Foundations of Dynamic Macroeconomic Analysis and the Neoclassical ...yinchiwang.weebly.com/uploads/8/1/4/1/8141722/lec3_neoclassical... · Foundations of Dynamic Macroeconomic Analysis

Foundations of Dynamic Macroeconomic Analysis andthe Neoclassical Growth Model

Yin-Chi Wang

The Chinese University of Hong Kong

September, 2012

Yin-Chi Wang (CUHK) Foundations of DGE and Neoclassical growth models September, 2012 1 / 33

Preliminaries Introduction

Neoclassical Growth Models

Solow-Swan Growth Model

Use variable ky to resove Harrod�s knife-edge problemConstant saving rate

Neoclassical growth model

Individual optimizationEndogenous saving rateEnables better understanding of the factors that a¤ect savings decisionsEnables to discuss the optimality of equilibria: can the (competitive)equilibria of growth models be improved upon?Notion of improvement: Pareto optimality.



How to write down a model that is an abstract of the real world

How to model households? Preferences? Problems of aggregation

How to model �rms?

Tools that we need

This set of lecture notes provides foundations of macro analysis, focusingparticularly on the following issues:

1 Fundamentals of Dynamic General Equilibrium2 Optimal Growth in Discrete Time3 Optimal Growth in Continuous Time

Basic references:

1 Acemoglu (2009): chs. 5-72 Aghion and Howitt (1998): chs. 1-23 Barro and Sala-i-Martin: ch. 2, secs. 4.1-4.34 Ljungqvist and Sargent (2000): chs. 3 and 115 Stokey and Lucas with Prescott (1989): chs. 3-5



Preliminaries

Consider an economy consisting of a unit measure of in�nitely-livedhouseholds

An uncountable number of households: e.g., the set of households Hcould be represented by the unit interval [0, 1]Emphasize that each household is in�nitesimal and will have no e¤ecton aggregatesCan alternatively think of H as a countable set of the formH = f1, 2, ...,Mg with M = ∞Advantage of unit measure: averages and aggregates are the same

Time horizon:1 �In�nitely lived�or consisting of overlapping generations with fullaltruism linking generations: In�nitely planning problem

2 Overlapping generations: Finite planning horizon (generally)



Time Separable Preferenceses

Standard assumptions on preference orderings so that they can berepresented by utility functionsEach household i has an instantaneous utility function

ui (ci (t))

ui : R+ ! R, increasing and concave; ci (t) :consumptionIn�nite horizon

Ei0

∞∑t=0

βti ui (ci (t))

Exponential discounting (βi 2 (0, 1)) and time separability: ensuretime-consistent behaviorA solution fx (t)gTt=0 (possibly with T = ∞) is time consistent ifwhenever fx (t)gTt=0 is an optimal solution starting at time t = 0,fx (t)gTt=t 0 is an optimal solution to the continuation dynamicoptimization problem starting from time t = t 0 2 [0,T ]Violation of time-consistency: hyperbolic preference, self-controlproblem...etc.


Representative Household and Firms Representative Household

Representative Household I

An economy admits a representative household if preference side canbe represented as if a single household made the aggregateconsumption and saving decisions subject to a single budgetconstraint (positive notion)

Stronger notion of �normative� representative household: if we canalso use the utility function of the representative household for welfarecomparisons

Simplest case that will lead to the existence of a representativehousehold: suppose each household is identical



Representative Household II

Representative household is valid when the optimization of individualhouseholds can be represented as if there were a single householdmaking the aggregate decisions using a representative preferencesubject to aggregate constraints

Ignoring uncertainty. Consider a particular preferences representation

max∞∑t=0

βtu (c (t))

Admits a representative household rather trivially

The above representative household�s preferences can be used forpositive and normative analysis



Representative Household III

If instead households are not identical: see whether can model as ifdemand side is generated by the optimization decision of arepresentative household

More realistic, but:

1 The representative household will have positive, but not always anormative meaning

2 Models with heterogeneity: often not lead to behavior that can berepresented as if generated by a representative household

Whether can be modeled as if there is a representative household

Let the excess demand of the economy be x(p)Key: whether this excess demand function x(p) can be obtained as asolution to the single household optimization problemAnswer at the �rst glance: it cannot be so obtained in general as theweak axiom of revealed preferences for individuals need not hold for theaggregate.



Representative Household IV

Theorem H1 (Debreu-Mantel-Sonnenschein Theorem) Let ε > 0 andN < ∞ be a positive integer. Consider a set of pricesPε = fp 2 RN

+ : pj/pj 0 � ε for all j and j 0g and any continuous functionx : Pε ! RN

+ that satis�es Walras�Law and is homogeneous of degree 0.Then there exists an exchange economy with N commodities and H < ∞households, where the aggregate demand is given by x(p) over the set Pε.

To yield a positive answer ensuring the excess demand function x(p)to be obtained as a solution to the single household optimizationproblem, we need to impose further restrictions, in particular, toremove strong income e¤ects.



Gorman Aggregation

A special but useful case for such a representation is to have linear value(indirect utility) functionAn indirect utility function for household i , vi

�p, y i

�, speci�es (ordinal)

utility as a function of the price vector p = (p1, ..., pN ) and householdincome y i

vi�p, y i

�:homogeneous of degree 0 in p and y .

Theorem H2 (Gorman�s Aggregation Theorem) Consider an economy with a�nite number N < ∞ of commodities and a set H of households. Suppose thatthe preferences of household i 2 H can be represented by an indirect utilityfunction of the form

v i�p, y i

�= ai (p) + b (p) y i

then these preferences can be aggregated and represented by those of arepresentative household, with indirect utility

v (p, y) =Zi2H

ai (p) di + b (p) y

where y �Ri2H y

idi is aggregate income.Yin-Chi Wang (CUHK) Foundations of DGE and Neoclassical growth models September, 2012 10 / 33


Representative Household V

The class of preferences described in Theorem H2 is referred to as�Gorman preferences� (1959 Econometrica).

In this class, the Engel curve of each household for each commodity islinear and its slope is identical to all individuals for the samecommodity.

By Roy�s Identity,

xhj�p,wh

�= � 1

b (p)∂ah (p)

∂pj� 1b (p)

∂b (p)∂pj

wh

Therefore, for each household, a linear relationship exists betweendemand and income and the slope, � 1

b(p)∂b(p)

∂pj, is independent of the

household�s identity (h).



Representative Household VI

Even under class of �Gorman preferences�a representative householdexists, typical macro models require further restrictions on:

the abstract of distribution e¤ects from the representative household�sconcern (strong representation);the use of the representative household�s preference as the welfarefunction of the aggregate economy (normative representation).



Representative Household VII

Normative representation requires convexity, interiority and commonbasic value (otherwise, one can transfer ε from low to high-valuationhouseholds)

Gorman preferences also imply the existence of a normativerepresentative household.

Recall an allocation is Pareto optimal if no household can be madestrictly better-o¤ without some other household being made worse-o¤



Representative Household VIII

Theorem H3 (Existence of a Normative Representative Household)Consider an economy with a �nite number N < ∞ of commodities, a setH of households and a convex aggregate production possibilities set Y .Suppose that the preferences of each household i 2 H take the Gormanform, v i

�p, y i

�= ai (p) + b (p) y i with p = (p1, ..., pN ) and that each

household i 2 H has a positive demand for each commodity

1 Then any allocation that maximizes the utility of the representativehousehold, v (p, y) = ∑i2H a

i (p) + b (p) y , with y � ∑i2H yi , is

Pareto optimal.2 Moreover, if ai (p) = ai for all p and all i 2 H, then any Paretooptimal allocation maximizes the utility of the representativehousehold


Representative Household and Firms Representative Firm

Representative Firm I

Representative �rm is valid when the optimization of individual �rmscan be represented as if there were a single �rm making the aggregatedecisions using an aggregate production function subject to aggregateconstraints.

Consider price and output vectors: p = (p1, ..., pN ) andy = (y1, ..., yN ), with py = ∑N

j=1 pjyj .

Let F be the countable set of �rms in the economy and

Y ��

∑f 2F

y f : y f 2 Y f for each f 2 F�

be the aggregate production possibility set (PPS).



Representative Firm II

When there is no price dependent �xed cost (ensured under perfectcompetition in the absence of externalities), we have:

Theorem F (Representative Frim Theorem) Consider a competitiveproduction economy with N 2 N[ f+∞g commodities and a countableset F of �rms, each with a convex production possibilities set Y f 2 RN .Let p 2 RN be the price vector in this economy and denote the set ofpro�t maximizing net supplies of �rm f 2 F by Y f (p) � Y f (so that forany y f 2 Y f (p), we have p � y f � p � y f for all y f 2 Y f ). Then thereexists a representative �rm with production possibilities set Y � RN andset of pro�t maximizing net supplies Y (p) such that for any p 2 RN

+,y 2 Y (p) if and only if y (p) = ∑f 2F y

f for some y f 2 Y f (p) for eachf 2 F .



Representative Firm III

Why such a di¤erence between representative household andrepresentative �rm assumptions? Income e¤ects.

Changes in prices create income e¤ects, which a¤ect di¤erenthouseholds di¤erently.

No income e¤ects in producer theory, so the representative �rmassumption is without loss of any generality.

Does not mean that heterogeneity among �rms is uninteresting orunimportant.

Many models of endogenous technology feature productivitydi¤erences across �rms, and �rms�attempts to increase theirproductivity relative to others will often be an engine of economicgrowth.


Representative Household and Firms Equilibrium

Equilibrium

An economy E is described by preferences, endowments, productionsets, consumption sets, and allocation of shares, that is,E �(H,F ,U,ω,Y,X, θ).An allocation (x, y) in E , x 2 X, y 2 Y, is feasible if∑h2H x

hj � ∑h2H ωh

j +∑f 2F yfj for all j 2 N.

De�nition E (Competitive Equilibrium) A competitive equilibrium foreconomy E �(H,F ,U,ω,Y,X, θ) is given by a feasible allocation(x� = fxh�gh2H, y� = fy f �gf 2F ) and a price system p� such that

1. (Firm optimization) For every �rm f 2 F , y f � maximizes pro�ts:p� � y f � � p� � y f for all y f 2 Y f .2. (Household optimization) For every household h 2 H, xh� maximizesutility: U h

�xh�� U h

�xh�for all x such that xh 2 X h and

p� � xh � p��

ωh +∑f 2F θhf yf�.


Representative Household and Firms Pareto Optimum

Pareto Optimum

The standard concept of optimality is Pareto optimum, though socialoptimum is often used in macroeconomics.

De�nition O (Pareto Optimum) A feasible allocation (x, y) for economyE �(H,F ,U,ω,Y,X, θ) is Pareto optimal if there exists no other feasibleallocation (x0, y0) such that x 0h 2 X for all h 2 H, y 0f 2 Y f for all f 2 F ,and U h

�x 0h�� U h

�xh�for all h 2 H with U h0

�x 0h

0�� U h0

�xh

0�for at

least one h0 2 H.


Representative Household and Firms Welfare Theorems

Welfare Theorems I



Welfare Theorem II



Welfare Theorem III

The Second Welfare Theorem can be readily used to prove theexistence of a competitive equilibrium based on Brouwer/Kakutani.

Theorem W3 (Existence of Competitive Equilibrium) Consider aneconomy E as described in Theorem W3, if p� � wh� > 0 for each h 2 H,then there exists a competitive equilibrium (x�, y�, p�).


Representative Household and Firms Dynamic General Equilibrium

Dynamic General Equilibrium I

To generalize the standard Arrow-Debreu-McKenzei generalequilibrium (GE) analysis to an in�nite-horizon dynamic framework(DGE), one needs to impose further restrictions, particularly in thefollowing aspects:

�nite value: bounded valuation of households/�rmsin�nite dimensional space: Banach Space, Hilbert Space, Polish Spacewith weak or weak* topology rather than standard product topologyin�nite dimensional �xed point: Schauder and othersinteriority in in�nite dimensioninformation and Arrow-Debreu trade.


Representative Household and Firms Dynamic General Equilibrium

Dynamic General Equilibrium II

In the context of optimal growth, such DGE frameworks are given by

Discrete time:

maxfct ,ktg∞t=0

∑∞t=0 βtu (ct )

s.t. kt+1 = f (kt ) + (1� δ) kt � ctct , kt � 0, k0 > 0

Continuous time:

maxfct ,ktg∞t=0

R ∞t=0 exp (�ρt) u (c (t)) dt

s.t. k (t) = f (k (t))� c (t)� δk (t)ct , kt � 0, k0 > 0


Dynamic Programming

Dynamic Programming I

Discrete-time in�nite-horizon optimization problem:

supfxt ,ytg∞t=0

∑∞t=0 βt Ut (t, xt , yt )

s.t. yt 2 G (t, xt ) 8t � 0xt+1 = f (t, xt , yt ) 8t � 0x0 > 0 given

Eliminate yt and rewrite the optimization problem:

V �0 (x0) = supfxtg∞t=0

∑∞t=0 βt Ut (xt , xt+1)

s.t. xt+1 2 G (t, xt ) 8t � 0x0 > 0 given


Dynamic Programming

Dynamic Programming II

Under stationarity, Ut = U and Gt = G , yielding the followingrecursive problem:

V �0 (x0) = supfxtg∞t=0

∑∞t=0 βtU (xt , xt+1)

s.t. xt+1 2 G (t, xt ) 8t � 0x0 > 0 given

The functional problem is (Bellman equation):

V (x) = supy2G (x )

fU (x , y) + βV (y)g 8x 2 X


Dynamic Programming

Dynamic Programming II

Feasible set:

Φ (xt ) =�fxsg∞

s=t : xs+1 2 G (xs ) , s = t, t + 1, ....

Let X be a compact subset of RK and

XG = f(x , y) 2 X � X : y 2 G (x)g .Assumption A1:

(a) G (x) 6= ∅ for all x 2 X ;(b) G is compact-valued and continuous;(c) G is an increasing set s.t. x � x 0 =) G (x) � G (x 0)

Assumption A2:(a) 8x0 2 X and 8x 2 Φ (x0) ,limn!∞ ∑nt=0 βtU (xt , xt+1) = V∞ < ∞;(b) U : XG ! R is continuous;(c) U (x , y) is strictly increasing in the �rst K elements for each y 2 X ;(d) U (x , y) is strictly concave in (x , y);(e) U is continuously di¤erentiable over intXG .


Dynamic Programming

Dynamic Programming IV

Applying sandwich theorem, we can show

Theorem DP1 (Equivalence of Values) Under A1(a) and A2(a), thesolution to the recursive problem V �(x0) is equivalent to the solution tothe functional problem V (x), i.e., V �(x) = V (x) for all x 2 X .

By recursive substitution, it is straightforward to obtain:

Theorem DP2 (Principle of Optimality) Under A1(a) and A2(a),consider a feasible plan x� 2 Φ (x0) that attains V �(x0) in the recursiveproblem. Then

V �(x�t ) = U (x�t , x�t+1) + βV �(x�t+1)

fort = 0, 1, ... with x�0 = x0. Moreover, if any x� 2 Φ (x0) attains V(x) in

the functional problem, then it attains the optimal value in the recursiveproblem.


Dynamic Programming

Dynamic Programming V

With continuity as well as compactness of the constraint set, we canapply Beige�s Theorem of Maximum and Weierstrass Theorem toobtain:

Theorem DP3 (Existence of Solutions) Under A1(a,b) and A2(a,b),there exists a unique continuous and bounded function V : X ! R thatsatis�es the Bellman equation. Moreover, for any x0 2 X , an optimal planx� 2 Φ (x0) exists.

With concavity and di¤erentiability, we have:

Theorem DP4 (Di¤erentiability of the Value Function) UnderA1(a,b) and A2(a,b,d,e), consider an optimal plan x� and de�ne π as thepolicy function satisfying x�t+1 = π(x�t ). Further assume that x 2intX andπ (x) 2intG (x). Then V (�) is di¤erentiable at x , with gradientDV (x) = DxU (x ,π(x)).


Dynamic Programming

Dynamic Programming VI

Let (s, d) be a norm space and T : S ! S be an operator mapping Sinto itself. If for some β 2 (0, 1),

d (Tz1,Tz2) � βd (z1, z2) 8z1, z2 2 S

Then T is a contraction mapping (with modulus β).

Theorem DP5 (Contraction Mapping Theorem) Let (s, d) be acomplete norm space and suppose that T : S ! S is a contraction. ThenT has a unique �xed point, z , i.e., there exists a unique z 2 S such thatTz = z .


Dynamic Programming

Dynamic Programming VII

In practice, the following theorem provide su¢ cient conditions for acontraction map:

Theorem DP6 (Blackwell�s Su¢ cient Conditions for a Contraction)Let X � Rk , and B(X ) be the space of bounded functions f : X ! Rde�ned on X equipped with the sup norm jj � jj. Suppose thatB 0(X ) � B(X ), and let T : B 0(X )! B 0(X ) be an operator satisfying thefollowing two conditions:

1. (Monotonicity) For any f , g 2 B 0(X ), f (x) � g(x) 8x 2 X ,(Tf )(x) � (Tg)(x)8x 2 X ;2. (Discounting) 9β 2 (0, 1) s.t. [T (f + c)](x) � (Tf )(x) + c 8f 2B(X ), c � 0, and x 2 X .Then T is a contraction (with modulus β) on B 0(X ).


Dynamic Programming

Dynamic Programming VIII

Applying Contraction Mapping Theorem to the value function V , wecan establish its properties:

Theorem DP7 (Value Function Properties) Under A1(a,b,c) andA2(a,b,c,d), V (x) in the Bellman equation is strictly increasing in all of itsarguments and strictly concave.

Theorem DP8 (Necessity and Su¢ ciency) Under A1(a,b,c) andA2(a,b,c,d,e), a sequence fx�t g∞

t=0 such that x�t+1 2intG (x�t ), t = 0, 1...,

is optimal for the recursive problem given x0 if and only if it satis�es thefollowing:1. (Euler Equations) DyU (x ,π(x)) + βDxU (π(x),π(π(x))) = 02. (Transversality Conditions) lim t!∞βtDxU (x�, x�t+1)x�t = 0


Dynamic Programming

Dynamic Programming IX

In standard optimal growth models, A1(a,b,c) and A2(a,b,c,d,e) areall met.

Example: consider U = ln(ct ) and G = kat � ct , so the Bellmanequation is: V (x) = maxy�0fln (xα � y) + βV (y)g

(Euler) 1x α�y = βV 0 (y)

(Benveniste-Scheinkman) V 0 (x) = αx α�1x α�y

(Policy function) kt+1 = αβkαt , ct = (1� αβ) kα

t


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Foundations of Dynamic Macroeconomic Analysis and the Neoclassical ...yinchiwang.weebly.com/uploads/8/1/4/1/8141722/lec3_neoclassical... · Foundations of Dynamic Macroeconomic Analysis