topic 2: economic growth i - university of hong kongyluo/teaching/6012_2019/lecture2a.pdftopic 2:...

Topic 2: Economic Growth I

Yulei Luo

Economics HKU

September 25, 2019

Luo, Y. (Economics HKU) ECON6012: Macro Theory September 25, 2019 1 / 49

The General Questions about Economic Growth

Robert Lucas (1988): �Once one starts to think about [economic growth],it is hard to think about anything else.�For example, if a country achieved 1% growth in its real GDP per capita,it will take about 70 years to double its real GDP per capita. If it achieved10%, the time will be reduced to 7 years. (The 70 rule.)

What are the determinants of long-run economic growth?

How can we explain the signi�cant di¤erences in both output levelsand growth rates across countries and over time?


Historical Note: The Solow Model

Solow (1956): What do simple neoclassical assumptions imply aboutgrowth? The model is the starting point for almost all analyses of growth.

Key assumptions include:The production function F has three factors: capital K , labor L, andtechnology A:

Y = F (K ,AL) (K , L,A > 0) , (1)

where FK ,FL > 0, and FKK ,FLL < 0 (diminishing returns to capitaland labor). AL means e¤ective labor.F is assumed to be constant return to scale in K and L:

Y = ALF�KAL, 1�= ALf (k) or y = f (k) , (2)

where y = Y / (AL) and f (k) = F (k, 1). F also satisfy:

limK!0

FK (K ,AL) = ∞, limL!0

FL (K ,AL) = ∞,

limK!∞

FK (K ,AL) = 0, limL!∞

FL (K ,AL) = 0,

F (0,AL) = 0 for all A and L.


Solving the Solow Model

Assume that there is only a representative agent in the economy. TheSolow model can be formulated as follows:

Kt+1 = (1� δ)Kt + sF (Kt ,AtLt ) , (3)

At+1 = (1+ g)At , (4)

Lt+1 = (1+ n) Lt , (5)

given K0, A0, and L0. Note that in the continuous-time version, weget a system of di¤erential equations.

De�ne k = KAL and y =

YAL , we have the following �rst-order nonlinear

di¤erence equation about capital per e¤ective labor unit (k):

kt+1 =(1� δ) kt + sf (kt )(1+ g) (1+ n)

.


Steady States

There exist two steady states (SS): k = 0 and k = k > 0 thatsatis�es:

k =(1� δ) k + sf (k)(1+ g) (1+ n)

.

When f (k) = kα,

k =�

s(1+ g) (1+ n)� (1� δ)

�1/(1�α)

.

Graphic Analysis [see the phase diagram.]

Policy implications: The e¤ects of s, α, and δ on the steady statestock of capital per e¤ective labor unit.

Alternatively, we can use the linearization method to approximate theoriginal nonlinear di¤erence equation around the intertemporalequilibrium (the steady state) and then solve the resulting lineardi¤erence equation.


Linear Approximation

How to linearize a di¤erence equation (DE): xt+1 = f (xt ), given aninitial condition, x0? Typically, linearize the equation around a SS xsatisfying x = f (x):

xt+1 ' x + f 0 (x) (xt � x) . (6)

If we treat this approximation as exact, we have the following�rst-order DE:

xt+1 = axt + b, (7)

where a = f 0 (x) and b = (1� a) x .The solution to this DE is

xt =�1� at

�x + atx0.

If jaj < 1, then at ! 0 as t ! ∞ so that xt ! x . Hence, the originalnonlinear DE is locally stable as it is approximated around the SS.For the Solow model, we have

kt =�1� at

�k + atk0, where a = α+

(1� α) (1� δ)

(1+ g) (1+ n)2 (0, 1) .


Speed of Convergence

How does the initial level of capital per capita a¤ect growth rates?

Convergence: Poor countries grow faster than rich countries.Divergence: Rich countries grow faster than poor countries.

The Solow model predicts that poor countries with low k will growfast because of decreasing returns to capital:

gt ,t+1 =kt+1 � kt

kt=

s(1+ g) (1+ n)

f (kt )kt

+

�(1� δ)

(1+ g) (1+ n)� 1�,

where

d�f (kt )kt

�/dkt =

f 0 (kt ) kt � f (kt )k2t

< 0

because f (kt ) = kαt is concave (i.e., f

0 (kt ) is decreasing).

This convergence is only observed among U.S. states, Canadianprovinces, European regions, etc, but not observed among thecountries of the world.


Summary

k is called �globally asymptotically stable� as kt converges to it fromany initial positive capital stock, k0.

What does this imply about the aggregate variables:

Since KtAtLt

is constant, Kt and AtLt grow at the same rate(1+ g) (1+ n)� 1 ' g + n.Since F (�, �) is homogeneous of degree 1, Yt also grows at the samerate g + n.Per capita output YtLt grows at rate g .The steady state for kt corresponds to a balanced growth path (asequence in which all of the variables grow at a constant rate) for theoriginal variables.


The Ramsey-Cass-Koopmans Model (Optimal GrowthModel)

The key di¤erence bw the RCK model and the Solow model is thatthe saving rate is endogenized in the RCK model. We posit a singlerepresentative agent in the economy and has the following preference

maxfct ,kt+1g

∞

∑t=0

βtu (ct ) , (8)

where u (ct ) also satis�es the same conditions as before.Abstracting from population growth or technological progress, theresource constraint facing the representative consumer at any t are

kt+1 = (1� δ) kt + f (kt )� ct , (9)

where k0 is given. δ 2 (0, 1) is the depreciation rate, the productionfunction f (kt ) satis�es the usual conditions: f 0 (kt ) > 0 andf 00 (kt ) < 0.


Solving the Model

As usual, we �rst set up the Lagrangian:

L =∞

∑t=0

βt fu(ct ) + λt [(1� δ) kt + f (kt )� ct � kt+1]g (10)

where λt � 0 denote the multiplier on the resource constraint at timet.

The FOCs w.r.t. ct and kt+1 are

u0(ct ) = λt (11)

β�1� δ+ f 0 (kt+1)

�λt+1 = λt (12)

Combining both FOCs can generate the Euler equation

β�1� δ+ f 0 (kt+1)

�u0(ct+1) = u0(ct ) (13)


(conti.) We also need the transversality condition (TVC) to guaranteethat asymptotically the shadow value of more capital is zero:

limt!∞

βtλtkt+1 = limt!∞

βtu0(ct )kt+1 = 0, (14)

where βtλt is the present-value utility evaluation of an additional unitof resources in period t.

Hence, the TVC says that the value (discounted into PV utilities) ofeach additional unit of capital at in�nity times the actual amount ofcapital has to be zero. Otherwise, the consumer can modify such acapital path and increase consumption for an overall increase in utilitywithout violating feasibility. In the �nite horizon case, thecorresponding condition is that kT+1 = 0.


(conti.) The nPg condition discussed in the last lecture and the TVChere play a very similar roles in dynamic optimization in a purelymechanical sense. In fact, they are the same condition when theFOCs are satis�ed:

limt!∞

βtλtkt+1 = 0() limt!∞

R�tt λ0kt+1 = λ0 limt!∞

R�tt kt+1 = 0,

where Rt = 1� δ+ f 0 (kt+1) and λ0 = u0(c0) is �nite.

However, the two conditions are conceptually very di¤erent: The nPgis a restriction on the choices of the agent. In contrast, the TVC is aprescription how to behave optimally, given a choice set.


Intertemporal Equilibrium (Steady State)

We have formed a nonlinear di¤erence equation system in terms of ctand kt that can characterize the economy completely:

u0(ct ) = β�1� δ+ f 0 (kt+1)

�u0(ct+1) (15)

kt+1 = (1� δ) kt + f (kt )� ct

with two boundary conditions: the initial condition k0 = k (0) andthe transversality condition (TVC) (14).

Now we can determine the intertemporal equilibrium (the steadystate) of the above dynamic system, i.e., kt+1 = kt = k andct+1 = ct = c :

1 = β�1� δ+ f 0

�k��

(16)

0 = �δk + f�k�� c (17)


(conti.) Note that the �rst equation is independent of c and caneasily determine k :

f 0�k�=1β� 1+ δ = ρ+ δ,

where ρ = 1β � 1 is called the time discount rate. Note that if

f (kt ) = Akαt ,

k =�Aα

ρ+ δ

�1/(1�α)

.

When k is determined,

c = f�k�� δk.


Qualitative Analysis (The Phase Diagram)

For this nonlinear system, we can use the phase diagram to analyzeits qualitative dynamics. More speci�cally, assume that the utilityfunction is ln ct , and rewrite the dynamic system as:

ct+1ct

= β�1� δ+ f 0 (kt+1)

�, (18)

kt+1 � kt = f (kt )� δkt � ct , (19)

which means

d�ct+1ct

�dkt+1

= βf 00 (kt+1) < 0 andd (kt+1 � kt )

dct= �1 < 0

and from which we can have two demarcation lines separating thespace. It turns out the intertemporal equilibrium is saddle-pointequilibrium. [Insert Figure here]



Alternatively, we can use the linearization method to approximate theoriginal nonlinear system around the intertemporal equilibrium andthen solve the resulting linear di¤erence equation system.Linearizing the above dynamic system around

�k, c�gives

ct+1 = c + β�1� δ+ Aαk

α�1�(ct � c) + (α� 1) βcAαk

α�2 �kt+1 � k

�,(20)

kt+1 = k � (ct � c) +�1� δ+ Aαk

α�1� �kt � k

�, (21)

where 1β = 1� δ+ Aαk

α�1and which can be written (denoteext+1 = xt+1 � x , x = k, c)"

1 � (α� 1) βcAαkα�2

0 1

#| {z }

J

� ect+1ekt+1�

| {z }u

+

� �1 01 � 1

β

�| {z }

M

� ectekt�

| {z }v

=

�00

�| {z }d

(22)


(conti.) Multiplying J�1 on both sides gives�1 00 1

�| {z }

I

� ect+1ekt+1�

| {z }u

+ J�1� �1 01 � 1

β

�| {z }

M

� ectekt�

| {z }v

=

�00

�| {z }d

Iu +Kv = 0 (23)

where we have substituted k =�Aα

ρ+δ

�1/(1�α)into the matrix M and

K = J�1M =

"1 (α� 1) βcAαk

α�2

0 1

# � �1 01 � 1

β

�

=

"�1+ (α� 1) βcAαk

α�2 � (α� 1) cAαkα�2

1 � 1β

#.


(conti.) The characteristic roots b can be obtained by solving theequation:

jbI +K j = 0 =)�� b� 1+ (α� 1) βcAαk

α�2 � (α� 1) cAαkα�2

1 b� 1β

�� = 0b2 �

h1+ (1� α) βcAαk

α�2+ (1+ ρ)

ib

+h1+ (1� α) βcAαk

α�2i(1+ ρ) + (α� 1) cAαk

α�2= 0 =)

trace = b1 + b2 = 1+ (1� α) βcAαkα�2

+1β> 2,

det = b1b2 =h1+ (1� α) βcAαk

α�2i 1β+ (α� 1) cAαk

α�2

=1β> 1


(conti.) Hence, the discriminant should be positive because

∆ = trace (JE )2 � 4 det (JE ) =

�1+ (1� α) βcAαk

α�2+1β

�2� 4 1

β

=

�1+ (1� α) βcAαk

α�2 � 1β

�2> 0

which means that both roots are real. Also, because det = 1β > 1 and

trace > 2, the two roots must individually be positive. We can alsojudge the magnitudes of the two roots as follows:

jbI +K j = 0() p (b) = (b� b1) (b� b2) = 0 =)p (1) = (1� b1) (1� b2) = 1� trace+ det

= � (1� α) βcAαkα�2

< 0

This can only be true if one root (say b1) is less than 1 and the otherroot is greater than 1. We can then conclude (and con�rm thepredictions of the PD) that the equilibrium is saddle-point.


Transitional Dynamics

After obtaining b1 < 1 and b2 > 1, we can have the general solutionfor this system: � ectekt

�=

�K1A1bt1 +K2A2b

t2

A1bt1 + A2bt2

�. (24)

where k1 and k2 are two constants determined by the roots (here weignore their detailed values). Given k0,ek0 = k0 � k = A1 + A2ec0 = c0 � c = K1A1 +K2A2Note that bt2 ! ∞ as t ! ∞ because b2 > 1. To guarantee aconvergent time path to the I.E. (i.e., to kill the explosive path), theendogenous ec0 need to be set in the right way and make A2 be 0:

c0 � c = K1�k0 � k

�. (25)

Hence, given any k0, we can �nd c0 s.t. the economy �jump� to thepair of stable branches and then move to the saddle point equilibrium.


Deriving the Transitional Dynamics (Optional)

Given the model parameters and steady state values, we candecompose the 2-by-2 matrix K as:

K = QΛQ�1, (26)

where Q has the eigenvectors of K as its columns, and Λ haseigenvectors of K down its diagonal:

Λ =

�b2 00 b1

�. (27)

The solution to system (24) can be written as:� ect+1ekt+1�= Q

�bt2 00 bt1

�Q�1

� ec0ek0�, (28)

where ek0 is given. Since b2 > 1, we have one explosive eigenvalueand the system will blow-up as t ! ∞ unless the endogenous variableec0 �jumps� in the right way so as to kill the explosive dynamics.


(Conti.) To see how this works, we can write the system as follows� ect+1ekt+1�

=1

det (Q)

�q11 q12q21 q22

� �bt2 00 bt1

� �q22 �q12�q21 q11

� � ec0ek0�,

=1

det (Q)

�q11bt2 q12bt1q21bt2 q22bt1

� �q22ec0 � q12ek0�q21ec0 + q11ek0

�,

=1

det (Q)

24 q11bt2 �q22ec0 � q12ek0�+ q12bt1 ��q21ec0 + q11ek0�q21bt2

�q22ec0 � q12ek0�+ q22bt1 ��q21ec0 + q11ek0�

35 ,where we assume that Q =

�q11 q12q21 q22

�.

Since bt2 ! ∞ as t ! ∞, we have to shut these explosive paths downby setting ec0 just right (this is our one degree of freedom, it is theonly endogenous variable not pinned down yet).


(Conti.) In other words, we choose consumption so that we are noton the explosive path that violates the transversality condition (TVC).Apparently, we should set

ec0 = q12q22ek0 (29)

to rule out the unstable dynamics.Hence our complete solution is� ect+1ekt+1

�=

1det (Q)

24 q12bt1 ��q21 q12q22 ek0 + q11ek0�q22bt1

��q21 q12q22 ek0 + q11ek0�

35 ,=

1det (Q)

24 q12 ��q21 q12q22 + q11�q22��q21 q12q22 + q11

� 35 bt1ek0. (30)

In practice we can �rst compute the matrix K , then compute itseigenvalues and eigenvectors to get the matrix Q and pick the stableeigenvalue so that we have a bounded solution.


A Special Case of the Optimal Growth Model

If we assume that the utility function is logarithmic (u (c) = ln c),the production function is Cobb-Douglas (f (k) = Akα), andcomplete depreciation (δ = 100%), we can solve the followingoptimal growth model explicitly without resort to linearization:

maxfct ,kt+1g

∞

∑t=0

βt ln ct (31)

subject to kt+1 = Akαt � ct .

We conjecture that the optimal solution takes the form: ct = ωAkαt

and then check whether we can �nd a constant ω that works.Substituting it into ct+1

ct= βαAkα�1

t+1 yields

ωAkαt+1

ωAkαt= βαAkα�1

t+1 ! kt+1 = αβAkαt .

Since kt+1 = Akαt � ct = (1�ω)Akα

t ,

βαA = (1�ω)A! ω = 1� αβ.


This case can also be solved explicitly using dynamic programming.The Bellman equation is

V (k) = maxeknln�Akα � ek�+ βV

�ek�o . (32)

First we guess that V (k) = E + F ln k. The FOC:

� 1

Akα � ek + βF1ek = 0! ek = βF

1+ βFAkα.

Substituting it into the Bellman equation:

E +F ln k = ln��1� βF

1+ βF

�Akα

�+ β

�E + F ln

�βF

1+ βFAkα

��.

(33)Matching the coe¢ cients:

F =α

1� αβand E =

lnA (1� αβ) +αβ1�αβ lnAαβ

1� β! ek = αβAkα.


Continuous-time Optimal Growth Model

The standard neoclassical production function is

Y = F (K , L) , (34)

which is a linearly homogeneous function with

YL > 0,YK > 0,YLL < 0,YKK < 0. (35)

We can rewrite the production function in per capita terms

y = f (k) where f 0 (k) > 0, f 00 (k) < 0, (36)

where y = Y /L and k = K/L.Assume that the utility function u (ct ) satis�es the usual conditions(just like those discussed in the discrete-time case).


(conti.) Then the equation of capital accumulation can be written as

K 0 = I � δK = Y � C � δK (37)

Denoting per person consumption as c = C/L, we have

K 0

L= y � c � δk. (38)

Suppose that the population growth rate is L0L = n. Since

k 0

k=K 0

K� L

0

L,

K 0

L=

KL

�k 0

k+ n�= k 0 + nk = y � c � δk =)

k 0 = f (k)� c � (n+ δ) k (39)


(conti.) If ρ the discount rate and the initial population is normalizedto L0 = 1, the objective function can be expressed as

V =Z ∞

0u (ct ) exp (�ρt) Ltdt =

Z ∞

0u (ct ) exp (�ρt) L0 exp (nt) dt

=Z ∞

0u (ct ) exp (� (ρ� n) t) dt =

Z ∞

0u (ct ) exp (�rt) dt

in which utility is weighted by a population that grows continuouslyat a rate of n. If r = ρ� n > 0, then the model is very similar to onewithout population weights but with a positive discount (as wediscuss in the discrete time case).

The optimal growth problem can now be stated as

maxZ ∞

0u (ct ) exp (�rt) dt (40)

s.t. k 0t = f (kt )� ct � (n+ δ) kt (41)

given k0.


Optimum Conditions

The Hamiltonian for this problem is

H = u (ct ) exp (�rt) + λt [f (kt )� ct � (n+ δ) kt ] (42)

From the maximum principle,

λt = u0 (ct ) exp (�rt) , (43)

k 0t =∂H∂λ

= f (kt )� ct � (n+ δ) kt , (44)

λ0t = �∂H∂λ

= �λt�f 0 (kt )� (n+ δ)

�. (45)

The TVC islimt!∞

exp (�rt) λtkt = 0. (46)


Construct a Phase Diagram

Note that

λt = u0 (ct ) exp (�rt) =) λ0t = u00 (ct ) exp (�rt) c 0t � ru0 (ct ) exp (�rt) .

(47)Substituting it into (45) gives the following Euler equation (aboutoptimal consumption path)

c 0t = �u0 (ct )u00 (ct )

�f 0 (kt )� (n+ δ+ r)

�, (48)

which combined with the resource constraint gives us a di¤erentialequation system that can characterize the economy completely.To construct the PD, we �rst need to draw two demarcation lines(c 0t = 0 and k

0t = 0):

f�k�� (n+ δ) k = c

f 0�k�� (n+ δ+ r) = 0


(Conti.)

∂k 0

∂c= �1 < 0

∂c 0

∂k= � u

0 (ct )u00 (ct )

f 00 (kt ) < 0

[Insert �gure here]

Hence, the intertemporal equilibrium (the steady state) is a saddlepoint. If we have an initial point that lies on one of the two stablebranches of the saddle point (given k0, pin down c0), the dynamics ofthe system will lead us to the IE. For any other point, the economywill eventually end up either with k = 0 (exhaustion of capital) orwith c = 0 (per capita consumption is 0) -both of which areeconomically unacceptable (i.e., contradicts the TVC).



For simplicity, assume that u (ct ) = ln ct and f (kt ) = kαt . The

original dynamic system can be rewritten as

c 0t =�f 0 (kt )� (n+ δ+ r)

�ct , (49)

k 0t = f (kt )� ct � (n+ δ) kt .

Approximating the nonlinear di¤erential equation system around�c , k�gives

c 0t = cf 00�k� �kt � k

�and k 0t = r

�kt � k

�� (ct � c) =)�

c 0

k 0

��0 cf 00

�k�

�1 r

�(c ,k)| {z }

JE

�ck

�=

�00

�


(conti.) The Jacobian matrix JE can determine the local stability ofthe equilibrium:

tr (JE ) = r1 + r2 = r > 0, (50)

det (JE ) = r1r2 = cf 00�k�< 0, (51)

where r1 and r2 are the characteristic roots with opposite signs (say,r1 < 0 and r2 > 0):

r1, r2 =r �

qr2 � 4cf 00

�k�

2.

Hence, the dynamic system yields a saddle point. Speci�cally, we canwrite the explicit solution as follows

kt = A1 exp (r1t)+A2 exp (r2t) and ct = k1A1 exp (r1t)+ k2A2 exp (r2t)

If initial conditions are such that A2 = 0 (that is, given k0, c0 need tobe pinned down such that A2 = 0), r2 > 0 will drop out of thediagram, leaving it to r1 < 0 to make the IE stable, i.e., the economyconverges to the IE along the pair of stable branches.


An Endowment Economy with Date-0 Trade

Consider an economy with only one in�nitely-lived consumer. Noproduction but the consumer is endowed with ωt units of the singleconsumption good at each date t.The absence of production and other storage technology means thatthe consumer is unable to move consumption goods across time: hemust consume all his endowment in each period or dispose of anybalance. Such an economy is called an exchange economy since theonly activity besides consumption is trading.Let the consumer�s utility from any given consumption path fctg∞

t=0be:

∞

∑t=0

βtu (ct ) . (52)

Since there is no other agent in the economy, market clearing requiresthat prices (pt ) for ct are such that the consumer is willing to haveexactly ωt at every t. We can normalize p0 = 1 such that the priceswill be relative to t = 0 goods.


Given these prices, the value of the consumer�s endowment is∞

∑t=0ptωt , (53)

the value of his expenditure is∞

∑t=0ptct (54)

and the budget constraint requires that∞

∑t=0ptct �

∞

∑t=0ptωt , (55)

which means that trading in all commodities takes place at the sametime: purchases and sales of consumption goods for every period arecarried out at t = 0.This market structure is called an Arrow-Debreu-McKenzie market, asopposed to a sequential market structure in which trading for eachperiod�s consumption good is undertaken in the corresponding period.


De�nitionA competitive equilibrium is a vector of prices fptg∞

t=0 and a vector ofquantities fc�t g∞

t=0 such that: (1)

fc�t g∞t=0 = arg max

fctg∞t=0

∞

∑t=0

βtu (ct ) (56)

s.t.∞

∑t=0ptct �

∞

∑t=0ptωt , (57)

ct � 0, 8t. (58)

(2) The market clearing condition:

c�t = ωt , 8t. (59)


In this model the agent consumes exactly his endowment in eachperiod, and this determines equilibrium prices. Note that herequantities are trivially determined but prices are not.To �nd the prices that support the quantities as a competitiveequilibrium, we use the FOC from the consumer�s optimizationproblem:

βtu0 (ωt ) = λpt , 8t, (60)

where we use the second equilibrium condition, c�t = ωt , and λ is theLagrange multiplier for the budget constraint.For any two consecutive periods, we have

ptpt+1

=1β

u0 (ωt )

u0 (ωt+1), (61)

which states the relative price of today�s consumption in terms oftomorrow�s consumption - the de�nition of the gross real interest rate- has to be equal to the marginal rate of substitution (MRS) betweenthese two goods that is inversely proportional to the discount factorand to the ratio of marginal utilities. We now have a completesolution for the competitive equilibrium for this economy (p0 = 1).


The Same Endowment Economy with Sequential Trade

Consider the same exchange economy but with a sequential marketsstructure. We allow 1-period loans and

Rt = 1+ rt (62)

denotes the interest rate on a loan between t and t + 1. Let at be thenet asset position of the agent at time t.We now allow the agent to transfer wealth from one period to thenext by lending 1-period loans to other agents. As there is only oneagent in the economy, there cannot be any outstanding loans.Therefore, the asset market will only clear if a�t = 0, 8t.With this market structure, the agent faces not a single but asequence of budget constraints:

ct + at+1 = R�t at +ωt , (63)

where R�t is the equilibrium interest rate that the agent takes as given.


De�nition

A competitive equilibrium is a set of sequences fc�t g∞t=0,

�a�t+1

∞t=0, and

fR�t g∞t=0 such that: (1)

fc�t , a�t+1g∞t=0 = arg max

fct ,at+1g∞t=0

∞

∑t=0

βtu (ct ) (64)

s.t.ct + at+1 = R�t at +ωt , (65)

ct � 0, 8t, (66)

limt!∞

at+1

t

∏i=0Ri

!�1= 0 (nPg condition), (67)

where a0 is given. (2) The asset market clearing condition:

a�t = 0, 8t. (68)

(3) the good market clearing condition:

c�t = ωt , 8t. (69)Luo, Y. (Economics HKU) ECON6012: Macro Theory September 25, 2019 39 / 49

Note that the third condition for the competitive equilibrium mustfollow from the �rst and second conditions due to the Walras�law: Ifn� 1 markets clear in each period, then the n-th one will clear as well.As in the date-0 trade economy, equilibrium quantities are trivial andprices, interest rates, are determined by the consumer�s Eulerequation evaluated at c�t = ωt :

u0 (ωt ) = βRt+1u0 (ωt+1) or Rt+1 =u0 (ωt )

βu0 (ωt+1). (70)


Optimal Growth vs. Competitive Equilibrium Growth

We have so far discussed optimal growth models with a centralplanner (or a representative agent), and we now move on to discussthe implications of competitive equilibrium growth with sequentialtrade.

The key di¤erence between the two models is that in the equilibriumgrowth model we need to model consumers/households�behavior and�rms�s behavior separately and obtain equilibrium prices at which allthe markets (labor, capital, and goods) will clear.

The markets can coordinate the decisions of households and �rms:

The goods market coordinates households�consumption decisions and�rms�output and investment decisions.The labor market coordinates �rms�demand for labor and households�supply of labor with the real wage rate equating labor demand andsupply.The capital market coordinates households�savings decisions and �rms�borrowing decisions through the real interest rate.


Firm�s Optimization Problem

Assume that there is a representative �rm in the economy and it usesan aggregate production function to produce goods. For simplicity,we use the same assumptions as before:

Y = F (K , L,A) (K , L,A > 0) (71)

where FK ,FL > 0, and FKK ,FLL < 0 (diminishing returns to capitaland labor). F is assumed to be constant return to scale in K and L:

Y = LF�KL, 1,A

�= Lf (k) or y = f (k) , (72)

where y = Y /L and f (k) = F (k, 1,A). (For simplicity, set A = 1.)F also satisfy the inada conditions:

limK!0

FK (K , L,A) = ∞, limL!0

FL (K , L,A) = ∞,

limK!∞

FK (K , L,A) = 0, limL!∞

FL (K , L,A) = 0,

F (0, L,A) = 0 for all A and L.


(conti.) For a given level of and given factor prices, we assume thatthe �rm maximizes the following pro�t function and factor markets(capital and labor markets) are clear:

maxKt�0,Lt�0

F (Kt , Lt ,At ) � RtKt � wtLt . (73)

Given the properties of F (�, �), the �rm should make zero pro�ts(otherwise, it would wish to hire arbitrarily large amounts of capitaland labor) and the factor prices should be equal to the marginalproducts

Rt = FK (Kt , Lt ,At ) , (74)

wt = FL (Kt , Lt ,At ) . (75)

It is straightforward to verify that given these factor prices, the �rmmakes zero pro�ts. Note that Rt and wt can also be written as:

Rt = f 0 (kt ) > 0, (76)

wt = f (kt )� kt f 0 (kt ) . (77)


Competitive Equilibrium

Suppose that the representative household can store capital andinvest (i.e., its wealth takes the form of capital), and rents capital andsupplies labor to the �rm. In addition, assume that it starts with anendowment of capital stock k0.

De�nitionA competitive equilibrium consists of paths of consumption, capital stock,wage and interest rates, fct , kt+1,wt ,Rtg∞

t=0, such that the representativehousehold maximizes its lifetime utility given k0 and factor pricesfwt ,Rtg∞

t=0, and the path of fwt ,Rtg∞t=0 is such that given the path of

fct , kt+1g∞t=0, all markets clear.


The household receives the competitive rental price:

Rt = f 0 (kt ) ,

which means that the gross rate of return for renting one unit ofcapital at time t in terms of date t + 1 goods:

1+ rt = f 0 (kt ) + (1� δ) . (78)

The optimization problem of the household is thus

maxfct ,at+1g

∞

∑t=0

βtu (ct ) ,

subject to the �ow budget constraint

at+1 = (1+ rt ) at � ct + wt , (79)

with a0 > 0, where at is asset holdings at t, and wt is the wageincome of the household (note that the labor supply is normalized to1.)


As before, we impose the non-Ponzi-game condition. Market clearingmeans that

at = kt (80)

because there is only one representative agent in the economy, noborrowing or lending happens and asset holdings (wealthaccumulation) will determine capital accumulation.

Of course, if you assume that there are many households in theeconomy, private debt must also be zero in the aggregate, and againwealth accumulation equals capital accumulation.


Using the same solution method we discussed before, we can obtainthe following Euler equation for the household�s problem:

β (1+ rt+1) u0(ct+1) = u0(ct ). (81)

The capital market clearing implies that:1+ rt+1 = f 0 (kt+1) + (1� δ). Combining (81) with this marketclearing condition, we obtain the same Euler equation we obtained inthe central planner growth model:

β�f 0 (kt+1) + (1� δ)

�u0(ct+1) = u0(ct ). (82)

In addition, substituting (77) and (78) into (79), we have

kt+1 =�f 0 (kt ) + (1� δ)

�kt � ct + f (kt )� kt f 0 (kt )

= (1� δ) kt � ct + f (kt ) ,

which is just the resource constraint we used to solve the centralplanner growth problem.


Remarks

In the above competitive equilibrium, when the household solves itsoptimization problem, we assume that it knows the whole sequence ofthe factor prices fwt ,Rtg∞

t=0. Here is the logic:

Consider an arbitary path of wages and rental rates. This sequence willlead the household to choose a path of consumption and wealthaccumulation.Given that wealth accumulation equals capital accumulation in theaggregate, the path of capital will in turn imply a path of wages andrental rates.The equilibrium paths of wages and rental rates are those paths thatreproduce themselves given optimal decisions by the �rm andhousehold.


Remarks (2)

This is called rational expectations (RE) or perfect foresightequilibrium. You may regard perfect foresight as a special case of RE,as RE can be applied to more general settings with uncertainty.

Note that the household makes forecasts of future factor prices andoptimizes based on those forecasts, and in equilibrium the forecastsare correct. The household cannot plan without knowing the wholepath of both wage and interest rates.

We will discuss more details about RE under uncertainty in the futurelectures.


topic 2: economic growth i - university of hong kongyluo/teaching/6012_2019/lecture2a.pdftopic 2:...

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