Market | General Equilibrium 1
Endowments and Allocations
Consider flrst the case of a pure exchange economy,(One with no production).
Suppose there are two consumers,A and B,in a two good economy,A starts with an endowment of !A =?!1A;!2A￠
and B starts with an endowment of !B =?!1B;!2B￠.
A actually consumes x1A of good 1 and x2A of good 2,B actually consumes x1B of good 1 and x2B of good 2.
These bundles are called allocations,The initial allocation is (!A;!B),The flnal allocation is (xA;xB).
The consumers cannot purchase more than there is of a particular good,Hence an allocation is feasible whenever:
x1A +x1B? !1A +!1B and x2A +x2B? !2A +!2B
Market | General Equilibrium 2
Edgeworth Boxes
The information on the previous slide can be summarised in an Edgeworth box.
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A
B
!1A
!1B
!2A !2B
x1A
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x1B
x2B
The total width of the box is !1A +!1B and the total height is !2A +!2B.
Consumer A consumes nothing at the bottom left corner,consumer B consumes nothing at the top right corner.
Label the endowment allocation ! and the flnal allocation x,Remember these represent bundles for all consumers.
Market | General Equilibrium 3
Preferences
Indifierence curves can be drawn in on an Edgeworth box,Consumer A prefers bundles to the north-east of the
diagram and consumer B prefers bundles to the south-west of the box,Hence the diagram looks like:
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A
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x
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Clearly many more indifierence curves can be drawn onto the diagram.
Hence the Edgeworth box can be used to represent preferences and feasible allocations,These are the economically
relevant characteristics of the consumers for the ensuing analysis.
Market | General Equilibrium 4
Pareto E–ciency
A Pareto e–cient allocation is one where it is not possible to make one consumer better ofi without making another
worse ofi,The contract curve is the curve that connects all the Pareto e–cient points (the dashed line).
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A
B
x
y
A point like y cannot be Pareto e–cient,It is possible to make consumer A better ofi without making B worse ofi,
by moving to an allocation like x,A has moved onto a strictly higher indifierence curve,whilst B is just as well ofi.
Only allocations like x can be Pareto e–cient,where the two indifierence curves are tangential,It is not possible to
make either consumer better ofi without making the other worse ofi.
Therefore,the contract curve connects all the indifierence curve tangency points.
Market | General Equilibrium 5
General Equilibrium
A single budget line can be drawn for both consumers,It is a downwardly sloping straight line,which must pass
through the endowment and have slope?p1=p2 (the price ratio).
What is an equilibrium in this exchange economy? (Often called general,competitive or Walrasian equilibrium).
Deflnition,A general equilibrium is an allocation x and a set of prices p such that:
1,Each consumer maximises their utility given their budget constraint.
2,Total demand for each good is no more than the total endowment.
The second condition is often called market clearing,Alternatively,the allocation must be feasible.
Market | General Equilibrium 6
Walras’ Law
Equilibrium (allocation x and prices p1 and p2) can be illustrated in the Edgeworth box:
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A
B
x
Budget line
Slope =?p1=p2
!
Both consumers are maximising utility given their budget constraints and the market clears for both goods.
Deflne aggregate excess demand for good 1 as z1(p1;p2)=(x1A(p1;p2)?!1A)+(x1B(p1;p2)?!1B),This is the net
amount of good 1 demanded by each consumer,At equilibrium prices,this is zero.
Following from the deflnitions of budget lines,Walras’ law states that the value of aggregate excess demand is zero:
p1z1(p1;p2)+p2z2(p1;p2)=0
So if the market clears exactly for one of the goods,it must clear exactly for the other.
Market | General Equilibrium 7
Production
Pure exchange economies have no production,All of the above analysis can be extended to the case of production.
There is one consumer,one producer and two goods,(This is the simplest possible case | it can be generalised).
In fact,the flrm and the consumer are the \same person",This is not unreasonable,most of us are both consumers
and producers simultaneously | everyone has to work for a living,including Robinson Crusoe.
The goods are coconuts,C,and leisure,Recall labour supply (L) can be derived from leisure demand,The price of
leisure is w and coconuts are the numeraire,There is a production technology which turns labour into coconuts.
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0 L
C
Production function
Indifierence curves
C?
L?
Market | General Equilibrium 8
Robinson Crusoe Economy
Economists call this model the Robinson Crusoe economy.
The flrm maximises proflt | pushing the isoproflt line upward until it is tangent to the production function.
The consumer maximises utility | pushing the indifierence curve upward until it is tangent to the budget line.
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0 0L
C
L
C
Production function
Isoproflt line
Indifierence curve
Budget lineC?
L?
C?
L? L
The slope of the budget line (and the isoproflt line) is w,The marginal product is set equal to w at the optimum
(as usual) and the MRS is set equal to the price ratio (w) as usual.
Putting these two pictures together yields the diagram on the last slide,The budget line and the isoproflt line must
coincide | any proflts that are made during production are all that is available to be spent.
Market | General Equilibrium 9
General Equilibrium Revisited
General equilibrium when there is production in the economy is a natural extension of the earlier deflnition.
Deflnition,A general equilibrium is an allocation x,a production plan y and a set of prices p such that:
1,Each consumer maximises utility subject to their budget constraint.
2,Each flrm maximises proflt given the market prices.
3,The market for each good (whether input or output) clears.
Each consumer and flrm acts \selflshly" in the sense that they only take an interest in their own well-being.
This is a static concept,There is no mention of how a general equilibrium might arise,whether it would arise,
indeed whether it even exists in a given economy (is there a set of prices and an allocation such that all three
conditions hold?),So far it is just a deflnition.
Market | General Equilibrium 10
Production Possibility Sets
A production possibility set is the set of bundles of goods that it is possible to produce.
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...0 0x1
x2
x1
x2
Production possibility set
Isoproflt lines
Slope =?p1=p2
Production possibility frontier
Slope =MRT
x?2
x?1
The above diagram illustrates the production possibility set,The outermost line represents the production possibility
frontier,A proflt maximising flrm will produce at the point at which the isoproflt line is tangential to the frontier.
The frontier slope is the marginal rate of transformation | the rate at which one good can be turned into another.
The slope of the isoproflt line is the price ratio between the two goods (C is equilibrium cost):
… = p1x1 +p2x2?C =) x2 = …+Cp
2
p1p
2
x1
Market | General Equilibrium 11
Comparative Advantage
Why is the production possibility set this shape?
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1
x2
x1
x2
x1
x2
Suppose flrm 1 was particularly good at making good 1,For every good 2 it makes it could make 2 of good 1,On
the other hand flrm 2 is an expert in good 2,For every good 1 it makes it could make 2 of good 2.
Firm 1 has a comparative advantage in making good 1,Firm 2 has a comparative advantage in good 2.
The above picture shows their production possibility sets and the joint production possibility set | notice the shape.
Market | General Equilibrium 12
MRS and MRT
In the below picture,an Edgeworth box is drawn under the production possibility set.
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0 x1
x2
x?
x1
x2
The general equilibrium is illustrated,The production plan is y =(x1;x2),The allocation is x? and the set of prices
is given by the slope of the budget line that separates the two indifierence curves.
Notice that the prices are also given by the slope of the isoproflt line tangential to the production frontier.
Hence,in a general equilibrium the MRS of each consumer and the MRT of the technology are equal and given by
the price ratio,This is true of a general equilibrium no matter how many flrms,consumers and goods there are.