alemayehu geda 1 final... · • lecture 7: investment • lecture 8: ... alemayehu geda (2008) ......
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Advanced and Contemporary Topics in
Macroeconomics I
Alemayehu Geda Email: [email protected]
Web Page: www.alemayehu.com
Lecture 1
Introduciton & The Solow Swan/Neoclassical
Model
Addis Ababa University
Departement of Economics
PhD Program
2014
Course Content/Outline 2014 • Lecture 0*: Review of Advance [Dynamics]
Mathematics [Optional]
• Lecture 1: The Solow-Swan/Neoclassical
Grwoth Model [Exogenous Saving]
• Lecture 2: The Ramsey-Cass-Koopmans
Growth Model [Endogenous Saving]
• Lecture 3: The Diamond/ OLG Model
• Lecture 4: The Endogenous Growth Models
Course Content….Cont’d
• Lecture 5: Real Business Cycle Models
• Lecture 6: The Political Economy of Grwoth
(in Ethiopia and Africa)
• Micro-Foundation
• Lecture 6: Consumption
• Lecture 7: Investment
• Lecture 8: The Labour Market
• Lecture 9*: Dynamic General Macroeconomic
Equilibrium Models & DSGE [Optional]
Literature For the Course
Main Readings/Books
Romer, David (2012). Advanced Macroeconomics.4th Edition
Barro, Robert Jr. and Xavier Sala-i-Martin (2004). Economic Growth.
Valdes, Benigno(1999).Economic Growth: Theory ,Empirics and Policy
Acemoglu, Daron (2008).Introduction to Economic Growth.
Nudulu et al (2008). The Political Economy of Grwoth in Africa, 2 volumes
(AERC and Cambridge University Press.
Alemayehu Geda (2008) The Political Economy of Grwoth in Ethiopia (In
the same book above/Nudulu et al, Vol 1, Ch 4)
• Chiang, Alpha C (1992). Elements of Dynamic Optimization. New York,
McGraw-Hill/Or
• Sydsaeter et al (2005). Further Mathematics for Econ Analysis (PTO)
• Taylor, Lance (2004). Reconstructing Macroeconomics. Princeton: Princeton
University Press.
• Weeks, John (2013). False Paradigm: The Irrelevance of Neoclassical
Macroeconomics. London: Edward Edgar
. .
Relevant Articles
• Solow, Robert M. (1956). “A Contribution to the Theory of Economic
Growth”, Quarterly Journal of Economics, 70:65-94.
• Baumol, William (1986). “Productivity Growth, Convergence, and
Welfare," American Economic Review, 76:1072-85
• DeLong, J. Bradford (1988). “Productivity Growth, Convergence, and
Welfare: Comment," American Economic Review, 78:1138-54.
• Mankiw, Gregory N., David Romer, and David N. Weil (1992). “A
Contribution to the Empirics of Economic Growth," Quarterly Journal
of Economics, 107:407-37..
Klenow, Peter J. and Andres Rodriguez-Clare (1997). “The
Neoclassical Revival in Growth Economics: Has It Gone Too Far?"
NBER Macroeconomics Annual, 12:73-103.
Hall, Robert E. and Charles I. Jones (1999). “Why Do Some Countries
Produce So Much More Output per Worker than Others," Quarterly
Journal of Economics,114:83-116
.
. Phelps, Edmund S., “The Golden Rule of Accumulation: A Fable for Growth
men,” American Economic Review, September 1961, pp. 638-643
Oded Galor, Unified Growth Theory (Princeton, NJ: Princeton University Press,
2011).
Romer (1986) „Increasing Returns and Long run Growth‟ Journal of Political
Economy, 94:1002-1037.
Romer (1990) „Endogenous Technological Change’, Journal of Political
Economy, 98: 71-102.
Frankel (1962)
Domar, Evsey D. (1946), “Capital Expansion, Rate of Growth and Employment”,
Econometrica 14: 137-147.
Harrod, Roy (1939),“An Essay in Dynamic Theory”,Economic Journal 49: 14-33.
Kaldor, Nicholas (1963), “Capital Accumulation and Economic Growth”, In
Proceedings of a Conference Held by the International Economics Association,
Friedrich A. Lutz and Douglas C. Hague (editors). London: Macmillan.
Swan, Trevor W. (1956), “Economic Growth and Capital Accumulation”,
Economic Record 32: 334-361.
.
Class schedule:
Lucas, Robert E. Jr., “Why Doesn‟t Capital Flow from Rich to Poor
Countries?” American Economic Review, May 1990, pp. 92-96.
Uzawa, Hirofumi (1961), “Neutral Inventions and the Stability of the
Growth Equilibrium!”, Review of Economic Studies 28: 117-124
William A. Brock and M. Scott Taylor, “The Green Solow Model,”
Journal of Economic Growth, June 2010, pp. 125-153.
Aghion, P and P. Howitt (1992) „A Model of Growth through Creative
Destruction‟ Econometrica, 60: 323-351.
– To be worked out/ Flexible Modular
– 3 to 6 hours lecture per week
Additional articles are given on the Course CD that Contains the
lecture slides
Course Assesement Course Assessment
A) Each Candidate will write a review of literature in 4/5 topical areas of the work
based on 5 to 10 latest working papers/and Presentation. (20%)
B) There will be bi-weekly/Weekly drill on each of the lectures (to be handled by a
teaching assistant) (25%)
C) A Group paper on African Business Cycle (A group of 2 in each team) (20%)
D) Class Presentation and participation (10%)
E) Individual term paper on grwoth in Ethiopia or Africa on the following topics, 20
pages in a model of AER, Journal of African Economies etc (30%)
A) A Solow-Swan Model for Ethiopia OR
B) Human Capital Augmented Solow-Swan Model for Ethiopia
C) Ramesy-Cass-Koopman Model for Ethiopia
D) Endogenous Grwoth Model for Ethiopia
E) The Political Economy of Grwoth in Ethiopia: Political, Institutions and
Grwoth
F) A Dynamic Stochastic General Equilibrium Model for Ethiopia/Kenya
G) Modelling Investment (or Public Private Investment Interaction) in Ethiopia
H) Modelling Consumption/Saving in Ethiopia
.
Chapter One
The Solow –Swan Model
[The Neoclassical Model]
Lecture One contents I. Introduction and Review of Advanced Mathematics
II. The Solow Model
Assumptions
Inputs and outputs
Production function
Evolution of inputs into production
Solution
III. The Dynamics of the Model
The Dynamics of k
The Balanced Growth Path
IV. Comparative Dynamics: Impact of a Change in Savings
Rate
The Impact on Output
The Impact On Consumption
. V. Quantitative Implication
Steady State: Quantitative Importance of Savings Rate in Affecting
Income Per Capita in the Long Run
Transition Dynamics: The speed of Convergence
VI. The Central Questions of Growth in the Solow Model
VII. Empirical Applications
Growth Accounting
Convergence
VIII. Conclusion
.
General Introduction about
Growth: Some basic facts about
economic growth
Method: Sherlock Holems’
Method!
• Sherlock Holems‟ 4 Rules of Scientific inquiry
Rule 1: Begin with the Data
Rule 2: Build a theory (or thoeries) capable of covering the facts that are know to u
Rule 3: Do not take for grants that your theory is correct because it cover all the facts (as new facts may come and other theories may cover that too)
Rule 4: If new evidence can not be accommodated with exiting theories reconsider your theories.
{See Valdes, Page 8-10)
Kaldor’s “Stylized Facts” about
Growth • In 1961 article he outlined empirical regularities: • “Capital Accumulation and Economic Growth” in F.A. Lutz and D.C. Hague
(eds). The Theory of Capital. New York: St Martin Press.
• SF1: Standard of living always increase from one generation to the next – Per capita income and labour productivity (y=Y/L) is increasing
(y grows at positive rate )
• SF2: The capital output ratio has no upward or downward trend (K/Y -no change in the long run )
• SF3: The functional distribution of income remains constant in the long run Π/K=profit, constant and hence Wage share is total income less profit share
• SF4: There are a variety of growth rates of percaptia income across the world
Kaldor’s “Stylized Facts”
• SFacts (1) and (2) imply the following:
• SFacts (2) and (3) together imply the following:
oo
oo
oo
ykL
Y
L
K
L
Y
Y
K
Y
K
Y
K
0
0/constant
/Y)( -1 W/Ydefinationby 1)W/Y/ (hence YWGiven
constant. be alsomust / capital physicl of share income
theS3by constant also is / and S2by constant is K/Y since
; ..
Y
Y
K
Y
K
YK
Y
YK
Some basic facts about economic growth
Standards of living in industrialized economies have
increased dramatically over the centuries
In the U.S and Western Europe real incomes is 10 – 30 times larger
than 100 years ago and 50 – 300 times larger than 200 years ago
Worldwide growth has not been constant
Average growth rates in industrialized countries were higher in the
twentieth century than in the nineteenth century, still growth rates
were higher in the nineteenth century than in the eighteenth
century…
Productivity growth has slowed down
In many industrialized countries annual growth in output per
person has slowed down since the 1970s
Standards of living vary enormously across countries
Real income is more than 20 times higher in the U.S than in
Bangladesh
Large variations in growth rates
Growth miracles, e.g. Japan, NICs
Growth disasters, e.g. Sub-Saharan Countries, Argentina
Over the modern era, cross-country income differences has
widened – enormous differences in human welfare across
different part of the world.
Some basic facts about economic growth:
(cont..)
An important quote fro Lucas(1988):
– When you think of the implication of a
solution to grwoth problem for mankind…..
“Once one start to think about economic
growth, it is hard to think about anything
else.”
(Robert E. Lucas, 1988)
.
Chapter One
The Solow –Swan Model
I. Introduction
As the first step, in order understand the role of
proximate causes (see nxt Slide) of economic
growth we develop a simple framework. We take
the Solow-Swan model as our starting point.
The model is named after Robert Solow and
Trevor Swan, who published two seminal papers
in the same year (1956).
Introduction.......cont’d
Robert Solow developed many applications of the
model, and was later awarded the Nobel prize in
economics.
This model has not only become the centerpiece of
growth theory but has also shaped the modern
macro theory.
Introduction.......cont’d
The central model of macroeconomics before the
Solow model came along was the Harrod-Domar
model, which was named after Roy Harrod and
Evsey Domar (Harrod (1939) and Domar (1946)).
The Harrod-Domar model focused on
unemployment and growth.
The distinguishing feature of the Solow model is
the neoclassical aggregate production function.
II. The model
The Solow model focuses on four variables: –Output (Y)
–Capital (K)
–Labour (L)
–Effectiveness of labour (A)
The Production function takes the form:
where t denotes time, which enters indirectly into the production function through K, L and A.
)1.1.(.........., tLtAtKFtY
Some properties of the production function:
Output changes over time only if input changes over
time
The amount of output obtained from given quantities
of capital and labour rises over time only if there are
technological progress, i.e. the effectiveness of labour
increases over time.
A and L enter multiplicatively into the model such that
the term AL is referred to as „effective labour‟ meaning
that technological progress is labour-augmenting
(Harrod-neutral).
Some critical assumptions regarding the production
function:
CRS – doubling input doubles output (A held constant):
• Implicitly assumes that the economy is sufficiently large that
any gains from specialization has been exhausted
• Implicitly assumes that other production factors (e.g. land) is
relatively unimportant
)2.1.........(..........0,, cALKcFcALcKF
With CRS, an intensive form of the production
function is easily specified:
Set
Hence (1.3) can be rewritten in intensive form production
function as:
ALc /1
)3.1........(..........,1
1,AL
YALKF
ALAL
KF
AL
KkWhere
kfyAL
KFALKF
ALAL
Y
)4.1......().........()1,(),(1
The intensive form production function is assumed
to satisfy the following conditions:
0)('lim
)('lim
0)(''
0)('
00
0
kf
kf
kf
kf
f
k
k
f(k)
k
. Since it follows that the
marginal product of capital,
{see equation 1.3}
Thus the assumptions that is positive and
is negative imply that the marginal product of
capital is positive ,but that it declines as capital
(per unit of effective labor) rises.
)(),(AL
KALfALKF
).()1
)((),(
kfALAL
KfAL
K
ALKF
)(kf )(kf
. These conditions (which are stronger than needed
for the model‟s central results) states that the
marginal product of capital is very large when the
capital stock is sufficiently small and that it
becomes very small as the capital stock becomes
larger;
their role is to ensure that the path of the economy
does not diverge.
Example:
Cobb-Douglas production function:
First order condition:
Second order condition:
kkfy
kAL
K
AL
ALK
AL
Y
ALKY
)(
)(
)5.1 ...(........................................ 10,)(
1
1
0)1(
1)(''
2
2
k
kkf
0)()( 1
kkfk
k
k
kf
The Evolution of the Inputs into Production
The remaining assumptions of the model concern how
the stock of labour, knowledge, and capital change over
time.
The model is set in continuous time(i.e., the variables
are defined every point in time)
Labour and knowledge (technology) is assumed to
grow at a constant rate over time:
where n and g are exogenously given constant growth
rates and a dot over a variable denotes a derivative w.r.t
time
)9.1(..............................).........(
)8.1.(..............................).........(
tAgA
tLnL
)/)()(( dttdLtL
. The growth rate of a variable refers to its
proportional rate of change .(i.e., the phrase the
growth rate of X refers to the quantity .
Thus equation (1.8) implies that the growth rate of
L is constant and equal n , and equation (1.9)
implies that A‟s growth rate is constant and equal
to g .
)(
)(
tX
tX
. A key fact about growth rates is that the growth
rate of a variable equals the rate of change of its
natural log. i.e.,
To see this: since lnX is a function of X and X is a
function of t, using the chain rule to write:
.)(ln
)(
)(
dt
tXd
tX
tX
)10.1.(....................).........()(
1)(
)(
)(ln)(lntX
tXdt
tdX
tdX
tXd
dt
tXd
. Applying the result that a variable‟s growth rate
equals the rate of change of its log to (1.8) and
(1.9) tells us that the rate of change of the logs of
L and A are constant and that they equal n and g,
respectively. Thus,
)12.1.........(....................,.........)]0([ln)(ln
)11.1.........(....................),........)]0([ln)(ln
gtAtA
ntLtL
. • Where L(0) and A(0) are the values of L and A at
time 0. Exponentiating (or taking the ant-log) both
sides of these equations gives us
Thus our assumption is that L and A each grow
exponentially.
)14.1.(..................................................)0()(
)13.1.(..................................................)0()(
gt
nt
eAtA
eLtL
. Coming to our model: Output is used to either
consumption or investment (saving).
• The saving rate (s) is assumed to be constant and
exogenously given.
– For simplicity one can assume that one unit of
investment is equal to 1 unit of new capital.
• Existing capital depreciates at a rate δ.
These assumptions imply that the capital stock
grows according to:
)15.1..(..........).........()()( tKtsYtK
. Although no restrictions are placed
,their sum is assumed to be positive .i.e.,
This completes the model.
The Solow model is grossly simplified in a
number of ways.
There is only one good
No government
Fluctuation in employment are ignored
, andgn
0 gn
. Production is described an aggregate production
function with just three inputs
The rate of saving, deprecation, population growth
and technological progress are constant
Etc….
The model omits many obvious features of the real
world which are important for growth
But the purpose of a model is not to be realistic
(NB Positivist Method!). After all, we already
possess a model that is completely realistic-the
world itself (see my 2nd Trade book Ch4
Appendix)
III. The Dynamics of the Model
Here we want to determine the behavior of the
economy we have just described in the previous
slides.
The evolution of two of the three inputs into
production, labor and knowledge, is exogenous.
Thus to characterize the behavior of the economy,
we must analyze the behavior of the third input,
Capital.
a) The Dynamics of k
Because the economy may grow over time, it is much easier
to focus on the capital stock per unit of effective labour, k,
than on the unadjusted capital stock K.
Since k(t)=K(t)/A(t)L(t), i.e. a function of K, L and A, which
are all functions of t, the chain rule applies and we can find
the intensive form of the capital growth equation from
.
kAL
Kwhere
tA
tA
tLtA
tK
tL
tL
tLtA
tK
tLtA
tK
tLtAtLtAtLtA
tK
tLtA
tK
dtdKtKtAtA
tktL
tL
tktK
tK
tktk
t
tLtA
tK
tk
simply is
)(
)(
)()(
)(
)(
)(
)()(
)(
)()(
)(
(1.16)
)()()()([))()((
)(
)()(
)(
etc...../)( NB )()(
)()(
)(
)()(
)(
)()(
)(
)()(
)(
)(
2
Since from (1.8) and (1.9) ,
respectively
Substituting these facts into (1.16) &using (1.15)
yields:
ntL
tL
)(
)(g
tA
tA
)(
)(
)18.1...(........................................).........()(
))(()()(
)(fact that the UsingAnd
)()()()()(
)(
(1.17)
)()()()(
)()()()(
)()(
)()(
tkgntksf
tkftLtA
tY
tgktnktktLtA
tYs
gtkntktLtA
tKtsYgtkntk
tLtA
tKtk
Break-even investment Total investment
Equation (1.18) is the key equation in the Solow model.
It states that, the rate of change in the capital stock per unit of
effective labour is the difference between total investment
per effective labour unit and the amount of investment
needed to keep the capital-to-effective labour-ratio constant.
The first term, sf(k), is actual investment per unit of effective
labor
The second term, (n + g + )k is break-even investment, i.e.
the amount of investment that must be done to keep k at its
existing level. Investment is needed to prevent k from falling
because the existing capital deprecates at rate , which is
captured by k term, and also because the quantity of effective
labor is also growing at rate (n+g), which is captured by the
term (n+g)k.
.
Therefore, when the actual investment per unit of
effective labor exceeds the break-even investment, k
rises, and vice-versa.
Moreover, when the two are equal, k is constant.
This is depicted in figure 2 below.
• The figure plots the two terms of the right-hand side of
the fundamental law of motion as functions of k. Since
F(0,L,A) = 0 it implies that f(0) = 0, and therefore
actual investment and break-even investment are equal
at k = 0.
. Furthermore, the Inada conditions imply that as k
goes to zero f’(k) becomes very large. Therefore
sf(k) is steeper than break even investment line
around k = 0; and actual investment is larger than
break-even investment.
• The Inada conditions also imply that f’(k) falls to
zero as k becomes very large. As a result, at some
point the slope of he actual investment curve
becomes less than the slope of the break-even
investment line, implying that the two lines must
eventually cross
Figure 2: Actual investment and break-even
investment
. Finally, due to diminishing returns, fkk < 0, the
two lines intersect only once. Let k* be the value
where actual investment equals break-even
investment, or in other words there is no change in
capital per unit of effective labor, .
This value of k is also called the steady state value
of k.
The next figure shows that the economy converges
to k regardless of where it starts.
0k
Figure 3: Steady State
When k < k* actual investment exceeds break-
even investment as a result .
When k > k* the actual investment is less than
break-even investment and therefore .
Finally for k =k*, .
0k
0k
0k
b) The Balanced Growth Path
How are output, capital and consumption growing
in this economy when k = k*.
We know that L and A are growing at
exogenously given rates - n and g, respectively.
The capital stock K = ALk, and since k is constant
at k*, the aggregate capital stock of the economy
is growing at the rate(n+g).
. With both capital and effective labor growing at the same
rate (n+g), the assumption of constant returns to scale
implies that aggregate output is also growing at the rate (n
+ g). Since consumption is (1 − s)Y , where s is constant,
consumption also grows at the same rate as output.
Finally, capital per unit of labor, K/L, and output per unit
of labor, Y/L, grow at the rate g.[Nb:! at (n+g)-n, n for the
percpita handling!)
Thus, the Solow model implies that regardless of its
starting point, the economy converges to a balanced
growth path, where each variable grows at a constant rate
. At this point we also need to discuss our assumption that
technological change is labor augmenting.
This is a restriction that is required for the existence of a
balanced growth path.
NB: Other types of technological change - Hicks-neutral
(unbiased technological change) and capital augmenting-
technical change - are not consistent with a balanced
growth path. For a proof look at Uzawa (1961).
Notice also that off the balanced growth path technological
change is no longer required to be labor-augmenting. (Your
reading assignment begin with Valdez)
. The idea of balanced growth though seemingly
abstract has a parallel in the data.
The Kaldor‟s stylized facts, Kaldor (1963), show
that while output per capita grew, the capital-
output ratio (K(t)/Y (t)), the interest rate(r(t)), and
the distribution of income between labor
(w(t)L(t)/Y (t)) and capital (R(t)K(t)/Y (t)) remain
roughly constant.
The figure below shows the factor shares for the
US since 1929.
Figure 4: Labor and capital shares in value added in the U.S.
IV. Comparative Dynamics: Impact of a Change in
Savings Rate
The parameter of the Solow model that policy is
most likely to affect is the savings rate.
• What is the effect of (unanticipated) change in the
savings rate s?
• Consider an economy that is on a balanced growth
path, and suppose that there is a permanent
increase in s. The increase in s shifts the actual
investment curve upwards, thereby resulting in an
increase in k*.
A. The Impact on Output
• Initially, when s increases and the curve shifts up,
at the initial steady-state value of k the actual
investment exceeds break-even investment.
• Thus is positive resulting in an accumulation of
k, which continues till it reaches the new steady-
state value of k.
• This is depicted in the figure below.
k
Figure 5: Effect of change in savings rate on investment
.
• Y/L, we concluded grew at rate g when k = k*.
• However, when k is increasing, as the economy moves
from one steady-state to another, Y/L grows at a rate
higher than g (because the source of growth is not
only the growth rate of A[=g], but also growth rate
of k (which was constant when k=k* hence y*=f(k*)
before s increased to s*).
• Once k reaches its new steady-state value, growth rate
of Y/L falls back to g.
• Thus a permanent increase in s produces temporary
increase in in the growth rate of output per worker, k
rises for some time but eventually it reaches a level at
which additional savings are devoted to maintaining
the higher level of k.
Figure 6: Effect of an increase in savings rate
Figure 6: Effect of an increase in savings rate
Effect of increase in saving, 2nd diagram explained
• After the transition period of “T” the new steady
state log y will follow the trajectory {At+T=log
y*t+T, At+T+1=log y*t+T+1,…}
– This is above ¶llel to the previous steady state
– In the transition period (T) the slop of log y [which is
grwoth rate] is greater than both stead states. (ie
during transition percapita income growth at a rate
greater than “g”
– Note that y=(Y/L) is one-to-one linked to y~= (Y/AL)
through A, y=A(y~), Thus as y~ growth at growth
rate of A=g (since it is f(k*) in which k is constant at
steady state), so is y will be growth (at “g‟ rate).
– The reason behind this dynamics is the diminishing MPK
. • At the end of the day a change in s has a level
effect but not a growth rate effect:
– ie. it changes the balanced growth path of the economy
and its level of output per worker, but it does not affect
the growth rate of output of per worker on the new
balanced growth path.
• In fact in the Solow model only changes in the rate
of technological progress have growth effects of
percaipita income; all other changes have level
effects.
B. The Impact on consumption
• Since consumption per unit of effective labor c =
(1 − s)f(k), an increase in s at the initial steady-
state level of k results in an initial decrease in c
and then as k rises to its new level c also rises.
• Whether or not c exceeds its original level can be
seen by writing down the expression for
consumption per unit of effective labor.
Steady-state consumption is given by:
• An increase in s raises k. Thus, c will rise in
response to an increase in s if the marginal product
of capital, fk, is greater than (n + g + δ).
• Intuitively when k rises investment must increase
by (n + g + δ) times k in order to sustain the new
level of k.
)20.1.(..........),,,(*
)]()),,,(*([*
)19.1...(..................................................*,........)(*)(*
s
gnskgngnskf
s
c
kgnkfc
k
. • If fk is less than (n + g +δ ), then the additional output
from a higher k is not enough to support the higher
level of k.
• As a result c must decline in the long run to maintain
the stock of capital.
• On the other hand, if fk exceeds (n+g+δ) there is more
than enough output to support the higher level of k,
and therefore c increases in the long run.
• However, if the steady-state value of k to start with is
changed duet to s in such a way that fk = (n + g + δ)
then a marginal change in s does not change c.
. • This value of k is called the golden rule level of the
capital stock (& the associate saving “saving gold”).
• At the golden rule level of capital, consumption is at
its maximum level {see the next 2 slides).
• Since s is exogenous in the Solow model, there is no
guarantee that k will be at its golden rule level.
• This cases are depicted in the figures below.
– See the implications of this for the choice of either
current or future consumption (hence Ramsey
model) & OLG models!! see Barro &Sala-i-
Martin, p. 34 on dynamic inefficiency [also briefly
in few slides ahead]
…
Figure 7: Output, investment and consumption on the
balanced growth path
V. Quantitative Implications of the Model
• A) Effect of saving on per capita income
• B) A little bit about Dynamic inefficiency
• C) Transition dynamics – the speed of
convergence from y to y* /or from k to k*
A. Steady State: Quantitative Importance of Savings
Rate in Affecting Income Per Capita in the Long Run
• In our discussions we saw that income per capita
varies significantly across countries. Can the savings
rate have a quantitatively important impact on income
per capita so as to explain such large income
differences?
• The long run effect of a change in savings rate on
output per unit of effective labor is given by: (see
derivation of 1.21, next slide)
)()(
)()(
)21.1.....(........................................),,,(
)(
*
**
**
*
kfsgn
kfkf
s
gnskkf
s
y
. • Where is the level of output per unit of effective
labor on the balanced growth path.
• Thus to find , we need to find
• To do this, note that k* is defined by the condition
; thus k* satisfies
• Equation (1.22) holds for all values of s(and of n, g, and δ)
*)(* kfy
s
y
*
s
k
*
0k
)22.1......(..........).........,,,(*)()),,,(*( gnskgngnsksf
. • Thus the derivatives of the two sides with respect to s
are equal:
Where the arguments of k* are omitted for simplicity.
• Rearranging (1.23) gives :
)23.1..(..............................*
)(*)(*
*)(s
kgnkf
s
kkfs
)25.1..(..............................*)()(
*)(*)(**
*
*
obtained be could (1.21) rulechain and ) (1.24 using
)24.1..(........................................*)()(
*)(*
kfsgn
kfkf
s
y
s
k
k
y
kfsgn
kf
s
k
. Two changes help in interpreting this expression.
• First convert (1.25) into an elasticity by
multiplying both sides by s/y*
• Second use the fact that to
substitute for s .
*)(*)( kgnksf
Making these changes results:
*)](/*)(*[1
*)(/*)(*
(1.26)
*)](/*)(*)()*)[((
*)(*)(*))(
*)(
*)('
*)()*)((
*)(
*)()(
*)(*)(
*)(
*
*
kfkfk
kfkfk
kfkfkgngnkf
kfkfkgn
kf
kf
kfsgnkf
ksf
kfsgn
kfkf
kf
s
s
y
y
s
. • is the elasticity of output with
respect to capital at k=k*.Denoting this by
• If markets are competitive and there are no externalities,
capital earns its marginal product . In this case, the total
amount received by capital (per unit of effective labor) as
the share of output on the balanced growth path is
*)(/*)('* kfkfk
have we*),(kk
)27.1....(..................................................*)(1
*)(*
* k
k
s
y
y
s
k
k
*).( *),(/*)('* korkfkfk k
• The elasticity of output w.r.t. saving depend on capital share of
income, . ie – With competitive markets and no externalities capital earns its marginal
product.
– & On the balanced growth path the share of income attributed to capital
must be
• Capital share of income is found to be about 1/3 in most
countries, which implies that (from eqn 1.27 above) the
elasticity of output w.r.t. saving is about 0.5:
– Thus, a 10% increase in savings rate increases per worker output in the
long run by 5% relative to the path it would have followed. For a 50%
change in savings rate y rises by only 25%.
– . Thus, big changes in savings rate have only a moderate effect on the
level of output on the balanced growth path
*)(kK
*)(*)(/*)('* kkfkfk K
5.03/11
3/1
*)(1
*)(*
*
k
k
s
y
y
s
k
k
B. Saving, Dynamic inefficiency and the prelude
to Ramesy-Cas-Koopman Model
• A little bit about Dynamic inefficiency here:
– If higher saving leads to higher level of income (higher
steady state) [though no higher growth rate], the best saving
rate will be 100% - but this is tautological [ie higher s,
higher k, higher Y etc….]
– Specifically in SS model a saving rate above the golden rule
reduces per capita consumption at steady state [called
dynamically inefficient/over saving – hence the need to
reduce it from s2 (the need to increase it form s1 in the next
diagram)
– From a societies welfare perspective high saving is not
necessarily good and low saving necessarily bad when once
see it dynamically either -partly because higher saving
today means lower consumption today (may be higher
consumption in the future!).
– .
Saving, Dynamic inefficiently and the prelude to
Ramsy-Cas-Koopman Model
– If we follow Keynes: “consumption is the sole end and
objective of all economic activity” so we can use it for
societal welfare indications
– However the welfare depends on hh valuation of future
versus current consumption (or themseleves versus future
generation).
– Thus, the optimal saving of a nation (hence investment)
should be inferred from the optimal level of
consumption/utility it generates over time
– Thus the importance of inter-temporal view and hence the
Ramesy-Kass-Koopman model: !!how much should a
nation save?!!Ramesy
– See Barro and Saal-i-Martin, p. 34, & nxt slide
Saving, Dynamic inefficiently and the prelude to
Ramsy-Cas-Koopman Model
• A little bit about Dynamic efficiently here:
– See Barro and Saal-i-Martin, p. 34, nxt slide
C. Transition Dynamics: Speed of Convergence
How rapidly does k approach k* when s changes?
– The growth of capital per unit of effective labour depends on
the size of the capital-to-effective labour ratio:
– At the point where we know that .
– These conditions imply that a first order Taylor-series
approximation of around the point give:
)(kkk
*kk 0k
)(kk *kk
*/)(*)()( kkkkkktktk
In the neighbourhood of the balanced growth path, k converges to
k* at a speed that is approximately proportional to its distance from
k*. Hence, the gap between k(t) and k* narrows at a rate that is
approximately constant and equal to :
To find an expression for we differentiate
w.r.t. k and evaluate the resulting equation at the point k=k*:
*)0(*)( kkektk t
)()()( tkgntksftk
gnk
kf
kfkgngn
gnksfk
kk
K
kk
*)(1
*)(
)('*
*)(')(
*
Using the expression for we can conclude that the speed of convergence of the capital stock is:
A reasonable assumption is that . With we have .
k moves 4% of the remaining distance toward k* per year
It takes approximately 17 years for k to get half the way to its balanced growth path if k(t) is in the vicinity of k* to begin with.
Since output per effective worker only depend on k, k and y converges to k* and y* at the same speed.
The impact of a change in the saving rate on output is both quite small and fairly slow!
*)0(*)( *)(1 kkektk tgnkK
%6 gn 3/1K
%41 gnK
VI. The Central Questions of Solow model the
growth theory
– Why are some economies much richer than others?
– Are income levels converging across nations?
The Solow model identifies two potential source to why output per
worker varies across countries and over time:
1. Differences in capital per worker (K / L)
2. Differences in the effectiveness of labour (A)
Due to convergence of k to k* changes in the effectiveness of labour is
the only factor that lead to permanent changes in the growth rate
• There are two ways to see that the Solow model implies that differences in
capital accumulation cannot account for large differences in income:
1. Required differences in output per worker**
• Differences in incomes between rich and poor countries roughly
corresponds to a factor of 10. This implies that the capital stocks in rich
and poor countries differs by a factor of .
In the real world one observes that the capital-to-labour ratio in rich
countries are 20 to 30 times larger than in the poor countries. Moreover,
capital-to-output ratios tend to be fairly constant over time.
2. Required differences in the rate of return to capital
• If markets are competitive the rate of return to capital equals its marginal
product minus depreciation.
– A tenfold difference in capital per worker implies a difference in the rate of
return to capital by a factor of 100.
– MPK in poor countries would be so high that there would be no reasonable
answer to why not all capital in the world relocates to the poor countries.
)3/1(100010 /1 KK
VII. Empirical Application
A. Growth Accounting
In the Solow model ,long run growth of output per worker
depends only on technological progress.
But short run growth can result from either technological
progress or accumulation.
Thus the model implies that determining the sources of
short run growth is an empirical issue
Thus the growth accounting relates to an empirical
extension that allows to distinguish between different
sources of growth.
. Growth accounting is pioneered by Abramovitz(1956) and
Solow(1957)
To see how it works, consider the production function
given as
).()(
)()(
)(
)()(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
yeilds side handright on the terms therewriting and )(by sides bothe Dividing
.ly respective,][ and ][ denote and where
)()(
)()(
)(
)()(
)(
)()(
is derivative totalits implies, This).()(),(()(
tRtL
tLt
tK
tKt
tA
tA
tA
tY
tY
tA
tL
tL
tL
tY
tY
tL
tK
tK
tK
tY
tY
tK
tY
tY
tY
L(AL)
YA
(AL)
Y
A
Y
L
Y
tAtA
tYtL
tL
tYtK
tK
tYtY
tLtAtKFtY
LK
. Note that
is the elasticity of output with respect to labor
at time t,
is the elasticity of output with respect to
capital at time t and
.
)(tL
)(tK
. Subtracting from both sides and using
the fact that results an expression for the
growth of output per worker as:
Note that:
The growth rate of Y,K and L are straight forward
to measure
If capital earns its marginal product, can be
measured using data on the share of income that
goes to capital
)(
)(
tL
tL
1)()( tt KL
)35.1..(..........).........(])(
)(
)(
)()[(
)(
)(
)(
)(tR
tL
tL
tK
tKt
tL
tL
tY
tYK
K
. R(t) then can be measured as the residual in
equation (1.35) above.
Thus equation (1.35) provides a way of
decomposing the growth of output per worker into
the contribution of growth of capital per worker
and a remaining term, the Solow Residual
The Solow residual –some times interpreted as a
measure of the contribution of technological
progress (TFP).
But as derivation shows, it reflects all sources of
growth other than the contribution of capital
accumulation via its private return
. Growth Accounting has been applied to many
issues:
Yong(1995)-used detailed growth accounting to argue that
the higher growth in the newly industrialized countries of
East Asian than the rest of the world is mainly Due to
rising investment, increasing labor force participation, and
improving labor quality(in terms of education),and not to
rapid technological progress other forces affecting the
Solow residual
See:Hsieh(1998a); Denison(1985); Bailyand
Gordon(1988); Griliches(1988);and Jorgeson(1988) for a
classic application of growth accounting
Alemayehu (2008; 2013) used it to examine Ethiopia‟s
growth “Miracle” 2000-2013.
Table 2.3: A Growth Accounting Exercise for Ethiopia in the Last Decade (2000-2010)
Note: The growth accounting is based on the result of econometric estimation reported in Alemayehu and
Befekadu (2005) and Alemayehu et al (2008). Using various models (both macro and micro) these
studies have come up with the capital share coefficient (β) that ranges from 0.28 to 0.36. We have used
0.30. The capital stock is generated using the perpetual method (Geda, 2013/14).
Year
GDP Growth The Contribution of
Capital
The Contribution of
Labour
The Contribution of Total
Factor Productivity (TFP)
2000/2001 7.4 0.6 2.6 4.2
2001/2002 1.6 0.8 2.7 -1.9
2002/2003 -2.1 1.0 2.6 -5.7
2003/2004 11.7 0.7 2.7 8.3
2004/2005 12.6 1.2 2.6 8.8
2005/2006 11.5 1.1 2.7 7.7
2006/2007 11.8 1.5 2.2 8.1
2007/2008 11.2 2.1 2.2 6.9
2008/2009 9.9 1.8 2.3 5.9
2009/2010 10.4 2.7 2.2 5.5
Average(2003/04-2009/10) 11.3 1.6 2.4 7.3
B. Convergence
Are poor countries growing faster than the rich ones, i.e. is
there convergence?
Are income levels converging across nations?
• Solow model suggests three reasons why countries are
expected to converge: 1. The model predicts that countries converge to their balanced growth
paths – to the extent that differences in output per worker depend on countries being at different stages relative to their balanced growth path, cross-country income differentials are expected to decrease.
2. The model predicts that the rate of return to capital is lower in countries with more capital per worker, which induce capital to flow from rich to poor countries and result in convergence.
3. Lags in technology diffusion can account for income differentials between countries and to the extent that poor countries gain access to state-of-the art technologies poor countries would catch up on rich countries.
.
• For classic application on the convergence
hypothesis see Baumol(1986), De
Long(1988); also Findely, AER, 1996. The
first two are discussed below
• Reading Assignment on Beta and Sigma
convergence/ Presentation - Seminar (form
the convergence folder)
Convergence… Cont‟d
Convergence… Cont‟d
Convergence… Cont‟d
• De Long (1988) noted Baumol‟s Finding is
largely spurious. Cause problems of:
– (a) sample selection (countries that have long data
series are those that are the most industrialized).
Countries not rich 100 years ago and in the sample
must be there if they grow fast
– He included countries as rich as the 2nd poorest in
Baumol‟s sample, Finland. This led to add more
countries (see Diagram below)
Convergence… Cont‟d
Convergence… Cont‟d
• De Long (1988) 2nd problem:
– (b) measurement error: estimates of 1870 are
imprecise. Measurement errors bias result
towards convergence [if 1870 is overstated,
growth 1870-79 is understated by an equal
amount and vice versa (growth tends to be low
where measured initial income is high even if no
relation between initial income actual growth).
– He then estimated the following model (see
Diagram below)
– This reduced the estimate of “b” to -0.566
Convergence… Cont‟d
Convergence… Cont‟d
• For example: if we find measured growth is
negatively related to initial income this could be
– Either measurement error is not important and there
is convergence OR
– Measurement is important and there is no
convergence
– This is what is called “model identification problem”
• However, De Long argues we can have an idea
of the accuracy of the 1870 data
Convergence… Cont‟d • In term of standard value (SD), an SD of 0.01
implies, we measured initial income with 1%
error [-implausibly low] and SD=0.50 is 50%
error [-Implausibly high]
• He found even moderate SD has substantial
impact on the result.
– For unbiased sample an SD of 0.15 makes the estimate
of “b” reaches 0 (no convergence); AND an SD of
0.20 makes it 1 (convergence big time!)
• Thus, a moderate measurement error could take
most of the remaining Baumol‟s estimate of
convergence (as we noted was already-0.566)
C. Human capital Augmented Solow-Swan Model
• Attributed to Mankiew et al (1992)….
• Good summary in Heijira (2009) P.411
The Augmented Solow Swan (Human
Capital) Neoclassical Model
1) The Puzzle :
Emperical estimates of the speed of adjustment shows that values around
(this is actually for the US & say 2% of the gap from the steady state is closed each year (i.e., the gap between
However ,having this values and bench mark value for US and other DC‟s
0226.0
tt yy and
estimate share capital theis 68.0)(
of valuecomputable the,04.0 and 032.0
gn
gnα
ygn
The augmented Solow Swan (Human
Capital) Neoclassical Model ….. Cont’d
However, the capital share for most DC‟s is about 30% .The question is what is happening. Why the SS model gives α=68% [see our Diagram bfr]
2) Mankiw, Romer and Weil(1992) suggested a simple solution to this problem/ puzzle
→we are not thinking about K in a right way. We take it as physical capital but it also incorporates human capital (skill)-this is because all output not consumed is not used to increase physical capital only; but also skill (health education, etc)
Human capital ….Cont’d • Key idea: add human capital to the model.
• Technology:
Y(t)= 𝐾(𝑡)αk+𝐻(𝑡)αH+[𝑍 𝑡 𝐿 𝑡 ]1−αk−α𝐻
( 0 < αK+ αH < 1)
where H(t) is the stock of human capital and αK and αH are
the efficiency parameters of the two types of capital
(0 < αK, α H < 1).
• In close accordance with the Solow-Swan model, productivity
and population growth are both exponential (𝑍 (t)/Z(t) = nZ
and 𝐿 (t)/L(t) = nL.
• The accumulation equations for the two types of capital can
be written in effective labour units as:
Human capital ….Cont’d
• Stability: The phase diagram is give next
Human capital ….Cont’d
• In Summary the model is given by:
• Stability: The phase diagram is give next
Humanĸ capital ….Cont’d
• Stability: The phase diagram is give next
k
Human capital ….Cont’d
Human capital ….Cont’d
Human capital ….Cont’d
VIII. Conclusion
Central conclusion from the Solow model:
Only differences in the productivity of labour
can account for vast differences in wealth across
space and time (see slide 84&85 in this lecture).
The „technology factor‟ is, however, exogenous to the
Solow model – the model makes no prediction of what this
factor really is, how it behaves or how it grows.
END… END…. END