Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Advanced Topics in Consumer Theory

Juan Manuel Puerta

November 9, 2009 Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Introduction

In this section, we will focus on some selected advanced topics in consumer theory. These are: 1 Aggregation: Could we construct aggregate demand functions out of individual maximization? 2 Evaluating Welfare: How do we assess the welfare effect on changes in prices? 3 Choice under Uncertainty: How does utility maximization looks like when outcomes are uncertain? Topics Aggregation Evaluating Welfare Choice Under Uncertainty

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 could be written as, PI x(p, m1, m2, ..., mI) = i=1 xi(p, mi) The main question is under which conditions we can write the aggregate demand as, PI x(p, m1, m2, ..., mI) = x(p, i=1 mi) In a word, under which conditions can we write the aggregate demand as a function of prices and aggregate income? Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Conditions for Aggregation

It is apparent that in order for the aggregate demand to be written as a function of aggregate income alone, it is necessary that for 0 0 0 any two income vectors (m1, m2, ..., mI) and (m1, m2, ..., mI) that PI PI 0 yield equal aggregate income (i.e. i=1 mi = i=1 mi ), aggregate P P 0 demands are equal (i.e. i x(p, mi) = i x(p, mi ) P Assume a disturbance vector dm such that i dmi = 0. Consider the effect of this change in the aggregate demand for a given good `, PI ∂x`i(p,mi) dm = 0 i=1 ∂mi i for every ` Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Conditions for Aggregation (cont.)

Recall that this has to be true for any initial m with dm satisfying P i di = 0. Then, the only way in which the equation above can be fulfilled is if, ∂x`i(p,mi) = ∂x`j(p,mj) ∂mi ∂mj for every ` and every two individuals i and j, and income distributions (m1, m2, ..., mI) †. Implication. This condition just means that the effect of income on demand has to be equal for every individual and income level. Geometrically, this implies that the IEP of the consumers are parallel linear paths. This property holds in particular with homothetic and quasilinear demands. But there is even a more general result applying (next slide) Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Conditions for Aggregation (cont.)

Proposition: A necessary and sufficient condition for the set of consumers to exhibit parallel, straight income expansion paths (IEP) at any price vector (p) is that preferences admit indirect utility 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). Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Final Considerations on Aggregation

Note that there are a few properties that carry over from individual demands to aggregate demands regardless of the income effects. Notably, continuity and homogeneity hold for the aggregate demand. Notice though that even when continuity of the individual demands is sufficient for continuity of the aggregate demand, it is not necessary (†). That is, non-continuous demand functions may yield a continuous aggregate demand function. Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Welfare

Motivation: We often want to measure how certain policies affect consumer welfare. A first question would be, how do we measure welfare. Assume that we moved from (p0, m0) to (p1, m1) as a consequence of a new policy. An intuitive way of checking how the welfare changed would be to look at the indirect utility function and check 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” difference that may be hard to interpret. In particular, it is difficult to answer to the following question: How much better/worse is the consumer as a consequence of the new policy? Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Money Metric Indirect Utility Functions

A way about this problem would be to use a money metric indirect utility function µ(q; p, m) Recall that µ(q; p, m) measures how much income the consumer needs at prices q in order to attain the same utility he had when prices 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 this expression? Topics Aggregation Evaluating Welfare Choice Under Uncertainty

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 with respect to the initial prices. This just asks how much money would be equivalent, at current prices, to the proposed change in terms of utility. Under the Compensating Variation, everything is measured with respect to the new prices. In this case, the second term asks in which amount you have to be compensated in order for you to accept the price change. ( Draw figure †) Topics Aggregation Evaluating Welfare Choice Under Uncertainty

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 new policy, it would seem more appropriate to use EV. If there are more than 2 policies, it would seem reasonable to compare them all with respect to the same benchmark prices. In this case, also, EV seems more reasonable. Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Empirical Issues and the “Integrability Problem”

How can we measure µ in practice? This is related to the integrability problem Assume that we observe demand functions x(p, m). Can we recover the underlying preferences from the observed behavior? This is the so-called “integrability problem”. We have established that if the expenditure function exists and is differentiable, it should fulfill the following conditions: ∂e(p,u¯) = h (p, u¯) = x (p, e(p, u¯)) ∂pi i i for i = 1, 2, .., k We could rewrite these conditions using the identityu ¯ = υ(q, m), ∂e(p,υ(q,m)) = x (p, e(p, υ(q, m))) ∂pi i for i = 1, 2, .., k Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Empirical Issues and the “Integrability Problem”

Using the definition of the Money metric indirect utility function, we can write these conditions as, ∂µ(q;p,m) = x (p, µ(q; p, m)) for i = 1, 2, .., k ∂pi i And the boundary condition, µ(q; q, m) = m These are the integrability equations. The solution to this system of partial differential equations allows you to find µ which you can then evaluate to compute EV and CV Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Empirical Issues and the “Integrability Problem”

The classic tool for measuring welfare changes is the consumer’s surplus. The consumer surplus associated to a change in prices between p0 and p1 is, R p1 CS = p0 x(t)dt. This is simply the area to the left of the demand curve between p0 and p1 It turns out that for some particular preferences, these measures coincide. In particular, if preferences are quasilinear (U(x0, x1, ..., xk) = x0 + u(x1, ..., xk)), then CS=CV=EV. But let’s talk some more about quasilinear utility functions. Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Quasilinear Utility

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

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

FOC for the interior solution yield: u(x1) = p1 and from substitution into the budget constraint x0 = m − p1x1(p1). Note that x1 does not depend on m.The indirect utility function can be written as,

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

where υ(p1) = u(x1(p1)) − p1x1(p1) Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Quasilinear Utility

Note that x1 cannot possibly be independent of income for every income level. If income is low enough, the “implicit” constraint x0 ≥ 0 may be binding. In that case x1 = m/p1 and V(p1, m) = u(m/p1). In what follows we will assume out this possibility. Note that in the standard case, demand depends only on prices and there are no income effects to worry about (How would IEP look like?). This property makes Quasilinear preferences particularly convenient for welfare analysis. However, are QL preferences a correct description of consumer behavior? It turns out that in cases in which the demand of the good is relatively independent from income, they are. Imagine the demand for paper or pencils, beyond the minimum income, it is unlikely that further increases in income would trigger increases in demands. Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Integrability under Quasilinear Utility

It turns out that the integrability problem is greatly simplified under quasilinear preferences. Since the inverse demand function 0 is given by p1(x1) = u (x1), it is very easy to recover the utility associated with a particular consumption level x1 R x R x − 1 0 1 u(x1) u(0) = 0 u (t)dt = 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 the consumption of x0

U(x0(m, p1), x1(p1)) = u(x1(p1)) + m − p1x1(p1) = R x 1 − 0 p1(t)dt + m p1x1(p1) Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Integrability under Quasilinear Utility

It is particularly important to note that it is very easy to solve for the money metric indirect utility function in the case of quasilinear (QL) preferences. To see this, rewrite the integrability equations defined above. dµ(t;q,m) dt = x1(t, µ(t; q, m)) = x1(t) µ(q; q, m) = m R p − Direct integration yields, µ(p; q, m) µ(q; q, m) = q x1(t)dt, or R p µ(p; q, m) = q x1(t)dt + m In other words, the MMIUF for the change between q and p is just the consumer surplus associated with a change from q to p plus income. Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Compensating and Equivalent Variations with Quasilinear Utility

From the result above, it is easy to show that, R p0 0 1 1 − 0 0 0 1 − 0 EV = µ(p ; p , m ) µ(p ; p , m ) = p1 x1(t)dt + m m R p1 1 1 1 − 1 0 0 1 − 0 CV = µ(p ; p , m ) µ(p ; p , m ) = m ( p0 x1(t)dt + m ) = R p0 1 − 0 p1 x1(t)dt + m m R p0 0 1 0 1 1 − 0 Let A(p , p ) = p1 x1(t)dt, then EV = CV = A(p , p ) + m m The intuition for this result (EV=CV) is the following: Since the compensation function is linear on m, the marginal utility of income is constant. Then, it doesn’t matter at which prices we evaluate the function. Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Consumer Surplus as an Approximation

With QL preferences, the consumer’s surplus (CS) is an exact measure 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) and u1 = υ(p1, m). Using the definition of the MMIUF again we can rewrite 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. R p0 1 R p0 0 1 − 1 1 ∂e(p,u ) 1 EV = e(p , u ) e(p , u ) = p1 ∂p dp = p1 h(p, u )dp R p0 0 R p0 0 0 − 1 0 ∂e(p,u ) 0 CV = e(p , u ) e(p , u ) = p1 ∂p dp = p1 h(p, u )dp Topics Aggregation Evaluating Welfare Choice Under Uncertainty

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 cannot observe (Hicksian). However, using the Slutsky equation we can find some relation between CV/EV that are related to the hicksian demand and CS that depends on the marshallian demand. Recall the slutsky equation, ∂x(p, m)/∂p = ∂h(p, m)/∂p − ∂x(p, m)/∂mx(p, m). If the good is normal so that ∂x(p, m)/∂m > 0, then |∂h(p, m)/∂p| > |∂x(p, m)/∂p|, i.e. the hicksian demand is steeper than the marshallian demand.

1 It is customary in economics to draw prices on the vertical axis. “left” should be interpreted in this context Topics Aggregation Evaluating Welfare Choice Under Uncertainty

From the chart, it is evident that for normal goods and p0 > p1, EV>CS>CV

Source: Varian, Microeconomic Analysis, 3rd ed, p. 168 Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Welfare and Aggregation

We saw that when the indirect utility function has the Gorman P form, vi(p, mi) = ai(p) + b(p)mi aggregation is yields x(p, i mi). The aggregate indirect utility function will have the form P V(p, M) = i ai(p) + b(p)M P where M = i mi Note that the QL preferences are a particular case with b(p) = 1. ∂υ0(p) Roy’s identity can be used to prove that x (p) = − i , or i ∂pi similarly R ∞ υi(p) = p xi(t)dt It follows that Pn Pn R ∞ R ∞ Pn V(p) = i=1 υi(p) = i=1 p xi(t)dt = p i=1 xi(t)dt That is, if all consumers have quasilinear utility functions, then the aggregate indirect utility function is simply the integral of the aggregate demand function. Furthermore, the aggregate demand function maximizes aggregate consumer surplus. Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Uncertainty and Lotteries

So far, we have assumed choices over certain outcomes. How does the analysis change if there is uncertainty? The building block for introducing uncertainty is the concept of lottery Definition: Let p ◦ x ⊕ (1 − p) ◦ y be a lottery between goods x and y. p ◦ x ⊕ (1 − p) ◦ y reads “the consumer will receive prize x with 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 Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Assumptions about lotteries

We will find it useful to make some assumptions about the consumer’s perception of lotteries 1 L1. 1 ◦ x ⊕ (1 − 1) ◦ y ∼ x. The consumer is indifferent between the lottery that gives x with probability 1 and y with probability 0 and getting x with certainty. 2 L2. p ◦ x ⊕ (1 − p) ◦ y ∼ (1 − p) ◦ y ⊕ p ◦ x. The consumer is indifferent about the ordering of the lottery. 3 L3. q ◦ (p ◦ x ⊕ (1 − p) ◦ y) ⊕ (1 − q) ◦ y ∼ qp ◦ x ⊕ (1 + q(1 − p) − q) ◦ y The consumer’s perception depends just on the “net” probabilities of obtaining a prize. Assumptions 1 and 2 are fairly obvious and innocuous. Assumption 3 is intuitive but there is some empirical evidence against that people treat simple and compound lotteries equally. Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Assumptions about lotteries

Let L be the space of lotteries available to the consumer. As usual we assume the consumer has complete, reflexive, and transitive preferences in L. Note that the fact that lotteries are defined over 2 outcomes is not restrictive. Using compound lotteries we can extend this to n outcomes. 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 have defined many other compound lotteries that equally yield each outcome with probability (1/3). Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Expected Utility

Under some additional assumption, it is possible to prove that there exists a continuous utility function representing the preferences defined over lotteries. That is it is possible to find a function 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 that under some additional assumptions, it is possible to find a particular monotonic transformation that has a very convenient property: the expected utility property u(p ◦ x ⊕ (1 − p) ◦ y) = pu(x) + (1 − p)u(y) Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Expected Utility: Interpretation

The expected utility property just says that the utility of a lottery could be obtained by multiplying each outcomes utility by the corresponding probability and summing the results. This utility is additively separable with respect to the two outcomes and linear in probabilities. The additional assumptions that we need to get a utility function that had the expected utility property are the following: 1 U.1 Continuity for lotteries: C+ = {p ∈ [0, 1] : p ◦ x ⊕ (1 − p) ◦ y  z} and C− = {p ∈ [0, 1] : z  p ◦ x ⊕ (1 − p) ◦ y} are closed sets for all x,z and z in L. 2 U.2 If x ∼ y, then p ◦ x ⊕ (1 − p) ◦ z ∼ p ◦ y ⊕ (1 − p) ◦ z Finally, 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 x in L, b  x  w 2 U.4 p ◦ b ⊕ (1 − p) ◦ w q ◦ b ⊕ (1 − q) ◦ w ⇔ p > q Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Expected Utility Theorem

If (L, ) satisfy the above axioms (L.1-L.3and U.1-U.4), there is a utility function u defined on L that satisfies the expected utility property: u(p ◦ x ⊕ (1 − p) ◦ y) = pu(x) + (1 − p)u(y) Topics Aggregation Evaluating Welfare Choice Under Uncertainty

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 is + − connected, it follows that C ∩ C , ∅. Therefore, pz exists. 2 0 Is pz unique? This follows from property (U.4). If both pz and pz 0 are the solution to (1) and they are not the same, then pz > pz or 0 pz < pz. In either case, property (U.4) implies that the lottery with the biggest probability of getting the best price is strictly preferred. Uniqueness follows. 3 We need to check that the u obtained has the expected utility property (next slide) 4 Finally, we need to check that u is indeed a utility function Topics Aggregation Evaluating Welfare Choice Under Uncertainty

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 the preferences over the space of the lotteries, i.e u(p ◦ x ⊕ (1 − p) ◦ y) = pu(x) + (1 − p)u(y) Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Proof (III)

Finally, we check that this is indeed a utility function, i.e that x 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 and y ∼ py ◦ b ⊕ (1 − py) ◦ w. 2 Axiom U.4 ensures that px > py. But then our definition of utility function implies that u(x) > u(y). 2

2 Strictly speaking, this property follows from the other assumptions, so it is not an axiom but rather a property you could prove. Cf. MWG Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Uniqueness of the Utility Function

We saw that in the deterministic case, any monotonic transformation of the utility function is a utility function representing 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 a function 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. An expected utility function is unique up to an affine transformation Proof: See the book (Homework!) Topics Aggregation Evaluating Welfare Choice Under Uncertainty

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 carry over to the n-outcome case. In that case, Xn piu(xi) (2) i=1 Subject to some minor technical details, the expected holds for continuous probability distributions. Let p(x) be the probability density function defined over outcomes x, then the expected utility of the gamble is, Z p(x)u(x)dx (3)

Since X is a random variable, so is u(X). Then (2) and (3) are the expectations of Eu(X) in the continuous and discrete cases. Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Risk Aversion

Assume the outcomes are in terms of money. Assume that the consumer can choose between a lottery that gives him x with probability 1/2 and y with probability 1/2. The expected utility of the gamble is 0.5u(x) + 0.5u(y) and this could be lower or bigger than the utility of the expected outcome u(0.5x + 0.5y) Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Risk Aversion

For t ∈ (0, 1), if u(tx + (1 − t)y) > tu(x) + (1 − t)u(y) we say the individual is risk averse. If u(tx + (1 − t)y) < tu(x) + (1 − t)u(y), the individual is risk loving. Finally, if u(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 terms of 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 risk aversion? Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Absolute Risk Aversion

The second derivative of the utility function is a natural candidate as it gives us an idea of the “curvature” of the utility function. The problem is that multiplicative transformations of the utility would yield different degrees of risk aversion. In order to avoid that, we normalize according to the first derivative. This is the Arrow Pratt Absolute Risk Aversion u00(w) r(w) = − u0(w) The concept of Absolute Risk is particularly useful for understanding the attitude towards risk of projects that imply absolute gains. In many economic applications, we are interested about risk attitudes when the losses or gains refer to a proportion of total income. In that case, the relevant measure would be Relative Risk Aversion wu00(w) ρ = − u0(w) Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Note that wealth-decreasing absolute risk aversion seems plausible. Decreasing relative risk aversion is not so obvious? Why? Topics Aggregation Evaluating Welfare Choice Under Uncertainty

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 changes income around w, that is locally around some wealth levels. Sometimes, we are interested in measures that of “global” risk aversion. The first way to formalize this is to say that an individually is globally more absolute-risk averse than other when his absolute risk aversion is higher than that of the other for every wealth level. Let A() and B() be the respective utility functions, A00(w) B00(w) − A0(w) > − B0(w) (2) Alternatively, one could think of A() utility being "more concave" is he’s more risk averse A(w) = G(B(w)) where G() is strictly concave. Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Local vs.Global Risk Aversion

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

A(w − πA(˜)) = E[A(w + ˜)] It sounds logical to say that A is (globally) more absolute-risk averse than B is πA(˜) > πB(˜) for all w. Which of the three should we choose? Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Local vs.Global Risk Aversion

It turns out that all these three are equivalent! Proposition:Pratt’s Theorem Let A(w) and B(w) be two differentiable, increasing and concave expected utility functions of wealth. Then the following properties are equivalent A00(w) B00(w) 1 − A0(w) > − B0(w) for all w 2 A(w) = G(B(w)) for some strictly concave G. 3 πA(˜) > πB(˜) for all random variables ˜ with E˜ = 0 Topics Aggregation Evaluating Welfare Choice Under Uncertainty

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 A0(w) = G0(B(w))B0(w) and A00(w) = G00(B(w))B0(w)2 + G0(B(w))B00(w) take the ratio of these two expressions, A00(w) G00(B)B0 G0(B)B00 A0(w) = G0(B) + G0(B)B0 Strict concavity of G together with positive marginal utility of G00(B)B0 B00 A00 wealth implies that G0(B) < 0 so that − B0 < − A0 . Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Proof: (2) ⇒ (3)

In order to prove this we will have to use a property of the expectation of concave functions. 1 Jensen’s Inequality: Let X be a non-degenerate random variable and f(X) be a strictly concave function of this random variable. Then Ef (X) < f (E(X)) Now, let prove that (2) implies (3).

def .π prop.(2) z}|{ z}|{ A(w − πA) = EA(w + ˜) = EG(B(w + ˜)) < G(EB(w + ˜)) = G(B(w − π )) |{z} |{z} B Jensen def .π < A(w − π ) (4) |{z} B prop.(2)

From these inequalities follows that πA > πB establishing the result. Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Proof: (3) ⇒ (1)

Fix ˜ and consider the family of random variables t˜ with t ∈ [0, 1]. Let π(t) be the risk premium as a function of t. Second order taylor expansion around 0: 1 π(t) ≈ π(0) + π0(0)t + π00(0)t2 (5) 2 The definition of π(t), A(w − π(t)) ≡ EA(w + t˜), implies that π(0) = 0 Differentiate twice the definition with respect to t, −A0(w − π(t))π0(t) = E[A0(w + t˜)˜] A00(w − π(t))π02(t) − A0(w − π(t))π00(t) = E[A00(w + t˜)˜2] Evaluating the first expression when t=0, −A0(w)π0(0) = E[A0(w)˜] = A0(w)E(˜) = 0 ⇒ π0(0) = 0 Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Proof: (3) ⇒ (1)

Evaluating the second expression at t=0, −A0(w)π00(0) = A00(w)E[˜2] Let σ denote the variance of ˜, σ2 = E(˜ − E˜)2 = E˜2. Then, 0 00 00 2 00 A00(w) 2 −A (w)π (0) = A (w)σ ⇒ π (0) = − A0(w) σ Use these results in the Taylor expansion, equation (5) above, 0 1 00 2 1 A00(w) 2 2 π(t) ≈ π(0) + π (0)t + 2 π (0)t = 0 + 0t − 2 A0(w) σ t But then for arbitrarily small t, we get that if A00 B00 πA > πB ⇒ − A0 > − B0 . And this is what proves this part. Since (1) ⇒ (2) ⇒ (3) ⇒ (1), the equivalence is established. Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Some Applications

Insurance Problem †. Comparative Statics of the Portfolio Problem † Asset Pricing† Topics Aggregation Evaluating Welfare Choice Under Uncertainty

Some Applications

Insurance Problem †. Assume a risk averse consumer has wealth W. There is a risk that he will lose an amount L. The consumer can buy insurance that will pay him q if the event occurs. Let π be the premium charged per unit of wealth insured so that if he takes insurance he would have to pay qπ, if p is the probability of incurring a loss, how much q would the consumer want to buy? If the problem is actuarilly fair, that is π = p, then a risk averse individual would choose to fully insured, i.e. q∗ = L