1. General Equilibrium
Dr. Andrew McGee
Simon Fraser University

2. Interconnectedness of markets
• Suppose new research is released showing that wine
consumption is good for one’s overall health.
• Demand for wine increases. Price of wine increases.
Marginal revenue product of wine workers increases.
More workers and firms enter this market.
• Cloth, used in wine production, also sees its demand
increase, and similarly its price will go up and the
marginal revenue product of workers in the cloth
industry will go up.
• The new firms and workers entering these
industries, however, come from somewhere—namely
other markets. Hence the developments in one market
will affect many other markets.

3. General equilibrium model
• Models all markets at once
• Markets include not just markets for
consumption goods but factor markets
(e.g., capital and labor)
• In the equilibrium of a general equilibrium
model, Qs=Qd for all markets simultaneously
– There is some vector of equilibrium prices such
that all markets clear

4. Assumptions of general equilibrium
model
• All markets are perfectly competitive (i.e.,
selling homogenous products, many buyers,
many sellers, zero transaction costs, firms are
price-takers)
– Law of one price must hold
• Consumers maximize their utility
• Firms maximize their profits

5. General equilibrium model with 2
goods
• Assume that all individual have identical
preferences concerning goods X and Y.
• Assume that there are fixed amount of inputs
(capital and labor) available for the production
of X and Y.
• Edgeworth box: represents all possible
allocations of inputs to the production of X
and Y with dimensions given by the total
amounts of capital and labor available

6. Edgeworth box
Total
capital
Total labor
0x
0y
Where point A is in the
Edgeworth box tells you
how much labor and
capital are being devoted
to the production of X and
how much to Y
A
Labor in X production
Capital in X production
Labor in Y production
Capital in Y
production
At any point in the Edgeworth box, all of
the capital and labor in the economy is
fully employed
All possible allocations of K & L are
depicted in the Edgeworth box

7. Efficient allocations
• Many allocations of K & L to the production of
X and Y are inefficient insofar as more of both
X and Y could be produced by allocating K & L
to the production of X and Y differently
• Assume that perfectly competitive markets
result in efficient allocations of inputs
• Overlay isoquant maps on top of the
Edgeworth box

8. Efficient allocations
0x
0y
Red: isoquants for Y
Blue: isoquants for X
A
Points such as A are clearly
inefficient because more X
and more Y can be
produced given society’s
endowments of K & L.
The green line represents the locus of all efficient allocations of K & L—allocations
at which there exists no allocation at which more X and more Y can be produced
relative to the current outputs. Notice that the isoquants are tangent to each
other at these efficient allocations, meaning that MRTSx=MRTSy.
Contract curve

9. Production possibility frontier (PPF)
Y
X
0x
0y
Note that at each efficient allocation in an Edgeworth
box, there are corresponding output levels of X & Y.
The graph below depicts those output levels. Note
that the bottom left-hand corner of an Edgeworth box
corresponds to the case in which no X is produced and
only Y is produced.
The production possibility frontier shown here depicts
the alternation combinations of two outputs that can
be produced with fixed quantities of inputs if those
inputs are employed efficiently.
The output bundle A corresponds to the inefficient
input combination A in the Edgeworth box.
A
The slope of the PPF reflects the rate at which X can be substituted for Y when total
resources are held constant. For instance, the slope might be -1/4 near 0x, meaning 1
unit of Y can be sacrificed in order to produce 4 more units of X. Near Oy, this slope
might be -5, meaning that 5 units of Y must be sacrificed to produce 1 more unit of X.
The negative of the slope is known as the rate of product transformation (RPT).

10. Shape of the PPF

11. Diminishing Returns
• If the production processes of both X and Y
exhibit diminishing returns to scale, then
increasing X output will increase MCx, while
reducing Y output will decrease MCy, which
would cause the production function to
exhibit increasing RPT.
• But economists don’t find DRS theoretically
compelling….

12. Specialized inputs
• Suppose some inputs are better suited for X
and others better suited for Y
(e.g., marshlands for the production of rice
and prairies for the production of wheat).
Producing more X requires using more inputs
ill-suited for the production of X and better
suited for the production of Y and vice versa.
• But we have assumed inputs are homogenous
(i.e., land is land is land).

13. Differing factor intensities

14. Why is the PPF bowed outward?
Y
X
0x
0y
y4
y2
x1 x3
If we assume CRS, then this output combination
is possible (because ½ the inputs used to
produce x1 will produce ½ of x1 and likewise for
the other output levels). We also know that all
K& L will be used at this new allocation (though
we can’t see that in this graph).
B
We know that bundle B uses ½ of the K used in x1 and ½ the K used
in x3 to produce the new amount of X and similarly ½ the L used in x1
and ½ the L used in x3 to produce the new amount of X. Similarly for
the amounts of K & L used to produce Y.

15. Why is the PPF bowed outward?
0x
0y
x1
y4
x3
y2
This input bundle, C, involves using ½ the
K used to produce output combination P1
and ½ the K used to produce the output
combination P2.
P1
P2
Notice that C is not on the contract curve; it is not efficient. It is possible to produce
more of both X and Y. Thus there must exist output bundles to the northeast of B on
the PPF in the preceding slide. If the K/L mix (factor intensity) does not vary
between production of X and Y, the contract curve and PPFs will be linear
C
Total
capital
Total labor

16. PPF & Opportunity cost
• For a society with finite resources, producing
more X requires producing less Y if all
resources are fully employed.
• The slope of the PPF gives the opportunity
cost of producing an additional unit of X in
terms of units of Y foregone and vice versa

17. Equilibrium prices
• Given that society faces tradeoffs in the
production and consumption of X and Y, how
does it decide how much X & Y to produce
(and consume)?
• The prices of X and Y must adjust so that the
quantity demanded of each equals the
amount produced by the society.

18. Determination of equilibrium prices
Y
X
0x
0y
U4
U3U2
U1
PPF
At price ratio px/py, firms produce (x1, y1), but
consumers demand (x1’, y1’)—supply does not
equal demand and markets do not clear.
x1
y1
x1’
y1’
px/py
x*
y*
px*/py*
Why is the tangency between the PPF and the price
ratio the profit maximizing choice for firms? Because
at this bundle the price ratio equals the ratio of
marginal costs (all firms are earning zero profit by
assumption).
Note that the red lines here
are budget constraints. X
and Y sell for pxX+pyY, but in
this economy the producers
are also the consumers, so
whatever they earn in
revenues is returned to
consumers through wages
and rents from capital.
Given that these red lines are budget constraints, the point of tangency with the
indifference curves is the utility maximizing consumption bundle (from 1st ½ of course).

19. Determination of equilibrium prices
Y
X
0x
0y
U4
U3U2
U1
PPF
x1
y1
x1’
y1’
px/py
x*
y*
px*/py*
At px/py, there will be excess demand for X and
excess supply of Y. Markets do not clear. In
order for the markets to clear, the price of X
must go up, and the price of Y must go down
(or some combination) such that the price ratio
gets steeper. A higher price for X both reduces
demand for X and increases its supply.
At px*/py*, the supply of X
equals the demand for X and
similarly for Y. This is a general
equilibrium: all markets clear
simultaneously.

20. Persistence of general equilibria
• Consistent with the definition of an
equilibrium, a GE will tend to persist
unchanged without a change in the underlying
economic fundamentals
• The economic fundamentals that could lead to
a change in the equilibrium prices and
quantities are preferences and technologies

21. Technological Change
Y
X
0x
0y
U2
PPF1
PPF2
U3
Here, producers get better at producing
X as evidenced by the fact that they can
produce more X along with any given
amount of Y still using the same fixed
quantities of K & L. This is what
technological progress looks like!
Notice that the market clearing
price ratio changes following
this technological change.
Society gets better at
producing X; its relative price
falls in equilibrium.

22. Change in preferences
Y
X
0x
0y
U2
Original
preferences Suppose people decide that they like X
less. This would lead to “flatter”
indifference curves. Notice that the price
of X declines relative to that of Y in
equilibrium (the price ratio gets flatter)
and less X is consumed.
New equilibrium price
ratio
Old equilibrium price
ratio
xoldxnew
yold
ynew
New preferences, U2’

23. A simple general equilibrium model:
Production

24. A simple general equilibrium model:
Consumption

25. A simple general equilibrium model:
Equilibrium

26. A simple general equilibrium model:
Consumers’ budget constraint

27. Corn Laws
• 19th century British laws placing high tariffs on
grain imports (1815-1846)
• Served the interests of landowners, who
benefits from higher grain prices
• Disliked by manufacturers
• Assuming Corn Laws prevented trade (which
they didn’t), repealing the Corn Laws would
have an unambiguously positive effect on
consumers, but what about the effects on
factor owners (landowners and capitalists)?

28. An argument for trade
Y
X
0x
0y
PPF1
U1
U2
A
E B
yA
yE
yB
xA xE xB
In the absence of trade, a society is limited to
consuming only those bundles which it can
produce. In this economy under autarky (i.e., a no
trade regime), the GE would occur at point E. If
trade with other countries/economies is
allowed, then this country could consume bundle
B while producing bundle A. It would be a net
exporter of good Y while being a net importer of
good X.

29. Winners & losers from trade?
0x
0y
x1
y4
x3
y2
P1
P2
The movement from E to A corresponds to a
move from P2 to P1;it causes the K to L ratio
to increase in both industries. This means that
the relative price of capital must fall (or the
relative price of labor increases).
Total labor Thus in this simple, stylized model, trade
benefits laborers while hurting capital
owners.
Total
capital
Total labor

30. Walrasian Equilibrium
• Suppose there is no production in the economy.
The economy is just endowed with a certain
amount of goods X and Y.
• Individuals hold endowments of X & Y and then
trade among themselves.
• Once trade among individuals occurs, a Walrasian
equilibrium is an allocation of resources and an
associated price vector (i.e., set of prices for all
goods in the economy) such that the demand for
each good equals the economy’s total
endowment for that good (i.e., markets clear).

31. Trading: how do we get from an initial
allocation to an equilibrium allocation
0B
0A
(4,4)
(8,2)
Price vector px/py has
slope -1/2: you can
trade on unit of x for 2
units of y
Initial allocation
Equilibrium
allocation
Here, A gives B 4 units of X in
exchange for 2 units of Y. In
this sense the price vector
defines the trades that can be
made and the set of possible
final allocations

32. Mathematical model of exchange
(no production)
Vector notation

33. Mathematical model of exchange
(no production)

34. Exchange

35. Walrasian Equilibrium

36. Normalizing prices

37. Existence of a general equilibrium

38. Existence “proof”

39. Fixed point theorem
1
1
x
f(x)
0
45-degree line
x=f(x)f(x*)
x*
Fixed point
Any continuous function that maps from a
closed compact set (here the interval [0,1])
onto that same set ([0,1]) will have a fixed
point where x=f(x).

40. Existence of GE proof
• Existence proof relied on a number of non-
trivial assumptions:
1. Price-taking by all individuals
2. Homogeneity of degree 0 demand functions
3. Continuity of demand functions
4. Budget constraints
5. Walras’ law: value of all quantities demanded
equals value of all endowments

41. First Fundamental Theorem of Welfare
Economics
• FFWT: Every Walrasian equilibrium is Pareto
efficient
• Pareto efficient allocation: An allocation of the
available goods in an exchange economy is
Pareto efficient if it is not possible to devise an
alternation allocation in which at least one
person is better off and no one is worse off.
• FFWT does not imply that every Walrasian
equilibrium is socially desirable.

42. FFWT
• Given initial endowments, individuals trade. They
move from their initial endowment to some final
allocation of goods. The equilibrium that emerges (at
the equilibrium prices) will be Pareto efficient insofar
as none of the traders can be made better off without
making another trader worse off.
• Pareto efficient equilibria can be highly inequitable
(points A or C, for instance)
• The location of the WE depended substantially on the
initial endowment, E
• The initial endowment constrains the set of possible
WE (between A & C). There may exist other far more
socially desirable but not attainable Pareto efficient
allocations.

43. FFWT in a 2 person, 2 good exchange
economy
0B
0A
Initial
endowment
Px/Py
Walrasian
equilibrium
(WE): after
trade, A and B
end up here
given prices Px
and Py
Notice that the WE is Pareto
efficient (FFWT)
A
C
Region between A & C is the Pareto-
improving set.

44. Proof of FFWT

45. Second Fundamental Theorem of
Welfare Economics
• SFWT: For any Pareto optimal allocation of
resources, there exists a set of initial
endowments and a related price vector (Px/Py
in the 2-good case) such that this allocation is
also a Walrasian equilibrium
• Implication: if you can adjust initial allocations
(i.e., through taxes and other forms of
coercion), then any Pareto efficient allocation
can be sustained as a Walrasian equilibrium

46. SFWT in a 2-good, 2-person exchange
economy
0B
0A
Initial
endowment
Px/Py
A
C
Q
Q is thought to
be more socially
desirable than
any WE between
A and C
If one can re-allocate goods from the initial endowment to any point such as Z on the price line
passing through the tangency between UA and UB, then Q can be sustained through trade as a
WE
Z
UA
UB

47. Example of SFWT in a 2-person (A & B)
exchange economy

48. Example of SFWT in a 2-person (A & B)
exchange economy

49. Example of SFWT in a 2-person (A & B)
exchange economy

50. Example of SFWT in a 2-person (A & B)
exchange economy

51. Social welfare functions

52. Insights from GE models
• In economic models, we take preferences and production
technologies as given. With these in hand, all prices are
necessarily determined.
• Firms and productive inputs are owned by households. All
income (profits and returns to inputs) accrues to
households.
• Bottom line in policy evaluations is the utility of
households. Firms and governments are just
intermediaries.
• All taxes distort economic decisions along some
dimensions. The welfare costs of such distortions must be
weighed against the benefits of such taxes (whether from
public good provision or equity-enhancing transfers)