Expected Utility Solutions
1. Exercise 4.2 from Lengwiler.Consider a binary risk: you will either win H with probability (e.g., the jackpot of the national lottery), or lose L (the price of participating in the lottery) with probability 1 . You can expose yourself in a continuous manner to this risk, meaning that you can buy x tickets. As- sume that you have preferences that can be represented with a risk-averse von NeumannñMorgenstern utility function. Prove that you will take some of this risk (x > 0) if the expected payo§, H (1 ) L > 0.
The solution in the book is a bit terse, I add some missing steps of the derivation here. The agent either stands to lose tL with probability 1 or gain tH with probability : Her choice is t: Her expected payo§
EU =(1 )v(w tL)+v(w+tH) !max: t
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v is a concave function, hence EU is a concave function of t: The Örst order condition L(1 )v0(w tL)+Hv0(w+tH) = 0
L(1 )v0(w tL) = Hv0(w+tH) crossmultiply L(1 )v0(w tL) = HsubtractL(1 ) onbothsides
v0 (w + tH)
v0(w tL) 1L(1 ) = H L(1 )
v0 (w + tH)
The rest of the discussion is in the textbook.
2. Consider the following ìportfolio choiceî problem. The investor has initial wealth w and utility v(x) = ln x: There is a safe asset that has a payo§ of 1 in both states. There is also a risky asset with a random payo§, it pays Rh with probability p and Rl < Rh with probability 1 p. Let a be the amount invested in the risky asset, so that w a is invested in the safe asset. Both assets cost 1, i.e. q = (1; 1) : Rh and Rl are such that there is no arbitrage.
(a) Find a as a function of w. Does the investor put more or less of his portfolio into the risky asset as his wealth increases?
(b) Another investor has the utility function v(x) = e x. How does her investment in the risky asset change with wealth?
(c) Find the coe¢ cients of absolute risk aversion A(x) = v00 (x) for the two v0 (x)
investors. How do they depend on wealth? How does this account for the qualitative di§erence in the answers you obtain in parts a: and b:?
Solution. a. If the investor allocates a to the risky asset and hence w a to the safe asset her payo§is aRh +w a in state h and aRl +w a in state l: Her expected utility
EU(a)=pv(aRh +w a)+(1 p)v(aRl +w a):
She maximizes that with respect to a: Assume v (x) = ln (x) the Örst order condition
p Rh 1 + (1 p) Rl 1 = 0 a(Rh 1)+w a(Rl 1)+w
We need to solve this for a as a function of w
p Rh 1 =(1 p) 1 Rl crossmultiply
a(Rh 1)+w a(Rl 1)+w
pa(Rh 1)(Rl 1)+p(Rh 1)w=(1 p)a(1 Rl)(Rh 1)+(1 p)(1 Rl)w
change the sign on the right to collect the common terms
pa(Rh 1)(Rl 1)+p(Rh 1)w=(p 1)a(Rl 1)(Rh 1)+(1 p)(1 Rl)w a(Rl 1)(Rh 1)=(1 p)(1 Rl)w p(Rh 1)w
a= w(1 p)(Rl 1)+p(Rh 1) (1) (Rl 1)(Rh 1)
To understand the sign of dependence of a on w; lets invoke the no arbitrage conditions. The investor faces a market with payo§ matrix
r = h ;thenr 1= Rh Rl Rh 1 Rl Rh R R R R
with q = (1;1) the Arrow security prices are 1 Rl ; Rh 1 Rh Rl Rh Rl
This implies Rl < 1; Rh > 1; hence the denominator in (1) is negative. If (1 p) (Rl 1) + p (Rh 1) < 0 the investor only invests into the safe asset. Hence we assume (1 p)(Rl 1)+p(Rh 1) > 0: Call this Assumption 1. But then a increases with w:
b. With v (x) = e x; v0 (x) = e x; hence the Örst order condition for EU(a)=pv(aRh +w a)+(1 p)v(aRl +w a):
with v (x) = exp (x) the Örst order condition is
p(Rh 1)exp(a w aRh)+(1 p)(Rl 1)exp(a w aRl)=0
Now we need to solve this for a as a function of w:
p (Rh 1) e (aRh+w a) = (1 p) (1 Rl) e (aRl+w a) cross multiply
e (aRh+w a)e(aRl+w a) = (1 p) (1 Rl) collect and take ln p(Rh 1)
a(Rl Rh) = ln(1 p)(1 Rl) p(Rh 1)
By the no arbitrage condition the inside of the ln is positive. By Assumption 1 the inside of the ln is also less than 1. Hence the right hand side is negative, so is (Rl Rh) : Hence a is positive but independent of w:
c. Forv(x)
hence A (w) Forv(x) hence A (w)
As we see a DARA investor increases his investment into risky assets when her wealth increases, a CARA investor does not, when her w goes up she buys more riskless bonds. The last four slides prove this point more generally and properly.
3. Consider the following lotteries on the outcomes {10,2,0}
p = (0:00; 1:00; 0:00); q = (0:10; 0:89; 0:01); r = (0:10; 0:00; 0:90); s = (0:00; 0:11; 0:89):
(a) Show that there are lotteries on the outcomes {10,2,0}, say x and y, and a number 2 [0; 1] such that
p = p+(1 )p; q= x+(1 )p r = x+(1 )y; s= p+(1 )y
(b) Show that an agent who satisÖes the expected utility hypothesis will rank these lotteries as follows:
p q () s r:
There is a separate Öle with the solution posted. The notes here help
to place this question in the right perspective.
= ln(x); v0(x)= 1; v00(x)= 1; x x2
= v00 (w) = 1 ; i.e. risk aversion decreases with wealth (DARA) v0 (w) w
= e x; v0 (x) = e x; v00 (x) = e x v00 (w) e w
= v0 (w) = e w = 1, risk aversion is independent of wealth (CARA)
1. Note that lotteries p; q; r; s are exactly those that appear on slides 13 and 14 on the Allais Paradox. According to this paradox the Experimental subjects reveal
p q BUT for the same people r s:
According to b: (notice the di§erence in the preferences) this means that the preferences of these experimental subjects DO NOT satisfy the expected utility hy- pothesis.
2. On the other hand the theoretical result by von Neuman and Morgenstern, on page 12 of the slides, states that if the preferences of the agents satisfy Axioms 1, 2, 3 listed on page 12 of the slides then their preferences MUST satisfy the expected utility hypothesis.
3. The only way to reconcile 1 and 2 above is to realize that the preferences of the experimental subjects in 1 fail to satisfy one (or more) of Axioms 1, 2, 3. This question helps to recover which one.
In a: x = (x1; x2; x3) and y = (y1; y2; y3) are the probabilities attached to the same outcomes {10,2,0}.
q= x+(1 )p implies ( and x are the choices) 0:1 = x1
0:89= x2+1 0:01 = x3
x+(1 )y implies ( ;x and y are the choices) 0= x2+(1 )y2
0:1 = x1+(1 )y1
0:9 = x3+(1 )y3
s = p+(1 )y implies ( and y are the choices) 0 = (1 )y1
0:11 = 0:89 =
+(1 )y2 (1 )y3
This system of 9 equations in 7 unknowns has a solution with > 0, see the uploaded Öle.
b. If the expected utility hypothesis holds and p q then there exists function v and three numbers v (10) ; v (2) ; v (0) such that
v (2) 0:1 v (10) + 0:89 v (2) + 0:01 v (0) 4
If the expected utility hypothesis holds and s r then for the same three numbers v(10); v(2); v(0)
0:11 v (2) + 0:89 v (0) 0:1 v (10) + 0:9 v (0)
Cancel the weights on v (0), add 0:89 v (2) to both sides and you have the same inequality as the one above for p q: This means that preferences over lotteries p q and s r are compatible with the expected utility hypothesis. In the Allais paradox the preferences are p q and s r; hence you cannot Önd the numbers v (10) ; v (2) ; v (0) to satisfy the corresponding inequalities simultaneously.
Given our work in a:
s r is equivalent to p+(1 )y x+(1 )y
The independence of the irrelevant alternatives allows removing the common term (1 ) y on both sides of this preference relation. Hence
px=) p xforany >0
The independence of the irrelevant alternatives also allows adding the common
term (1 ) p to both sides of this preference relation. Hence p= p+(1 )p x+(1 )p=q:
Thus the failure of the independence of the irrelevant alternatives axiom can lead to the preferences as in the Allais Paradox.
4. Consider the expected utility function of a risk-averse decision-maker:
s u(xs) s=1
For S = 2 and given , draw the indi§erence curves of the expected utility function in a state-contingent outcome diagram.
Solution. Fix the indi§erence curve
1 u ( x 1 ) + 2 u ( x 2 ) = u
Calculate the MRS as usual
MRS (x1; x2) = 1 u0 (x1): 2 u0 (x2)
Notethatonthecertaintylinex2 =x1 =x;MRS(x;x)= 1:Notealsothat 2
for the risk neutral agent with u (x) = x; MRS (x1; x2) = 1 at every point of the 2
indi§erence curve. Hence the risk-neutral agentís indi§erence curve is linear with the slope 1 : In general, the indi§erence curve is downward sloping and convex due to
the risk-aversion, i.e. u (x) < 0: Consider u (x) = ln (x) and 1 = p: Then
MRS(x1;x2)= p x2: 1 px1
The (x1; x2) here are not arbitrary, they satisfy
plnx1 +(1 p)lnx2 = u or equivalently
xpx1 p = Const = v 12
5. In a two-period economy, a consumer has Örst-period initial wealth w. The consumerís utility level is given by
u (c1; c2) = u (c1) + v (c2) ;
where u and v are concave functions and c1 and c2 denote consumption levels in the Örst and the second period, respectively. Denote by x the amount saved bytheconsumerintheÖrstperiod(sothatc1 =w xandc2 =x),andletx0 be the optimal value of x in this problem.
We now introduce uncertainty in this economy. If the consumer saves x in the Örst period, his wealth in the second period is given by x + y, where y is distributed according to . In what follows, E[] always denotes the expectation with respect to . Assume that the vNM utility function over realized wealth levels in the two periods (w1; w2) is u(w1) + v(w2). Hence, the consumer now solves
max u (w x) + E [v (x + y)] : x
Denote the solution to this problem by x.
(a) Show that if the coe¢ cient of absolute risk aversion of v (w), A (w) ; is decreasing with wealth w, then v000 (w) > 0; that is P (w) > 0. Hence DARA implies prudent behaviour.
(b) Show that if E[v0(x0 + y)] > v0(x0), then x > x0:
(c) Show that if v000 > 0, and E[y] = 0, then E[v0(x + y)] > v0(x) for all values
of x. Solutions. a:
A(w) = v00 (w); P (w) = v000 (w) v0 (w) v00 (w)
A (w) decreasing implies
0 v000 v0 (v00)2 v00 v000 v00
Aw= v0(w)2 =v0 v00+v0 <0
The Örst multiplier is negative by risk-aversion, then the second must be positive.
multiply both sides by v00 which is negative
000 (v00)2
max u (w x) + v (x) x
v00 > v000 v0 v00
H e n c e A 0w < 0 i m p l i e s P ( w ) > 0 : b: With no uncertainty
determines the optimal savings
u0 (w x0) = v0 (x0) : Now letís have uncertainty over the future. Let
(x)=u(w x)+Ey [v(x+y)]!max x
the Örst order condition
0(x)= u0(w x)+Ey[v0(x+y)] attheoptimum 0(x)=0:
00 (x) = u00 (w x) + Ey [v00 (x + y)] < 0 for any x by concavity of u and v:
Hence 0 (x) is decreasing in x; this implies that if 0 (x) > 0 then x < x: At x0
Eyv0 (x0 + y) > v0 (x0)
0 (x0) = u0 (w x0) + Ey [v0 (x0 + y)] = v0 (x0) + Ey [v0 (x0 + y)] > 0:
This proves b:
c:Let= v0 andletv000 >0;then00 <0henceisaconcavefunction. If
E[y] = 0 then E[(x+y)] < E[(x)] by concavity of and then E[v0 (x+y)] > E [v0 (x)] for any x:
by assumption in b: Hence
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