Presentation on theme: "Overview... Consumption: Basics The setting"— Presentation transcript:
1Overview...Consumption: BasicsThe settingThe environment for the basic consumer optimisation problem.Budget setsRevealed PreferenceAxiomatic Approach
2A method of analysisSome treatments of micro-economics handle consumer analysis first.But we have gone through the theory of the firm first for a good reason:We can learn a lot from the ideas and techniques in the theory of the firm……and reuse them.
3Reusing results from the firm What could we learn from the way we analysed the firm....?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 .Begin with notation
4Notation xi x = (x1, x2 , ..., xn) X pi p = (p1 , p2 ,..., pn) y Quantitiesxia “basket of goodsamount of commodity ix = (x1, x2 , ..., xn)commodity vectorXconsumption setx Î X denotes feasibilityPricespiprice of commodity ip = (p1 , p2 ,..., pn)price vectoryincome
5Things that shape the consumer's problem 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...)
6The consumption setThe set X describes the basic entities of the consumption problem.Not a description of the consumer’s opportunities.That comes later.Use it to make clear the type of choice problem we are dealing with; for example:Discrete versus continuous choice (refrigerators vs. contents of refrigerators)Is negative consumption ruled out?“x Î X ” means “x belongs the set of logically feasible baskets.”
7The set X: standard assumptions Axes indicate quantities of the two goods x1 and x2.x2Usually assume that X consists of the whole non-negative orthant.Zero consumptions make good economic senseBut negative consumptions ruled out by definitionno points here…Consumption goods are (theoretically) divisible……and indefinitely extendable…But only in the ++ directionx1…or here
8Rules out this case...Consumption set X consists of a countable number of pointsx2Conventional assumption does not allow for indivisible objects.But suitably modified assumptions may be appropriatex1
9... and thisConsumption set X has holes in itx2x1
10... and thisConsumption set X has the restriction x1 < xx2ˉConventional assumption does not allow for physical upper boundsBut there are several economic applications where this is relevantx1ˉx
11Overview... Budget constraints: prices, incomes and resources Consumption: BasicsThe settingBudget constraints: prices, incomes and resourcesBudget setsRevealed PreferenceAxiomatic Approach
12The budget constraint x2 x1 Two important subcases determined by The budget constraint typically looks like thisx2Slope is determined by price ratio.“Distance out” of budget line fixed by income or resourcesTwo important subcases determined by… amount of money income y.…vector of resources Rp1– __p2Let’s seex1
13Case 1: fixed nominal income y . .__p2Budget constraint determined by the two end-pointsx2Examine the effect of changing p1 by “swinging” the boundary thus…Budget constraint isnS pixi ≤ yi=1y . .__p1x1
14Case 2: fixed resource endowment Budget constraint determined by location of “resources” endowment R.x2Examine the effect of changing p1 by “swinging” the boundary thus…Budget constraint isn nS pixi ≤ S piRii= i=1ny = S piRii=1Rx1
15Budget constraint: Key points Slope of the budget constraint given by price ratio.There is more than one way of specifying “income”:Determined exogenously as an amount y.Determined endogenously from resources.The exact specification can affect behaviour when prices change.Take care when income is endogenous.Value of income is determined by prices.
16Overview... Deducing preference from market behaviour? Consumption: BasicsThe settingDeducing preference from market behaviour?Budget setsRevealed PreferenceAxiomatic Approach
17A basic problemIn 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?
18The Axiomatic Approach We could “invent” an objective function.This is more reasonable than it may sound:It is the standard approach.See later in this presentation.But some argue that we should only use what we can observe:Test from market data?The “revealed preference” approach.Deal with this now.Could we develop a coherent theory on this basis alone?
19Using observables only 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
20“Revealed Preference” Let market prices determine a person's budget constraint..x2Suppose the person chooses bundle x...For example x is revealed preferred to x′x is revealed preferred to all these points.Use this to introduce Revealed Preferencex′xx1
21Axioms of Revealed Preference Axiom of Rational Choicethe consumer always makes a choice, and selects the most preferred bundle that is available.Essential if observations are to have meaningWeak Axiom of Revealed Preference (WARP)If x RP x' then x' not-RP x.If x was chosen when x' was available then x' can never be chosen whenever x is availableWARP is more powerful than might be thought
22graphical interpretation WARP in the marketSuppose that x is chosen when prices are p.If x' is also affordable at p then:Now suppose x' is chosen at prices p'This must mean that x is not affordable at p':graphical interpretationOtherwise it would violate WARP
23WARP in actionTake the original equilibriumx2Now let the prices change...Could we have chosen x° on Monday? x° violates WARP; x does not.WARP rules out some points as possible solutionsTuesday's pricesTuesday's choice:On Monday we could have afforded Tuesday’s bundlex°Clearly WARP induces a kind of negative substitution effectBut could we extend this idea...?x′Monday's pricesMonday's choice:xx1
24Trying to Extend WARP x'' x' x x2 x1 Take the basic idea of revealed preferencex2x″ is revealed preferred to all these points.Invoke revealed preference againInvoke revealed preference yet againx''x' is revealed preferred to all these points.Draw the “envelope”x'x is revealed preferred to all these points.Is this an “indifference curve”...?No. Why?xx1
25Limitations of WARP x′ x x″ x″′ Is RP to WARP rules out this pattern...but not thisIs RP toxx′WARP does not rule out cycles of preferenceYou need an extra axiom to progress further on this:the strong axiom of revealed preference.x″x″′
26Revealed Preference: is it useful? You can get a lot from just a little:You can even work out substitution effects.WARP provides a simple consistency test: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:But that's what we're going to try right now.It’s time to look at the mainstream modelling of preferences.
27Overview... Standard approach to modelling preferences Consumption: BasicsThe settingStandard approach to modelling preferencesBudget setsRevealed PreferenceAxiomatic Approach
28The Axiomatic Approach Useful for setting out a priori what we mean by consumer preferences.But, be careful......axioms can't be “right” or “wrong,”...... although they could be inappropriate or over-restrictive.That depends on what you want to model.Let's start with the basic relation...
29The (weak) preference relation The basic weak-preference relation:x x'"Basket x is regarded as at least as good as basket x' ..."From this we can derive the indifference relation.x x'“ x x' ” and “ x' x. ”…and the strict preference relation…x x'“ x x' ” and not “ x' x. ”
30Fundamental preference axioms CompletenessTransitivityContinuityGreed(Strict) Quasi-concavitySmoothnessFor every x, x' X either x<x' is true, or x'<x is true, or both statements are true
31Fundamental preference axioms CompletenessTransitivityContinuityGreed: local non-satiation(Strict) Quasi-concavitySmoothnessFor all x, x' , x″ X if x x' and x‘ x″ then x x’’
32Fundamental preference axioms CompletenessTransitivityContinuityGreed(Strict) Quasi-concavitySmoothnessFor all x' X the not-better-than-x' set and the not-worse-than-x' set are closed in X
33Continuity: an example The indifference curveTake consumption bundle x°.Construct two other bundles, xL with Less than x°, xM with Moredo we jump straight from a point marked “better” to one marked “worse"?Better than x ?There is a set of points like xL, and a set like xMDraw a path joining xL , xM.xMIf there’s no “jump”…“No jump” means no inverse preference, that is to say, a sequencethenx°but what about the boundary points between the two?xLWorse than x?
34Axioms 1 to 3 are crucial ... completeness transitivity continuity The utilityfunction
35A continuous utility function then represents preferences... x x'U(x) ³ U(x')
36Notes on utilityThat is ,U measures all the objects of choice on a numerical scale, and a higher measure on the scale means the consumer like the object more. It’s typical to refer to a function . Utility is a fn. for the consumer .Why would we want to a know whether preference has a numerical representation ? Essential, it is convenient in application
37to work with utility fn. It is relatively easy to specify a consumer’spreference by writing down a utilityfunction. And we can turn a choiceproblem into a numerical maximizationproblem. That is, if preference hasnumerical representation U, then the“best” alternatives out of a setaccording to are precisely those
38element of A that has the maximum utility. If we are luckyenough to know that the utilityfunction U, and the set A fromwhich choice is made are “nicelybehaved” (eg. U is differentiable andA is a convex compact set), then wecan think of applying the technique
39of optimization theory to solve this choice problem.
40Tricks with utility functions U-functions represent preference orderings.So the utility scales don’t matter.And you can transform the U-function in any (monotonic) way you want...
41Irrelevance of cardinalisation U(x1, x2,..., xn)So take any utility function...This transformation represents the same preferences...log( U(x1, x2,..., xn) )…and so do both of theseAnd, for any monotone increasing φ, this represents the same preferences.exp( U(x1, x2,..., xn) )( U(x1, x2,..., xn) )U is defined up to a monotonic transformationEach of these forms will generate the same contours.Let’s view this graphically.φ( U(x1, x2,..., xn) )
42A utility function u U(x1,x2) x2 x1 Take a slice at given utility levelProject down to get contoursU(x1,x2)The indifference curvex2x1
43Another utility function By construction U* = φ(U)Again take a slice…U*(x1,x2)Project down …The same indifference curvex2x1
44Assumptions to give the U-function shape CompletenessTransitivityContinuityGreed(Strict) Quasi-concavitySmoothness
45The greed axiom B Bliss! x' x2 x1 Greed: utility function is monotonic Pick any consumption bundle in X.x2increasingpreferenceGreed implies that these bundles are preferred to x'.IncreasingpreferenceGives a clear “North-East” direction of preference.Bliss!BWhat can happen if consumers are not greedyIncreasingpreferenceincreasingpreferenceGreed: utility function is monotonicincreasingpreferencex'x1
46A key mathematical concept We’ve previously used the concept of concavity:Shape of the production function.But here simple concavity is inappropriate:The U-function is defined only up to a monotonic transformation.U may be concave and U2 non-concave even though they represent the same preferences.So we use the concept of “quasi-concavity”:“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 (x1,x2)-space, common parlance describes these as “convex to the origin.”It’s important to get your head round this:Some examples of ICs coming up…Review Example
47Conventionally shaped indifference curves x1x2Slope well-defined everywherePick two points on the same indifference curve.Draw the line joining them.AAny interior point must line on a higher indifference curveCICs are smooth…and strictly concaved-contouredI.e. strictly quasiconcaveincreasingpreferenceB(-) Slope is the Marginal Rate of SubstitutionU1(x)——U2 (x) .sometimes these assumptions can be relaxed
48Other types of IC: Kinks x1x2Strictly quasiconcaveBut not everywhere smoothABCMRS not defined here
49Other types of IC: not strictly quasiconcave Slope well-defined everywherex2Not quasiconcaveQuasiconcave but not strictly quasiconcaveutility here lower than at A or BABCIndifference curves with flat sections make senseBut may be a little harder to work with...Indifference curve follows axis herex1
50Summary: why preferences can be a problem Unlike firms there is no “obvious” objective function.Unlike firms there is no observable objective function.And who is to say what constitutes a “good” assumption about preferences...?
51Review: basic concepts Consumer’s environmentHow budget sets workWARP and its meaningAxioms that give you a utility functionAxioms that determine its shapeReviewReviewReviewReviewReview
52What next? Setting up consumer’s optimisation problem Comparison with that of the firmSolution concepts.