Topic 3: Economic Growth II (OLG Models and
Endogenous 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 the
economy when discussing in…nite-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 old
generations.
Decisions by older generations will a¤ect the prices faced by younger
generations. We therefore need overlapping generations models (OLG)
to better capture these observations.
The OLG models are more suitable to address some insights about the
role of national debt and social security in the economy.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
2 / 38
The Baseline OLG Model
Time is discrete and runs to in…nity. Each individual lives for two
periods: young and old. The utility function for individuals born at t is
u (c1 (t )) + βu (c2 (t + 1)) ,
(1)
where u ( ) satis…es the regular conditions. Factors markets are
competitive. Individuals only work in the …rst 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 we
set A = 1):
Y (t ) = F (K (t ) , L (t )) ,
where L (t ) increases as follows: L (t ) = L (0) (1 + n )t . For
simplicity, assume δ = 1 such that
1 + r (t ) = R (t ) = f 0 (k (t )) and w (t ) = f (k (t ))
f 0 (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
fc (t ),c (t +1 ),s (t )g
u (c1 (t )) + βu (c2 (t + 1)) ,
s.t.c1 (t ) + s (t )
c2 ( t + 1 )
w (t )
(3)
,
R (t + 1) s (t ) ,
(4)
(5)
where we assume that young individuals rent their savings as capital
to …nal good producers at the end of t and receive the return at
t + 1. Since u 0 ( ) > 0, both constraints hold as equalities.
Optimization means that
u 0 (c1 (t )) = βR (t + 1) u 0 (c2 (t + 1)) .
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
(6)
4 / 38
(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)) .
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
(9)
5 / 38
Equilibrium
De…nition
A competitive equilibrium in the OLG economy can be represented by
sequences: fc1 (t ) , c2 (t ) , K (t ) , w (t ) , R (t )gt∞=0 , such that the factor
prices are given by (2), individual consumption decisions are given by (6)
and (7), and the aggregate capital stock evolves according to (9).
De…nition
The steady state equilibrium is de…ned as an intertemporal equilibrium in
which k = K /L is constant.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
6 / 38
The fundamental law of motion of the OLG economy:
k (t + 1) =
=
s (w (t ) , R (t + 1))
(10)
1+n
s (f (k (t )) f 0 (k (t )) k (t ) , f 0 (k (t + 1)))
. (11)
1+n
In the steady state,
k =
s (f (k )
f 0 (k ) k , f 0 (k ))
,
1+n
(12)
which depends on the form of the saving function s ( ).
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
7 / 38
Restrictions on Utility and Production Functions
Assume that
u (c (t )) =
c (t )1 θ 1
and f (k ) = k α ,
1 θ
(13)
where θ > 0. The production function is f (k (t )) = k (t )α .
The Euler equation for consumption is then
c2 ( t + 1 )
= [ βR (t + 1)]1/θ ,
c1 ( t )
(14)
which means that the saving function is
s (t ) =
where
h
ψ (t + 1) = 1 + β
w (t )
,
ψ (t + 1)
1/θ
R (t + 1)
which ensures that positive consumption.
Luo, Y. (SEF of HKU)
Macro Theory
(15)
(1 θ )/θ
i
> 1,
September 27, 2013
(16)
8 / 38
The E¤ects of Factor Prices on Saving
sw
=
sR
=
∂s (t )
1
=
2 (0, 1) ,
∂w (t )
ψ (t + 1)
1 θ
∂s (t )
=
[ βR (t + 1)]
∂R (t + 1)
θ
(17)
1/θ
s (t )
,
ψ (t + 1)
which means that sR < 0 if θ > 1, sR > 0 if θ < 1, and sR = 0 if
θ = 1. The e¤ects are determined by the interactions of income and
substitution e¤ects of a change in the interest rate.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
9 / 38
The Canonical OLG Model
Here we consider a special case in which θ = 1, i.e., we have log
utility. In this case
c2 ( t + 1 )
c1 ( t )
= βR (t + 1) ,
(18)
β
w (t ) ,
1+β
(19)
s (t ) =
which means the capital accumulation equation should be
k (t + 1) =
s (t )
β w (t )
β (1 α ) k (t ) α
=
=
,
1+n
1+β1+n
1+β
1+n
where we use the fact that w (t ) = (1
factor market.
Luo, Y. (SEF of HKU)
Macro Theory
(20)
α) k (t )α in the competitive
September 27, 2013
10 / 38
(conti.) It is straightforward to show that there is a unique steady
state in which
1/(1 α)
β (1 α )
k =
.
(21)
(1 + n ) (1 + β )
Just like the Solow model, this OLG model can lead to globally stable
steady state equilibrium. [Insert …gure here.]
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
11 / 38
Capital Overaccumulation Problem in the OLG Model
We now can compare the competitive equilibrium of the OLG
economy to the choice of a social planner who maximizes a weighted
average of all generations’utilities:
∞
∑ ξ t [u (c1 (t )) + βu (c2 (t + 1))] ,
(
t =0
s.t.F (K (t ) , L (t )) = K (t + 1)
K ( t ) + L ( t ) c1 ( t ) + L ( t
1 ) c2 (
where ξ t is the weight that the planner places on generation t 0 s
utility. 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 0 (c1 (t )) = βf 0 (k (t + 1)) u 0 (c2 (t + 1)) ,
(23)
which is the same as that obtained in the equilibrium OLG model
because R (t + 1) = f 0 (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 = c1 +
c2
=c ,
1+n
which means that
∂c
= f 0 (k )
∂k
which determines the golden rule k:
n,
(24)
f 0 (kgold ) = n.
(25)
∂c
It is clear that if k > kgold , then ∂k
< 0, which means that reducing
savings can increase total consumption for everybody. If this is the
case, the economy is said to be dynamically ine¢ cient (i.e., it
overaccumulates capital).
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
13 / 38
(conti.) Imagine that introducing a social planner into the OLG
economy that is on its balanced growth path with k > kgold .
If the planner does not change k , the available consumption for per
worker each period is just: f (k ) nk .
Suppose now that in some period (t0 ), the planner allocates more
resources to consumption and fewer on savings so that capital per
worker 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 is
greater 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 de…nition of the Golden rule
of consumption.
This policy can thus make more resources available for consumption
in every period and improve everyone’s welfare.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
14 / 38
An OLG Model with In…nitely-lived Agents
Weil (1989): All agents are in…nitely-lived, but newborns from their
own households and do not receive any bequests. In the Weil model,
agents of di¤erent vintages turn out to own di¤erent amounts of
capital in equilibrium and thus e¢ ciency production requires trade in
factor services. For simplicity, we assume log utility and no
technological progress.
An individual born on date v (the individual’s vintage) maximizes:
Utv =
∞
∑ βs
t
ln (csv ) ,
(27)
s =t
on t. Total population grows at n: Lt +1 = (1 + n ) Lt . We normalize
the size of the initial vintage v = 0 (i.e., initial population) at L0 = 1.
Each period, agents earn income from wages and from renting out
capital. The period budget constraint for a family of vintage v is:
ktv+1 = (1 + rt ) ktv + wt
Luo, Y. (SEF of HKU)
Macro Theory
ctv .
(28)
September 27, 2013
15 / 38
Maximization yields:
ctv+1
= β (1 + rt +1 ) .
ctv
(29)
Next, we need to aggregate the capital accumulation and Euler
equations across di¤erent vintages to drive the dynamic equations
governing aggregate per capita capital and consumption:
kt +1
kt
=
ct + 1 =
f (kt ) ct
nkt
,
1+n
1+n
1 + f 0 (kt +1 ) [ βct n (1
(30)
β) kt +1 ] ,
(31)
+1
where we use the facts that ktt+
1 = 0, rt kt + wt = f (kt ) by Euler’s
theoren, and
xt =
xt0 + nxt1 + n (1 + n ) xt2 +
(1 + n )t
Luo, Y. (SEF of HKU)
Macro Theory
+ n (1 + n )t
1
xtt
.
September 27, 2013
(32)
16 / 38
(conti.) Note that to calculate aggregate consumption or capital per
capita, we must sum them of all vantages born since t = 0. Vintage
v = 0, born at t = 0, has L0 = 1 members. Total population on
t = 1 is L1 ; of this population, L1 L0 = (1 + n ) 1 = n are of
vintage v = 1. Similarly, vintage v = 2 contains
L2 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 total
consumption divided by the total population: Lt = (1 + n )t .
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
17 / 38
Basic Ideas
The neoclassical growth models we have discussed attribute economic
growth to exogenous technology progress, and they say nothing about
the factors that drive technological progress itself. The rate of
technology progress is assumed to be beyond the control of a country
– It just happens.
However, in reality, countries can do something to increase their
technology level.
New growth theories (or endogeoneous growth theories) extend
neoclassical growth theory to incorporate market-driven innovation
and therefore allow for endogeneously driven growth. The pioneers
include Romer (1986, 1990), Lucas (1988), Rebelo (1990), and
Aghion and Howitt (1992).
We now consider three types of endogenous growth models: the AK
model, the Romer externality model, and the human capital model.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
18 / 38
The AK Model
In the Solow or RCK models, there are diminishing returns to scale in
capital, holding e¢ ciency labor constant. Consequently, the economy
eventually settles down to a steady state growth path in which
capital-labor ratio is constant.
The AK model assume that at the aggregate level, output is linear in
capital:
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 technical
progress and there are constant returns to scale w.r.t. Kt .
Output per capita should be
yt = Akt .
Luo, Y. (SEF of HKU)
Macro Theory
(34)
September 27, 2013
19 / 38
Model Setting
Consider an economy in which the standard engines of neoclassical
growth are absent: there is no technological progress and the
population size is constant. The in…nitely-lived representative
consumer-manager with the standard isoelastic utility solves:
∞
max ∑ βt
fc t g t =0
ct1 1/σ
,
1 1/σ
(35)
where σ > 0 is the EIS. Given that the interest rate is rt +1 , at
optimum the following Euler equation must hold:
β (1 + rt +1 ) =
ct + 1
ct
1/σ
.
(36)
Each worker manages his own …rm, and the production technology is
yt = Akt . In each period, …rms invest up to the point where the net
marginal product of capital equals the interest rate
rt +1 = A.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
20 / 38
Note that for any other interest rate, …rms invest either an in…nite
amount or 0. Finally, the model is closed by the following goods
market equilibrium condition
ct + it = yt = Akt ,
(37)
where it = kt +1
(1 δ) kt is investment.
Equilibrium growth is determined by
ct + 1
= [ β (1 + A)]σ , 1 + g .
ct
(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 the
initial level of capital stock, which means that all countries with
di¤erent starting points, can achieve persistent growth, and the
growth rate only depends on model parameters: β, σ, A, and δ.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
21 / 38
Some Remarks
Such a steady state is feasible provided capital stock and output grow
at the same constant growth rate, g . If k grows at g , we have
it = kt +1
kt = gkt =
g
yt
A
where for simplicity assume that δ = 0. Since ct + it = yt ,
ct =
A
g
A
yt
Note that the model can also be formulated in the following central
planner economy:
kt +1 =
given k0 .
Luo, Y. (SEF of HKU)
1| + {z
A
e
A
Macro Theory
δ} kt
ct ,
September 27, 2013
22 / 38
Some Remarks (2)
A key di¤erence between this model with the neoclassical model is
that a change in the saving rate (e.g., an increase in β) now has a
permanent e¤ect on the growth rate.
A second di¤erence is that the economy reaches its steady state
growth path immediately; there is no transition period. The intuition
is that the linear production function ties down the interest rate
independently 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 signi…cant component of factor input.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
23 / 38
Romer’s Externality Model
Romer (1986): There are externalities to capital accumulation, so that
individual savers do not realize the full return on their investment.
Each individual …rm adopts the following production function:
F K , L, K = AK α L1
α
ρ
K ,
(39)
where K is individual …rm’s capital stock and K is the aggregate
capital stock in the economy. We assume that ρ = 1 α such that a
central planner faces an AK model (for the central planner, K = K ).
Note that if we assume that α + ρ > 1, balanced growth path would
not be possible.
The rationale behind this speci…cation is that the production process
generates knowledge externalities. The higher the average level of
capital intensity in the economy, the greater the incidence of
technological spillovers that raise the marginal productivity of capital
throughout the economy.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
24 / 38
In the competitive equilibrium, the wage rate is determined by
wt = (1
1 α
α) AKtα Lt α K t
.
(40)
For simplicity, we now assume that leisure is not valued and normalize
Lt = 1. Assume that there is a measure one of the …rms, so that the
equilibrium wage is
wt = (1 α) AK t .
(41)
Similarly, the rental rate of capital is given by
Rt = αA.
Luo, Y. (SEF of HKU)
Macro Theory
(42)
September 27, 2013
25 / 38
(conti.) Therefore, the Euler equation in the decentralized economy is
gCE =
ct + 1
= ( βRt +1 )1/γ = ( βαA)1/γ ,
ct
(43)
while the growth rate in the corresponding central planner economy is
gCP =
ct + 1
= ( βA)1/γ > gCE ,
ct
(44)
which is consistent with the fact that capital accumulation has
externality.
This model overcomes the labor is irrelevant shortfall of the AK
model. However, it is little evidence in support of a signi…cant
externality to capital accumulation. Note that here ρ = 1 α = 2/3
means signi…cant externality (α = 1/3).
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
26 / 38
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 and
suppose that the production function per capita
y = Ak α h1
α
, α 2 [0, 1] .
(45)
No exogenous technical progress. Real resources are also needed for
human capital accumulation:
∆kt +1 = itk
δkt ,
∆ht +1 =
δht ,
ith
where itk and ith are the levels of investment in physical and human
capital, respectively. Assume no population growth.
The national resource constraint satis…es
yt = ct + itk + ith .
1 γ
And the utility function is isoelastic: u (ct ) = ct
Luo, Y. (SEF of HKU)
Macro Theory
1 / (1
September 27, 2013
γ ).
27 / 38
Solving the Model
Set up the Lagrangian as follows
(
1 γ
∞
ct
1
Aktα ht1 α ct (kt +1 + ht +1 )
+ λt
L = ∑ βt
+ (1 δ) (kt + ht )
1 γ
t =0
)
.
(46)
The FOCs with respect to ct , kt +1 , and ht +1 are
ct
γ
λt
λt
= λt ,
= β αAktα+11 ht1+1α + 1 δ λt +1 ,
= β (1 α) Aktα+1 ht +α1 + 1 δ λt +1 ,
respectively. Combining them, we have
"
γ
ct + 1
kt +1 α 1
αA
+1
β
ct
ht + 1
Luo, Y. (SEF of HKU)
Macro Theory
#
δ = 1,
(47)
(48)
(49)
α
kt +1
=
.
ht + 1
1 α
September 27, 2013
28 / 38
Economic Implications
Note that since hktt ++11 is a constant, the growth rates of the two types
of capital are the same. In equilibrium we have
kt +1
k
= ,
ht + 1
h
(50)
and the Euler equation is
ct + 1
ct
γ
"
k
h
#
α 1
+1
δ = 1.
The rate of growth of consumption is thus
( "
ct + 1
k α 1
+1
gc =
1 = β αA
ct
h
#)1/γ
β
αA
Assume that the growth is balanced,
gc = gk = kkt +t 1 1 = gh = hht +t 1
Luo, Y. (SEF of HKU)
Macro Theory
δ
(51)
1.
(52)
1 .
September 27, 2013
29 / 38
(conti.) If we now substitute hktt ++11 =
function, we obtain the AK model:
yt = Aktα ht1
α
k
h
=
α
1 α
α
=A
1 α
{z
|
A
into the production
α 1
kt .
(53)
}
In the human capital model labor is treated more seriously.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
30 / 38
A Model of Endogenous Innovation and Growth
Consider an alternative endogeneous growth model proposed by
Romer (1990), in which invention is a purposeful economic activity
that requires real resources. By explicitly modeling the research and
development (R&D) process, one can gain insights about the e¤ects
of government policy on growth.
Key assumption: Ideas are nonrival. That is, there are no
technological barriers preventing more than one …rm from
simultaneously using the same idea. Romer assumes that inventors
can obtain patent licenses on the blueprints for their innovations.
Final goods production:
Yt = L1Y ,tα
At
∑ Kjα,t ,
(54)
j =1
where j 2 f1, , At g indexes the di¤erent types of capital goods Kj
that can be used in production, and At captures the number of types
of capital that have been invented as of t.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
31 / 38
Note that in this speci…cation, an increase in Kj has no e¤ect on the
marginal productivity of Ki , i 6= j. For simplicity, we assume the
depreciation rate of capital goods is 100%.
R&D production:
At + 1
At = θAt LA,t ,
(55)
where θ is a productivity shift parameter and LA,t is the amount of
labor employed in R&D.
Assume there exists a third sector that intermediates between the
R&D sector and the …nal goods production sector. Firms in the R&D
sell blueprints to an intermediate capital goods sector that
manufactures the designs in t and then sells the machines to …rms
that in the …nal goods production sector in t + 1. That is, once an
intermediate goods producer buys the blueprint to produce capital
good j it becomes the monopoly supplier of that type of capital to
the …nal goods sector.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
32 / 38
To solve the model, we guess that its equilibrium involves a constant
interest rate, constant relative prices, and a constant allocation of
labor across the two sectors. (We will con…rm that this guess is
correct.)
The demand for immediate capital goods by the …nal goods sector is
determined by:
max L1Y
fK j g
At
α
∑ Kjα
j =1
At
∑ pj Kj ,
(56)
j =1
where pj is the price of capital Kj in terms of …nal goods. Maximizing
implies
pj = αL1Y α Kjα 1 .
(57)
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
33 / 38
The intermediate goods producer sets Kj to
max
fK j g
1
pj Kj
1+r
Kj =
1
pj αL1Y α Kjα
1+r
1
Kj
(58)
where we assume that capital sold at t must be produced at t 1
and 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
, p.
α
(60)
Given that the cost of producing the capital good is 1 + r (in terms of
the …nal consumption good), this expression means that optimal price
is a constant markup over cost. This is just the usual formula for a
monopolist facing a constant price elasticity of demand.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
34 / 38
(conti.) The present value pro…t on capital produced in t
in t:
1
Π=
pK
1+r
K =
1
α2
1+r
α
α
1 for sale
1/(1 α)
LY .
The next question is what a blueprint will sell for. Since there is a
free entry into the intermediate goods sector, the value of a blueprint
must equal the entire discounted present value of the pro…t stream an
intermediate goods producer will enjoy after purchasing it:
∞
pA =
Π
∑ (1 + r )s
s =t
Luo, Y. (SEF of HKU)
Macro Theory
t
=
(1 + r ) Π
.
r
September 27, 2013
(61)
35 / 38
The …nal step is to …nd the equilibrium rate of growth. If LA is
constant over time, the growth rate of A is
g,
At + 1 At
= θLA ,
At
which means that in steady state the number of capital good types
grows at g , whereas the quantity of each types of capital good
remains constant at K .
To solve for the optimal allocation of labor across the two sectors, we
equalize the marginal product of labor in the two sectors:
∂ (pA θALA )
= pA θA = (1
∂LA
α) LY α AK α =
∂Y
,
∂LY
(62)
where we use the fact that in the symmetric equilibrium,
∑Aj=1 Kjα = AK α . After using the expressions for p, K , and pA , we
have
r
LY =
.
(63)
θα
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
36 / 38
(conti.) We then obtain a technology-determined relationship
between growth and the interest rate:
r
.
α
g = θL
So far we have only dealt with the supply side of the economy. To
close the model, we model the demand side as usual:
∞
max ∑ βs
1 1/σ
t cs
fc s g s =t
1
1/σ
1
,
(64)
which means that
ct + 1
1
= [ β (1 + r )]σ , 1 + g or 1 + r = (1 + g )1/σ .
ct
β
(65)
Because capital depreciates by 100%, the economy jumps
immediately to a steady state in which K , Y , C , and A grow at the
same constant rate, which verify our guess on constant equilibrium
interest rate, relative price, and labor allocations.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
37 / 38
(conti.) In the special case in which σ = 1,
r
=
g
=
α (1 + θL β)
,
1 + αβ
αβθL (1 β)
.
1 + αβ
(66)
(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 for
this condition to be met, the pro…ts from invention are insu¢ cient to
pay for the labor costs, and there will be no innovation growth.
Luo, Y. (SEF of HKU)
Macro Theory
September 27, 2013
38 / 38