Macroeconomia I
General Equilibrium Notes
Paulo Henrique Vaz
Programa de Pós-Graduação em Economia - PIMES Universidade Federal de Pernambuco
Outline
1 Basic GE Setup
2 Agents
3 Firms
4 Endowments and Property Rights
5 Competitive Equilibrium
Basic GE Setup
Basic GE Setup
Basic GE Setup
As one can see, there is no much doubt about general equilibrium representing the language of modern macroeconomics.
Basic GE Setup
As one can see, there is no much doubt about general equilibrium representing the language of modern macroeconomics.
We need to set up the language we will be working with. More specifically, it is useful to formally describe our Model Economy: (We will follow Parente GE notes)
Basic GE Setup
As one can see, there is no much doubt about general equilibrium representing the language of modern macroeconomics.
We need to set up the language we will be working with. More specifically, it is useful to formally describe our Model Economy: (We will follow Parente GE notes)
1 Commodities (goods and services)
Basic GE Setup
As one can see, there is no much doubt about general equilibrium representing the language of modern macroeconomics.
We need to set up the language we will be working with. More specifically, it is useful to formally describe our Model Economy: (We will follow Parente GE notes)
1 Commodities (goods and services) 2 Agents (households and firms)
Basic GE Setup
As one can see, there is no much doubt about general equilibrium representing the language of modern macroeconomics.
We need to set up the language we will be working with. More specifically, it is useful to formally describe our Model Economy: (We will follow Parente GE notes)
1 Commodities (goods and services) 2 Agents (households and firms) 3 Preference
Basic GE Setup
As one can see, there is no much doubt about general equilibrium representing the language of modern macroeconomics.
We need to set up the language we will be working with. More specifically, it is useful to formally describe our Model Economy: (We will follow Parente GE notes)
1 Commodities (goods and services) 2 Agents (households and firms) 3 Preference
4 Technology
Basic GE Setup
As one can see, there is no much doubt about general equilibrium representing the language of modern macroeconomics.
We need to set up the language we will be working with. More specifically, it is useful to formally describe our Model Economy: (We will follow Parente GE notes)
1 Commodities (goods and services) 2 Agents (households and firms) 3 Preference
4 Technology
5 Endowments and property rights
Commodities
Formal Description
A commodity is a good or service completely specified physically, temporally, and spatially. (Arrow 1959)
Commodity Space - Case 1 - n consumption goods and Euclidian norm
S =nX ∈RL | kXk= [x12+...+xL2]1/2o
Commodity Space - Case 2 - Infinity Commodities and Sup Norm
S ={X ∈R∞
| kXk∞=supt|xt|}
In words: An element x of the commodity space S is a list of the quantities of every commodity. An element of S, denoted by s ∈S is a commodity vector.
Commodities
If the price vector is given byP = (P1, ...,PL)∈RL+, then the cost of a consumption choice will be given by:
<P.X >= L
X
i=1
pixi
Consumers
The economy is composed of people types, with measure µi, of type i ∈I.
Typically, the number of people types is finite, and often there will be only one type. Individuals of the same type are identical and their preferences and endowments.
Consumption Set (A closed and convex set)
The set of technologically feasible consumption vectors for the consumer.
Xi ≤S ∀i ∈I
Note: the consumption set is not the budget set! The consumption set is in no way defined by the consumer’s income or prices.
Preference
Assume that each consumer type has a preference ordering given by a utility function:
ui :Xi −→R
Consumer’s Preferences
It is common to impose restrictions so that the utility function has some desirable properties
1 Continuity
2 Concavity (strict)- diminishing marginal utility 3 Differentiability
4 Increasing in its arguments
5 Marginal utility approaches infinity as consumption tends to zero
Mathematical Aside: Concavity
A set X is convex if∀x,y∈X andλ∈[0,1],λx+ (1−λ)y ∈X
Mathematical Aside: Concavity
A set X is convex if∀x,y∈X andλ∈[0,1],λx+ (1−λ)y ∈X
A function f :X →Ris concave, asX ⊂Rn is convex, if∀x,y∈X
andλ∈[0,1],f(λx+ (1−λ)y)≥λf(x) + (1−λ)f(y)
Mathematical Aside: Concavity
A set X is convex if∀x,y∈X andλ∈[0,1],λx+ (1−λ)y ∈X
A function f :X →Ris concave, asX ⊂Rn is convex, if∀x,y∈X
andλ∈[0,1],f(λx+ (1−λ)y)≥λf(x) + (1−λ)f(y)
Why do we desire this property of functions?
1 First order necessary conditions for a maximum are sufficient
conditions. (The same is true for the Kuhn-Tucker Theorem with restrictions on the constraint functions.)
2 Additionally, concavity guarantees a global maximum. The assumption
of strict concavity of the objective function ensures that the global maximum is unique.
Firms
Let us assume there is a number J firms in our model economy.
Production Set
The set of possible production plans
Yj ⊂S
Aggregate Production Set
Y =
J
X
j=1
yj
where yj might be positive (output) or negative (inputs)
Firms
Desired Properties on the Technology set of Firms:
The aggregate production set is convex. (Rules out increasing returns to scale)
Either S is finite dimensional or Y has a non-empty interior. Y is non-empty
Y is closed
No free lunch (it is not possible to produces something from nothing) Inaction (0∈Y)
Free disposal (the absorption of any additional amount of inputs without any reduction in output is always possible).
Endowments and property rights
Economy Endowments
¯
ω= ( ¯ω1, ...,ω¯L)∈RL
Property rights
Setting individual endowments:
I
X
i=1
ωi = ¯ω
Claims to profits of firm j by individual i:
θij ∈[0,1],
I
X
i=1
θij =1 ∀j ∈J
Feasibility
Definition of (type i identical) Feasible Allocation: An allocation (x,y) is feasible if
xi ∈Xi for each consumer typei ∈I
yj ∈Yj for each firm j =1,2, ...,J
PI
i=1µixi = ¯ω+
PJ
j=1yj
where µis the measure of each type in the model
Some comments
1 If utility is quasi-concave, then restricting attention to type-identical
allocations makes sense. This will be proved later.
2 Thus, far nothing has been said about an allocation mechanism.
Prices are one way be which allocations are made. Prices are the mechanism in aCompetitive Equilibrium.
Competitive Equilibrium
The C.E. is a way to choose among feasible allocations through a system of property rights and a price system [p≡(p1, ...,pL)]
Technical comment: For S infinite dimensional, the appropriate concept of C.E. is a Valuation Equilibrium. See Debreu 1954 "The Valuation Equilibrium and Pareto Optimum"
Competitive Equilibrium
A Competitive Equilibrium is an allocation (x∗
,y∗
) and a price vector p∗
such that:
Utility maximization: given the price vector, p∗
,x∗
i maximizes
consumeri’s utility subject to his budget constraint. That is,
ui(x∗
i )≥ui(xi)∀xi ∈Xi such that p ∗
xi ≤p ∗
ωi+ J
X
j=1
θijp∗
y∗ j
Profit maximization: given the price vector, p∗
,y∗
j maximizes profit:
p∗
y∗ j ≥p
∗
y∗ j ∀j
Market Clearing Conditions: Markets shall clear, that is,
I
X
i=1
x∗ ih=
J
X
j=1
y∗ jh+
I
X
i=1
ωih ∀h =1, ...,L
Competitive Equilibrium Properties
An allocation is Pareto Optimal if it is feasible and there exists no other feasible allocation that makes at least one person better off without making no individual worse.
Definition of Pareto Optimal Allocation
A feasible allocation x∗
Pareto Dominates feasible allocationx if:
1 ui(x∗
i )>ui(xi) for some i
2 ui(x∗
i )>ui(xi) forall i
Thus,x∗
is Pareto Optimal if there is no feasible allocationx that Pareto dominates it.
Competitive Equilibrium Properties
First Welfare Theorem
If Preferences are locally non-satiated, then any Walrasian Competitive Equilibrium is Pareto Optimal.
FWT may be violated in many economic models
The above theorem is true for Arrow-Debreu economies, in which it is assumed that there is market for every good. Simple modifications such as the introduction of externalities, overlapping generation or even
distortionary taxes may cause some trouble on the FWT.
Competitive Equilibrium Properties
Assumptions:
1 For each i,Xi is convex 2 Strict concavity of utility 3 Utility is continuous
4 The aggregate production possibility set is convex (there are no increasing
returns to scale)
5 Either the commodity space is finite-dimensional, or Y has a non-empty
interior
Second Welfare Theorem
Given an economy specified by ({Xi,ui}Ii=1,{Yj}
J
j=1,ω¯ and assume assumptions
1-5 hold. Let (x∗,y∗)be a Pareto Optimal Allocation and assume that for at
least one consumex∗
i is not a satiation point. Then there existp6=0 such that (x∗,y∗,P)is a Price Quasi-Equilibrium with Transfers.
As a practical matter, whenever the model is well-behaved in way that satisfies both WFTs, one can solve a social planner problem to find competitive equilibrium allocations and price.
