possible solutions to the workouts, chp 1-15. chapter 1 1 · possible solutions to the workouts,...
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Possible solutions to the workouts, chp 1-15.
Chapter 1
1.1
Reservation price = Price where the consumer is just indifferent between consuming the
next unit of the good and not consuming it, ie; the maximum price the consumer is
willing to pay for the good.
a. 15-18 (they don't have to pay the reservation price).
b. P = 15-18 (does not alter the supply curve)
Gross: 15*5 = 75 18*5 = 90
Revenue after tax: 75 - 5*5 = 50 90 - 5*5 = 65
c. Price: 10 – 13
Net: 50 - 65
(The demand curve facing the landlord will shift 5 units lower => price 10-13.
d. Pareto efficiency means that there is no way to make some individual better off
without hurting some other.
This is not a Pareto efficient allocation since F is willing to pay more. We can give the
apartment to F and compensate E.
e. E, who is willing to pay only 10 kronas for an apartment would sublet to F, who is
willing to pay 18 kronas.
f. We have the following relationship:
P Q TR
40 1 40
35 2 70
30 3 90
25 4 100
18 5 90
The optimal price is then 25 and A, B, C and D will get apartments.
0
5
10
15
20
25
30
35
40
A D C B F G E H
Supply (5
units)
This is not Pareto efficient since F is willing to pay 18 kronas for an apartment. Given
that the monopolist does not have to charge the same price for all a Pareto improvement
is possible.
g. 40 + 35 + 30 + 25 + 18 = 148
A D C B F
5 apartments will be rented and the maximum revenue is 148.
h. 18 kr TR=18*5=90
Chapter 2
2.1
PA = 1, 2 ration cards
PB = 1, 4 ration cards.
Total income ind 1= 9 kr, Rationing cards=24
a. We can solve this by seeing that we have two restrictions, one for income so that
1A+1B=9 income budget line A=9-B
and one for ration cards
2A+4B=24 A=12-2B
Both restrictions must apply, if we draw these then;
Because both restrictions must apply the budget set is bounded by the inner (darkened)
line.
b. Individual 2 will not spend his entire income since the budget set is completely
bounded by the ration card restriction.
c. PA = 0,5.
We know that he spends all of his income and uses all his ration cards. This means that
the choice must lie on the ration card restriction. According to this restriction he can (at
the extreme), consume either 6 units of B or 12 units of A. Thus his income must be:
PA * A = 0,5*12 = 6 or
PB * B = 1*6 = 6.
Thus his income must be equal to 6. (Any point, not only extremes on the ration card
restriction line may be taken)
A
9
12
Rationing card
constraint
Budget constraint
9 6
A
12 Rationing card constraint
Budget constraint
16
6
16
Chapter 3
3.1
Work is a ’bad’. This means that if we give Jens more work, we have to have to give
him some extra income to compensate him.
Income
Work
3.2
a. If we give her more than 8 pancakes, the pancakes are ’bads’ for Johanna. This means
that we have to give her some extra jam to compensate her (jam being always a ’good’).
Jam
Pancake
8
M ore jam compensate
b. In this case more than 8 pancakes doesn't change Johannas' utility. (she just leaves
them on her plate)
Jam
Pancake
8
3.3
Draw a 45° line that reflects the situation where the consumption of avocado is equal to
the consumption of
grapefruit. To the left of this line the slope of the indifference curve dA
dG is -2 and to the
right of this line the slope is - 1
2.
45°
Avocado
Grapefruit
She has convex preferences, but not strictly convex preferences. Since she is indifferent
to some of the weighted averages (when taken on only one side of the 45o line).
Convex preferences:
Averages are preferred to extremes.
If (x1, x2) (y1, y2) ==>
(tx1 + [1-t]y1, tx2 + [1-t]y2) (x1, x2) where 0 < t < 1.
Strict convexity means that the weighted average of the two indifferent bundles is
strictly preferred to the two extreme bundles.
3.4
If we change the lunch time from 12 noon, we have to compensate him with other
goods.
Other goods (income)
Lunch time
12
3.5
The slope (dBurgers
dCoke) is currently 0,5 (2 burgers for 1 Coke).
a. Yes, since he is willing to pay with 2 burgers for 1 Coke.
b. No, he would require at least 2 cheeseburgers to give up one coke.
c. 20 -10 = 10 burgers. (to the level when he has 10 burgers and 25 cokes)
d. 2 burgers for 1 Coke.
3.6
a. We have three different points: (cookies, milk)
(8,4) (13,1) (4,6)
We have a case of satiation: there is some overall best bundle, at (8,4).
b.
M C
D = 0 if 6 4
D = 4 if 8 4 (6-8)2 + (4-4)2 = 4
4 4
6 2
6 6
D = 9 9 4 (6-9)2 +(4-4)2 = 9 3 4
6 1
6 7
Alternately, D is the equation for a circle around point (4,6).
Milk
Cookie8
4
13
1
6
4
Mother
Chapter 4
4.1
U(x,y) = xy
a. U = 4*12 = 48 ; which implies that the indifference curve must move through this
point and all others where U(x,y)=xy=48; so that the equation for the indifference curve
in the x/y plane will be
==> y = x
48 , ex: U(1,48)=U(2,24)=U(8,6)=(48,1)=48 etc…. graphically the
indifference curve for U=48 is:
0 20 40 600
20
40
60
y( )x
x
b. If we have that (x0,y0) (x1,y1), then we also must have that x0 y0 = x1 y1 (all
according to the utility function). Now we double the amount of each good:
U(2xo,2y0) = 2xo 2y0 = 4 x0 y0
U(2x1,2y1) = 2x1 2y1 = 4 x1 y1
==> 4 x0 y0 = 4 x1 y1 (the utility function is homogeneous)
4.2 U = x + y
a. Quasilinear preferences. The indifference curves are vertical transfers of one
another.
b. The general equation for the indifference curves is: y=U- x; When x=9, y=10, then
U(9,10)=13 and the indifference curve for U=13 is y=13- x, while for U(16,10)=14 the
indifference curve is y=14- x
x y U
4 11 13
9 10 13
16 9 13
16 10 14
9 11 14
4 12 14
c. Martin is indifferent between (9,10) and (4,11). Now we double the amount of each
good:
U(18,20) = 18 +20 24,24 and U(8,22) 24,83
==> The same does not hold as for Thomas.
Observe that the utility functions are not homogeneous.
4.3
UM = xy
100 UB = 1000x2y2
==> y = x
U100 M ==> y = 2
B
x1000
U
U x y U x y
0,16 8 2 256' 8 2
4 4 4 4
10 1,6 10 1,6
0,24 6 4 576' 6 4
3 8 3 8
8 3 10 1,6
ie: for positive values of U as is in this case, the utility functions are monotonic
transformations and have the same indifference curves even though the ’numbers’ of the
curves are different.
UB= 72M 10*U
4.4
U = x2 + y2
==> y = U - x2
Set U to 2 arbitrary values, graphing the indifference curves we get for example:
b. No, concave preferences.
4.6
For Sture the utility is given by the combination (x + 2y or y + 2x) that gives the lowest
value.
x + 2y y + 2x U
U(6,3) 6 + 2*3 = 12 3 + 2*6 = 15 12 (the first)
U(4,4) 4 + 2*4 = 12 4 + 2*4 = 12 12 (both)
U(8,2) 8 + 2*2 = 12 2 + 2*8 = 18 12 (the first)
U(2,8) 2 + 2*8 = 18 8 + 2*2 = 12 12 (the second)
An easy way to solve this is to sketch the curves for U=12 separately, ie y=6-0,5x and
y=12-2x. The outer bound must be the relevent indifference curve (because more is
better than less, then increasing either x or y from the lower curve will lead to a utility
>12) y
x6
4
3
3
6
4
Chapter 5
5.1
The problem can be written as
Max U = x(y+2)
s.t
m = Py y + Px x
Can be solved using ex: Lagrange or the necessary condition that MUxMUy
= PxPy
, for the
latter:
y
x
First we derive the marginal utilities:
dx
dU = MUx = y + 2
dy
dU = MUy = x
Set the marginal rate of substitution equal to the slope of the budget line:
MRS = dx
dy =
MUxMUy
= PxPy
y+2
x =
PxPy
(solve for x)
x = Py(y+2)
Px (insert this into the budget restriction)
m = Py y + Px Py(y+2)
Px (Px is cancelled out)
m = Py y + py y + 2Py = 2Py y + 2Py (solve for y)
y = m - 2Py
2Py =
yP2
m- 1 (insert this into the expression for x)
x = X
yy
P
)21P2
m(P
= m
2Px +
PyPx
Thus demand for y is:
y =
yP2
m - 1
The demand for x is:
x = m
2Px +
PyPx
5.2
Income = 15
Price at 12 = 5
Discount: t kronas per hour.
5.3
Cheese: 50 kr/kg
Marmalade: 40 kr/kg
a. The slope of the budget line M
c
P
P
dC
dM ==>
50
40 = -1,25
If Leif wishes to buy 1 kg cheese, he must abstain from 1,25 kg marmalade. But as he is
not willing to buy any cheese, his marginal rate of substitution must be is less than 1,25
(indifference curves are ’flatter’ than the budget line).
In this case we have a boundry (corner) solution, and it need not be so that the slope of
the budget line is equal to the slope of the indifference curve as is generally the case
with an interior solution.
12
14
cheese
marmelad
e
2
2,
5
Budget line
Max indifference curve
obtainable
Lunch time
kr
15
5.4
A: max 350' other and 350' computers.
B: max 300' other and 350' computers
min 50' computers.
C: max 200' other and 350' computers
min 150' computers.
D: max 300' other and 450' computers
E: max 300' other and 350' computers
A kink at 150' computers and 200' other (the maximum grant)
5.5
Bill: u(x,y) = x2y3
Use for example Lagrange.
Max x2y3
s.t
Pxx+Pyy=m
L= x2
y3+ ( Pxx+Pyy-m)
Set
0d
Ld
dy
Ld
dx
Ld and solve
From the first two we get
y = 3 Px x
2 Py (substitute this into the budget restriction)
m = Py 3 Px x
2 Py + Px x =
3Px x
2 + Px x =
5
2 Px x (solve for x)
x = 2m
5Px (substitute this into the expression for y)
y = 3 Px 2m
2 Py 5Px =
6m
10Py =
3m
5Py
Buster: u(x,y) = x2/5y3/5
Same demand functions as for Bill. The utility function is just a monotonic
transformation of Bill’s. Check for yourself! What you put into the budget constraint is
the same as above.
Ben: u(x,y) = (x+1)2 (y+2)3
This may be solved using Lagrange or by using the necessary condition.
dx
dU = 2(x+1) (y+2)3
dy
dU = (x+1)2 3(y+2)2
2(x+1) (y+2)3
(x+1)2 3(y+2)2 =
PxPy
(simplify)
2(y+2)
3(x+1) =
PxPy
(solve for y)
2y = PxPy
3(x+1) - 4
y = PxPy
3(x+1)
2 -
4
2 (substitute this into the budget restriction)
m = Px x + Py (PxPy
3(x+1)
2 -
4
2) (simplify)
m = Px x + Px 3(x+1)
2 - Py 2
m = Px x + 3
2 Px x +
3
2 Px - 2 Py (solve for x)
Px x + 3
2 Px x = m -
3
2 Px + 2Py
5
2 Px x = m -
3
2 Px + 2Py (divide both sides by
5
2 Px)
x = 2
5 m
Px -
3
2 2
5 PxPx
+ 2 2
5 PyPx
x = 2
5 m
Px -
3
5 +
4
5 PyPx
(substitute this into the expression for y)
y = PxPy
3(2
5 m
Px -
3
5 +
4
5 PyPx
+ 1)
2 - 2
y = PxPy
3
2 2
5 m
Px -
PxPy
3
2 3
5 +
PxPy
3
2 4
5 PyPx
+ PxPy
3
2 1 - 2 (simplify)
y = m
Py 6
10 -
PxPy
9
10 +
12
10 +
PxPy
3
2 - 2
y = m
Py 3
5 +
6
10 PxPy
- 8
10
y = m
Py 3
5 +
3
5 PxPy
- 4
5
Barbara perfect substitutes
Two cases: i) If the budget line is flatter than the indifference curve
she consumes only x1 (when p1/p2<3/2 or 2p1<3p2).
ii) If the budget line is steeper than the indifference curve she consumes only x2
132
132
132
2
2
21
21
21211
1
pp0
pppmand0betweennumberany
ppp
m
x
p5.1p0
p5.1ppmand0betweennumberany
p5.1pp3p2p
m
x
2
21
2
1
when
when /
en wh
when
when /
or en wh
Beth perfect complements
always consumes yx pp
myx
Chapter 6
6.1
q = 10 - 2p
a. _q
_p = - 2
p = 3 ==> q = 4
p = - 2 3
4 = -1,5
b. p = - 2 p
10 - 2p = -1 ==>
-(10-2p) = -2p
4p = 10
p = 2,5 ==> q = 5
c. q = a - bp
_q
_p = -b
= - b p
a-bp =
-bp
a - bp
6.2
y = 400
10 food stamps per week, price 5 kr. Can be exchanged for 10 kr worth of food.
Old budget line: max 400 other and 400 food.
New budget line: max 400 other and 450 food
kink at 100 food and 350 other.
Homothetic preferences: The income expansion path is a straight line through origin.
Since Jan spent half his income on food, he will continue doing so. His income
increases by 50 kr, thus he spends 25 kr more on food.
6.3
Solved in class!
Chapter 7
7.1
b. Price Set Bundle Cost
A B C D E
A 40 451 25* 20* 20* A f C,D,E
B 75 55 40* 35* 30* B f C,D,E
C 40 45 25 20* 20* C f D,E
D 50 115 45 30 40
E 75 55 40 35 30
The entries in each row measure how much the consumer would have spent if he had
purchased the bundle. Thus: 1) PA
1 XB
1 + PA
2 XB
2 = 1*35 + 1*10.
The combinations marked with a star was affordable when another bundle was
purchased. Thus that combination is revealed not preferred to the original bundle.
c. All points that could have been chosen by the consumer but were not are directly
revealed as worse. We also know that all combinations on and under the budget set D
and E are indirectly revealed as worse.
d. The consumer is consistent with WARP. No direct inconsistency.
7.2
1 2 3
1 28 27 30 12
2 33 32 30 23
3 13 17 16 31
Does not violate WARP (directly OK), but does violate SARP (indirectly not OK:
123 but 31).
7.3&7.4
Solved in class!
Chapter 8
8.1
a. Increased income:
Wine unaffected. w
m = 0
Cigar is a luxury good. c
m > 1.
Luxury good: if the demand for a good goes up by a greater proportion than income.
This must be the case if Rudolf consumes both wine and cigars.
b. Decreased income:
Decreased consumption of wine, normal good, w
m > 0
Increased income when he sells the bottle:
Increased consumption of cigars, luxury good.
c. Increased price of wine:
Decreased consumption of cigars:
The substitution effect: increased consumption of cigars
The income effect: decreased consumption of cigars.
The income effect > substitution effect.
Cigars is a normal good.
d. If all prices change with the same degree this is exactly the same as a change in
income:
m = 1,2 p1 x1 + 1,2 p2 x2
m = 1,2 (p1 x1 + p2 x2)
Fel! = p1 x1 + p2 x2
Both goods change with the same amount: = 1 for both.
e. Decreased price of cigars:
Decreased consumption of cigars: cigar is a Giffen good.
Increased consumption of wine: wine is a luxury good.
f. This can also be seen as an increased income (wine is his endowment)
Cigars is unaffected: m = 0
Everything of his increased income goes to wine, luxury good: > 1.
Chapter 9
9.1
a. The monetary value of the endowment:
PE WE + PT WT = 25*25 + 25*5 = 750
b. The 45° line shows the combinations where Hans consumes in a 1:1 ratio.
See overhead.
He will consume 15 kg of each good. ==> He will sell 10 kg eggplants.
c. Value of endowment: 25*25 + 75*5 = 1.000
The new budget line: max 40 eggplant and max 1.000
75 _ 13,3 tomato.
See overhead.
He will consume 10 kg of each good.
d. Consumption if PT = 75 and m = 750.
New budget line: max 30 eggplant and 10 tomato.
See overhead.
He will now consume 7,5 kg of each good. Note that this is the ordinary income effect
since we have held the value of the endowment constant.
e. _T = _Ts + _Tm + _TE
_Ts = 0 since eggplants and tomatoes are complements.
The total change _T is (10 - 15) = - 5 kg.
The ordinary income effect _Tm is (7,5 - 15) = -7,5 kg.
==> _T = 0 - 7,5 + 2,5 = - 5
9.2
a. Maximal consumption: 500 + 50*50 = 3.000. Minimum consumption 500.
See overhead.
b. Value of endowment: 500 + 50*50 = 3.000
c. R = Leisure
L = Labour supplied
Budget: pC = M + wL
pC - wL = M (add w L on both sides)
pC + w( L - L) = M + w L ( L - L = R and L = R )
pC + wR = M + w L
Maximise U(C,L) = CR
s.t. pC + wR = M + w L
Assumption:
Pc = 1.
First we find the MRS:
_U
_C = R
_U
_R = C ==> MRS =
MUC
MUR =
R
C
Set MRS equal to the slope of the budget line:
MUC
MUR =
PC
PR ==>
R
C =
1
w
R
C =
1
50
C = 50 R
Insert this into the budget restriction:
pC + wR = 3.000 (p = 1 and C = 50R)
1 * 50R + 50R = 3.000
R = 30 hours
Labour supplied: L = L - R = 50 - 30 = 20 hours.
Consumption = C = 50R = 50*30 = 1.500
d. The budget restriction is
pC + wR = M + w L
In optimum we have that: C = wR (insert this into the budget restriction)
wR + wR = M + w L
2wR = M + w L
R = M + w L
2w (insert this into the expression for consumption)
C = w M + w L
2w =
M + w L
2
e. Labour supplied:
L = L - R = L - M + w L
2w
= 2w L
2w -
M + w L
2w =
2w L - M - w L
2w
= w L - M
2w
f. Labour supplied if M = 0:
L = w L - M
2w =
50*50 - 0
2*50 = 25
9.3
a. Max consumption: 168*100 = 16.800.
See overhead.
b. See overhead.
c. Sven will work more. There is no income effect in the case of over-time, only a
substitution effect, since there is no higher payment for all hours, only for the extra
hours.
9.4
a. Straight line: max consumption 80*40 = 3.200
b. He can work for 25 hours with no tax.
Two kinks: 80 hours, 1.000 and 55 hours, 2.000
max consumption 1.200 + 2.000 = 3.200
c. Max consumption: 40*0,5*80 = 1.600
d. A kink at 60 hours, 800
Max consumption 2.000
e. A jump at 60 hours from 400 to 1.400
Max consumption 2.600.
See overhead.
Chapter 10
10.1
a. Wrong: the consumption (c1) can decrease.
_C1
_r =
_Cs
1
_r + (m1 - C1)
_Cm
1
_m
Saver means that (m1 - C1) > 0
Normal good means that _C
m
1
_m > 0 (for a saver the income increases when the interest
rate increases)
But the substitution effect is negative: _C
s
1
_r < 0.
We can view the interest rate as the price of consumption. If the interest
increase this means that the price of consumption today increases.
==> We cannot say what sign _C1
_r will have.
b. True: the consumption (c2) increases.
_C2
_r =
_Cs
2
_r + (m1 - C1)
_Cm
2
_m
The general Slutsky equation can be written as:
_xi_pj
= _x
s
i
_pj +
_xm
i
_m (wj - xj)
Saver means that: m1 - C1 > 0
Normal good means that _C
m
2
_m > 0 (for a saver the income increases when the interest
rate increases)
The substitution effect is also positive since the price of consumption tomorrow
decreases.
==> _C2
_r > 0.
10.2
m1 = 200 000 and m2 = 200 000
Interest rate borrower: 200%
Interest rate lender: 0%
a. C2 = m2 + (1+r)(m1 - C1)
Maximum consumption in period 1:
200 000 +
r1
200000 = 200 000 +
21
200000 = 266 667
If she borrows 66 667 she will have to pay (1+r) 66 667 in period 2.
Maximum consumption in period 2:
200 000 + 200 000 = 400 000
See overhead.
b. New endowment: m1 = 300.000 and m2 = 150.000.
Maximum consumption in period 1:
300.000 + 150.000
3 = 350.000
Maximum consumption in period 2:
300.000 + 150.000 = 450.000
See overhead.
The consumer will be better off. The consumption that she did choose before the change
in the endowment is still affordable.
c. m1 = 150.000 and m2 = 300.000
Maximum consumption in period 1:
150.000 + 300.000
3 = 250.000
Maximum consumption in period 2:
150.000 + 300.000 = 450.000
See overhead.
b is better or at least as good as c.
If he is a saver then c is better than a. Better otherwise we cannot say if a or c is the
better alternative.
10.3
m1 = 200.000 m2 = 110.000
r = 10%
a. PV = 200.000 + Fel! = 300.000
b. Future value = 200.000 (1+0,1) + 110.000 = 330.000
c. U = C1 C2
_U
_C1 = C2
_U
_C2 = C1
MRS = _C2
_C1 = -
MU1
MU2 = -
C2
C1
d. The budget constraint:
C2 = m2 + (1+r)(m1 - C1)
C2 = m2 + (1+r)m1 - (1+r)C1
Slope: _C2
_C1 = -(1+ r) = -(1 + 0,1)
Set this equal to the MRS:
- C2
C1 = -(1+0,1) (solve for C2)
C2 = 1,1 C1 (insert this into the budget constraint)
1,1C1 = 110.000 + (1+0,1)200.000 - (1+0,1)C1 (solve for C1)
2,2C1 = 330.000
C1 = 150.000 (insert this into the expression for C2)
C2 = 1,1 * 150.000 = 165.000
She will save (200.000 - 150.000) 50.000 in the first period.
e. If r = 20% then C2 = 1,2C1 (insert this into the budget constraint)
1,2C1 = 110.000 + (1+0,2)200.000 - 1,2C1 (solve for C1)
C1 _ 145.833
C2 = 1,2 * C1 = 175.000
Utility in d: U = 150.000 * 165.000 = 24.750.000
Utility in e: U = 145.833 * 175.000 = 25.520.775
She is better off if the interest rate is 20%.
10.4
a. No interest income tax
C2 = m2 (1-t) + (1+r)[(1-t)m1 - C1]
b. Interest income tax
C2 = m2 (1-t) + {1+[r(1-t)]}[(1-t)m1 - C1]
c. Investment in pension plan = X = m1 - t(m1 - X) - C1 (only tax on the amount that
is not put into the pension plan)
(solve for X)
X(1-t) = m1 - tm1 C1
X = m1(1-t) - C1
(1-t) (insert this into the budget constraint)
C2 = (1-t)m2 + (1-t)[X(1+r)]
= (1-t)m2 + (1-t)(1+r) m1(1-t) - C1
(1-t)
= (1-t)m2 + (1+r) [m1(1-t) - C1]
The same budget constraint as in a.
Chapter 11
11.1
a. Price = Fel! = 1.100.000
b. Price = Fel! = 1.000.000
c. a) 110' or 10% b) 100' or 10%
11.2
W(t) = 100 e0.20t - 0,001t2
r = 5%
a. We want to find the rate of growth of the market value and set this equal to the
interest rate.
W'(t) = (0,2 - 0,002t) e0.20t - 0,001t2
Rate of growth = W'(t)
W(t) = Fel! = 0,2 - 0,002t
Set this equal to r:
0,05 = 0,2 - 0,002t
t = 75
The trees should be 75 years.
b. When is the rate of growth equal to zero
W'(t)
W(t) = 0
0,2 - 0,002t = 0
t = 100
11.3
a. If we mine today we receive PG - 25 and if we mine next year we receive Fel!. (150 -
25 is the revenue if we mine next year).
Sellers must be indifferent between these two alternatives:
PG - 25 = Fel!
PG = 138,6
b. Sellers must be indifferent between selling today or in ten years:
PG - 25 = Fel!
PG = 73,2
What will happen with the price when we come closer to the year when the gold will be
exhausted? The price will be rising.
Chapter 12
12.1
U(_1, _2, C1, C2) = _1 C1 + _2 C2
a. UNo game = 100.000 = 316,2
UGame = 0,5 90.000 + 0,5 110.000 = 315,8
Jonas doesn't take the bet.
Declining marginal utility for money ==> risk aversion.
b. C1 = 0 C2 = 200.000
UGame = 0,5 200.000 = 223,6
Jonas doesn't take the bet, since UNo game > UGame
Nytta
U(15)
U(10)
U(5) 0,5 U(5) + 0,5 U(15)
Koncave utilityfunction
Wealth
c. C1 = 0 C2 = 600.000
UGame = 0,5 600.000 = 387,3
Jonas take the bet, since UGame > UNo game
d. C1 = 0 C2 = X
UGame = 0,5 0 + 0,5 X
UNo game = 100.000
Set this two equal to find the value of X where Jonas is indifferent between taking the
bet or not:
0,5 0 + 0,5 X = 100.000
X = Fel!
X1/2 = Fel!
X = Fel!
X = 400.000
That is: "If you win I will give you 300.000."
12.2
U = _1 C2
1 + _2 C2
2
a. UNo game = 1.0002 = 1.000.000
C1 = 800 C2 = 1.200 _1 = 0,75 _2 = 0,25
UGame = 0,75 * 8002 + 0,25 * 1.2002 = 840.000 < 1.000.000
b. UGame = 0,75 * 02 + 0,25 * 2.0002 = 1.000.000
He is indifferent between taking the bet or not.
c. UGame = UNo game
0,75 * C2
1 + 0,25 * C2
2 = 1.000.000
C2
1 = Fel! - Fel! CFel!
C1 = Fel!
C2 C1
0 1154,7
500 1118
1.000 1.000
1.500 763,8
2.000 0
See overhead.
d. UGame = UNo game
0,5 * C2
1 + 0,5 * C2
2 = 1.000.000
C2
1 = Fel! - Fel! CFel!
C1 = 2.000.000 - C2
2
C2 C1
0 1.414,2
500 1.323
1.000 1.000
1.414,2 0
See overhead.
12.3
a. E(U) = 0,9 1.000.000 + 0,1 40.000 = 920
b.
1000000 920p
1.000.000 - p = 9202
p = 1.000.000 - 9202
p = 153.600= K
This is the maximum price he is willing to pay for an insurance. We can also find (the
price per insured krona) and K:
Completely insured when C1 = C2
1.000.000 - K = 40.000 + K - K
1.000.000 - 153600 = 40.000 + K - 153.600
K = 960.000
= 153.600
960.000 = 0,16
Let us also take a look at the insurance company. Their expected profit is:
Profit = K - B K
Profit = K( - B)
At break-even we have:
= B ==> = 0,1
The difference between 0,16 and 0,1 is the insurance companies' expected profit.
Chapter 14
14.1
a. P = 30 - 2B
b. See overhead.
Willingness to pay: 10*10 + (30-10) * 10
2 = 200
Consumer surplus: 30-20
2 10 = 100
14.2
U = x0.1
1 x20.9
2 . This is a Cobb-Douglas utility function.
m = 100 and Px1 = 1 and Px2 = 1.
Since this is a Cobb-Douglas utility function we use earlier information and state that
the demand functions are:
x1 = m
P1 x2 =
(1- )m
P2
The demand before the price changes is
x1 = Fel! = 10
x2 = Fel! = 90
The utility is approximately:
U = 100,1 * 900,9 =72,247
The demand changes when the price changes:
x1 = Fel! = 5
x2 = 90
The utility is now:
U = 50,1 * 900,9 =67,409
Compensating Variation
CV gives the minimum amount of money that must be given to a household while
leaving it just as well off as it was before a rise in prices.
We want to find the income (=m') when P1 = 2 and P2 = 1, where U = 72,247.
The demand for x1 and x2 are
x1 = Fel! x2 = Fel!
Insert these into the utility function:
U = ( Fel!)0,1 ( Fel!)0,9 (factor out m')
U = (m')0,1+0,9 ( Fel!)0,1 ( Fel!)0,9
U = m' 0,050,1 0,90,9 (U = 72,247 = original utility)
m'= 107,178
CV = m' - m0 = 107,178 - 100 = 7,178
If Birgitta receives 7,178 she is just as happy as before the price increase.
Equivalent Variation
EV gives the maximum amount of money that must be taken from a household to make
it as well of as it would have been after a rise in prices. (N.B. utility decreases when the
price increases)
We want to find the income (m'') when P1 = 1 and P2 = 1, where U = 67,409 (the utility
if the price increase occur).
The demand for x1 and x2 are:
x1 = Fel! x2 = Fel!
Insert these into the utility function:
U = ( Fel!)0,1 ( Fel!)0,9 (factor out m')
U = (m'')0,1+0,9 ( Fel!)0,1 ( Fel!)0,9 (U = 67,409)
m''= 93,304
EV = m0 - m'' = 100 - 93,304 = 6,696
If we take 6,696 from Birgitta she is just as happy as after the price increase.
Consumer Surplus
The demand function: x1 = Fel! = Fel!
P1 Q1
1 10
1,2 8,33
1,4 7,14
1,8 5,56
2 5
The change in consumer surplus is the area under the demand curve between the price
change.
We can also find the exact value (using the ln rule):
1
2
10
P1 dP1 = 10 ln P1 |21 = 10 ln 2 - 10 ln 1 = 6,931
The change in the consumer surplus lies between CV and EV.
Not sure:
Price increase:
EV < CS < CV
Price decrease:
CV < CS < EV
Chapter 15
15.1
U = 75w2 + 0.01y2
Budget restriction:
pv + y = 500 ==> y = 500 - pv
Utility if v = 0:
U = 0,01 * 5002 = 2.500
Utility if v = 1:
U = 75 + 0,01 (500 - p)2
The reservations price is by definition the price where:
U(0,500) = U(1,500 - p)
75 + 0,01 (500 - p)2 = 2.500 [(x - y)2 = x2 + y2 - 2xy]
75 + 0,01(5002 + p2 - 1.000p = 2.500
2.575 + 0,10p2 - 10 p = 2.500
0,01p2 - 10p + 75 = 0 [ax2 + bx +c
x = -b± b2 - 4ac
2a ]
p = Fel!
p1 = 992,4 p2 = 7,56
His reservation price is 7,56.
Alternative solution:
75 + 0,01 (500 - p)2 = 2.500
(500 - p)2 = Fel!
500 - p = 242.5000,5
p = 500 - 492,44
p = 7,56
15.2
q(p) = (p+1)-2
a. = _q
_p p
q
_q
_p = -2(p+1)-3 ==> = - 2(p+1)-3
p
(p+1)-2 = - 2(p+1)-1 p = - 2p
p+1
b.
- 1 = - 2p
p+1
p +1 = 2p
p = 1
c. TR = pq = p (p+1)-2 = p
(p+1)2
d. q(p) = (p+a)b a > 0 b < -1
a. = -b(p+1)b-1 p
(p+a)b = -b p
p+a
b. -b p
p+a = -1
bp = p + a
(b-1)p = a
p = a
b-1
c. TR = p(p+a)b
Chapter 16
16.1
a. D(p) = S(p)
200 - p = 150 + p
p = 25
D(p) = 200 - 25 = 175
b. When is S = 160:
160 = 150 + p
p = 10
c. Willingness to pay at q = 160:
160 = 200 - p
p = 40
Maximum price - minimum price = 40 - 10 = 30 = price of the coupons.
d. See overhead.
Dead-weight loss of restricting the supply:
DW = 30 (175-160)
2 = 225.
Dead-weight loss if market we have a market for coupons will be lower (comparison to
exercise 1.1f). We do not know whether the people with the largest willingness to pay
receives the coupons initially.
16.2
qD = 100 - 5p qS = 5p
pS = qS
5
No tax:
100 - 5p = 5p
p = 10
q = 50
Tax:
pS = qS
5 + 2 ==> q = 5p - 10
100 - 5p = 5p - 10
p = 11
q = 45
The revenue per unit is (according to the old supply curve):
p = 45
5 = 9
N.B. the price increase is not equal to the tax.
See overhead.
Dead-weight loss:
(50-45)* (11-9)
2 = 5
16.3
= -1 ==> TR = pq is constant.
p = 10 and q = 6.000
a. TR = 10*6.000 = 60.000
==> q = TR
p =
60.000
p This is the demand function.
If p = 12: q = 60.000
12 = 5.000
b. S: 6.000 * 1,05
D: 60.000
p * 1,1
S = D: 6.000 * 1,05 = 60.000
p * 1,1
p = Fel! _ 10.476
Percentage change:
Fel! _ 5%
More generally:
S = D 6.000 * 1,05 = 60.000
p * 1,1 (insert p0)
6.000 * 1,05 = 60.000
p
p0p0
* 1,1 (p0 = 10 and x = p
p0)
6.000 * 1,05 = 60.000
x 10 * 1,1
x = Fel! = Fel!
p1 - p0
p0 =
p1
p0 - 1 = Fel! - 1 = Fel! _ 0,1 - 0,05 = 0,05
c. y% - x%
Chapter 17
17.1
a. See overhead.
b. TRS = Hay
_Grain = -
MPGrain
MPHay
2. 500
-700 = - 0,71
3. 500
-562 = - 0,89
4. 500
-469 = - 1,07
5. 500
-394 = - 1,27
6. 500
-335 = - 1,49
c. Yes.
d. The slope of the isocost: 1:1
Draw this in the diagram and we find that the minimum-cost combination is Hay: 6.000
and Grain: 4.892.
17.2
f(x) = Axa
1 xb
2
Constant Returns to Scale (CRS):
If we use twice as much of each input, then we will get twice as much output.
Increasing Returns to Scale (IRS):
If we use twice as much of each input, then we will get more than twice as much output.
Decreasing Returns to Scale (DRS):
If we use twice as much of each input, then we will get less than twice as much output.
Homogenous Functions
Definition:
f is homogenous of the degree k if f(tx) = tk f(x) t > 0.
If k = 1 then we have CRS.
f(tx) = A(tx)a
1 (tx)b
2
= Ata+b xa
1 xb
2
= ta+b f(x)
If a+b = 1: CRS
If a+b > 1: IRS
If a+b < 1: DRS.
17.3
a. _Y
_L = 200LKT
_Y
_K = 100L2T
_Y
_T = 100L2K
Y
L = 100LKT
Y
K = 100L2T
Y
T = 100L2K
b. _Y
_L = 0,67 L-0,33 K0,43
_Y
_L = 0,43 L0,67 K-0,57
Y
L = L-0,33 K0,43
Y
K = L0,67 K-0,57
17.4
20
10
10 20
x 1
x 2
y = 100
= 220 I RS
y = 200 CRS
= 190 DRS
CRS: We just replicate all inputs.
IRS: In Varian: if we double the diameter of a pipe, we use twice as much materials, but
the volume of the pipe goes up by a factor of 4.
Suppose that we have a spherical container. The volume of the sphere is 4 r3
3 , where r
is the radius. The cost of the sphere depends upon how much steel it takes to make it.
That cost is related to the surface area, which equals 4 r2. Doubling the radius raises the
volume (and output) by a factor of 8, but raises surface area by only a factor of 4.
(V = 23 * 4 r3
3 ).
DRS: we are unable to replicate some inputs. In a formal sense, it can always be
assumed that DRS is due to the presence of some fixed input.
Chapter 18
18.1
a. Three things that can occur if the firm expand:
1. It becomes to big. The firm could get so large that the returns to scale become
decreasing.
2. It becomes to big. The firm could dominate the market. There is no competition.
3. If one firm can expand then any other firm can also expand. But if all firms expand
then the price will decrease and so the profits.
b. When the economic profits are zero we say the firm is making normal profits. Its
accounting profit just pay the opportunity cost of the owner's money and time.
c. Two different views on the role of the profit:
1. Above-normal profits serve as a valuable signal that firm or industry output should be
increased, thus there is a signal for expansion and entry. In the long run economic
profits are zero for every industry. This is the neo-classical view.
2. We can also view profit as a reward for innovation and efficiency. Innovations in turn
leads to further profits. It is not the equilibrium that are of interest, probably this
equilibrium will never be obtained. This is the Schumpeterian or Austrian view.
18.2
f(L) = 6L2/3 w = 6 p = 3.
a. At the optimum: MP*P = MC or MP = MC
P
_Y
_L = f'(L) =
2
3 6L-1/3
= 4L-1/3
In optimum:
MP*P = MC
4L-1/3 * 3 = 6
L-1/3 = 2
L = 23 = 8
8 units of labour will be employed.
b. f(L) = 6*82/3 = 24
Chapter 19
19.1
f(x) = x
a. f(tx) = (tx)1/2 = t1/2 f(x) DRS
b. f(x) = 10
x1/2 = 10
x = 102 = 100 units of the input.
c. TC = Price of inputs * The use of inputs = w*10
d. f(x) = y
x1/2 = y
x = y2
e. If we want to produce y units we need y2 units of the input.
TC = w y2
f. AC = TC
y =
wy2
y = wy
19.2 In the long run this proposition should be true. If we had superior management we
would probably have to pay a high wage to him or offer him fringe benefits. Of course
this is more troublesome in the short run, since there is a difficulty with knowing the
true alternative cost.
Chapter 20
20.1
c(y) = 4y2 + 16
AC = c(y)
y = 4y +
16
y = 4y + 16y-1
a. AC is minimised when _AC
_y is equal to zero.
_AC
_y = 4 - 16y -2
Set this equal to zero:
4 - 16y -2 = 0
y -2 = 4
16
y = (4
16)-1/2 (=
16
4 )
y = 2
AC is minimised when y = 2.
b. AVC = VC
y =
4y2
y =4y
_AVC
_y = 0 ==> 4 = 0
Thus the level of output that minimises the level of average variable cost is y = 0.
20.2
a.
1. MC = _TC
_Q = b ATC =
TC
Q =
a + bQ
Q = aQ-1 + b
2. MC = b + 2c ATC = a + bQ + cQ2
Q
3. MC = 60 - 10Q + 3Q ATC = 50 + 60Q - 5Q2 + Q3
Q
b.
Profit maximising output level is found by setting:
MC = MR
1. MC = _TC
_Q = 2 + 2Q MR =
_TR
_Q = 3
Profit maximum:
2 + 2Q = 3
Q = 1
2 = 0,5
TR = 3*0,5 = 1,5
FC = 0
VC = 2Q + Q2 = 1,25
_ = TR - TC = 1,5 - 1,25 = 0,25
2. MC = 10Q MR = 6
Profit maximum:
10Q = 6
Q = 6
10 = 0,6
TR = 6*0,6 = 3,6
FC = 200
VC = 5Q2 =5*0,62 = 1,8
_ = 3,6 - 1,8 - 200 = - 198,2
20.3
Y = 2L + 5K w = 2 (the price of L) r = 3 (the price of K)
The cost function is:
C = 2L + 3K
We want to produce 10 units at the lowest possible cost. The general condition for cost
minimisation is
- w2
w1 = -
MP1
MP2
r
w =
MPK
MPL (that is: the slope of the isocost = the slope of the isoquant)
Isocost:
C = wL + rK
L = C
w -
r
w K
MPK = _Y
_K = 5 MPL = 2
MPK
MPL =
5
2 = 2,5 but
r
w =
3
2 = 1,5
That is we cannot find a optimal solution according to this. What we have is a corner
solution. Since the two inputs are perfect substitute we will only use one of the inputs.
In this case we will only use capital.
The cost is: PK K = 3*2 = 6
If we use only labour the cost will be: 2*5 = 10
See overhead.
Chapter 21
21.1
1. P*y - VC = Revenues - variable costs
C
Q
MC
AT C
AVC
P0
2. The area above the marginal cost curve. This is true since the area under the marginal
cost curve measures the total variable cost. So we subtract that from the total revenue.
3. Or combining these to measures: the area to the left of the supply curve. Use (1) until
MC = AVC and then (2).
21.2
c(y) = 10y2 + 1.000
The supply curve is given by
MC = P
c'(y) = 20y ==> 20y = p (inverse supply curve)
y = p
20
21.3
S(p) = 100 + 20p
Ps = y - 100
20
21.4
Since the fixed cost doesn't change we can measure the change in profit by the change in
producers' surplus.
See overhead.
Change in PS: 10*40 + 10*40
2 = 600
If p = 10:
S(p) = 4*10 = 40
TR = 40*10 = 400
Producers' surplus =10*40 -
0
10
4p dp = 2p2 |10
0 = 2*102 - 2*02 = 200
If p = 20
S(p) = 4*20 = 80
TR = 80*20 = 1.600
Producers' surplus = 80*20 -
0
20
4p dp = 2p2 |20
0 = 2*202 - 2*02 = 800
The change in producers' surplus:
800 - 200 = 600
Chapter 22
22.1
S1(p) = p - 10 p = 10 + y
S2(p) = p - 15 p = 15 + y
A kink at price p = 15
See overhead.
22.2
In the short run the consumers pay the whole tax. In the long run the producers pay the
whole tax.
D SR
D LR
y
p
22.3
Value tax (ad valorem tax) - tax levied on money spent: p1 + p1
Equilibrium: PD = (1+ )PS
The short-run supply curve is given by
p = MC(y,k)
The long-run supply curve is given by
p = MCl(y,k(y))
In the long run k (for example plant size) is variable and depends on output.
The curves will coincide at least one time, that this the output where the level of the
fixed factor associated with the Short-run MC is the optimal choice also in the long run.
In the long run the firm is more flexible, and this suggests that the long-run supply curve
is more elastic than the short-run supply curve.
P
Q
S kor t sikt
S lång sikt
LRS
LRS tax
SRS
SRS tax
p 0
p 1
p 2
Short run: Higher prices and lower quantity
Long run: even higher prices and lower quantity.
22.4
MC = 5
D(p) = 1.100 - 20p ==> p = 55 - 0,05q
a. Equilibrium:
MC = D(q)
5 = 55 - 0,05q
q = 1.000 The number of taxicabs: 1.000
20 = 50 units.
p = 5
b. We now that the demand cannot be greater than 1.000. At what price is the demand
equal to this:
1.000 = D(p)
1.000 = 1.120 - 20p
p = 6
c. _Per ride = $1
_Per license = 20 * $1 = $20
_Per year = 365 * $20 = $7.300
d. The market price of the license must in equilibrium be equal to the expected profit
having the license. If we assume infinite horizon we find the discounted value by
dividing with the interest rate:
Price = Fel! = $73.100
If the payments is forever:
PV = x
1+r +
x
(1+r)2 + . . . (factor out 1
1+r )
PV = 1
1+r [ x +
x
1+r +
x
(1+r)2 + . . ] (The terms in the bracket is x + PV)
PV = 1
1+r [x + PV]
PV(1 + r) = x + PV
PV + PVr - PV = x
PV = x
r
e. D(p) = 1.120 - 20p ==> D(q) = 56 - 0,05q
In equilibrium:
MC = D(q)
5 = 56 - 0,05q
q = 1020
==> We need 20
20 = 1 more license.
f. Since the profit is zero, the market value will be zero.
g. Every cab owner is willing to pay $73.000, so together the cab owners would be
willing to pay 50*$73.000 = $3.650.000 to prevent any new license from being issued.
The consumers are (at maximum) willing to pay the increase consumer surplus. This is
given by the area ABCD
See overhead.
ABCD = 1*1.000 + 20*1
2 = 1.010
or:
5
6
1.120 - 20p dp = 1.120p - 10p2|6
5 = 1.120*6 - 10*62 - (1.120*5 - 10*52) = 1.010
Total willingness to pay:
Fel! = $3.686.500
Thus the consumers' willingness to pay for another license is greater than the producers'
willingness to pay for a prevention. But we do not know whether this will occur or not.
The cab drivers might be better organised, they may be potential voters and so on.
Chapter 23
23.1
D(p) = 100 - 2p ==> D(y) = 50 - 0,5y
c(y) = 2y ==> MC = c'(y) = 2
TR = py = 50y - 0,5y2 MR = _TR
_y = 50 - y
Profit maximisation implies:
MC = MR
2 = 50 - y
y = 48
p = 50 - 0,5*48 = 26
23.2
MC = 5 + 0,3*10 = 8
a. E(Return) = p - 8
Equilibrium: p = MC
p = 8
b. Q = A - Bp
i and ii) Since the marginal cost is constant the equilibrium consumption will stay the
same regardless of the government sells the confiscated marijuana or not.
c. The price will also stay the same.
d. Consumption will increase if the marijuana were sold, because the supply curve will
shift to the right, lowering the price.
MS Sell
MC Destroy
Q
P
23.3
Markup = 1
1-1
| (y)|
= 1
1-1
2
= 2
Amoroso-Robinson: MR = P(1 + 1
) (set MC = MR)
MC = P(1 + 1
) (insert the value of )
MC = P(1 + 1
-2)
P = 2MC
23.4
D(p) = 10p-3
c(y) = 2y ==> MC = c'(y) = 2
= _y
_p p
y = - 30 p-4
p
10p-3 = -30p-3
10p-3 = -3
We know since before that:
MR = p(1 + 1) (MC = MR ==>)
MC = p(1 + 1)
2 = p(1+1
-3)
p = 2
(1+1
-3)
= 3
D(p) = 10*3-3 = 0,37
23.5
D(p) = 100
p ==> D(y) =
100
y
c(y) = y2 ==> MC = 2y
TR = py = 100
y y = 100 ==> MR = 0
Profit maximum:
MC = MR
2y = 0
y = 0
The firm will not sell anything.
23.6
It should not be any problem regulating a monopolist and set price equal to marginal
cost. One problem is that it might be the case that the monopolist does not break even at
that price. This is the case for some cases of natural monopolies.
The traditional definition:
The existence of economies of scale is important. This means that the average cost for
the firm declines as market output increases.
Kahn: a single large firm serving the entire market would have a lower average cost than
any smaller rival.
The new view:
The most models focus on the multiproduct nature of regulated firms.
It is not necessary with economies of scale throughout the range of production in the
market. A single supplier might even without this be able to serve the entire market at a
lower unit cost than any industry configuration with two or more firms.
Panzar and Willig: An industry is said to be a natural monopoly if, over the entire
relevant range of output, the firm's cost function is subadditive.
Subaddivity of the cost function means that the total output can be produced more
cheaply by one firm producing all alone than it would be for a collection of two or more
firms whose individual output vectors sum to the same industry output.
But we leave this aside for now and look at the case with economies of scale over the
entire range of demand.
MC
AT C
P
Q
C
P D
Välfärdsförlust
If:
P = MC : Optimal
But nonpositiv profit:
- Subsidies --> efficiency looses if we have to collect these with
taxes
- 1. A fixed charge that the consumer pays and
2. P = MC
This is called a two-part tariff.
P = AC: Zero profit
Welfare loss
This is often called a second-best solution.
Chapter 24
24.1
To minimise the differences the should locate as given in the figure below
Right Left
x x
But both parties has an incentive to move to the middle and "steal" voters from the other
party. In equilibrium both parties will be located in the middle of the scale.
b. It need not be an equilibrium in the middle.
c. No equilibrium.
24.2
U.S. Q1 = 50.000 - 2.000P1
England Q2 = 10.000 - 500P2
C(Q) = 50.000 + 2Q ==> MC = 2
a. The aggregated demand is:
Q = 50.000 - 2.000P1 + 10.000 - 500P2
Q = 60.000 - 2.500P ==> P = 24 - 1
2.500 Q
TR = 24Q - 1
2.500 Q2 ==> MR = 24 -
1
1.250Q
Profit maximum requires:
MR = MC
24 - 1
1.250Q = 2
Q* = 27.500
P* = 24 - 1
2.500 27.500 = 13
The profit:
_ = 27.500*13 - 50.000 - 2*27.500 = 252.500
b. MC = 2
P1 = 25 - 1
2.000 Q1 MR1 = 25 -
1
1.000 Q1
P2 = 20 - 1
500 Q2 MR2 = 20 -
1
250 Q2
Profit maximum requires:
MR1 = MC
25 - 1
1.000 Q1 = 2
Q1 = 23.000
P1 = 25 - 1
2.000 23.000 = 13,5
MR2 = MC
20 - 1
250 Q2 = 2
Q2 = 4.500
P2 = 20 - 1
500 4.500 = 11
_ = 23.000*13,5 + 4.500*11 - 50.000 - 2*27.500 = 255.000
c. We have that Q = Q1 + Q2 in the case of linear demand curves and constant marginal
cost.
Proof:
Q1 = a1 - b1p1
Q2 = a2 - b2p2
MC = c
TR1 = a1
b1Q1 -
Q1
b1Q1
TR2 = a2
b2Q2 -
Q2
b2Q2
Third degree price discrimination:
__1
_Q1 =
a1
b1 - 2
Q1
b1 - c = 0
==> Q1 = a1
2 -
b1c
2
Q2 = a2
2 -
b2c
2
Monopoly:
Q = Q1 + Q2 = (a1 + a2) - (b1 + b2) p
__
_Q =
a1 + a2
b1 + b2 - 2
Q
b1 + b2 - c = 0
==> Q = a1 + a2
2 -
b1 + b2c
2
==> Q = Q1 + Q2
Chapter 25
25.1
a. One buyer and one seller.
b. NHL: after the draft the club and the player negotiates.
Employer and employee.
c. Divide the analysis in two steps.
1. What will the outcome be if there are one buyer (monopsony)?
The demand curve is the marginal value of product. We also have to find the cost curve.
If the supply is given by S(q), then the total cost is S(q)q:
S: p = a + bq
TC = pq = aq + bq2
MCmonopsony = a + 2bq
Profit maximum:
MC = MRP ==> P1 and Q1
2. What will be the outcome if there are one seller (monopoly)?
MCmonopoly = MR ==> P2 and Q2
P1
Q1Q2
Q
P
MC monopson
D = MRP
MR
S P2 = MC monopoly
We do not know the equilibrium in the case of a bilateral monopoly. The outcome
depends on negotiations.
Chapter 26
26.1
Cournot: The firms moves at the same time, and they have to forecast what the other
firm's output will be.
Q = 100 - P
2 firms, x and y.
a. We can use the expression (and derivations) from the text book. If the demand curve
is linear, p = a - b(Qx + Qy), and MC = 0, then the reaction function is given by
Qy = a - bQx
2b
Qy = 100 - Qx
2
Firm x has the same reaction function.
We can also think in the following way:
1. Suppose that x sells 60 units. The demand facing firm y is then
Qy = 100 - p - 60 = 40 - p ==> py = 40 - Qy
TRy = 40Qy - Q2
y ==> MRy = 40 - 2Qy
Profit maximum:
MRy = MCy
40 - 2Qy = 0
Qy = 20
2. Suppose that x sells 80 units. The demand facing y is then
Qy = 100 - p - 80 = 20 - p ==> py = 20 - Qy
TRy = 20Qy - Q2
y ==> MRy = 20 - 2Qy
Profit maximum:
MRy = MCy
20 - 2Qy = 0
Qy = 10
3. Suppose that x sells 0 units. The demand facing y is then
Qy = 100 - p - 0 = 100 - p ==> py = 100 - Qy
TRy = 100Qy - Q2
y ==> MRy = 100 - 2Qy
Profit maximum:
MRy = MCy
100 - 2Qy = 0
Qy = 50
See overhead.
b. In equilibrium each firm produces what the other firm expects it to produce, thus we
insert the reaction function of firm x into the reaction function of firm y:
Qy =
100 - (100-Qy
2)
2 (solve for Qy)
2Qy = 100 - (100-Qy
2)
2Qy = 100 - 50 + 1
2Qy
Qy = Fel! = 33Fel!
Qx = 331
3
Q = 662
3 ==> p = 100 - 66
2
3 = 33
1
3
Total profit: QP = 662
3 33
1
3 = 2.222
c. If the two firms collude they act as a monopolist:
P = 100 - Q ==> TR = 100Q - Q2 ==> MR = 100 - 2Q
MC = MR
0 = 100 - 2Q
Q = 50 ==> P = 50
_ = QP = 50*50 = 2.500
d. The total profit is higher if the two firms co-operate. This may lead to co-operation,
however we need to know more about the market structure, before saying anything
about the possibilities of co-operation.
26.2
In a Stackelberg model there is one leader and many followers, we call this a sequential
game. The Cournot model is a simultaneous game.
26.3
No, the Cournot equilibrium is always a possible Stackelberg equilibrium.
26.4
D(p) = 200 - p ==> D(y) = 200 - y
Small firms:
S(p) = 100 + p
Large firm:
c(y) = 25y ==> MC = c'(y) = 25
a. The residual demand curve facing the leader:
R(p) = D(p) - S(p)
R(p) = 200 - p - (100 + p)
R(p) = 100 - 2p ==> R(y) = 50 - 1
2y
MRR = 50 - y
Profit maximum for the leader requires:
MRR = MC
50 - y = 25
y = 25
p = 50 - 1
2 25 = 37,5
_ = 37,5*25 -25*25 = 312,5
The following firms take the price as given and sell:
y = 100 + 37,5 = 137,5
Total output is
137,5 + 25 = 162,5
See overhead.
b. TR = py = 200y - y2 ==> MR = 200 - 2y
Profit maximum requires:
MR = MC
200 - 2y = 25
y = 87,5
p = 200 - 87,5 = 112,5
The profit is
_ = 112,5*87,5 - 25*87,5 = 7656,25
Chapter 27
27.1
a. High output and High output.
b. Yes and No! The total profit is maximised if both firms produce low. But given the
beliefs of what the other firm may do it is profit maximising to produce high.
c. The firms can agree on producing low, but then we have to create an incentive not
breaking the agreement, for example a penalty. If we repeat the games many times the
outcome might be that both produce low. The problem is that if the game is played a
finite times, there is an incentive to cheat the last time. But if there is an incentive to
cheat the last time (and if they are aware of that) then they will cheat the time before
that, and so on. This problem will no occur if we repeat the game an infinitely times.
27.2
a. If you know that everybody is playing Dove, then it pays off for you playing Hawk.
The opposite is also true if you know that everybody else is playing Hawk.
b. Fraction of Hawks = p
Fraction of Doves = 1-p
If you are a Hawk the average payoff must be:
_Meeting a Hawk * Pay-off + _Meeting a Dove * Pay off
p (-5) + (1-p) 10 = 10 - 15 p
If you are a Dove the average pay-off must be:
_Meeting a Hawk * Pay-off + _Meeting a Dove * Pay off
p * 0 + (1-p) 4 = 4 - 4p
If the males are supposed to be indifferent between being a Hawk and Dove, then the
average pay-off being a Hawk should be equal to the average Pay-off being a Dove:
10 - 15p = 4 - 4p
11p = 6
p = 6
11
27.3
"Tit-for-tat" strategy: You do what your opponent did the last round.
In order to punish the opponent the other player also defects. But then the other player
also defects in response to that, and each player will continue to defect in response to
the other's defection.
In this case the "tit-for-tat" strategy doesn't work out well.
27.4
Dominant strategy equilibrium are always Nash equilibria. But Nash equilibrium are not
always dominant strategy equilibria, there can be many Nash equilibria.
Chapter 28
28.1
Plot the different alternatives in an Edgeworth box. Note that the box is bounded by 40
caramels and 40 lollipops. Connect the indifferent alternatives, they represent an
indifference curve. Also note that this indifference curve goes through the endowment.
See overhead.
a. Yes, all allocations that lies between the two indifference curves are better for both
individuals.
b.
Kalle
He is prepared to give 6 lollipops if he gets 10 caramels. Or he require 10 caramels if he
must give away 6 lollipops.
The price is: 6 lollipops for 10 caramels
0,6 lollipops / 1 caramel, or
10 caramels for 6 lollipops
1,7 caramels / 1 lollipop.
Lisa
She require 5 lollipops if she must give away 10 caramels. Or she is prepared to give 10
caramels is she gets 5 lollipops:
The price is: 5 lollipops for 10 caramels
0,5 lollipops / 1 caramel, or
10 caramels for 5 lollipops
2 caramels / 1 lollipop.
Thus the price must lies between
1,7 caramels / 1 lollipop _ p _ 2 caramels / 1 lollipop, or
0,5 lollipops / 1 caramel _ p _ 1,7 caramels / 1 lollipop.
c. Yes. Kalle gives Pelle 6 lollipops and receives, after a while, 10 caramels. Lisa gives
Pelle 10 caramels and receives 5 lollipops. ==> Pelle keeps 1 lollipop.
28.2
Yes.
x
Pareto
U 2
U 1
28.3
First Theorem of Welfare Economics:
All Walrasian equilibria are Pareto efficient.
Assumptions:
- No consumption externalities (the agents only cares about his own consumption)
- Agents behave competitively (which really is not realistic in our context)
Second Theorem of Welfare Economics:
Suppose that x is a Pareto efficient allocation and all agents have convex preferences.
Then x is a Walrasian (market) equilibrium for an certain assignment of endowments.
X2
X1A
B
Pareto efficient
allocation
Change the
endowment Assumptions:
- No consumption externalities
- Agents behave competitively
- Convex preferences. In this case the Pareto efficient allocation is not a Walrasian
(market) equilibrium since agent A prefers the other allocation.
X2
X1A
B
B
A
Pareto efficient
allocation
A prefers this
This theorem simply says that every Pareto efficient allocation can be achieved.
The political implication is that perfect competition is desirable. The second theorem
also says that we can separate the choice of allocation from the choice of effectiveness.
If we can affect the initial endowment (by a lump-sum tax, for example that we tax the
initial endowment of labour, the important thing is that we do not disturb the economy)
then we can reach the Pareto-efficient allocation that we desire.
Chapter 29
29.1
Both theorems holds in an economy with production.
Assumption for the first theorem:
Same as before plus: - no production externalities
Assumptions for the second theorem
Same as before plus: - no production externalities
- Convex production sets: this means no IRS.
If IRS were the case in equilibrium then the firm (at the competitive prices) would want
to produce more output.
29.2
PF = 3 PC = 6
In equilibrium:
MRS = PF
PC
_C
_F =
PF
PC (if we give up one C: set _C = 1)
1
_F =
PF
PC (solve for _F)
_F = 2
If we give up 1 C we can produce 2 F.
29.3
MRSC,F = - 2 (2 coconuts for 1 fish)
MRTC,F = -1 (1 coconut for 1 fish)
He should produce more fish. He is willing to give up 2 coconuts for 1 fish, but he only
has to sacrifice 1 coconut.
29.4
Page 520-520:
Robinson:
10 pound fish / h
20 pound coconuts / h
Friday
20 pound fish / h
10 pound coconuts / h
They both want to produce 60 pounds of each good.
Without co-operation
Robinson:
60 fish
10 +
60 coconuts
20 = 9
Friday:
60 fish
20 +
60 coconuts
10 = 9
Total amount of hours: 18 hours.
Co-operation
Robinson produce coconuts and Friday produce fish.
Robinson:
120 coconuts
20 = 6
Friday:
120 fish
20 = 6
Total amount of hours: 12 hours.
This is a case where comparative advantage is present:
Robinson:
MRT
R
C
F = - 2
Friday:
MRT
F
C
F = -
1
2
Robinson has a comparative advantage in coconut production and Friday has a
comparative advantage in fish production.
29.5
a. See overhead.
We draw the Edgeworth box for the production set 30 fish and 10 coconuts.
When production did not exist the condition for Pareto efficiency was that:
MRSAx,y = MRSB
x,y (the line of mutual tangency's)
This condition still must hold. We can say that this is the condition for trade that must
hold. But we could now also think of another way to trade; we can produce less of one
good and more of another. The question now is which choices from the production
possibilities set will be Pareto efficient? The condition must be:
MRT
F1
F2
x
y
= MRS
A
B
x
y
b. We will probably have another allocation.
c. MRSAx,y = MRSB
x,y
MRT
F1
x
y = MRT
F2
x
y (if not it is possible to change the production)
MRT
F1
F2
x
y
= MRS
A
B
x
y
Chapter 30
30.1
a. Y wins, and this is independent of the ordering
b. A: x,y,z
B: y,z,x
C: z,x,y
We do not know it depends on the ordering of the voting:
y wins if we start by voting between x and z:
x vs. z ==> z vs. y ==> y
z wins if we start by voting between x and y
x vs. y ==> x vs. z ==> z
x wins if we start by voting between y and z
y vs. z ==> y vs. x ==> x
c. Intransitive preferences: Y is better than X and X is better than Z, but Z is better than
Y.
The preferences are not single-peaked in the second case. Suppose for example that
x = low expenditures
y = moderate expenditures
z = high expenditures
A prefers low to moderate and moderate to high. He has single-peaked preferences. B
prefers moderate to high and high to low. He has single-peaked preferences. But C
prefers high to low and low to moderate, even though moderate expenditure is closer
than low expenditure to the best outcome of high expenditure. Voter C does not have
single-peaked preferences.
30.2
The 45° line show different combinations where the income distribution is symmetric.
a. The income distribution will be perfectly symmetric.
b. The income distribution will be worse.
c. The would look like an L, this is the minimax case. Social welfare depends only on
the welfare of the worst off agent - the person with the minimal utility
W(u1, ... , un) = min{u1, ... , un }
d. According to Rawls D is better than both A, B and C, because both agents are better
off. But D must also be better than E, since Ingvar is the worst-off agent in D and E we
must maximise his outcome and this is point D.
e. "Theory of Justice". When we are born we do not know whether we are going to be
successful or not. There is a "veil of ignorance", therefore it is in our own interest that
also those who fail are well off.
f. The maximax solution. Point E.
30.3
First we define another
- Equitable allocation: if no agent prefers any other agent's bundle of goods to his or her
own. Note that this doesn't imply that the allocation is perfectly symmetric since it is
utility that are off interest.
A fair allocation is an allocation that is Pareto-efficient and equitable
Here we start with some specific moral judgement, that is the definition of fair. If we are
going to use this we have to justify why this is a relevant approach.
Another problem is what we mean with equitable, do we refer to the endowment or the
final allocation? Could we instead talk about possibilities, meaning that everybody
should have the same possibilities?
30.4
If there is no one that does not envy anyone, then everybody wants to exchange. This
must imply that this allocation cannot be a Pareto-efficient allocation.
A envies B envies C envies D envies A.
==> Exchange will occur.
Chapter 31
31.1
a. The total profit if x boats trap are
_ = 1.000(10x - x2) - 2.000x
At what value of x is this equal to zero? This is the equilibrium since boats will enter
until the profit is equal to zero.
_ = 0 ==> 1.000(10x - x2) - 2.000x = 0
8.000x - 1.000x2 = 0 (divide both sides with x)
8.000 - 1.000x = 0
x = 8.000
1.000 = 8 boats.
b. The total profit if x boats trap are
_ = 1.000(10x - x2) - 2.000x
Profit maximum requires that MR = MC or
__
_x = 0 ==> 10.000 - 2.000x - 2.000 = 0
x = 4 boats
c. With a license fee of F thousand dollars per month, the marginal cost of operating a
boat for a month would be 2.000 + F. Boats (or firms) enter until the profit is equal to
zero:
_ = 1.000(10x - x2) - 2.000x - Fx = 0 (we set x = 4)
1.000(10*4 - 42) - 2.000*4 - F*4 (solve for F)
4F = 40.000 - 16.000 - 8.000
F = 16.000
4 = 4.000 per month and boat.
31.2
a. If property rights are clearly defined then we have to pay for externalities, for example
if someone owns the air, the firm has to pay if the pollute the air. Another problem is
that there are difficulties constructing a market, there can be uncertainty about the
effects, there might be to many or to few agents.
b. Yes, the implication of the Coase theorem is that if
- property rights are clearly defined and
- transactions costs are low
then problem of externalities can be solved by the market.
31.3
Airport: _A = 48X - X2
Developer: _D = 60Y - Y2 - XY
a. The airport maximises its profit
__A
_X = 0 ==> 48 - 2X = 0
X = 24
_A = 48*24 - 242 = 576
The developer maximises his profit
__D
_Y =0 ==> 60 - 2Y - X = 0 (set X = 24)
60 - 2Y - 24 = 0
Y = 18
_D = 60*18 - 182 - 24*18 = 324
b. The airports profit function now becomes:
_A = 48X - X2 - XY
The developers profit function now becomes
_D = 60Y - Y2 - XY + XY
The developer maximises his profit:
__D
_Y =0 ==> 60 - 2Y = 0
60 - 2Y = 0
Y = 30
_D = 60*30 - 302 = 900
The airport maximises its profit
__A
_X = 0 ==> 48 - 2X - Y = 0 (set Y = 30)
48 - 2X - 30 = 0 (solve for X)
X = 9
_A = 48*9 - 92 - 9*30 = 81
c. Total profits would be
_ = _A + _D = 48X - X2 + (60Y - Y2 - XY)
Profit maximum requires that:
__
_X = 0 and
__
_Y = 0
__
_X = 48 - 2X - Y = 0 ==> X = 24 -
Y
2
__
_Y = 60 - 2Y - X = 0 ==> X = 60 - 2Y (set these two equal)
24 - Y
2 = 60 - 2Y
Y = 36
3
2
= 24
X = 24 - 24
2 = 12
Total profit is
48*12 - 122 + 60*24 - 242 - 12*24 = 1.008
Chapter 33
33.1
a. In contrast to the private good case it may not be the case that the market provides an
Pareto efficient level of the public good. Since everyone must consume the same
amount of the pubic good there is an incentive for the consumer of not providing
anything of the public good.
b. The easiest way to solve the problem is to create some sort of possibility of excluding
or making everybody pay (for example by letting the government run the utility and
collection money through the tax system).
33.2
a. A particular kind of externality - the same amount of the good has to be provided to
all consumers. There is no way of excluding individuals from consumption of the good.
b. Everyone must consume the same amount of the good.
c. In the figure we add the individual MRS curves vertically to get the sum of the MRS
curves.
Public Good
MRS
MRS A
MRS B
MRS A + MRS B
MC
Q*
MRSA + MRSB = MC
If we know the individual MRS curves we can find each agent's willingness to pay. If
we want we can use this information to tax the individual consumers in order to pay for
the public good. Then each agent would be paying exactly his or her true valuation of
the public good, this type of taxation scheme is called a Lindhal equilibrium.
Private goods: X2 = X1a + X2
b
Public goods: X2 = X2a
X2 = X2b
33.3
a. The outcome depends on the ordering.
1. Fine vs. Prison ==> Fine vs. Death ==> Death
2. Fine vs. Death ==> Death vs. Prison ==> Prison
3. Death vs. Prison ==> Prison vs. Fine ==> Fine
Person C has not single-peaked preferences.
b. If you are smart you require that we first vote between Fine and Prison.
Chapter 34
34.1
a. The buyers would be willing to pay
_Good WTPGood + _Bad WTPBad
100
200 2.500 +
100
200 300 = $1.400
But the owners of a good car is not willing to sell at this price. Owners of a bad car is
willing to sell at this price. The equilibrium is that owners of good cars would not sell
them. Owners of lemon would sell them. The price of a used car would be $300.
b. The buyers would be willing to pay
_Good WTPGood + _Bad WTPBad
120
200 2.500 +
80
200 300 = $1.620
At this price the owners of the good cars are willing to sell. One equilibrium has all cars
sold at a price of $1.620. Perhaps surprisingly, there also exists an equilibrium in which
only the lemons are sold. The price of used cars is $300 and no owner of a good used
car will want to sell.
34.2
a. When different agents have different information. This arise because information is
not costless. As we have seen this can create problems with the efficient functioning of
a market.
b. Signalling of quality and so on.