Notes on
Modeling the Aggregate Economy:
Competitive Equilibrium and
Welfare Theorems
Intermediate Macroeconomics
Spring 2006
Guido Menzio
University of Pennsylvania
Competitive Equilibrium
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To this point, we have studied decision theoretic problems:
– household’s optimal choice of leisure and savings, given the price of leisure w
and the price of future consumption (1+r)-1
– firm’s optimal choice of labor and investment, given the wage rate w and the
price of future output (1+r)-1
– government’s fiscal policy, given the price of future output (1+r)-1
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Next, we want to incorporate into the analysis the mechanism through which prices
are determined
Competitive Equilibrium
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How are prices determined?
– Auction. The auctioneer names a price for a certain good. Given the price,
buyers report to the auctioneer the amount of good they would like to buy.
Given the price, sellers report to the auctioneers the amount of good they would
like to sell. If aggregate demand is greater than aggregate supply, the
auctioneer increases the price. If aggregate supply is greater than aggregate
demand, the auctioneer reduces the price. If aggregate supply and demand are
equal, the price clears the market and all the transactions take place.
Competitive Equilibrium
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A Competitive Equilibrium for the aggregate economy is
– a vector of quantities { l, l’, C, C’, N, N’, I , K’}
– a vector of fiscal policy {G, G’, T, T’}
– a vector of profits {П, П’}
– a vector of prices {w, w’, (1+r)-1}
such that the following conditions are satisfied
Competitive Equilibrium
Household’s Optimality
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Given the prices {w, w’, (1+r)-1} , the taxes {T, T’} and the profits {П, П’ }, the
vector {l, l’, C, C’} is a solution to the household’s optimization problem
max U(C, C’, l , l’) subject to
C + (1+r)-1 C’ ≤ w (L – l) + (П – T) + (1+r)-1 (w’ (L – l’) + (П’ – T’) )
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The HH optimality condition is equivalent to
w = Ul / Uc
(1+r)-1 = Uc’ / Uc
w’ = Ul’ / Uc’
C + (1+r)-1 C’ = w (L – l) + (П – T) + (1+r)-1 (w’ (L – l’) + (П’ – T’) )
Competitive Equilibrium
Firm’s Optimality
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Given the prices {w, w’, (1+r)-1} , the vector {N, N’, I} is a solution to the firm’s
optimization problem
max I , K’, N, N’
[z F (K, N) – I – w N ] + (1+r)-1 [z’ F (K’, N’) – w’ N’ + (1 – d) K’]
subject to
K’ = K (1 – d) + I
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Moreover, given the prices {w, w’, (1+r)-1} and the vector {N, N’, I}, the profits of
the firm are
П = z F (K, N) – I – w N
П’ = z’ F (K’, N’) – w’ N’ + (1 – d) K’
Competitive Equilibrium
Firm’s Optimality (ctnd...)
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The Firm’s optimality condition is equivalent to
w = MPN
r + d = MPK’
w’ = MPN’
K’ = K (1 – d) + I
together with….
П = z F (K, N) – I – w N
П’ = z’ F (K’, N’) – w’ N’ + (1 – d) K’
Competitive Equilibrium
Government’s Budget is Balanced
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Given the prices {w, w’, (1+r)-1} and the vector {T, T’, G’}, the government
expenditure in the current period is such that the government’s budget is balanced
G = T + (1+r)-1 T’ - (1+r)-1 G’
Competitive Equilibrium
Market Clearing Conditions
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The current labor market clears
N=L–l
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The current consumption good market clears
z F(K, N) = C + G + I
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The future labor market clears
N’ = L – l’
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The future consumption good market clears
z’ F(K’, N’) + K’ (1 - d) = C’ + G’
Competitive Equilibrium
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In the competitive equilibrium, agents are free to buy and sell goods at the
equilibrium market prices. In turn, the equilibrium market prices guarantee that, for
every agent that wants to sell one unit of a good, there is another agent that wants to
buy it.
Competitive Equilibrium
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From the mathematical point of view, the competitive equilibrium is a system of
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15 equations:
4 HH optimality conditions,
6 Firm optimality conditions,
1 government’s balanced budget,
4 market clearing
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14 unknowns:
{ l, l’, C, C’, N, N’, I , K’},
{П, П’},
{w, w’, (1+r)-1}, G
–
7 parameters:
technology {z, z’, d}
initial capital stock K
fiscal policy {G’, T, T’}
Competitive Equilibrium
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A system of 15 distinct equations in 14 unknowns has no solution. What gives?
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Walras’ Law: “Consider an economy with N markets. If N -1 markets clear and all
agents respect their budget constraint, then the Nth market clears as well.”
In other words, one of the market clearing equations is just a reformulation of some
of the other 14 equations are satisfied. The system has only 14 distinct equation and
there exists a unique solution.
Graphical Solution of the Competitive Equilibrium
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Before embarking into the analysis of the competitive equilibrium of the model
economy, we are going to make some simplifying assumptions
– the marginal utility of current leisure Ul is only a function of current leisure l
– the marginal utility of current consumption Uc is only a function of current
consumption c
– the marginal utility of future leisure Ul’ is only a function of future leisure l’
– the marginal utility of future consumption Uc’ is only a function of future
consumption c’
Graphical Solution of the Competitive Equilibrium
Current Labor Market
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In the {N, w} space, draw two curves:
– labor demand curve MPN (N, K)
– labor supply curve Ul( L – N) / Uc(C)
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•
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The labor demand curve is downward sloping (because of decreasing marginal
productivity of labor), shifts upwards when z or K increase
The labor supply curve is upward sloping (because of the concavity of the utility
function), shifts downwards when C increases
The intersection between supply and demand pins down the equilibrium wage w,
the equilibrium labor input N and the equilibrium leisure L - N
Graphical Solution of the Competitive Equilibrium
Future Labor Market
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In the {N’, w’} space, draw two curves:
– future labor demand curve MPN’ (N’, K’)
– future labor supply curve Ul’( L – N’) / Uc’(C’)
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•
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The future labor demand curve is downward sloping (because of decreasing
marginal productivity of labor), shifts upwards when z’ or K’ increase
The future labor supply curve is upward sloping (because of the concavity of the
utility function), shifts upwards when C’ increases
The intersection between supply and demand pins down the equilibrium wage w’,
the equilibrium labor input N’ and the equilibrium leisure L – N’
Graphical Solution of the Competitive Equilibrium
Future Consumption Good Market
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In the {C’, (1+r)-1} space, draw two curves:
– future consumption demand curve Uc’(C’) / Uc(C)
– future consumption supply curve 1 / (MPK’ (K’, N’) + (1-d) ), where
C’ = z’ F(K’, N’) + (1-d) K’ – G’
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•
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The future consumption demand curve is downward sloping (because of the concavity of the
utility function), shifts upwards when C increases
The future consumption supply curve is upward sloping (because K’ is increasing in C’, and
MP’K is decreasing in K’), shifts downwards when N’ or z’ increase, shifts upwards when G’
increases
The intersection between supply and demand pins down the equilibrium interest rate 1+r,
future consumption C’, future capital K’ and investment I = K’ – (1-d) K
Graphical Solution of the Competitive Equilibrium
Current Consumption Good Market
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So far, we have derived {l, l’, C’, K’, I, N, N’}and {w, w’, (1+r)-1}as a function of
current consumption C. To conclude the analysis, we need to pin down C
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The market clearing condition requires that
C = z F(K, N(C, z , K)) – I (C, z’, G’) – G
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Current consumption is increasing in z, K, decreasing in G and G’
Properties of the Competitive Equilibrium: Welfare Theorems
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The competitive equilibrium is a model of the free-market economy.
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We can use such model to answer two normative questions:
– Does the free-market mechanism lead to an efficient allocation of time between
leisure and work and to an efficient allocation of output between consumption
and investment?
Properties of the Competitive Equilibrium: Welfare Theorems
Pareto Efficiency
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An allocation is {l, l’, N, N’, C, C’, I}
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An allocation is feasible if
– L–l =N
– L – l’ = N’
– C + G + I = z F(K, N)
– C’+ G’ = z F(K’, N’) + (1-d) K’
– K’ = (1-d) K + I
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A feasible allocation {l, l’, N, N’, C, C’, I} is Pareto efficient if there is no other allocation
{l*, l’*,N*, N’*, C*, C’*, I*} such that
– {l*, l’*, N*, N’*, C*, C’*, I*} is feasible
– {l*, l’*, N*, N’*, C*, C’*, I*} makes all the households in the economy better-off than
the allocation {l, l’, N, N’, C, C’, I}
Graphical Solution of the Competitive Equilibrium
The Benevolent Social Planner Problem
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The Pareto efficient allocations can be found by maximizing the utility of the
representative household
U(C, C’, l , l’)
over the set of allocations {l, l’ N, N’, C, C’, I} that satisfy the feasibility constraints
– L–l =N
– L – l’ = N’
– C + G + I = z F(K, N)
– C’+ G’ = z F(K’, N’) + (1-d) K’
– K’ = (1-d) K + I
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This maximization problem is known as the benevolent social planner problem
Graphical Solution of the Competitive Equilibrium
The Benevolent Social Planner Problem
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Once the constraints are substituted into the household’s utility function, the social
planner problem reads
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The Pareto efficient allocations can be found by maximizing the utility of the
representative household
U( z F(K, N) – G – I, z F(K’, N’) + (1-d) K’ – G’, L - N , L - N’)
where K’ = (1-d) K + I
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Because of the concavity of the utility and production function, the maximization
problem above is concave in N, N’ and I
Graphical Solution of the Competitive Equilibrium
The Benevolent Social Planner Problem
The optimality conditions are:
1. The product between the marginal utility of current consumption and the marginal
productivity of current labor is equal to the marginal utility of leisure
Uc (C) MPN (K, N) = Ul (L - N)
2. The marginal utility of current consumption equals the product between the
marginal utility of future consumption and the number of units of output that can be
produced in the future by investing one extra unit today
Uc (C) = Uc’ (C’) ( MPK’ (K’, N’) + 1 - d )
Graphical Solution of the Competitive Equilibrium
The Benevolent Social Planner Problem
The optimality conditions are:
3. The product between the marginal utility of future consumption and the marginal
productivity of future labor is equal to the marginal utility of future leisure
Uc’(C’) MPN’ (K’, N’) = Ul’ (L – N’)
4. Current, future consumption and future capital are
C = z F(K, N) – G – I
C’ = z F(K’, N’) + (1-d) K’ – G’
K’ = K(1-d) + I
Graphical Solution of the Competitive Equilibrium
The Benevolent Social Planner Problem
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Mathematically, the solution to the social planner problem is a system of 6
equations in 6 unknowns. Because of the concavity assumptions, the solution is
unique.
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The solution is the Pareto efficient allocation.
Graphical Solution of the Competitive Equilibrium
How does the competitive equilibrium allocation compares to the efficient allocation?
1. In the competitive equilibrium, supply and demand of current labor are equal
Ul( L – N) / Uc(C) = MPN (K, N)
2. In the competitive equilibrium, supply and demand of future labor are equal
MPN (K’, N’) = (1+r) Ul’( L – N’) / Uc(C)
or equivalently
MPN (K’, N’) = Ul’( L – N’) / Uc’(C’)
Graphical Solution of the Competitive Equilibrium
How does the competitive equilibrium allocation compares to the efficient allocation?
3. In the competitive equilibrium, supply and demand of future consumption are equal
Uc’(C’) / Uc(C) = 1 / (MPK’ (K’, N’) + (1-d) )
or equivalently
Uc(C) = Uc’(C’) [MPK’ (K’, N’) + (1-d)]
4. In the competitive equilibrium, the consumption good markets clear
C = z F(K, N) – G – I
C’ = z F(K’, N’) + (1-d) K’ – G’
Graphical Solution of the Competitive Equilibrium
How does the competitive equilibrium allocation compares to the efficient allocation?
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Self-interested actors freely trading at the market clearing prices achieve the
socially efficient outcome that a benevolent dictator would choose.
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Adam Smith understood that the free market would lead to socially efficient
outcomes:
“the owner of a private business by directing that industry in such a manner as its
produce may be of the greatest value, he intends only his own gain, and he is in
this, as in many other cases, led by an invisible hand to promote an end which was
no part of his intention”
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Equilibrium prices reflect the social value of goods and they “invisibly drive” the
self-interest actions towards efficiency
Graphical Solution of the Competitive Equilibrium
What properties of the model are necessary for this equivalence result?
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Three properties are critical:
– government expenditure is financed through lump-sum taxes
– the economic choices of one agent affect other agents only through prices
– all agents take prices as given
Graphical Solution of the Competitive Equilibrium
Example 1: Distortionary Taxes
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Suppose that the government finances its expenditure through proportional taxes
on the labor income earned by the households. In particular, suppose that the tax
rates in the first and second periods are respectively t > 0, t’ > 0.
Graphical Solution of the Competitive Equilibrium
Example 1: Distortionary Taxes
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With a proportional tax on labor earnings, the household chooses l, l’, C and C’ to
solve the following maximization problem
max U(C, C’, l , l’) subject to
C + (1+r)-1 C’ ≤ w (1-t) (L – l) + (П – T) + (1+r)-1 (w’ (1-t’) (L – l’) + (П’ – T’) )
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The worker’s optimal choice of current labor supply satisfies
Ul( L – N) / Uc(C) = (1-t) w
Graphical Solution of the Competitive Equilibrium
Example 1: Distortionary Taxes
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On the other hand, the firm’s optimal hiring decision rule is not affected by the
labor taxes, i.e.
MPN (N, K) = w
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The market clearing condition in the labor market is
(1-t) MPN (N, K) = Ul( L – N) / Uc(C)
Uc(C’) MPN (N, K) > Ul( L – N’)
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When the government’s expenditure is financed through distortionary taxes, the
market equilibrium is not socially efficient, because the worker’s retribution from
working is smaller than the social gain of work
Graphical Solution of the Competitive Equilibrium
Example 2: Externalities
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Together with units of consumption good, the production process generates a
certain amount of pollution. Specifically, if the firm employs K and N, then the
pollution output is
p = P(K, N)
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The level of pollution p in the economy affects negatively the utility of the
household. Specifically, the household’s utility function is
U(C, C’, l , l’) – D(p, p’)
where D(p,p’) is strictly increasing in both p and p’
Graphical Solution of the Competitive Equilibrium
Example 2: Externalities
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The firm does not pay for the amount of pollution generated in the production process.
Therefore, the optimal choice of {N, N’, I, K’} remains unchanged, i.e.
w = MPN (K, N)
r + d = MPK’ (K’, N’)
w’ = MPN’ (K’, N’)
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Given the specification of the household’s utility function with respect to pollution, the
optimal choice of {l, l’, C, C’} remains unchanged, i.e.
w = Ul (L – l) / Uc (C)
(1+r)-1 = Uc’ (C’) / Uc (C)
w’ = Ul (L – l’) / Uc’ (C’)
Graphical Solution of the Competitive Equilibrium
Example 2: Externalities
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On the contrary, the social planner takes into account that the production process has a
negative externality of household’s welfare. Now, the optimality conditions for the planner’s
problem are
Uc (C) MPN (K, N) – Dp PN (K, N) = Ul (L – l)
Uc (C) = (MPK’ (K’, N’) + 1 – d) Uc (C’) – Dp’ PK’ (K’, N’)
Uc (C’) MPN (K’, N’) – Dp’ PN’ (K’, N’) = Ul (L – l’)
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Because there is not a market for clean air, the firm cannot be made to internalize the cost of
pollution on the household and the market outcome is inefficient.
Graphical Solution of the Competitive Equilibrium
Example 3: Price Setters
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Suppose that the household can set the price for labor w and then the firm decides how many
hours to buy. The optimality condition for the household’s labor supply becomes
Uc (C) w + Uc (C) MPNN (K, N) = Ul (N)
When the household sets the price of labor, an increase in the number of hours worked has
three effects:
– HH gets w extra units of consumption good in exchange for the extra unit of labor
– HH has to lower the wage w to convince the firm to buy the extra units of labor
– HH has one less unit of time to leisurely enjoy
The second effect is new and leads to a lower supply of labor for each wage rate
Graphical Solution of the Competitive Equilibrium
Example 3: Price Setters
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The equilibrium on the labor market is given by
Uc (C) MPN (K, N) + Uc (C) MPNN (K, N) = Ul (N)
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When some agents have the power to set prices in the market, the decentralized
equilibrium is not efficient.
Graphical Solution of the Competitive Equilibrium
The Welfare Theorems
First Welfare Theorem: Consider an economy with N goods and I households where
every agent only cares about its own consumption and there are no distortionary
taxes. Then the competitive equilibrium is Pareto efficient.
Second Welfare Theorem: Consider an economy with N goods and I households
where every agent only cares about its own consumption and there are no
distortionary taxes. Moreover, suppose that every household has a monotonic and
concave utility function and the aggregate production set is concave. Then, for
every Pareto efficient allocation A*, there exists an initial distribution of property
rights across households such that A* is a competitive equilibrium.
Graphical Solution of the Competitive Equilibrium
References
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Master the material on these slides. Think about it.
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It might help reading Chapters 5 and 9 in Williamson, which are related to what we
have done. Nevertheless, knowledge of the textbook does not make up for
knowledge of the slides (not even close).
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Check out the classic treatment of general equilibrium in Debreu’s “Theory of
Value”