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# Consumption basics

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### Consumption basics

1. 1. Frank Cowell: Microeconomics Consumption Basics MICROECONOMICS Principles and Analysis Frank CowellOctober 2006
2. 2. Overview... Consumption: BasicsFrank Cowell: Microeconomics The setting The environment for the basic Budget sets consumer optimisation problem. Revealed Preference Axiomatic Approach
3. 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. 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. 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. 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. 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. 8. ... and thisFrank Cowell: Microeconomics Consumption set X has x2 holes in it x1
9. 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. 10. Overview... Consumption: BasicsFrank Cowell: Microeconomics The setting Budget constraints: Budget sets prices, incomes and resources Revealed Preference Axiomatic Approach
11. 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. 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. 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. 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. 15. Overview... Consumption: BasicsFrank Cowell: Microeconomics The setting Deducing preference from Budget sets market behaviour? Revealed Preference Axiomatic Approach
16. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 26. Overview... Consumption: BasicsFrank Cowell: Microeconomics The setting Standard approach to Budget sets modelling preferences Revealed Preference Axiomatic Approach
27. 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. 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. 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. 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. 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. 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. 33. Axioms 1 to 3 are crucial ...Frank Cowell: Microeconomics completeness transitivity The utility continuity function
34. 34. A continuous utility function then represents preferences...Frank Cowell: Microeconomics x < x U(x) U(x)
35. 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. 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. 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. 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. 39. Assumptions to give the U-function shapeFrank Cowell: Microeconomics  Completeness  Transitivity  Continuity  Greed  (Strict) Quasi-concavity  Smoothness
40. 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. 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. 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. 43. Other types of IC: KinksFrank Cowell: Microeconomics Strictly quasiconcave x2 But not everywhere smooth A  C MRS not defined here B x1
44. 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. 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. 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. 47. What next?Frank Cowell: Microeconomics  Setting up consumer’s optimisation problem  Comparison with that of the firm  Solution concepts.
48. 48. PrerequisitesAlmost essentialFirm: OptimisationConsumption: Basics Frank Cowell: Microeconomics Consumer Optimisation MICROECONOMICS Principles and Analysis Frank CowellOctober 2006
49. 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. 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. 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. 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. 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. 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. 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. 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. 57. Overview... Consumer: OptimisationFrank Cowell: Microeconomics Primal and Reusing results Dual problems on optimisation Lessons from the Firm Primal and Dual again
58. 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. 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. 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. 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. 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. 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. 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. 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. 66. Overview... Consumer: OptimisationFrank Cowell: Microeconomics Primal and Exploiting the Dual problems two approaches Lessons from the Firm Primal and Dual again
67. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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