1.
There are two households h=1,2 and two goods i=1,2. The preference orders of the
households are described by the following utility functions.
1
1
Household 1: u1 ( x11 , x12 ) = ( x11 ) 2 ( x12 ) 4
1
1
Household 2: u 2 ( x12 , x22 ) = ( x12 ) 4 ( x22 ) 3
Moreover, the goods are allocated to both households in two different ways:
x = (2,4,3,5) and y = (1,5,6,1)
Calculate the utility vectors of both allocations, u(x) and u(y). Describe exemplarily the
meaning of each component in x and y.
Solution:
Allocation x: x11 = 2, x12 = 4, x12 = 3, x22 = 5,
Household 1 obtains 2 units of good 1 and 4 of good 2 and household 2 obtains 3 units of
good 1 and 5 of good 2. Analogously for allocation y.
Utility vectors:
u ( x) = (u1 (2, 4), u 2 (3,5)) = (2; 2.25)
u ( y ) = (u1 (1,5), u 2 (6,1)) = (1.5; 1.57)
Seitenumbruch
2.
There are two households (H=2) and three goods (N=3). The preference orders of the
households are described by the following utility functions.
u 1 ( x 11 , x 12 , x 13 ) = x 11x 12 x 32
u 2 ( x 12 , x 22 , x 32 ) = x 12 + x 22 + x 32
The quantities of the goods in the economy are:
Good 1: 10 units
Good 2: 15 units
Good 3: 20 units
Find an allocation x of these goods, so that the quantities allocated to both households exactly
utilize the available quantities. Write down x and u(x).
Solution:
e.g.: x = (0,0,0,10,15,20)
u(x) = (0,35)
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3.
Consider the allocations x and y of example 1. Are these allocations Pareto-comparable? (Is x
a Pareto-improvement compared to y or vice versa?)
If yes, write down two allocations which are Pareto-incomparable. Otherwise, find two
allocations that are Pareto-comparable.
Solution:
Allocation x is a Pareto-improvement compared to allocation y, because both households are
better off with allocation x. Consequently, the allocations are Pareto-comparable.
The following allocations are Pareto-incomparable: x = (2,4,5,3); y = (1,5,5,6)
u ( x) = (u1 (2, 4), u 2 (3,5)) = (2; 2.25)
u ( y ) = (u1 (1,5), u 2 (5, 6)) = (1.5; 2.72)
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4.
Let the quantities a1 = 10, a2 = 12 of two goods be given. The utility functions of two
households are as in example 1.
Consider the allocation x = (6, 2, 4,10).
a)
Is this allocation feasible? (What has to be proved?)
b)
h
Is this allocation Pareto-optimal? (In order to answer compute the MRS1,2
for both
households.)
c)
If not, suggest a change which improves the welfare of both households.
Solution:
a) yes, this allocation is feasible, for both households together don’t receive more of each
good than the available quantities (6+4=10, 2+10=12)
1
2
= MRS1,2
b) Condition for the Pareto-optimum: MRS1,2
1
MRS1,2
∂u1
∂u 2
∂x
∂x
2
2
= − 11 = − , MRS1,2
= − 12 = −1,875 ;
∂u
∂u
3
∂x2
∂x2
The marginal rate of substitution of both households is not identical and for that
reason this allocation is not Pareto-optimal.
c) We know from b) that household 1 accepts to forgo one unit of good 1 in exchange for
0.66 more units of good 2. Household 2, however, would abandon 1,875 units of good
2 in exchange for one unit of good 1.
Suggestion of change which makes both better off:
Household 2 obtains one unit of good 1 from household 1, which gets one unit of good
2 from household 2.
As a consequence, both households obtain one unit of that good which is relatively
more valuable for them. In exchange, the have to hand over one unit of that good,
which is relatively less valuable, to the other household.
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5.
There are two firms and two resources, i. e. M=F=2. Derive the first order conditions from the
optimisation problem for efficient production (P1-P4). (Hint: The Lagrangean for P1-P4 is
stated in Appendix II.1) Try to transform the first-order conditions to derive the formula
∂f 1
∂f 2
∂v11 ∂v12
2
= 2 (i. e. that MRtS112 = MRtS12
).
∂f 1
∂f
∂v12 ∂v 22
Solution:
Optimisation-Problem for M=F=2:
max f 1 (v11 , v 12 )
s.t. v 1m + v m2 ≤ rm
m = 1, 2
f (v , v ) ≥ y k k = 2
k
k
1
k
2
v mk ≥ 0
k = 1, 2; m = 1, 2;
Lagrangean:
L = f 1 (v11 , v 12 ) − λ1 (v11 + v12 − r1 ) − λ 2 (v 12 + v 22 − r2 ) + α 2 ( f 2 (v12 , v 22 ) − y 2 )
1)
∂L ∂f 1
:
− λ1 = 0
∂v11 ∂v11
2)
∂L ∂f 1
:
− λ2 = 0
∂v 12 ∂v 12
3)
∂f 2
∂L
:
−
λ
+
α
=0
1
2
∂v12
∂v12
∂L
∂f 2
4) 2 : −λ 2 + α 2 2 = 0
∂v 2
∂v 2
Equating conditions 1 and 3 gives:
∂f 2
∂f 1
−
=
−
+
λ
λ
α
1
1
2
∂v11
∂v12
∂f 1
∂v11
= α2
∂f 2
∂v12
Equating conditions 2 and 4 gives:
∂f 2
∂f 1
−
=
−
+
λ
λ
α
2
2
2
∂v12
∂v 22
∂f 1
∂v12
= α2
∂f 2
∂v 22
Furthermore, this results in:
∂f 1 ∂f 2
∂v12
∂v11
=
∂f 1 ∂f 2
∂v 22
∂v 12
MRtS11, 2 = MRtS12, 2
The marginal rates of technical substitution between both factors have to be equal for both
firms.
HENCE:
∂f 1
∂f 1
∂v11
∂v 12
=
∂f 2
∂f 2
∂v12
∂v 22
The MRT2,1 (the marginal rate of transformation between both firms) has to be equal for all
factors.
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6.
Let the quantity r of one factor of production be given. This factor can be employed for the
production of two goods x1 and x2 in accordance with following production functions:
x1 = 2v1
x2 = 0,5 (v2)1,5
where v1 + v2 = r holds. Calculate and draft the transformation curve. (Hint: Invert the
production function for good 1 and use v2 = r - v1 in the production function for good two to
derive the transformation curve: x2 as a function of x1 and r.)
Solution:
Calculation of the transformation curve:
v1 =
x1 2
x
, v = r − v1 → v 2 = r − 1
2
2
3
x ⎞2
1⎛
Transformation curve: x2 = ⎜ r − 1 ⎟
2⎝
2⎠
MRT = slope of the transformation curve:
1
∂x
x ⎞2
3 ⎛
MRT = 2 = − . ⎜ r − 1 ⎟
∂x1
8 ⎝
2⎠
Seitenumbruch
7.
There are two households h=1,2 and two goods i=1,2. The households’ utility functions are
u1 (x11 , x12 ) = (x11 )α (x12 )β , u 2 (x12 , x 22 ) = (x12 ) γ (x 22 )δ . The available budget b is derived from the
households’ initial endowments x11 , x12 , x12 , x 22 . .
a) Calculate the demand functions of the households (for arbitrary values of the
parameters) and then specify them by using the following information:
h = 1: α = 0, 4; β = 0, 6;
h = 2: γ = 0, 6; δ = 0, 4;
initial endowment: x11 = x21 = x12 = x22 = 10
h
h
Hint: Let prices p1 and p2 be given and compute xhi as a function of x1 , x 2 , p1, p2, and
of the parameters of the utility function.
b) Compute the general equilibrium for the numerically specified demand functions.
Solution:
a) optimisation problem:
max u ( x1 , x 2 ) = x1α x 2β
x1 , x2
s.t. p1 x1 + p 2 x 2 ≤ p1 x1 + p 2 x 2
x1 , x 2 ≥ 0
L = α ln x1 + β ln x2 − λ ( p1 x1 + p2 x2 − p1 x1 − p2 x2 )
∂L α
= − λ p1
∂x1 x1
∂L β
= − λ p2
∂x2 x2
∂L
= − p1 x1 − p2 x2 + p1 x1 + p2 x 2
∂λ
Setting these equations equal zero gives:
α x2 p1
=
β x1 p2
Converting the budget constaint gives:
p1 x1 + p2 x2 − p1 x1 = p2 x2
p1 x1
px
+ x2 − 1 1 = x2
p2
p2
The demand functions result from the budget constraint and the optimality conditions:
α(
p1 x1
px
+ x2 − 1 1 )
p2
p2
p
= 1
p2
β x1
x1 =
x2 =
α
α+β
β
α+β
*
p1 x1 + p 2 x2
p1
*
p1 x1 + p 2 x2
p2
For household 1:
x11 = 4
p + p2
p1 + p 2
und x 12 = 6 1
p2
p1
For household 2 (setting α = γ, β = δ in the derived conditions for household 1):
x12 = 6
p1 + p 2
p + p2
und x 22 = 4 1
p1
p2
b) Conditions for a general equilibrium:
Supply of good 1 = demand for good 1:
x11 + x12 = x11 + x12
20 = 10
p1 + p 2
p1
Supply of good 2 = demand for good 2:
x 21 + x 22 = x 12 + x 22
20 = 10
p1 + p 2
p2
p1 = p 2 = 1
According to these price relations there are following quantities demanded:
x11 = 8; x12 = 12; x 12 = 12; x 22 = 8
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8.
Let the intertemporal utility function:
u(c t , c t +1 ) ≡ U(c t ) +
1
U(c t +1 ),
1+ δ
(1)
be given, where U: IR → IR describes periodical utility out of consumption and U is identical
in both periods. For this type of utility function δ > 0 is explicitly denoted as the rate of time
preference.
a)
Calculate generally the MRSc c for the type of utility function described in (1)
t t+1
(i.e. express the partial derivatives of numerator and denominator in
MRSc c
t t+1
≡ −[∂u(c t , c t +1 ) / ∂c t ] /[∂u(c t , c t +1 ) / ∂c t +1 ] by U'(ct), U'(ct+1) and δ).
What is the MRSc c for a consumption bundle with ct = ct+1? Is its value higher,
t t+1
lower or equal to one? Assuming a consumption bundle with ct = ct+1, if one gets
one more unit of present consumption, what would be the maximal amount of
future consumption one in prepared to give up?
b)
Set U(c) = c0,8, δ = 0,2 and calculate the MRSc c for
t t+1
(i) (ct, ct+1) = (5,6),
(ii) (ct, ct+1) = (7,7).
Solution:
a)
MRSct ,ct+1
∂u (ct , ct +1 )
∂u (ct )
∂ct
∂ct
U '(ct )
=−
=−
= −(1 + δ )
∂u (ct , ct +1 )
1 ∂u (ct +1 )
U '(ct +1 )
∂ct +1
1 + δ ∂ct +1
MRS for ct=ct+1:
Because u(.) is equal in both periods,
U '(ct )
= 1 holds. I.e.: MRSct ,ct +1 = −(1 + δ )
U '(ct +1 )
For δ>0, the MRS is higher 1 (or lower -1 if one considers the negative sign). For one
more unit of present consumption the household is prepared to give up 1+ δ units of
future consumption.
b)
MRSct ,ct +1 = −(1 + 0, 2)
0,8 ⋅ ct −0,2
0,8 ⋅ ct−+0,2
1
(i)
MRS5,6 = -1,244
(i)
MRS7,7 = 1,2 (analogously to a))
Seitenumbruch
9.
Consider a household endowed with wealth wh = 10. With a probability of π2 = 0.2 there will
be a catastrophe, which damages the entire wealth ( w 2h = 0 ). An insurance premium costs p =
0.4 per Euro of coverage.
Draw an indifference curve, for an expected utility function, defined by
Uh (x1h ,xh2 ) ≡ π1uh (x1h ) + π2uh (xh2 ) with uh (xih ) = (xih )0,5 . (Calculate some points on the
indifference curve). Draw the household’s budget line and draft its optimal insurance choice.
Do the same for a premium of p = 0.2.
Solution:
Expected utility function of the household:
U h ( x1h , x2h ) = 0,8 ⋅ ( x1h )0,5 + 0, 2 ⋅ ( x2h )0,5
h
=
The corresponding MRS is: MRS1,2
0,8 ⋅ x1−0,5
0, 2 ⋅ x2−0,5
Budget line: in the state of normality the household can forgo p and obtains in the case of a
catastrophe 1-p (because the insurance premium has to be paid in both cases).
h
Hence, the optimisation condition is: ( MRS1,2
=)
0,8 ⋅ x1−0,5 1 − p
=
0, 2 ⋅ x2−0,5
p
For p = 0.4:
h
( MRS1,2
)=
0,8 ⋅ x1−0,5 0, 6
=
→ x1−0,5 < x2−0,5 → x1 > x2
0, 2 ⋅ x2−0,5 0, 4
In this case the premium is higher than the probability of loss. Due to the risk aversion
(decreasing marginal utility – arises from the underlying utility function), the household is
going to insure itself, but only a part of its wealth. (I.e., in the case of loss the household
owns, in spite of insurance, less than in the normal case)
For p = 0.2:
h
=
MRS1,2
0,8 ⋅ x1−0,5 0,8
=
→ x1−0,5 = x2−0,5 → x1 = x2
0, 2 ⋅ x2−0,5 0, 2
The premium equals exactly the probability of loss and as a consequence the household is
going to take a complete insurance. Hence, the household owns exactly the same wealth in the
case of loss as in the normal case.
Seitenumbruch
10.
Let H=2 households and N=2 goods be given. The preferences of the households can be
described by following utility functions:
1
1 2
1
u (x , x ) = (x ) (x )
1
1
1
1
2
1
2
1
2
u 2 ( x12 , x 22 ) = x12 + ln x 22
Consider the allocations x = (5,3,2,4) and y = (1,2,7,4)
a)
Compute u(x) = (u1(x1), u2(x2)) and u(y). Are the two allocations comparable
according to the Pareto-criterion?
b)
Try to find (by trial and error), a new allocation x , for which social welfare is the
same as with x, given a utilitarian social welfare function, but individual utility
positions u( x ) are different from u(x).
Compare x and y according to the social welfare functions
c)
according to Bentham (utilitarian welfare function)
d)
according to Nash
e)
according to Rawls
Now consider the utility function u 2 instead of u2, which is defined as
1
u 2 (x12 , x 22 ) = (x12 + ln x 22 ) 2 .
f)
Do u 2 and u2 describe the same (ordinal) preferences (= same indifference
curves)?
g)
Solve c) and e) with u 2 instead of u2.
Solution:
a) The two allocations are Pareto-incomparable.
u(x) = (3.87, 3.38) and u(y) = (1.41, 8.38)
b) Social Welfare with allocation x: W = 7.259
One possible allocation x = (3.37,1,3.63, 6)
W(3.37,1,3.63,6)=7.257
c) Bentham:
W = u1 + u 2
W ( x) = u 1 (5,3) + u 2 (2,4)
W ( x) = 3,873 + 3,386 = 7,259
W ( y ) = u 1 (1,2) + u 2 (7,4)
W ( y ) = 1,414 + 8,386 = 9,8
Allocation y is superior to x.
d) Nash:
W = u1 * u 2
W ( x) = u 1 (5,3) * u 2 (2,4)
W ( x) = 3,873 * 3,386 = 13,114
W ( y ) = u 1 (1,2) * u 2 (7,4)
W ( y ) = 1,414 * 8,386 = 11,858
Allocation x is superior to y.
e) Rawls:
W = min{u 1 , u 2 }
W ( x) = min{u 1 (5,3), u 2 (2,4)}
W ( x) = min{ 3,873; 3,386} = 3,386
W ( y ) = min{u 1 (1,2), u 2 (7,4)}
W ( y ) = min{1,414; 8,386} = 1,414
Allocation x is superior to y.
f) Because it is just a matter of monotonic transformation the ordinal preferences remain the
same:
2
2
if u 2 ( x12 , x22 ) > u 2 ( y12 , y22 ) then u ( x12 , x22 ) > u ( y12 , y22 )
g) According to Bentham: W ( x ) = 5, 71;W ( y ) = 4,31
According to Rawls: W ( x) = 1,84;W ( y ) = 1, 41
Seitenumbruch
12.
Let a set of 3 alternatives: A={x,y,z} be given. The preference ordering (binary relation) R1
over A of person 1 is described by the set:
R1 ≡ {x ≥1 y, y ≥1 z, x ≥1 z, y ≥1 x, ¬ (z ≥1 y), ¬ (z ≥1 x)},
where x ≥1 y means that person 1 does not consider x inferior to y.
The sign ¬ characterizes a negation, so ¬(z ≥1 y) means: it is not true that alternative z is not
considered inferior to y by that person.
(y ≥1 z und ¬(z ≥1 y)) can be abbreviated by y >1 z, which means that that person considers y
strictly superior to z.
Furthermore (x ≥1 y and y ≥1 x) can be abbreviated by x ~1 y, which means that person 1 is
indifferent between x and y.
a) Write the relation (=set) R1 with these abbreviations. (Hint: Summarize x ≥1 y and y ≥1
x, y ≥1 z and ¬(z ≥1 y), x ≥1 z and ¬(z ≥1 x).)
b) Is the relation R1 complete? Why or why not? (Hint: Completeness of the relation
means that of any two elements in A either the first is not considered inferior to the
second or vice versa.)
A binary relation is called transitive, if for 3 arbitrary elements a, b, c ∈ A the following
always holds:
If a ≥1 b and b ≥1 c are in the relation (=set), then a ≥1 c also belongs to that relation (=set).
Remark: One may extend the definition of transitivity to strict superiority a >1 b.
c) Is the relation R1 transitive?
(Hint: Use the detailed relation (=set) instead of the abbreviated one.)
A binary relation is called reflexive, if for every element a ∈ A, a ≥1 a belongs to that relation.
We neglect this issue. A reflexive, transitive and complete binary relation is called
(preference) ordering.
Now there are two more persons with the following preferences over A:
R2 ≡{y ≥2 x, ¬ (x ≥2 y), z ≥2 y, ¬ (y ≥2 z), z ≥2 x },
R3 ≡{y ≥3 x, y ≥3 z, z ≥3 x}.
d) Are R2 and R3, respectively, complete and transitive relations? Write R2 with the
abbreviations ≥2, ~2.
We define majority voting between a pair of alternatives a, b ∈ A as follows (where R
without index denotes the social ordering derived from majority voting):
a ≥ b holds, if the number of those persons h, for which a ≥h b occurs in their preference
ordering, is not less than the number of those persons k, for which b ≥k a occurs.
e) Determine the result (the social relation R) of majority voting for all possible pairs in
the set A ={x,y,z}.
For example, majority voting results in y ≥ x because all persons exhibit the relation y
≥h x (h = 1,2,3) while only person 1 exhibits the ordering x ≥1 y. We look for the set R
= {…} containing all relations analogously to y ≥ x, which result from majority voting
over all pairs of alternatives in A.
Solution:
a) R1 = {x ~1 y, y >1 z, x >1 z}
b) The relation is complete and transitive, for all possible pairs (x,y) (y,z) (x,z) are
ranked and their rankings are not inconsistent.
c) R2 abbreviated = {y >2 x, z >2 y, z ≥2 x }
R3 = {y ≥3 x, y ≥3 z, z ≥3 x}
d) Both relations are complete, i.e. all possible pairs are ranked. There even aren’t any
inconsistencies in their rankings and thus, both relations are transitive.
e) Majority voting::
x against y:
person 1: x ~1 y
person 2: y >2 x
person 3: y ≥3 x
2 persons prefer y, one person is indifferent. Hence, y wins.
x against z:
z wins
y against z:
y wins
Thus, the outcome of the majority voting ist:
R={y>x, z>x, y>z}
Seitenumbruch
13.
The preferences of a household over two goods with quantities x and y can be represented by
the utility function u(x,y) = 0,1 x +ln y.
a) What is the type of this utility function called?
The household’s budget is b = 100.
b) Determine the (uncompensated) demand function of the household for both goods (in
dependency of the prices px and py, respectively).
Hint: For the calculation of the demand function for good y one just needs the
marginal-rate-condition but not the budget constraint.
Furthermore, let are p1x =4 and p1y =5.
c) Determine the gain in consumer surplus if the price of the second good decreases to
p 2y =2.
Hint: To do so the inverse demand function MWP(y) (= py) = ... has to be integrated
appropriately.
d) Show that the compensated demand function for good y equals its uncompensated
demand function.
Hint: Solve the expenditure minimizing problem for that.
min x , y px x + p y y
s.t.
0,1x + ln y ≥ u
x, y ≥ 0
and determine the compensated demand function out of the first order conditions.
Analogously to b), where the uncompensated demand function for good y can be
determined solely by the marginal-rate-condition MRSxy = −px/ py without utilizing
the budget constraint, one can calculate the compensated demand function for good y
without using the utility condition, which has to be fulfilled in equality u(x,y) = u in
the optimum.
Solution:
a)
This is a quasi-linear utility function. There is a linear good (x) and a non-linear
good (y).
b)
Marshallian demand:
marginal-rate-condition: MRS x , y
demand for y: y =
∂u
px
p
0,1
=
→ ∂x =
= 0,1 y = x
∂u
1
py
py
dy
y
10 px
py
demand for x:
marginal-rate-condition: 0,1 y =
the insertion gives: x =
px
, budget constraint: px x + p y y = 100
py
100
− 10
px
Hence, the demand for the linear good just depends on own price but not of the
other good’s price.
c)
Because the demand for x just depends on px, it is satisfying to investigate the
demand for y (otherwise, the consumer surplus changing due to a variation of y
will also be changed by an alteration of the demand for x).
demand for y: y =
10 px
10
, inverse demand function: p y = ⋅ px
py
y
consumer surplus differential (initial demand: 8 units, new: 20 units):
d)
L = px x + p y y − λ (0,1x + ln y − u )
First order conditions:
∂L
∂L
λ
: px − 0,1λ = 0; : p y − = 0
y
∂x
∂y
The insertion gives: y c =
10 px
, Thus, Marshallian and compensated demand f
py