Advanced and Contemporary Topics in Macroeconomics I AG.pdf · Advanced and Contemporary Topics in...
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Advanced and Contemporary Topics in
Macroeconomics I
Alemayehu Geda
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
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
The Lucas-Uzawa model…cont’d
– 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
The Lucas-Uzawa model…cont’d
• 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
The Lucas-Uzawa model…cont’d
• 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
The Lucas-Uzawa model…cont’d
• 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
The Lucas-Uzawa model…cont’d
• 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(
The Lucas-Uzawa model…cont’d
• 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 Lucas (1988) model…cont’d
• 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 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)**.
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
• 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
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
• 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 Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
• 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
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
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)(
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
• 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 Lucas (1988) model…cont’d
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
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
Thus, given that gH = gC = g, from eqn[9] it
follows that
(11) ))(1(
)(1*
B
B
B
Bu
H
HH
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
• 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
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
• 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)
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
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*
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
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
:
,:
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
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
The Lucas (1988) model…cont’d
• 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
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
• 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 Lucas (1988) model…cont’d
• 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(
)(
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
• 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
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
• 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
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
• 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
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
• 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
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
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
The Lucas (1988) model…cont’d
The Lucas (1988) model…cont’d
• 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
The Lucas (1988) model…cont’d
• 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