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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
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 29 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 30 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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 .
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 31 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 32 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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!
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 33 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 34 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 35 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 36 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 37 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 38 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 39 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 40 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 41 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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 .
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 42 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 43 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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)
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 44 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 45 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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).
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 46 / 221
Some Basic Issues Comparing different Degrees of Riskiness
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 47 / 221
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
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 48 / 221
Some Basic Issues Attitudes towards Risk – Measuring Risk Aversion
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 49 / 221
Some Basic Issues Attitudes towards Risk – Measuring Risk Aversion
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 ].
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 50 / 221
Some Basic Issues Attitudes towards Risk – Measuring Risk Aversion
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 51 / 221
Some Basic Issues Attitudes towards Risk – Measuring Risk Aversion
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 52 / 221
Some Basic Issues Attitudes towards Risk – Measuring Risk Aversion
Thus, we can distinguish three scenarios
I yc < y : risk averse household
I yc = y risk neutral household
I yc > y risk loving household
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 53 / 221
Some Basic Issues Attitudes towards Risk – Measuring Risk Aversion
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 54 / 221
Some Basic Issues Attitudes towards Risk – Measuring Risk Aversion
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 .
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 55 / 221
Some Basic Issues Attitudes towards Risk – Measuring Risk Aversion
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 ])
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 56 / 221
Some Basic Issues Attitudes towards Risk – Measuring Risk Aversion
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 57 / 221
Some Basic Issues Attitudes towards Risk – Measuring Risk Aversion
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.
Jorg Lingens (WWU Munster) Advanced Microeconomics October 8, 2012 58 / 221
Some Basic Issues Attitudes towards Risk – Measuring 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.
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