answers to problem set 3
TRANSCRIPT
Spring 2007 John RustEconomics 425 University of Maryland
Answers to Problem Set 3
0. Jack has a utility function for two perfectly divisible goods, x and y. Jack’s utility function isu(x,y) = (x + y)2. Derive Jack’s demand function for the two goods as a function of px (the price ofgood x), py (the price of good y), and I, (Jack’s total income to be allocated to the 2 goods).
Answer As I showed in class the answer to this is easy if you perform a montonic transformation, f (u) =√u to get v(x,y) = f (u(x,y)) =
√
u(x,y) =√
(x+ y)2 = x + y. The transformed utility functionhas constant marginal utility for both goods, with marginal utility equal to 1 for both goods. Thus,the indifference curves for this consumer are straight lines with a slope of −1. You can see thissince the slope of the indifference curve is the negative of the ratio of the marginal utilities ofthe two good, which are both 1 in this case. Or you can solve for an indifference curve directly,treating the consumption of good y as an implicit function of the consumption of good x, i.e. tosolve for y(x) in the equation
u(x,y(x)) = u = x+ y(x) (1)
so we see thaty(x) = u− x (2)
which is a straight line with a slope of −1, as claimed. Then, with such indifference curves, itis easy to see that the demand for the two goods has a “bang-bang” type of solution, i.e. Jackis almost always at a “corner solution” where he/she consumes either entirely x or entirely ydepending on which one is cheaper. Thus, if py > px, then the slope of the budget line I = pxxpyyis −px/py > −1 (i.e. less steep than the indifference curves), so that Jack will get the highestutility by spending his entire budget on x and consume none of y as in figure 1 below.We can see from figure 1 that when py > px, it is better for Jack to only buy good x, and so withan income of 10, Jack can afford 10 units of x. If py < px, then we have the opposite situationwhere the budget line is steeper than Jack’s indifference curves and so Jack only buys good y andnone of good x. The “”knife-edge” case occurs when px = py. Then the indifference curve andthe budget line both have the same slope, −1. In this knife-edge case, any combination of x and yon the budget line is utility maximizing, and Jack is indifferent about consuming from any pointon the budget line.Writing all of this down mathematically we have
x(px, py, I) =
I/px if py > px
0 if py < px
[0, I/px] if py = px
y(px, py, I) =
0 if py > px
I/py if py < px
[0, I/py] if py = px
(3)
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Figure 1: Jack’s utility maximization solution when I = 10, px = 1 and py = 2
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
Quantity of Good x
Qua
ntity
of G
ood
y
Jacks demand for x and y when I=10 and px=1 and p
y=2
Indifference curveIndifference curveBudget line
In the demand equations above, the notation y(px, py, I) = [0, I/px] denotes the knife-edge casewhere y could be any value between 0 and the number I/py, which happens when Jack spends hiswhole budget on good y. In the knife edge case, if Jack consumes y units in this interval, then tosatisfy his budget constraint, his consumption of x is given by x = (I− pyy)/px, i.e. he spends therest of his budget on x.It is possible to arrive at the same conclusion by writing down the Lagrangian for this problem asI did in class. We have
L(x,y,λ) = (x+ y)+λ(I− pxx− pyy) (4)
Taking the first order conditions (for maximization) with respect to x and y we have
∂L∂x
(x∗,y∗,λ∗) = 1−λ∗px ≤ 0
∂L∂y
(x∗,y∗,λ∗) = 1−λ∗py (5)
The inequalties reflect the possibility of corner solutions: if x∗ = 0, (a corner solution for x),then we have ∂L(x∗,y∗,λ∗) ≤ 0. However if x∗ > 0 (and interior solution for x∗), then we have∂L(x∗,y∗,λ∗) = 0. Suppose that px < py. Then we conjecture that x∗ > 0, and solving the firstfirst order condition (for an interior solution for x∗), ∂L(x∗,y∗,λ∗) = 0, we deduce that λ∗ = 1/px,i.e. the marginal utility of income in this case is 1/px. From the other first order condition, wehave
∂L∂y
(x∗,y∗,λ∗) = 1−λ∗py = 1− py
px< 0 (6)
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which implies that we have a corner solution for y∗, i.e. y∗ = 0. Using the budget equation, itfollows that when y∗ = 0 we have x∗ = I/px (i.e. Jack spends his entire budget on good x, nothingon y), and this is the same solution as we got above. Thus, by following through the various cases,we can see that solving the Lagrangian problem gives us the same solution, i.e. the same demandfunctions for goods x and y as we derived above by intuitive means.
1. Provide an example of a utility function that leads to at least one good being an inferior good. Providea general proof that all goods cannot be inferior goods.
Answer Consider the utility function u(x1,x2) = x1/21 + x3/2
2 . The Lagrangian for the utility maximizationproblem is
L(x1,x2,λ) = x1/21 + x3/2
2 +λ(y− p1xx − p2x2) (7)
The first order conditions for the maximization of the Lagrangian with respect to x1 and x2 are
∂L∂x1
(x∗1,x∗2,λ
∗) =12 [x∗1]
−1/2 −λ∗p1 = 0
∂L∂x2
(x∗1,x∗2,λ
∗) =32[x∗2]
1/2 −λ∗p2 = 0 (8)
Soving these two equations we get
x∗1 =1
4λ2 p21
x∗2 =49λ2 p2
2 (9)
Substituting the above solutions into the budget constraint, y = p1x∗1 + p2x∗2 and solving for λ∗ weget
14λ2 p1
+49
λ2 p32 = y (10)
Multiply both sides of this equation by λ2 and rearrange terms to get
49 p3
2λ4 − yλ2 +1
4p22
= 0 (11)
Let γ ≡ λ2. Then we can rewrite the equation above as a quadratic equation for γ
aγ2 +bγ+ c = 0 (12)
where a = 49 p3
2, b = −y and c = 14p2
2. Recall the quadratic formula: there are two solutions for γ
from the equation above. They are
γ =−b±
√b2 −4ac
2a(13)
Plugging in the formulas for a, b and c above we get
λ ≡√γ =
√
√
√
√
y±√
y2 − 49
p32
p189 p3
2(14)
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Regardless of whether we take the + or − root in the formula above, λ∗ is an increasing functionof y, as we would expect. But since x∗1 = 1
4λ2 p21, it follows that x∗1 is a decreasing function of y, i.e.
x1 is an inferior good.
Another example of a utility function that leads to an inferior good is
u(x1,x2) = log(x1)+ ex2 (15)
Guess which of the two goods is inferior. Yes, you’re right, it is x1, the good with the diminishingmarginal utility. The other good, x2 has increasing marginal utility, just like the previous example.Setting up the Lagrangian as in the previous case and taking first order conditions, you should beable to show that
x∗1 =1
λp1
ex∗2 = λp2 (16)
So we just need to show that λ is an increasing function of y in this case and we will have anotherexample of a utility function for which one of the goods is an inferior good.Plugging into the budget constraint to solve for λ∗ we get
y = p1x∗1 + p2x∗2 =1λ
+ p2 log(λp2) (17)
Rearranging this equation, we get
λ = exp{
y− p2 log(p2)− 1λ
p2
}
(18)
Now, by totally differentiating the above equation (more exactly, using the implicit function the-orem), we get
∂λ∂y
=A
1+A> 0 (19)
where
A =1p2
exp{
y− p2 log(p2)− 1λ
p2
}
(20)
Thus, we see that λ is an increasing function of y, so that x∗1 = 1λp1
is a decreasing function of yand is thus and inferior good.
2. Define mathematically what a homothetic utility function is.
a. Show that homothetic utility functions lead to demand functions that are linear in income, y.
Answer A homothetic function is defined to be a monotonic transformation of a homogeneous of degree1 function. Thus, a utility function is homothetic if it can be represented as
u(x) = f (l(x)) (21)
where f : R → R is a monotonic increasing function (i.e. f ′(u) > 0 for all u ∈ R) and l : Rn → Ris a linearly homogenous function of the vector x (i.e. for any positive scalar λ > 0 we have
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l(λx) = λl(x)). Since we showed in class that the gradient of a linearly homogeneous function ishomogenous of degree zero, this implies that the slopes of the indifference curves of a homotheticutility function are the same on any ray through the origin. To see this, recall that the slope of anindifference curve — the marginal rate of substitution between two goods x i and x j — is give by
MRSi j = −∂u∂xi
(x)∂u∂x j
(x)= −
f ′(l(x)) ∂l∂xi
(x)
f ′(l(x)) ∂l∂x j
(x)= −
∂l∂xi
(x)∂l
∂x j(x)
(22)
But since ∂l∂x j
is homogenous of degree 0 we have for any λ > 0
∂l∂x j
(λx) =∂l∂x j
(x) (23)
This implies that the MRSi j is constant at any point on the ray throught the origin, R ={y ∈ R|y = λx for some λ > 0}.
This property provides the intuitive basis for the argument that a homothetic utility function has ademand function of the form
x(p,y) = x(p,1)y (24)where x(p,1) is the demand for the good when y = 1. The reason that this is the case is that ifx(p,1) maximizes the utility function when y = 1, this means that the MRSi j equals the price ratio− pi
p jfor every pair of goods {i, j}. But then if we consider demand for some other income y, we
know that as long as the prices are fixed, the budget “hyperplane” for income y will be parallelto the budget hyperplane (or budget line in the two good case) when y = 1. But the property ofhomothetic functions implies that the MRSi j is constant along any ray through the origin, thisimplies that by scaling up the demand bundle x(p,1) which makes the budget plane tangent tothe indifference surface when y = 1, then by scaling x(p,1) by the factor y, the bundle x(p,1)y(which is on a ray from the origin that contains the point x(p,1)) will be optimal at income y. Itis also budget-feasible since x(p,1) is budget-feasible at income y = 1, i.e. p ′x(p,1) = 1 so thenwe must have y = p′x(p,1)∗ y = 1∗ y, so that x(p,1)∗ y is also budget-feasible at income y.To prove this rigorously, we use a proof by contradiction. Let x(p,1) maximize the utility functionwhen y = 1. That is,
x(p,1) = argmaxx
l(x) subject to: p′x ≤ 1 (25)
Notice that we have used the property of homotheticity and the fact that the solution to a utilitymaximization problem is unchanged if you take a monotonic transformation of the utility func-tion. So we have conveniently chosen a transformation to make the utility function u(x) a linearhomogeneous utility function l(x).Now, we claim that for any y 6= 1, the utility maximizing bundle must be x(p,y) = x(p,1) ∗ y.Suppose this is not the case. Then there is some bundle x̂ which is budget-feasible, i.e. p ′x̂ ≤ yand for which we have l(x̂) > l(x(p,1)∗ y). However since l is linear homogeneous we have
l(x̂/y) > l(x(p,1)) (26)
Notice that if p′x̂ ≤ y then p′x̂/y ≤ 1 so the bundle x̂/y is budget-feasible for an income of y =1. Then the equation above tells us that this alternative bundle produces higher utility than thesupposed utility maximizing bundle x(p,1). This is a contradiction.
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b. Is the converse true, i.e. if a utility function leads to demand functions that are linear in income,must the utility function be homothetic? Provide a proof if you answer yes, or a counterexampleif you answer no.
Answer The converse is true. Once again we will prove this by contradiction, but first let’s understand theintuition. If the demand function has the form
x(p,y) = x(p,1)∗ y (27)
This means that holding prices fixed as we vary income, the demanded bundle lies on a ray fromthe origin going through the vector x(p,1), which is the demanded bundle when y = 1. Thisimplies that the slopes of the indifference curves along this ray through the origin must be thesame at every point on the ray, otherwise we can find a point on this ray, call it x(p,1) ∗ y forsome y > 0, where the slope of the indifference curve is different than the slope of the budgethyperplane. But if this is the case, x(p,1) ∗ y cannot satisfy the necessary condition for utilitymaximization, and thus cannot be the utility-maximizing (i.e. demanded) bundle, contradictingthe assumption that x(p,1)∗ y is the demanded bundle for every p and y.This is the basic idea of the proof. To make it completely rigorous we need to show a) a function ishomothetic if and only if the slopes of its indifference curves are constant along every ray throughthe origin, b) the “unit demand function” x(p,1) varies sufficiently as p varies such that it crossesany ray from the origin in the positive orthant, i.e. given any strictly positive vector x there is ap ∈ Rn and an income y such that x = x(p,1)∗ y. We are going to assume that the latter conditionholds, so we only need to prove part a). We do this by contradiction. So suppose we have autility function u(x) that has indifference curves whose slopes are the same along every ray fromthe origin (in the positive orthant) but which is not homothetic. This means that u(x) cannot bewritten as a monotonic transformation of a linear homogeneous function l(x). I leave it to you tocomplete the proof that this implies that along some ray through the origin the indifference curvesfor u(x) do not have the same slope for every x along this ray. This is a contradiction.
3. Prove that if (x∗,λ∗) are a saddlepoint pair for the Lagrangian function
L(x,λ) = u(x)+λ[K − f (x)] (28)
then x∗ is a solution to the constrained optimization problem
maxx
u(x) subject to: f (x) ≤ K (29)
Answer Suppose x∗ does not solve the constrained optimization problem. Then either x∗ is not feasible(i.e. f (x∗) > K), or there is some other x̂ which is feasible that gives a higher objective value, (i.e.∃ x̂ such that u(x̂) > u(x∗) and f (x̂) ≤ K). We show that in either case, there is a contradictionof the hypothesis that (x∗,λ∗) is a saddlepoint. If x∗ is not feasible there is a contradiction sincethen we would have that K − f (x∗) < 0 but in this case we can drive the Lagrangian to −∞ bydriving λ∗ → +∞. This contradicts the assumption that (x∗,λ∗) is a solution to the Lagrangiansaddlepoint problem, i.e. that x∗, λ∗ and L(x∗,λ∗) are finite quantities.Now suppose that x∗ is feasible but there is a feasible x̂ for which u(x̂) > u(x∗). Then we alsohave a contradiction since by the complimentary slackness condition λ∗[K− f (x∗)] = 0 we have
L(x∗,λ∗) = u(x∗) (30)
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Now consider the value of the Lagrangian at the point (x̂, λ̂) where λ̂ = 0. We have
L(x̂, λ̂) = u(x̂) > u(x∗) = L(x∗,λ∗) (31)
But this contradicts the assumption that (x∗,λ∗) is a saddlepoint solution to the Lagrangian, i.e.that it maximizes the Lagrangian in x and minimizes it in λ.
1. Intertemporal utility maximization with certain lifetimes. Suppose a person has an additivelyseparate, discounted utility function of the form
V (c1, . . . ,cT ) =T
∑t=1
βtsu(ct ) (32)
where βs is a subjective discount factor and u(ct) is an increasing utility function of consumption ct inperiod t. Let the market discount factor is βm = 1/(1+ r) where r is the market interest rate.
a. If βs = βm show that the optimal consumption plan in a market where there are no borrowingconstraints (i.e. the consumer has unlimited ability to borrow and lend subject to an intertemporalbudget constraint) is to have a constant consumption stream over time, i.e. c1 = c2 = · · · = ct =ct+1 = · · · = cT .
b. If βs < βm will the optimal consumption stream be flat, increasing over time, or decreasing overtime, or can’t you tell from the information given?
c. How does your answer to part b change if I tell you that the utility function u(c) is convex in c?
Answers: We did this in class, and it is also in the lecture notes. See pages 23 onward in the lecturenotes on intertemporal choice. For part c, note that if the utility function is convex, then u ′′(c) > 0 andthe answers to parts b is reversed, iv βs < βm, then optimal consumption will be increasing over time,the opposite of the case if utility is concave (diminishing marginal utility), in which case consumptionwould be decreasing over time.2. Expected Discounted Utility with Uncertain Lifetimes Consider the intertemporal utility maxi-mization problem, but extended to allow for uncertain lifetimes. Let T̃ denote the (random) lifetime ofa person, in years. Let f (t) denote the probability density function for the person’s lifetime. Thus, wehave
f (t) = Pr{T̃ = t}, (33)
i.e. f (t) is the probability that the person lives for t −1 years and dies when they reach t years old.
a. What does the sum ∑∞t=1 f (t) equal?
b. Show that the person’s expected discounted lifetime utility, allowing for the possibility of dying,is given by
E{U} =∞
∑t=1
[
t
∑s=1
βsu(cs)
]
f (t) (34)
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c. Show that the person’s expected discounted lifetime utility can also be written as
E{U} =∞
∑t=1
[1−F(t −1)]βt u(ct), (35)
where F(t) is the cumulative probability distribution corresponding to the probability densityf (t), i.e.
F(t) = Pr{T̃ ≤ t} =t
∑s=1
f (s). (36)
d. In words, what is the interpretation of the quantity [1−F(t −1)]?
e. Suppose that the person’s random age of death T̃ is geometrically distributed, i.e.
f (t) = pt(1− p), t = 1,2, . . . (37)
where p ∈ (0,1) is the probability of surviving in any given year. Show that the expected dis-counted lifetime utility in this case is
E{U} =∞
∑t=1
f (t)t
∑s=1
βsu(cs) =∞
∑t=1
[pβ]t u(ct ). (38)
Hint: Use the rules from calculus on interchanging the orders of integration of an integral over a trian-gular region,
Z ∞
0
[
Z x
0f (x,y)dy
]
dx =Z ∞
0
[
Z ∞
yf (x,y)dx
]
dy (39)
and show that the same reasoning leads to the following analogous formula for interchanging the orderof summations in a summation over a triangular region
∞
∑t=1
[
t
∑s=1
f (t,s)
]
=∞
∑s=1
[ ∞
∑t=s
f (t,s)
]
. (40)
Answers:
a. ∑∞t=1 f (t) = 1 since it is a probability distribution so it must sum to 1.
b. Show that the person’s expected discounted lifetime utility, allowing for the possibility of dying,is given by
E{U} =∞
∑t=1
[
t
∑s=1
βsu(cs)
]
f (t) (41)
The expected lifetime utility is just the weighted summation of the probability of living to age t,f (t), times the discounted utility of living t years, ∑t
s=1 βsu(cs). We assume that person dies atthe end of their tth year of life, so they actually get to enjoy the consumption ct in their last yearbefore they die.
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c. Show that the person’s expected discounted lifetime utility can also be written as
E{U} =∞
∑t=1
[1−F(t −1)]βt u(ct), (42)
where F(t) is the cumulative probability distribution corresponding to the probability densityf (t), i.e.
F(t) = Pr{T̃ ≤ t} =t
∑s=1
f (s). (43)
Using the Hint above, equation (40), we reverse the order of the summations to get
E{U} =∞
∑t=1
[
t
∑s=1
βsu(cs)
]
f (t)
=∞
∑s=1
∞
∑t=s
f (t) [βsu(cs)]
=∞
∑s=1
[ ∞
∑t=s
f (t)
]
[βsu(cs)]
=∞
∑s=1
[1−F(s−1)] [βsu(cs)] (44)
This last equation follows because we have
1 =∞
∑t=1
f (t)
=s−1
∑t=1
f (t)+∞
∑t=s
f (t) (45)
or∞
∑t=s
= 1−s−1
∑t=1
f (t) = 1−F(s−1), (46)
since by definition we have
F(s−1) = Pr{
T̃ ≤ s−1}
=s−1
∑t=1
f (t). (47)
d. In words, what is the interpretation of the quantity [1−F(t −1)]?
Answer: this is the survival probability, i.e. the probability a person will live to at least age t orlonger.
e. Suppose that the person’s random age of death T̃ is geometrically distributed, i.e.
f (t) = pt−1(1− p), t = 1,2, . . . (48)
where p ∈ (0,1) is the probability of surviving in any given year. Show that the expected dis-counted lifetime utility in this case is
E{U} =∞
∑t=1
f (t)t
∑s=1
βsu(cs) =∞
∑t=1
[pβ]t u(ct ). (49)
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Answer: Using the result from part c, we just need to figure out what F(s−1) is for the geometricdistribution. We have
F(s−1) =s−1
∑t=1
pt−1(1− p)
=s
∑j=0
p j(1− p)
= (1− p)s
∑j=0
p j
= (1− p)1− ps
(1− p)
= 1− ps (50)
Thus we have 1−F(s−1) = ps and using formula (44) above we get
E{U} =∞
∑s=1
[1−F(s−1)] [βsu(cs)]
=∞
∑s=1
[ps] [βsu(cs)]
=∞
∑s=1
[pβ]s u(cs)
=∞
∑t=1
[pβ]t u(ct). (51)
3. Recursive Representation of Lifetime Utilities Consider a discounted sum of utilities for a personwith a known lifespan of T years
V0 =T
∑t=0
βtu(ct ) = u(c0)+βu(c1)+ · · ·βT u(cT ). (52)
Thus, V0 represents the discounted utility of a person at age t = 0, looking ahead over the rest of theirlife. Now let Vt denote the discounted utility of an age t person, looking forward from age t onwards.
a. Write a formula for Vt . What is VT ?
VT = u(cT )
VT−1 = u(cT−1)+βVT = u(cT−1)+βu(cT )
· · · = · · ·
Vt = u(ct )+βVt+1 =T
∑s=t
βs−tu(cs)
· · · = · · ·
V1 = u(c1)+βV2 =T
∑s=1
βs−1u(cs)
V0 = u(c0)+βV1 =T
∑s=0
βsu(cs)
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=T
∑t=0
βtu(ct ). (53)
b. Show that the utilities Vt and Vt+1 are connected recursively via the formula
Vt = u(ct)+βVt+1 (54)
Answer: see the answer to part a above.
c. Show that by age t = 0, the recursive representation of V0, i.e.
V0 = u(c0)+βV1 (55)
gives the same value V0 as the non-recursive representation of discounted lifetime utility as in theoriginal formula, (52).
Answer: see the answer to part a above.
d. Now let’s extend this recursive way of thinking about discounted utilities to expected discountedutilities when there are uncertain lifetimes. Let V0 be given by
V0 =T
∑t=0
[1−F(t −1)]βtu(ct ), (56)
the same formula for expected lifetime utility when there is uncertain mortality as you derived inequation (42) above, where F(t) = Pr{T̃ ≤ t} = ∑t
s=0 f (t) is the cumulative probability of dyingon or before age t and f (t) = Pr{T̃ = t} is the probability of dying exactly on age t. DefineVt analogously to Vt in the case where there is no uncertainty about age of death, i.e. Vt is theexpected discounted utility from age t onwards, to whatever random age T̃ the person dies. Showthat the appropriate form for the recursive representation for expected discounted utilities is inthis case
Vt = u(ct )+β1−F(t)
1−F(t −1)Vt+1 (57)
Answer. If the oldest any person can possibly live, the maximal lifespan is T , then if you surviveto this age, it must be your last year of life. Thus
VT = u(cT ) (58)
since there is zero probability that the person could live to age T + 1 or longer. Now considersomeone who survived to age T −1. I claim that the value function at this age, VT−1, is given by
VT−1 = u(cT−1)+β1−F(T −1)
1−F(T −2 VT . (59)
Thus, VT−1 equals the sum of the utility at time T − 1, u(cT−1), plus the expected discountedutility in the last period, T . This is the product of the value in the last period, VT (which equalsu(cT )), the discount factor β, and the conditional probability of surviving to age T given that theperson survived to age T −1. This probability is [1−F(T −1)]/[1−F(T −2)]. To see why thisis, note that there are two possibilities: either this person dies at the end of their T −1st year, or
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they live to enjoy their final year T too. The conditional probability that a person who survived toage T −1 will survive to age T is defined as
Pr{
T̃ > T −1|T̃ ≥ T −1}
=1−F(T −1)
1−F(T −2). (60)
This is a specific example of the general formula for conditional probability given by
Pr{A|B} =Pr{A∩B}
Pr{B} . (61)
Where A∩B is the event that “A and B occurs.” So for this example, event A is the event thatthe person lives to age T , i.e. A = {T̃ = T} and event B is the event that the person lives to atleast age T − 1, i.e. B = {T̃ ≥ T − 1}. Notice that since T is the maximal possible lifespan, wecan write A = {T̃ > T − 1} and similarly we can write B = {T̃ > T − 2}. Now the event A∩Bis clearly the same as event A since A ⊂ B (i.e. the event that someone lives to at least age T is asubset of the event that someone lives to at least age T −1). Thus we have in this case
Pr{A|B} =Pr{A∩B}
Pr{B} =Pr{A}Pr{B} . (62)
But if A = {T̃ > T −1}, clearly Pr{A} = Pr{T̃ > T −1} = 1−F(T −1), since F(T −1) is thecumulative probability that a persion will die at age T −1 or before,
F(T −1) =T−1
∑t=1
f (t). (63)
Similarly, we have Pr{B} = Pr{T̃ > T −2} = 1−F(T −2). Thus, the conditional probability ofliving to age T given that you have lived to age T −1 is
Pr{A|B} = Pr{
T̃ > T −1|T̃ > T −2}
=Pr
{
T̃ > T −1}
Pr{
T̃ > T −2}
=1−F(T −1)
1−F(T −2)(64)
More generally, we can see that if Vt is the expected utility from age t onwards, we have
Vt = u(ct )+β1−F(t)
1−F(t −1)Vt+1 (65)
since [1−F(t)]/[1−F(t − 1)] is the conditional probability of surviving to age t + 1 or longer,given that the person already survived to age t. Now let’s see why, when we work backward toperiod t = 0 and write
V0 = u(c0)+β1−F(0)
1−F(−1)V1 (66)
we get the same answer as the “direct” way of writing the expected utility, i.e.
V0 =T
∑t=0
[1−F(t −1)]βtu(ct ), (67)
12
Note first that F(−1) = 0, i.e. there is zero chance of dying in year t = −1, the year before theperson is born. Thus we can write the recursive formula for V0 as
V0 = u(c0)+β[1−F(0)]V1. (68)
But notice that by the backward recursion process, V1 is the expected discounted utility of a personwho survived birth and made it to age t = 1. This utility would be
V1 = u(c1)+β1−F(1)
1−F(0)V2
=T
∑t=1
βt−1 1−F(t −1)
1−F(0)u(ct ), (69)
since [1−F(t − 1)]/[1−F(0)] is the conditional probability of surviving to age t given that theperson survived to age 1. So when be multiply V1 by [1−F(0)] we get
[1−F(0)]V1 =T
∑t=1
βt−1[1−F(t −1)]u(ct ), (70)
so we have
V0 = u(c0)+β1−F(0)
1−F(−1)V1
= u(c0)+β[1−F(0)]V1
= u(c0)+T
∑t=1
βt [1−F(t −1)]u(ct )
=T
∑t=0
βt [1−F(t −1)]u(ct ). (71)
e. Show that1−F(t)
1−F(t −1)= 1−h(t) (72)
where h(t) is the hazard rate, i.e. the conditional probability of dying at age t given that one hassurvived to age t −1:
h(t) ≡ f (t)1−F(t −1)
=f (t)
f (0)+ f (1)+ · · · f (t −1)(73)
and thus, 1− h(t) is the survival rate, i.e. the conditional probability that a person who lives toage t −1 will survive another year, to be at least age t or older before they die.
Answer: Note thatF(t) = F(t −1)+ f (t) (74)
So we have1−F(t) = 1−F(t −1)− f (t) (75)
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It follows that1−F(t)
1−F(t −1)=
1−F(t −1)− f (t)1−F(t −1)
= 1− f (t)1−F(t −1)
= 1−h(t). (76)
4. Annuities with Uncertain Lifetimes
a. Suppose that a person’s lifetime is uncertain, so that the random variable T̃ denotes the randomage of death, but that we know that the probability distribution of T̃ is geometric with parameterp ∈ (0,1). That is, as noted above, f (t) = pt(1− p), t = 0,1, . . .. If a person consumes a flatamount of $10000 per year until they die, and if the discount factor is β ∈ (0,1), write a formulafor the expected discounted amount that this person will consume over their lifetime.
Answer: Let Va denote the expected value of an annuity that pays the holder an annual amounta until they die, with a discount factor of β = 1/(1 + r). Using the ideas from problem 2, wecan either write a direct formula for Va, or an indirect or recursive formula for Va. The recursiveformula is much easier, so let’s start with this. We have
Va = a+ pβVa. (77)
That is, the expected value of an annuity equals the current payment, a, plus the expected dis-counted value of the annuity from tomorrow onward, pβVa. This latter term is the product of theprobability you survive p, the discount factor β times the expected discounted value of the annuityfrom tomorrow onward, Va. The expected value of the annuity does not change over time due tothe particular memoryless property of the geometric distribution. That is, the survival probabil-ity of a person with a geometric distribution of lifetimes is always p regardless of how old theycurrently are. This means that we do not have to keep track of the person’s age to compute theexpected value of the annuity, and this makes the problem stationary. The stationarity results in athe simple equation above, (77) for the expected present value of the annuity. Now if your initialwealth is W , we equate the value W you hand over to the annuity company to the expected valueof the annuity and solve for a
Va =a
1−βp=
100001−βp
. (78)
Now consider the direct way to compute Va. We have
Va =∞
∑t=0
f (t)t
∑s=0
βsa
=∞
∑s=0
∞
∑t=s
f (t)βsa
=∞
∑s=0
[1−F(s−1)]βsa
=∞
∑s=0
psβsa
=a
1−βp, (79)
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where the latter formula is a consequence of the formula for an infinite geometric series, ∑∞j=0 r j =
1/(1− r) when r ∈ (0,1).
b. An annuity is a contract such that if a person pays a given amount W up front at the start of theirlifetime, the annuity company will in return provide that person with a constant payment of $c peryear over their entire lifetime. Using the result from part 1 above, if a person has endowment of$1,000,000 when they are born and an annuity is purchased for them, how much will this annuitypay the person if their probability of survival is p = .98 and the discount factor is β = .95?
Answer: The maximal annuity that would be paid in a competitive annuity market will satisfy
W = Va (80)
That is, competition by different annuity companies will result in bidding up the the present valuethey offer the person until the present value, Va, equals the amount of the person’s wealth W to beinvested in the annuity. Solving for a we get
W = Va =a
1−βp(81)
ora = W (1−βp) = 1000000∗ (1− .95× .98) = 69000. (82)
This annuity enables the annuity company to just break even, not earning a profit but also notmaking any losses.
c. Suppose there are no annuity markets and that a person has a lifetime utility function (conditionalon living T years) equal to
U =T
∑t=1
βt log(ct) (83)
Describe the optimal consumption strategy for this person using dynamic programming, assumingthat they are born with an initial endowment of W = 1,000,000 and β = .95 and p = .98.
Use the hint (below) that the maximized expected discounted utility function
V (W ) = max{ct}
∞
∑t=0
pt(1− p)t
∑s=0
βt log(cs) subject to: initial wealth = W (84)
satisfies the Bellman equation
V (W ) = max0≤c≤W
[
log(c)+βpV
(
W − cβ
)]
(85)
and the additional hint (conjecture) that V (W ) takes the form
V (W ) = a+b log(W ) (86)
for coefficients (a,b) to be determined. Plugging this conjecture on both sides of the Bellmanequation (85) we get
a+b log(W ) = max0≤c≤W
[
log(c)+βp[a+b log(
W − cβ
)
]
]
(87)
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Taking first order conditions for c∗ on the right hand side of this equation, and using the fact thatlog(x/y) = log(x)− log(y) we get
0 =1c− βpb
W − c. (88)
Solving this for c∗ we getc∗(W ) =
W1+βpb
. (89)
Now, substitute this expression for c∗ back into the Bellman equation (85) to get
a+b log(W ) = log
(
W1+βpb
)
+βp
[
a+b log(
pbW1+βpb
)]
, (90)
Where we used the fact that
W − c∗(W )
β=
W − W1+βpb
β=
W pb1+βpb
. (91)
Now consider the left and right hand sides of the substituted version of the Bellman equation,(90). If this equation is to hold for all (positive) values of W then we need the coefficient oflog(W ) on the left hand side, b, to equal the coefficients of log(W ) on the right hand side. Usinglog(x/y) = log(x)− log(y) and gathering the coefficients of log(W ) on the right hand side of (90),we get the following equation for b
b = 1+βpb (92)
orb =
11−βp
. (93)
Substituting this into the equation for the optimal consumption function c∗(W ) in equation (89),we get
c∗(W ) =W
1+ βp1−βp
= W (1−βp). (94)
This is a version of the permanent income hypothesis. That is, the person consumes a constantfraction of their wealth, where the fraction is a mortality adjusted “interest earnings”. Now weneed to solve for the a coefficient to complete the problem. Using the finalized version of the con-sumption function above, and using the formula for the b coefficient, we get another expressionfor the Bellman equation
a+log(W )
1−βp= log(W )+ log(1−βp)+βp
[
a+log(W )+ log(p)
1−βp
]
. (95)
Gathering the terms that are “constants” (i.e. do not have log(W ) in them) on the left and righthand sides of equation (95), we get the following equation for a
a = log(1−βp)+βp
[
a+log(p)
1−βp
]
, (96)
where we used the fact that
W − c∗(W )
β=
W −W (1−βp)
β= pW. (97)
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Solving the equation for a above, we bet
a =log(1−βp)
1−βp+
log(p)
(1−βp)2 . (98)
We conclude the the utility of a person who gradually consumes their initial endowment (as op-posed to buying an annuity with it) is
V (W ) =log(1−βp)
1−βp+
log(p)
(1−βp)2 +log(W )
1−βp. (99)
d. Now suppose that there are annuity markets. The person now has the option, at the start of theirlife, to exchange their entire initial endowment of wealth W for a lifetime annuity. Which optionwould the person prefer: 1) to exchange W and take the annuity, or 2) not buy the annuity andfollow the optimal consumption plan described in part 3 above?
Answer: Let Va(W ) be the expected discounted utility to the person if he/she chooses the annuity,a = W (1−βp), that they can afford by converting all of their initial wealth W into the annuity.Then we have the following recursive formula for Va(W )
Va(W ) = log(a)+βpVa(W ) (100)
or solving, we get
Va(W ) =log(a)
1−βp=
log(W )+ log(1−βp)
1−βp. (101)
Comparing this formula with the formula for expected discounted utility associated with not buy-ing an annuity, (99), we get
Va(W )−V (W ) =− log(p)
(1−βp)2 > 0, (102)
since log(p) < 0 since p ∈ (0,1). Thus, we conclude that the person is better off taking theannuity. Why is this? It might appear that the person should be indifferent between taking theannuity and managing their own consumption saving since in the first period of their life theoptimal consumption (no annuity) is
c∗(W ) = W (1−βp) = a, (103)
i.e. the optimal consumption in the first period is the same as the annuity payment. Howeverconsider the second period. The person’s wealth in the second period is
W2 = W − c∗(W ) = W −W(1−βp) = pW < W. (104)
Thus, the consumption for the person in period 2 is
c∗(W2) = W2(1−βp) = pW (1−βp) < a. (105)
Thus, the second period consumption is less than the annuity. Continuing forward we see that inperiod t we have
c∗(Wt) = Wt (1−βp) = pt−1W (1−βp) < a. (106)
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Thus, since 0 < p < 1, the consumption of the person who did not choose the annuity is tend-ing to zero over time, whereas the consumption of the person who chose the annuity is alwayconstant and equal to the initial consumption W (1−βp) of the person who did not choose the an-nuity. Thus, since the consumption stream under the annuity clearly “dominates” the consumptionstream a person can get from “self-insuring” their own mortality risk by only gradually consum-ing their wealth as they age, it is clear that in this example it would always be a better choice topurchase the annuity rather than to self-insure mortality risk.
Hint: To solve part 3, use the method of dynamic programming with the following Bellman equation
V (W ) = max0≤c≤W
[log(c)+ pβV (W − c)] (107)
Conjecture that V (W ) is of the form
V (W ) = a+b log(W ) (108)
and using the Bellman equation above, solve for the coefficients a and b so that the Bellman equationwill hold. From this solution you should be able to derive the associated optimal consumption function,c(W ), which specifies how much the person will consume each year given that they start that year withtotal savings of W .
5. Consumption and Taxes Suppose a consumer has a utility function u(x1,x2) = log(x1)+ log(x2) andan income of y = 100 and the prices of the two goods are p1 = 2 and p2 = 3.
a. In a world with no sales or income taxes, tell me how much of goods x1 and x2 this consumer willpurchase.
b. Now suppose there is a 10% a sales tax on good 1. That is, for every unit of good 1 the personbuys, he/she has to pay a price of p1(1+ .1) = 2.2, where the 10% of the price, or 20 cents, goesto the government as sales tax. How much of goods 1 and 2 does this person buy now?
c. Suppose instead there is a 5% income tax, so that the consumer must pay 5% of his/her incometo the government. If there is no sales tax but a 5% income tax, how much of goods 1 and 2 willthe consumer consume?
d. Which would the consumer prefer, a 10% sales tax on good 1, or a 5% income tax? Explain yourreasoning for full credit.
e. How big would the sales tax on good 1 have to be for the government to get the same revenueas a 5% income tax? Which of the two taxes would the consumer prefer in this case, or is theconsumer indifferent because the consumer has to pay a total tax of $5 (5% of $100) in eithercase?
Answer I answer each part separately below. Notice that the utility function is a monotonic transforma-tion of a Cobb-Douglas utility function l(x1,x2) = x1/2
1 x1/22 , so demands are x1(p1, p2,y) = y/2p1 and
x2(p1, p2,y) = y/2p2. With these, it is very easy to answer this question.
a. x1 = 100/(2∗2) = 25 and x2 = 100/(2∗3) = 16.66667
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b. With the tax in place, the price of good 1 increases to 2.2 so quantities demanded are x1 = 100/(2∗2.2) = 22.727273 and x2 = 100/(2 ∗ 3) = 16.66667. The total taxes the person pays are .2x1 =.2100/(2∗2.2) = 4.54.
c. With a 5% income tax, the consumer has after-tax income equal to $95 (100(1−τ) where τ = .05).So the consumption of goods 1 and 2 is given by x1 = 95/(2∗2) = 23.75 and x2 = 95/(2∗3) =15.8333
d. With the sales tax, the consumer consumes less of good 1 and more of good 2, and pays lessin tax overall. With the income tax the consumer consumes more of good 1 but less of good 2and pays more overall in tax ($5.00 versus $4.54). But the only way to see which alternativethe consumer prefers is to plug the consumption bundles into his/her utility function and seewhich one give more utility. The utility under the sales tax is log(22.727273)+ log(16.66667) =5.93699764. The consumer’s utility under the income tax is log(23.75)+ log(15.8333) = 5.9297so the consumer prefers the sales tax to the income tax.
e. Now we want to set the sale tax rate α so that we raise tax revenue of $5, the same revenue thatwe collect under an income tax of 5%. The equation for the necessary tax rate is
5 = α100
2(2+α)(109)
Solving this for α we get α = 2/9 = .22222. Under this tax rate, consumption of good 1 falls tox1 = 100
2(2+α) = 22.5 and the tax revenue collected is 22.5∗2/9 = 5. Now the person’s utility underthe sales tax is log(22.5)+ log(16.66667) = 5.926926, so that now, the consumer slightly prefersto have the income tax over the sales tax.
6. Risk Neutrality and Risk Aversion An person is said to be risk neutral if when offered a gamble,their maximum willingness to pay to undertake the gamble equals the expected value of the gamble.That is, if G̃ denotes a random payoff from a gamble, the maximum “entry fee” F that a risk neutralperson would be willing to pay to get the gamble payoff G̃ is
F = E{G̃}. (110)
A person is risk averse if F < E{G̃} and risk loving if F > E{G̃}.
a. Suppose a person has a utility function u(W ) = W , and suppose that initially (before consideringtaking the gamble) the person has W = 1000000 of wealth. Suppose the gamble under consid-eration is to flip a coin, and if it lands heads the person wins $100, and if tails the person getsnothing. What it is the maximum amount F this person would be willing to pay for this gamble?Is this person risk neutral, risk loving, or risk averse?
A risk neutral person will be willing to pay up to the expected value of the gamble, i.e. F = E{G̃}.In this case it is
F = E{G̃} = p100+(1− p)0 = p100 = 50. (111)
b. Suppose a person has a utility function u(W ) = log(W ) and this person also has W = 1000000 ininitial wealth before considering taking the gamble. What is the maximum amount F this personwould be willing to pay for the gamble?
19
Answer: The general equation for the maximum amount someone would be willing to pay is
u(W ) = E{u(W −F + G̃)} (112)
since the left hand side is the utility of not participating in the gamble, and the right hand side isthe expected utility of paying the fee F and then getting the gamble payoff G̃. Note that if youhave a linear utility function u(W ) = a+bW with b > 0, then
a+bW = E{a+b(W −F + G̃)} = a+bW −bF +bE{G̃}, (113)
using the properties of expectations (i.e. that for a linear function E{a+bX̃} = a+bE{X̃}). Wesee that the value of F that equates both sides of the above equation is F = E{G̃}, which was theanswer we got in part a for a risk neutral person. Now for a person with log utility, we get
log(W ) = E{
log(
W −F + G̃)}
= p log(W −F +100)+(1− p) log(W −F). (114)
This is the equation that needs to be solved for F to determine the person’s maximal willingnessto pay. Note that in general F is a function of W so we can write F(W ) as the amount a person ofwealth level W would be willing to pay. It is not difficult to show that in general for this person,
F(W ) < E{G̃}, (115)
so that the person is risk averse since their maximal willingness to pay is less than the expectedvalue of the gamble.
c. Suppose a third person has a utility function u(W ) =W 2 and also has initial wealth W = 1000000before considering the gamble. What is the maximum amount this third person would pay for thegamble?
Answer: In this case the equation for F is
W 2 = E{
(
W −F + G̃)2
}
= p(W −F +100)2 +(1− p)(W −F)2. (116)
In this case it turns out that the solution F(W ) satisfies
F(W ) > E{G̃} (117)
and so this person is risk loving.
d. Are the persons in cases b and c above risk neutral, risk averse or risk loving?
Answer: In part a the person is risk neutral, in part b the person is risk averse, and in part c theperson is risk loving.
e. Prove that a person is risk neutral if their utility function is linear, u(W ) = a+bW for b > 0, andrisk averse if their utility function is concave, u′(W ) > 0 and u′′(W ) < 0, and risk loving if theirutility function is convex, u′(W ) > 0 and u′′(W ) > 0.
Hint: If a person does not take the gamble, they will have utility u(W ) from consuming their wealth W .If the person pays an amount F for a gamble G̃, their expected utility would be E{u(W −F + G̃)}. The
20
maximum willingness to pay for the gamble G̃ would be the amount F∗ that makes the person indifferentbetween paying F∗ for the gamble and not taking the gamble, i.e. it is the solution to
U(W ) = E{u(W −F∗ + G̃)}. (118)
You can use Jensen’s Inequality which states that for a concave function u and any random variable X̃we have
E{u(X̃)} ≤ u(E{X̃}). (119)
You should be able to use Jensen’s inequality to show that people with concave utility functions are riskaverse.
Answer: We already showed above that a person with a linear utility function is risk neutral, i.e.they would be willing to pay the expected value of the gamble, F = E{G̃}. Now consider a riskaverse person. Jensen’s inequality tells us that net of the fee F we have for a risk averse person
E{
u(W −F + G̃)}
≤ u(W −F +E{G̃}). (120)
It follows that the fee F(W ) must be less than or equal to E{G̃} to induce the risk averse personto take the gamble since if F > E{G̃} we have
u(W ) > u(W −F +E{G̃}) ≥ E{u(W −F + G̃)} (121)
and this says that at the fee F the person would not want to take the gamble. For a risk lovingperson, Jensen’s inequality is reversed,
E{
u(W −F + G̃)}
≥ u(W −F +E{G̃}). (122)
Thus, for this person the fee F(W ) must be greater than E{G̃} since otherwise if F < E{G̃} wehave
u(W ) < u(W −F +E{G̃}) ≤ E{u(W −F + G̃)} (123)
and thus the person would be strictly better off taking the gamble. The fee can therefore beraised above the expected payoff E{G̃} and still induce the risk loving person to take the gamble.The specific answers for F(1000000) in the logarithmic utility case are F(1000000) = 49.9988,which is just slightly less than the expected value of the gamble, and in the quadratic utility case,F(1000000) = 50.00125, which is just slightly more than the expected value. This is becauseat such high wealth levels, both the logarithmic and quadratic utility functions look pretty “lin-ear” and thus individuals with these utility functions at high wealth levels act approximately riskneutrally, even though they are technically risk-averse and risk-loving, respectively. If you recal-culate the maximal willingness to pay at lower levels of wealth, you get bigger deviations fromthe expected value. For example when W = 1000, F(1000) = 48.75 for the logarithmic utilityfunction, and F(1000) = 51.25 for the quadratic utility function.
7. St. Petersburg Paradox Consider the following gamble G̃. You flip a fair coin until it lands on tails.Let h̃ denote the number of heads obtained until the first tail occurs and the game stops. Your payofffrom playing this game is
G̃ = 2h̃ (124)
21
a. Suppose you are risk neutral. What is the maximum amount F ∗ that you would be willing to payto play this game?
Answer: A risk neutral person would be willing to pay up to the the expected value
E{G̃} =∞
∑h=0
2h[12]h
12
=12
∞
∑h=0
1 = ∞, (125)
since the probability of getting h heads until the first tail is geometrically distributed with densityf (h) given by
f (h) =
[
12
]h 12 . (126)
b. Suppose you have a utility function u(W ) = log(W ) and W = 1000000 in initial wealth. What isthe maximum amount you would be willing to pay to play this gamble in this case?
Answer: As we saw above, the logarithmic utility has a negative second derivative (diminishingmarginal utility of wealth) and is thus concave, and therefore the person is risk averse. Themaximal willingness to pay for a gamble by a risk averse person is less than the expected value ofthe gamble. In this case, the willingness to pay satisfies
log(W ) = E{
log(W −F + G̃)}
=∞
∑h=0
log(W −F +2h)[12 ]h
12 . (127)
Figure 1 plots the maximal willingness to pay function calculated by the Matlab functioncertaintyequiv.m which I have posted also with these answers. This program finds a zero of theMatlab function eu.m which is defined by
eu(F) =∞
∑h=0
log(W −F +2h)[12]h
12− log(W ), (128)
where I numerically approximate the infinite summation in the equation for eu(F) above. The programcertaintyequiv.m uses Matlab’s fsolve routine to find an F such athat eu(F) = 0.
Figure 1 plots F(w) for W ∈ [100,1000]. We see that the amount that this person is willing to payfor this gamble is faily small: less than $7 for wealth up to $1000. If I solve for the certainty equivalent(i.e. F(W )) when W = 1000000, using certaintyequiv.m I get F(1000000) = 10.93. Thus, even avery rich person with a logarithmic utility function would not be willing to pay very much to undertakethis gamble. In this sense, risk aversion is the resolution of the St. Petersburg Paradox.
22
100 200 300 400 500 600 700 800 900 10004.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6Certainty Equivalents for Log Utility, Bernoulli Paradox
Initial Wealth, W
Cer
tain
ty E
quiv
alen
t, C
Figure 1 F(W ): Maximal Willingness to Pay for the Gamble G̃
23