Exercises of Microeconomics

Exchange
Production
Exercises of Microeconomics
Exchange - Production (Ch. 17-18 Varian)
Fabio Tramontana (University of Pavia)
slides available at: http://tramontana.altervista.org/teaching.html
PhD in Economics at L.A.S.E.R.
Tramontana
Exercises Micro
Exchange
Production
Outline
1
2
Exchange
Exercise
Exercise
Exercise
Exercise
17.4
17.6
17.9
17.11
Production
Exercise 18.2
Tramontana
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Outline
1
2
Exchange
Exercise
Exercise
Exercise
Exercise
17.4
17.6
17.9
17.11
Production
Exercise 18.2
Tramontana
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Exercise 17.4
There are two consumers A and B with the following utility
functions and endowments:
u (x , x
u (x , x
A
B
) = a ln xA1 + (1 − a) ln xA2 ωA = (0, 1)
) = min(xB1 , xB2 )
ωB = ( 1 , 0 )
B
1
2
A
1
A
2
B
Calculate the market clearing prices and the equilibrium allocation.
Let us start by calculating the income of the two consumers, given
their endowments, that is:
m
m
= p2
B = p1
A
Tramontana
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Solution
The rst consumer has a Cobb-Douglas type utility functions. So,
the demands are:
m
= a pA
xA2 = (1 − a) mp A
x
1
A
1
2
In our case:
x
= a pp
xA = (1 − a)
1
2
A
1
2
Concerning the second consumer, the utility function is such that
the demands for the two goods will be the same:
x
1
B
Tramontana
= xB2
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Solution
The budget constraint of the second consumer takes the form:
(p1 + p2 ) xB1 = p1
that is:
x
1
B
= xB2 =
p
p
1
1 + p2
We can solve the problem only by setting one price as numeraire.
For example p1 = 1.
Tramontana
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Solution
Now, consider the demands for good 1.
We know that the sum of the two demands must be equal to the
sum of the endowments.
In our case only the second consumer already owns one unit of
good 1, whose value is 1.
So we have:
xA1 + xB1 = 1
⇓
ap
+ 1+1p = 1
2
2
Form which we obtain the value of
p
2
=
Tramontana
p:
2
1−a
a
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Solution
Now we can calculate the demands:
x
x
= 1−a
=a
B
1
A
1
Tramontana
x
x
= 1−a
=a
B
2
A
2
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Outline
1
2
Exchange
Exercise
Exercise
Exercise
Exercise
17.4
17.6
17.9
17.11
Production
Exercise 18.2
Tramontana
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Exercise 17.6
We have two agents with indirect utility functions:
v (p , p , y ) = ln y − a ln p
v (p , p , y ) = ln y − b ln p
1
1
2
1
2
1
2
1
− (1 − a) ln p2
− (1 − b) ln p2
and initial endowments
ω1 = (1, 1) ω2 = (1, 1).
Calculate the market clearing prices.
The Roy's identity is the direct way to obtain the demand functions
from the indirect utility functions.
Tramontana
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Solution
Remember that:
Roy's identity
x (p, m) =
i
∂ v (p ,m)
∂p
− ∂ v (p,im)
∂m
So we need to calculate six partial derivatives:
∂ v1 (p ,y )
∂ p1
∂ v2 (p ,y )
∂ p1
= − pa
1
= − pb
1
∂ v1 (p ,y )
∂ p2
∂ v2 (p ,y )
∂ p2
Tramontana
= − (1p−a)
2
= − (1p−b)
2
Exercises Micro
∂ v1 (p ,y )
∂y
∂ v2 (p ,y )
∂y
=
=
1
y
1
y
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Solution
Now we can calculate the demand functions:
x
x
1
1
=
ay
1
2
=
by
p1
p1
x
x
2
1
=
(1−a)y
2
2
=
(1−b )y
p2
p2
The two consumers have the same endowments, that is:
y =p
1
+ p2
and we also know that:
x
x
1
1
2
1
+ x21 = ω11 + ω12 = 2
+ x22 = ω12 + ω12 = 2
Tramontana
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Solution
We can substitute the demand functions in these equations.
Consider for instance good 1:
ay by
+
p p
1
Given that
y =p
1
+ p2 and setting
a(1 + p
2
=2
1
p
1
= 1 as numeraire we have:
) + b(1 + p2 ) = (1 + p2 )(a + b) = 2
and nally:
p
2
=
Tramontana
2
a+b
−1
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Outline
1
2
Exchange
Exercise
Exercise
Exercise
Exercise
17.4
17.6
17.9
17.11
Production
Exercise 18.2
Tramontana
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Exercise 17.9
Consider an economy with 15 consumers and 2 goods. Consumer 3
has a Cobb-Douglas utility function u3 (x31 , x32 ) = ln x31 + ln x32 . At a
certain Pareto ecient allocation x ∗ , consumer 3 holds (10,5).
What are the competitive prices that support the allocation x ∗ ?
Utility maximization requires that each
consumer's marginal rate
.
of substitution be equal to the common price ratio
This must also hold for consumer 3:
∂ u3 (x ∗)
∂ x1
∂ u3 (x ∗)
∂ x2
Tramontana
=
p
p
1
2
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Solution
After the calculations of the derivatives we obtain:
x
x
2
3
1
3
=
p
p
1
2
By using the Pareto ecient allocation we have:
p
p
1
2
Tramontana
1
2
= .
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Outline
1
2
Exchange
Exercise
Exercise
Exercise
Exercise
17.4
17.6
17.9
17.11
Production
Exercise 18.2
Tramontana
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Exercise 17.11
Person A has a utility function of uA (x1 , x2 ) = x1 + x2 and person B
has a utility function uB (x1 , x2 ) = max {x1 , x2 } . Agent A and agent
B have identical endowments of (1/2,1/2).
(a) Illustrate this situation in an Edgeworth box diagram.
(b) What is the equilibrium relationship between p1 and p2 ?
(c) What is the equilibrium allocation?
The maximization of utility condition gives the realtionship between
prices:
1=
p
p
1
2
=⇒ p1 = p2
Tramontana
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Solution
In our situation there is a total of 1 unit of good 1 and 1 unit
of good 2 available.
In the initial situation the goods are equally distributed
between the consumers.
Given that in equilibrium the prices are the same, we can
consider this situation a barter.
The initial situation give to the consumer 1 an utility of 1, and
to the consumer 2 an utility of 1/2.
If consumers change a good with the same amount of the
other, nothing changes for consumer 1.
Tramontana
Exercises Micro
Exercise 17.4
Exchange
Exercise 17.6
Production
Exercise 17.9
Exercise 17.11
Solution
Consumer 2 has convenience in giving the total amount of 1 good
in change of the same amount of the other one. In fact in this case
he/she reaches an utility of 1 while the utility of consumer 1 does
not vary.
The consequence is the we will have two possibile equilibrium
solutions:
A : (1, 0) B : (0, 1)
or
A : (0, 1) B : (1, 0)
Tramontana
Exercises Micro
Exchange
Production
Exercise 18.2
Outline
1
2
Exchange
Exercise
Exercise
Exercise
Exercise
17.4
17.6
17.9
17.11
Production
Exercise 18.2
Tramontana
Exercises Micro
Exchange
Production
Exercise 18.2
Exercise 18.2
Consider an economy with two rms and two consumers. Firm 1 is
entirely owned by consumer 1. It produces guns from oil via the
production function g = 2x . Firm 2 is entirely owned by consumer
2; it produces butter from oil via the production function b = 3x .
Each consumer owns 10 unit of oil. Consumer 1's utility function is
u (g , b) = g .4 b.6 and consumer 2's utility function is
u (g , b) = 10 + .5 ln g + .5 ln b.
(a) Find the market clearing prices for guns, butter, and oil.
Let us take the price of oil as numeraire:
Tramontana
p
o
= 1.
Exercises Micro
Exchange
Exercise 18.2
Production
Solution
Both the product functions display constant resturns to scale, so, in
equilibrium, they make zero prots.
If we consider rm 1 we have:
π1 = pg g − po x = pg 2x − x
The prot is zero provided that:
p
g
=
1
2
With a similar argument for the second rm we obtain:
π2 = pb b − po x = pb 3x − x = 0
⇓
pb = 13
Tramontana
Exercises Micro
Exchange
Exercise 18.2
Production
Exercise 18.2(b)
(b) How many guns and how much butter does each consumer
consume?
Remember that demand functions with cobb-Douglas utility
functions are in general:
Cobb-Douglas
Given an utility function of the form:
u (x , x
1
2
) = x1α x2
β
the demands functions for the two goods are:
x
1
= α pm
1
Tramontana
;
x
2
= β pm
2
Exercises Micro
Exchange
Production
Exercise 18.2
Solution
In our case both consumers have the same income given by their
endowments of 10 units of oil:
m
1
= m2 = 10po = 10
Given the already calculated prices of guns and butter we have:
x
x
;
= (0.4) 110
/2 = 8
10
2
= (0.5) 1/2 = 10 ;
g
1
g
Tramontana
x
x
= (0.6) 110
/3 = 18
10
2
= (0.5) 1/3 = 15
b
1
b
Exercises Micro
Exchange
Production
Exercise 18.2
Exercise 18.2(c)
(c) How much oil does each rm use?
Firm 1 produces guns. In total it produces a number of guns equal
to:
xg = xg1 + xg2 = 18
From the production function we obtain the quantity of oil the rm
needs to produce 18 guns:
g = 2x ⇒ 18 = 2x
x
Tramontana
⇓
=9
Exercises Micro
Exchange
Production
Exercise 18.2
Solution
Similarly, rm 2 produces butter. The quantity produced is:
x
b
= xb1 + xb2 = 33
From the production function we obtain the quantity of oil the rm
needs to produce 33 units of butter:
b = 3x ⇒ 33 = 3x
x
Tramontana
⇓
= 11
.
Exercises Micro