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Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Advanced Topics in Consumer Theory

Juan Manuel Puerta

November 9, 2009


Introduction

In this section, we will focus on some selected advanced topicsin consumer theory. These are:

1 Aggregation: Could we construct aggregate demand functionsout of individual maximization?

2 Evaluating Welfare: How do we assess the welfare effect onchanges in prices?

3 Choice under Uncertainty: How does utility maximizationlooks like when outcomes are uncertain?


The problem of aggregation

Assume we have I consumers with Walrasian demands xi(p,mi)for i = 1, 2, ..., I. The aggregate demand of the I consumers couldbe written as,

x(p,m1,m2, ...,mI) =∑I

i=1 xi(p,mi)

The main question is under which conditions we can write theaggregate demand as,

x(p,m1,m2, ...,mI) = x(p,∑I

i=1 mi)

In a word, under which conditions can we write the aggregatedemand as a function of prices and aggregate income?


Conditions for Aggregation

It is apparent that in order for the aggregate demand to be writtenas a function of aggregate income alone, it is necessary that forany two income vectors (m1,m2, ...,mI) and (m′1,m

′2, ...,m

′I) that

yield equal aggregate income (i.e.∑I

i=1 mi =∑I

i=1 m′i), aggregatedemands are equal (i.e.

∑i x(p,mi) =

∑i x(p,m′i)

Assume a disturbance vector dm such that∑

i dmi = 0. Considerthe effect of this change in the aggregate demand for a givengood `, ∑I

i=1∂x`i(p,mi)

∂midmi = 0

for every `


Conditions for Aggregation (cont.)

Recall that this has to be true for any initial m with dm satisfying∑i di = 0. Then, the only way in which the equation above can

be fulfilled is if,∂x`i(p,mi)

∂mi=

∂x`j(p,mj)∂mj

for every ` and every two individuals i and j, and incomedistributions (m1,m2, ...,mI) †.

Implication. This condition just means that the effect of incomeon demand has to be equal for every individual and income level.Geometrically, this implies that the IEP of the consumers areparallel linear paths.

This property holds in particular with homothetic and quasilineardemands. But there is even a more general result applying (nextslide)


Conditions for Aggregation (cont.)

Proposition: A necessary and sufficient condition for the set ofconsumers to exhibit parallel, straight income expansion paths(IEP) at any price vector (p) is that preferences admit indirectutility functions of the Gorman form with the coefficient on mi

the same for every consumer i. That is,

vi(p,mi) = ai(p) + b(p)mi

Proof: Omitted (Straightforward application of Roy’s identity).


Final Considerations on Aggregation

Note that there are a few properties that carry over fromindividual demands to aggregate demands regardless of theincome effects. Notably, continuity and homogeneity hold for theaggregate demand.

Notice though that even when continuity of the individualdemands is sufficient for continuity of the aggregate demand, it isnot necessary (†). That is, non-continuous demand functionsmay yield a continuous aggregate demand function.


Welfare

Motivation: We often want to measure how certain policies affectconsumer welfare.A first question would be, how do we measure welfare. Assumethat we moved from (p0,m0) to (p1,m1) as a consequence of anew policy. An intuitive way of checking how the welfarechanged would be to look at the indirect utility function andcheck whether

υ(p1,m1) − υ(p0,m0)is positive/negative, so that welfare increased/decreased.Note that the result of such calculation is a “utility” differencethat may be hard to interpret. In particular, it is difficult toanswer to the following question:How much better/worse is the consumer as a consequence of the

new policy?


Money Metric Indirect Utility Functions

A way about this problem would be to use a money metricindirect utility function µ(q; p,m)

Recall that µ(q; p,m) measures how much income the consumerneeds at prices q in order to attain the same utility he had whenprices were p and income was m. More precisely,

µ(q; p,m) = e(q, υ(p,m))

So letting again the superscript (1) denote “after” and (0)“before”, the utility difference above can be rewritten as,

µ(q; p1,m1) − µ(q; p0,m0)

Now the question is, at which prices q should we evaluate thisexpression?


Equivalent and Compensating Variations

Two obvious choices are: p0 and p1,

EV = µ(p0; p1,m1) − µ(p0; p0,m0) = µ(p0; p1,m1) − m0

CV = µ(p1; p1,m1) − µ(p1; p0,m0) = m1 − µ(p1; p0,m0)

Under the Equivalent Variation, everything is measured withrespect to the initial prices. This just asks how much moneywould be equivalent, at current prices, to the proposed change interms of utility.

Under the Compensating Variation, everything is measuredwith respect to the new prices. In this case, the second term asksin which amount you have to be compensated in order for you toaccept the price change. ( Draw figure †)


Which Measure is more appropriate?

If you are trying to arrange a compensation for a welfare change,it seems appropriate to use the new prices, and consequently, CV.

If instead you want to measure willingness to pay for a newpolicy, it would seem more appropriate to use EV.

If there are more than 2 policies, it would seem reasonable tocompare them all with respect to the same benchmark prices. Inthis case, also, EV seems more reasonable.


Empirical Issues and the “Integrability Problem”

How can we measure µ in practice? This is related to theintegrability problem

Assume that we observe demand functions x(p,m). Can werecover the underlying preferences from the observed behavior?This is the so-called “integrability problem”.

We have established that if the expenditure function exists and isdifferentiable, it should fulfill the following conditions:

∂e(p,u)∂pi

= hi(p, u) = xi(p, e(p, u))

for i = 1, 2, .., k

We could rewrite these conditions using the identity u = υ(q,m),∂e(p,υ(q,m))

∂pi= xi(p, e(p, υ(q,m)))

for i = 1, 2, .., k



Using the definition of the Money metric indirect utility function,we can write these conditions as,

∂µ(q;p,m)∂pi

= xi(p, µ(q; p,m)) for i = 1, 2, .., k

And the boundary condition,

µ(q; q,m) = m

These are the integrability equations. The solution to thissystem of partial differential equations allows you to find µwhich you can then evaluate to compute EV and CV



The classic tool for measuring welfare changes is theconsumer’s surplus. The consumer surplus associated to achange in prices between p0 and p1 is,

CS =∫ p1

p0 x(t)dt.

This is simply the area to the left of the demand curve betweenp0 and p1

It turns out that for some particular preferences, these measurescoincide. In particular, if preferences are quasilinear(U(x0, x1, ..., xk) = x0 + u(x1, ..., xk)), then CS=CV=EV. But let’stalk some more about quasilinear utility functions.


Quasilinear Utility

As stated before, a utility function is quasilinear if it can bewritten as U(x0, x1, ..., xk) = x0 + u(x1, ..., xk). For the rest of thediscussion, we will consider the case of 2 goods (without loss ofgenerality). We usually assume that u(.) is strictly concave.

The maximization problem is: maxx0,x1 x0 + u(x1) such thatx0 + p1x1 = m

FOC for the interior solution yield: u(x1) = p1 and fromsubstitution into the budget constraint x0 = m − p1x1(p1). Notethat x1 does not depend on m.The indirect utility function can bewritten as,

V(p1,m) = u(x1(p1)) + m − p1x1(p1) = υ(p1) + m

where υ(p1) = u(x1(p1)) − p1x1(p1)


Quasilinear Utility

Note that x1 cannot possibly be independent of income for everyincome level. If income is low enough, the “implicit” constraintx0 ≥ 0 may be binding. In that case x1 = m/p1 andV(p1,m) = u(m/p1). In what follows we will assume out thispossibility.Note that in the standard case, demand depends only on pricesand there are no income effects to worry about (How would IEPlook like?). This property makes Quasilinear preferencesparticularly convenient for welfare analysis.However, are QL preferences a correct description of consumerbehavior? It turns out that in cases in which the demand of thegood is relatively independent from income, they are. Imaginethe demand for paper or pencils, beyond the minimum income, itis unlikely that further increases in income would triggerincreases in demands.


Integrability under Quasilinear Utility

It turns out that the integrability problem is greatly simplifiedunder quasilinear preferences. Since the inverse demand functionis given by p1(x1) = u′(x1), it is very easy to recover the utilityassociated with a particular consumption level x1

u(x1) − u(0) =∫ x1

0 u′(t)dt =∫ x1

0 p1(t)dt

But this is easily computed as the inverse demand is observable.

Total utility would consist of the consumption of x1 plus theconsumption of x0

U(x0(m, p1), x1(p1)) = u(x1(p1)) + m − p1x1(p1) =∫ x1

0 p1(t)dt + m − p1x1(p1)


Integrability under Quasilinear Utility

It is particularly important to note that it is very easy to solve forthe money metric indirect utility function in the case ofquasilinear (QL) preferences. To see this, rewrite theintegrability equations defined above.

dµ(t;q,m)dt = x1(t, µ(t; q,m)) = x1(t)

µ(q; q,m) = m

Direct integration yields, µ(p; q,m) − µ(q; q,m) =∫ p

q x1(t)dt, or

µ(p; q,m) =∫ p

q x1(t)dt + m

In other words, the MMIUF for the change between q and p isjust the consumer surplus associated with a change from q to pplus income.


Compensating and Equivalent Variations with QuasilinearUtility

From the result above, it is easy to show that,

EV = µ(p0; p1,m1) − µ(p0; p0,m0) =∫ p0

p1 x1(t)dt + m1 − m0

CV = µ(p1; p1,m1) − µ(p1; p0,m0) = m1 − (∫ p1

p0 x1(t)dt + m0) =∫ p0

p1 x1(t)dt + m1 − m0

Let A(p0, p1) =∫ p0

p1 x1(t)dt, then EV = CV = A(p0, p1) + m1 − m0

The intuition for this result (EV=CV) is the following: Since thecompensation function is linear on m, the marginal utility ofincome is constant. Then, it doesn’t matter at which prices weevaluate the function.


Consumer Surplus as an Approximation

With QL preferences, the consumer’s surplus (CS) is an exactmeasure of welfare. It turns out that even with other preferences,CS could be interpreted as an approximated effect.

Consider a change in the price of good 1 from p0 to p1.Furthermore, assume m0 = m1 = m. Let u0 = υ(p0,m) andu1 = υ(p1,m). Using the definition of the MMIUF again we canrewrite EV and CV as functions of the expenditure function.

EV = e(p0, u1) − e(p0, u0) = e(p0, u1) − e(p1, u1)

CV = e(p1, u1) − e(p1, u0) = e(p0, u0) − e(p1, u0)

Where we have used the fact that e(p1, u1) = e(p0, u0) = m.

EV = e(p0, u1) − e(p1, u1) =∫ p0

p1∂e(p,u1)∂p dp =

∫ p0

p1 h(p, u1)dp

CV = e(p0, u0) − e(p1, u0) =∫ p0

p1∂e(p,u0)∂p dp =

∫ p0

p1 h(p, u0)dp


Consumer Surplus as an Approximation

It follows that EV(CV) are the area to the left1 of the “hicksian”demand function at the final(original) level of utility respectively.

The “true” welfare effect is the integral of a function we cannotobserve (Hicksian). However, using the Slutsky equation we canfind some relation between CV/EV that are related to thehicksian demand and CS that depends on the marshalliandemand.

Recall the slutsky equation,∂x(p,m)/∂p = ∂h(p,m)/∂p − ∂x(p,m)/∂mx(p,m). If the good isnormal so that ∂x(p,m)/∂m > 0, then|∂h(p,m)/∂p| > |∂x(p,m)/∂p|, i.e. the hicksian demand is steeperthan the marshallian demand.

1It is customary in economics to draw prices on the vertical axis. “left” should be interpreted in this context


From the chart, it is evident that for normal goods and p0 > p1, EV>CS>CVSource: Varian, Microeconomic Analysis, 3rd ed, p. 168


Welfare and Aggregation

We saw that when the indirect utility function has the Gormanform, vi(p,mi) = ai(p) + b(p)mi aggregation is yields x(p,

∑i mi).

The aggregate indirect utility function will have the formV(p,M) =

∑i ai(p) + b(p)M

where M =∑

i mi

Note that the QL preferences are a particular case with b(p) = 1.Roy’s identity can be used to prove that xi(p) = −

∂υ′i (p)∂pi

, orsimilarly

υi(p) =∫ ∞

p xi(t)dtIt follows that

V(p) =∑n

i=1 υi(p) =∑n

i=1

∫ ∞p xi(t)dt =

∫ ∞p

∑ni=1 xi(t)dt

That is, if all consumers have quasilinear utility functions, thenthe aggregate indirect utility function is simply the integral of theaggregate demand function. Furthermore, the aggregate demandfunction maximizes aggregate consumer surplus.


Uncertainty and Lotteries

So far, we have assumed choices over certain outcomes. Howdoes the analysis change if there is uncertainty? The buildingblock for introducing uncertainty is the concept of lotteryDefinition: Let p ◦ x ⊕ (1 − p) ◦ y be a lottery between goods xand y. p ◦ x ⊕ (1 − p) ◦ y reads “the consumer will receive prize xwith probability p and prize y with probability (1 − p)

Prizes could be goods, bundles of goods or even other lotteries.We will often put uncertainty in the framework of lotteries


Assumptions about lotteries

We will find it useful to make some assumptions about theconsumer’s perception of lotteries

1 L1. 1 ◦ x ⊕ (1 − 1) ◦ y ∼ x. The consumer is indifferent betweenthe lottery that gives x with probability 1 and y with probability 0and getting x with certainty.

2 L2. p ◦ x ⊕ (1 − p) ◦ y ∼ (1 − p) ◦ y ⊕ p ◦ x. The consumer isindifferent about the ordering of the lottery.

3 L3.q ◦ (p ◦ x⊕ (1− p) ◦ y)⊕ (1− q) ◦ y ∼ qp ◦ x⊕ (1 + q(1− p)− q) ◦ yThe consumer’s perception depends just on the “net” probabilitiesof obtaining a prize.

Assumptions 1 and 2 are fairly obvious and innocuous.Assumption 3 is intuitive but there is some empirical evidenceagainst that people treat simple and compound lotteries equally.


Assumptions about lotteries

Let L be the space of lotteries available to the consumer. Asusual we assume the consumer has complete, reflexive, andtransitive preferences in L.

Note that the fact that lotteries are defined over 2 outcomes is notrestrictive. Using compound lotteries we can extend this to noutcomes. For example, assume a lottery that gives x, y and z,each with probability 1/3. Then,

(2/3) ◦ ((1/2) ◦ x ⊕ (1/2) ◦ y) ⊕ (1/3)z

will give each outcome with probability (1/3).

Note: The previous representation is not unique. One could havedefined many other compound lotteries that equally yield eachoutcome with probability (1/3).


Expected Utility

Under some additional assumption, it is possible to prove thatthere exists a continuous utility function representing thepreferences defined over lotteries. That is it is possible to find afunction u such that p ◦ x ⊕ (1 − p) ◦ y � q ◦ w ⊕ (1 − q) ◦ z⇔u(p ◦ x ⊕ (1 − p) ◦ y) > u(q ◦ w ⊕ (1 − q) ◦ z)

Of course that this utility function is not unique. It turns out thatunder some additional assumptions, it is possible to find aparticular monotonic transformation that has a very convenientproperty: the expected utility property

u(p ◦ x ⊕ (1 − p) ◦ y) = pu(x) + (1 − p)u(y)


Expected Utility: Interpretation

The expected utility property just says that the utility of a lotterycould be obtained by multiplying each outcomes utility by thecorresponding probability and summing the results.This utility is additively separable with respect to the twooutcomes and linear in probabilities.The additional assumptions that we need to get a utility functionthat had the expected utility property are the following:

1 U.1 Continuity for lotteries:C+ = {p ∈ [0, 1] : p ◦ x ⊕ (1 − p) ◦ y � z} andC− = {p ∈ [0, 1] : z � p ◦ x ⊕ (1 − p) ◦ y} are closed sets for all x,zand z in L.

2 U.2 If x ∼ y, then p ◦ x ⊕ (1 − p) ◦ z ∼ p ◦ y ⊕ (1 − p) ◦ zFinally, we will be making 2 other assumptions for convenience,although they are not strictly needed.

1 U.3 There is a best (b) and a worst (w) lottery. That is, for any xin L, b � x � w

2 U.4 p ◦ b ⊕ (1 − p) ◦ w � q ◦ b ⊕ (1 − q) ◦ w⇔ p > q


Expected Utility Theorem

If (L,�) satisfy the above axioms (L.1-L.3and U.1-U.4), there isa utility function u defined on L that satisfies the expected utilityproperty:

u(p ◦ x ⊕ (1 − p) ◦ y) = pu(x) + (1 − p)u(y)


Proof (I)

Define u(b) = 1 and u(w) = 0. For an arbitrary z, set u(z) = pz,with pz defined by:

pz ◦ b ⊕ (1 − pz) ◦ w ∼ z (1)

For the proposed probability pz there are 2 things to check:1 Does pz exist? Continuity of lotteries (U.1) means that C+ and C−

are closed. Since [0, 1] ⊂ (C+ ∪ C−) and the real line isconnected, it follows that C+ ∩ C− , ∅. Therefore, pz exists.

2 Is pz unique? This follows from property (U.4). If both pz and p′zare the solution to (1) and they are not the same, then pz > p′z orpz < p′z. In either case, property (U.4) implies that the lottery withthe biggest probability of getting the best price is strictlypreferred. Uniqueness follows.

3 We need to check that the u obtained has the expected utilityproperty (next slide)

4 Finally, we need to check that u is indeed a utility function


Proof (II)

This follows from the following substitutions:

p ◦ x ⊕ (1 − p) ◦ y ∼

∼ p ◦ (px ◦ b ⊕ (1 − px) ◦ w) + (1 − p)(py ◦ b ⊕ (1 − py) ◦ w)

∼ ppx + (1 − p)py ◦ b ⊕ p(1 − px) + (1 − p)(1 − py) ◦ w

∼ [pu(x) + (1 − p)u(y)] ◦ b ⊕ [�p − pu(x) + 1 − u(y) − �p + pu(y)] ◦ w

By construction of the utility function, it represents thepreferences over the space of the lotteries, i.eu(p ◦ x ⊕ (1 − p) ◦ y) = pu(x) + (1 − p)u(y)


Proof (III)

Finally, we check that this is indeed a utility function, i.e thatx � y⇐⇒ u(x) > u(y).It is quite easy to see this given our assumptions,

1 Assume x � y, then we know that x ∼ px ◦ b ⊕ (1 − px) ◦ w andy ∼ py ◦ b ⊕ (1 − py) ◦ w.

2 Axiom U.4 ensures that px > py. But then our definition of utilityfunction implies that u(x) > u(y). 2

2Strictly speaking, this property follows from the other assumptions, so it is not an axiom but rather a property you couldprove. Cf. MWG


Uniqueness of the Utility Function

We saw that in the deterministic case, any monotonictransformation of the utility function is a utility functionrepresenting the same preferences. This is so because utility is“ordinal” rather than “cardinal”It turns out that only certain transformations will preserve the“expected utility property”. In particular it is easy to see that afunction v(.) = au(.) + c would do the trick.

v(x ∼ p ◦ x ⊕ (1 − p) ◦ y) = au(x ∼ p ◦ x ⊕ (1 − p) ◦ y) + c

= a(pu(x) + (1 − p)u(y)) + c

= p(au(x) + c) + (1 − p)(au(y) + c)

= pv(x) + (1 − p)v(y)

Proposition: Uniqueness of the expected utility function. Anexpected utility function is unique up to an affine transformationProof: See the book (Homework!)


Other notation for expected utility

We have so far worked with lotteries defined over 2 outcomes.All the proofs and, in particular, expected utility theorem carryover to the n-outcome case. In that case,

n∑i=1

piu(xi) (2)

Subject to some minor technical details, the expected holds forcontinuous probability distributions. Let p(x) be the probabilitydensity function defined over outcomes x, then the expectedutility of the gamble is, ∫

p(x)u(x)dx (3)

Since X is a random variable, so is u(X). Then (2) and (3) are theexpectations of Eu(X) in the continuous and discrete cases.


Risk Aversion

Assume the outcomes are in terms of money. Assume that theconsumer can choose between a lottery that gives him x withprobability 1/2 and y with probability 1/2. The expected utilityof the gamble is 0.5u(x) + 0.5u(y) and this could be lower orbigger than the utility of the expected outcome u(0.5x + 0.5y)


Risk Aversion

For t ∈ (0, 1), if u(tx + (1 − t)y) > tu(x) + (1 − t)u(y) we say theindividual is risk averse. If u(tx + (1 − t)y) < tu(x) + (1 − t)u(y),the individual is risk loving. Finally, ifu(tx + (1− t)y) = tu(x) + (1− t)u(y), the individual is risk neutral.It is evident that these definitions could be reinterpreted in termsof the concavity/convexity of the utility function. (Strict)Concavity is related with risk averse behavior while (strict)convexity is related to risk loving.

What are the ways we have of measuring the degree of riskaversion?


Absolute Risk Aversion

The second derivative of the utility function is a naturalcandidate as it gives us an idea of the “curvature” of the utilityfunction. The problem is that multiplicative transformations ofthe utility would yield different degrees of risk aversion. In orderto avoid that, we normalize according to the first derivative. Thisis the Arrow Pratt Absolute Risk Aversion

r(w) = −u′′(w)u′(w)

The concept of Absolute Risk is particularly useful forunderstanding the attitude towards risk of projects that implyabsolute gains. In many economic applications, we areinterested about risk attitudes when the losses or gains refer to aproportion of total income. In that case, the relevant measurewould be Relative Risk Aversion

ρ = −wu′′(w)u′(w)


Note that wealth-decreasing absolute risk aversion seemsplausible. Decreasing relative risk aversion is not so obvious?Why?


Local vs.Global Risk Aversion

In this section we will focus on absolute risk aversion.(1) The measures of risk aversion are valid for small changesincome around w, that is locally around some wealth levels.Sometimes, we are interested in measures that of “global” riskaversion. The first way to formalize this is to say that anindividually is globally more absolute-risk averse than otherwhen his absolute risk aversion is higher than that of the otherfor every wealth level. Let A() and B() be the respective utilityfunctions,

−A′′(w)A′(w) > −

B′′(w)B′(w)

(2) Alternatively, one could think of A() utility being "moreconcave" is he’s more risk averse

A(w) = G(B(w))where G() is strictly concave.



(3) Finally, a third way to see it is as A(.) being “more willing”to pay in order to get rid of risk. Let ε be a random variable withexpectation 0, i.e E(ε) = 0. Then let define πA(ε) as themaximum amount of his wealth A is willing to give up in orderto avoid facing the variable ε

A(w − πA(ε)) = E[A(w + ε)]

It sounds logical to say that A is (globally) more absolute-riskaverse than B is πA(ε) > πB(ε) for all w.

Which of the three should we choose?



It turns out that all these three are equivalent!Proposition: Pratt’s Theorem Let A(w) and B(w) be twodifferentiable, increasing and concave expected utility functionsof wealth. Then the following properties are equivalent

1 −A′′(w)A′(w) > −

B′′(w)B′(w) for all w

2 A(w) = G(B(w)) for some strictly concave G.3 πA(ε) > πB(ε) for all random variables ε with Eε = 0


Proof: (1)⇒ (2)

Define implicitly G(B) implicitly from A(w)=G(B(w)).Monotonicity of utility functions implies G is well defined.Differentiation yields

A′(w) = G′(B(w))B′(w)

and

A′′(w) = G′′(B(w))B′(w)2 + G′(B(w))B′′(w)

take the ratio of these two expressions,A′′(w)A′(w) =

G′′(B)B′G′(B) + ��G′(B)B′′

��G′(B)B′

Strict concavity of G together with positive marginal utility ofwealth implies that G′′(B)B′

G′(B) < 0 so that −B′′B′ < −

A′′A′ .


Proof: (2)⇒ (3)

In order to prove this we will have to use a property of theexpectation of concave functions.

1 Jensen’s Inequality: Let X be a non-degenerate random variableand f(X) be a strictly concave function of this random variable.Then Ef (X) < f (E(X))

Now, let prove that (2) implies (3).

A(w − πA)def .π︷︸︸︷= EA(w + ε)

prop.(2)︷︸︸︷= EG(B(w + ε))

<︸︷︷︸Jensen

G(EB(w + ε)) =︸︷︷︸def .π

G(B(w − πB))

<︸︷︷︸prop.(2)

A(w − πB) (4)

From these inequalities follows that πA > πB establishing theresult.


Proof: (3)⇒ (1)

Fix ε and consider the family of random variables tε witht ∈ [0, 1]. Let π(t) be the risk premium as a function of t. Secondorder taylor expansion around 0:

π(t) ≈ π(0) + π′(0)t +12π′′(0)t2 (5)

The definition of π(t), A(w − π(t)) ≡ EA(w + tε), implies thatπ(0) = 0

Differentiate twice the definition with respect to t,

−A′(w − π(t))π′(t) = E[A′(w + tε)ε]

A′′(w − π(t))π′2(t) − A′(w − π(t))π′′(t) = E[A′′(w + tε)ε2]

Evaluating the first expression when t=0,−A′(w)π′(0) = E[A′(w)ε] = A′(w)E(ε) = 0⇒ π′(0) = 0


Proof: (3)⇒ (1)

Evaluating the second expression at t=0,

−A′(w)π′′(0) = A′′(w)E[ε2]

Let σ denote the variance of ε, σ2 = E(ε − Eε)2 = Eε2. Then,

−A′(w)π′′(0) = A′′(w)σ2 ⇒ π′′(0) = −A′′(w)A′(w) σ

2

Use these results in the Taylor expansion, equation (5) above,

π(t) ≈ π(0) + π′(0)t + 12π′′(0)t2 = 0 + 0t − 1

2A′′(w)A′(w) σ

2t2

But then for arbitrarily small t, we get that ifπA > πB ⇒ −

A′′A′ > −

B′′B′ . And this is what proves this part.

Since (1)⇒ (2)⇒ (3)⇒ (1), the equivalence is established.


Some Applications

Insurance Problem †.

Comparative Statics of the Portfolio Problem †

Asset Pricing†


Some Applications

Insurance Problem †.

Assume a risk averse consumer has wealth W. There is a risk thathe will lose an amount L. The consumer can buy insurance thatwill pay him q if the event occurs. Let π be the premium chargedper unit of wealth insured so that if he takes insurance he wouldhave to pay qπ, if p is the probability of incurring a loss, howmuch q would the consumer want to buy?

If the problem is actuarilly fair, that is π = p, then a risk averseindividual would choose to fully insured, i.e. q∗ = L

advanced topics in consumer theory - central … · advanced topics in consumer theory juan manuel...

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