This preview shows pages 1–2. Sign up to view the full content.
Econ 51: Final Exam  Suggested Solutions
1. General Equilibrium (27 points)
a.
i. (3 points) These are log utilities, so all we need to do is to equate MRSs. This
gives us
b
i
a
i
=
b
j
a
j
for each
i; j
. In addition, to be feasible, we must have the sum
of the bananas equal to 5
N
and the sum of the apples equal to 10
N
. These two
allocation that satisfy
b
i
a
i
=
1
2
for all
i
.
ii. (3 points) Normalize
p
a
= 1 and denote
p
b
=
p
. We will clear the apple market.
Excess demand by each individual (they are all identical) is given by
z
i
a
(
p
) =
1
2
10+5
p
1
10 = 2
:
5
p
5. Total excess demand in the market is simply
z
a
(
p
) =
N
(2
:
5
p
5). To clear the market, we must have
p
= 2. Allocation for each
agent is then given by (10
;
5), that is the initial endowment. The solution does
not depend on
N
. This is because, essentially, the general equilibrium already
assumes very many identical agents of each type, so having
N
types that are all
identical is exactly like an economy with a single individual.
b.
i. (3 points) Now the MRSs must also be equal the MRT, which is 1. Thus, we
must have
b
i
a
i
= 1 for all
i
. To achieve this, the production plan for the ±rm must
make total number of apples equal total number of bananas. This is achieved by
a production plan of (
y
a
; y
b
) = (
2
:
5
N;
2
:
5
N
).
ii. (3 points) We know that the ±rm will be in business (due to the shortage in
bananas), so prices must be
p
a
=
p
b
= 1 (we normalized the price of apples as
before). At these prices, allocations to each individual are (7
:
5
;
7
:
5). The ±rm’s
production plan is (
y
a
; y
b
) = (
2
:
5
N;
2
:
5
N
). For the same reason as in part a.ii,
the answer doesn’t depend on
N
.
iii. (2 points) It won’t change any of the answers. The overall production plan in
the industry will remain the same (for both parts), and it can be arbitrarily split
between the two ±rms, as they make zero pro±ts anyway. That is, production
plans must satisfy (
y
A
a
; y
A
b
) = (
x; x
) and (
y
B
a
; y
B
b
) = (
w; w
) such that
x; w
±
0
and
x
+
w
= 2
:
5
N
.
iv. (2 points) It won’t change any of the answers. The overall production plan in
the industry will remain the same (for both parts), and it can be arbitrarily split
between the two ±rms, as they make zero pro±ts anyway. That is, production
plans must satisfy (
y
A
a
; y
A
b
) = (
x; x
) and (
y
B
a
; y
B
b
) = (
w;
w
) such that
x; w
±
0
and
x
w
= 2
:
5
N
.
c.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 03/19/2011 for the course ECON 51 taught by Professor Tendall,m during the Winter '07 term at Stanford.
 Winter '07
 Tendall,M

Click to edit the document details