ECON 121 Winter 2011
Professor Faingold
Noam Tanner
Homework Assignment 7
Solutions to Homework Assignment #7
Economics is common sense made di
ffi
cult
Anonymous
Congratulations everyone! You have o
ffi
cially graduated from the simple supply
demand equations of intro micro and entered the Competitive Equilibrium
framework used to by pros! In this assignment, you will solve for an equilibrium
in a model economy with much more detail than possible with the supply and
demand line approach. Now you can not only discuss price e
ff
ects (and compar
ative statics) on outputs, you can discuss price e
ff
ects (and comparative statics
on inputs.
This assignment can get computationally intense if you don’t proceed the
right way.
Because of this, it might be a good idea to set
β
=
1
2
in for the
function
f
1
. Throughout the problem set I will say the solutions for the case
where
β
=
1
2
.
1. The first thing to do is to write what you know about this economy:
We have two consumers, A and B, one firm (with production function
f
),
and two consumption goods (1 and 2).
•
Technology:
The firm’s production functions are:
y
1
=
f
1
(
k
1
,l
1
)
=
k
β
1
l
1

β
1
and
y
2
=
f
2
(
k
2
,l
2
)
=
√
k
2
l
2
where
β
∈
(
1
2
,1] is a technology parameter and
y
1
,
k
1
,
and
l
1
denote the output
and input levels in the production of good 1, respectively, and likewise for good
2.
•
Preferences:
Consumers derive utility functions as in HW #6:
u
A
(
x
A
1
,x
A
2
)=
(
x
A
1
)
α
(
x
A
2
)
1

α
and
u
B
(
x
B
1
,x
B
2
)=
x
B
1
x
B
2
•
Endowments:
One important detail to pick up here is that each con
sumer gets L units of labor and K units of capital.
Remember this
when it comes to writing the budget constraints!
Now for the
feasibility constraints:
x
A
1
+
x
B
1
=
y
1
x
A
2
+
x
B
2
=
y
2
k
1
+
k
2
=
2
K
l
1
+
l
2
=
2
L
1
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Now let’s get down to business:
•
(a) The question asks why the production bundle (
y
1
,y
2
) in a Pareto op
timal allocation (
x
A
1
,x
A
2
,
x
B
1
,x
B
2
,
y
1
,y
2
) is on the
production possibili
ties frontier (PPF)
. Notice that the allocation must specify the output
quantities of goods 1 and 2
consumed by A
and
B in addition to the
production bundle
). This is an easy way to lose points on homeworks
and exam.
MAKE SURE YOU INCLUDE ALL OF THE QUANTITIES
(
x
A
1
,x
A
2
,
x
B
1
,x
B
2
,
y
1
,y
2
)
WHEN SPECIFYING A PARETO OPTIMAL ALLO
CATION IN AN ECONOMY WITH PRODUCTION!
Back to (a):
Let’s say that (
y
1
,y
2
) is NOT on the PPF, in other word’s
y
1
=
Y
1
(
y
2
) where
Y
1
(
y
2
) =
max
k
1
,l
1
,k
2
,l
2
f
1
(
k
1
, l
1
)
subject to
f
2
(
k
2
, l
2
) =
y
2
k
1
+
k
2
=
2
K
l
1
+
l
2
=
2
L
In words, this means that
Y
1
(
y
2
)
is the largest amount of good 1 that can be
produced given that the amount of good 2 produce id
y
2
and that the fea
sibility conditions are satisfied (all the inputs are used and the economy is
not using more than this amount of inputs).
Thus, if
y
1
=
Y
1
(
y
2
) then this
means that
MORE
y
1
CAN BE PRODUCED WHILE KEEPING THE
AMOUNT OF
y
2
CONSTANT.
Thus, since both utility functions are strictly
increasing, if we produce
Y
1
(
y
2
)
we can take the extra amount of good 1 pro
duced (
y
1

Y
1
(
y
2
)
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 Spring '09
 SAMUELSON
 Economics, k2, xa

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