Download ECMC02 – Week 10

ECMC02 – Week 10
General Equilibrium Analysis – continued
Objectives this week:
- review contract curve, pareto efficient
points, pareto efficient trades (barter)
- how competitive markets lead to trading
along the contract curve in an exchange
economy (Edgeworth Exchange Box)
- the first and second welfare theorems
about competitive markets
1
Initial endowment, pareto-preferred
allocations, pareto-efficient trades (barter)
Mike’s
Guinness
Pat’s
lamb
60
80
20
40
0 for Mike
50
10
40
30
20
20
30
10
40
50
0 for
Pat
10
50
30
70
90
100
Pat’s
Guinness
2
Mike’s
lamb
The contract curve of all possible paretoefficient allocations
Mike’s
Guinness
Pat’s
lamb
60
80
20
40
0 for Mike
50
10
40
30
20
20
30
10
40
50
0 for
Pat
10
50
30
3
70
90
100
Pat’s
Guinness
Mike’s
lamb
Will consumers have incentives to trade
towards pareto-efficient allocations? Is
trade welfare-improving?
4
Allocation J and allocation K are the only two
possible allocations of goods amongst a group of
potential consumers. We are told that allocation K
is Pareto-preferred to allocation J. We may
therefore conclude that:
A) every consumer must be better off with K than
with J
B) a majority of consumers are better off with K
than J
C) a move from K to J will make some consumers
better off and none worse off
D) a move from J to K will not make any consumers
better off
E) K is pareto optimal
F) none of the above
5
Bill has 6 glasses of scotch and 3 glasses of
water. Ann has 3 glasses of scotch and 6
glasses of water. Bill regards scotch and
water as perfect 1-for-1 complements while
Ann regards them as perfect 1-for-1
substitutes.
(a) Is this initial endowment paretoefficient?
(b) If not, what is the feasible set of
pareto-efficient allocations? Give an
example of a feasible trade that could
be made to reach a pareto-optimal
outcome.
6
Charlie has an initial endowment of 3 apples
and 12 bananas while Doris has 6 apples and 6
bananas. Charlie’s utility function is U = ACBC
where AC is the amount of apples Charlie
consumes and BC is the amount of bananas he
consumes and ACBC is the product of these
two numbers. Doris’ utility function is U = AD
BD ,where AD is the amount of apples Doris
consumes and BD is the amount of bananas she
consumes.
Given their initial allocation, at every Pareto
optimal allocation in the interior of the
Edgeworth Exchange Box, does Doris consume
the same number of apples as she does
bananas?
7
What is a competitive market in an EEB? In
barter, the outcome of trade can depend on
the bargaining power of the two parties.
Competitive markets have many buyers and
sellers, so no one has power (individually) to
change price.
Define excess demand by an individual as a
positive difference between the amount
desired at a particular price and the
endowed amount. Excess supply is a
negative difference. Aggregate excess
demand is net excess demand over all
individuals. Aggregate excess supply is net
excess supply over all individuals.
8
Aggregate excess demand for a product will
make its price rise. Aggregate excess
supply of a product will make its price fall.
A competitive equilibrium will be one where
the quantity demanded of each product is
equal to the quantity supplied of each
product. (a general equilibrium). (Remember
there is no production, so consumers supply
from endowment).
In other words, a competitive equilibrium is
a set of prices (a price ratio) at which
aggregate excess demand for each product
is zero.
9
How does the price respond to excess
demand or excess supply? To ensure there
are no disequilibrium trades, economists
invent auctioneer to call out bids. Bids will
only be accepted if demand = supply.
10
What if PX/PY is too shallow? Too steep?
Mike’s
Guinness
80
60
40
0 for Mike
20
Pat’s
lamb
10
40
20
30
20
10
40
Mike’s
lamb
0 for
Pat
10
30
50
80
100
Pat’s
Guinness
11
Competitive equilibrium will exist, will be
unique, and will be pareto-optimal.
Mike’s
Guinness
80
40
60
0 for Mike
20
Pat’s
lamb
10
40
20
30
20
10
40
Mike’s
lamb
0 for
Pat
10
30
50
80
100
Pat’s
Guinness
12
Assume that the initial endowment of good
X and good Y between person A (Alfie) and
person B (Bernice) is XA = 60, YA = 20, and
XB = 40, YB = 20.
Assume further that the utility functions
for Alfie and Bernice are Cobb-Douglas, so
UA = XA0.5YA0.5
UB = XB0.75YB0.25
Will a price ratio of 1-1 (i.e., PX = 1 and PY =
1) be a competitive equilibrium?
Will a price ratio of 1/2 (i.e., PX = 0.5 and PY
= 1) be a competitive equilibrium?
What is the competitive equilibrium? How
much trade will occur at this equilibrium?
13
Work this out in class
14
In competitive equilibrium, the ratio of the
prices (PX/PY) will equal the ratio of the
marginal utilities = MRS.
Invisible Hand leads to pareto-efficient
equilibrium
First Theorem of Welfare Economics
Every competitive equilibrium allocation is
pareto-efficient
Second Theorem of Welfare Economics
Any allocation on the contract curve can be
sustained as a competitive equilibrium
Separability of efficiency and equity
15
Two consumers, Bill and Fred, each consume only
two goods, X and Y. At the initial endowment point
in the Edgeworth Box, Fred's MRS is 1/2 while
Bill's MRS is 1 (in both cases, the MRS is defined
as -dY/dX holding U constant). Then:
A) both benefit if Fred trades 3 units of X to Bill
in exchange for 1 unit of Y
B) both benefit if Bill trades 3 units of X to Fred
in exchange for 1 unit of Y
C) both benefit if Fred trades 3 units of Y to Bill
in exchange for 1 unit of X
D) both benefit if Bill trades 3 units of Y to Fred
in exchange for 1 unit of X
E) neither benefits from a trade because they are
on the contract curve
F) to make a definitive statement about a trade,
we would need more information
G) none of the above
16
Answer:
MRSXY (= -dy/dx) means the rate at which
the individual is willing to substitute X for Y.
Fred’s MRS is ½, so he is willing to trade ½
unit of Y for one unit of X. Bill’s MRS is 1, so
he is willing to trade 1 unit of Y for 1 unit of
X. Relatively speaking, Fred regards Y as
valuable, while Bill regards X as valuable. So,
Fred will seek to get Y from Bill and Bill will
seek to get X from Fred. However, the deal
must be a good one. Fred will want to get Y
at a rate that is better than his MRS (more
than 1 unit of Y for every 2 units of X). And
Bill feels the same way. He will want to get X
at a rate that is better than his MRS (more
than 1 unit of X for every 1 unit of Y). Any
deal that satisfies both Fred and Bill’s
demands will result in a trade.
17
The options with Bill giving up X or Fred giving
up Y are not welfare-improving, so options B
and C are not possible. Option A would make
Bill happy, but Fred would be worse off.
Option D would make Fred happy, but Bill
would be worse off. A possible trade would
be something like Bill gives up 1 unit of Y for
1½ units of X. In this case, both Bill and Fred
are better off.
18
Consider a simple economy with two goods food and clothing - and two consumers (Bert
and Ernie). There is an initial endowment in
this simple “exchange” economy, and the
initial price ratio is price of food/price of
clothing = 3/1. At this initial price ratio, Bert
wants to buy 6 units of clothing while Ernie
wants to sell 2 units of food. We can
conclude that this initial price ratio is:
(A) an equilibrium price ratio
(B) too low (in other words, the ratio of the
price of food to the price of clothing will have
to rise to give us equilibrium
(C) too high (in other words, the ratio of the
price of food to the price of clothing will have
to fall to create an equilibrium
(D) none of the above
19
Conclusions from Edgeworth Exchange Box
Competitive markets can lead to a general
equilibrium across all markets that exhausts
all potential gains from trade (i.e., is
efficient).
We do not need to give up efficiency in
order to get equity.
20
Think about the production decision for
firms that operate in competitive input
markets – Edgeworth Production Box
What is the contract curve in an Edgeworth
Production Box?
Are there forces in competitive input
markets that will act to efficiently
reallocate resources between firms?
What is the relationship between the
contract curve in the Edgeworth Production
Box and a production possibilities frontier?
21