3. Neoclassical Demand Theory. 3.1 Preferences

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1 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page Neoclassical Demand Theory 3.1 Preferences Not all preferences can be described by utility functions. This is inconvenient. We make the assumptions about preferences that utility functions require. Not very realistic No attention to psychology EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 82 Preference relations % are defined on X R n +. In order to use set notation, we describe % by the upper contour sets G(), y G() y %, or by the lower contour sets B(), y B() G(y) or y - We also define indifference sets I () =G() B(). If y I () we write y. Both set and preference notation can be used interchangeably.

2 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 83 Notation 3.1. Vector inequalities: À Strictly Greater means > in all components > Greater means in all components but > in some GreaterorEqual means in all components À strictly greater > greater greater or equal Definition 3.1. % is complete if for every and y, y G() or G(y). Definition 3.2. % is transitive if for every, y and z,if z % y and y %, thenz %. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 84 Definition 3.3. % is (strictly) conve if for every, G() is a(strictly)conveset. Definition 3.4. % is homothetic if y % = for all α > 0, αy % α (that is, if every indifference curve is a constant multiple of every other indifference curve.

3 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 85 Definition 3.5. Monotonicity (more is better): % is strongly monotone if > y = Â y % is monotone if À y = Â y % is locally nonsatiated if every neighborhood of contains a vector y such that y Â. [there is always a better point close by] EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 86 Illustration of monotonicity: { y } G() B()

4 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 87 Let G 0 () ={z z G(), z / B()} = the strictly-preferred set Monotonicity: Moreofall goods increases utility. thebluedarkarea not including G0 (). or the dotted lines Strong monotonicity: Moreofany goods increases utility. thebluedarkarea including the dotted lines but not G0 (). EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 88 Local nonsatiation: you can always increase utility by making a small change in your consumption bundle. Eamples?

5 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 89 Eample 3.1. Leicographical preferences: Commodities are ordered from most important to least important. At first, only the most important commodity is used to determine whether one bundle is more preferred than another If the amounts of the most important commodity in two bundles are eactly equal, then the net most important commodity is used to break the tie, etc. If 2 is most important and 4 is net then: Â Â EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 90 Illustration of Leographical preferences. Important Good ( ) G X X ( ) B X Unimportant Good With leicographical preferences, indifference curves have only one point. Why?

6 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page Some informal topology. Definition 3.6. (Informal) The set N is a neighborhood of a point if N contains a sphere (or circle or interval) around. A neighborhood of must include all points very close to. A neighborhood can be as small as you like. If N is a neighborhood of and N N 0,thenN 0 is a neighborhood of. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 92 N D C N is a neighborhood of C is not a neighborhood of. D is not a neighborhood of. Why not?

7 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 93 Definition 3.7. is a boundary point of asetd if every neighborhood of (no matter how small) contains a point in D and a point outside of D. N D C is a boundary point of D, but not of N or C. Boundary points are the points at the edge of a set. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 94 Definition 3.8. A set is open if it contains none of its boundary points. Definition 3.9. A set is closed if it contains all of its boundary points. The complement of a closed set is open. The complement of an open set is closed. An open set is a neighborhood of each of its points. Why?

8 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 95 Definition A function f maps a space X into a space Y (written f : X Y )iff assigns a point in Y to each point in X.ThespaceX is called the domain of the f,andy is called the codomain. The part of Y that is used is called the range. f y f () f () y X z f () z Y Functions can do things to spaces (transform them). Eamples: shrinking, stretching, flattening, etc. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 96 If f : X Y and S X,thentheimage of S (written f (S)) is the set of all points in Y that come from S. If f : X Y and R Y, then the inverse image of R (written f 1 (R)) is the set of all points in X that go into R. U f V X Y V is the image of U. U f 1 (V ). Why not U = f 1 (V )?

9 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 97 Definition (Informal) A function is continuous if all points that are near each other have images that are near each other. Definition (Informal) Afunctionisdiscontinuous if some points that are near each other have images that are far apart [see drawing below]. f U V X Y Define a function that fits this drawing. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 98 f U V X Y Proposition 3.1. A function is continuous if and only if the inverse image of every open set is an open set. Above, the inverse image of V (set with green outline) is U, which is not open.

10 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 99 Definition A preference relation % is continuous if for all, thesetsg() and B() are closed. Proposition 3.2. If % is continuous and if y G() and z B() then for some α [0, 1 ] the point αy +(1 α)z is indifferent to. y G() ~ α y + (1 ~ α )z B() z EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 100 Proof. By the continuity of %, B() and G() are closed. Let q(α) =αy +(1 α)z. We have q(0 )=z B(), q(1 )=y G(). Let α =sup{α q (α) B()}. q ( α) is a boundary point of B() and of B c () [the complement of the set B()], because every neighborhood of q ( α) contains points in both sets [why?]. But B c () G() [why?] q( α) B() and q( α) G(). [If A B, B is closed and is a boundary point of A, then B. ] Therefore q( α).

11 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 101 Leicographical preferences are not continuous, so the proposition about indifference doesn t apply. Why not? Important Good ( ) GX X ( ) B X Unimportant Good EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 102 Proposition 3.3. If % is complete, transitive, monotonic and continuous, then it can be represented by a continuous utility function. e { y } ~ α e αe G() 0 45 o B()

12 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 103 Proof (Construction). 1. Choose an À 0. We will define U () Let e be the diagonal vector Choose ᾱ, such that ᾱe À. 4. By monotonicity, ᾱe G() and 0 B(). 5. By continuity and the previous intermediate-value proposition, there is an α such that αe. 6. Define the value of U () to be α. 7. We have: U ()e. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 104 Proof of Representation. U () U (y) % y. We show 1. To show: U () U (y) = % y: (a) U () =U (y) = U ()e = U (y)e y = [by transitivity] y. (b) U () > U (y) = U ()e À U (y)e = [by monotonicity] U ()e  U (y)e = [by transitivity]  y. 2. To show: U () U (y) = % y: (a) U () > U (y) and y % contradict statement 1, (b) so it follows that y % = U (y) U () (c) Switch and y in the above statement.

13 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 105 Proof of the Continuity of U. To show U is continuous, we show that the inverse image of any open interval (a, b) is an open set. e B(ae) ae be 1 U [( a, b )] G(be) 0 45 o EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page For this proof, we will need to use the following propositions. Let f be any function and let Y and Z be any sets in the codomain of f.lets C denote the complement of S. (a) If f 1 (Y ) and f 1 (Z ) are closed sets, then f 1 (Y Z ) is a closed set. (b) f 1 (Y C ) f 1 (Y ) C Prove these propositions as an eercise! 2. (a, b) C =[0, a] [b, ). 3. But U 1 ([0, a]) = B(ae), which, by the continuity of %, must be closed, 4. and U 1 ([b, )) = G(be), which must also be closed for the same reason. 5. Therefore, by 1a above, we know that U 1 ([0, a] [b, )) is closed.

14 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page It follows that U 1 ([0, a] [b, )) C is open. 7. But by 1b, wehave U 1 ((a, b)) U 1 (([0,a] [b, )) C )=U 1 ([0,a] [b, )) C, so that U 1 ((a, b)) must be open. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 108 Eample 3.2. The utility function U ( 1, 2 )= 2 1 is continuous, but it represents preferences that lack monotonicity [why?]. However, those preferences can be represented by a utility function. Why is this possible? Can we usetheaboveprooftodemonstratethisfact?

15 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page Sequences The open sets of a space define the idea of closeness in a space. y X z Think of it this way: open sets determine neighborhoods. And a point s is closer to than other points are if it is in more (and smaller) neighborhoods of. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 110 Definition A sequence is an ordered list of points, usually an infinite number of them. The list 1 2, 1 3, 1 4, 1 5,... is an eample of an infinite sequence. Definition A sequence is eventually in a set U if it starts either inside or outside of U, then goes inside and then doesn t come out again. Definition A sequence converges to a point if the sequence is eventually in every neighborhood of.

16 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 111 In the drawing below, the sequence {a 1, a 2, a 3,...} is converging to, because it is eventually inside every neighborhood of. ia 2 ia 1 i ia 4 ia 3 i a a 5 i EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page Compact Sets Compact sets have a special kind of finiteness property. If you travel around a compact set long enough there will be a point in the set that you are frequently close to. If a set isn t compact, you can travel inside it forever, without being close to any point frequently. Definition A set C has the convergent-subsequence property if every sequence of points in C has a subsequence that converges to a point in C. Definition A set C is compact if has the convergent-subsequence property. Eample 3.3. The set R + { 0 } is not compact. Why not? Consider the sequence {1, 2, 3, 4...}

17 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 113 Eample 3.4. The interval (0, 1 ) isn t compact. Why not? Definition A set C has the finite-subcover property if the following is true: whenever an infinite number of open sets covers C,a finite collection of those open sets will be sufficient to cover C (the remaining sets are unnecessary). C In the illustration, only the red sets (which are finite in number) are needed to cover C. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 114 Proposition 3.4. A set is compact if and only if it has the finite-subcover propery. Proposition 3.5. If C is compact and f is a continuous function, then f (C ) is compact. Proposition 3.6. Suppose C is compact and suppose F is closed and F C. Then F is compact.

18 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 115 Definition A set X is bounded if it is contained in a sphere (of finite radius). Proposition 3.7. (Bolzano-Weierstrass Theorem) In Euclidean vector spaces (including the real line), a set is compact if and only if it is closed and bounded. Proposition 3.8. If the image of a set S under a continuous function is unbounded, then S is NOT compact. Proposition 3.9. Any compact set C R contains a maimum point (and a minimum point). Proposition (Maimum value theorem) Suppose C R n is compact and suppose f : C R is continuous. Then f takes a maimum value on C. Proof. Because C is compact and f is continuous, we know that f (C ) is a compact subset of R, sothatf (C ) has a maimum point. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page Utility Maimization Finding the most preferred point in budget set: Definition The constrained utility-maimization problem (UMP) is given by: ma u() s.t. 0 p w.

19 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 117 Proposition If u is continuous, then for any p, w, UMP has a solution. We write the set of solutions for each p, w as (p, w) [the demand correspondence]. Proof. The budget set B p,w { p w} is compact. Because u is continuous, u(b p,w ) is also compact. u(b p,w ) R. Therefore u(b p,w ) contains its maimum value, ū. The set u 1 (ū) argma {U () p w} (p, w). EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 118 Proposition The demand correspondence (p, w) is homogeneous of degree 0 Proof. Budget set is unchanged when p, w is multiplied by a positive constant. Proposition If utility u is continuous, locally nonsatiated (lns), then (p, w) eists and satisfies Walras Law. Proof. By the continuity of u, weknowthat(p, w) eists. Suppose (p, w) and p < w. Then there is a neighborhood N of such that for any N, p < w (why?). By local non-satiation there is ˆ N, with u(ˆ) > u( ), a contradiction to utility maimization.

20 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page Conveity, Concavity and Quasiconcavity Definition A set X in a vector space is conve if for any, y X, the line y X,thatisifα +(1 α)y X whenever 0 α 1. y EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 120 Definition A function f : X R is conve if for all, y X,andα [0, 1 ], f (α +(1 α)y) αf () +(1 a)f (y) that is, the line between any two points on the graph of the function, lies on or above the graph. f () α f ( ) + (1 α ) f ( ) > f ( α + (1 α ) ) α + ( 1 α ) Proposition A function is conve if and only if the set above its graph, F = {(, z ) X, z f ()}, is conve.

21 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 121 Definition A function g : X R is concave if and only if g is a conve function. Definition A function U is quasiconcave if for all the sets G() ={y U (y) U ()} are conve. A function V is quasiconve if for all p the sets B(p) ={q V (q) V (p)} are conve. Proposition All conve functions are quasiconve, and all concave functions are quasiconcave (but not the other way round). Proof. Try it as an eercise. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 122 Concavityisobservedinthegraphofafunction. Quasiconcavity is observed in the domain of the function. Concave and quasiconcave. U(z) U() 0 z G(z)

22 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 123 Quasiconcave but not concave. U(z) U() 0 z G(z) EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 124 The following are 3-dimensional graphs of two utility functions The graph below represents U ( 1, 2 ) 3 ln( )( 2 + 1). This utility function is concave, therefore, quasiconcave. The surface curves downwards not only from side-to-side as above, but also along a ray from the origin.

23 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page U ( 1, 2 ) 1 0 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 126 The graph below represents U ( 1, 2 ) ( 1 2 ) 1.5. The function is quasiconcave, because the G sets are conve sets. However, the function is not concave, because the set under the surface is not conve. Thereasonisthatthesurfacecurvesupasyoumoveawayfrom the origin along any ray.

24 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page U ( 1, 2 ) 1 0 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 128 Notice that both utility functions represent very similar preferences. Quasiconcave functions can be made concave by monotonic transformations, so the preferences that they represent need not change.

25 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 129 Proposition If u is strictly quasiconcave, (p, w) is a function. u ( ) > u u ( ) = u EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 130 Proof. Suppose 0, 00 (p, w). Then is in B p,w (budget sets are conve). Butbystrictq.c.u( ) > u( 0 )=u( 00 ) remember: with strict q.c., lines lie above indifference curves This implies 0 and 00 are not utility-maimizing, a contradiction.

26 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page Multivariate Optimization and Utility Maimization Definition Let f : R n R be differentiable. Then f () f () [ f () 1,..., f () n ] is called the gradient of f at. Definition Let z be a vector of unit length (kz k = 1 ) and let be a point in R n.define h(α) =f ( + αz ). The directional derivative of f at in the direction z,isgivenby h 0 (0 ). EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 132 z f + α z f ( + α z) f () n

27 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 133 Proposition If z is any vector of unit length, then f () z is the directional derivative of f in the direction z,so that f ( + αz )=f () +( f () z )α + (second-order-small error) The derivative of f in the direction f () is the maimum directional derivative for all directions, which means that f () is the direction of the maimum rate of change of f at. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 134 Proof. First, we show that f () z is the directional derivative: By the chain rule, h 0 (α) = X i X i f ( + αz ) ( i + αz i ) i α f ( + αz ) i z i f ( + αz ) z, so that h 0 (0 ) f () z. h(α) =(appro) h(0 )+h 0 (0 )α, Therefore, f( + αz) =(appro) f()+( f() z)α.

28 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 135 Now we show that the directional derivative is greatest in direction f u v = kukkvk cos θ, whereθ is the angle between u and v. Let z = 1 k f k f, direction of z = direction of f k z k = 1 so that f z = k f kk z k cos 0 = k f k If kz k = 1, f z = k f kkz k cos θ k f k which means that the derivative in the direction z (= direction f ) gives the maimum directional derivative. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 136 Proposition (Kuhn-Tucker Necessary Conditions) Let f : R n R be twice differentiable and let B R n with B. Then f ( )=ma{f () B} only if the directional derivative of f ( ) z 0 for every direction z within B (that is, for every direction z such that for all positive and sufficiently small α, wehave + αz B). * *+ α z f f (*) B z f ( * + α z )

29 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 137 Proof. We prove the equivalent contrapositive proposition. Suppose that for some z we have + αz B for all α sufficiently small. Then if f ( ) z > 0,wehavef ( + αz ) > f ( ) for small α. Therefore f ( ) is not a maimum for B. Now we can apply this to the UMP. Assume U is well-behaved (continuous and twice differentiable). Then we have... EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 138 Proposition If À 0 is the solution to ma U () such that 0 p = w then for some λ > 0, U ( )=λp or, equivalently U = λp U L = λp L [Economic interpretation: If a positive quantity of every good is demanded, then marginal utility per dollar must be the same for every good.]

30 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 139 Informal economic proof: Suppose MU i /p i > MU j /p j. Decrease demand for j by j. Utility loss = MU j j Money saved = p j j Use money save to by i can buy i = p j j /p i Utility gain = MU i p j j /p i > utility loss. Utility was not maimized. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 140 Proof. w 2 p p 2 [ p U ( *)] p z θ U(*) * z B p, w U ( *) [ p U ( *)] p w 1 p 1

31 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 141 Suppose U ( ) is not a multiple of p. Assume kpk = p p = 1 (also p p = 1 ). The projection of U ( ) onto p is [p U ( )]p So z = U ( ) [p U ( )]p is orthogonal to p. [Why?] [See illustration] EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 142 For α small enough, + αz À 0 p( + αz )=p = w Therefore + αz is in the budget set. But the derivative of U in the direction z is U ( )z = U ( ) U ( ) (p U ( )) 2 > 0 [why?] Conclusion: does not maimize U.

32 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 143 Eample 3.5. Construct a utility function that corresponds to the following demand schedule: 1 = 2w 3p 1 2 = w 3p 2. Solution strategy for eample above: The proposition implies that any indifference curve must be orthoginal to p at the quantities demanded. Why? We will try to find a function 2 = 2 ( 1 ) that describes a curve with that property. This will be our candidate indifference curve. We then find a utility function U ( 1, 2 ) that is constant everywhere on the candidate indifference curve. Finally, we check to see if that utility function produces the desired demand curve. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 144 Note that 1, 2 À 0 at any positive prices, so that the UMP must have only interior solutions. Solving for prices we find the demand-price function: p 1 = 2w 3 1 p 2 = w 3 2. Theslopeofthepricevectorisp 2 /p 1,soifthecurve 2 ( 1 ) is orthoginal to p, then the slope of 2 ( 1 ) must be p 1 /p 2.

33 EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 145 If we apply the demand-price function, we have the equation d 2 = p 1 = 2 2. d 1 p 2 1 This can be represented as the differential equation d 2 = 2 d 1, 2 1 whichmustbetruefor 2 ( 1 ), our candidate indifference curve. Integrating both sides gives us the following implicit formula for 2 ( 1 ): log 2 ( 1 )= 2log 1 + C, or C =2log 1 +log 2 ( 1 ), where C is the constant of integration. EC 701, Fall 2005, Microeconomic Theory September 28, 2005 page 146 If we define a utilty function by u ( 1, 2 ) 2log 1 +log 2, then u ( 1, 2 ( 1 )) = C for all combinations ( 1, 2 ( 1 )), so that this definition of u satisfies the requirement that it be constant on the candidate indifference curve. Using a monotonic transformation, we can transform u ( 1, 2 ) to U ( 1, 2 )= 2/3 1 1/3 2, which is a Cobb-Douglas function. Finally, it is clear that this utility function yields the desired demand function.