Advanced and Contemporary Topics in Macroeconomics I AG.pdf · Advanced and Contemporary Topics in...

Advanced and Contemporary Topics in

Macroeconomics I

Alemayehu Geda

[email protected]

PhD Program/2014

(Teaching Assistant Addis Yimer)

Chapter 3: Endogenous Growth Models/Part II Based on the materials by: Vahagn Jerbashian (2014) , David Romer

(2009/2012) & Aghion and Howitt (2009)

Class lecture Note on

Human capital accumulation:

The Uzawa-Lucas/Lucas (1988) model

mailto:[email protected]

The Lucas-Uzawa model

Inspired by Becker‟s (1964) theory of human capital and Uzawa‟s (1961) human capital based model two decades earlier, Lucas (1988) developed an endogenous growth model.

The model assums that there are two types of assets endogenously accumulated in the economy:

physical and human capital.

The idea is very simple: it says, in addition to producing, for instance, more infrastructure, we also "produce” better (or more) educated workers.

The better educated workers, then, produce more, while using the same amount of labor. Therefore, the labor productivity increases, and this, together with capital accumulation, may enable long run growth.

A. The Conceptual/Basic Lucas-Uzawa

General Model

The Lucas-Uzawa model…cont’d

• The biggest difference between Romer (1986) and Lucas‟ (1988) models is that the latter endogenizes the process of labor productivity growth through human capital accumulation, while the former thinks of spillover effects/ externalities (learning-by doing).

• Lucas (1988) assumed infinitely living individuals who chose at each date how to allocate their time b/n current production and skill acquisition (or schooling).

– The skill increase productivity in future periods.

• If h denotes the current human capital stock of the representative agent and u the fraction of her time currently allocated to production. Then the two basic equations of the Lucas model are give by

[1] )( 1 uHAky


– In Eq [1] “k” denotes physical capital stock which evolves

over time according the same differential equation as

SS/RCK model by

and

• In Eqn [2] the current schooling time (1-u) affects

the accumulation of human capital.

• NB, if “learning by doing” rather than education

were the source of human capital formation as in

Romer (1986), Eqn[2] would have been something

like

– Note also that Eqn[2] resembles “A” in SS/RCK model

[2] 0 )1( HuH

cyk

huh


• However, in contrast to “nonrival” technological knowledge/spillover, human capital doesn‟t necessarily involve externalities/spillover in Lucas-Uzawa model.

• Yet, the assumption that human capital accumulation involves constant return to scale in existing stock of human capital allows a positive growth in the steady state given by

• u* is the optimal allocation of individual‟s time b/n production and education. Thu, u* maximizes the individual‟s intertemporal utility:

– Subject to

*)1( ugH

H

dtec

U t

t

t

0

1

1

cyk


• u* can be shown to depend negatively on the rate of time preference (ρ) and the coefficient of relative risk aversion (σ)

– Thus giving comparative static properties as in the R&D model discussed before.

• The Lucas (Uzawa-Lucas) model is elegant but individual‟s return to education over time remains constant which is at odd with stylized facts/empirical facts about education.

• One way to handle this is to model it in the context of OLG where individuals inherit the human capital accumulated by their parents (see d‟Autume and Michel (1994) for a systemic analysis of the Lucas model in an OLG framework).

• Another interesting work in this genre is Nelson and Phelp‟s (1966) model where they modeled growth as generated by productivity-improving adaptions whose arrival rate would depend upon the stock of human capital:

y technolgoorldfrontier w theis A re whe)A)(( AHfA

B. The One Sector AK-version

The Uzawa-Lucas one Sector AK version of the model

• If presented in one sector form, the final good production side and the asset accumulation processes of Lucas (1988) model can be written as

• where H is the human capital input, IK and IH are the investments for physical and human capital accumulation, respectively.

• That is,

HK IICHAKY 1


• where δK is the depreciation rate of physical

capital and δH is the depreciation rate of human

capital.

• For the current exercise let δK = δH = δ. This

and,

HHI

KKI

HH

KK


• given that we consider an equilibrium where both assets are accumulated, the returns to both assets should be equal.

• Thus,

KAY

KH

H

Y

K

Y

H

Y

K

Y

11

)1(

)1(


• Thus, in terms of algebra, the ideas behind the Romer (1986) and Lucas (1988) [Uzawa-Lucas] models are quite similar.

– Both end up having an aggregate production function which is linear in capital (hence the name AK/aK, linear[A] in K[Capital]).

• This one sector model was a simple representation of Lucas (1988) [Uzawa-Lucas] model. The model with two sectors and corresponding assumptions is the following.

C. The Two Sector Lucas (1988) model –

Detailed Presentation Main assumptions

• This is a two-sector model of growth, where the physical capital is still produced with the same technology as the consumption good, but human capital is produced with a different technology.

• Human capital is the essential input for the production of new human capital.

– The motivation for this is that the human capital of one generation is an important factor in affecting the formation of human capital of the later generations. • If the production of human capital is within a household, that would be

the human capital "embodied" in the parents. If its production is through formal education, then that would be the human capital of the teachers with their methodologies.

– The accumulation (production) of human capital H follows a law of motion

– where HH is the human capital stock used for its own production. Every unit of human capital produces B > 0 new units of human capital. This stock depreciates at a rate δH > 0 (e.g., due to "aging").

HBHH HH

The Lucas (1988) model…cont’d

• Note: There are no diminishing returns to the production of human capital with this type of production function for the human capital.

• The non-decreasing returns to the production of human capital will be the engine of long-run growth in this model.

• The increasing stock of human capital drives the accumulation of physical capital and the economy grows indefinitely.

• If, instead, the production of human capital had decreasing returns to its input, this model would have the same predictions as the Solow-Swan model and it would not be able explain growth in the long-run.


• The production of final output combines physical capital stock and human capital HY , i.e.,

– where HY is the human capital used in production of final good.

Standard neoclassical assumptions apply.

• From the consumption-side, the representative HH chooses its consumption

path, the assets (physical and human capital) in the next period and the allocation of its human capital input between final good and human capital production, in order to maximize its lifetime utility

•

• subject to standard budget constraint and the law of motion of human capital, where u(C) is given by

1

YHAKY

0

)( dteCuU t

1

1)(

1CCu

The Lucas (1988) model…cont’d Market equilibrium

• Define the fraction of human capital used in the production of final output as: u =HY/H .

• There are no externalities involved in the input and output markets. – By the first welfare theorem it is known that the

competitive equilibrium will achieve the first-best allocations.

– The second welfare theorem implies that one can directly solve for the optimal allocations, as there are prices that will support the competitive equilibrium that achieves such intertemporal and intertemporal allocations.


The Lucas (1988) model…cont’d • The intertemporal allocation problem has two

controls, consumption (c) and allocation (u) of human capital in the two sectors of production that compete for it.

• There are two state variables, human and physical capital.

• Physical capital accumulation requires the (exogenous, as in SS/RCK, saving of output (less consumption), while the human capital accumulation requires investments in terms of real resources (human capital needs to be driven out of the production of human capital stock/say Education, H, at the rate B)**.



• Thus in this model the representative households

problem is **

• Let qK and qH be the shadow prices for the physical

and human capital, respectively.

.0)0(),0(

(2) )1(

(1) )(

toSubjected

1

1)(

1

0

1

,

givenHK

HHuBH

CKuHAKK

dteC

Cu

H

K

t

CuUMax



• This problem/the RA program, if written in terms of

current value Hamiltonian, is given by**

(6) ))1([)1( :

)(

(5) )( :

(4) ,)1( :

(3) , :

are rules Optimal The

11

1 11

,

HHKKH

KK

KKKK

HK

K

HHKK

Cu

uBqK

YqqqH

K

Yq

K

YqqqK

BHqu

Y qu

qCC

HHuBqCKuHAKqC

LCHMax



• The standard TVCs apply, one for each of the state variables,

i.e.,**

(8) )(

])1([

])1([)1(

)1(

thatfollows [6] and [4] Eqns From

(7) 1

path,n consumptio optimal thefollows [5] and [3] Eqns From

0)()(lim

0)()(lim

HH

HHHH

HHHHH

K

Bq

uBqBuqq

uBqH

Y

u

Y

BHqqq

K

Y

C

C

tetHt

Hq

t

tetKt

Kq

t



Balanced growth path

• From the optimal consumption path (7) and from the fact that

„the growth rate of consumption at steady-state should be

constant‟, it follows that the aggregate output Y and capital

stock K grow at the same rate, i.e., gK = gY .**

• From the resource constraint (or the law of motion of capital)

follows that in steady-state the consumption and capital grow

at the same rate, i.e., gC = gK = gY . ****

K

C

K

Y

K

C

K

uHAK

K

KKK

1)(



• From the production of human capital, given that B; δH = const and in steady-state = const; follows that the share of human capital in production of final good is constant, i.e.,

• From the production of final good follows that

• Given that gK = gY and A ,u = const the growth rates of physical and human capital are equal, i.e., gK = gH = gC = gY g: From [4] and given that gH = gY and u,α ,B = const follows that

1)(uHAKY

11)(

K

HuA

K

uHAK

K

Y

K

K

H

H

q

q

q

q

(9) .1

)1(

constB

gu

HHuBH

HH

K

H

H


The above result should not be a surprising result. Given that

both human and physical capital should be accumulated in

balanced growth path equilibrium, the rates of return on their

accumulation are equal.

This equality implies then that

),( , KHiq

q

i

i

(10) )(11

thatfollowsit (7)path optimal thefromTherefore,

)()(

HK

K

KKH

H

H

BK

Yg

q

q

K

YB

q

q



Thus, given that gH = gC = g, from eqn[9] it

follows that

(11) ))(1(

)(1*

B

B

B

Bu

H

HH



• In order to show that u* > 0, consider, for instance, the TVC

for human capital

0 *u

0 )()1(

)( )(1

0 )(1

)(

hold toTVC the

inorder )(1

and )( state-steadyin Given that

H

HH

HH

HHH

H

H

B

BB

BB

BgBq

q

0)()(lim

t

Ht

etHtq



• Meanwhile, from (10) follows that in steady-state**

KH

KH

KHH

KK

HH

B

B

K

Y

K

C

K

Y

Y

CY

B

K

Y

K

YC

B

K

Y

*

**

*

***

*

s*

is rate savings theTherefore

)1()(1

1

K

state-steadyin that follows capital ofmotion of law theFrom

(12)



Comparative statics

Increase in B increases g*. Ambiguous effects on s* and u*

(for (1/θ)≥ 1, s* increases and u* decreases)

Increase in θ (or ρ) decreases both g* and s*, while it

increases u*

Increase in α increases s* but has no effect on g* and u*



Transition dynamics

• In order to describe the transition dynamics of this model consider variables that do not grow in the steady-state

and u. χ is control like variable. Given the level of K it can change within period. In contrast, ω is a state like variable. It cannot change within a period.

ωH

K

χK

C

:

,:



Rewrite the model in terms of these variables. From the

definitions of χ and ω and (1) and (7) it follows that

)(

)(

,1

thatfollowsit (41) and (38), (37), from In turn,

(14) )()1()1(1

thatfollowsit (2) and (1) and and ofde.nition thefrom Further,

])1([1

(13) ][][1

)1(1

)1(1)1(1

HB

Hq

Hq

KK

Y

Kq

Kq

H

Y

Hq

Kq

Bu

KHuBAu

Hg

Kg

Au

AuAugg

K

KKKC


• Therefore,

. and , functionsunknown 3for solved becan that

equations aldifferenti 3 give (15) and (14), (13), Equations

(15) )(11

asrewritten becan it (14) Using

)1()1()()(

//

u

BuBu

u

u

uB

K

Y

ggq

q

q

q

u

u

HK

HK

HKKY

H

H

K

K



• It is further convenient to define and use

(19) )]([1

(18) ])1([1

(17) )1()]([)1(

equations aldifferenti of system

following thegivesit of instead using and Dropping

)1()]([1

)1(

is of rategrowth The

(16) )1(1

BuBu

u

Az

AzBz

z

z

AzB

u

u

z

z

z

uz

HK

K

HK

HK


• The first differential equation depends on z only. Integrating it

gives

A

B

b

atz

bza

zbe

bza

za

tz

HK

t

at

)(1)(lim

thatimplieswhich

'

)0(

)0(

)0(

)0(

)(



• In general one could attempt to solve also for χ(t) and u (t).

For the current exercise, however, it is sufficient to analyze

their dynamics. In order to do that notice that and

depend on z and χ only. Therefore, the dynamics of z and χ are

characterized by the Jacobian

• The determinant of J1 is negative. Therefore, we have saddle-

path stability. Along this path z (t) and χ(t) increase (or

decrease) to their steady-state values.

z

z

1

0)1(

1

A

J



• Next, note that . Therefore, in order to

have stability it has to be the case (i.e.,

parameters have to be such) that when χ is

greater/less than its steady-state value

• In such a circumstance, u and χ decline/increase to their

steady-state values.

For any of the variables using * to denote the steady-

state value of, in terms of phase diagram this

corresponds to (next slide)

0

u

u

u

0/0 u

u

u

u





• Returning to ω

*)(*)(

)]([1

])1([1

)]([*

*)(*)(

*)()]([1

])1([1

)]([1

:asrewritten becan (19) of rategrowth theusing(18)In turn .)(1

*

*)(

uuBzzAu

u

Therefore

BBBu

uuBzzA

uuBAzB

AzBuBu

u

uA

BzWhere

zzAu

u

HKKHK

HK

KHK

HK



• Now, if z (0) < z then z increases over-time to its steady-state

value. Assuming this implies from (18) that also

increases toward its steady-state value (i.e., ). Therefore, u

increases toward its steady-state value according to (19).

• This implies that and the system can be on stable path

only if ω(0)>ω* .

• If, however, z (0) > z* then and u > u*.Therefore,

and ω(0)<ω* .

0

0

0

0



In terms of the original variables in the model, we have

• This implies that gC increases/decreases together with z ; which is negatively correlated with ω . Therefore, if ω=K/H is higher/lower than its steady-state value then gC increases/decreases over time.

• If u is higher/lower than its steady-state value then gH declines/increases over time. The analysis of the behavior of gK is not so straight-forward, however.

*)()(*(1

)

)(1

uuBAzg

Azg

Azg

KH

KK

KC



• This analysis applies to the close neighborhood of

steady-state [i.e., u ≠ u* but u ε (0; 1)]. In case the

economy in terms of K/H ratio is very far away from

the steady-state then during some part of the transition

only one of the types of capital will be accumulated

Kaldor’s Stylized facts and the Model

• Assuming:

– Aggregate human capital is distributed uniformly, ie, H=hL.

– No population grwoth

– Then human capital can be taken as labour-augmenting and

hence

*h

h

H

Hg

)( 1 uhLAKY


• Thus we have

increases at the rate g:

• Thus, – Growth rate differs due to difference in technology and

preference parameters

– Initial conditions (human & physical capital) have permanent effect on the level of welfare • This leads to no convergence in levels of Y/L /////////END/////////

g rate at the increasesh H

Y-1

(uL)

Y wrate wageThe

constant is -Br rateinterest real The

constant is K

Y

g rate at the increases also L

K

H

1 hhL

KAu

L

Y

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