A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
General Equilibrium and Welfare
Lectures 2 and 3, ECON 4240 Spring 2017
Summary of Snyder et al. (2015)
University of Oslo
24.01.2017 and 31.01.2017
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Outline
General equilibrium: look at many markets at the same time. Here
all prices determined in the model. Goal:
Study effects that occur when changes in one market have
repercussions in other markets
Study connections between markets for goods and markets for
factors of production
Make general welfare statements about how well a market
economy performs
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Outline
We proceed by steps:
A graphical model of General Equilibrium with 2 consumption
goods and 2 factors of production
A mathematical model of exchange with n goods
A mathematical model of production and exchange with n
goods
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Perfectly Competitive Price System
In all the general equilibrium models we borrow many assumptions
from the partial equilibrium analysis:
All individuals are price-takers, and utility-maximizers
All firms are price-takers and profit-maximizers
Zero transaction costs
Perfect information
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
A graphical GE Model with 2 goods
Many identical individuals, many identical firms
2 goods (x and y ) and 2 inputs (capital and labor)
The endowments of capital and labor are fixed
For simplicity, all individuals have identical endowments of
capital and labor, and they all own equal shares of each firm
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
A graphical GE Model with 2 goods
We first focus on the supply side. As firms are profit maximizers, in
equilibrium inputs will be used in an efficient ways to produce the
two goods (efficient = it is not possible to produce more of one
good without producing less of the other)
We draw an "Edgeworth box", where the horizontal side
measures total labor endowment, the vertical side total capital
endowment
Any point in the box is a full-employment allocation of inputs
Each point in the Edgeworth box measures how much of each
factor is devoted to the production of x and how much is
devoted to the production of y
Nothing in the Edgeworth about how much of each factor is
used in each single firm (for now it does not matter)
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
A graphical GE Model with 2 goods
In the Edgeworth box we can draw isoquants for good x and
isoquants for good y
Definition
An isoquant (for good x) shows those combinations of capital and
labor that can produce a given level of good x.
Note that these are aggregate isoquants, not at single firm
level
The efficient allocations are the ones where the isoquants for
the production of the two goods are tangent
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
A graphical GE Model with 2 goods
The information about all possible efficient allocations can be used
to construct a production possibility frontier in the graph of
combinations of good x and good y
Definition
The production possibility frontier is the curve in the x-y space
that delimits all combinations of x and y that can be produced,
given the initial endowment of capital and labor
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Efficient Allocations of Inputs
Definition
The rate of product transformation (RPT) between two outputs
is the negative of the slope of the production possibility frontier for
those outputs:
RPT = − dy
dx (along the production possibility frontier)
Pick any 2 prices for the inputs
For any point (xa , ya ) on the production possibility frontier the
total cost for producing xa units of good x and ya units of y is
the same.
MCx
Note that RPT = − MC
y
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Efficient Allocations of Inputs
Concave frontier ↔
MCx
MCy
increases as x increases and y decreases
Possible reasons behind concavity of frontier:
Diminishing returns
Specialized inputs
Differing factor intensities
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Determination of Equilibrium Prices
At any given (good) price ratio
px
py
firms altogether produce
some amount (x f , y f ) such that
(a) (x f , y f ) lies on the production possibility frontier,
(b) at (x f , y f ) the production possibility frontier has slope
− ppyx .
In order to find the equilibrium price ratio (denote it
need to consider the demand side
px∗
py∗ )
we
Note that the competitive markets only determine equilibrium
relative prices, not absolute prices (more on this in a bit)
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Determination of Equilibrium Prices
In the x − y space, we can draw individuals’ indifference curves
(aggregating individuals’ indifference curve is not a trivial
issue, here for simplicity say that if (xb , yb ) is consumed
altogether and there are n individuals, then each individual
gets xb /n units of good x and yb /n units of good y )
For any pair of prices px and py , individuals altogether demand
some amount (x c , y c ) such that the indifference curve passing
through the point (x c , y c ) has slope − ppyx
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Determination of Equilibrium Prices
A pair of prices are equilibrium prices if the overall quantities
chosen by firms coincide with the overall quantities chosen by
individuals: (x f , y f ) = (x c , y c ) = (x ∗ , y ∗ )
What about the budgets of individuals? How do individuals
pay for x and y ?
In equilibrium total revenues (=sum of revenues of all firms)
are x ∗ px + y ∗ py .
These revenues cover the costs of inputs and if anything is left
they become profits. Whether the revenues are used to cover
costs or they are profits they all go to individuals anyway
For individuals, the revenue from selling labor and capital,
together with the profits, add up to x ∗ px + y ∗ py
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
General Equilibrium Models in International Trade
So far we learned that the equilibrium price ratio px∗ /py∗
persists as long as technologies or preferences change.
We can apply this model to questions of International Trade
Note that the model can be made more realistic (and more
useful) by assuming that there are more inputs (e.g. high and
low skilled labor) and individuals differ in the endowments of
the inputs
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
General Equilibrium Models in International Trade
The Corn Law Debate: focus on tariffs on grain imports.
Grain = x, manufactured goods = y
With tariffs high enough to completely prevent trade:
E
equilibrium "E ", domestic price ratio: ppxE
y
Removing tariffs:
pxA
,
pyA
grain imports of xA − xE , financed with
export of manufacturing of yA − yE
Result of tariff removal: lower (relative) price of grain, less
grain production, less farmers, less rent for land owners
(different from textbook story)
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
General Equilibrium Models In International Trade
US elections: trade was an important issue
Idea: trade affects the relative incomes of various factors of
production
In the US exports use intensively skilled labor, while imports
are products that require low skilled labor
More free trade: increasing relative wage of high skilled
workers (more inequality (in the US - worldwide not clear))
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
A Mathematical Model of Exchange: Setting
Model of exchange = no production, goods already exist
n goods, m individuals
x i : vector of consumption of individual i = 1..m (vector of size
n)
x i : vector of endowment of individual i (vector of size n)
Here the same individuals can be on both sides of different
markets; for simplicity imagine an individual sells all her
endowment and uses the revenues to buy whatever she
consumes
Market value of individual’s endowment: px i
Budget constraint of individual i: px i ≤ px i
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Equilibrium and Walras’ Law
Vector of demand of individual i: x i (p, px i )
These demands are homogeneous of degree 0:
x i (p, px i ) = x i (tp, tpx i ) ∀t > 0
Definition
Walrasian equilibrium is an allocation of resources and an
associated price vector p ∗ such that quantity demanded and
quantity supplied of each good coincide:
i
i ∗ ∗ i
m
Σm
i=1 x (p , p x ) = Σi=1 x
(1)
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Equilibrium Existence
Does an equilibrium exists? Not obvious as:
(1) from last slide corresponds to n equations
The equations in (1) need not be linear (x i (·) is a vector of n
equations, not necessarily linear)
These equations are not independent: they are related by
Walras’ Law
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Walras’ Law
Definition
Walras’ law :
i
m
i
Σm
i=1 px = Σi=1 px
Walras Law is a direct consequence of the individual’s budget
constraints: for every individual i = 1, ..m we have px i = px i :
adding over the m individuals we have Walras’ law
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Existence of Equilibrium
Intuitive evidence seems to suggest equilibrium exists
Proof of existence:
As demands are homogeneous of degree 0, and budget
constraints are not affected if all prices are multiplied by a
constant t > 0, we can "normalize" prices
One way to normalize prices is, for example, to multiply every
price by Σn 1 pk , thus getting pi0 = Σn pi pk
k=1
k=1
This is called in jargon a normalization as Σni=1 pi0 = 1 (note
that relative prices are unaffected by the normalization)
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Existence of Equilibrium
Proving existence ↔ showing there exists a vector p ∗ such that
m
i
i ∗ ∗ i
z(p ∗ ) := Σm
i=1 x (p , p x ) − Σi=1 x = 0
Starting from an arbitrary vector of prices p 0 , define p 1 as
(textbook expression is wrong)
pn0 + αzn (p 0 )
p10 + αz1 (p 0 )
p = f (p ) :=
,
..,
Σnk=1 (pk0 + αzk (p 0 ))
Σnk=1 (pk0 + αzk (p 0 ))
1
0
where α is some positive constant
Function f is continuous and maps normalized prices onto
other normalized prices (= if p0 is a vector of normalized
prices, so is p1 ) (no he second part of the statement)
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Brouwer Fixed Point Theorem
Function f meets the conditions of Brouwer fixed point
theorem for existence of fixed point p ∗ (see next slide)
A fixed point of function f is a vector p ∗ such that p ∗ = f (p ∗ )
Theorem
Brouwer Fixed Point Theorem: Any continuous function from a
closed compact set onto itself has a fixed point such that f (x) = x.
(No proof)
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Existence of Equilibrium
Definition
S is a compact subset of Rn if S is closed (= contains its
boundaries) and bounded (= pick any point in Rn : along any "ray"
starting from that point and going in any direction there are points
outside S)
In our case, we consider a subset of Rn : [0, 1]n
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Welfare Properties of Competitive Equilibria
After establishing existence of competitive equilibria, we should
ask ourselves whether these equilibria lead to socially desirable
divisions of resources (divisions of resources= allocations)
Welfare economics studies criteria for choosing among
alternative allocations
Aside: when considering 1 consumer, any allocation that
maximizes his utility is clearly the best one: definition of
efficiency is UNIQUE. When considering n consumers, there
are many different definitions of efficiency
25/33
Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Welfare Properties of Competitive Equilibria
Definition
An allocation of the available goods in an exchange economy is
Pareto efficient if it is not possible to devise an alternative
allocation in which at least one person is better off and no one is
worse off.
Other measures of efficiency require to formally define a social
welfare function:
Definition
A social welfare function is a scheme from ranking potential
allocations of resources based on the utility they provide to
individuals.
Social Welfare = SW [U1 (x 1 ), U2 (x 2 ), .., Um (x m )]
26/33
Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Welfare Properties of Competitive Equilibria
Note: for any "reasonable" SW (.) any allocation that
maximizes SW is Pareto Efficient
Pareto Efficiency is a VERY limited way to rank allocations:
e.g. crazily unequal allocations can be Pareto Efficient
(My) view: Pareto efficiency is a necessary but not sufficient
requirement for an allocation to be acceptable
Utilitarian SW function
SW (U1 , U2 , .., Um ) = U1 + U2 + ... + Um
Maximin SW function
SW (U1 , U2 , .., Um ) = min [U1 , U2 , ..., Um ]
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
First Theorem of Welfare Economics
Theorem
First Welfare Theorem. Every Walrasian equilibrium is Pareto
Efficient.
Proof by contradiction. (LOTS of typos in the textbook proof)
Consider a Walrasian equilibrium with prices p ∗ and allocation
x k (k = 1, .., m).
Suppose there exists an alternative feasible allocation, x̃ k ,
such that a consumer i is better off and everyone is at least as
well off as with x k . Then:
p ∗ x̃ i > p ∗ x i (Suppose instead p ∗ x̃ i ≤ p ∗ x i : then individual i could
afford x̃ i , and as we assume individual i is better off with x̃ i then with x i
we should conclude that x i is not the optimal choice for consumer i. We
reached a contradiction as x i IS optimal. So it must be the case that
p ∗ x̃ i > p ∗ x i
)
Summary of Snyder et al. (2015)
28/33
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
First Theorem of Welfare Economics
p ∗ x̃ k ≥ p ∗ x k for k = 1..m (Again, suppose instead that p ∗ x̃ k ≤ p ∗ x k
for some k: then consumer k could ensure the same utility obtained from
consuming x k at a lower cost by purchasing x̃ k , and with what she saves
she could buy more of some good thus increasing overall utility, so x k is
not her optimal choice. This is a contradiction as x k is the optimal
choice for k = 1..m.)
i
i
m
Σm
i=1 x̃ = Σi=1 x (This is the definition of feasibility.)
Multiply both sides of the last equality by p ∗ :
∗ i
m
∗ i
Σm
i=1 p x̃ = Σi=1 p x
But the previous 2 inequalities, together with Walras Law,
∗ i
m
∗ i
imply: Σm
i=1 p x̃ > Σi=1 p x , hence we reach a contradiction
29/33
Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Second Welfare Theorem
Theorem
Second Welfare Theorem. For any Pareto Optimal allocation of
resources, there exists a set of initial endowments and a related
price vector such that this allocation is also a Walrasian Equilibrium.
We provide a graphical proof of the theorem using an
Edgeworth box.
A model of exchanges is sufficient to talk about welfare
properties of different allocation, but if we think of policy
instruments, such as taxes and subsidies, used to obtain the
desired allocation, then we need to account for the role of
these policies on the incentive to produce
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Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
A Mathematical Model of Production and Exchange
n goods (including factors of production).
Goods consumed and used as factor of production can come
from the household endowments or can be produced by some
firm
r profit-maximizing firms. Production function of firm j: n × 1
column vector y j .
Positive entries in y j : outputs, negative entries: inputs
Profit of firm j: πj (p) = py j
Firms can choose y j = 0: exit (long run perspective) (TYPOS in
book)
Firms are owned by individuals. Simplification: each individual
i owns a share si of every firm (simplification: same share of
each firm)
31/33
Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Budget Constraints and Walras’ Law
Budget constraint of individual i:
px i (p) = si Σrj=1 py j (p) + px i (p)
Note slight inconsistency: we allow for positive profits (like in
short run) but we allow for firms exit (like in the long run)
Walras law:
i
px(p) = py (p) + pΣm
i=1 x ,
where:
i
x(p) = Σm
i=1 x (p),
r
y (p) = Σj=1 y j (p) and
i
x = Σm
i=1 x .
32/33
Summary of Snyder et al. (2015)
General Equilibrium
A graphical GE Model with 2 goods
A Model of Exchange
A Model of Exchange and Production
Walrasian Equilibrium
Equilibrium price vector p ∗ at which quantity demanded equals
quantity supplied in all markets simultaneously:
x(p ∗ ) = y (p ∗ ) + x.
(One) use of these models: look at effect on wages of chances
in exogenous factors
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Summary of Snyder et al. (2015)
General Equilibrium