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- 1. Frank Cowell: Microeconomics Consumption Basics MICROECONOMICS Principles and Analysis Frank CowellOctober 2006
- 2. Overview... Consumption: BasicsFrank Cowell: Microeconomics The setting The environment for the basic Budget sets consumer optimisation problem. Revealed Preference Axiomatic Approach
- 3. NotationFrank Cowell: Microeconomics Quantities a ―basket xi of goods •amount of commodity i x = (x1, x2 , ..., xn) •commodity vector X •consumption set x X denotes feasibility Prices pi •price of commodity i p = (p1 , p2 ,..., pn) •price vector y •income
- 4. Things that shape the consumers problemFrank Cowell: Microeconomics The set X and the number y are both important. But they are associated with two distinct types of constraint. Well save y for later and handle X now. (And we havent said anything yet about objectives...)
- 5. The consumption setFrank Cowell: Microeconomics The 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 X ‖ means ―x belongs the set of logically feasible baskets.‖
- 6. The set X: standard assumptionsFrank Cowell: Microeconomics Axes indicate quantities of x2 the two goods x1 and x2. Usually assume that X consists of the whole non- negative orthant. Zero consumptions make good economic sense But negative consumptions ruled out by definition no points Consumption goods are here… (theoretically) divisible… …and indefinitely x1 extendable… …or here But only in the ++ direction
- 7. Rules out this case...Frank Cowell: Microeconomics Consumption set X x2 consists of a countable number of points Conventional assumption does not allow for indivisible objects. x1 But suitably modified assumptions may be appropriate
- 8. ... and thisFrank Cowell: Microeconomics Consumption set X has x2 holes in it x1
- 9. ... and thisFrank Cowell: Microeconomics Consumption set X has x2 ˉ the restriction x1 < x Conventional assumption does not allow for physical upper bounds x1 But there are several ˉ x economic applications where this is relevant
- 10. Overview... Consumption: BasicsFrank Cowell: Microeconomics The setting Budget constraints: Budget sets prices, incomes and resources Revealed Preference Axiomatic Approach
- 11. The budget constraintFrank Cowell: Microeconomics The budget constraint x2 typically looks like this Slope is determined by price ratio. “Distance out” of budget line fixed by income or resources Two important subcases determined by 1. … amount of money income y. p1 – __ p2 2. …vector of resources R x1 Let’s see
- 12. Case 1: fixed nominal incomeFrank Cowell: Microeconomics y Budget constraint __ . x2 . p2 determined by the two end- points Examine the effect of changing p1 by “swinging” the boundary thus… Budget constraint is n pixi ≤ y i=1 y __ . . p1 x1
- 13. Case 2: fixed resource endowmentFrank Cowell: Microeconomics Budget constraint x2 determined by location of “resources” endowment R. Examine the effect of changing p1 by “swinging” the boundary thus… n Budget constraint is y= piRi n n i=1 pixi ≤ piRi i=1 i=1 R x1
- 14. Budget constraint: Key pointsFrank Cowell: Microeconomics 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.
- 15. Overview... Consumption: BasicsFrank Cowell: Microeconomics The setting Deducing preference from Budget sets market behaviour? Revealed Preference Axiomatic Approach
- 16. A basic problemFrank Cowell: Microeconomics 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?
- 17. The Axiomatic ApproachFrank Cowell: Microeconomics 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?
- 18. Using observables onlyFrank Cowell: Microeconomics 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
- 19. ―Revealed Preference‖Frank Cowell: Microeconomics Let market prices x2 determine a persons budget constraint.. Suppose the person chooses bundle x... x is example x is For revealed preferred to all revealed Use this to introduce these points.x′ preferred to Revealed Preference x′ x x1
- 20. Axioms of Revealed PreferenceFrank 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 If x was chosen when x was Preference (WARP) 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
- 21. WARP in the marketFrank Cowell: Microeconomics Suppose 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: Otherwise it would graphical violate WARP interpretation
- 22. WARP in actionFrank Cowell: Microeconomics Take the original equilibrium x2 Now let the prices change... Could we have chosen x on Monday? x violates WARP rules out some points WARP; x does not. as possible solutions Tuesdays choice: On Monday we could have x afforded Tuesday’s bundle Clearly WARP x′ induces a kind of Mondays negative substitution choice: effect x But could we extend x1 this idea...?
- 23. Trying to Extend WARPFrank Cowell: Microeconomics Take the basic idea of x2 revealed preference x″ is revealed preferred to all Invoke revealed preference these points. again Invoke revealed preference yet again x x is revealed Draw the “envelope” preferred to all these points. x x is revealed Is this an “indifference preferred to all these points. curve”...? x No. Why? x1
- 24. Limitations of WARPFrank 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.
- 25. Revealed Preference: is it useful?Frank Cowell: Microeconomics 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 consumers motives: But thats what were going to try right now. It’s time to look at the mainstream modelling of preferences.
- 26. Overview... Consumption: BasicsFrank Cowell: Microeconomics The setting Standard approach to Budget sets modelling preferences Revealed Preference Axiomatic Approach
- 27. The Axiomatic ApproachFrank Cowell: Microeconomics Useful for setting out a priori what we mean by consumer preferences But, be careful... ...axioms cant be ―right‖ or ―wrong,‖... ... although they could be inappropriate or over-restrictive That depends on what you want to model Lets start with the basic relation...
- 28. The (weak) preference relationFrank Cowell: Microeconomics The basic weak-preference "Basket x is regarded as at relation: 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 “ x < x ” and not “ x < x. ” relation… x Â x
- 29. Fundamental preference axiomsFrank Cowell: Microeconomics Completeness For every x, x X either x<x is true, or x<x is true, or both statements are true Transitivity Continuity Greed (Strict) Quasi-concavity Smoothness
- 30. Fundamental preference axiomsFrank Cowell: Microeconomics Completeness Transitivity For all x, x , x″ X if x<x and x<x″ then x<x″. Continuity Greed (Strict) Quasi-concavity Smoothness
- 31. Fundamental preference axiomsFrank Cowell: Microeconomics Completeness Transitivity Continuity For all x X the not-better-than-x set and the not-worse-than-x set are closed in X Greed (Strict) Quasi-concavity Smoothness
- 32. Continuity: an exampleFrank Cowell: Microeconomics Take consumption bundle x . x2 Construct two other bundles, xL with Less than x , xM with More Better There is a set of points like than x ? xL, and a set like xM do we jump straight from Draw a path joining xL , xM. a point marked ―better‖ to M x one marked ―worse"? If there’s no “jump”… x but what about the boundary points between the two? The indifference xL curve Worse than x ? x1
- 33. Axioms 1 to 3 are crucial ...Frank Cowell: Microeconomics completeness transitivity The utility continuity function
- 34. A continuous utility function then represents preferences...Frank Cowell: Microeconomics x < x U(x) U(x)
- 35. Tricks with utility functionsFrank Cowell: Microeconomics 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...
- 36. Irrelevance of cardinalisationFrank Cowell: Microeconomics 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 these And, for any monotone increasing φ, this represents exp( U(x1, x2,..., xn) ) the same preferences. ( U(x1, x2,..., xn) ) U is defined up to a monotonic transformation φ( U(x1, x2,..., xn) ) Each of these forms will generate the same contours. Let’s view this graphically.
- 37. A utility functionFrank Cowell: Microeconomics Take a slice at given utility level Project down to get contours U(x1,x2) The indifference curve x2 0
- 38. Another utility functionFrank Cowell: Microeconomics By construction U* = φ(U) Again take a slice… U*(x1,x2) Project down … The same indifference curve x2 0
- 39. Assumptions to give the U-function shapeFrank Cowell: Microeconomics Completeness Transitivity Continuity Greed (Strict) Quasi-concavity Smoothness
- 40. The greed axiomFrank Cowell: Microeconomics Pick any consumption x2 bundle in X. Greed implies that these bundles are preferred to x. Gives a clear “North-East” direction of preference. Bliss! B What can happen if consumers are not greedy Greed: utility function is monotonic x x1
- 41. A key mathematical conceptFrank Cowell: Microeconomics 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 Review Example 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…
- 42. Conventionally shaped indifference curvesFrank Cowell: Microeconomics Slope well-defined x2 everywhere Pick two points on the same indifference curve. Draw the line joining them. A Any interior point must line on a higher indifference curve C ICs are smooth …and strictly concaved- contoured B (-)I.e. strictly quasiconcave Slope is the Marginal Rate of Substitution sometimes these x1 U1(x) —— .. assumptions can U2be relaxed (x) .
- 43. Other types of IC: KinksFrank Cowell: Microeconomics Strictly quasiconcave x2 But not everywhere smooth A C MRS not defined here B x1
- 44. Other types of IC: not strictly quasiconcaveFrank Cowell: Microeconomics Slope well-defined x2 everywhere Not quasiconcave Quasiconcave but not strictly quasiconcave utility here lower than at A or B A C B Indifference curves Indifference curve follows axis here with flat sections make sense x1 But may be a little harder to work with...
- 45. Summary: why preferences can be a problemFrank Cowell: Microeconomics 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...?
- 46. Review: basic conceptsFrank Cowell: Microeconomics Review Consumer’s environment Review How budget sets work Review WARP and its meaning Review Axioms that give you a utility function Review Axioms that determine its shape
- 47. What next?Frank Cowell: Microeconomics Setting up consumer’s optimisation problem Comparison with that of the firm Solution concepts.
- 48. PrerequisitesAlmost essentialFirm: OptimisationConsumption: Basics Frank Cowell: Microeconomics Consumer Optimisation MICROECONOMICS Principles and Analysis Frank CowellOctober 2006
- 49. The problemFrank Cowell: Microeconomics Maximise consumer’s utility U assumed to satisfy the U(x) standard “shape” axioms Subject to feasibility constraint Assume consumption set X is x X the non-negative orthant. and to the budget constraint The version with fixed money n income pixi ≤ y i=1
- 50. Overview... Consumer: OptimisationFrank Cowell: Microeconomics Primal and Two fundamental Dual problems views of consumer Lessons from optimisation the Firm Primal and Dual again
- 51. An obvious approach?Frank Cowell: Microeconomics We now have the elements of a standard constrained optimisation problem: the constraints on the consumer. the objective function. The next steps might seem obvious: set up a standard Lagrangean. solve it. interpret the solution. But the obvious approach is not always the most insightful. We’re going to try something a little sneakier…
- 52. Think laterally...Frank Cowell: Microeconomics In microeconomics an optimisation problem can often be represented in more than one form. Which form you use depends on the information you want to get from the solution. This applies here. The same consumer optimisation problem can be seen in two different ways. I’ve used the labels ―primal‖ and ―dual‖ that have become standard in the literature.
- 53. A five-point plan The primalFrank Cowell: Microeconomics problem Set out the basic consumer optimisation problem. The dual problem Show that the solution is equivalent to another problem. Show that this equivalent problem is identical to that of the firm. The primal problem again Write down the solution. Go back to the problem we first thought of...
- 54. The primal problemFrank Cowell: Microeconomics Contours of The consumer aims to x2 maximise utility... objective function Subject to budget constraint Defines the primal problem. Solution to primal problem Constraint set max U(x) subject to n x* pixi y i=1 But theres another way x1 at looking at this
- 55. The dual problemFrank Cowell: Microeconomics Alternatively the consumer x2 z q could aim to minimise cost... Constraint Subject to utility constraint set Defines the dual problem. Solution to the problem Cost minimisation by the firm minimise n pixi i=1 x* z* subject to U(x) Contours of x1 z But where have we seen objective function the dual problem before?
- 56. Two types of cost minimisationFrank Cowell: Microeconomics The similarity between the two problems is not just a curiosity. We can use it to save ourselves work. All the results that we had for the firms ―stage 1‖ problem can be used. We just need to ―translate‖ them intelligently Swap over the symbols Swap over the terminology Relabel the theorems
- 57. Overview... Consumer: OptimisationFrank Cowell: Microeconomics Primal and Reusing results Dual problems on optimisation Lessons from the Firm Primal and Dual again
- 58. A lesson from the firmFrank Cowell: Microeconomics Compare cost-minimisation for the firm... ...and for the consumer z2 q x2 The difference is only in notation So their solution functions and response functions must be the same z* x* Run through z1 x1 formal stuff
- 59. Cost-minimisation: strictly quasiconcave UFrank Cowell: Microeconomics Use the objective function Minimise Lagrange ...and output constraint n multiplier ...to build the Lagrangean pi xi + [ – U(x)] U(x) Differentiate w.r.t. x1, ..., xn and i=1 set equal to 0. ... and w.r.t Because of strict quasiconcavity we Denote cost minimising have an interior solution. values with a * . A set of n+1 First-Order Conditions U1 (x ) = p1 one for U2 (x ) = p2 each good … … … Un (x ) = pn = U(x ) utility constraint
- 60. If ICs can touch the axes...Frank Cowell: Microeconomics Minimise n pixi + [ – U(x)] i=1 Now there is the possibility of corner solutions. A set of n+1 First-Order Conditions U1 (x ) p1 U2 (x ) p2 … … … Un(x ) pn Interpretation = U(x ) Can get ―<‖ if optimal value of this good is 0
- 61. From the FOCFrank Cowell: Microeconomics If both goods i and j are purchased and MRS is defined then... Ui(x ) pi ——— = — Uj(x ) pj MRS = price ratio “implicit” price = market price If good i could be zero then... Ui(x ) pi ——— — Uj(x ) pj MRSji price ratio “implicit” price market price Solution
- 62. The solution... Solving the FOC, you get a cost-minimising value forFrank Cowell: Microeconomics each good... xi* = Hi(p, ) ...for the Lagrange multiplier * = *(p, ) ...and for the minimised value of cost itself. The consumer’s cost function or expenditure function is defined as C(p, ) := min pi xi {U(x) } vector of goods prices Specified utility level
- 63. The cost function has the same properties as for the firmFrank Cowell: Microeconomics Non-decreasing in every price. Increasing in at least one price Increasing in utility . Concave in p Jump to “Firm” Homogeneous of degree 1 in all prices p. Shephards lemma.
- 64. Other results followFrank Cowell: Microeconomics Shephards Lemma gives demand H is the “compensated” or as a function of prices and utility conditional demand function. Hi(p, ) = Ci(p, ) Properties of the solution Downward-sloping with respect function determine behaviour of to its own price, etc… response functions. ―Short-run‖ results can be used For example rationing. to model side constraints
- 65. Comparing firm and consumerFrank Cowell: Microeconomics Cost-minimisation by the firm... ...and expenditure-minimisation by the consumer ...are effectively identical problems. So the solution and response functions are the same: Firm Consumer m n Problem: min wizi + [q – (z)] min pixi + [ – U(x)] z i=1 x i=1 Solution function: C(w, q) C(p, ) Response z * = Hi(w, q) xi* = Hi(p, ) function: i
- 66. Overview... Consumer: OptimisationFrank Cowell: Microeconomics Primal and Exploiting the Dual problems two approaches Lessons from the Firm Primal and Dual again
- 67. The Primal and the Dual…Frank Cowell: Microeconomics There’s an attractive symmetry about the two approaches to the n problem pixi+ [ – U(x)] i=1 In both cases the ps are given and you choose the xs. But… n U(x) + [y– pi xi ] …constraint in the primal i=1 becomes objective in the dual… …and vice versa.
- 68. A neat connectionFrank Cowell: Microeconomics Compare the primal problem of the consumer... ...with the dual problem x2 x2 The two are equivalent So we can link up their solution functions and response functions x* x* Run through x1 x1 the primal
- 69. Utility maximisationFrank Cowell: Microeconomics Lagrange Use the objective function Maximise multiplier ...and budget constraint n ...to build the Lagrangean U(x) + [ y – i=1 p x ] ii ii Differentiate w.r.t. x1, ..., xn and set equal to 0. i=1 ... and w.r.t If U is strictly quasiconcave we have Denote utility maximising an interior solution. values with a * . A set of n+1 First-Order Conditions U1(x ) = p1 If U not strictly one for quasiconcave then U2(x ) = p2 each good replace ―=‖ by ― ‖ … … … Un(x ) = pn budget n constraint Interpretation y = pi xi i=1
- 70. From the FOCFrank Cowell: Microeconomics If both goods i and j are purchased and MRS is defined then... Ui(x ) pi (same as before) ——— = — Uj(x ) pj MRS = price ratio “implicit” price = market price If good i could be zero then... Ui(x ) pi ——— — Uj(x ) pj MRSji price ratio “implicit” price market price Solution
- 71. The solution...Frank Cowell: Microeconomics Solving the FOC, you get a utility-maximising value for each good... xi* = Di(p, y) ...for the Lagrange multiplier * = *(p, y) ...and for the maximised value of utility itself. The indirect utility function is defined as V(p, y) := max U(x) { pixi y} vector of money goods prices income
- 72. A useful connectionFrank Cowell: Microeconomics The indirect utility function maps The indirect utility function works prices and budget into maximal utility like an "inverse" to the cost = V(p, y) function The cost function maps prices and The two solution functions have utility into minimal budget to be consistent with each other. y = C(p, ) Two sides of the same coin Therefore we have: Odd-looking identities like these = V(p, C(p, )) can be useful y = C(p, V(p, y))
- 73. The Indirect Utility Function has some familiar properties...Frank Cowell: Microeconomics (All of these can be established using the known properties of the cost function) Non-increasing in every price. Decreasing in at least one price Increasing in income y. quasi-convex in prices p Homogeneous of degree zero in (p, y) But what’s this…? Roys Identity
- 74. Roys IdentityFrank Cowell: Microeconomics = V(p, y)= V(p, C(p, )) ―function-of-a- Use the definition of the function‖ rule optimum Differentiate w.r.t. pi . 0 = Vi(p,C(p, )) + Vy(p,C(p, )) Ci(p, ) Use Shephard’s Lemma Rearrange to get… So we also have… 0 = Vi(p, y) + Vy(p, y) xi* Marginal disutility of price i Vi(p, y) Marginal utility of xi* = – ———— money income Vy(p, y) Ordinary demand function xi* = –Vi(p, y)/Vy(p, y) = Di(p, y)
- 75. Utility and expenditureFrank Cowell: Microeconomics Utility maximisation ...and expenditure-minimisation by the consumer ...are effectively two aspects of the same problem. So their solution and response functions are closely connected: Primal Dual n n Problem: max U(x) + [y – pixi ] min x i=1 pixi + [ – U(x)] x i=1 Solution function: V(p, y) C(p, ) Response x * = Di(p, y) xi* = Hi(p, ) function: i
- 76. SummaryFrank Cowell: Microeconomics A lot of the basic results of the consumer theory can be found without too much hard work. We need two ―tricks‖: 1. A simple relabelling exercise: cost minimisation is reinterpreted from output targets to utility targets. 2. The primal-dual insight: utility maximisation subject to budget is equivalent to cost minimisation subject to utility.
- 77. 1. Cost minimisation: two applicationsFrank Cowell: Microeconomics THE FIRM THE CONSUMER min cost of inputs min budget subject to output subject to utility target target Solution is of the Solution is of the form C(w,q) form C(p, )
- 78. 2. Consumer: equivalent approachesFrank Cowell: Microeconomics PRIMAL DUAL max utility min budget subject to budget subject to utility constraint constraint Solution is a Solution is a function of (p,y) function of (p, )
- 79. Basic functional relationsFrank Cowell: Microeconomics Utility Review C(p, ) cost (expenditure) H is also known as "Hicksian" demand. Compensated demand Review Hi(p, ) for good i Review V(p, y) indirect utility ordinary demand for Review Di(p, y) input i money income
- 80. What next?Frank Cowell: Microeconomics Examine the response of consumer demand to changes in prices and incomes. Household supply of goods to the market. Develop the concept of consumer welfare

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