1
Chapter 1
1.1 Suppose we have eight people who want to rent an apartment. Their reservation prices are
given below.
Person:
Price:
A
40
B
25
C
30
D
35
E
10
F
18
G
15
H
5
a. Suppose that the supply of apartments is fixed at five units. What is the range of prices that
will be the equilibrium prices?
b. Suppose that a 5-krona tax is levied on the landlords. Determine the equilibrium price and
the revenue after tax.
c. How does this compare to what happens when the tax is imposed on renters?
d. Suppose instead that the maximum rent is set to nine kronas by the government. Further,
suppose that individuals A, B, C, D and E manage to get an apartment. Is this allocation
Pareto efficient?
e. Same conditions as in d, but subletting is legal. Who will sublet to whom in equilibrium?
f. Suppose that a monopolist owns all five apartments. How many apartments will be rented
and what is the maximum revenue? Suppose that price discrimination is not allowed.
g. Suppose again that a monopolist owns all five apartments, but now that it is possible with
perfect price discrimination. How many apartments will be rented and what is the maximum
revenue?
h. If the monopolist was required to rent exactly five apartments, what price would he charge
to maximise his revenue? Assume that price discrimination is not allowed.
Chapter 2
2.1 To buy a commodity one has to pay with money and a certain amount of ration cards.
Suppose that we have two commodities A and B. The price on each commodity is 1 krona,
but in order to buy A you also need two ration cards and to buy B you need 4 ration cards.
a. Show the budget set containing all of the commodity bundles that the following individuals
can afford.
Individual 1
Individual 2
Income
9 kr
16 kr
Ration cards
24
24
b. Show that individual 2 will not spend his entire income.
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c. Another person also has an endowment of 24 ration cards, but he is allowed a discount on
commodity A. The price of A is 0.5 kr. He spends all his income and uses all his ration cards.
What is his largest possible income?
Chapter 3
3.1 Draw some indifference curves for Jens, who loves money but hates work. Assume
strictly convex preferences. (Horizontal axis: work and vertical axis: income)
3.2 Johanna likes pancakes with jam. After eight pancakes, she gets tired of pancakes and
eating more pancakes makes her less happy. However, she always prefers more jam to less.
Show Johanna’s preferences (horizontal axis: jam and vertical axis: pancakes) if:
a. Johanna dislikes her parents complaining about her not eating everything put on her plate.
b. She does not care about her parents complaints.
3.3 Mary Granola consumes two goods, grapefruits and avocados. If she has more grapefruits
than avocados, her marginal rate of substitution is 2 (2 grapefruits for 1 avocado), otherwise it
is 1/2. Draw two indifference curves for Mary. (Horizontal axis: avocado, vertical axis:
grapefruit)
Does Mary have convex preferences? Strictly convex preferences?
3.4 Ronny Rigid likes to eat lunch at 12 noon, but he is prepared to eat earlier or later if he is
sufficiently compensated for it. Draw a few of Ronny's indifference curves for money spend
on "all other goods" and dining time.
3.5 Paul is currently consuming 20 cheeseburgers and 20 Cokes a week. A typical
indifference curve for Paul is depicted below.
Coke
30
20
10
10
20
30
Cheeseburgers
a. Would Paul accept to trade one cheeseburger for one extra Coke?
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b. Would he accept to trade one Coke to get an extra cheeseburger?
c. What is the maximum number of cheeseburgers he would give up at an exchange rate of 2
cheeseburgers for 1 Coke?
d. At what rate of exchange would Paul be willing to stay put at his current consumption
level?
3.6 Markus is happiest when he has 8 cookies and 4 glasses of milk per day. When he has
more of either food, he gets sick. When he has less than his favourite combination, giving him
more makes him better off. His mother makes him drink 6 glasses of milk and only allows
him 4 cookies per day. One day, Markus's sadistic sister made him eat 13 cookies and only
gave him 1 glass of milk. Although Markus complained later to his mother, he had to admit
that he liked the diet that his sister forced on him better than what his mother demanded.
a. Draw some indifference curves for Markus that are consistent with this story.
b. Markus mother believes that the optimal amount for him to consume is 6 glasses of milk
and 4 cookies. She wishes D = (6-M)2 + (4-C)2 to be as small as possible, where M = glasses
of milk and C = numbers of cookies. Sketch a few of her indifference curves.
Chapter 4
4.1 Tomas Brolin's utility function is U(x, y) = xy
a. Suppose that Tomas originally consumed 4 units of x and 12 units of y. Draw an
indifference curve through this point.
b. Show that if he is indifferent between the bundles (x0, y0) and (x1, y1) then if you doubled
the amount of each good in each bundle, the new bundles will also be regarded as indifferent.
4.2 Martin Dahlin has the utility function U = x + y
a. What is the name for this kind of preferences?
b. Draw an indifference curve passing through the point x = 9, y = 10 and an indifference
curve passing through the point x = 16, y = 10.
c. Does the same hold as for Tomas in 4.1.b?
xy
4.3 Martha Modest has preferences represented by the utility function U = 100 and Bertha
Brassy has preferences represented by U = 1000x2y2. Draw two indifference curves for each
one and compare their preferences.
The first curve must pass through the point x = 8 and y = 2.
The second curve must pass through the point x = 6 and y = 4.
4
4.4 Bo Rund has preferences represented by the utility function U(x, y) = x2 + y2
a. Draw a few of his indifference curves.
b. Does Bo have convex preferences?
4.5 Sture has the utility function U(x, y) = min(x + 2y, y + 2x). Draw the indifference curve
that passes through the point (6,3).
Chapter 5
5.1 Fredrik's utility function is U = x(y+2). Derive the demand for y as a function of income
and prices.
5.2 Now return to Ronny Rigid in chapter 3. His favourite diner has adopted the following
policy: if you show up for lunch t hours before or after 12 noon, you get to deduct t kronas
from your bill.
Assume that Ronny has 15 kr and that lunch at noon costs 5 kr. Illustrate Ronny's budget set
in a graph, where the horizontal axis measures the time of the day he eats lunch, and the
vertical axis measures the amount of money that he will have to spend on things other than
lunch. Also, draw in some indifference curves for Ralph that will lead to 2 P.M. as being his
optimal choice of dining time.
5.3 The market price for cheese is 50 kr/kg, and the market price for marmalade is 40 kr/kg.
Leif consumes only marmalade at these prices.
Suppose that Leif has an income of 100 kr. Draw a picture (horizontal axis: cheese, vertical
axis: marmalade) that illustrates Leif's budget set and indifference curve consistent with the
above description.
What can we say about his marginal rate of substitution? Is it greater/less than?
Is it necessary for Leif's indifference curve to have the same slope as his budget line?
5.4 The Ministry of Education wants to encourage "computer literacy" in the schools.
Currently the average school devotes 100.000 kr of its 300.000 kr budget to this subject. The
following five projects have been proposed:
A. A grant of 50.000 kr to each school to spend in whatever way they see fit.
B. A grant of 50.000 kr to each school, but also require each school to spend at least 50.000 kr
on computer instruction.
C. A grant of 50.000 kr, but also require that each school to spend at least 50.000 kr more
than they are currently spending on computer instruction.
D. For each krona spent on computer education, a school will receive half a krona from the
state.
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E. Same as D, but the amount is limited to 50.000 kr.
a. Show how the different plans affect the budget set of a typical school. The horizontal axis
measures computer education and the vertical axis measures "other things".
b. Which plan would probably lead to the largest increase in utility for the school? Draw some
indifference curves to illustrate your answer.
c. Which plan would probably lead to the largest increase in money spent on computers?
Draw some indifference curves to illustrate your answer.
5.5 Find the demand functions for
Bill:
U(x1,x2) = x12 x23
Buster:
U(x, y)=x2/5y3/5
Ben:
U(x, y) = (x+1)2 (y+2)3
Barbara:
U(x1,x2) =3x1 +2x2
Beth:
U(x, y)=min{x , y}
Chapter 6
6.1 Bengt Nordlund receives 500 kr in a birthday present. He then buys one bottle of wine (w)
to celebrate his birthday. He spends the rest of the money on other things (y). His utility
function is given by U = 75w2 + 0.01y2. Determine Bengt's reservation price for a bottle of
wine. Does Bengt have quasilinear preferences?
6.2 Jan has fallen on hard times. His income per week is 400 kr, spending 200 kr on food and
200 kr on all other goods. However, he is also receiving a social allowance in the form of 10
food stamps per week. The coupons can be exchanged for 10 kr worth of food, and he only
has to pay 5 kr for such coupons. Show the budget line with and without the food stamps. If
Jan has homothetic preferences, how much more food will he buy when he receives the food
stamps?
6.3 Bengt has two passions in his life: Stamps (x1) and beer (x2). His preferences are
represented by the utility function U(x1, x2) = x1 + ln x2.
a. Solve the demand functions for beer and stamps.
b. Bengt's girlfriend Lena complains that whenever Bengt gets an extra krona, he always
spends it all on stamps. Is she right?
c. What is the price and income elasticity of demand for beer and stamps?
Chapter 7
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7.1 Here is a table of prices and choices made by a consumer.
Obs
A
B
C
D
E
p1
1
1
1
3
1
p2
1
2
1
1
2
x1
5
35
10
5
10
x2
35
10
15
15
10
a. Sketch each of his budget lines and label the point chosen in each case.
b. Indicate all of the points that you are certain are worse for him than bundle C
c. Suppose that you know that he has convex and monotonic preferences. Indicate all of the
points that you are certain are at least as good as bundle C.
d. Is his behaviour consistent with the Weak Axiom of Revealed Preference?
7.2 A table illustrates some observed prices and choices for three different goods at three
different prices
Obs
1
2
3
p1
1
4
3
p2
2
1
1
p3
8
8
2
x1
2
3
2
x2
1
4
6
x3
3
2
2
Check for WARP and SARP.
7.3 Family McEmpty spends 500 kronas on food and 250 kronas on other things. A new
welfare program has been introduced which gives them a choice between two alternatives
A. Receiving a grant of 250 kronas per week that they can spend any way they want.
B. Buying any number of 10 kr food coupons for 5 kronas apiece.
a. Draw their old budget line and the two new budget lines.
b. Suppose that food is a normal good; darken the portion of the "250 kr grant"-budget line
where their consumption bundle could possibly be.
c. Which choice would you recommend the family?
d. Would you have been able to tell the family what to do if you had not known whether food
was a normal good?
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7.4 In 1994, Anders Limpar tried to figure out if he was better or worse off playing in Everton
than playing in Cremonese in 1986. In 1986, he made 2,5 million and he earned 5,4 million in
1994. A Laspeyre Consumer Price Index, with base 1986 = 100, was 180 in 1994.
a. Assuming that Anders spent all of his income in 1986, was Anders' 1994 income big
enough to buy his 1986 bundle at 1994 prices or can't you tell?
b. What is the smallest amount of money that Anders could have made in 1994 without
changing your answer to Part a?
c. How, if at all, would your answer to part a change if the CPI were a Paasche index?
Chapter 8
8.1 Rudolf spends all of his income on wine and cigars. In each event try to figure out what it
has told you about Rudolf's preferences, i.e. is it a normal, inferior, Giffen, or luxury good.
Rudolf has convex preferences and likes both goods.
a. Rudolf finds a fifty-krona note on the street. He immediately goes out and buys 50 kr worth
of cigars.
b. Rudolf drops a one-hundred-krona note. He decides to sell the rest of his bottle of wine and
spend the money on cigars. (Hint: think step-by-step)
c. The price of wine goes up and Rudolf decides to cut back his purchases of cigars.
d. The price of wine and cigars both rise by 20%. Rudolf cuts his expenditures on both items
by the same proportion.
e. The price of cigars falls by 50%. Rudolf lowers his consumption of cigars by 5% and uses
all the extra money to buy more wine.
f. Rudolf finds an only slightly used bottle of wine. He drinks it down and does not change his
purchases of wine or cigars.
Chapter 9
9.1 Hans has a small garden where he raises eggplants and tomatoes to consume and to sell in
the market. He always consumes eggplants and tomatoes in a 1:1 ratio. One week his garden
yielded 25 kg of eggplants and 5 kg of tomatoes. At that time, the price of each vegetable was
25 kr/kg.
a. What is the monetary value of Hans's endowment? Draw his budget set. (Horizontal axis:
tomatoes, vertical: eggplants)
b. How much will Hans consume and sell/buy?
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c. Suppose that the price of tomatoes rises to 75 kr/kg. What is the value of his endowment
now? Decide his new consumption bundle.
d. What would his consumption bundle be if the price of tomatoes were 75 kr and his money
income were fixed at the original level given in a?
e. The change in demand can be decomposed into three effects. Decide the values of these
effects.
9.2 Tomas is trying to figure out how to supplement the study allowances of 500 kr a week.
He is considering a part-time job at a gas station. The wage is 50 kr per hour. His utility
function is U(C, R) = CR. The amount of leisure time that he has left after allowing for
necessary activities is 50 hours a week.
a. What is the monetary value of Tomas' endowment?
b. Draw Tomas' budget set (horizontal axis: leisure and vertical axis: consumption).
c. Set up the maximisation problem and decide optimal consumption and leisure.
d. Let M = study allowance and L = total amount of leisure time. Express his demand for
consumption as a function of study allowance and wage.
e. Express his supply function for labour as a function of study allowance and wage.
f. How many hours would Tomas work if he did not receive any study allowance?
9.3 Sven Karlsson is a plumber. He charges 100 kr per hour and he can work as many hours
as he likes.
a. Draw Sven's budget set, showing the various combinations of weekly leisure and income.
(Horizontal axis: leisure (maximum 168 hours) and vertical: income)
b. Sven chose to work 40 hours per week. A construction firm offered Sven 200 kr per hour
but Sven still chose to work only 40 hours per week. Draw Sven's new budget line and
indifference curves that are consistent with his choice of working hours.
c. Now the construction firm gave him an offer he could not refuse. They would pay him only
100 kr an hour for the first 40 hours and 200 kr for every hour of "overtime". Draw Sven's
new budget line. Will Sven work more or less than 40 hours per week with this pay schedule?
9.4 Use the Slutsky equation to explain whether Sweden would be better or worse off from
price increases in bananas and cars?
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Chapter 10
10.1 Decide whether the following statement is True or False.
"If both current and future consumption are normal goods, an increase in the interest rate will
necessarily make a saver save more."
10.2 Suppose that a consumer has an endowment of 200.000 kr each period (period 1 and 2).
He can borrow money at an interest rate of 200%, and he can lend money at a rate of 0%.
a. Illustrate his budget set.
b. The consumer is offered an investment that will change his endowment to m1 = 300.000
and m2 = 150.000. Would the consumer be better or worse off, or can't you tell?
c. If he is offered m1 = 150.000 and m2 = 300.000, is he better or worse off?
10.3 Mainy Landin has an income of 200.000 kr this year and she expects an income of
110.000 kr next year. She can borrow and lend money at an interest rate of 10%.
Consumption goods cost 1 kr and there is no inflation.
a. What is the present value of Mainy's endowment?
b. What is the future value of Mainy's endowment?
c. Suppose that Mainy has the utility function U = C1 C2. Write down Mainy's marginal rate
∆C2
of substitution
.
∆C1
d. Set this slope equal to the slope of the budget line and solve for the consumption in period
1 and 2. Will she borrow or save in the first period.
e. = d, but the interest rate is 20%. Will Mainy be better or worse off?
Chapter 12
12.1 Jonas Thern maximises expected utility:
U(π1, π2, C1, C2) = π1 C1 + π2
C2
Jonas's friend Stefan Schwarz has offered to bet him 10.000 kr on the outcome of the toss of a
coin. If the coin comes up head, Jonas must pay Stefan 10.000 kr, and if the coin comes up
tails, Stefan must pay Jonas 10.000 kr. If Jonas doesn't accept the bet, he will have 100.000 kr
with certainty. Let Event 1 be "coin comes up heads".
a. What is Jonas’s utility if he accepts the bet and if he decides not to bet? Does Jonas take the
bet?
b. Answer the question in a, if the bet is 100.000 kr.
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c. Answer the question if Jonas must pay Stefan 100.000 kr it he coin comes up head, but if
the coin comes up tails Stefan must pay Jonas 500.000 kr.
d. Klas Ingesson would also like to gamble with Jonas. He is very intelligent and realises the
nature of Jonas' preferences. He offers him a bet that Jonas will take. Klas says: "If you loose
you will give me 10.000 kr. If you win, I will give you ........?
12.2 Gabriel likes to gamble and his preferences are represented by the expected utility
function
2
2
U = π1 C1 + π2 C2
Gabriel has not worked out very well, he only have 1.000 kr. Thomas shuffled a deck of cards
and offered to bet Gabriel 200 kr that he would not cut a spade from the deck.
a. Show that Gabriel refuses the bet.
b. Would Gabriel accept the bet if they would bet 1.000 kr instead of 200 kr?
c. Sketch one of Gabriel's indifference curves (let Event 2 be the event that a card drawn from
a fair deck of cards is a spade)
d. On the same graph, sketch the indifference curve when the gamble is that he would not cut
a black card from the deck.
12.3 Thomas Ravelli has signed a 1.000.000 kr contract for playing professional football. If
an injury ends his career, he will receive a 40.000 kr contract as a cheerleader. He has a von
Neumann-Morgenstern expected utility function of the form U = C . There is a 10% chance
that Thomas will be injured.
a. What is Thomas Ravelli's expected utility?
b. If Thomas pays p kr for an insurance policy that would give him 1.000.000 kr if he is
injured then he would be sure to have an income of 1.000.000 - p kr no matter what happened
to him. Find the largest price that Thomas would be willing to pay fur such an insurance
policy. Let the cost of the insurance be equal to γK, where K is the amount that Tomas
receives from the insurance company if he is injured.
Chapter 14
14.1 Mattias has quasilinear preferences and his demand function for books is B = 15 – 0.5p.
a. Write the inverse demand function
b. Mattias is currently consuming 10 books at a price of 10 kr. How much money would he be
willing to pay to have this amount, rather than no books at all? What is his level of consumer's
surplus?
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14.2 Suppose Birgitta has the utility function U = x10.1 x20.9. She has an income of 100 and
P1= 1 and P2= 1. Calculate compensating and equivalent variation when the price of x1
increases to 2. Also, try to estimate the change in consumer's surplus measured by the area
below the demand function.
14.3 Again, consider Bengt in ex. 6.3. Calculate compensating and equivalent variation if his
income is 4 and the price of x2 increases from 1 to 2, while the price of x1 does not change.
Chapter 15
15.1 Linus has a demand function q = 10 - 2p
a. What is the price elasticity of demand when the price is 3?
b. At what price is the elasticity of demand equal to -1?
c. Suppose that his demand function takes the general form q = a - bp. Write down an
algebraic expression for his elasticity of demand at an arbitrary price p.
15.2 The demand function is q(p) = (p+1)-2
a. What is the price elasticity of demand?
b. At what price is the price elasticity of demand equal to minus one?
c. Write an expression for total revenue as a function of the price.
d. Answer a-c when the demand function takes the more general form q(p) = (p+a)b where a
> 0 and b < -1.
Chapter 16
16.1 Suppose we have the following demand and supply equations
D(p) = 200 - p
S(p) = 150 + p
a. What is the equilibrium price and quantity?
b. The government decides to restrict the industry to selling only 160 units by imposing a
maximum price and rationing the good. What maximum price should the government impose?
c. The government doesn't want the firms in the industry to receive more than the minimum
price that it would take to have them supply 160 units of the good. Therefore, they issue 160
ration coupons. If the ration coupons were freely bought and sold on the open market, what
would be the equilibrium price of these coupons?
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d. Calculate the dead-weight loss from restricting the supply of the goods. Will the deadweight loss increase or decrease if the government would not allow the coupons to be sold on
the open market?
16.2 The demand curve is qD = 100 - 5p and the supply curve is qS = 5p.
a. A quantity tax of 2 kr per unit is placed on the good. Calculate the dead-weight loss of the
tax.
b. A value (ad valorem) tax of 20 % is placed on the good. Calculate the dead-weight loss of
the tax.
Chapter 18
18.1 Suppose 5000 litres of milk can be produced by a cow fed the following combinations of
hay and grain:
Hay (kg)
5 000
5 500
6 000
6 500
7 000
7 500
Grain (kg)
6 154
5 454
4 892
4 423
4 029
3 694
a. Draw the isoquant.
b. Determine the technical rate of substitution (TRS).
c. Is the isoquant convex?
d. Suppose the price of 1 kg hay is equal to the price of 1 kg grain. Determine the minimumcost combination.
18.2 The Cobb-Douglas production function is given by f(x1,x2)=Ax1ax2b. Which values of
a+b will be associated with the different returns to scale?
18.3 Illustrate constant-, increasing- and decreasing returns to scale with the help of
isoquants. Also, give an example of each case.
18.4 Do the following production functions have constant-, increasing- or decreasing returns
to scale? Which ones fail to satisfy the law of diminishing returns?
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a.
y=4K1/2L1/2
b.
y=aK2+bL2
c.
y=min(aK, bL)
d.
y=4K+2L
e.
y=K0.5L0.6
f.
y=K0.3L0.3T0.3
Chapter 19
19.1 " ... the only reasonable long-run level of profits for a competitive firm that has constant
returns to scale at all levels of output is a zero level of profits" Varian, 324.
a. Explain why they only get zero profits in the long run.
b. What is the definition of profit in this case?
c. What role does the profit play in a market-based economy?
19.2 The short-run production function of a competitive firm is given by f(L) = 6L2/3 and w
= 6 and p = 3.
a. How many units of labour will the firm hire?
b. How much output will it produce?
19.3 If p*MP>w1 should the firm increase or decrease the amount of factor 1?
Chapter 20
20.1 "Given the economist's definition of cost as opportunity cost, the average total costs of
different firms in an industry cannot differ. If one firm, for example, had superior
management or land, it would have lower average variable costs than its competitors, but its
average fixed costs would be correspondingly higher." Discuss.
20.2 For each of the following situations, calculate the profit-maximising output level, Q*, as
well as the totals of revenue, fixed cost, variable cost, marginal cost and profit at Q*.
1. TC = 2Q + Q2
2. TC = 2 + 5Q2
TR = 3Q
TR = 6Q
Chapter 21
21.1 The cost function of a firm is C(y) = 4y2 + 16.
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a. At what level of output is average costs minimised?
b. At what level of output is variable average cost minimised?
c. At what level of output is marginal cost minimised?
21.2 For the production function Y=f(K,L)=K0.5L0.5 with PK=4 and PL=2, find the values of K
and L that minimise the cost of producing 2 units of output?
21.3 A competitive firm has a production function of the form
Y = 2L + 5K
If the cost of L, w, is 2, and the cost of K, r, is 3, what will be the minimum cost of producing
10 units of output?
Chapter 22
22.1 A firm has a cost function given by c(y)=10y2 + 1000. What is its supply curve?
22.2 If the supply curve is given by S(p)= 100 + 20p, what is the formula for the inverse
supply curve.
22.3 Earl produces lemonade using lemons, x1, and labour, x2. The production function is
f(x1,x2)=x11/3x21/3.
a. What is his cost function, c(w1,w2, y)? (Hint: Solve the factor demand functions first, then
substitute into cost equation [c=w1x1(.) + w2x2(.)])
b. If lemons cost 1 krona each and the cost of labour is 1 krona, what is his marginal cost
function, MC (y)?
c. What is his supply curve, S(p)?
d. If lemons instead cost 4 kronas and the wage rate is 9 kronas, then what is his supply
curve?
Chapter 23
23.1 If S1(p)= p – 10 and S2(p)= p - 15, then at what price does the industry supply curve have
a kink in it?
23.2 Consider a competitive market where the government imposes an ad valorem tax. What
will happen with price and quantity
a. in the short run?
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b. in the long run?
23.3 In 1990 a town had a more-or less free market in taxi services (assume perfect
competition). The constant marginal cost per trip of a taxi ride is $5, and the average taxi has
a capacity of 20 trips per day. The demand function for taxi rides is given by D(p) = 1.100 20p, where demand is measured in rides per day.
a. What is the competitive equilibrium price and quantity.
b. In 1990 the government created a taxicab licensing board and issued a fixed number of
licences to each of the existing cabs. The board stated that it would continue to adjust the taxi
cab fares so that the demand for rides equals the supply, but that they will issue no new
licenses in the future. In 1995 costs had not changed, but the demand curve for taxicab rides
had become D(p) = 1.120 - 20p. What was the equilibrium price of a ride in 1995?
c. What was the profit per ride, per taxicab license per day and per taxicab license per year.
Assume that the taxi operated every day.
d. If the interest rate was 10%, and costs, demand, and the number of licenses were expected
to remain constant forever, what would be the market price of a taxicab license?
e. Suppose that the board in 1995 decided to issue enough new licenses to reduce the price per
ride to $5. How many licenses would this take?
f. If demand is not going to grow any more, how much would a taxicab license be worth at
this new fare?
g. What is the total amount that all taxicab owners together would be willing to pay to prevent
any new licenses from ever being issued?
h. What is the total amount that consumers of taxi rides would be willing to pay to have
another taxicab licenses issued?
Chapter 24
24.1 Varian RQ 23.1
24.2 Varian RQ 23.2
24.3 Varian RQ 23.3
24.4 a. What is a natural monopoly?
b. Is natural monopoly a problem?
c. Is it possible to "solve" the problem?
d. Many old natural monopolies, like the post- and telecom services, are opened up to
competition. Explain why with the help of a figure.
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24.5 Why is a monopoly solution not Pareto efficient?
Chapter 27
27.1 The market demand curve is given by Q = 100 - P. Suppose that TC = 0 and that two
firms operate in the market.
a. Illustrate the reaction functions, according to Cournot, for firm 1 and 2.
b. If the firms were operating at the Cournot equilibrium point, what would the output and
market price be? What is the total profit?
c. What would profit maximising output and price be if the two firms decided to collude.
d. Compare the result in b and c. Comments?
27.2 Compare the assumptions in the Stackelberg model and the Cournot model.
27.3 Varian RQ 27.1
Chapter 28
28.1 Consider the following game matrix. The pay-offs in the matrix are profits.
Y
Low output
High output
Low output
X
High output
X: 15
Y: 15
X: 2
Y: 20
X: 20
Y: 2
X: 8
Y: 8
a. What is the dominant strategy of this game?
b. Is this strategy profit maximising?
c. Is it possible to change the outcome of the game so that the total profit will increase? How?
28.2 Males of certain species frequently come into conflict with other males over the
opportunity to mate with females. If a male runs into a situation of conflict, he has two
alternative "strategies":
- Play "Hawk", fight until he wins or is badly hurt.
- Play "Dove", retreat if opponent starts to fight.
17
The animal cannot tell in advance whether another animal will behave like a Hawk or like a
Dove. Suppose the conflict involves food, that contains 12 calories. When two doves interact,
they share the food so each receives 6 calories. When a hawk and a dove interact the hawk
wins and get everything. When two hawks interact the winning hawk gets everything, but the
fight itself consumes 10 calories for each hawk. This means that the winning hawk get (1210=)2 calories and the losing hawk (0-10=) -10 calories. Thinking of hawks as a whole, then,
the expected payoff from a hawk-hawk fight is (2-10)/2=-4 calories per individual.
The payoffs in this game are depicted in the matrix below.
Animal B
Hawk
Dove
Hawk
A: - 4
B: - 4
A: 12
B: 0
Dove
A: 0
B: 12
A: 6
B: 6
Animal A
a. Why can there not be an equilibrium in which all males act like Doves or an equilibrium in
which all males act like Hawks?
b. Suppose that 4/5 of the population are hawks and 1/5 doves. What is the average payoff for
a hawk and a dove?
c. Suppose that 2/5 of the population are hawks and 3/5 doves. What is the average payoff for
a hawk and a dove?
d. As we have seen there is not an equilibrium in which everybody chooses the same
strategies, and situation b and c are not equilibria. Suppose that the fraction of a large
population that chooses the Hawk strategy is p. If we assume that the males should be
indifferent between being a Hawk and Dove, solve the value of p, which describes the
equilibrium.
e. Explain why the value in d is an equilibrium.
28.3 Varian RQ 28.2
28.4 Varian RQ 28.3
Chapter 29
29.1 Let us return to our lovely childhood. Kalle and Lisa love sweets, and their grandmother
has given them a bag of sweets each. In each bag, there are 20 lollipops and 20 caramels.
Their preferences for lollipops and caramels are given below.
18
Kalle
Caramels
Indifferent alternatives 20
14
29
Better alternative
30
Lollipops
20
24
14
14
Lisa
Caramels
Indifferent alternatives 20
10
30
Better alternative
10
Lollipops
20
24
16
25
a. Is it possible for Kalle and Lisa to be better off by trading with each other?
b. If this is the case, what prices (caramels / lollipop or lollipops / caramel) are possible?
c. Is it possible for Pelle to be a middleman if Kalle and Lisa don’t realise that they can be
better of by trading?
29.2 Varian RQ 29.1
29.3 Varian RQ 29.2
29.4 Varian RQ 29.4
29.5 Explain the First and Second Welfare Theorem: meaning, assumptions and political
implications.
29.6 Is there another way, also theoretical, a Pareto-efficient allocation can be achieved?
Chapter 30
30.1 Do the first and second Welfare Theorems hold in an economy with production?
30.2 Varian RQ 30.1
30.3 Varian RQ 30.4
30.4 Varian RQ 30.5
30.5 In the pure exchange, economy we had that the set of Pareto efficient allocations was
given by the contract curve. Now we introduce production into the general equilibrium
framework. Suppose that there are only two agents Robinson and Friday and only two goods
coconuts and fish.
19
a. Suppose that Friday and Robinson produce 30 kg fish and 10 kg coconuts. Illustrate (in an
Edgeworth box) the Pareto efficient allocation.
b. Now the production is changed to 20 kg fish and 20 kg fish. How is the Pareto efficient
allocation affected?
c. What are the requirements for an economy to be in a Pareto efficient allocation?
Chapter 31
31.1 Suppose that three voters (A, B, C) rank three different alternatives in the following way:
A:
B:
C:
x, y, z
y, z, x
z, y, x
a. Which of the alternative will win under the assumption of majority voting?
b. Suppose that C:s ranking changes to z, x, y
Which of the alternative will win under the assumption of majority voting?
c. Comment on the differences between a and b.
31.2 Suppose that we have an economy consisting only of two persons, Carl and Göran. In the
figure below, some different allocations are possible. Suppose that income is a good way of
measuring their utility.
Carl
45°
E
> > D
> C
F
> B
Utility
possibilities
set
Göran
A
a. If we move from point B to point C what will happen with the income distribution?
b. If we move from point C to point D what will happen with the income distribution?
c. How would a Rawlsian isowelfare curve look like?
d. Which of the points would be the best according to Rawls?
20
e. Why do you think Rawls recommends the minimax solution?
f. Which of the points would be best according to Nietzsche? (see Varian RQ 31.2)
31.3 What is a fair allocation? Do you think that "fair allocation" is a good approach to
describe social welfare?
31.4 Varian RQ 31.4
Chapter 32
32.1 A village is located on a bay that is inhabited by lobsters. The government issues permits
to trap lobsters and is trying to determine how many permits to issue. The economics of the
situation are this:
1. It costs $2.000 dollars a month to operate a lobster boat.
2. If there are x boats operation, the total revenue from the lobster catch per month will be
f(x) = $1.000(10x - x2).
a. If the permits are free of charge, how many boats will trap lobsters?
b. What number of boats maximises total profits?
c. If the government wants to restrict the number of boats to the number that maximises total
profits, how much should it charge per month for a lobster permit?
32.2 Two firms, X and Y, have access to five different production processes, each one of
which has a different cost and gives off a different amount of pollution, see table below. If
pollution is unregulated, each firm will use process A, for a total pollution of 8 tons per day.
The city council wants to cut smoke emissions by half. The first option is to require each firm
to reduce its emission by half. The second is to use taxes.
Process
(smoke)
A
4 tons/day
B
3 tons/day
C
2 tons/day
D
1 ton/day
E
0 tons/day
Cost to firm X
100
190
600
1200
2000
Cost to firm Y
50
80
140
230
325
a. What is the total cost for the firms if required to cut pollution by half?
b. What is the pollution and the total cost if the tax is set to 50 per ton and day?
c. What is the minimum tax to achieve the goal? What is then the total cost to society?
d. Compare the solutions from a and c.
21
32.3 Smith can operate his sawmill with or without soundproofing. Operation without
soundproofing results in noise damage to his neighbour Jones. The relevant gains and losses
for the two individuals are listed in the table below:
Without soundproofing With soundproofing
Gains to Smith
150 kr/week
34kr/week
Damage to Jones 125kr/week
6kr/week
a. If Smith is not liable for noise damage and there are no negotiation costs, will he install
soundproofing? Explain.
b. How, if at all, would your answer differ if the negotiation costs of maintaining an
agreement were 4kr/week. Explain.
c. Now suppose Jones can escape the noise damage by moving to a new location, which will
cost him 120kr/week. With negotiation costs again assumed to be zero, how, if at all, will
your answer to part (a) differ? Explain.
32.4 An airport is located next to a large tract of land owned by a housing developer. Let X be
the number of planes that fly per day and let Y be the number of houses that the developer
builds. Suppose that the profits will be:
Airport: πA = 48X - X2
Developer: πΒ = 60Y - Y2 - XY
a. Suppose that no bargains can be struck between the airport and the developer. What
number of planes maximises the profit for the airport? Given that the airport is landing this
number, what number of houses maximises the developer's profits? What will their profits be?
b. Suppose that a law is passed that makes the airport liable for all damages to the developer's
property values, the total amount of damages awarded to the developer will be XY. What
number of planes maximises the profit for the airport? What number of houses maximises the
profit for the developer. What will their profits be?
c. Suppose that a single firm bought the developer's land and the airport and managed both to
maximise joint profits? What number of houses and flights maximises the profit?
Chapter 35
35.1 Varian RQ 35.3
35.2 Bob and Ray are two students who have spotted an old sofa they want to buy to their
shared apartment. Bob’s utility function is UB(S, MB)= (1+S)MB, and Ray’s utility function is
UR(S, MR)= (2+S)MR, where S=1 if they buy the sofa, and M are the amounts of money they
spend on other goods. Bob has WB dollars to spend and Ray has WR dollars.
a. What is Bob’s and Ray’s reservation prices for the sofa?
b. What is the maximum price they will pay for the sofa if WB=100 and WR=75?
22
35.3 Suppose that three persons-Helms, Bradley and Nunn- have well defined willingness-topay for three alternative projects: a new missile, a medical research project, and more aid to
the poor.
missile
medical res.
aid
Helms
100
35
-20
Bradley
-25
90
60
Nunn
45
40
95
a. Which of the alternatives will win under the assumption of majority voting?
b. Suppose that you are Helms, and that you are the chairman. Being a chairman means that
you can decide the order in which the alternatives are presented. What order will you decide?
c. Which of the alternatives will win when doing a cost-benefit analysis? (same costs)
d. Suppose instead that we only rank how much of GNP each individual wants to spend on
national defence: Helms 50%, Bradley 6% and Nunn 8%. If the alternatives under
consideration are 5, 8, 11, 20, 40 and 60%, which alternative will win?
e. Which are the pros and cons of majority voting and cost-benefit analysis?
Chapter 36
36.1 In a town, 200 people want to sell their used cars. Everybody knows that 100 of these
cars are "lemons" and 100 of these cars are "good". However, nobody except the original
owner knows which are which. Owners of lemons will be happy to get rid of their cars for any
price greater than $200. Owners of good used cars will be willing to sell them for any price
greater than $1500. A large number of buyers who would be willing to pay $2500 for a good
used car, but only $300 for a lemon. When the buyers are not sure of the quality of the car
they buy, they are willing to pay the expected value of the car, given the knowledge they
have.
a. If all 200 used cars were for sale, describe the equilibrium that would take place.
b. Suppose that everyone is aware that there are 120 good cars and 80 lemons. Describe the
equilibrium or equilibria that would take place.