ECOS3022 Tutorials 4 - Expected Utility
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ECOS3022 Tutorials
Tutorial 4 - Expected Utility
Overview
• Last week we looked at how we construct BCs in the asset economy with uncertainty that we study.
• This week we look at the objective that agents maximise when they optimise their financial decisions under uncertainty.
• Risk aversity
• Q1,3,4 are briefly discussed, but the focus is on Q2,5
Q1 - Von Neumann-Morgenstern utility
The takeaway from this question is we need to understand when the expected utility repre- sentation makes sense. That is, when the utility can be written as
V(π,x) =工 πsv(xs )
s
we must have the following conditions
• State independence
• Only the final result matters, not how we get that result
• Irrelevance of common alternatives
Then, if the expected payoff is positive and larger than your risk premium you will take on the risk.
Q2 - Risk Aversity and Wealth
We have that state utility is v(x) = lnx and there are two assets, the risk free asset always pays out 1 and a risky asset that pays Rh with probability p and Rl < Rh with probability 1 − p. Both assets cost 1, the agent has ω wealth and we assume there is no arbitrage. We let a be the amount the agent demands of the risky asset.
(a) We want a(ω), that is, we want the amount she invests in the risky asset as a function
of wealth. We have our expected utility
U =工 πiv(x) = pln(aRh + (ω − a)) + (1 − p)ln(aRl + (ω − a))
i
We can then get our F.O.C
U\ = p + (1 − p) = 0
which we can solve for our optimal a
(1 − p)(Rl − 1) + p(Rh − 1)
(Rl − 1)(Rh − 1)
you may have noticed that if we wanted we equivalently could have constructed a BC here and used the Lagrangian.
What happens if ω increases? In order to answer such a question we need to know if the fraction in our expression for a is positive or negative. Intuitively, arbitrage would be possible if the risky asset paid either above 1 or below 1 in both states. We know we must have Rh > 1 and Rl < 1
We can actually use the law of one price to confirm this. We have r = [R(R)l(h)
q = (1, 1). Next we know
1(1)] and
αr = q =⇒ α = qr−1 = (11) [R(R)l(l) R(R)R(R)l(l)] = ( , )
Next, if we use the fact that α 1 ,α2 > 0 we can say that
0 < Rl < 1
Rh > 1
If we now look at our expression for a(ω), we now have that the denominator is negative
(1 − p)(Rl − 1) + p(Rh − 1)
(Rl − 1)(Rh − 1)
the numerator is the expected value of the asset which we know must be positive for a risk averse investor to want to invest. Thus if the individual does invest in the risky asset, the fraction must be negative and the amount she invests increases with wealth.
(b) What if we change the utility to v(x) = −e−x . We can do the same process as before
to find an expression for the optimal level of investment
1 (1 − p)(1 − Rl )
We can see that a is independent of wealth ω
(c) Do we have a measure that can explain these results and explain why some individuals change their investment with wealth while other don’t. We have the the measure of Absolute Risk Aversion
v\\ (x)
v\ (x)
Let’s calculate it for the investor in part (a)
v(x) = lnx,v\ (x) = ,v\\ (x) = − =⇒ A(x) =
so risk aversion decreases with wealth, this makes sense she invests more as she gets more wealth. When A\ (x) < 0 we say she has DARA (decreasing absolute risk aver- sion).
Similarly for the investor in part (b) we can find
A(x) = 1
Her risk aversion does not change with wealth, which again makes sense in that her investments do not depend on wealth. We say she has CARA (constant absolute risk aversion)
Q3 - Allais Paradox - Please do in your own time
The idea is that when the axioms of the expected utility representation are violated, you can find inconsistencies in the results. This is mostly for interest so is not focused on here.
Q4 - Indifference curves
When we have expected utility representations, how can we expect indifference curves to look when we say know they are risk averse. Suppose we have a risk averse individual with the following expected utility
U =工 πsu(xs )
s
let us fix a single iso-utility and look at two states with p being the probability of state 1
pu\ (x1 ) = pu(x1 ) + (1 − p)u(x2 ) =⇒ MRS = −
as you increase your demand for good 1 risk aversity means u\ (x1 ) decreases and so the iso- utility flattens. Similarly, as you increase x2 , u\ (x2 ) decreases and the iso-utility steepens. If you are risk neutral, then your iso-utility will be a straight line. See the worked solutions for a specific example.
Q5 - Prudence and more on risk aversion
Note: Please try and follow the proofs and understand why each step is taken. We do not expect you to be able to reproduce the proofs, but we do expect you to understand the concepts used within them.
There are two periods of consumption. Let the initial wealth be ω and let the utility be U = u(c1 ) + v(c2 ) where u and v are both concave i.e. risk averse. Consumption in the first period is wealth less savings, c1 = ω − x. When there is no uncertainty second period consumption is then c2 = x. Let the optimal level of savings be x0 .
Now introduce uncertainty: In the second period she consumes the amount she saved x plus y which is distributed according to π . The problem is then given by
max u(ω − x) + E[v(x + y)]
x
Let the optimal level of savings when there is uncertainty be given by x∗
We also have a measure of prudence for a utility v which is given by
v\\\ (x)
v\\ (x)
We say the individual has prudent behaviour is P(ω) > 0.
(a) Claim:
A\ (ω) < 0 =⇒ v\\\ (ω) > 0 =⇒ P(ω) > 0
Let’s first start by finding A\ (ω)
A(ω) = − =⇒ A\ (ω) = − = ( − + )
Now, what let’s see what is implied by our assumption A\ (ω) < 0. We first note that < 0 as v\\ < 0 by risk aversity.
( − + ) < 0 =⇒ − + > 0 =⇒ v\\\ > > 0
In the last implication, the direction of inequality changes because we are multiplying both sides by v\\ which is less than zero. Next, let’s look at P(ω). v\\\ is positive by what we found above and v\\ is negative by risk aversity. Hence, DARA (A\ (ω) < 0) implies prudence.
v\\\ (ω)
P(ω) = −
(b) Claim:
E[v\ (x0 + y)] > v\ (x0 ) =⇒ x∗ > x0
Recall that x0 is the optimal savings level when there is no uncertainty in the second period and x∗ is the optimal savings when there is uncertainty.
Let’s first find x0 :
max u(ω − x) + v(x)
F.O.C
u\ (ω − x0 ) = v\ (x0 )
Now let’s find x∗ :
max ϕ(x) = u(ω − x) + E[v(x + y)]
F.O.C
ϕ\ (x∗ ) = 0
ϕ\ (x) is decreasing in x due to concavity, so if ϕ\ (x) > 0 then that implies x < x∗ . This is effectively how we will prove that x0 < x∗ .
So we need to prove that ϕ\ (x0 ) > 0. Remember our assumption is that E[v\ (x0 +y)] > v\ (x0 ).
ϕ\ (x0 ) = −u\ (ω − x0 ) + E[v\ (x0 + y)] = −v\ (x0 ) + E[v\ (x0 + y)] > 0
(F.O.C for x0 ) (assumption)
Then, ϕ\ (x0 ) > 0 implies that x0 < x∗ by what we proved earlier.
(c) Claim: If we have v\\\ > 0 and E[y] = 0 then we have E[v\ (x + y)] > v\ (x) Ax
v\\\ > 0 means that v\ is convex. So someone who’s utility is given by v\ is risk seeking.
As a risk seeker you prefer playing the game to receiving expected value, that is E[v\ (x + y)] > v\ (E[x + y])
Then, as E[y] = 0 we have that E[x + y] = E[x] = x and thus
E[v\ (x + y)] > v\ (x)
as required.
What is done in the worked solutions? They effectively do the same thing by defining η = −v\ and thus someone with η as a utility is risk averse, that is
E[η(x + y)] < η(x) =⇒ E[v\ (x + y)] > v\ (x)
2022-09-19