Universidad Carlos III Microeconomics CONSUMER THEORY I. Preferences

Universidad Carlos III
Microeconomics
CONSUMER THEORY
I. Preferences
1. Explain why, under our axioms A1-A3 about consumer preferences, the following
statements are necessarily true:
a) Indi¤erence curves are downward sloping.
b) There must be strictly one indi¤erence curve corresponding to every bundle.
c) There can not be ”thick”indi¤erence curves.
d) Two indi¤erence curves can not cross.
2. Assume that commodity x is a “good”and commodity y is a “bad”for individual
A.
a) Draw several indi¤erence curves for this individual.
b) Consider an individual B to whom the same property applies, but he dislikes
commodity y more than individual A. Draw a few indi¤erence curves for B and explain
the di¤erence.
3. “The more Jodie Foster’s movies I see, the more I like them”. For a person with
such preferences, how does the MRS (marginal rate of substitution) between that
good and the rest of commodities vary as the consumption of movies starred by that
actress increases?.
4. “I always need 1000 milligrams of Tylenol to obtain the same relief to my pain I
get with 500 milligrams of Aspirin.”Represent some indi¤erence curves of this person
for the commodities Tylenol and Aspirin.
5. “I like my martinis with one dose of vermouth and 5 doses of gin”. Represent
some indi¤erence curves of this person for the commodities vermouth and gin.
6. Exercise 2 from Chapter 3 of PR, page 101.
II. Utility Functions, Budget Constrain
7. Assume the preferences of an individual are represented by the utility function
p
u(x; y) = xy
a)Graph the indi¤erence curve corresponding to 9 utils.
p
b) Consider the utility function, in utils, v(x; y) = 2 xy: Graph the indi¤erence
curve corresponding to 18 utils.
p
c) Consider the utility function, in utils, r(x; y) = 4 + 3 xy: Graph the indi¤erence curve corresponding to 31 utils.
d) Consider the utility function, in utils, s(x; y) = xy: Graph the indi¤erence
curve corresponding to 81 utils.
e) Compute the MRS between both goods for the previous functions.
8. a) Two items have been weighted. The …rst one weighs 50 Kilos and the second
55 Kilos. As 55/50=1.1, we say that the second item is 10% heavier than the …rst.
Is that statement still true if the weight is measured in pounds?
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b). The temperature of 2 objects has been measured. The temperature of the …rst
one is 50 degrees Fahrenheit and that of the second is 55 degrees Fahrenheit. We can
say: “the second object is 10% hotter than the …rst”. Can we keep that statement if
the measurement is made in Celsius? The temperature of a third object is 65 degrees
Fahrenheit. We say: “the di¤erence between the temperature of the third and the
second object is twice the di¤erence between the second and the …rst.” Is this true
independently of the scale of measurement?
c) For a consumer whose preferences satisfy axioms A1-A3, we have the utility
levels corresponding to 2 di¤erent bundles. The level of the …rst is 50 utils and that
of the second is 55 utils. We can say: ”the second bundle conveys 10% more utility
than the …rst one”. Is that statement still true if the measurement is made in Utils?
(note: in order to obtain Utils, we raise to the power 2 the measurement in utils).
d) The utility of a third bundle is 65 utils. We can say: ”the utility di¤erence
between the third and the second bundle doubles the di¤erence between the second
and the …rst”. Is that statement still true if the measurement is made in Utils?
Finally, we can say that the third bundle conveys a higher welfare than the second;
and the second, in turn, conveys a higher welfare than the …rst. Can we keep that
statement if the measurement is made in Utils?
9. Assume that the price of natural gas is 0.05 euros/m3 and the price of electricity
is 0.06 euros/Kilowatt per hour. However, after buying 1000 Kilowatts per hour it
falls to 0.03 euros. If the consumer has 120 euros at his disposal to spend on energy,
draw his budget set.
10. RENFE is selling a special pass that allows students to obtain a per cent discount
over the normal railway tari¤.
a) Draw a student’s budget constraint before and after buying the pass (the good
”number of travels”must be in the horizontal axis and ”the rest of all goods”in the
vertical axis).
b) Discuss whether it is false or true the following statement: ”If a student is
indi¤erent between buying the pass or the ordinary tari¤, he will spend more on
train commuting if he decides to buy the pass”.
11. In some autonomous communities, the water tari¤s have a scheme as follows:
in order to receive any water supply at all, the consumer must pay an initial tax
T , which allows the individual to consume a certain quantity of water without extra
costs. Between that quantity and x, he must pay every liter at px units, whereas for
quantities bigger than x he pays p0x < px .
a) Graph the corresponding budget constraint.
b) Do you think there can be an individual who consumes no water at all under
this scheme?
c) Given that any consumer has paid the tax T , would you expect him to consume
less than x liters of water?
d) Do you think it is possible that, under the axioms A1-A4, the individual will
be indi¤erent between two di¤erent bundles of water consumption (and the rest of
things)?
12. Exercises 3, 4, 6, 7 and 8 from Chapter 3 of P.R., page 102
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III. Consumer Choice, Demand Functions
13. A consumer has positive quantities of 2 goods x and y, and his MRS is 4. If in
the market one unit of good x can be purchased at 4 euros and good y is sold at 2
euros per unit, explain exactly what this consumer could do to raise his utility.
14. The preferences of a consumer about goods x and y are represented by the utility
function u(x; y) = x2 y:
a) Calculate his optimal consumption bundle for prices and income (px ; py ; I) =
(3; 3; 100) :
b) Compute the system of demand functions x (px ; py ; I) and y (px ; py ; I) :
c) Determine and represent the Engel curves for x and y when the prices are
(px ; py ) = (3; 3) and (px ; py ) = (1; 2). Are x and y normal or inferior goods
d) Calculate and represent the demand curve of good x for (I; py ) = (100; 3) and
(I; py ) = (500; 3): Are x and y gross substitutes or gross complements?
15. A consumer preferences over x and y are described by the utility function u(x; y) =
y + 10 ln(x): His monetary income is I = 80 and the prices of the goods are px = 1
and py = 2.
a) Calculate his optimal consumption bundle.
b) Compute the system of demand functions x (px ; py ; I) and y (px ; py ; I) :
c) Obtain and explain the income and substitution e¤ects when p0x = 2. What
kind of good is x? How would you call the relationship between both goods?.
16. A consumer preferences over x and y are described by the utility function u(x; y) =
2x + y: His monetary income is I = 15 euros.
a) Calculate his optimal consumption bundle when prices of the goods are (px ; py ) =
(1; 2); (p0x ; p0y ) = (3; 1) and (p00x ; p00y ) = (2; 1). .
b) Obtain the ordinary demand functions, x (px ; py ; I) and y (px ; py ; I) :
17. An individual owns an income I = 200 that she devotes to buying water (x) and
food (y), whose prices are px = 4 and py = 2. Her preferences over these goods are
represented by the utility function u(x; y) = min fx; yg :
a) Graph some of his/her indi¤erence curves, the budget constraint and the optimal choice, and identify her optimal consumption bundle.
b) Assume now that she must pay a tax t = 1 euro for every unit in excess of 10;
that is, if she consumes 12 units of water, for example, the …rst 10 units are charged
a price px = 4 euros per unit, and the 2 remaining units are charged px + t = 5 euros
per unit. Repeat exercise a).
18. A consumer-worker, who receives a non-wage rent of 360 euros every day, has the
following preferences over consumption (c) and leisure (h), represented by the utility
function U (h; c) = c3 h:
a) Determine which is the lowest wage per hour for which he is willing to work a
positive amount of time.
b) How many hours will he work at a wage of 4 euros per hour?
c) How much time will he work at a wage of 9 euros per hour? And at 11.25 per
hour?
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d) Determine the income and substitution e¤ect of an increase in the wage per
hour from 9 to 11.25 euros.
19. A consumer preferences de…ned over 2 goods x and y satisfy the usual properties.
Assume that the good x is inferior for him/her.
a) Given the monetary income and the price of y, use 2 graphs to derive the
demand curve of good x as a function of its own price, from the optimal solutions in
the plane (x; y).
b) Assume that the monetary income increases and derive the new demand curve
in the 2 previous graphs.
20. Jaimito, who is 5 years old, hates beets and loves chocolate. He is allowed to have
2 chocolate bars a day and, moreover, his parents o¤er him an additional chocolate
bar for every 20 grams of beets. We know that, in equilibrium, Jaimito consumes
positive quantities of both goods. Represent graphically such situation.
b) Assume that the child’s parents don’t allow him to have the 2 chocolate bars
for free a day, but keep o¤ering him the possibility of acquiring chocolate in the same
terms as in the last paragraph. Under the assumption that chocolate is a normal
good for Jaimito, can you say whether his beet consumption is bigger or smaller than
in the previous situation?. Would your answer vary is you are informed that beets
are an “inferior bad”for him?.
c) Assume now that, keeping the initial situation, his parents o¤er him one chocolate bar and a half for every 20 grams of beets. Represent the substitution e¤ect and
the income e¤ect of this change in the price of chocolate in terms of beets.
d) Would Jaimito consume more beets in situation c) rather than in situation a)
if chocolate is a normal good? Would your answer vary if you are informed that beets
are an inferior good for him?.
21. Consider the following information about the prices, income and quantities of
goods x and y consumed by a certain individual.
px
py
I
x
y
4
2 1 2 3 2
4
7 2 1 2 1
5
2 1 2 2 1
2
7 5 2 2 5
28 16 18 20 24 20 28 28 20 16 24 29
3.5 4 9 5 4 5 3.5 2 5 8 6 10
2.8 4 9 5 6 10 7
2 2 4 6 2
a) Use these observations to represent the ordinary demand of good y:
b) Represent the income-consumption curve and determine whether x and y are
normal or inferior goods.
22. Consider an individual whose preferences over 2 goods x and y satisfy the usual
properties. When px = py = 1 he consumes the quantities (x; y), whereas if px = 1:3
and py = 1 he consumes the quantities (^
x; y^), where x^ = y^. Therefore x is a normal
good. Is that true or false? Justify your answer graphically.
23. Assume that the preferences for goods x and y by two people a and b - a married
couple - are given by the utility function u(x; y) = xy: These people obtain an income
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r(a) = 200 euros and r(b) = 100 euros, in such a way that the total income of the
couple is denoted by r = 300 euros. The prices of both goods are px = py = 1.
a) Assume that the common preferences are maximized subject to the marriedcouple budget constraint. Verify that when r(a) or r(b) increase by 50 units the
married couple will make the same decision.
b) Assume now that the preferences of person a are represented by the utility
function ua (x; y) = xy; whereas those of b are given by ub (x; y) = x2 : Assume also
that the couple makes its decisions according to the weighted sum of both utility
functions, with weights equal to 1 and r(b)=r; respectively. That is, the marriedcouple utility function is now equal to u(x; y) = xy + (r(b)=r)x2 : Verify that when
r(a) increases by 50 euros the couple will make di¤erent choices than when r(b)
increases by 50 euros.
24. A consumer has preferences described by the utility function u(x; y) = 2xy:
a) Determine and represent the optimal consumption bundle if his income is I=15
and the prices of the goods are px = 2 and py = 3.
b) Compute the demand curve for good y. Which is the price-elasticity of this
good?
c) Compute the income and substitution e¤ects over good x of an increase to
0
px = 3.
d) Is good x inferior or normal? Is x a Gi¤en-good?
e) Represent the Engel-curve of x for px = 2 and py = 3.
25. A consumer considers goods x and y perfect complements. Assume that px
decreases and, as a result of it, the demand for x increases by 1 unit. Determine the
income and substitution e¤ects over the quantity demanded of good x.
26. The preferences of a consumer about goods x and y are represented by the utility
function u(x; y) = x2 y:
a) Compute the system of demand functions x (px ; py ; I) and y (px ; py ; I) :
b) Represent the budget set for the consumer and compute his optimal bundle for
prices (px ; py ) = (2; 1) and income I = 36.
c) Compute the income and substitution e¤ects over the quantity demanded of x
of an increase of it price to px = 4:
27. Exercise 12 from Chapter 3 of P.R., page 102; and exercises 1,2,8 from chapter
4, pages 138-139.
28. Lucas preferences for every day leisure and consumption are described by the
utility function u(h; c) = c (10 h)2 : His daily endowment of these goods is (24; 0).
a) Obtain the functions of labor supply and consumption demand.
b) If the real wage is 20 euros per hour, how many hours will he work and how
much consumption will he demand every day?
c) Obtain the income and substitution e¤ects over the demand for leisure of a
20% tax on wage income.
29. Suppose that a consumer-worker owns 15 hours a day available for leisure and
work, and has preferences over consumption (c) and leisure (h) represented by the
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utility function u(h; c) = ch + 2h: Assume that the price of the consumption good is
1 euro/unit.
a) Determine this individual’s equilibrium when the wage per hour of work is 4
euros.
b) Suppose now that the wage per hour is 2 euros and determine the new equilibrium.
c) Decompose the total e¤ect on the number of hours worked in terms of the
substitution and income e¤ects from the wage-change.
30. Consider a person who works 2000 hours per year at a wage of 12 euros per hour,
and therefore obtains an annual income of 24000 euros.
a) Graph his equilibrium choice and indicate which is the annual unemployment
subsidy that would induce him not to work at all.
b) We often hear that the unemployment subsidy induces people not to work.
Represent the situation of a person for which that is the case. That is, choose a wage
and a subsidy so that the person prefers not to work.
c) In the light of your answers to the previous questions, how is it possible that
in a country with unemployment subsidy there is somebody willing to work?
31. Comment and analyze graphically the three following situations:
a) Alberto devotes 40 hours per week to working and 10 to leisure. If he were
o¤ered a 500-euro-per-week unemployment subsidy he would decide not to work.
b) Marta earns a wage of 12 euros per hour, works for 20 hours and devotes
30 hours to leisure. Her …rm wants her to work longer. With that purpose she is
o¤ered the following plan: if she works for more than 30 hours, she will be given a
…xed quantity of 30 euros apart from her wage. However, Marta decides to continue
working for 20 hours.
c) Marta works for 10 hours no matter how much she is paid.
32. Pedro’s preferences for consumption of goods (c) and leisure (h) are described by
the utility function u (h; c) = h(c + 2): His total time endowment is T = 18 hours,
which he can devote either to leisure or working for a wage of w euros per hour.
Suppose that the price of the consumption good is equal to one.
a) Graph some of his indi¤erence curves.
b) Compute and represent Pedro’s labor-supply curve.
c) How much time would he work and how much would he devote to leisure if the
wage is w = 1=6?
d) A relative of Pedro’s died, leaving him an inheritance of H = 1 euro. What will
happen with the labor-supply curve and the number of hours he is willing to work if
the wage is still w = 1=6? Comment the result.
33. Robinson Crusoe lives by means of his e¤ort. He
p obtains wheat (x) from e¤ort
according to the production function x = f (e) = e: As a consumer, his preferences for wheat and leisure (h) are represented by the utility function u(h; x) = xh3 :
Robinson has 28 hours a week to distribute between leisure and work.
a) Which is his optimal combination of leisure, work and wheat?
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b) Assume now that Robinson’s technology improves, in such a way that he is
able to obtain wheat according to the production function c = g(e) = e: Will he work
more or less than before?
c) Which is the income and the distribution e¤ect derived from such a technological
improvement?
34. Miguel, a second-year student of economics, has preferences for leisure (h) and
consumption (c) described by the utility function u(h; c) = (2c + 2)h2 : The price of
the consumption good is 1 euro per unit. In order to a¤ord his expenses, Miguel
works in the library as long as he wants at a wage w = 19 euros per hour. His total
time endowment is 24 hours.
a) The Ministry of Education decides to award Miguel with a scholarship, B =
13. Determine the optimal choice for Miguel before and after being awarded the
scholarship, and represent graphically the problem.
b) Alternatively, the education authorities design a new scholarship system: they
o¤er Miguel the same number of monetary units (B = 13), but for every euro he gets
working in the library, the scholarship will be reduced by the same quantity. Compute
Miguel’s optimal choice with this new scholarship and represent it graphically.
c) Which is the cheapest system for the government? What system does the
individual prefer? Reason your answers.
35. An employee of the metal industry wants to distribute his available time between
two activities: either he works - in order to obtain the consumption good (c)- or he
enjoys leisure (h). This worker’s preferences are represented by the utility function
u(h; c) = h + ch: His income comes only from labor and we assume that the price of
the consumption good is one euro. The maximum amount of time he can work per
day is 16 hours.
a) Derive and represent graphically the labor-supply curve of this worker. For
which wage range would be optimal not to work at all?
b) The metal union decides to establish an unemployment insurance that the
unemployed workers would receive along the day. Determine graphically the budget
constraint of the average employee under this situation, and also the amount of the
unemployment insurance from which this worker would stop working if the wage were
equal to 3 euros per hour worked.
36. Pedro lives in Valladolid and his preferences for leisure (h) and consumption (c)
are represented by the utility function u(h; c). Pedro enjoys a non-labor income of M
euros and a daily time endowment of T hours. Every day Pedro decides how many
hours to work in a …rm placed at the outskirts of the city. In order to get to the …rm
he must take a free bus which takes t hours to get to the workplace. For every hour
of work he receives a wage of w euros and the consumption price is one euro.
a) Set up the problem faced by Pedro when he chooses between consumption and
leisure. Represent graphically his budget constraint and his optimal bundle of leisure
and consumption.
b) Pedro receives a labor o¤er in Madrid for an employment similar to the current
one. The o¤er him a wage w0 > w, although the daily commuting costs are also
higher (t0 > t). Represent the new Pedro’s budget constraint and his choice. Will he
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always accept the job in Madrid? Whenever this is the case, decompose graphically
the change in the optimal choice into the income e¤ect, the substitution e¤ect and
-the so-called- “city e¤ect”.
37. A consumer preferences for leisure (h) and consumption (c) are represented by
the utility function u(h; c) = h0:7 c0:3 : The wage equals 14 euros per hour of work
and the price of the consumption good is 3 euros per unit. Taking into account the
hours needed to sleep and for domestic tasks, the consumer has 15 hours per day to
distribute between leisure and work.
a) Determine his optimal leasure-consumption bundle and his utility level u .
b) Let’s assume now that there is a minimum subsistence level of the consumption
good of 28 units. That is, it is necessary for survival to consume at least 28 units.
Determine the hours worked under this restriction, u: Is u larger or smaller than u ?
(It is convenient to represent graphically the problem.)
c) Which is the non-labor income we would have to give the consumer for him to
reach both the subsistence level of consumption and the utility level u ?
38. Every consumer-worker has 1 unit of time (one day, for example) he can devote to
leisure or work, and some preferences over leisure (h) and consumption (c) described
by the utility function u(h; c) = h + ln c: Apart from his wage income, the consumer
owns a non-labor income of M euros (that is, the quantity M is independent of the
wage and the time he works). The price of the consumption good is p = 1 and the
wage is w.
a) Derive the labor supply of each individual as a function of w and M .
b) Compute the labor supply of each individual as a function of w and M .
c) Compute the aggregate labor supply, with the assumption that there are 10
identical individuals, 5 of whom with a non-labor income M = 3, and the remaining
; compute the
5 with M = 0. Given that the aggregate labor demand is LD (w) = 20
w
wage, the employment level and the consumer surplus of workers in the competitive
equilibrium. Represent the aggregate-demand and aggregate-supply curves and the
equilibrium situation in a diagram.
39. Carlos, a student of management, has received a scholarship to study the last
year of his career abroad. That scholarship allows him to own a daily quantity of
18 euros. Given that Carlos considers that this amount is not enough to cover his
expenses, he has looked for a job in which he can work at a wage per hour of 0.65
euros. Carlos’preferences for leisure (h) and consumption (c) are represented by the
utility function u(h; c) = hc2 : Normalize the price of consumption to be p = 1.
a) How many hours will he work?
b) Carlos’ parents consider that his child should not waste his time working.
For that reason, they decide to help him with a quantity that, together with the
scholarship, will only support his studies. What quantity should they give Carlos for
him not to work at all? Represent graphically this situation.
IV. Normative Aspects of Consumer Theory
40. Assume that x and y represent housing services, measured in squared meters per
year, and the rest of goods, respectively. A representative consumer has preferences
8
for those goods represented by the utility function u(x; y) = xy 2 : Prices are px = 3
and py = 1. The government proposes a subsidy of 1 euro per square meter consumed.
The opposition complains arguing that the value of the subsidy to the individual is
inferior to the cost incurred by the State. What would you recommend? Why?
41. Exercise 10 in Chapter 3 of P.R., page 102.
42. Consider a consumer whose budget set is I in the …gure and whose preferences
satisfy the assumptions A1-A4. As a consequence of an economic policy measure, his
budget set turns to II in the …gure. Observer A says: “If in the …rst situation the
individual has chosen the best consumption bundle within the subset OFGH, then
necessarily his welfare will worsen after the policy change.”Observer B says: “Not at
all. If in the …rst situation the consumer proceeds as you note, then he can worsen,
improve or stay indi¤erent after the policy change.” Who is right: A, B, neither or
both? Reason your answer.
Y
I
F
II
G
X
O
H
43. Let’s classify goods into two groups: clothes and shoes x and food y. The
preferences of a retired person who earns a pension of I = 250 euros are represented
by a utility function u(x; y) = x0:4 y 0:6 : At the prices of 1975 p0 = (1; 1), which will be
taken as a base year, he chooses the bundle q0 = (100; 150). In 1986 the prices were
p1 = (2; 3=2) and our pensioner consumed the combination q1 = (50; 100).
a) How much should the Government increase the pension to guarantee that the
pensioner keeps the welfare level reached in 1975? We will denote by I1 the new
income.
b) A true ”price index”, to summarize in a scalar the price evolution between
both dates, could be de…ned in the following way: I(p1 ; p0 ; u0 ) = I1 =I0 , where in this
case I0 = 250. Verify that the Laspeyres price index estimated in the usual way is an
upper bound for this expression.
44. It has been observed that 2 consumers spend their available income in the following way:
9
Total Expenditure Food Durable Goods Other Goods and Services
640
256
192
192
1.280
320
320
640
One year later, the growth rate of the prices of each good has been 10, 5% and 20%,
respectively.
a) Compute the Consumption Price Index (IPC) for that economy with the usual
formula.
b) Indicate who has been the most damaged person by in‡ation and why.
45. Consider the following situation of a pensioner who consumes two goods, food
(x) and clothes (y). When he got retired in 1997, the Social Security awarded a
pension of 15000 euros to him. In that year, the prices of food and clothes were 8
euros and 50 euros, respectively. Suppose that the utility function of the pensioner
p
is u(x; y) = x y:
a) Determine and represent the pensioner’s choice under those conditions.
b) Suppose that in 1998 the prices of food and clothes are 10 euros and 75 euros,
respectively. Determine and represent the choice of the pensioner in case that his
pension is not updated.
c) Which pension should we give the pensioner for him to recover his initial utility
level with the minimum cost for the social security?.
46. Let’s assume that the government wants to obtain a revenue of G euros from a
consumer whose preferences satisfy axioms A1 A3. They can create a direct tax
over individual income, or an indirect tax over the consumption of a good x. Prove
that the welfare loss su¤ered by the consumer is bigger with the second tax.
47. The preferences of an individual over leisure (h) and consumption (c) are represented by the utility function u(h; c) = hc3 : The price of the consumption good is
equal to 1 euro, the wage is 4 euros per hour and the individual can allocate 16 hours
a day between consumption and leisure.
a) How much leisure does the individual consume every day?
b) If the individual had to pay one third of his labor income as taxes, which would
be his daily consumption of leisure? Which e¤ect is bigger, the income e¤ect or the
substitution e¤ect?
c) If instead of a proportional tax over income the individual must pay 16 euros
per day, which would be his daily consumption of leisure?.
d) Which taxation system will the individual prefer?
48. The preferences of a consumer between 2 goods x and y are described by the
utility function u(x; y) = ln x+ln y: The prices of these goods are px = 1 and py = 1=2.
a) Determine the equilibrium solution for the consumer at those prices for any
income R.
b) Because of an ecological disaster, the supply of good x decreases and its price
doubles. As a consequence, the welfare of the consumer decreases. Trying to mitigate
the disaster, the local authority is willing to subsidize the consumer. Compute the
monetary quantity S that must be given to the consumer to keep the same utility
level he had before the disaster.
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c) If the authority did not know the preferences of the consumer but had observed
the quantities of both goods consumed before the disaster, they could compensate the
consumer giving him the income variation that would allow him to purchase those
quantities at the new prices. Which would be the cheapest solution for the local
authority?
V. Consumer Theory under uncertainty
49. An individual su¤ers a severe illness. The options are to follow a medication or
to have surgery. The medication permits to eliminate reasonably well the majority of
symptoms but he will have a probability of 2/3 to live 20 more years and a probability
of 1/3 to live only 10 more years. Surgery eliminates with probability 0:7 the illness
and allows him to live 30 more years. However, with probability 0:3 he will die
during the surgery.
p The patient’s preferences are represented by the Bernoulli utility
function u(x) = x, where x represents the additional years the patient is expected
to live. Which decision will he take?
50. A student has just graduated. He has just received an inheritance of 4 millions of
euros and is considering whether to invest 2 millions of euros in a start up business.
If the business is successful, he expects a gross pro…t of 6 millions of euros, but if it
fails he will lose the investment. The probability of success is p = 1=2.
a) Supposing that the student’s preferences are represented by the Bernoulli utility
functionpu(x) = x, would he try this investment? What if u(x) = x2 ? And if
u(x) = x?
b) Suppose now the student’s preferences are represented by the utility function
u(x) = x2 : A study that costs a millions of euros predicts with certainty if the
investment will be lucky or not. Should the student buy the mentioned study if
a = 1? And if a = 0:5?
c) Suppose …nally
that the student’s preferences are represented by the utility
p
function u(x) = x: Let’s suppose that, to a this student; it is o¤ered the possibility
to implement the the previous investment if he receives a tax break of b millions of
euros. Should the student accept this tax break if b = 1? And if the tax break is
b 6= 1?
51. The NBA team Memphis Grizzlies has the objective of playing the playo¤s this
season. The team’s managers are considering alternative strategies to achieve this
goal. As the club is in a …nancially stable situation – its estimated value is $2500
millions – they can a¤ord to hire Kobe Bryant. If they do so, the probability of
classifying for the playo¤s is 0.6, while without Kobe this probability is only 0.1.
The cost of hiring Kobe is $196 millions. If the team hires Kobe and makes it
to the playo¤s, then it expect to have a chance to hire Shaquille O’Neil for $228
millions. With Shaq, the Grizzlies would have a probability of winning the …nal of
0.9, while without him this probability is only of 0.3. If the team did not hire Kobe
but classi…ed for the playo¤s nevertheless, then they do not they will be able to hire
anybody before playing the matches, and the probability of winning the …nal under
these circumstances is only 0.01. Classifying for the playo¤s would increase the teams
value by $525 millions for the club, and winning the …nal would further increase the
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value by an additional $420 millions.
p The teams preferences are represented by the
Bernoulli utility function u(x) = x:
a) Describe the problem using a decision tree.
b) Suppose that the team have hired Kobe and has classify for the playo¤s, and
is now considering whether to hire Shaq or not. What should they do it?
c) Determine whether the Grizzlies should contract Kobe, or Kobe-and-Shaq, or
none.
52. Pedro Banderas has a wealth of 100 thousand euros and is considering whether
to produce a movie whose budget is 250 thousand euros. A …lm company is willing to
…nance the movie but wants Pedro to share some of the risk (and pro…ts); speci…cally
it is willing to …nance 80% of the budget. Assuming that the distributors like the
movie, Pedro expects the movie to generate box o¢ ce revenue of 250 thousand euros
if the reviews are bad, and as much as 1,5 million euros it the reviews are good. It is
known that distributors like 8 out of 10 movies that are produced, and that 1 out of
10 movies that are distributed get good p
reviews. Pedro’s preferences are represented
by the Bernoulli utility function u(x) = x.
a) Represent the decision problem and determine whether or not Pedro should
produce the movie.
b) Determine whether Pedro may be willing to …nance 40% (instead of 20%) of
the movies’budget.
53. You have to guess the result of tossing a coin (heads or tails). If you win (you
guess right), you get 10 euros and you have the choice of playing again for a maximum
of three times (in total). If you loose, you give all your earnings back and you cannot
play again. At the end of the game you have to pay 2 euros for each bet. Represent
this decision problem using a decision tree. Determine the decision that maximizes
expected utility for u(x) = x.
54. An investment may lead to the following pro…ts (in millions of euros) with the
following corresponding probabilities:
20
Pro…ts -20 -10 0
:
Pr
0.2 0.2 0.4 0.2
If the result of the investment is 20 millions, then the investor has the possibility of
making a second investment that can lead to the following pro…ts with the following
corresponding probabilities:
Pro…ts 50 -10
:
Pr
0.8 0.2
The investor preferences are represented by a Bernoulli utility function u(x) =
that satis…es the following:
p
x
x
-20 -10 0
10
20
30
40
45
50
60
70
u (x) 0 0.3 0.5 0.65 0.75 0.825 0.9 0.93 0.95 0.975 1
a) Draw the decision tree corresponding to this problem taking into account that
the pro…ts are 0 if the …rst investment is not made.
12
b) Determine whether, in accordance with the criterion of maximizing the expected utility, the investor should make each of the two investments.
c) What is the certainty equivalent and the risk premium for the second investment?
55. The oil company Tibitrol has bought some deserted land in Monegros. The
company’s geologist estimates that the probability that they will …nd oil in this land
is 0.2. The drilling of the land in order to check whether or not it really has oil costs
100 millions of euros. If they …nd oil, then the company will make revenues of 300
millions of euros. If they do not …nd oil, then the drilling will be completely useless.
The company has to decide whether or not it will do the drilling. If the company is
risk neutral what will it decide to do in order to maximize its expected utility? And
if the company is risk averse?
56. A consumer has a house with a 250,000
p euros value and her preferences are
represented by the utility function u (x) = x; where x is the wealth of the consumer
at the end of the year. The probability that the house will be totally destroyed by
an accidental …re (in which case it will loose all its value) is 0.01.
a) Would she accept to pay 3,000 euros for a full insurance of his house? What
is the maximum premium she is willing to pay for this insurance? What is the
relationship between these premium, the certainty equivalent and the risk premium
of the lottery she faces?
b) Assuming that the risk of …re is the same for all the consumers (and it is
independent among them), is 3000 euros an acceptable insurance premium for a risk
neutral insurance company? What is the minimum premium the company is willing
to o¤er?
57. The owner of a shop (which is worth 64 million euros) thinks that his shop will
be destroyed by a …re during the year with probability 1%. pThe preferences of the
shop’s owner are represented by the utility function u (x) = x; where x represents
his wealth at the end of the year.
a) Calculate the expected utility of the lottery faced by the shop’s owner and its
certainty equivalent. Would he agree to sell his shop for 60 millions of euros? What
about for 63 millions?
b) An insurance company o¤ers an annual contract that covers all the risk for 1
million of euros. Should the shop’s owner accept this contract?
c) Assume now that the shop’s owner has (in addition to his shop) 1 million of
euros in cash. A …rm proposes him to rent a …re team that will reduce the probability
of …re to 0.5%. Should the shop’s owner pay 50,000 euros for renting this …re team?
What is the maximum annual amount that he should pay in order to rent this team?
58. An p
individual whose preferences are represented by the Bernoulli utility function
u(x) = x has just insured his new motorcycle — whose value is 2,500 euro— against
theft. Motorcycle theft is very common in the town where he lives, and when a
motorcycle is stolen, it never appears afterwards. Knowing that if the price of the
insurance policy increases then the individual will not purchase insurance, and that
its actual price is 99 euro, you are asked to:
a) Calculate the certainty equivalent of the lottery faced by the individual.
13
b) If this individual received a prize of 3,600 euro, would this individual still buy
insurance?
59. A participant in a television show who is risk neutral has answered correctly to all
the questions asked so far. In the last question there remain two possible answers but
he is completely indecisive (that is, he thinks that the probabilities of each answer
of being correct are equal). Until now his accumulated gains are 361 euros. If he
abandons he wins these gains. If he decides to answer and is correct he obtains in
addition 315 euros, …nishing with a total of 676 euros. But if his answer is wrong, he
loses 261 euros, leaving with only 100 euros.
a) Should he continue playing if he is risk
p neutral? Should he continue playing
when his utility function is given by u(x) = x?
b) What is p
the certainty equivalent of the lottery when his Bernoulli utility function is u(x) = x? What is the risk premium?
60. In the market of car insurance, there are two kinds of drivers, the good ones (they
have one accident per year with probability 0.1 and no accident with probability 0.9)
and the bad ones (they have one accident per year with probability 0.1, two accident
with probability 0.05 and no accident with probability 0.85). The costs of repairing
a car are on average 2,000 euros. The proportion of good and to bad drivers is 2 to
1.
a) Assume that the insurance companies are risk neutral and they cannot distinguish between good and bad drivers. What is the minimum price that these companies
would be willing to o¤er in order to cover the risk of an accident?
b) Imagine
p that the preferences of the drivers are represented by the utility function u (x) = x; and that their initial wealth is 5,000 euros. Which type of drivers
(good and/or bad) will subscribe to an insurance policy with the minimum price
determined in part (a)?
61. A salesman preferences are represented by the Bernoulli utility function u(x) = x.
He makes sales by phone, and received a phone list of potential customers. Each day
he can make a limited number of phone calls. Each phone call costs one euro and for
each successful sale he receives 20 euros for commission. According to his experience,
he manages to speak to the right person in 3 out of 10 phone calls. Moreover, when
he manages to speak to the right person, he succeeds in making the sale in 2 out of
10 cases.
a) Draw the decision tree. Which is the expected monetary value of each phone
call?
b) The phone company also o¤ers a service called “person-to-person”. With this
service you only pay the cost of the phone call p only if you reach the person you
want. Which is the maximum price p that the salesman is willing to pay per phone
call in this new service?
62. A risk neutral person needs to put a mortgage on one of his buildings in order to
get 200,000 euros. He has to pay back this amount in 2 annual payments of 100,000
euros, each one with the corresponding interest rate. The mortgage credits among
which he could choose are:
14
Fixed interest rate: 10% per year.
Interest rate 9% in the …rst year which can increase to 14%, decrease to 8% or
remain the same in the second year.
Interest rate 7% in the …rst year which can increase to 20%, decrease to 6% or
remain the same in the second year.
a) Determine the decision which maximizes the expected pro…ts knowing that the
interest rate increases with probability 0.6 and decreases with probability 0.2.
b) How much is this person willing to pay in order to learn whether the interest
rate will increase, decrease or remain the same?
63. A consumer must choose between buying an apartment in Madrid or a house in
the suburb. Both choices would cost him 120,000 euros. He is indi¤erent between the
two options, except for his expectation regarding revaluation. If the housing prices
keep on increasing (event E1 ), the price of the apartment will reach 140,000 euros,
while the price of the house will reach 340,000 euros. The probability that this will
happen is 0.3. If the opposite thing (decrease in the housing prices) happens (event
E2 ), the price of the apartment will be 70,000 euros and the price of the house 20,000.
p
The preferences of the consumer are represented by the utility function u(x) = x,
where x is the wealth expressed in euros. The initial wealth of the consumer is 140,000
euros.
a) Represent the decision problem and determine whether the consumer should
buy the house or the apartment.
b) Should he pay 20,000 euros in order to learn whether the housing prices will
decrease or increase?
64. The introduction of a new product in the market takes includes three stages:
Design, Experimentation, and Production. 7 out of 10 products do not pass the
design stage. From those that do pass it, only 10% pass the experimentation stage
and are being produced. Only 1 out of 5 products produced has success in the market.
For each new product the costs of each stage are 100,000, 20,000, and 200,000 euros,
respectively. The expected pro…ts from a product that passed successfully the three
stages are 60 millions of euros.
a) Which is the expected value of constructing a new product?
b) For 15,000 euros a consultant can predetermine (without any uncertainty)
whether or not a product that has already passed the design stage will pass the
experimentation stage. What is the value of the consultant’s services, assuming the
entrepreneur is risk neutral?
65. The marketing chief of a big computer producer has to decide whether to launch
a new campaign before (d1 ) or after the month of May (d2 ). If he does before, he will
manage to obtain 100 millions Euros of sales. If he does after, there is a risk that its
competitor launches its own campaign before (C), which will occur with probability
0.4. Moreover, the sales also depend on the predictions of the state of the economy,
which can be good (A) with probability 0.5, stable (E) with probability 0.3, and bad
(R). If the economy is good, and the competitor has not launched its campaign,
15
sales can reach 150 millions Euros, and if its competitor did launch its campaign,
sales would reach a value of 120 millions Euros. If the economy is stable, sales would
reach 90 millions of Euros if the competitor launches its campaign and 110 millions
if it does not. Finally, when the economy is bad, and if the competitor launches its
campaign, sales will reach 70 millions Euros while they would go up to 80 millions
Euros if the competitor does not. Assuming that the producer is risk-neutral, what
is the best decision? How much would be the marketing chief ready to pay in order
to know with certainty all the uncertain variables of the problem? How much would
he be willing to give to an industrial spy who would tell him with certainty whether
the competitive …rm will launch its own campaign or not?
66. A professional has an annual wage of 250; 000 and his income tax rate is 50%.
He considers whether he should declare his full income, declaring half of his income,
or declaring nothing at all. It is known that the probability of a Hacienda inspection
is 0:1. If the inspection detects that he misdeclared his income, he will have to pay
the amount of the missing tax plus the same amount
p as a fee. His preferences are
r, where r is his income. (For
represented by the Bernoulli utility function
u(r)
=
p
negative amounts of r it is u(r) = 2
r:)
a) Draw the decision tree that corresponds to this problem.
b) Suppose now that he decides not to declare anything and that after doing so he
gets afraid of a possible inspection and asks a friend to help him. In such a situation,
he has to pay m euros in order to be sure that he will not have any problems with
the inspection. How much is he willing to pay for the service of his friend (m)?
c) Would
your answer in part (a) change if the utility function was given by
p
u(r) = r? And if it was given by u(r) = 2r?
d) Suppose now that Hacienda has already decided (before the professional makes
the declaration) the list of persons that will be inspected. His friend o¤ers to check
whether his name is on the list for 20,000 euros. Will the professional accept? Draw
the conditions for …nding out the maximum amount that the professional is willing
to pay for this information.
67. In a region there are …ve risk neutral contractors who regularly go to auctions of
electricity projects. The contract for the project is given to the one that o¤ers the
lowest price. Suppose that you are the owner of one of companies that operates as
an electricity contractor. The name of your company is Los Muhonestos. Today you
have a work lunch with the other contractors of your region. One knows that in only
10% of these meetings people talk about the “allocation”of the electricity contracts
- that is, about which contractor will make the lowest o¤er for the future contracts
(There are only 4 more contractors in the region and the decision about who is going
to be the contractor that will make the lowest o¤er is decided through a lottery
- all the contractors have the same probability of winning this lottery). Winning
the contract (i.e. being the contractor which makes the lowest o¤er) corresponds to
winning 100,000 euros in average. Alternatively, you can decide not to go to the work
lunch and to spend the afternoon playing golf in the local golf club. One knows that
there is a 50% probability of meeting someone important in the club. The average
bene…ts of such type of contacts are equal to 6,000 euros.
16
a) Determine which act (going to the work lunch or playing golf) will maximize
your expected pro…ts.
b) Which is the value of knowing in advance whether or not they will talk about
the “allocation”of the electricity contracts during the lunch?
c) Ralph Sonrisas (a business friend) has o¤ered to provide you with precise information about the topics that will be discussed during the work lunch (this way you
will be able to decide whether or not to go). Moreover, he guarantees that, in the
case that they will talk about the contract, you will be the one wining the contract.
How much are you willing to pay Ralph for his services?
17