Advanced Microeconomics I: Consumers, Firms and Markets ... · appendix of Jehle/Reny). 19 Advanced...

1 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J1 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J

Advanced Microeconomics I: Consumers, Firmsand MarketsChapters 1+2

Prof. Dr. Oliver Gürtler

Winter Term 2012/2013


1. Introduction� Microeconomic theory covers the analysis of individual economic behavior(e.g., a consumer or �rm) and the aggregation of individuals�actions in an in-stitutional framework (e.g., a price mechanism in an impersonal market place).

� Doing this, we intend to get a better understanding of economic activity andoutcomes. This is useful in two distinct senses:

� positive sense: we obtain a better understanding of individual behaviorin certain situations.

� normative sense: we understand when to intervene, both at the govern-ment level and at the institutional level.

� However, the models we will analyze are highly simpli�ed and sometimes toosimple to be realistic.

� Still, they have some general predictive power and represent the building blocksof more complex and realistic testable models.


Structure of the course:

2. Consumer Theory

3. Theory of the Firm

4. Partial Equilibrium

5. General Equilibrium

6. Social Choice and Welfare

Literature:

� Jehle, Reny (2011): Advanced Microeconomic Theory, Pearson Education.

� MasColell, Whinston, Green (1995): Microeconomic Theory, Oxford UniversityPress.


Organizational matters:

� The course consists of a lecture (10am) and an integrated exercise course(2pm; starting in the �fth week).

� Slides and exercise sets are available at ILIAS.

� Students are expected to prepare the exercises and to present their solutions.

� Some mathematical concepts are needed, most of which should be known toeveryone. The concepts that are probably not known to everyone are brie�yaddressed when they are encountered for the �rst time.

� The �nal exam takes place on Jan 31, 2013 at 10 am.


2. Consumer Theory� In consumer theory, we focus on an individual�s (consumer�s) decision to con-sume a set of commodities (goods and services).

� There are four building blocks in any such model:

� consumption set

� feasible set

� preference relation

� behavioral assumption

� By specifying each of these in a given problem, many di¤erent situations in-volving choice can be formally analyzed.


� We focus on n commodities i = 1; :::; n.

De�nition 1 The consumption setX represents the set of all consumption bundlesthat the consumer can conceive.

De�nition 2 A consumption bundle (or consumption plan) x = (x1; :::; xn) is avector specifying the amounts of each of the di¤erent commodities.

� Note: Time and location are included in the de�nition of a commodity.


Assumption 1 The consumption set X satis�es:

1. X � Rn+ (nonnegativity)2. X is closed, it includes its own boundary.3. X is convex: if x 2 X and y 2 X then z = �x + (1� �)y 2 X (for every� 2 [0; 1]).4. 0 2 X:

De�nition 3 The feasible set B � X represents those consumption plans that theconsumer can conceive and a¤ord.

� The behavioral assumption speci�es the ultimate objectives in choice (typi-cally utility maximization).


2.1 Preferences and Utility

� Each consumer is endowed with a binary relation, %, de�ned on the con-sumption set X.

� The expression x % y means that �x is at least as good as y�for the consumer.

� This relation gives us information about the consumer�s tastes for the di¤erentobjects of choice.

� We impose several axioms that the relation should ful�ll.

� These axioms formalize the view that the consumer can choose and thatchoices are consistent in a particular way. Moreover, they serve to characterizethe consumer�s tastes over the consumption bundles.


Axiom 1 Completeness. For all x;y 2 X, either x % y or y % x (or both).

! Consumer can always compare two consumption bundles and decide which is(weakly) preferred.

Axiom 2 Transitivity. For any three elements x;y;z 2 X, if x % y and y % z,then x % z.

! Axiom links pairwise comparisons in a consistent way.

! Consumer can rank any �nite number of elements in X from best to worst(ties are possible).

De�nition 4 The binary relation % on the consumption set X is called a preferencerelation if it satis�es Axioms 1 and 2.


De�nition 5 The binary relation � on the consumption set X is de�ned as follows:x � y if and only if x % y and not y % x.The relation � is called the strict preference relation induced by %.

De�nition 6 The binary relation � on the consumption set X is de�ned as follows:x � y if and only if x % y and y % x.The relation � is called the indi¤erence relation induced by %.

� Note: � and � do not necessarily have the same properties as %. For instance,while both are transitive, neither is complete.


De�nition 7 Let x0 be any point in X. Relative to this point, we can de�ne thefollowing subsets of X:

1. %�x0�=�x��x 2 X;x % x0, called the "at least as good as" set.

2. -�x0�=�x��x 2 X;x0 % x, called the "no better than" set.

3. ��x0�=�x��x 2 X;x0 � x, called the "worse than" set.

4. ��x0�=�x��x 2 X;x � x0, called the "preferred to" set.

5. ��x0�=�x��x 2 X;x � x0, called the "indi¤erence" set.

� Note: For any bundle x0, the three sets ��x0�, �

�x0�and �

�x0�partition

X.


� We introduce some additional axioms to put more structure on preferences(from now on X = Rn+).

Axiom 3 Continuity. For all x 2 Rn+, the sets % (x) and - (x) are closed in Rn+.

� Technical assumption ensuring that there are no sudden preference reversals.

! Rules out "open" area in the indi¤erence set.

Axiom 4 Strict Monotonicity. For all x;y 2 Rn+, if x � y, then x % y, while ifx� y, then x � y.

� Consumer always prefers a consumption bundle involving more to one involvingless.

! Rules out the possibility of having "zones of indi¤erence".

! Ensures that indi¤erence sets have negative slope.


Axiom 5 Strict Convexity. If x 6= y and x % y, tx+(1� t)y � y for all t 2 (0; 1).

� The consumer prefers "balanced" bundles to more extreme ones.

! The (absolute value of the) slope of the indi¤erence curve (the marginal rateof substitution) is decreasing.

� (Weak) convexity is de�ned analogously.


De�nition 8 A real-valued function u : Rn+ ! R is called a utility function repre-senting the preference relation % if for all x;y 2 Rn+, u (x) � u (y), x % y.

� Utility function represents consumer�s preference relation if it assigns highernumbers to preferred bundles.

� Utility function is a device for summarizing the information contained in thepreference relation.

� Is it always possible to represent the preference relation by a continuous real-valued function?

Theorem 1 If the binary relation % is complete, transitive, continuous and strictlymonotonic, there exists a continuous real-valued function, u : Rn+ ! R, whichrepresents %.


Proof.

� Idea: For given preferences, �nd one function representing these preferences!

� Let e = (1; :::; 1) 2 Rn+ and consider the mapping u : Rn+ ! R de�ned by

u (x) e � x

� In words: assign to any bundle x the number u (x) such that the consumer isindi¤erent between x and a bundle with u (x) units of every commodity.

� Does such number always exist? If so, is it uniquely determined?


� To tackle the �rst question, de�ne

A � ft � 0 jte % xgB � ft � 0 jte - xg

� We have to show that A \B is always nonempty.

� By continuity of %, both A and B are closed in R+.

� By strict monotonicity, t 2 A) t0 2 A for all t0 � t.

! A is a closed interval of the form�t̂;1

�.

� Similarly, B is a closed interval of the form�0; ~t�.


� By completeness, te - x or te % x; i.e., t 2 A [B.

! A [B = R+.

� Since A [B =�0; ~t�[�t̂;1

�, it must be that t̂ � ~t.

! A \B is nonempty.

� To tackle the second question, suppose there were two numbers t1 and t2satisfying t1e � x and t2e � x.

� By transitivity, we have t1e � t2e.

� By strict monotonicity, t1 = t2.


� It remains to show that the function we have constructed represents % and iscontinuous.

� Consider two bundles x and y and their associated numbers u (x) and u (y).Let x % y.

� By the de�nition of u, u (x) e � x and u (y) e � y.

� By transitivity, u (x) e � u (y) e.

� By strict monotonicity, u (x) � u (y).

! The function we have constructed represents % :

� To show that the function is continuous, we show that the inverse image underu of every open ball in R is open in Rn+ (by Theorem A1.6 in the mathematicalappendix of Jehle/Reny).


� Since open balls in R are open intervals, we must show that u�1 ((a; b)) isopen in Rn+ (for a < b).

� By de�nition, the inverse image u�1 ((a; b)) is�x 2 Rn+ ja < u (x) < b

, or

��x 2 Rn+ jae � u (x) e � be

(by strict monotonicity), or

��x 2 Rn+ jae � x � be

(by u (x) e � x), or

� � (ae)\ � (be)

� By continuity of �, the sets % (be) and - (ae) are closed in Rn+.

! � (ae) and � (be), being the complements of closed sets, are open.

� u�1 ((a; b)) is open. Q.E.D.


� According to the theorem, we can describe the consumer�s preferences in twoways, either by the preference relation or a continuous utility function.

� The latter approach is often more convenient.

� The preference relation ranks consumption bundles.

! There is more than one utility function representing a particular ranking.

Theorem 2 Let � be a preference relation on Rn+ and suppose u (x) is a utilityfunction that represents it. Then v (x) also represents � if and only if v (x) =f (u (x)) for every x, where f : R ! R is strictly increasing on the set of valuestaken on by u.

� Utility function is invariant to positive monotonic transforms.

� Intuition: positive monotonic transform does not change the ranking of bun-dles.


Theorem 3 Let � be represented by u : Rn+ ! R. Then

1. u (x) is strictly increasing if and only if � is strictly monotonic.2. u (x) is quasiconcave if and only if � is convex.3. u (x) is strictly quasiconcave if and only if � is strictly convex.

� The proof follows easily from the de�nitions involved. Consider part 2, forexample.

� By de�nition a function u (�) is quasiconcave if and only if the set�y 2 Rn+ ju (y) � k

is convex for every k 2 R.

� If x is chosen such that u (x) = k, the de�nition of quasiconcavity of u (�)coincides with the de�nition of convexity of %.

� In the following we additionally assume u (�) to be di¤erentiable.


De�nition 9 The �rst-order partial derivative of u (x) with respect to xi is calledthe marginal utility of good i.

� Assume n = 2.

� Let x2 = f (x1) be the function describing an indi¤erence curve.

! u (x1; f (x1)) = const:

� Di¤erentiating with respect to x1 yields:

@u

@x1+@u

@x2f 0 (x1) = 0

,MRS = �f 0 (x1) =@u@x1@u@x2

� The marginal rate of substitution equals the ratio of the marginal utilities ofthe two goods.

� As indicated before, convexity of % may be interpreted as diminishing MRS.


2.2 The Consumer�s Problem

� Formally, the consumer seeks

x� 2 B such that x� % x for all x 2 B

� Consumer is assumed to operate within a market economy.

� His impact on the market is negligible.

! The vector of prices p = (p1; :::; pn) � 0 is �xed from the consumer�s pointof view.

� The feasible set (or budget set) B is de�ned by

B =�x 2 Rn+ jpx � m

where m � 0 denotes the consumer�s �xed money income.


� The consumer�s maximization problem can be stated equivalently as

maxx2Rn+

u (x) s.t. px � m

� Under the assumptions made u (x) is continuous and real-valued, while B isnonempty and compact (closed and bounded).

! By the Weierstrass theorem, a solution to the maximization problem alwaysexists.

� Since B is convex and u (x) strictly quasiconcave, the solution is unique.

� The solution vector x� depends on p and m:

� The optimal quantities, viewed as functions of p and m, are known as ordinalor Marshallian demand functions.

� We write as x (p;m) the vector of these quantities.


Derivation of the solution:

� To solve this nonlinear programming problem with inequality constraint, weform the Lagrangian

L = u (x) + � (m� px)

� We assume x� � 0. Then the solution satis�es the following optimalityconditions:

@L

@xi=@u (x�)

@xi� ��pi = 0, i = 1; :::; n

m� px� � 0�� (m� px�) = 0

�� 0


� Suppose �� = 0. Then @u(x�)@xi

= 0; 8i, contradicting strict monotonicity.

! �� > 0 and, hence, m� px� = 0.

� Combining the conditions for di¤erent commodities i 6= j yields

@u(x�)@xi

@u(x�)@xj

=pipj

! At the optimum, the marginal rate of substitution between any two goodsmust be equal to the ratio of the goods�prices.

� But: the conditions we stated are merely necessary. What about su¢ cientconditions?

� Under the assumptions imposed, the conditions are also su¢ cient for optimal-ity.


Proof:

� Because u (�) is quasiconcave, we haveru (x) (y � x) � 0 whenever u (y) �u (x) and x;y � 0.

� The optimality conditions can be restated as

ru (x�) = ��ppx� = m

� If x� is not utility-maximizing, there must be some z � 0 such that

u (z) > u (x�)

pz � m


� Because u (�) is continuous and m > 0, there exists some t 2 [0; 1] closeenough to one such that

u (tz) > u (x�)

ptz < m

� Setting y = tz, it follows

ru (x�) (y � x�) = ��p (y � x�)= �� (py � px�) < 0

� This condition contradicts the condition from the beginning of the proof.Q.E.D.


2.3 Indirect Utility and Expenditure2.3.1 The Indirect Utility Function

� The ordinary utility function u (x) represents the consumer�s preferences di-rectly.

! It is called the direct utility function.

De�nition 10 The function obtained by substituting the Marshallian demands in theconsumer�s utility function is the indirect utility function: v(p;m) = u(x(p;m))

� Alternatively: v(p;m) = maxx2Rn+

u (x) s:t: px � m


Theorem 4 If u (x) is continuous and strictly increasing on Rn+, then v (p;m) is

1. continuous on Rn++ � R+;2. homogeneous of degree zero in (p;m) ;3. strictly increasing in m;4. decreasing in p,5. quasiconvex in (p;m).Moreover, it satis�es Roy�s identity: If v (p;m) is di¤erentiable at

�p0;m0

�and

@v(p0;m0)@m 6= 0, then

xi�p0;m0

�= �

@v(p0;m0)@pi

@v(p0;m0)@m

, i = 1; :::; n:


Proof of parts 2, 3, 5, 6:

� To prove 2, we must show that v (p;m) = v (tp; tm) for all t > 0.

� By de�nition,

v(tp; tm) = maxx2Rn+

u (x) s:t: tpx � tm

= maxx2Rn+

u (x) s:t: px � m = v (p;m)

� To prove 3, note that

@v (p;m)

@m=@u (x (p;m))

@m=Xi

@u

@xi

@xi@m

� Using the optimality conditions @u@xi

= ��pi and px (p;m) = m)P

i pi@xi@m =

1, we obtain@v (p;m)

@m= �� > 0


� The proof of part 5 is more elaborate.

� Let us consider the following three sets

B1 =�x��p1x � m1

B2 =

�x��p2x � m2

Bt =

�x��ptx � mt

with pt � tp1 + (1� t)p2 and mt � tm1 + (1� t)m2, t 2 [0; 1].

� v (p;m) is quasiconvex in (p;m) if and only if

v�pt;mt

�� max

�v�p1;m1

�; v�p2;m2

�;8t 2 [0; 1]

� It su¢ ces to show that every choice a consumer can make when facing Btcould also be made when facing either B1 or B2.


� The latter condition holds trivially for t = 0 or t = 1. So, let us assumet 2 (0; 1).

� The proof is by contradiction. Therefore, assume that there exists some t 2(0; 1) and x 2 Bt such that x =2 B1 and x =2 B2. Then

p1x > m1 and p2x > m2

� Multiplying the �rst inequality by t and the second by (1� t) yields

tp1x > tm1 and (1� t)p2x > (1� t)m2

� Adding the two conditions yields the desired contradiction x =2 Bt:

! If x 2 Bt, then x 2 B1 or x 2 B2;8t 2 [0; 1].


� To prove the last part, note that

@v (p;m)

@pi=@u (x (p;m))

@pi=Xj

@u

@xj

@xj@pi

� Using the optimality conditions @u@xj

= ��pj and px (p;m) = m ) x�i +Pj pj

@xj@pi

= 0 yields@v (p;m)

@pi= ��x�i

� Roy�s identity is obtained by solving the condition for x�i and inserting@v(p;m)@m

for ��. Q.E.D.


2.3.2 The Expenditure Function

� The expenditure function is derived from a di¤erent kind of optimization prob-lem.

� We ask: What is the minimum level of money expenditure the consumermust make to achieve a given level of utility?

� Formally, the expenditure function is de�ned as

e (p; u) = minx2Rn+

px s:t: u (x) � u

� Let U =�u (x)

��x 2 Rn+ denote the set of attainable utility levels.! The domain of e (�) is Rn++ � U .


� The solution to the minimization problem is denoted by xh (p; u).

! e (p; u) = pxh (p; u)

� Idea: We imagine a process where the consumer�s income is changed tocompensate him for changes in prices such that he can achieve exactly thesame utility level as before.

� Still, the optimal quantities of the goods will typically change.

! The optimal quantities, viewed as functions of p and u, are known as com-pensated or Hicksian demand functions.

� xh (p; u) is the vector of these demands.


Theorem 5 If u (x) is continuous and strictly increasing on Rn+, then e (p; u) is

1. zero when u takes on the lowest level of utility in U ,2. continuous on its domain Rn++ � U;3. for all p� 0; strictly increasing and unbounded above in u;4. strictly increasing in p;5. homogeneous of degree one in p,6. concave in p.Moreover, if u (x) is strictly quasiconcave, we have Shephard�s lemma: e (p; u) isdi¤erentiable in p at

�p0; u0

�with p0 � 0 and

@e�p0; u0

�@pi

= xhi�p0; u0

�, i = 1; :::; n:


Proof of parts 1, 4, 6, 7:

� To prove 1, note that the lowest value in U is u (0) (since u (x) is strictlyincreasing). x = 0 requires an expenditure of 0, hence e (p; u (0)) = 0.

� Part 4 follows from the proof of part 7.

� To prove 6, note that the expenditure function will be concave in prices if

te�p1; u

�+ (1� t) e

�p2; u

�� e

�tp1 + (1� t)p2; u

�for any two price vectors p1 and p2 and t 2 [0; 1].


� Denote the bundles minimizing expenditure to achieve u in the three situationsby x1, x2 and xt.

� By de�nition,

p1x1 � p1xp2x2 � p2x

for any other bundle x achieving utility u:

� In particular

p1x1 � p1xt

p2x2 � p2xt


� Multiplying the �rst inequality by t and the second by (1� t) yields

tp1x1 � tp1xt and (1� t)p2x2 � (1� t)p2xt

� Adding the two conditions, we obtain

tp1x1 + (1� t)p2x2 � tp1xt + (1� t)p2xt

, tp1x1 + (1� t)p2x2 ��tp1 + (1� t)p2

�xt

� This is the same as

te�p1; u

�+ (1� t) e

�p2; u

�� e

�tp1 + (1� t)p2; u

�


� To prove 7, we di¤erentiate e (p; u) with respect to pi.

@e (p; u)

@pi=@�pxh (p; u)

�@pi

= xhi +Xj

pj@xhj@pi

� From u (x) = u, we haveP

j@u@xj

@xhj@pi

= 0.

� Deriving the optimality conditions and combining them yields @u@xj

=pjpi

@u@xi.

� Together, these conditions implyP

j pj@xhj@pi

= 0 completing the proof of part7.

� Since xhi � 0, part 4 follows immediately. Q.E.D.


2.3.3 Relations Between The Two

� There are close relations between the indirect utility function and the expen-diture function.

� From the de�nitions of e and v, we know

e (p; v (p;m)) � m;8 (p;m)� 0;

v (p; e (p; u)) � u; 8 (p; u) 2 Rn++ � U:

� One can further show that both of these inequalities, in fact, must be equal-ities.

� This means that knowing the indirect utility function enables us to calculatethe expenditure function (and vice versa).

� Accordingly, we do not have to solve both optimization problems to derive theindirect utility function and the expenditure function.


� There are also close relations between the Marshallian and Hicksian demandfunctions.

Theorem 6 We have the following relations between the Hicksian and Marshalliandemand functions for p� 0; m � 0, u 2 U and i = 1; :::; n :

1. xi (p;m) = xhi (p; v (p;m))2. xhi (p; u) = xi (p; e (p; u))

� If x� solves the utility-maximization problem at (p;m), it also solves theexpenditure-minimization problem at (p; u), where u = u (x�).

� Conversely, if x� solves the expenditure-minimization problem at (p; u), it alsosolves the utility-maximization problem at (p;m), where m = px�.

� In this sense x� has a dual nature.


2.4 Properties Of Consumer Demand

� The theory developed so far leads to a number of predictions about behaviorin the marketplace.

� In the following, we present some of these predictions.

Theorem 7 The consumer demand function xi (p;m) ; i = 1; :::; n; is homoge-neous of degree zero in all prices and income, and it satis�es budget balancedness,px (p;m) = m for all (p;m).

Proof:

� We have shown before that v (p;m) is homogeneous of degree zero in (p;m),i.e.,

v (p;m) = v (tp; tm) ;8t > 0, oru (x (p;m)) = u (x (tp; tm)) ;8t > 0.


� Since budget sets at (p;m) and (tp; tm) are the same, x (p;m) was feasiblewhen x (tp; tm) was chosen (and vice versa).

� From the last equality on the previous slide and strict quasiconcavity of u itthen follows that x (p;m) = x (tp; tm).

� This condition says that xi (p;m) ; i = 1; :::; n; is homogeneous of degree zeroin prices and income.

� The property of budget balancedness has been proven before. Q.E.D.

� Homogeneity implies that we could focus on relative prices and real income.

� For instance, if good n serves as numéraire, we have

x (p;m) = x (tp; tm) = x

�p1pn; :::;

pn�1pn

; 1;m

pn

�


� An important consideration is the response in quantity demanded when priceschange.

� Typically, we expect a consumer to buy more of a good when its price declines,but this is not always true.

� To get an understanding of the relevant e¤ects, we consider the Slutsky equa-tion.

Theorem 8 Let x (p;m) be the consumer�s Marshallian demand system. Let u� bethe level of utility the consumer achieves at prices p and income m. Then,

@xi (p;m)

@pj=@xhi (p; u

�)

@pj� xj (p;m)

@xi (p;m)

@m; i; j = 1; :::; n:

� The substitution e¤ect is the �rst, the income e¤ect the second term onthe RHS.


Proof:

� Start from the identity

xhi (p; u�) = xi (p; e (p; u

�))

� Di¤erentiating both sides with respect to pj , yields

@xhi (p; u�)

@pj=@xi (p; e (p; u

�))

@pj+@xi (p; e (p; u

�))

@m

@e (p; u�)

@pj

� From Shephard�s lemma, we have @e(p;u�)@pj

= xhj (p; u�) = xj (p; e (p; u

�)).

� Moreover, we know u� = v (p;m) and m = e (p; v (p;m)) = e (p; u�).


� Using these conditions, we obtain

@xhi (p; u�)

@pj=@xi (p;m)

@pj+@xi (p;m)

@mxj (p;m)

() @xi (p;m)

@pj=@xhi (p; u

�)

@pj� xj (p;m)

@xi (p;m)

@m

Q.E.D.

Intuition:

� Substitution e¤ect: (hypothetical) change in consumption that would occurafter prices were set to the new level but where income was adapted suchthe maximum level of utility the consumer can achieve were kept the same asbefore the price change (Hicks).

� Income e¤ect: change in consumption if prices were kept at the new levelbut income was set back to its real value.


� The decomposition of the total e¤ect of a price change into the substitutionand income e¤ect is useful since we are able characterize the single e¤ects inmore detail.

Theorem 9 Let xhi (p; u) be the Hicksian demand for good i. Then@xhi (p;u)@pi

� 0,i = 1; :::; n:

� In words, own-substitution terms are negative.

Proof:

� From Shephard�s lemma, we have xhi (p; u) =@e(p;u)@pi

.

� Di¤erentiating with respect to pi, yields

@xhi (p; u)

@pi=@2e (p; u)

@p2i

� We have shown before that e (p; u) is concave in p, hence @2e(p;u)@p2i

� 0.Q.E.D.


De�nition 11 A good is called normal if consumption of it increases as incomeincreases, holding prices constant.

De�nition 12 A good is called inferior if consumption of it declines as incomeincreases, holding prices constant.

� From the two theorems derived before and the de�nitions, it is easy to derivethe so-called "law of demand".

Theorem 10 A decrease in the own price of a normal good will cause quantitydemanded to increase. If an own price decrease causes a decrease in quantity de-manded, the good must be inferior.

� The law of demand connects concepts about the reactions of quantity de-manded to income changes with concepts about the reactions of quantitydemanded to price changes.


� In the remainder of this section, we derive some useful elasticity relations.

� These relations can be derived from the budget balancedness constraint,px (p;m) = m.

De�nition 13 Let xi (p;m) be the consumer�s Marshallian demand for good i.Denote as

1. �i � @xi(p;m)@m

mxi(p;m)

the income elasticity of demand for good i;

2. �ij � @xi(p;m)@pj

pjxi(p;m)

the price elasticity of demand for good i and

3. si =pixi(p;m)

m � 0 (withPn

i=1 si = 1) the income share spent on purchases ofgood i.


Theorem 11 Let x (p;m) be the consumer�s system of Marshallian demands. Thenthe following relations must hold among income shares, price, and income elasticitiesof demand:

1. Engel aggregation:Pn

i=1 si�i = 1:2. Cournot aggregation:

Pni=1 si�ij = �sj ; j = 1; :::; n:

Proof:

� Di¤erentiate both sides of the budget constraint px (p;m) = m with respectto income to obtain

nXi=1

pi@xi@m

= 1

� The LHS can be rewritten asnXi=1

pixim

xim

@xi@m

=

nXi=1

pixim

@xi@m

m

xi=

nXi=1

si�i


� Di¤erentiating both sides of the budget constraint px (p;m) = m with respectto pj , yields

xj +nXi=1

pi@xi@pj

= 0

� This condition is equivalent tonXi=1

pipjm

@xi@pj

= �pjxjm

,nXi=1

pixim

pjxi

@xi@pj

= �pjxjm

� Using the respective de�nitions, gives us the second condition of the theorem.Q.E.D.

� Because of budget balancedness, all consumer demand responses to price andincome changes must add up in a way that preserves the budget constraint.


2.5 Uncertainty

� Until now, we have assumed that consumers act in a world of perfect certainty.

� In what follows, we relax this assumption and consider uncertain situations.

! Consumers are assumed to have a preference relation over gambles (insteadof consumption bundles).

� Let A = fa1; :::; ang denote a �nite set of outcomes.

� The ai�s involve no uncertainty, but it is not clear which of these outcomes isrealized.

� Let pi denote the probability that outcome ai is realized.


De�nition 14 Let A = fa1; :::; ang be the set of outcomes. Then GS , the set ofsimple gambles (on A), is given by

GS �((p1 � a1; :::; pn � an)

��pi � 0;nXi=1

pi = 1

):

� Note: When one or more of the pi�s is zero, we can drop those componentsfrom the expression.

� It is sometimes the case that gambles have prizes that are themselves gambles.

� Such gambles are called compound gambles.

� Example: In some lotteries, one can win monetary prizes, but also tickets forthe lottery.

� For simplicity, we rule out in�nitely layered compound gambles.


� Let G denote the set of all (simple and compound) gambles.

� A gamble g 2 G can then be written as g =�p1 � g1; :::; pk � gk

�, k � 1,

where gi might be a compound gamble, a simple gamble or an outcome.

� As indicated before, the consumer (or decision maker) is assumed to havepreferences, %, over G.

� We impose certain axioms the preference relation is assumed to ful�ll, calledaxioms of choice under uncertainty.

Axiom 6 Completeness. For any two gambles g; g0 2 G, either g % g0 or g0 % g (orboth).

Axiom 7 Transitivity. For any three gambles g; g0; g00 2 G, if g % g0 and g0 % g00,then g % g00.


� Note that each ai 2 A is represented in G as a degenerate gamble.

! The two axioms imply that the elements in A can be ordered by %.

� Assume that these elements have been indexed such that a1 % a2 % ::: % an.

Axiom 8 Continuity. For any gamble g 2 G, there is some probability � 2 [0; 1]such that g � (� � a1; (1� �) � an).

Axiom 9 Monotonicity. For all probabilities �; � 2 [0; 1], (� � a1; (1� �) � an) %(� � a1; (1� �) � an) if and only if � � �.

� Monotonicity implies a1 � an.

! Decision maker is not indi¤erent between all outcomes in A.


Axiom 10 Substitution. If g =�p1 � g1; :::; pk � gk

�and h =

�p1 � h1; :::; pk � hk

�are in G, and if gi � hi for every i, then g � h.

� If decision maker is indi¤erent between the realizations of one gamble andanother, and the realizations occur with the same probabilities, then he isindi¤erent between the two gambles.

� In a compound gamble, one can calculate the e¤ective probabilities of thesingle outcomes in A:

� For any gamble g 2 G, if pi denotes the e¤ective probability assigned to ai byg, then we say that g induces the simple gamble (p1 � a1; :::; pn � an) 2 GS .

� The �nal axiom states that the decision maker cares only about the e¤ectiveprobabilities a gamble assigns to the single outcomes in A.

Axiom 11 Reduction to Simple Gambles. For any gamble g 2 G, if (p1 � a1; :::; pn � an)is the simple gamble induced by g, then (p1 � a1; :::; pn � an) � g.

� By this axiom and transitivity, a decision maker�s preferences over all gamblesare completely determined by the preferences over simple gambles.


� We can show that preferences, obeying the above axioms, can be representedwith a continuous, real-valued function.

� Let u : G! R describe a utility function representing % on G.

! For every g 2 G, u (g) denotes the utility number assigned to g.

� Similarly, u (ai) denotes the utility number assigned to the degenerate gamble(1 � ai) (called the utility of outcome ai).

De�nition 15 The utility function u : G ! R has the expected utility propertyif, for every g 2 G,

u (g) =nXi=1

piu (ai) ;

where (p1 � a1; :::; pn � an) is the simple gamble induced by g.

! u assigns to each gamble the expected value of utilities that might result!


� A utility function possessing the expected utility property is referred to as vonNeumann-Morgenstern (VNM) utility function.

Theorem 12 Let preferences % over gambles in G satisfy the above axioms. Thenthere exists a utility function u : G! R representing % on G, such that u has theexpected utility property.

� Implication: Under the axioms imposed, one can assign utility numbers to theoutcomes in A such that the decision maker prefers one gamble over anotherif and only if it has a higher expected utility.

Proof:

� We construct a function and show that it possesses the desired properties.

� Consider an arbitrary gamble g 2 G and let u (g) be the number satisfying

g � (u (g) � a1; (1� u (g)) � an)


� By continuity, such number exists and, by monotonicity, it is unique.

! We have derived a real-valued function u on G.

� To show that u represents %, consider two arbitrary gambles g; g0 2 G andsuppose g % g0.

� By transitivity and the de�nition of u, this is equivalent to

(u (g) � a1; (1� u (g)) � an) % (u (g0) � a1; (1� u (g0)) � an)

� By monotonicity, this is equivalent to u (g) � u (g0).


� It remains to show that u has the expected utility property.

� To do this, let g 2 G be an arbitrary gamble and gs � (p1 � a1; :::; pn � an) 2GS the simple gamble it induces.

� By reduction to simple gambles, we have g � gs and, hence, u (g) = u (gs).

� We therefore must show that u (gs) =Pn

i=1 piu (ai).

� By de�nition, u (ai) satis�es

ai � (u (ai) � a1; (1� u (ai)) � an) � qi

� By substitution, gs � (p1 � a1; :::; pn � an) ��p1 � q1; :::; pn � qn

�� g0.

� The simple gamble induced by g0 is

g0s =

nXi=1

piu (ai)

!� a1;

1�

nXi=1

piu (ai)

!� an

!


� By reduction to simple gambles and transitivity, gs � g0 � g0s, or

gs �

nXi=1

piu (ai)

!� a1;

1�

nXi=1

piu (ai)

!� an

!

� By de�nition, u (gs) is the unique number satisfying

gs � (u (gs) � a1; (1� u (gs)) � an)

� Comparing the two expressions, we obtain

u (gs) =nXi=1

piu (ai)

Q.E.D.


� VNM utility functions are not unique.

� However, because of the expected utility property not every positive monotonictransformation is possible.

Theorem 13 Suppose the VNM utility function u (�) represents %. Then the VNMutility function v (�) represents those same preferences if and only if for some scalar� and some scalar � > 0

v (g) = �+ �u (g) ;

for all gambles g.

� In words: VNM utility functions are unique up to positive a¢ ne transfor-mations.

Proof:

� For simplicity, we suppose g � (p1 � a1; :::; pn � an) is a simple gamble andprove only necessity.


� Since u (�) represents %, we have

u (a1) � u (a2) � ::: � u (an) and u (a1) > u (an) :

� Hence, for any i there is a unique �i 2 [0; 1] such that

u (ai) = �iu (a1) + (1� �i)u (an) :

� Because of the expected utility property, it follows that

u (ai) = u (�i � a1; (1� �i) � an) ; orai � (�i � a1; (1� �i) � an) :

� If v (�) represents % as well, we have

v (ai) = v (�i � a1; (1� �i) � an) :

� If v (�) has the expected utility property, this condition can be transformed into

v (ai) = �iv (a1) + (1� �i) v (an) :


� Combining the expressions for u (ai) and v (ai), yields

u (a1)� u (ai)u (ai)� u (an)

=1� �i�i

=v (a1)� v (ai)v (ai)� v (an)

for every i such that ai � an and, thus, �i > 0.

� Rearranging, we obtain

(u (a1)� u (ai)) (v (ai)� v (an)) = (v (a1)� v (ai)) (u (ai)� u (an)), v (ai) = �+ �u (ai) ;

with � � u (a1) v (an)� u (an) v (a1)u (a1)� u (an)

; � � v (a1)� v (an)u (a1)� u (an)

> 0.


� Using this condition, it follows that

v (g) =nXi=1

piv (ai) =nXi=1

pi (�+ �u (ai))

= �+ �nXi=1

piu (ai) = �+ �u (g) :

Q.E.D.


� VNM utility functions are not completely unique, nor are they entirely ordinal.

� Still, we should not attach undue signi�cance to the absolute level of a gamble�sutility.

� It is, for instance, not possible to use VNM utility functions for interpersonalcomparisons of well-being.

� In the remainder of this section, we analyze the relationship between a VNMutility function and the decision maker�s attitude toward risk.

� We con�ne attention to gambles whose outcomes consist of di¤erent amountsof wealth w.

� We assume A = R+ and, hence, wealth levels to be nonnegative.

� Moreover, we consider only gambles giving �nitely many outcomes a strictlypositive e¤ective probability.

� Finally, we assume u (�) to be strictly increasing and twice di¤erentiable for allwealth levels.


De�nition 16 Let u (�) be a decision maker�s VNM utility function for gambles overnonnegative levels of wealth. Then for the simple gamble g = (p1 � w1; :::; pn � wn) ;the decision maker is said to be

1. risk averse at g if u (E (g)) > u (g) ;2. risk neutral at g if u (E (g)) = u (g) ;3. risk loving at g if u (E (g)) < u (g) :If for every nondegenerate simple gamble, g, the decision maker is, for example, riskaverse at g, then he is said simply to be risk averse. Similarly, a decision maker canbe de�ned to be risk neutral and risk loving.

� Note that

u (E (g)) = u

nXi=1

piwi

!and

u (g) =

nXi=1

piu (wi)

� Intuition: The gamble entails risk, while both options lead to the same ex-pected value.


� A decision maker is risk averse (risk neutral, risk loving) if and only if hisVNM utility function is strictly concave (linear, strictly convex) over theappropriate domain of wealth.

Illustration for two wealth levels w1 and w2:

� With two wealth levels, we have

u (E (g)) = u (p1w1 + (1� p1)w2) and u (g) = p1u (w1) + (1� p1)u (w2)

� Consider a graph, where w is measured on the abscissa and u (�) on the ordi-nate.

� If we connect the points (w1; u (w1)) and (w2; u (w2)), we obtain a line seg-ment.

� u (g) is the ordinate of a particular point on this line segment.

� If the decision maker is risk averse, we have u (E (g)) > u (g).

! The utility function must proceed above the line segment and, thus, be strictlyconcave.


De�nition 17 The certainty equivalent of any simple gamble g over wealth levelsis an amount of wealth, CE, o¤ered with certainty, such that u (g) � u (CE). Therisk premium is an amount of wealth, P , such that u (g) � u (E (g)� P ). Clearly,P � E (g)� CE.

� When a person is risk averse and prefers more money to less, we have CE <E (g) and, thus, P > 0.

! A risk averse decision maker will "pay" some positive amount of wealth toavoid the gamble�s inherent risk.

� Similarly, we have CE = E (g) and CE > E (g) for risk neutral and riskloving decision makers.

� Often, we do not only want to know whether a decision maker is risk averse,but also how risk averse he is.

De�nition 18 The Arrow-Pratt measure of absolute risk aversion is given byRa (w) =

�u00(w)u0(w) .


� Ra (w) is positive, zero or negative as the decision maker is risk averse, riskneutral or risk loving.

� Ra (w) is una¤ected by any positive a¢ ne transformation of the utility function(u00 (w) is not).

� Decision makers with a larger Arrow-Pratt measure have a lower certaintyequivalent and are willing to accept fewer gambles.

Proof of the last statement:

� Consider two decision makers and denote their VNM utility functions by u (w)and v (w), with u0 (w) ; v0 (w) > 0.

� Let R1a (w) =�u00(w)u0(w) > �v00(w)

v0(w) = R2a (w), for all w � 0.

� Assume that v (w) takes on all values in [0;1) and write

h (x) = u�v�1 (x)

�, for all x � 0.

� Di¤erentiating h twice with respect to x, one can show that h0 (x) > 0 andh00 (x) < 0.


� Consider a gamble (p1 � w1; :::; pn � wn) over wealth levels and denote by CE1and CE2 the two decision makers�certainty equivalents for the gamble.

� By de�nition,nXi=1

piu (wi) = u (CE1) ;

nXi=1

piv (wi) = v (CE2) :

� Substituting v (w) by x and using h (x) = u�v�1 (x)

�, we obtain

u (CE1) =nXi=1

pih (v (wi))

� By Jensen�s inequality (because h is strictly concave), this expression is strictlysmaller than

h

nXi=1

piv (wi)

!= h (v (CE2)) = u (CE2)


� Because u (�) is strictly increasing, it follows that CE1 < CE2.

� If both decision makers have the same initial wealth, this �nding implies that2 will accept any gamble that 1 will accept. Q.E.D.

� Note that h (x) = u�v�1 (x)

�implies u (w) = h (v (w)).

! u (�) is more concave than v (�), because it is a concave function of v (�).

� Arrow-Pratt measure of absolute risk aversion typically varies with wealth.

� A VNM utility function is said to display constant, decreasing or increasingabsolute risk aversion over some domain of wealth if, over that interval,Ra (w) remains constant, decreases or increases with an increase in wealth.

� Decreasing absolute risk aversion (DARA) is often a sensible assumption.

! The greater wealth, the less averse one becomes to accepting the same gamble.

Date post:	05-May-2020
Category:	Documents
Upload:	others
View:	79 times
Download:	3 times

Advanced Microeconomics I: Consumers, Firms and Markets ... · appendix of Jehle/Reny). 19 Advanced...

Documents