advanced microeconomics - the economics of uncertainty · j org lingens (wwu munster) advanced...

Some Basic Issues Comparing different Degrees of Riskiness

Tourguide

Introduction

General Remarks

Expected Utility Theory

Some Basic Issues

Comparing different Degrees of Riskiness

Attitudes towards Risk – Measuring Risk Aversion

Partial Equilibrium Models with Risk/Uncertainty

Optimal Household’s Behavior

The Firm’s Behavior in the Presence of Risk

General Equilibrium Models of Risk/Uncertainty

Risk Sharing within a Group – the Arrow-Lind Theorem

The endowment economy

The Production economy with many goods

Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 28 / 221


Measuring Riskiness

I Let us first of start with the question how to measure the riskiness of

different lotteries.

I Since this question is very important when thinking about lotteries it

has been the focus of statistical analysis for a long time.

I Consider different lotteries which are all characterized by the same set

of possible states of the world s, but that every state of the world has

a different realization probability.



I How would we rank these lotteries (or distributions) concerning their

riskiness?

I The first most intuitive measure for the riskiness of lotteries is

comparing their variance.

I A lottery with a higher variance would then be considered more risky

I The advantage of this measure are obvious: it is easy to calculate and

intuitive to understand.

I The downside is that it gives counterintuitive results.



I Consider the following situation: s ∈ 1, 2, 3, ., n are the different

states that an agent faces which generate payoff ys = y + zs .

I The probability of the different states is ps and the probabilities sum

to one.

I Let utility be denoted by u[ys ] which can be written as

u[ys ] = u[y ] + u′[y ]zs + 1/2u′′[y ]z2s + 1/3!u′′′[y ]z3

s + ...

I by means of a Taylor approximation around y .



I Now consider the expected utility of the household (which we have

shown to be the basis of the household’s behavior):

I Eu[ys ] = u[y ] + u′[y ]Ezs + 1/2u′′[y ]Ez2s + 1/3!u′′′[y ]Ez3

s + ...

I With this the problem becomes obvious.



I Consider that lotteries are ranked concerning their variance E[z2s ].

Then it would not be clear that riskier lottery (higher variance) would

also have lower expected utility.

I Thus, an individual (even a riskaverting individual!) would prefer the

’high’ risk lottery. The reason is that higher moments (skewness,

kurtosis..) play an important role in the ranking of lotteries.

I So a sensible ranking based on variance is only valid if a.) u′′′ and

higher derivatives are zero or b.) higher moments (third and higher)

are zero.

I This is an important qualification!



I Based on these shortcomings of the variance as a measure of the

riskiness of lotteries another concept has been invented: stochastic

dominance.

I This is a powerful concept which ensures that when comparing two

distributions (i.e. lotteries), the distribution with stochastic

dominance implies higher expected utility.

I Before we come to that, we have to define stochastic dominance.



I Consider some lottery that generates income Y ∈ [Ymin,YMax ] and is

distributed under some function (we consider a continuous

distribution of states!).

I For the ease of comparison we normalize the random variable

y = Y−YminYMax−YMin

such that y ∈ [0, 1].

I Take two lotteries with distribution function F 1 and F 2.



Stochastic Dominance [First Degree]

If F 2 ≤ F 1 for all y (with the inequality being strict for at least some y)

we say that distribution F 2 has stochastic dominance over distribution F 1.



I Intuitively we would say that the probability of getting an income of y

or less is smaller under F 2 than under F 1.

I Thus, getting a higher income is more probable under distribution 2

compared to 1.

I This implies that F 2 is somewhat better than F 1.



I This ’better’ can be formalized:

I Ei (y) =∫ 1

0 f iydy = [1F i (1)− 0F i (0)]−∫ 1

0 F idy =∫ 1

0 [1− F i (y)]dy

I Where the first manipulation follows from integration by parts:

[uv ] =∫uv ′ +

∫vu′ with u′ = f i



I Thus, it must be true that

E1(y)− E2(y) =∫ 1

0 [1− F 1[y ]]− [1− F 2[y ]]dy

I If F 2 stochastically dominates F 1 it will be true that the expected

income under lottery 2 will be larger than lottery 1.

I However, we were not talking about expected income but about

expected utility.

I Note that this also helps us when talking about expected utility.



I Let u(y) be the utility function of income and let us assume that

u′ > 0.

I Under lottery i , expected utility will be

Ei (u[y ]) =∫ 1

0 f iu[y ]dy = u[1]−∫ 1

0 u′[y ]F i [y ]dy

I Comparing lotteries 1 and 2 implies E1(u[y ])− E2(u[y ]) =

u[1]−∫ 1

0 u′[y ]F 1[y ]dy − u[1] +∫ 1

0 u′[y ]F 2[y ]dy < 0

I Thus, the lottery with stochastic dominance is preferred by utility

maximizing individuals.



I What we have shown is that (first-degree) stochastic dominance is a

rational measure of the riskiness of lotteries.

I The drawback, however, is that it is very demanding.

I Only very few comparisons are characterized by lotteries whose

probability distribution will be strict below that of all other lotteries.



I In standard comparisons the lotteries (or probability distributions)

which are involved cross at least once such that first degree stochastic

dominance cannot be applied.

I A second less demand concept measuring the riskiness is that of

second-degree stochastic dominance.

I In this concept not the absolute value of the probability distribution is

the focus of comparison, but the area under the distribution.

I Define T i (y) =∫ y

0 F (y)dy .



Stochastic Dominance [Second Degree]

If T 2 ≤ T 1 for all y (with the inequality being strict for at least some y)

we say that distribution F 2 has stochastic dominance (second degree) over

distribution F 1.



I Using very similar arguments as before we can show that a

distribution i which stochastically dominates some other distribution,

generates higher expected utility.

I We know that

E1(u[y ])− E2(u[y ]) = −∫ 1

0 u′[y ]F 1[y ]dy +∫ 1

0 u′[y ]F 2[y ]dy

I which we can write as

E1(u[y ])− E2(u[y ]) =

−((u′[1]T 1[1]− u′[0]T 1[0]) +

∫ 1

0u′′[y ]T 1[y ]dy)

+((u′[1]T 2[1]− u′[0]T 2[0]) +

∫ 1

0u′′[y ]T 2[y ]dy)



I Eventually, this is (note that T i [0] = 0):

E1(u[y ])− E2(u[y ]) =

u′(1)[T 2(1)− T 1(1)] +

∫ 1

0u′′[y ](T 1[y ]− T 2[y ])dy

I In the case that lottery 2 stochastically dominates (second order)

lottery 1, expected utility will be larger (with u′′ < 0).

I An individual with u′′ < 0 would hence prefer lottery 2 over lottery 1.



I A concept which is very close to that of second degree stochastic

dominance is that of a mean preserving spread (mps).

I Note that the concept of a mps is heavily applied in economics when

asking a model how the households behavior would change under a

more risky situation.

I The intuitive notion of a mps is that a lottery which is a mps of

another lottery has the same mean but more probability mass at the

tails (fat tails).



I Formalizing this notion implies the following definition: distribution

F 1[y ] is a mps of F 2[y ] if

I distribution 2 stochastically dominates 1 i.e. T 2[y ] ≤ T 1[y ] and

I T 1[1] = T 2[1]

I where the last equality implies that the means of the two distributions

are identical.


Some Basic Issues Attitudes towards Risk – Measuring Risk Aversion

Tourguide

Introduction

General Remarks

Expected Utility Theory

Some Basic Issues

Comparing different Degrees of Riskiness

Attitudes towards Risk – Measuring Risk Aversion

Partial Equilibrium Models with Risk/Uncertainty

Optimal Household’s Behavior

The Firm’s Behavior in the Presence of Risk

General Equilibrium Models of Risk/Uncertainty

Risk Sharing within a Group – the Arrow-Lind Theorem

The endowment economy

The Production economy with many goods



Measuring Risk Aversion

I We have shown how to measure the riskiness of different lotteries.

I All these measures have basically been based on properties of the

probability distribution of the lotteries.

I However, the second degree stochastic dominance also has shown

that the property of the utility function seems to be relevant, too

I Thus, in a second step we are going to discuss the households

attitudes towards risk which is based on the preferences of households.



I Consider a household who is characterized by some utility function

u[y ] where y denotes (monetary) income.

I For the ease of exposition consider that the household faces a simple

lottery with state space {y1, y2} and probabilities {p1, p2}.

I The average income of the household is hence y = p1y1 + p2y2 which

generates utility u[y ].

I Average utility is given by p1u[y1] + p2u[y2] which in general is not

equal to u[y ].



I To classify the households attitude towards risk, we ask how much

money we would have to give to the household such that he is

indifferent between the safe income and the lottery.

I Answer: u[yc ] = p1u[y1] + p2u[y2], where yc denotes the certainty

equivalent (of the specific lottery at hands!).

I Basically, yc is a measure of the ’value’ of the lottery.



I How can we classify the household? We just compare the average

income of the lottery y with the ’value’ of the lottery (attached by

the household).

I If the value was lower (higher) than average income the household is

labeled risk averse (risk loving).

I In the risk averse case the household would be willing to sacrifice

income to ’buy’ himself out of the lottery.



Thus, we can distinguish three scenarios

I yc < y : risk averse household

I yc = y risk neutral household

I yc > y risk loving household



I Using this we can define the risk premium r := y − yc .

I The risk premium is the amount of money that the household is

willing to sacrifice in order to get out of the lottery

I or (equivalently) the money we would have to pay to the household to

give up a safe income yc and participate in the lottery.



I The risk premium r is endogenous and a function of a) the utility

function of the household and b) of the form of the lottery.

I We would like to disentangle these effects on the risk premium in

order to understand the channels which shape r .

I Consider the following lottery: an individual with income/wealth y

receives (with prob. ps) a state dependent income flow of zs .

I Thus, state dependent income is ys = y + zs . Note that∑Ss=1 pszs = 0, i.e. average income is y .



I By definition it is true that u[yc ] = Eu[ys ] which is

u[y − r ] = Eu[y + zs ].

I Using a Taylor linearization (=approximation), we can write for the

left hand side u[y − r ] ' u[y ]− ru′[y ].

I With this we could derive an explicit expression for r . However, the

utility and lottery effects would remain hidden in E(u[y + zs ])



I Thus, we use another Taylor linearization for the right-hand side and

write

E(u[y + zs ]) ' u[y ] + E(zs)u′[y ] + 0.5E(zs)2u′′[y ]

I In this lottery it will be true that E(zs) = 0 and E(zs)2 denotes the

variance of the lottery.



I With both approximations, we are in a position to derive an explicit r

in which the lottery and the utility effect are disentangled

u[y ]− ru′[y ] ' u[y ] + 0.5E(zs)2u′′[y ]

r ' 0.5E(zs)2−u′′[y ]

u′[y ]

I where −u′′[y ]

u′[y ] is called the Arrow-Pratt coefficient of absolute risk

aversion.



I With the expression for r we have a direct interpretation/intuition

what absolute risk aversion implies.

I The coefficient of absolute risk aversion shows how much money the

household is willing to sacrifice to avoid a lottery with variance 2.

I Note that this is not only a function of the functional form of the

utility function, but also a function of the initial endowment of the

household.


advanced microeconomics - the economics of uncertainty · j org lingens (wwu munster) advanced...

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