Advanced Macroeconomics I Lecture 4 (1) Zhe Li SUFE Spring 2011 (SUFE) Advanced Macro Spring 2011 1 / 26
Advanced Macroeconomics ILecture 4 (1)
Zhe LiSUFE
Spring 2011
(SUFE) Advanced Macro Spring 2011 1 / 26
Competitive equilibrium -consumers
A "representative consumer"there are a large number of consumers in theeconomyall identical: same preference, sameendowment, rational
(SUFE) Advanced Macro Spring 2011 2 / 26
Prices
Prices of commodities will adjust so thatmarkets clear
prices will make all the identical consumersmake the same decisionsprices will have to adjust so that consumersdo not interact
(SUFE) Advanced Macro Spring 2011 3 / 26
Prices
ExampleThe dynamic model without production gives atrivial allocation outcome: the consumerconsumes the endowment of every product
The competitive mechanism ensures that thisoutcome is achieved by prices being set so thatthe consumer, when viewing prices as beyond hiscontrol, chooses to consume no more and no lessthan his endowments
(SUFE) Advanced Macro Spring 2011 4 / 26
Competitive equilibrium - �rms
The production factors (capital and labor)are owned by many individual households
Households�decisions consist of the amountof factors to provide to �rms, and theamount of consumption goods to purchasefrom them
The technology to transform those factorsinto consumption goods is operated by �rms
Firms have to choose their productionvolume and factor demand
(SUFE) Advanced Macro Spring 2011 5 / 26
Competitive equilibrium - markets
Markets: the device by which sellers andbuyers are driven togetherEquilibrium: for some given prices,individual households�and �rms�decisionsshow an aggregate consistency
the amount of factors that suppliers arewilling to supply equals the amount thatproducers are willing to takeconsumption goods: sypply = demandwe say that markets clear
(SUFE) Advanced Macro Spring 2011 6 / 26
Competitive equilibrium
The word "competitive" indicates that weare looking at the perfect competitionparadigm, as opposed to economies inwhich �rms might have some sort of"market power"A competitive equilibrium is a vector ofprices and quantities that satisfy certainproperties related to the aggregateconsistency of individual decisions(SUFE) Advanced Macro Spring 2011 7 / 26
Competitive equilibrium properties
1 Households choose quantities so as tomaximize the level of utility attained giventheir "wealth" (factor ownership evaluatedat the given prices)
When making decisions, households takeprices as given parametersThe maximum monetary value of goods thathouseholds are able to purchase given theirwealth is called the budget constraint
(SUFE) Advanced Macro Spring 2011 8 / 26
Competitive equilibrium properties
2 Firms chose the production volume thatmaximizes their pro�ts at the given prices
3 Markets clear: the quantity choice is"feasible".
demand can be produced with the availabletechnology using the amount of factors thatsuppliers are willing to supply
(SUFE) Advanced Macro Spring 2011 9 / 26
Dynamic competitive equilibriumHow trade takes place over time
Are the economic agents using assets?Di¤erent arrangements for the samephysical environment - same �nal allocation:
Firms rent their inputs from consumers everyperiod
do not need assets to ful�ll their pro�tmaximization objective
Firms buy and own the long-lived capitalthey use in production
need to consider the relative values of pro�ts indi¤erent periods
(SUFE) Advanced Macro Spring 2011 10 / 26
Dynamic competitive equilibriumProcesures
Mathematically there are two alternativeprocedures:
equilibria can be de�ned and analyzed interms of (in�nite) sequencesor they can be expressed recursively, usingfunctions
(SUFE) Advanced Macro Spring 2011 11 / 26
Date 0 arrangementArrow-Debreu-McKenzie
Let goods be datedfor example, in a one-good per date context,there is an in�nite sequence of commodities:consumption at t = 0, consumption att = 1, etc.like in a static model, let the trade in allthese commodities take place once and for all
In this arrangement, there is no need forassets(SUFE) Advanced Macro Spring 2011 12 / 26
Sequencial arrangement
Assets are traded every period
There are nontrivial decisions made in everyfuture period
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The neoclassical growth modelwith date-0 trade
The consumer is endowed with 1 unit of"time" each period, which he can allocatebetween labor and leisureThe utility derived from the consumptionand leisure stream fct, 1� ntg∞
t=0 is givenby
U�fct, 1� ntg∞
t=0
�=
∞
∑t=0
βtu (ct)
leisure is not valued; equivalently, laborsupply bears no utility costWe also assume that u is strictly increasingand strictly concave
(SUFE) Advanced Macro Spring 2011 14 / 26
The production
The consumer rents his labor services att to the �rm for a unit rental (or wage) rateof wtThe production function of theconsumption / investment good is F(K; n)
F is strictly increasing in each argument,concave, and homogeneous of degree 1
(SUFE) Advanced Macro Spring 2011 15 / 26
The prices
Price of consumption good ct at every tpt intertemporal relative pricesif p0 = 1, then pt is the price of ct at trelative to (in terms of) c0
Price of capital services at t: ptrt rental rateprice of capital services at t interms of c0
Price of labor: ptwt wage rateprice of labor at t interms of c0
(SUFE) Advanced Macro Spring 2011 16 / 26
De�nition of a competitiveequilibrium
A competitive equilibrium is a set of sequences:Prices: fp�t g
∞t=0 , fr�t g
∞t=0 , fw�t g
∞t=0
Quantities: fc�t g∞t=0 , fK�t+1g
∞t=0 , fn�t g
∞t=0 such
that
fc�t g∞t=0 , fK�t+1g
∞t=0 , fn�t g
∞t=0 solve the
consumer�s problem
(SUFE) Advanced Macro Spring 2011 17 / 26
Equilibrium de�nition - Consumer
Consumer�s problem
fc�t ,K�t+1, n�t g∞t=0
= arg maxfct ,Kt+1,ntg∞
t=0
(∞
∑t=0
βtu (ct)
)
(SUFE) Advanced Macro Spring 2011 18 / 26
Equilibrium de�nition - Consumer
Budget constraint:
s.t.∞
∑t=0
p�t (ct +Kt+1)
=∞
∑t=0
p�t [r�t Kt + (1� δ)Kt + ntw�t ]
ct � 0 8t, k0 givenKt+1 is quoted in the same price as ctLabor has no utility cost: wt > 0 implies thatthe consumer supplies all his time, n�t = 1
(SUFE) Advanced Macro Spring 2011 19 / 26
Equilibrium de�nition - Firm
fK�t+1g∞t=0 , fn�t g
∞t=0 solve the �rm�s
problem:
fK�t , 1g∞t=0
= arg maxKt , nt
�p�t F(Kt, nt)
�p�t r�t Kt � p�tw�t nt
�The �rm�s decision problem involves just aone-period choice
all of the model�s dynamics come from theconsumer�s capital accumulation problemfor example, we could imagine that �rms livefor just one period
(SUFE) Advanced Macro Spring 2011 20 / 26
Equilibrium de�nition - market
Firm�s condition may equivalently beexpressed as follows: 8t : (r�t ,w�t ) satisfy:
r�t = FK(K�t , 1)
w�t = Fn (K�t , 1)
If the production function F(K; n) isincreasing in n, then n�t = 1 followsFeasibility (market clearing):
c�t +K�t+1 = F(K
�t , 1) + (1� δ)K�t
(SUFE) Advanced Macro Spring 2011 21 / 26
Characterize the equilibrium
intertemporal �rst-order conditions
ct : βtu0(c�t ) = p�t λ�
where λ� is the Lagrange multipliercorresponding to the budget constraint
ct+1 : βt+1u0(c�t+1) = p�t+1λ
�
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Marginal rate of substitution
p�tp�t+1
=1β
u0(c�t )u0(c�t+1)
p�tp�t+1
as the real interest rate
1βu0(c�t )u0(c�t+1)
as the marginal rate of substitutionof consumption goods between t and t + 1
(SUFE) Advanced Macro Spring 2011 23 / 26
Technical substitution
FOC with Kt+1
Kt+1 : λ�p�t = λ�p�t+1 [r�t+1+ 1� δ]
p�tp�t+1
= r�t+1+ 1� δ
= FK(K�t , 1) + 1� δ
FK (K �t , 1) + 1� δ is the marginal return oncapitalthe marginal rate of technical substitution(transformation) between ct and ct+1
(SUFE) Advanced Macro Spring 2011 24 / 26
First welfare theorem
Same as the Euler Equation from theplanner�s problem
u0(c�t ) = βu0(c�t+1) [FK(K�t , 1) + 1� δ]
Therefore a competitive equilibriumallocation satis�es the optimality conditionsfor the centralized economy: thecompetitive equilibrium is optimal
the First Welfare Theorem(SUFE) Advanced Macro Spring 2011 25 / 26
Second welfare theorem
We have assumed that there is a singleconsumer, so in this case Pareto-optimalityjust means utility maximization
In addition, with the appropriateassumptions on F(K; n) (constant returnsto scale), an optimum can be supported asa competitive equilibrium, which is theresult of the Second Welfare Theorem
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