Topic 3: Economic Growth II (OLG Models and …yluo/teaching/econ6012/2013topic3a.pdf2013/09/27 · Topic 3: Economic Growth II (OLG Models and Endogenous Growth Models) Yulei Luo

Topic 3: Economic Growth II (OLG Models andEndogenous Growth Models)

Yulei Luo

SEF of HKU

September 27, 2013

Luo, Y. (SEF of HKU) Macro Theory September 27, 2013 1 / 38

Basic Ideas

We have assumed that there exists a representative agent in theeconomy when discussing infinite-horizon growth models. However,this assumption is not appropriate in some situations:

In reality, we observe new households arrive in an economy over time,which introduces a range of new interactions between new and oldgenerations.Decisions by older generations will affect the prices faced by youngergenerations. We therefore need overlapping generations models (OLG)to better capture these observations.The OLG models are more suitable to address some insights about therole of national debt and social security in the economy.


The Baseline OLG Model

Time is discrete and runs to infinity. Each individual lives for twoperiods: young and old. The utility function for individuals born at t is

u (c1 (t)) + βu (c2 (t + 1)) , (1)

where u (·) satisfies the regular conditions. Factors markets arecompetitive. Individuals only work in the first period of their lives(i.e., when they are young) and supply one unit of labor inelastically,earning the equilibrium wage rate w (t).The production side of the economy is the same as before (here weset A = 1):

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

where L (t) increases as follows: L (t) = L (0) (1+ n)t . Forsimplicity, assume δ = 1 such that

1+ r (t) = R (t) = f ′ (k (t)) and w (t) = f (k (t))− f ′ (k (t)) k (t) ,(2)

where k = K/L.Luo, Y. (SEF of HKU) Macro Theory September 27, 2013 3 / 38

Consumption Decisions

Savings by an individual of generation t, s (t), are determined by:

max{c (t),c (t+1),s(t)}

u (c1 (t)) + βu (c2 (t + 1)) , (3)

s.t.c1 (t) + s (t) ≤ w (t) , (4)

c2 (t + 1) ≤ R (t + 1) s (t) , (5)

where we assume that young individuals rent their savings as capitalto final good producers at the end of t and receive the return att + 1. Since u′ (·) > 0, both constraints hold as equalities.Optimization means that

u′ (c1 (t)) = βR (t + 1) u′ (c2 (t + 1)) . (6)


(conti.) Combining the Euler equation with the constraints yields

s (t) = s (w (t) ,R (t + 1)) . (7)

Total savings in the economy is

S (t) = s (t) L (t) , (8)

where denotes the size of generation t, who are saving for t + 1.

δ = 1 means that

K (t + 1) = L (t) s (w (t) ,R (t + 1)) . (9)


Equilibrium

DefinitionA competitive equilibrium in the OLG economy can be represented bysequences: {c1 (t) , c2 (t) ,K (t) ,w (t) ,R (t)}∞

t=0, such that the factorprices are given by (2), individual consumption decisions are given by (6)and (7), and the aggregate capital stock evolves according to (9).

DefinitionThe steady state equilibrium is defined as an intertemporal equilibrium inwhich k = K/L is constant.


The fundamental law of motion of the OLG economy:

k (t + 1) =s (w (t) ,R (t + 1))

1+ n(10)

=s (f (k (t))− f ′ (k (t)) k (t) , f ′ (k (t + 1)))

1+ n. (11)

In the steady state,

k∗ =s (f (k∗)− f ′ (k∗) k∗, f ′ (k∗))

1+ n, (12)

which depends on the form of the saving function s (·).


Restrictions on Utility and Production Functions

Assume that

u (c (t)) =c (t)1−θ − 11− θ

and f (k) = kα, (13)

where θ > 0. The production function is f (k (t)) = k (t)α.The Euler equation for consumption is then

c2 (t + 1)c1 (t)

= [βR (t + 1)]1/θ , (14)

which means that the saving function is

s (t) =w (t)

ψ (t + 1), (15)

whereψ (t + 1) =

[1+ β−1/θR (t + 1)−(1−θ)/θ

]> 1, (16)

which ensures that positive consumption.


The Effects of Factor Prices on Saving

sw =∂s (t)∂w (t)

=1

ψ (t + 1)∈ (0, 1) , (17)

sR =∂s (t)

∂R (t + 1)=

(1− θ

θ

)[βR (t + 1)]−1/θ s (t)

ψ (t + 1),

which means that sR < 0 if θ > 1, sR > 0 if θ < 1, and sR = 0 ifθ = 1. The effects are determined by the interactions of income andsubstitution effects of a change in the interest rate.


The Canonical OLG Model

Here we consider a special case in which θ = 1, i.e., we have logutility. In this case

c2 (t + 1)c1 (t)

= βR (t + 1) , (18)

s (t) =β

1+ βw (t) , (19)

which means the capital accumulation equation should be

k (t + 1) =s (t)1+ n

=β

1+ β

w (t)1+ n

=β

1+ β

(1− α) k (t)α

1+ n, (20)

where we use the fact that w (t) = (1− α) k (t)α in the competitivefactor market.


(conti.) It is straightforward to show that there is a unique steadystate in which

k∗ =[

β (1− α)

(1+ n) (1+ β)

]1/(1−α)

. (21)

Just like the Solow model, this OLG model can lead to globally stablesteady state equilibrium. [Insert figure here.]


Capital Overaccumulation Problem in the OLG Model

We now can compare the competitive equilibrium of the OLGeconomy to the choice of a social planner who maximizes a weightedaverage of all generations’utilities:

∞

∑t=0

ξt [u (c1 (t)) + βu (c2 (t + 1))] , (22)

s.t.F (K (t) , L (t)) = K (t + 1)−K (t) + L (t) c1 (t) + L (t − 1) c2 (t) ,where ξt is the weight that the planner places on generation t

′sutility. Note that in per capita term:

f (k (t)) = (1+ n) k (t + 1)− k (t) + c1 (t) +c2 (t)1+ n

,

The Euler equation is thus

u′ (c1 (t)) = βf ′ (k (t + 1)) u′ (c2 (t + 1)) , (23)

which is the same as that obtained in the equilibrium OLG modelbecause R (t + 1) = f ′ (k (t + 1)).Luo, Y. (SEF of HKU) Macro Theory September 27, 2013 12 / 38

Capital Overaccumulation

However, the competitive equilibrium is not Pareto optimal. Why?

In the steady state of the OLG economy, we have

f (k∗)− nk∗ = c∗1 +c∗21+ n

= c∗,

which means that∂c∗

∂k∗= f ′ (k∗)− n, (24)

which determines the golden rule k:

f ′ (kgold ) = n. (25)

It is clear that if k∗ > kgold , then ∂c ∗∂k ∗ < 0, which means that reducing

savings can increase total consumption for everybody. If this is thecase, the economy is said to be dynamically ineffi cient (i.e., itoveraccumulates capital).


(conti.) Imagine that introducing a social planner into the OLGeconomy that is on its balanced growth path with k∗ > kgold .If the planner does not change k∗, the available consumption for perworker each period is just: f (k∗)− nk∗.Suppose now that in some period (t0), the planner allocates moreresources to consumption and fewer on savings so that capital perworker the next period is just kgold , and thereafter remains k at kgold .Under this plan, in t0, consumption per worker could be

f (k∗) + k∗ − (1+ n) kgold , (26)

which can be rewritten as f (k∗) + (k∗ − kgold )− nkgold and isgreater than f (kgold )− nkgold , and in each subsequent period,consumption per worker could be

f (kgold )− nkgold ,which is greater than f (k∗)− nk∗ by the definition of the Golden ruleof consumption.This policy can thus make more resources available for consumptionin every period and improve everyone’s welfare.Luo, Y. (SEF of HKU) Macro Theory September 27, 2013 14 / 38

An OLG Model with Infinitely-lived Agents

Weil (1989): All agents are infinitely-lived, but newborns from theirown households and do not receive any bequests. In the Weil model,agents of different vintages turn out to own different amounts ofcapital in equilibrium and thus effi ciency production requires trade infactor services. For simplicity, we assume log utility and notechnological progress.An individual born on date v (the individual’s vintage) maximizes:

Uvt =∞

∑s=t

βs−t ln (cvs ) , (27)

on t. Total population grows at n: Lt+1 = (1+ n) Lt . We normalizethe size of the initial vintage v = 0 (i.e., initial population) at L0 = 1.Each period, agents earn income from wages and from renting outcapital. The period budget constraint for a family of vintage v is:

kvt+1 = (1+ rt ) kvt + wt − cvt . (28)


Maximization yields:

cvt+1cvt

= β (1+ rt+1) . (29)

Next, we need to aggregate the capital accumulation and Eulerequations across different vintages to drive the dynamic equationsgoverning aggregate per capita capital and consumption:

kt+1 − kt =f (kt )− ct1+ n

− nkt1+ n

, (30)

ct+1 =(1+ f ′ (kt+1)

)[βct − n (1− β) kt+1] , (31)

where we use the facts that k t+1t+1 = 0, rtkt + wt = f (kt ) by Euler’stheoren, and

xt =x0t + nx

1t + n (1+ n) x

2t + · · ·+ n (1+ n)

t−1 x tt(1+ n)t

. (32)


(conti.) Note that to calculate aggregate consumption or capital percapita, we must sum them of all vantages born since t = 0. Vintagev = 0, born at t = 0, has L0 = 1 members. Total population ont = 1 is L1; of this population, L1 − L0 = (1+ n)− 1 = n are ofvintage v = 1. Similarly, vintage v = 2 containsL2 − L1 = (1+ n)2 − (1+ n) = n(1+ n).Therefore, for any vintage v > 0, the number of their members is:n(1+ n)v−1, which implies (32). Note that (32) is simply totalconsumption divided by the total population: Lt = (1+ n)

t .


Basic Ideas

The neoclassical growth models we have discussed attribute economicgrowth to exogenous technology progress, and they say nothing aboutthe factors that drive technological progress itself. The rate oftechnology progress is assumed to be beyond the control of a country— It just happens.

However, in reality, countries can do something to increase theirtechnology level.

New growth theories (or endogeoneous growth theories) extendneoclassical growth theory to incorporate market-driven innovationand therefore allow for endogeneously driven growth. The pioneersinclude Romer (1986, 1990), Lucas (1988), Rebelo (1990), andAghion and Howitt (1992).

We now consider three types of endogenous growth models: the AKmodel, the Romer externality model, and the human capital model.


The AK Model

In the Solow or RCK models, there are diminishing returns to scale incapital, holding effi ciency labor constant. Consequently, the economyeventually settles down to a steady state growth path in whichcapital-labor ratio is constant.The AK model assume that at the aggregate level, output is linear incapital:

Yt = AKt ,A > 0. (33)

where Kt is interpreted to mean all capital including human capital.Here technical progress can be embodied in new capital investment,thereby making new capital more productive than old capital. (e.g.,computers)The key assumption here is that there is no exogenous technicalprogress and there are constant returns to scale w.r.t. Kt .Output per capita should be

yt = Akt . (34)


Model Setting

Consider an economy in which the standard engines of neoclassicalgrowth are absent: there is no technological progress and thepopulation size is constant. The infinitely-lived representativeconsumer-manager with the standard isoelastic utility solves:

max{ct}

∞

∑t=0

βtc1−1/σt

1− 1/σ, (35)

where σ > 0 is the EIS. Given that the interest rate is rt+1, atoptimum the following Euler equation must hold:

β (1+ rt+1) =(ct+1ct

)1/σ

. (36)

Each worker manages his own firm, and the production technology isyt = Akt . In each period, firms invest up to the point where the netmarginal product of capital equals the interest rate

rt+1 = A.


Note that for any other interest rate, firms invest either an infiniteamount or 0. Finally, the model is closed by the following goodsmarket equilibrium condition

ct + it = yt = Akt , (37)

where it = kt+1 − (1− δ) kt is investment.

Equilibrium growth is determined by

ct+1ct

= [β (1+ A)]σ , 1+ g . (38)

It is clear that there will be long-run growth, i.e., g > 0 provided that[β (1+ A)]σ > 1.

Unlike the neoclassical growth models, here g is independent of theinitial level of capital stock, which means that all countries withdifferent starting points, can achieve persistent growth, and thegrowth rate only depends on model parameters: β, σ, A, and δ.


Some Remarks

Such a steady state is feasible provided capital stock and output growat the same constant growth rate, g . If k grows at g , we have

it = kt+1 − kt = gkt =gAyt

where for simplicity assume that δ = 0. Since ct + it = yt ,

ct =A− gA

yt

Note that the model can also be formulated in the following centralplanner economy:

kt+1 =(1+ A− δ︸︷︷︸

)A

kt − ct ,

given k0.


Some Remarks (2)

A key difference between this model with the neoclassical model isthat a change in the saving rate (e.g., an increase in β) now has apermanent effect on the growth rate.

A second difference is that the economy reaches its steady stategrowth path immediately; there is no transition period. The intuitionis that the linear production function ties down the interest rateindependently of the economy’s capital stock.

The problem with the AK model is that labor is not productive.However, in the data labor is a significant component of factor input.


Romer’s Externality Model

Romer (1986): There are externalities to capital accumulation, so thatindividual savers do not realize the full return on their investment.Each individual firm adopts the following production function:

F(K , L,K

)= AK αL1−αK

ρ, (39)

where K is individual firm’s capital stock and K is the aggregatecapital stock in the economy. We assume that ρ = 1− α such that acentral planner faces an AK model (for the central planner, K = K ).Note that if we assume that α+ ρ > 1, balanced growth path wouldnot be possible.

The rationale behind this specification is that the production processgenerates knowledge externalities. The higher the average level ofcapital intensity in the economy, the greater the incidence oftechnological spillovers that raise the marginal productivity of capitalthroughout the economy.


In the competitive equilibrium, the wage rate is determined by

wt = (1− α)AK αt L−αt K

1−αt . (40)

For simplicity, we now assume that leisure is not valued and normalizeLt = 1. Assume that there is a measure one of the firms, so that theequilibrium wage is

wt = (1− α)AK t . (41)

Similarly, the rental rate of capital is given by

Rt = αA. (42)


(conti.) Therefore, the Euler equation in the decentralized economy is

gCE =ct+1ct

= (βRt+1)1/γ = (βαA)1/γ , (43)

while the growth rate in the corresponding central planner economy is

gCP =ct+1ct

= (βA)1/γ > gCE , (44)

which is consistent with the fact that capital accumulation hasexternality.

This model overcomes the labor is irrelevant shortfall of the AKmodel. However, it is little evidence in support of a significantexternality to capital accumulation. Note that here ρ = 1− α = 2/3means significant externality (α = 1/3).


The Human Capital Model

In this model we model physical capital and human capital separately.Let h denote human capital per capita and k physical capital andsuppose that the production function per capita

y = Akαh1−α, α ∈ [0, 1] . (45)

No exogenous technical progress. Real resources are also needed forhuman capital accumulation:

∆kt+1 = ikt − δkt ,

∆ht+1 = iht − δht ,

where ikt and iht are the levels of investment in physical and human

capital, respectively. Assume no population growth.The national resource constraint satisfies

yt = ct + ikt + iht .

And the utility function is isoelastic: u(ct ) =(c1−γt − 1

)/ (1− γ).


Solving the Model

Set up the Lagrangian as follows

L =∞

∑t=0

βt

{c1−γt − 11− γ

+ λt

[Akα

t h1−αt − ct − (kt+1 + ht+1)+ (1− δ) (kt + ht )

]}.

(46)

The FOCs with respect to ct , kt+1, and ht+1 are

c−γt = λt , (47)

λt = β(αAkα−1

t+1 h1−αt+1 + 1− δ

)λt+1, (48)

λt = β((1− α)Akα

t+1h−αt+1 + 1− δ

)λt+1, (49)

respectively. Combining them, we have

β

(ct+1ct

)−γ[

αA(kt+1ht+1

)α−1+ 1− δ

]= 1,

kt+1ht+1

=α

1− α.


Economic Implications

Note that since kt+1ht+1

is a constant, the growth rates of the two typesof capital are the same. In equilibrium we have

kt+1ht+1

=kh, (50)

and the Euler equation is

β

(ct+1ct

)−γ[

αA(kh

)α−1+ 1− δ

]= 1. (51)

The rate of growth of consumption is thus

gc =ct+1ct− 1 =

{β

[αA(kh

)α−1+ 1− δ

]}1/γ

− 1. (52)

Assume that the growth is balanced,gc = gk

(= kt+1

kt− 1)= gh

(= ht+1

ht− 1).


(conti.) If we now substitute kt+1ht+1

= kh =

α1−α into the production

function, we obtain the AK model:

yt = Akαt h1−αt = A

(α

1− α

)α−1

︸︷︷︸A∗

kt . (53)

In the human capital model labor is treated more seriously.


A Model of Endogenous Innovation and Growth

Consider an alternative endogeneous growth model proposed byRomer (1990), in which invention is a purposeful economic activitythat requires real resources. By explicitly modeling the research anddevelopment (R&D) process, one can gain insights about the effectsof government policy on growth.Key assumption: Ideas are nonrival. That is, there are notechnological barriers preventing more than one firm fromsimultaneously using the same idea. Romer assumes that inventorscan obtain patent licenses on the blueprints for their innovations.Final goods production:

Yt = L1−αY ,t

At

∑j=1K αj ,t , (54)

where j ∈ {1, · · ·,At} indexes the different types of capital goods Kjthat can be used in production, and At captures the number of typesof capital that have been invented as of t.Luo, Y. (SEF of HKU) Macro Theory September 27, 2013 31 / 38

Note that in this specification, an increase in Kj has no effect on themarginal productivity of Ki , i 6= j . For simplicity, we assume thedepreciation rate of capital goods is 100%.

R&D production:At+1 − At = θAtLA,t , (55)

where θ is a productivity shift parameter and LA,t is the amount oflabor employed in R&D.

Assume there exists a third sector that intermediates between theR&D sector and the final goods production sector. Firms in the R&Dsell blueprints to an intermediate capital goods sector thatmanufactures the designs in t and then sells the machines to firmsthat in the final goods production sector in t + 1. That is, once anintermediate goods producer buys the blueprint to produce capitalgood j it becomes the monopoly supplier of that type of capital tothe final goods sector.


To solve the model, we guess that its equilibrium involves a constantinterest rate, constant relative prices, and a constant allocation oflabor across the two sectors. (We will confirm that this guess iscorrect.)

The demand for immediate capital goods by the final goods sector isdetermined by:

max{Kj}

L1−αY

At

∑j=1K αj −

At

∑j=1pjKj , (56)

where pj is the price of capital Kj in terms of final goods. Maximizingimplies

pj = αL1−αY K α−1

j . (57)


The intermediate goods producer sets Kj to

max{Kj}

11+ r

pjKj −Kj =1

1+ rpj(

αL1−αY K α−1

j

)−Kj (58)

where we assume that capital sold at t must be produced at t − 1and future sales must be discounted by 1+ r . Maximizing implies

Kj =(

α2

1+ r

)1/(1−α)

LY , K , (59)

for any j .Combining (57) with (59):

pj =1+ r

α, p. (60)

Given that the cost of producing the capital good is 1+ r (in terms ofthe final consumption good), this expression means that optimal priceis a constant markup over cost. This is just the usual formula for amonopolist facing a constant price elasticity of demand.


(conti.) The present value profit on capital produced in t − 1 for salein t:

Π =1

1+ rpK −K =

(1− α

α

)(α2

1+ r

)1/(1−α)

LY .

The next question is what a blueprint will sell for. Since there is afree entry into the intermediate goods sector, the value of a blueprintmust equal the entire discounted present value of the profit stream anintermediate goods producer will enjoy after purchasing it:

pA =∞

∑s=t

Π(1+ r)s−t

=(1+ r)Π

r. (61)


The final step is to find the equilibrium rate of growth. If LA isconstant over time, the growth rate of A is

g , At+1 − AtAt

= θLA,

which means that in steady state the number of capital good typesgrows at g , whereas the quantity of each types of capital goodremains constant at K .

To solve for the optimal allocation of labor across the two sectors, weequalize the marginal product of labor in the two sectors:

∂ (pAθALA)∂LA

= pAθA = (1− α) L−αY AK

α =∂Y∂LY

, (62)

where we use the fact that in the symmetric equilibrium,∑Aj=1 K

αj = AK

α. After using the expressions for p, K , and pA, wehave

LY =r

θα. (63)


(conti.) We then obtain a technology-determined relationshipbetween growth and the interest rate:

g = θL− rα.

So far we have only dealt with the supply side of the economy. Toclose the model, we model the demand side as usual:

max{cs}

∞

∑s=t

βs−tc1−1/σs − 11− 1/σ

, (64)

which means that

ct+1ct

= [β (1+ r)]σ , 1+ g or 1+ r = 1β(1+ g)1/σ . (65)

Because capital depreciates by 100%, the economy jumpsimmediately to a steady state in which K , Y , C , and A grow at thesame constant rate, which verify our guess on constant equilibriuminterest rate, relative price, and labor allocations.


(conti.) In the special case in which σ = 1,

r =α (1+ θL− β)

1+ αβ, (66)

g =αβθL− (1− β)

1+ αβ. (67)

Since negative growth is not possible here, we require that

θL >1− β

αβ, (68)

which means that if the initial size of the economy is too small forthis condition to be met, the profits from invention are insuffi cient topay for the labor costs, and there will be no innovation growth.


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Topic 3: Economic Growth II (OLG Models and …yluo/teaching/econ6012/2013topic3a.pdf2013/09/27 · Topic 3: Economic Growth II (OLG Models and Endogenous Growth Models) Yulei Luo