DECISIONS AND GAMES. PART I

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1 DECISIONS AND GAMES. PART I 1. Preference and choice 2. Demand theory 3. Uncertainty 4. Intertemporal decision making 5. Behavioral decision theory DECISIONS AND GAMES. PART II 6. Static Games of complete information 7. Dynamic games of complete information 8. Static games of incomplete information 9. Dynamic games of incomplete information 10. Behavioral game theory References Gibbons, R Game theory for applied economists. Princeton University Press. Princeton, New Jersey. Varian, H Microeconomic analysis. Norton & Company. New York. Chapters 7, 8, 9, 10, 11, 19. Camerer, C., Loewenstein, G., and Rabin, M Advances in Behavioral Economics. Princeton University Press. Mas-Colell, A., Whinston, M., and Green, J Microeconomic theory. Oxford University Press. Chapters 1, 2, 3, 4, 6. 1

2 Two approaches to analyze behavior of decision makers. 1. From an abstract construction generate choices that satisfy rationality restrictions. 2. Directly from choice, ask for some structure to these choices and derive properties. Relation between the two approaches. 2

3 1. Preference and choice Consider a set of alternatives (mutually exclusives) X and we consider a preference relation on X. Preferences The preference relation at least as good as is a binary relation over the set of alternatives X. Preferences express the ordering between alternatives resulting from the objectives of the decision maker. Properties: Rational Preference Relation (or Weak order, or Complete preorder) if -Completeness, Transitivity (and Reflexivity) A relation is complete if x, y X, x y or y x. A relation is transitive if and only if x, yz, X, if x yand y z, then x z. 3

4 Transitiveness is more delicate than completeness. Cases where transitiveness may not hold are: -Just perceptible differences -Framing: Choice depends on how the problem is framed. Kahneman & Tversky. From can also be deduced the indifference relation ~ and the strict preference relation and the corresponding properties when is rational. ~ is reflexive, transitive and symmetric. is irreflexive and transitive. Utility A function relation if: u : X R is a utility function representing a preference x y u ( x) u( y) -Any monotone (strictly increasing) transformation of u(x) will be also a utility function representing the same preferences. That is, if f is an increasing function, v(x) = f(u(x)) represents the same preferences as u(x). 4

5 -A preference relation can be represented by a utility function only if it is rational. -Not all rational preference relations can be represented by a utility function. (It s enough if X is finite). Choice Another way to analyze decisions is via choices that we suppose satisfy some minimal properties to understand behavior. -We define a budget set Β X as a subset of X. Β is a family of nonempty sets of X. - A choice rule C() is a correspondence that selects a subset of B for every B defined in the problem. Since any choice is possible, we impose a reasonable restriction on choices that we call the weak axiom of revealed preference. If an individual chooses x when she can choose between x and y, then she shouldn t choose y when faced with x, y, z. 5

6 Weak Axiom of Revealed Preference If for some B Β, with, x y B we have x C( B) B Β with x, y B and y C( B ) we must also have x C( B ) Examples: X = X Β= { x, y, z} { xy, },{ xyz,, },{ x} { } Possible Choice Structure:, ({, }) {,, } ( Β C1 () ). C1 x y = x It does not satisfy the Weak axiom. C1 ( x y z ) = y = Β= { x, y, z} { xy, },{ xyz,, },{ x} { } Possible Choice Structure:, ({, }) { } ( Β C2 () ) C2 x y = x It does not satisfy the Weak axiom C2 ( x, y, z ) = { x, y} Revealed preference relation * x y B such that x, y B and x C B ( ) ( ) ( ) Β. such that, and and B * x y B Β x y B x C B y C, then WARP: If x is revealed at least as good as y, then y cannot be revealed preferred to x. 6

7 Relation between preference approach and choice approach If an agent has a rational preference relation, her decisions generate a choice structure satisfying the Weak Axiom of Revealed Preference. (, ) = { : for every } * C B x B x y y B * Suppose C ( B, ) is nonempty. Proof: if we have a rational preference relation, we must show ( * that Β, C (, )) satisfies the weak axiom. * Suppose we have, and (, ) x y B x C B x y. * Consider now ( ) B with x, y B and y C B, y z, z B. * By transitivity, since x y y z x z z B x C ( B ) and, then,,. 7

8 If choices from a choice structure satisfy the weak axiom, is there a rational preference relation consistent with them? ( ) Given a choice structure Β,C ( ), we say that a rational preference * relation rationalizes C() relative to Β if C( B) = C ( B, ) The Weak Axiom is not sufficient to guarantee that there exists a preference relation that rationalizes choices. Example: X = { x, y, z} { xy, },{ yz, },{ xz, } { } { } (, ) = {}, ({, }) = { }, ({, }) = {} Β= C x y x C y z y C x z z WARP is satisfied. Rationalizable preferences do not exist. ({, }) {} {, } C x y = x x y transitivity : x z but C x, z z C( y z ) = { y} y z ({ }) = {} 8

9 If a choice structure satisfies: -Weak axiom - Β includes all subsets of X of up to three elements. Then there is a rational preference relation that rationalizes C() relative to Β and it is unique. (the revealed preference relation). This result is positive but too restrictive for many situations that economists care about. The result needs choice defined over all subsets of X and many economic situations (consumer problem) define choice for certain subsets. Hence, in consumer theory the WARP is not enough to guarantee the equivalence. More conditions on choice will be needed. 9

10 Consumer choice Commodities: a commodity vector x (commodity bundle) is a list of the amount of the L commodities. L R is the commodity space. We will use commodity vectors also as consumption vectors. x1 x = x L Consumption set: L X R. The consumption set incorporates physical or institutional restrictions on the commodity space. Leisure hours 24 8 Bread x 2 X = R L + L { : l 0, 1,..., } L X = R+ = x R x l = L x 1 10

11 Competitive budgets: Consumers must choose facing economic constraints. 1. L commodities traded in the market at monetary prices publicly quoted (principle of completeness or universality of markets). We further assume, for convenience, that p 0, that is p l >0 2. Consumers are price-taking. Affordability depends on prices, p, and consumer s wealth, w. L x R is affordable if p x= p x + p x + p x w pw, L { R+ } Walrasian ( competitive) budget : B = x : p x w L L x 2 L { x R+ : p x= w} B pw, slope = p p 1 2 x 1 The Walrasian budget set is convex. (Show it) 11

12 Demand functions The Walrasian (market, ordinary) demand correspondence x( pw, ) assigns a set of consumption bundles from each pair ( pw., ) Assumptions: 1. Homogeneity of degree zero. ( ) ( ) xαp, αw = x pw,, pw, and α > 0. Homogeneity of degree zero allows us normalizations 2. Walras Law: (, ) satisfies Walras' Law if 0 and > 0, then =, (, ) x pw p w p x w x x pw ( ) Notice that we have a choice structure Β W, x( ) that does not include all possible subsets of X. { B, w : p 0, w 0} W Β = >. By homogeneity of degree zero, x() depends only on the budget set. 12

13 Comparative statics Wealth Wealth effects: Given p, x( pwis, ) the consumer s Engel Function. L Its image in R+, Ep = { x( p, w) : w> 0} is the Wealth Expansion Path. We call x l ( p, w) w the wealth effect for good l. x 2 Ep x 1 - Commodity l is normal if x l ( p, w) commodity is normal. - Commodity l is inferior if x l ( p, w) 0 w < 0 w.. Demand is normal if every It is common to assume normality for aggregates. Specific goods usually become inferior at some point by substitution. 13

14 Prices Offer curve p 2 x 2 p2 x2 x 1 x 2 p2 w p 2 x 2 is Giffen at p 2 x 1 A good can be Giffen only if it is inferior. 14

15 -If the Walrasian demand function is homogeneous of degree zero, ( x( αp, αw) = x( pw, ), pw, and α > 0. ) then for all p and w. L k = 1 (, ) (, ) xl p w xl p w pk + w= 0 for l = 1,..., L. p w k We define the elasticities as the percentage change in demand per marginal percentage change in prices or wealth: ε ε lk lw ( pw, ) ( pw, ) (, ) xl p w pk = p x p w k (, ) l l (, ) xl p w w = w x p w (, ) L k = 1 ε lk ( ) ε ( ) p, w + p, w = 0 for l = 1,..., L. lw 15

16 - If Walras law is satisfied (, ) satisfies Walras' Law if 0 and > 0, then =, (, ) x pw p w p x w x x pw L l= 1 ( p, w) xl pl + xk ( p, w) = 0 for k = 1,..., L. p k Total expenditure cannot change in response to a change in price. L l= 1 p l x l ( p, w) pw = 1. If we define the budget share of expenditure in commodity l as b l pl xl ( ) ( p, w) p, w =, we can also derive: w L bl( p, w) εlk ( p, w) + bk ( p, w) = 0 l= 1 L l= 1 l (, ) ε (, ) b p w p w lw = 1. 16

17 Weak axiom of Revealed Preference Assume that x( pw, ) is single valued, homogeneous of degree zero and satisfies Walras Law. x( pw, ) satisfies the WARP if the following condition holds for any two pairs ( ) ( p, w and p, w ): If (, ) and (, ) (, ) then (, ) px p w w x p w x pw px pw > w This means that if x( p, w ) was affordable at ( p, w ) and was not chosen, then x( pw, ) must be preferred and will not be chosen at ( p, w ) because is not affordable. x 2 B p, w x 2 B p, w B pw, B pw, (, ) x p w x( pw, ) x 1 (, ) x p w (, ) x p w x 1 17

18 WARP can also be expressed with aid of (Slutsky) compensated price changes. x 2 B pw, p1 B p, w x( pw, ) x 1 (, ) w = p x p w Compensated Law of Demand -If x( pw, ) is homogeneous of degree zero and satisfies Walras Law, then it satisfies WARP iff: For any compensated price change from ( p, w ) to ( p, w ) = ( p, p x( p, w) ) we have ( p p) x( p, w ) x( p, w) 0. With <0 if x( p, w ) x( p, w) 18

19 Compensated Law of Demand x 2 B pw, (, ) x p w B p, w x( pw, ) x 1 Uncompensated own price decreases may generate decrease in demand. The compensated law of demand leads to a differential expression that it will be useful: Consider a differential price change dx and a compensation dw = x( p, w) dp. dp dx 0 p Hence : (, ) (, ) dx = D x p w dp + D x p w dw w (, ) (, ) (, ) (, ) (, ) (, ) dx = Dpx p w dp + Dwx p w x p w dp = Dpx p w + Dwx p w x p w dp ( ) ( ) ( ) T dp Dx p pw, + Dx w pw, x pw, dp 0 The expression in brackets is the Substitution or Slutsky matrix: 19

20 (, ) (, ) s11 p w s1l p w S( p, w) = Substitution matrix sl1 ( p, w) sll( p, w) where : (, ) (, ) xl p w xl p w slk ( p, w) = + xk ( p, w). Substitution effect p w k When there is a price change with wealth unchanged the effect is xl ( p, w) dp k. If we adjust wealth to afford the original consumption, p k we change wealth by x ( pwdp., ) The effect of this wealth change is (, ) xl p w x k ( pwdp, ) k w. k k 20

21 - If a differentiable demand function x( pw, ) satisfies Walras Law, homogeneity of degree zero and the weak axiom, then at any ( p, w ) the Slutsky matrix satisfies ( ) L v S p, w v 0 for any v R. This means that means that: S( p, w ) is negative semidefinite. In particular, this ( ) s p, w 0 the substitution effect of good l with respect to its own ll price is nonnegative. Notice that this means that a good can be a Giffen good at ( p, w) only if it is inferior. (, ) (, ) xl p w xl p w sll ( p, w) = + xk p, w p w (, ) (, ) xl p w xl p w If > 0 < 0 pl w GIFFEN l INFERIOR ( ) 0 21

22 The substitution matrix is not necessarily symmetric (for L=2 it is). This technical condition will be essential when we compare the choice approach from this part to the preference approach in next section. Since the Walrasian budgets do not include all possible budgets there is nothing up to now that guarantees that we have rationalizable preferences. Preferences will require a symmetric substitution matrix that now we can t guarantee. 22

23 2. Demand theory Preferences: We define preferences on the consumption set L X R+. The preference relation is rational (complete and transitive). More assumptions on preferences: Desirability properties: Monotonicity or local non satiation (weaker) x 2 x x x x x 1 - In the previous graph local non satiation is not satisfied. - These properties will imply increasing utility functions. 23

24 Convexity: A preference is convex if its upper contour ({ y X : y x} set is convex. If y x z x αy+ ( α) z x α [ ] and, then 1, 0,1. ) Convexity means diminishing marginal rates of substitution. Convexity may also be interpreted as the tendency to diversify consumption. y x 2 ( 1 α ) α y+ x x z x 1 (Convexity will imply utility functions quasiconcave; strict convexity, strict quasiconcavity) ( ) and ( ) implies that ( α + ( 1 α) ), R,, and α [ 0,1] f x t f x t f x x t t x x A 24

25 Some types of preferences play an important role in economics because they allow applied economists infer the whole preference relation from a single indifference curve. Homothetic preferences: If x y α x αy, α 0. x 2 α x x α y y x 1 25

26 Quasilinear preferences: A preference is quasilinear with respect to 1 (numerarie) if: All indifference sets are parallel displacements along the axis of commodity 1. If x y ( x+ e ) ( y+ e ) e = ( ) then α α for 1,0,...,0 and α R Good 1 is desirable; x+ αe1 x, x, α > 0. x 2 x 1 26

27 Continuity: A preference relation is continuous if both its upper and lower contour sets are closed. Example: Lexicographic preferences are not continuous. Proposition: If the preference relation is continuous then there is a continuous utility function that represents. (there may be discontinuous functions representing ). -We usually assume that utility functions are twice continuously differentiable. 27

28 -Recall that convexity (strict convexity) implies quasiconcavity (strict quasiconcavity), but not concavity. - Monotonicity implies that the utility function will be increasing: u( x) > u( y) if x y. -There are continuous preferences that are not representable by differentiable utility functions: Leontief preferences are an example (x y iff Min { x } { 1, x2 Min y1, y }) 2 More results: -A continuous preference is homothetic iff admits a utility function that is homogeneous of degree one. u ( α x) = α u( x) α > 0. -A continuous preference is quasilinear with respect to the numeraire iff admits a utility function of the form u ( x) x + ω( x2,..., xl =. 1 ) 28

29 x 2 x = x 1 2 u u 0 Leontief preferences x 1 x 2 u 0 u u Quasilinear preferences x 1 x 2 u 0 u Homothetic preferences x 1 29

30 Utility maximization Assume rational, continuous and locally non satiated preferences and u(x) a continuous utility function representing these preferences. Take L X = R+ and p>>0. ( x) Max u * x 0 x, s.t. p x w ( p w) Proposition: If p>>0 and u() is continuous, then there exists a solution. (The budget set is closed and bounded if p>>0. A continuous utility function always has a maximum in a compact set.) The solution to the previous problem, L ( p, w) R is known as the x + Walrasian Demand (Funtion) Correspondence. x 2 u 1 u 0 x( pm, ) x 1 30

31 x 2 u 1 u 0 x( pm, ) x 1 x 2 u 0 u p > p (, 1 ) x p m 0 (, m 1 ) x p x 1 31

32 Properties: Assume u() is continuous and represents a locally non satiated preference relation over = R L X +. -Homogeneity of degree zero. -Walras Law. -Convexity / uniqueness: Convexity of preferences quasiconcavity of u convexity of x(p,w). Quasiconcavity: A function f defined in a convex A set is quasiconcave if its upper contour sets are convex sets: ( ) and ( ) implies that ( α + ( 1 α) ),,, and α [ 0,1] f x t f x t f x x t t R x x A. ( α + ( 1 α) ) { ( ), ( )},, and α [ 0,1] f x x Min f x f x x x A (A concave function is quasiconcave) Strict convexity of preferences strict quasiconcavity of u unicity of x(p,w) 32

33 -If preferences are continuous, strictly convex and locally non satiated continuity of x(p,w). - If u is continuously differentiable, Kuhn Tucker necessary conditions characterize the optimal solution to the utility maximization problem. u x * ( ) ( ) λ p = 0 * * x u x λ p 33

34 Indirect utiliy The utility value of the UMP is denoted v(p,m)=u(x * ) for any ( ) x * x p, w. -homogeneous of degree zero. -strictly increasing in w and nonincreasing in p l. -quasiconvex -continuous in p and w. 34

35 Expenditure minimization problem Min p x x h, s. t.: 0 u u ( x) ( p u) (Hicksian Demand Function) Duality: under certain conditions, both the UMP and the EMP yield the same solution. Expenditure function: e * ( p, u) = p x -Homogeneous of degree 1 in p. -Strictly increasing in u and nondrecreasing in p i. -Concave in p -Continuous in p and u. 35

36 Properties of the hicksian demand Homogeneity of degree zero in p No excess utility Convexity/uniqueness 36

37 Compensated law of Demand: Demand and price move in opposite directions for Hicksian wealth compensated price changes If u is continuous, the preference relation is locally non satiated and the hicksian demand is a singleton, for all p, p : ( p p ) h( p, u) h( ) p, u 0. Since h(p,u) is the optimal solution to the expenditure minimization, is less expensive than any other bundle with a utility of u. ( ) ( ) ph p, u ph p, u ( p p ) h( p, u) h( p, u) 0 ph ( p, u) ph ( p, u ) 37

38 Duality: ( p v( p, w) ) = w and v( p, e( p, u) ) u e, =. ( p, w) = h( p, v( p w) ) x l, h ( p, u) = x ( p, e( p u) ) l l, Relationships between demands, indirect utility and expenditure 1) h( p, u) = p e( p, u) Properties of the price derivatives of the Hicksian Demand Function - 2 ( p, u) = D e( p u) Dp h p, - D p h( p, u) is a negative semidefinite matrix. (Compensated law of demand) - D p h( p, u) is a symmetric matrix. - D p h( p, u) p = 0. (There is at least one substitute) 38

39 Slutsky equation We can compute ( p u) D p h, from the walrasian demand function, under certain circumstances. (continuous utility, locally non satiated and strictly convex preferences. h p ( p, u) x ( p, w) x ( p, w) = x ( p w) l l l k, k pk w + for all l,k. (Slutsky equation) Roy s identity The relation between the solution and the value function in the UMP problem is not as straightforward as in the EMP problem. The problem is that the indirect utility function is not invariant to increasing transformations of utility, hence the price derivative of the indirect utility can not equal the walrasian demand. If u is continuous, representing a locally non satiated and strictly convex preference relation and the indirect utility is differentiable at ( p, w) : x x 1, = v ( p w) l ( p, w) w v = v ( p, w) ( p, w) p ( p, w) w l p v ( p, w) 39

40 Comparative statics Engel curves Good Elasticity= Good Good Good 2 is Good Good Good 2 is 40 Good

41 Offer curve Good 1 A good can be Giffen only if it is inferior. 41

42 Integrability The problem of integrability consists of establishing the connection from demand to preferences. -If rational preferences, then x(p,w) is homogeneous of degree zero, Walras Law, and has a substitution matrix S(w,p) symmetric and negative semidefinite at all (p,w). -If we have a demand function x(p,w) homogeneous of degree zero, Walras Law, and with a substitution matrix S(w,p) symmetric and negative semidefinite at all (p,w), can find preferences that rationalize it? Yes - When the matrix S(w.p) is not symmetric, demand satisfying the weak axiom can not rationalized. Implications: Welfare analysis from demand. 42

43 Welfare analysis Instruments: Money metric utility functions: (Measures welfare changes in dollars) Min p z z m ( p, x) = e( p, u( x) ).. u( x) u( z) s t If x is given, then m() is an expenditure function and behaves as an expenditure function on p. If p is given, m() is a utility function. For given p, m() is strictly increasing in u, and hence a monotone transformation from the originative utility function, thus a utility function. Money metric indirect utility functions: Measures the amount of dollars at prices p necessary to get the same welfare as the one obtained at prices q and wealth w. µ ( p ; q, m) = e( p, v( q, m) ) To measure the effect on the welfare of the consumer from a change in prices (from p 0 to p 1 ) or wealth we could use any indirect utility function derived from the (known) preferences and compute v( p, m) v( p, m)

44 The money metric indirect utility function allows us to measure this change in welfare in dollars units. ( p; p1, m) ( p; p0, m) = e( p, v( p1, m) ) e( p, v( p0, m) ) µ µ 0 ( 0 ) 1 ( 1 ) ( 0 0) ( 1 1) ( 0, 1, ) = ( 0, 1) ( p0, u0) = e( p0, u1) m (,, ) = (, ) (, ) = e( p, u ) Let u = v p, m and u = v p, m and note e p, u = e p, u = m EV p p m e p u e CV p p m e p u e p u m EV is the amount in dollars that the consumer would accept instead of the price change. If EVZ>0, this means that the consumer is better off with the change. CV is the net revenue of a planner that would compensate the consumer for the price change. If CV>0 means that the planner would get money from the consumer that is better off with the change. 1 0 x 2 p = p = EV u 0 u 1 (, 1 ) x p m 0 (, m 1 ) x p x 1 44

45 x 2 p = p = u 0 u 1 (, 1 ) x p m CV 0 (, m 1 ) x p x Normal good or p < p, then EV>CV When researchers want to have a measure of willingness to pay, they usually use EV for two reasons: 1) EV uses present prices and, therefore, it is more useful for politicians than uncertain future prices. 2) If there are different alternatives, EV allows us to compare them with the same prices. 45

46 If there is no wealth effect for good 1: -If preferences are quasilinear with respect to some good k- the numeraire- different from 1) -If preferences are homothetic and wealth is predetermined. Then EV=CV. 0 1 (, 1, ) = (, 1, ) = (, 1, ) h p p u x p p w h p p u We call the common EV and CV the change in Marshallian consumer surplus. 46

47 Homothetic preferences Homogeneous of degree 1: ( ) = ( ( )) ( ) ( ) n f : R R is homogeneous of degree 1 if f tx = tf x t > 0. f is homothetic if f x g h x and h is homogeneous of degree 1., where g is strictly increasing A homothetic function is monotone transformation of a homogeneous of degree 1 function. Hence, a homothetic utility function represents the same preferences as a homogeneous of degree 1 utility function. This is interesting because if the utility fu nction is homogeneous of degree 1, then (, ) ( ) e p u = e p u (, ) ( ) (, ) ( ) v p m = v p m x p m = x p m Quasilinear utility: (,,..., ) = + (,..., ) U x x x x u x x 0 1 k 0 1 k Consider a simple case with two goods: x ( ) + u x 0 1 x0, x1 ( ) Max x + u x 0 1 st..: x + px = m FOC: ( ) u x = p 1 1 ( ) x m p x p 0 = > 0 Indirect utility function: (, ) ( ) ( ) ( ) ( ) v p m = m p x p + u x p = v p + m

48 Hence, demand depends on price for sufficiently high levels of wealth. (Suppose the case where x 0 =0, then x 1 =m/p 1 and v()=u(m/p 1 ) Quasilineal utility is useful because of the independence of demand from wealth; hence no wealth effects are present. Integrability is especially simple in the quasilinear case: From FOC, ( ) = ( ) (inverse demand function). Then, the utility from p x u x consuming x 1. x1 x1 1( 1) 1( 0 ) = ( ) = ( ) u x u u t dt p 0 0 t dt Total utility includes the utility derived from x 0 : 1 u ( x1( p1) ) + m px1( p1) = p ( t) dt + m px1( p1) x 0 48

49 Labor Supply Wealth was considered exogenous in the consumer problem, but a more realistic approach leads us to consider wealth as an endogenous variable and time as an exogenous variable from which the consumer allocates (optimally) part of it to labor in order to get wealth, and part of it to leisure (a consumption good). The main result to obtain in this section is that a change in the price of leisure (wage) has an ambiguous impact on the demand of leisure, hence on the supply of labor. Researchers rule out this complication by invoking specific utility functions that guarantee an unambiguous effect and simplify matters. 49

50 In order to get precise about the decision problem we analyze, consider a consumer with preferences over consumption goods x and labor l represented by a utility function u( x, l ). Both x and l are normal goods. Max xl, (, ) u x l st..: px+ wl m+ wt l+ t T x 0, l 0 If we concentrate in interior solutions, ul u x ( ) () = w p x. From the maximization program we obtain walrasian demand functions (,, ), (,, ) l * l w p M x * x w p M = =. M = m+ wt. We are interested in sign ( ) dl ( w, p, M ) dw. ( ) ( ) ( ) dlwpm,, lwpm,, lwpm,, M mw, = + dw w M w Using the Slutsky equation: M = cons. (,, ) (,, ) (,, ) (,, ) dlwpm lwpm lwpm lwpm = + dw w M M l T M= cons. < 0 (Substitution effect) < 0 (Wealth effect) > 0 (Extra wealth effect) > 0 > 0 < 50

51 Aggregate demand Economy with i=1, n consumers. Each one with a demand function ( ) 1 k over k goods: ( ) ( ) ( ) x pm, = x pm,,..., x pm,, i= 1,... n. i i i i i i Aggregate demand is defi ned as follows: (,,..., ) = n (, ). X pm1 mn x i 1 i pm = i Important characteristics of individual demand that we would like to see in aggregate demand. Express it as a function of prices and wealth. The main problem is aggregate wealth ( n i= 1 m i ). To express demand in terms of aggregate wealth means that distributions don t matter. This holds when wealth effects for every commodity for any two consumers and wealth levels are identical. This imposes a restriction on the preferences of consumers (identical homothetic preferences, quasilinear with respect to the same good, and more generally, of the Gorman Form. See below). If individual wealth levels are generated by a wealth distribution rule (depend on both prices and aggregate wealth, then we can write aggregate demand as a function ( i ) (, ), (, ) x pw = xi pw pw i 51

52 Individual demand generated by rational preferences necessarily satisfies weak axiom -Assume a wealth distribution rule -Recall x(p,w) satisfies Weak axiom iff it satisfies compensated law of demand. Then, aggregate demand satisfies the weak axiom for price-wealth changes that are compensated for every consumer. On the other hand, a price change compensated in the aggregate, need not be compensated at the individual level and may generate aggregate demand that does not satisfy WA. -If the individual demand functions satisfy the Uncompensated Law of Demand ( ) ( ) ( ) p ' p xi p', wi xi p, wi 0, p, p' and wi. Then so does the aggregate demand function (, ) = (, α, x pw x p w i i i ) as a consequence, the aggregate demand satisfies the weak axiom. -If preferences are homothetic, then x( pw, ) satisfies the uncompensated law of demand. Individual demand has welfare significance. 52

53 - If individual demands are continuous, aggregate demand is continuous. - Properties that aggregate demand gets: continuity, homogeneity of degree zero and Walras Law. - If the indirect utility function adopts the Gorman form (with equal wealth coefficients), then we can aggregate demands and obtain an aggregate demand in the sense of representative consumer. ( ) = ( ) + ( ) v p, m a p b p m. i i i (homothetic and quasilinear preferences are examples that generate Gorman indirect utility functions) (If preferences are quasilinear with respect to l, then there is an indirect utility of the form a i ( p) i w + ) p l 53

54 3. Uncertainty Many choices have uncertain outcomes. Strategies must be chosen without knowledge of the other players decisions, hence with uncertainty with respect to them. Consider a set of possible outcome or consequences C. A simple lottery L is a list ( ) probability of occurrence of outcome n. L= p,..., p, p 0 and p = 1 1 n n n n. P is the n A compound lottery is a combination of lotteries with certain probabilities associated to each lottery. The reduced lottery L of any compound lottery ( L,... L ; α,..., α ) is obtained by vector addition: 1 k 1 k L= α L + α L + α L k k 54

55 Preferences over lotteries First: Only reduced lotteries matter. The set of alternatives is Γ, the set of all simple lotteries over the set of outcomes C. Assume a rational preference relation (completeness and transitiveness) on Γ. Continuity The preference relation on Γ is continuous if for LL,, L Γ, the sets { α [ 0,1 ]: αl+ ( 1 α) L L } [ 0,1] α [ 0,1 ]: L αl+ ( 1 α) L 0,1 { } [ ] are closed. Small changes in probabilities do not alter orderings between two lotteries. Lexicographic preferences are ruled out. This ensures that preferences over lotteries can be represented by a utility function. 55

56 Independence If we mix two lotteries (each one) with a third one, the preference between these two does not depend on the third lottery. The preference relation on L satisfies independence if for LL,, L Γ, and α ( 0,1) ( ) ( ) L L in and only if αl+ 1 α L αl + 1 α L. The independence axiom is important because it allows us to construct the theory of choice under uncertainty. It says that the preference between two lotteries should determine his preference over two compound lotteries independent of the other outcome of this compound lottery. (notice that this is not as to say independence in consumption with other goods but instead of ) 56

57 Expected utility U : Γ R has an expected utility form if there exist numbers ( u,..., 1 u n ) for the N outcomes such that for every L ( p p ) ( ) U L = u p + + u p 1 1 n n =,..., 1 n Γ: The utility of a lottery is the expected value of the utilities outcomes. A utility function has the expected utility form iff it is linear. u n of the N -Expected utility is preserved only by increasing linear transformations. 57

58 Expected utility theorem If a rational preference over lotteries satisfy the continuity and the independence axioms, then can be represented by a utility function with expected utility form. We can assign a number un to each outcome such that for L ( p,..., p ) and L ( p,..., p ) = = : 1 n 1 n N L L up up N n n n n n= 1 n= 1 Steps of the proof: 1) 2) 3) 4) 5) Example: Construction of an expected utility function Importance: Easy to use Guide to action: If one accepts independence. 58

59 Allais paradoxe: Three prices: 1 st price: nd price: rd price: 0 L L ( 0,1,0 ) L (.1,.89,.01) ( 0,.11,.89 ) L (.10,0,.90 = = 1 1 = = ) 2 2 Many people reveals L 1 L 1 and L 2 L 2. L L u > 0.1 u u u L L 0.11 u u < 0.1 u u (By adding 0.89u u to the first inequality one can note the contradiction.) 59

60 1) Responses to Allais paradox: Even if its occurrence is common, we should teach against it. The paradox occurs because of extreme payoffs and probabilities close to 0. Consider a wider spectrum of objects as elements of choice (regret theory regret over a choice not made) The paradox exploits the independence axiom. Get a weaker requirement to construct the theory. 2) Other important paradoxes: Machina. It illustrates the importance of disappointment in choice, or the importance of counterfactuals in well-being and decisions. 3) There is a defense of the independence axiom based on the economic consequences in the market of behavior guided by the systematic violation of the independence axiom. ( α) L L and L L but (inconsistent with IND. AXIOM) αl + 1 L L Observe that this violation means that one person having L would pay to change to αl + ( 1 α) L, but once this lottery is resolved, obtaining L or L, she would pay to get L. 4) Induced preferences. Some examples show that when preferences are induced from an action to be taken before the uncertainty is resolved, then the independence axiom is not 60

61 satisfied. To satisfy the axiom, preferences should not be induced or derived form ex ante actions. Problem: In many circumstances preferences are induced. Intertemporal decisions are an example when preferences are not additively separable between periods. Risk aversion Define risk aversion and related concepts as fair choices, etc. We consider gambles over nonnegative amounts of wealth. When we define risk aversion we compare decisions between different lotteries: ( ) Lbp ( ) Lb ( ) Lbp ( ) ( ) L( b ) Risk averse: Lb,1, Risk neutral:,1, Risk lover: L b, p,1 Lotteries can be thought as random variables. We express amount of money by x. A lottery will be described as a cumulative distribution function : [ 0, 1] F R. If the requirements of the expected utility theorem are satisfied, we can assign values u(x) with the property that any F() can be evaluated by a utility function U() ( ) u( x) df( x). U F = Note that 61

62 -U is linear in F(). - The form of u (Bernoulli utility function) is open Restrictions on u: -increasing, continuous (and bounded, to avoid St. Petersburg paradox). Thus, if preferences over lotteries satisfy the requirements to represent it by a utility function in the expected utility form, Risk aversion: ( ) (). ( ) ( ) ( ) u x df x u xdf x F Risk aversion is equivalent to the concavity of u(). Strict Concavity means that the marginal utility of money is decreasing, hence a lottery in which one can get x or loose x with prob=1/2 is not worth taking the risk. 62

63 Certainty equivalent: CE is the amount of money that leaves indifferent the individual between playing the lottery and getting CE. ( ( )) u( x) df( x) u CE F, u =. u( w 2 ) u ( ) ( ) + ( 1 p) u( w ) pu w ( 1+ ( 1 p) w2) u pw 1 2 u( w 1) w CE pw ( 1 ) + p w 1 2 w w 63

64 Probability premium: For any fixed x and number ε, the probability premium π ( x, ε, u) is the excess in winning prob. over ½ that makes the individual indifferent between x and the gamble bet ween x ε and ( ) ( ) + x ε. ( 1 π, ε, ) ( ε) ( 1 π (, ε, )) ( ε) u x = + x u u x+ + x u u x 2 2 ( 1 π) u( x ε 2 ) ( 1 π 2 ) u( x ε) ( + ε ) u x = u x ( ) u ( ) ( ε ) u x x ε x x + ε w ( 1 + π )( x+ ε) + ( 1 π)( x ε)

65 Proposition: Suppose a decision maker is expected utility maximizer with u() defined on money, then the following pr operties are equivalent: -The decision maker is risk averse -u () is concave (Bernoulli) ( ) ( ) ( ) ( x u) x - CE F, u xdf x F. - π, ε, 0, ε. Demand for insurance (example page 187 Mas-Colell) Measures of risk aversion Arrow-Pratt Given a Bernoulli utility function Pratt coefficient of absolute risk aversion at x is: r A ( x) u ( ) (twice differentiable), the Arrow- u = u ( x) ( x) -It introduces the notion of curvature to measure risk aversion (more curvature, less CE, so more risk aversion). Recall that a linear u means risk neutrality. -To rule out changes in the measure due to linear transformations, it normalizes u by u. 65

66 Comparisons across individuals u2 () is more risk averse than u1 ( ) if (and all are equivalent) 1. ( ) ( ) r x, u r x, u x. A 2 A 1 2. u () is more concave than u ( ). Exists an increasing concave ψ 2 ( ) ( ) u ( x) = ψ u ( x) s.th CE ( F, u ) CE ( F, u ) F ( ) ( ) ( ) π x, ε, u π x, ε, u x, ε u x u x df x u x 1, F, x. 5. u ( x) df( x) ( ) ( ) ( ) ( ) Comparisons across wealth levels 1 - u () exhibits decreasing absolute risk avers ion if r ( x, u) A is a decreasing function of x. (individuals with this property take more risks as they become wealthier). Coefficient of relative risk aversion The coefficient or relative risk aversion at x is r R ( x, u) ( x) ( x) xu =. u Non increasing relative risk aversion (substitute for absolute risk aversion in applications) means that the individual becomes less risk averse with regard to gambles that are proportional to his wealth as 66

67 his wealth increases. It s stronger than decreasing absolute risk aversion: r ( x, u) = x r ( x, u). R Comparisons of payoff distributions A To compare two payoff distributions F() and G() we will use two criteria FOSD and SOSD. To define them we will assume distributions s.th. F ( ) F( x) 0 = 0 and = 1 for some x. 67

68 First order stochastic dominance: We say that F first-order stochastically dominates G when every expected utility maximizer (who prefers more to less) prefers F() over G() or, equivalently, when for every x the probability of getting at least x is higher under F() that under G(). F() first-order stochastically dominates G() if, for every nondecreasing function u :, ( ) ( ) u( x) dg( x). u x df x Proposition: F() first-order stochastically dominates G() iff F ( x) G( x) x. 1 G( x) F( x) x -If the mean of F is higher than the mean of G, this does not imply that F first-order stochastically dominates G. 68

69 Second-order stochastic dominance This concept refers to the idea of riskiness or dispersion. We say that a distribution second-order stochastically dominates another if every risk averter prefers the former. For any two distributions F ( x) and G( x ) with the same mean, F ( x ) second-order stochastically dominates G( x ) if for every nondecreasing concave function u : R + R ( ) ( ) u( x) dg( x) u x df x - Second-order stochastic dominance of F ( x) over G( x ) is equivalent to G( x ) being a mean preserving spread of F ( x ). - Second-order stochastic dominance can also be expressed by another expression. Recalling that F and G have the same mean, and that for some x, F(x)=G(x)=1, then () () x x G t dt F t dt, x

70 Extensions of expected utility - State-dependent utility For some situations it may be reasonable to assume that the Benoulli utilities change with the state of nature. If this is the case, then we have state-dependent utilities. When we model behavior according to state-dependent utilities we can observe optimal decisions that differ from state independent formulation. Hence, it is a matter of behavioral economics to analyze for which situations it is more convenient one type or the other. - Subjective probability In many circumstances, agents are not able to express probabilistic assessments of uncertain situations. Subjective probability theory allows us to deduce probabilities from choices if certain axioms are satisfied. In this case, we have an appropriate extension of expected utility, as in the case of state-dependent utilities. What it is imposed is that state dependent utilities differ only by increasing linear transformations. Risk attitudes towards money gambles are the same across states (state uniformity). 70