ECMC02 – Week 11
More General Equilibrium Analysis
THE PRODUCTION POSSIBILITIES
FRONTIER
From the contract curve in the production
box.
Tradeoff between X and Y at cost-minimizing
points along the contract curve.
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PPF - all technically efficient combinations of
X and Y that can be produced with a fixed
amount of resources (i.e., of L and K). Points
inside the PPF are inefficient; points outside
the PPF are unattainable.
2
The (absolute value of the) slope of PPF is
MRTXY = -(dY/dX) - the marginal rate of
transformation of good X for good Y.
It measures the rate at which the economy
can (efficiently) transform good Y into good
X, by reallocating resources from one to the
other.
It can also be described as the opportunity
cost of X in terms of Y.
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For example, we could have this PPF:
Y = 14 – (X/6) – (X2/12) ,
0<=X <= 12
Then dY/dX = - 1/6 – 1/6(X), so the MRT =
1/6 + 1/6(X)
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Values of X
0
1
2
4
6
8
12
Values of Y
14
13 ¾
13 1/3
12
10
7 1/3
0
5
MRT = -dY/dX
1/6
2/6
3/6
5/6
7/6
9/6
13/6
The Marginal Rate of Transformation of good
X for good Y is equal to the ratio of the
marginal costs of the two goods
MRTXY = MCX/ MCY.
As more Y is produced, the marginal cost of Y
increases and the marginal cost of X falls.
This gives us the increasing cost PPF, or
concave PPF (bowed outwards).
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The PPF shows us the many possible
combinations of X and Y that are technically
efficient.
How, then, do we find the efficient product
mix?
This is not technical efficiency. Instead
relationship between preferences and costs.
In other words, the efficient product mix
must have the MRS of every consumer equal
to the marginal rate of transformation (the
rate at which consumers just value X in terms
of Y is equal to the cost of transforming Y
into X.
Therefore, the product mix is not efficient
unless MRS for all consumers = MRT.
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Why must this condition be met? Assume it
isn’t. Then, perhaps, MRTXY = 1 while
consumers have MRSXY = 2. That means the
ratio of the marginal costs of the two goods
is equal to 1 (the opportunity cost of X = 1
unit of Y), but consumers are willing to give up
2 units of Y and take 1 unit of X in exchange
and stay equally happy. Therefore, the
economy could give up producing 2 units of Y,
produce 2 units of X instead, and make
consumers happier.
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Will a competitive equilibrium give us an
efficient product mix?
We are now looking at the decision of firms
about how much to produce. In perfect
competition a producer will keep on producing
until P = MC (profit maximization).
If each firm does this then:
PX/ PY = MCX/ MCY.
But consumers will each set MRSXY = PX/ PY .
So in the final competitive equilibrium:
MUX/MUY (=MRSXY) = PX/PY= MCX/MCY (=MRTXY)
A competitive general equilibrium gives
efficiency in consumption, in production, and
in product mix.
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REVIEW
(a) competitive equilibrium in exchange will
give MRSPXY = PX/PY = MRSMXY
(consumers will adjust consumption to
reach the contract curve)
(b) competitive input markets will give
MRTSXLK = MRTSYLK (firms will trade
inputs to reach contract curve)
(c) competitive producers will adjust output
until MCX = PX and MCY = PY (firms
profit maximize in choosing amount of
output).
(d) As a result, firms choose the mix of
output such that PX/ PY = MCX/ MCY
MUX/MUY = PX/PY = MCX/MCY
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Example: from Problem Set, Question #4
Assume that PPF is X2 + 12Y = 432
Utility function aggregated over all individuals
in society (or one person): U = XY
What is the pareto-optimal mix of products in
this economy?
What amount of utility (in utils) will society
get?
11
Set up Lagrangean
ℒ = XY + λ(432 – X2 – 12Y)
Take the partial derivatives of this function
and set equal to zero to find constrained
maximum
12
The optimal product mix is therefore X = 12
and Y = 24. Here the MRS = MRT (check).
At these values, U = XY = 12 x 24 = 288
utils
13
Second part of this question asks what
happens if there is a possibility of
international trade. Let’s look at this issue
graphically.
14
Second part of Question #4
Assume that X and Y can trade
internationally on a 1-for-1 basis. Will this
country trade? How much will it trade?
How much will it produce of X and Y? How
much will it consume of X and Y? What will
be the effect on utility?
Trade on a 1-1 basis says that this country
can trade its surplus X produced for an
equal number of units of Y (from
international trade). In mathematical
terms, XP – XC = YC – YP.
The two constraints to the problem are the
PPF and the set of international prices. The
PPF affects production and the
international prices affect consumption and
production.
15
Set up Lagrangean
ℒ = XCYC + λ(432 – XP2 – 12YP) + γ(XP – XC - YC
+ YP)
Take the partial derivatives of this function
and set equal to zero to find constrained
maximum
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HOW WILL AN EXCISE TAX AFFECT
EQUILIBRIUM?
V
Z
17
Of course, there will be a chain of effects,
but the immediate effect of the tax (let us
say a tax on good Y) will be to change the
relative price ratio which affects the
behaviour of producers. If there is a tax of t
(where t is a proportion) then the price ratio
which was PX/ PY will become PX/ (1 – t)PY.
This will be a steeper price line than before,
encouraging producers to change their output
mix until PX/ (1 – t)PY = MCX/ MCY . In other
words, producers will want to move around to
point Z from point V (i.e. away from the
original equilibrium). Point Z will have more X
and less Y than the original equilibrium.
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However, although producers see this new
price ratio, consumers will see a different
price ratio (there is a wedge driven between
the prices paid by consumers and those
received by sellers).
The marginal rate of transformation will be
different than the (common) marginal rates
of substitution. The result is not pareto
optimal.
There is too much X and too little Y in the
product mix, but it is an equilibrium.
Note the new conclusion. A tax on one
product will cause too little production in that
market, but too much in the other market!!!
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