Axiomatic Approach

The budget constraint Frank Cowell: Microeconomics §The budget constraint typically looks like this x 2 §Slope is determined by price ratio. §“Distance out” of budget line fixed by income or resources Two important subcases determined by p – __1 p 2 x 1 1. … amount of money income y. 2. …vector of resources R Let’s see

Case 1: fixed nominal income Frank Cowell: Microeconomics x 2 y __ p 2 Budget constraint determined by the two endpoints § Examine the effect of changing p 1 by “swinging” the boundary thus… § . . § Budget constraint is n y __ p 1. . x 1 S pixi i=1 ≤y

Case 2: fixed resource endowment Frank Cowell: Microeconomics Budget constraint determined by location of “resources” endowment R. § Examine the effect of changing p 1 by “swinging” the boundary thus… § x 2 y= § n S pi. Ri i=1 Budget constraint is n n S pixi ≤ S pi. Ri i=1 h. R x 1

Budget constraint: Key points Frank Cowell: Microeconomics n n Slope of the budget constraint given by price ratio. There is more than one way of specifying “income”: u u n Determined exogenously as an amount y. Determined endogenously from resources. The exact specification can affect behaviour when prices change. u u Take care when income is endogenous. Value of income is determined by prices.

Overview. . . Frank Cowell: Microeconomics Consumption: Basics The setting Deducing preference from market behaviour? Budget sets Revealed Preference Axiomatic Approach

A basic problem Frank Cowell: Microeconomics n n n In the case of the firm we have an observable constraint set (input requirement set)… …and we can reasonably assume an obvious objective function (profits) But, for the consumer it is more difficult. We have an observable constraint set (budget set)… But what objective function?

The Axiomatic Approach Frank Cowell: Microeconomics n n We could “invent” an objective function. This is more reasonable than it may sound: u u n But some argue that we should only use what we can observe: u u u n It is the standard approach. See later in this presentation. Test from market data? The “revealed preference” approach. Deal with this now. Could we develop a coherent theory on this basis alone?

Using observables only Frank Cowell: Microeconomics n n Model the opportunities faced by a consumer Observe the choices made Introduce some minimal “consistency” axioms Use them to derive testable predictions about consumer behaviour

“Revealed Preference” Frank Cowell: Microeconomics Let market prices determine a person's budget constraint. . § x 2 §Suppose the person chooses bundle x. . . x. For is example revealed x is preferred revealed to all these points. preferred to x′ l ′ Use this to introduce Revealed Preference § x x x 1

Axioms of Revealed Preference Frank Cowell: Microeconomics Axiom of Rational Choice Essential if observations are to have meaning the consumer always makes a choice, and selects the most preferred bundle that is available. Weak Axiom of Revealed Preference (WARP) If x was chosen when x' was available then x' can never be chosen whenever x is available If x RP x' then x' not-RP x. WARP is more powerful than might be thought

WARP in the market Frank Cowell: Microeconomics h. Suppose that x is chosen when prices are p. h. If x' is also affordable at p then: h. Now suppose x' is chosen at prices p' h. This must mean that x is not affordable at p': Otherwise it would violate WARP graphical interpretation

WARP in action Could we have chosen x° on Monday? x° violates WARP; x does not. y da es s Tu ice pr hx° l ′ § Take the original equilibrium § Now let the prices change. . . §WARP rules out some points as possible solutions Tuesday's choice: On Monday we could have afforded Tuesday’s bundle 's Frank Cowell: Microeconomics x 2 x Mo pr nda ice y s 's l §Clearly WARP induces a kind of negative substitution effect Monday's choice: x But could we extend this idea. . . ? § x 1

Trying to Extend WARP Frank Cowell: Microeconomics §Take the basic idea of revealed preference x 2 x″ is revealed preferred to all these points. Invoke revealed preference again § Invoke revealed preference yet again § l x'' x' is revealed preferred to all these points. l § Draw the “envelope” x ' l x x is revealed preferred to all these points. Is this an “indifference curve”. . . ? § §No. x 1 Why?

Limitations of WARP Frank Cowell: Microeconomics §WARP rules out this pattern §. . . but not this x′ x WARP does not rule out cycles of preference § You need an extra axiom to progress further on this: § x″′ x″ §the strong axiom of revealed preference.

Revealed Preference: is it useful? Frank Cowell: Microeconomics n You can get a lot from just a little: u n WARP provides a simple consistency test: u u n You can even work out substitution effects. Useful when considering consumers en masse. WARP will be used in this way later on. You do not need any special assumptions about consumer's motives: u u But that's what we're going to try right now. It’s time to look at the mainstream modelling of preferences.

Overview. . . Frank Cowell: Microeconomics Consumption: Basics The setting Standard approach to modelling preferences Budget sets Revealed Preference Axiomatic Approach

The Axiomatic Approach Frank Cowell: Microeconomics n An a priori foundation for consumer preferences u u n Careful! (1): axioms can’t be “right” or “wrong” u u n they could be inappropriate or over-restrictive depends on what you want to model Careful! (2): we blur some important distinctions u u u n axioms explain clearly what we mean provide a basis for utility analysis psychologists distinguish between… decision utility – explains choices experienced utility – “enjoyment” Let’s start with the basic relation. . .

The (weak) preference relation Frank Cowell: Microeconomics The basic weak-preference relation: "Basket x is regarded as at least as good as basket x'. . . " x < x' From this we can derive the “ x < x' ” and “ x' < x. ” indifference relation. x v x' …and the strict preference relation… x x' “ x < x' ” and not “ x' < x. ”

Fundamental preference axioms Frank Cowell: Microeconomics n Completeness n Transitivity n Continuity n Greed n (Strict) Quasi-concavity n Smoothness For every x, x' X either x<x' is true, or x'<x is true, or both statements are true

Fundamental preference axioms Frank Cowell: Microeconomics n Completeness n Transitivity n Continuity n Greed n (Strict) Quasi-concavity n Smoothness For all x, x' , x″ X if x<x' and x'<x″ then x<x'″.

Fundamental preference axioms Frank Cowell: Microeconomics n Completeness n Transitivity n Continuity n Greed n (Strict) Quasi-concavity n Smoothness For all x' X the not-better-than-x' set and the not-worse-than-x' set are closed in X

Continuity: an example Frank Cowell: Microeconomics §Take consumption bundle x°. § Construct two other bundles, x. L with Less than x°, x. M with More x 2 Better than x ? do we jump straight from a point marked “better” to l one marked “worse"? M x l l x° x. L Worse than x ? but what about the boundary points between the two? The indifference curve x 1 There is a set of points like x. L, and a set like x. M § § Draw a path joining x. L , x. M. § If there’s no “jump”…

Axioms 1 to 3 are crucial. . . Frank Cowell: Microeconomics lcompleteness ltransitivity lcontinuity The utility function

Frank Cowell: Microeconomics A continuous utility function then represents preferences. . . x < x' U(x) ³ U(x')

Tricks with utility functions Frank Cowell: Microeconomics U-functions represent preference orderings. n So the utility scales don’t matter. n And you can transform the U-function in any (monotonic) way you want. . . n

Irrelevance of cardinalisation Frank Cowell: Microeconomics l l U(x 1, x 2, . . . , xn) log( U(x 1, x 2, . . . , xn) ) § So take any utility function. . . This transformation represents the same preferences. . . § § …and so do both of these And, for any monotone increasing φ, this represents the same preferences. § l exp( U(x 1, x 2, . . . , xn) ) l ( U(x 1, x 2, . . . , xn) ) § l φ( U(x 1, x 2, . . . , xn) ) §Each U is defined up to a monotonic transformation of these forms will generate the same contours. §Let’s view this graphically.

A utility function Frank Cowell: Microeconomics u Take a slice at given utility level § Project down to get contours § U(x 1, x 2) The indifference curve x 2 0 x 1

Another utility function Frank Cowell: Microeconomics u § By construction U* = φ(U) Again take a slice… § Project down … § U*(x 1, x 2) The same indifference curve x 2 0 x 1

Frank Cowell: Microeconomics Assumptions to give the U-function shape n Completeness n Transitivity n Continuity n Greed n (Strict) Quasi-concavity n Smoothness

The greed axiom Frank Cowell: Microeconomics x 2 §Gives a clear “North-East” direction of preference. l nc cr in fere e pr !B x' any consumption bundle in X. §Greed implies that these bundles are preferred to x'. sing a e r e inc erenc f pre Incr e pref asing eren ce ing s a re e Inc ergenc rseif n p a e e §Pick Bliss §What inc pre reas fer ing en ce l x 1 can happen if consumers are not greedy Greed: utility function is monotonic §

A key mathematical concept Frank Cowell: Microeconomics n We’ve previously used the concept of concavity: u n But here simple concavity is inappropriate: u u Review Example n The U-function is defined only up to a monotonic transformation. U may be concave and U 2 non-concave even though they represent the same preferences. So we use the concept of “quasi-concavity”: u u u n Shape of the production function. “Quasi-concave” is equivalently known as “concave contoured”. A concave-contoured function has the same contours as a concave function (the above example). Somewhat confusingly, when you draw the IC in (x 1 , x 2 )-space, common parlance describes these as “convex to the origin. ” It’s important to get your head round this: u Some examples of ICs coming up…

Frank Cowell: Microeconomics Conventionally shaped indifference curves §Slope well-defined everywhere §Pick two points on the same indifference curve. x 2 §Draw the line joining them. § Any interior point must line on a higher indifference curve l. A ng si e a e nc cr in fere e pr l C § ICs are smooth §…and strictly concavedcontoured l. B x 1 I. e. strictly quasiconcave (-)§Slope is the Marginal Rate of Substitution sometimes these U 1(x) assumptions ——. . can U 2 be (x)relaxed.

Other types of IC: Kinks Frank Cowell: Microeconomics §Strictly x 2 §But l. A MRS not defined here l C l. B x 1 quasiconcave not everywhere smooth

Other types of IC: not strictly quasiconcave Frank Cowell: Microeconomics §Slope well-defined everywhere x 2 §Not quasiconcave §Quasiconcave but not strictly quasiconcave utility here lower than at A or B l. A l C l. B §Indifference curves with flat sections make sense Indifference curve follows axis here x 1 §But may be a little harder to work with. . .

Frank Cowell: Microeconomics Summary: why preferences can be a problem Unlike firms there is no “obvious” objective function. n Unlike firms there is no observable objective function. n And who is to say what constitutes a “good” assumption about preferences. . . ? n

Review: basic concepts Frank Cowell: Microeconomics Review Review Consumer’s environment n How budget sets work n WARP and its meaning n Axioms that give you a utility function n Axioms that determine its shape n

What next? Frank Cowell: Microeconomics Setting up consumer’s optimisation problem n Comparison with that of the firm n Solution concepts. n