Econ 452/551, spring 2007
Assignment 1
Economics 452/551, Spring 2007
Assignment 1, with suggested (sketched) answers
L. Welling
January 18, 2007
Total value: (40 marks))
Due: January 26, 2007
4:00 pm.
1. For the utility of wealth function U ( w) = ln w , derive the demand for insurance for an
individual with initial wealth w in the following cases:
i)(5 marks) tthe individual faces a loss of $2000 with probability 0.25, and can
purchase insurance at a rate of δ=$0.30 (per dollar of insurance). Calculate the
coefficient of absolute risk aversion, and show how the demand for insurance changes as
wealth increases.
U "( w)
Ans: Coefficient of ARA is defined as −
. With U ( w) = ln w , this coefficient has
U '( w)
1
; this is strictly positive and decreasing in wealth.
value
w
Demand for insurance is derived as follows:
1. differentiate EU with respect to (wrt) I, and set derivative =0; solve for I:
EU= 0.25ln( w − 2000 + (1 − .3) I ) + (1 − 0.25)ln( w − (1 − .3) I )
dEU
.25 × .7
.75 × .3
=
−
=0
*
dI
w − 2000 + .7 I
w − .3I *
15000 5
Solving for I* gives I * =
− w (or some variation on this)
7
21
Differentiating wrt I yields
∂I *
5
= − , so optimal insurance is linear and decreasing in wealth –
∂w
21
consistent with decreasing absolute risk aversion.
Thus
ii) (5 marks) the individual faces a loss of L with probability p, and can purchase
insurance at a rate of δ (per dollar of insurance). How does the demand for insurance
respond to changes in p and δ?
Ans: using same utility function as above, with general values; using same process:
EU = p ln( w − L + (1 − δ ) I ) + (1 − p)ln( w − δ I )
First order condition for maximum satisfies:
dEU
p (1 − δ )
(1 − p )δ
=
−
=0
*
dI
w − L + (1 − δ ) I
w −δ I*
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Econ 452/551, spring 2007
Assignment 1
Solving for I* gives (some variant of ) I * =
pw
δ
−
(1 − p )( w − L)
(1 − δ )
w ( w − L)
pw (1 − p )( w − L)
∂I
∂I *
= +
> 0 as w>L;
=− 2 −
<0
∂p δ (1 − δ )
(1 − δ ) 2
∂δ
δ
(both signs follow because as w>L and δ ∈ (0,1).
*
Solving for partials:
2. Jane owns a house worth $400,000.00. She cares only about her wealth, w,
which consists entirely of her house. In any given year, there is a 25% chance that her
house will burn down. If it does, its scrap value will be $40,000.00. Jane's utility of
wealth function is U ( w) = w0.5 .
(a) (2 marks) Sketch Jane's utility function. (Use about 1/2 page for your
diagram.)
Ans: this asks for a diagram in (w, utils space): U(w) is positively sloped and concave.
b) (4) How much would Jane be willing to pay to insure her home against fire?
Explain how you calculated this, and relate your numerical answer to the diagram
in (a).
Ans: the maximium Jane would be willing to pay for insurance would be that amount
which would leave her indifferent between buying no insurance and the insurance.
If she buys no insurance, she obtains EU= .25(40,000)0.5 + .75(400,000)0.5 . If she buys
insurance of $I she obtains EU = .25(40,000 + (1 − δ ) I )0.5 + .75(400,000 − δ I )0.5 . If she
faces full insurance, in each state her wealth will be (400,000-c), where c is the cost of
the insurance. The max c she will be willing to pay satisfies
.25(40,000)0.5 + .75(400,000)0.5 = (400,000 − c)0.5
On the diagram, this is the difference between 400,000 and the certainty equivalent of
the fire “lottery´she faces.
c) (4) Suppose Jane could purchase insurance at a price of $ δ per dollar of
coverage. Write down the equation determining her optimal amount of insurance,
I*, and solve for I* as a function of δ .
Ans: This is uglier, but comes from EU = .25(40,000 + (1 − δ ) I )0.5 + .75(400,000 − δ I )0.5
As before, differentiate this wrt I, to yield the first order condition determining the
(1 − δ )
3δ
optimal insurance:
−
=0
* 0.5
(40,000 + (1 − δ ) I )
(400,000 − δ I * )0.5
(1 − δ )
3δ
; squaring both sides, and
=
0.5
(40,000 + (1 − δ ) I )
(400,000 − δ I )0.5
solving for I * yields
(10 − 20δ + δ 2 )
I * = 40,000
. This is Jane’s demand function for insurance.
δ (1 − δ )(1 + 8δ )
Rearranging yields
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Assignment 1
d) (5) Sketch Jane's demand function for insurance (which you derived in (c)).
Explain your diagram, and indicate the "full insurance" point on this diagram.
Under what circumstances would Jane purchase full insurance? More than full
insurance? Less than full insurance? No insurance?
Ans: demand function for insurance…since the only variables in the above equation are
δ and I, it makes sense to plot the demand curve: I on the horizontal axis, and the unit
price, δ , on the vertical.
From the FOC, have that I * = L = 360,000 when δ =0.25. Slope of demand curve? If
take derivative of I* wrt δ , have (assuming I have done my math correctly)
*
dI *
20 − 2δ
(10 − 20δ + δ 2 )(1 + 2δ )(1 + 12δ )
=−
−
dδ
δ (1 − δ )(1 + 8δ )
[δ (1 − δ )(1 + 8δ )]2
So long as I*≥0, this expression will be <0.
3. (5) Two fast food restaurants are located next to each other, and offer different
procedures for ordering food. Each restaurant has five servers. The first offers five lines
each leading to a server; the second has a single line leading to the servers, with the next
person in line going to the first available server. Suppose the utility which hungry
individuals derive from a meal depends on the time spent waiting to order the meal. If
most people are risk averse in time, which restaurant will be preferred when the average
waiting times are equal? Explain.
Ans: If average waiting times are the same, then can view the first restaurant as a lottery
with an expected value (ie, wait time) equal to the certain outcome obtainable from the
second restaurant. For an individual who is risk averse in time, the variability associated
with the wait time for the first restaurant would reduce the expected utility derived from
it, compared to the certain outcome of the second. Hence we would expect the second
restaurant to be preferred.
As a consequence, we might expect that, in equilibrium, the average waiting time would
be longer at a business which operates with a single line leading to several servers,
compared to one where you get to play the lottery of choosing a single server.
4. (10 marks; each part worth 2 marks) Jonathan has an income of $60. He is offered the
following bet: a die is thrown. If a one comes up, Jonathan loses $1. If a two, three, four,
five, or six comes up, he wins $3. Jonathan can take either side of the bet, and he is risk
averse.
a) What are the contingent commodities in this problem?
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Assignment 1
Ans: consumption (or wealth) if a one appears (call this state 1), and consumption
(wealth) if one of {2,3,…6} appears (let this be state 2).
b) Sketch the budget constraint. Explain how you derived it.
Ans: The information in the question gives two points on the budget line: Jonathan’s
state-contingent consumption pair if he does not bet, (c1 , c2 ) =(60,60) – so a point on
the certainty line- and his consumption pair if he bets $1: (c1 , c2 ) =(59,63). What the
“budget line” looks like depends on your interpretation of the question:
Interpretation 1: his only choices are to bet $0 or $1. Then his budget “line”
consists of the two disjoint points
Interpretation 2: he can bet $0 or any amount up to $60. In this case his budget
line is the straight line through the two points above, and it extends to both axes.
c) Sketch the indifference curves. Why do they have the shape they do?
Ans: three properties to note: indifference curves are negative, strictly convex, and
where c1 = c2 the slope of the IC’s are equal to (minus) the ratio of the probability of
state 1 to the probability of state 2 . Why? Indifference curves join bundles of
contingent commodities which yield the same expected utility. The slope is negative
because individual prefers more consumption to less, in each state; thus to keep EU
constant, increased consumption in one state must be traded off against decreased
consumption in the other. An IC is strictly convex because a risk averse individual
prefers certain outcomes to uncertain ones with the same expected value, and the
greater the risk, as measured by the difference between the outcomes across states,
the lower the EU for any given expected value. That the slope of an IC is equal to the
ratio of the probabilities along the certainty line follows from the math.
d) Show on a diagram the amount that Jonathan bets. Explain why this is the correct
amount.
Ans: again, depends on your interpretation:
For interpretation 1, Jonathan will choose between the two points on the basis of
comparing the utility of (60,60) with the EU of betting $1. Since the expected gain is
positive, it is possible that he will prefer the uncertainty over the certain outcome, or
vice versa – whichever point lies on the higher indifference curve is the one he will
choose, and nothing in the information in the question pins this down.
For interpretation 2: Jonathan will choose the optimal bet by equating his MRS
between the two contingent commodities to the slope of the BL he faces. Since the
bet has a strictly positive expected gain, we know that the slope of the BL through his
endowment point is not equal to that of his indifference curve through (60,60): he
will be strictly better off betting something. How much? Depends on the degree of
risk aversion: greater the aversion, the less he will bet.
e) Define and sketch the fair odds line associated with this bet. How much does he
bet if the odds are fair?
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Assignment 1
Ans: the fair odds line: the budget line he would face if the expected win =0. This
budget line would have the same slope as his IC on the certainty line, so he would bet
nothing.
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