Download 1 Competitive Equilibrium

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Competitive Equilibrium
Each household and each firm in the economy act independently from each other,
seeking their own interest, and taking as given the fact that other agents will also seek
their best. In the previous section we have described the behavior of each agent in
the economy, here we show how all individual actions aggregate into the behavior of
the whole economy. For this purpose, we use the notion of equilibrium, meaning
that in each market aggregate demand equals aggregate supply, so the corresponding
equilibrium price “clears the market”.
We make a simplifying assumptions. We assume that all households are the same.
Identical consumers behave in identical ways, so it’s enough to analyze the behavior
of one consumer, the representative consumer of the economy. We make the same
assumption for firms. We assume all firms have the same CRS technology, and just
study the behavior of the representative firm. Representative household (who buys
good and sells labor) and firm (who sells goods and buys labor) play the role of a
stand-in for all consumers and firms in the economy. The third actor is the government
that buys goods and taxes agents to finance such purchases.
Definition of CE
We need to distinguish the exogenous variables determined outside the model, and
treated as parameters into the model from the endogenous variables determined in
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equilibrium. In our case, the exogenous variables are   ̄  and the endogenous
variables are the quantities {∗  ∗  ∗  ∗   ∗ }, the tax policy {∗ }, the wage {∗ }  We
are now ready to define a competitive equilibrium for the one-period economy we have
described:
Definition 1 A competitive equilibrium of this economy is a set of quantities {∗  ∗  ∗  ∗   ∗ },
a government policy {∗ }, a price {∗ } such that
1. Given {∗  ∗ }, the consumer chooses {∗  ∗ } optimally to solve ( )
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2. Given ̄ ∗    the firm chooses {∗ } optimally to solve (  )
3. The market for labor clears, i.e. ∗ = 1 − ∗ , and ∗ =  (∗  ̄) is the market
clearing wage
4. The firms distribute dividends ∗ = (∗  ̄) −  (∗  ̄)∗
5. The government sets the tax policy ∗ such that the budget constraint is balanced
in equilibrium:  = ∗
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6. The goods market clears: ∗ +  =  ∗
Therefore a competitive equilibrium is the combination of optimizing behavior
for firms and workers and a set of market clearing conditions. How do we solve
for the CE? We have 7 endogenous variables which are determined in equilibrium
{∗  ∗  ∗  ∗   ∗  ∗  ∗ }  Let’s count equations:
1. Budget constraint: ∗ (1 − ∗ ) + ∗ − ∗ = ∗
2. Household optimality:  = ∗
¢
¡
3. Firm’s optimality:  ∗  ̄ = 
¡
¢
4. Output determination:  ∗ =  ∗  ̄
¢
¡
5. Distributed dividends equal profits: ∗ =  ∗ =  ∗  ̄ − ∗
6. Labor market clearing: 1 − ∗ = ∗
7. Balanced government budget:  = ∗
8. Goods market clearing:  ∗ = ∗ + 
It seems we have eight equations, so one more than we have unknowns. But we
can show that one of the market clearing conditions (6-8) is redundant. This is Walras
Law: in an economy with  markets, if  − 1 are in equilibrium, also the  market
is in equilibrium. To prove that, for example, (8) is redundant, start from the budget
constraint
∗ (1 − ∗ ) + ∗ − ∗ = ∗ 
£ ¡
¢
¤
∗ (1 − ∗ ) +  ∗  ̄ − ∗ ∗ − ∗ = ∗ 
£ ¡
¢
¤
∗ ∗ +  ∗  ̄ − ∗ ∗ = ∗ + ∗ 
 ∗ = ∗ + 
where the first line is just the budget constraint; the second line uses the equilibrium
dividends, the third line uses the market clearing condition in the labor market; the
fourth line uses the government balanced budget condition.
Graphical Approach
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We now show that the competitive equilibrium lends itself naturally to a graphical
analysis in the ( ) space. First, we can rewrite the goods market clearing condition
as
¡
¢
¡
¢
 =   ̄ −  =  1 −  ̄ − 
which can be thought of as a function  =  () describing the production possibility frontier of the economy. This function describes the trade-off dictated by the
production technology between leisure and consumption: the economy can enjoy more
leisure at the cost of lower time devoted to work, lower production and therefore lower
consumption good . All the points within the PPF (included those on the frontier)
are feasible. The slope of the production possibility frontier is given by the marginal
product of labor, i.e.
¡
¢
 0 () = − 1 −  ̄ ≡ − 
This slope is also called marginal rate of transformation between consumption and
leisure: it is the rate at which leisure can be transformed into consumption through
work, given the existing technology. The slope is negative, but is the curve  () concave
or convex?
¡
¢
 00 () =  1 −  ̄  0
by decreasing marginal returns to labor. So the production possibility frontier is concave.
From the optimization conditions of the households, we have that
 (∗  ∗ )
=  (∗  ∗ ) 
 =
∗
∗
 (   )
∗
Using the market clearing wage
¢
¡
∗ =  ∗  ̄ =  (∗  ∗ )
into the equation above, we conclude that the equilibrium can be characterized as the
point (∗  ∗ ) where
¢
¡
 (∗  ∗ ) =  =  ∗  ̄ =  (∗  ∗ ) 
Thus, at the equilibrium point (∗  ∗ )  the rate at which households are willing to substitute between ( ) equals the real wage which equals, in turn, the marginal product
of labor, i.e. the rate at which firms can convert leisure (and work) into consumption
through the production technology.
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Graphically, (∗  ∗ ) is the point where the convex indifference curve is tangent to the
production possibility frontier. Finally, note that the budget constraint cuts exactly
at the point (∗  ∗ ) separating the indifference curve from the production possibility
frontier: to understand, recall that the slope of the wage is − and that the equilibrium
allocations have to be affordable (i.e. on the budget constraint).
Comparative Statics
Let’s do some comparative statics on the equilibrium allocations (∗  ∗ ) using this
graphical approach. We study the effect of a change in  and a change in  on the
equilibrium allocations.
TFP change: What happens when there is an increase in total factor productivity
? Think of this as the recent improvement in information and communication technologies that improved the productivity in the economy over the last 30 years. The term
 enters the production possibility frontier (PPF), so we need to understand how the
latter shifts. Again, let’s use the total differential on the equation  =  ( 1 − )−.
Differentiate with respect to both  and  keeping  fixed to see how the curve shifts:
¡
¢
¡
¢

=  1 −  ̄  0
 =  1 −  ̄  ⇒

so the curve shifts outward. But is the shift equal at every point? To understand,
differentiate one more time the expression above with respect to  This differential
measures how large is the shift at different levels of  :
¡
¢
2 
= − 1 −  ̄  0

which means that the shift is small for high  and large for low  The reason is that the
harder the household work, the more she can benefit from a higher level of productivity.
If no labor is supplied to the market, the larger productivity does not translate to any
output improvement.
¡ ¢
Remark 1 If we assume that labor is essential to production, then  0 ̄ = 0 and
¡ ¢
=  0 ̄ = 0 and the PPF does not
the PPF will tilt around the point  = 1 since 

shift at  = 1
The result of this increase in productivity is that consumption increases unambiguously: the economy is more productive and the level of consumption will go up. The
change in leisure is ambiguous and depends on income vs substitution effect, driven by
the preferences: a higher productivity level makes it a good time to work (substitution
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effect), but the fact that wages increase creates an income effect that leads workers
to decide to reduce their hours worked, since now they can earn the same by working
fewer hours.
Data on all developed countries for the last 60-70 years suggest that growth is
balanced, i.e., average hours worked by households have remained constant, roughly, so
income and substitution effects offset each other. It must be this way, if productivity
is what drives long run growth: if the substitution (income) effect dominated, for
example, hours worked would grow (fall) continuously over time, but at some point
would hit the time endowment (zero): it makes no sense, and it is not what the data
say.
Change in  : Consider now an increase in government expenditure  The PPF
shifts down and both leisure and consumption decrease. The explanation is as follows:
suppose leisure does not change, then production remains constant. But because 
grows, this means that  has to fall proportionately: government expenditure “crowds
out” consumption through higher taxes. Now, recall from the graph of the budget
constraint that a pure negative income effect (rise in  or fall in ) decreases leisure
because leisure is a normal good, i.e., a good whose consumption increases when (nonlabor) income rises. So hours worked will rise and this increases production, which
means that consumption will fall less than the rise in  i.e. crowding out is less than
one for one. For this reason, output goes up since ∆ = ∆ + ∆ and we just showed
that ∆  −∆
Does household welfare fall after a rise in government expenditures ? Clearly, it
depends on how the increase in the public good  is valued by the agents: if it is
strongly valued through  ()  then welfare —i.e. the total utility  ( ) +  ()— could
become larger, notwithstanding the decrease in both ( ) 
In the data, business cycles fluctuations are such that output and private consumption co-move (when one rises, the other rises too). Based on this observations, we
conclude that productivity shocks are a more likely force behind aggregate fluctuations
than government expenditures shocks.
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Social Efficiency of Equilibrium Allocations
The competitive allocation is the aggregate outcome the economy will settle upon,
through the self-interested behavior of individuals and the natural functioning of markets. One important question we should ask, in relation to the competitive allocation,
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is whether this allocation is the “best” possible allocations among the feasible ones, or
if there are better allocations than this one. In principle, all the allocations on the PPF
are feasible: what is that guarantees us that the point picked up by the competitive
equilibrium is the so called “first best”, i.e. is the one guaranteeing the highest utility
to the agents?
To answer this question, we introduce a new character, the benevolent dictator.
Consider a fictitious benevolent social planner who maximizes households’ welfare
 ( ) by choosing consumption and leisure on their behalf, subject to the aggregate feasibility constraint of the economy that tells us which combinations of ( ) are
attainable and which are not, i.e. the usual constraint given by the PPF.
¡
¢
 +  ≤  1 −  ̄
We can write the Social Planner Problem (SP) as
max  ( )
{}
(SP)

¡
¢
 +  ≤  1 −  ̄
The solution to this problem tells us what is the best feasible consumption and leisure
bundle that can be achieved in this economy, i.e. the solution to this problem is the
“socially efficient” allocation. We want to compare the socially optimal allocations to
the competitive equilibrium allocations.
Writing the Lagrangian of the (SP) problem and denoting with  the multiplier on
the feasibility constraint, we conclude that the solution of (SP) is
³ ´
 ̂ ˆ = 
³ ´
³
´
 ̂ ˆ =  1 − ˆ ̄
where we use hats to denote socially optimal allocations to distinguish them from the
“star” competitive equilibrium allocations. Dividing through those two equations, we
arrive at the familiar condition:
³ ´
³
´
 ̂ ˆ
³ ´ =  1 − ˆ ̄
 ̂ ˆ
³ ´
³ ´
ˆ
which implies that the  ̂  =  ̂ ˆ exactly as in the competitive
equilibrium. Since the equilibrium allocations (∗  ∗ ) also satisfy
³ the
´ same condition,
it means that they are equal to the socially optimal allocations ̂ ˆ 
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2.1
Socialism vs market economies
This result has some normative implications for competitive markets: Adam Smith
said that an unrestricted competitive market economy will behave as if there was
an invisible hand guiding and coordinating the actions of self-interested individuals
towards a state of affairs that is beneficial for all. What he meant is that the market
equilibrium is the first best, as we showed.
But we also showed that a planner can do as well as the market. So why did the
socialist planning economies fail so miserable? They should have behaved exactly as
market economies, according to our derivation above. The reason lies in the strong
informational requirements of the Planner’s solution: the Planner needs to have a lot
of information about preferences of every household in the economy, and production
technology of every firm. In the representative agent economy, this requirement is
minimal, but in a real economy with millions of different families and thousands of
firms, it is a huge requirement. In the decentralized competitive equilibrium, nobody
needs to have all this information: it is enough for every firm and individual to know
their own preferences and technology, then the market will take care of aggregation
towards the achievement of the common goods through prices. Prices signal scarcity.
For example, suppose that there is a rise in the demand for apples in the economy. In
the planning economy, the planner would need to be aware of this change in preferences
and would then command an increase in the production of apples. The market economy
reaches the same objective through prices. The rise in demand leads to a rise in prices.
Firms realize that there is a profit opportunity and enter the market, so the production
of apples increases.
There is another reason why socialist economies failed. The planner needs to be
benevolent, but we know that many dictators are corrupt and divert resources towards
their own welfare rather than towards the welfare of their citizens.
2.2
Pareto Optimality
The notion of social efficiency that we have just derived, as a solution to the (SP)
problem, is equivalent to the notion of Pareto Optimality. Therefore, we can use the
terms social efficiency and Pareto optimality interchangeably.
Definition 2 A feasible allocation is Pareto Optimal if there is no other feasible allocation that can improve the welfare of at least one agent in the economy without reducing
the welfare of some other agent.
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The concept of Pareto optimality takes its name from the 19th century Italian
economist Wilfredo Pareto. In economies with one representative agents, like the static
model we studied, a Pareto optimal allocation is the feasible allocation that makes the
agent as well off as possible, so it is the solution to the (SP) problem described above.
Note that Pareto optimality has to do with economic efficiency, not with fairness.
For example, an allocation where all resources are assigned to only one agent in the
economy and zero is given to everyone else is Pareto-efficient, but not “fair” according
to the common definition of fairness which is related to economic equality. There are
other ways to rank allocations based on fairness. For example, suppose that you have
 types in the economy, each one with different productive abilities. Then one can use
several criteria:
• Rawlsian criterion: maximizes the utility of the least able agent
• Elitist criterion: maximizes the utility of the most able agent
• Benthamian criterion: maximizes the weighted-sum of the utilities
• Democratic criterion: maximizes the ability of the class with the median ability
It is useful to look at an example of a planner’s problem with more than one agent.
Suppose there are two types of agents, 1 and 2 The Planner’s problem is
max  (1  1 ) + (1 − )  (2  2 )
{}
(SP2)

¡
¢
1 + 2 +  ≤  2 − 1 − 2  ̄
where  is the relative weight the planner puts on agent 1 relative to agent 2. The
solution to this problem is indexed by  For each value of  we have a different solution
and each one is a PO. When   05, the PO allocation is such that agent 1 consumes
more than agent 2 Try to characterize the solution of this problem on your own.
2.3
Welfare Theorems
Technically the equivalence between competitive equilibria and social efficient allocations is stated by the Welfare Theorems. The First Welfare Theorem states that
every CE is PO. The Second Welfare Theorem states that every PO allocation (i.e.,
every solution to a planner’s problem) can be decentralized as a CE allocations through
the right “initial transfers” to the agent in the economy. These transfers are going to
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be proportional to the welfare weights: an agent who has a lot of weight in the planner’s solution will consume a lot in the PO allocation. How do you get that agent to
consume a lot in the CE? By starting off that agent with a high level of wealth relative
to the others.
This equivalence result is also important in another way: we can compute an equilibrium just by solving the planner’s problem, which as you might have noticed is
simpler: for example, it has no prices. This is key: the planner dictates allocations of
consumption and leisure to the agents, he doesn’t have to worry about markets and
prices. There are no market and no prices in the fictional world of the social planner.
The Welfare Theorems hold under some technical conditions. Often these conditions
do not hold. When these conditions fail, the CE allocations feature a wedge between
the MRS and the MRT. Recall that  =  is always the solution to the
Planner’s problem.
Examples where socially efficient allocations differ from competitive equilibrium
and the Welfare Theorem fails are economies with:
• Distortionary taxes, like a proportional labor income tax in presence of an endogenous labor supply decision.
• Externalities not internalized by individuals and markets. These are cases where
the consumption choice of an individual will affect the consumption of another.
The individual does not take this interaction (externality) into account, but the
social planner does. There are negative externalities (e.g., pollution) and positive
externalities (e.g., keeping your backyard clean and cute-looking is a pleasure for
your neighbors too).
• Non-competitive markets, e.g., monopolies where prices (and profits) are too high
and quantities too low compared to the efficient benchmark.
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