econ 581. the solow growth model - university of … (solow model and models of endogenous growth)...
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ECON 581. The Solow Growth Model
Instructor: Dmytro Hryshko
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General info
e-mail: [email protected]
office: HM Tory Building, 9-19
office hours: Friday 2–3 PM, or byappointment
Teaching material available online
web-page: http://www.artsrn.ualberta.ca/econweb/hryshko/
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General info
Recommended text: Daron Acemoglu (2009).Introduction to Modern Economic Growth.Princeton University Press.
But there’re many other.
Grading: an in-class midterm (35%) and final(45%) exams, and two problem sets (20%).
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Scope of the course
Growth (Solow model and models ofendogenous growth)
Consumption and savings
Asset pricing
Policy issues
etc.
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Outline for today
Measures for standard of living
The importance of economic growth
Main long-run growth facts and theirimplications
A basic model of long-run growth (the Solowmodel)
Readings: Acemoglu, Chapters 1–3.
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What is Economic Growth?
Standard economists’ answer: Growth inGDP per capita (or per worker), Y/L
Does growth in per capita GDP reflectanything important about the improvement ofliving standards?–No consideration, for example, of thedistribution of income within the economy(inequality)
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Key Questions in Economic Growth
Why are there countries so rich and others sopoor?
Why do growth rates vary across countriesand over time?
What are the policies that can change growthin the short and long run?
Why do some countries “take off” whileothers fall behind
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Measures of Living Standards
Normally, the focus on two statistics of theaverage person’s well-being: income and GDPper worker (productivity measure), and incomeand GDP per capita (welfare measure).
They correlate with many other importantstatistics measuring well-being: infantmortality, life expectancy, consumption, etc.
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.1.2 Income and Welfare 7
1.2 Income and Welfare
Should we care about cross-country income differences? The answer is definitely yes. High income levels reflect high standards of living. Economic growth sometimes increases pollution or may raise individual aspirations, so that the same bundle of consumption may no longer satisfy an individual. But at the end of the day, when one compares an advanced, rich country with a less-developed one, there are striking differences in the quality of life, standards of living, and health.
Figures 1.5 and 1.6 give a glimpse of these differences and depict the relationship between income per capita in 2000 and consumption per capita and life expectancy at birth in the same year. Consumption data also come from the Penn World tables, while data on life expectancy at birth are available from the World Bank Development Indicators.
These figures document that income per capita differences are strongly associated with differences in consumption and in health as measured by life expectancy. Recall also that these numbers refer to PPP-adjusted quantities; thus differences in consumption do not (at least in principle) reflect the differences in costs for the same bundle of consumption goods in different countries. The PPP adjustment corrects for these differences and attempts to measure the variation in real consumption. Thus the richest countries are not only producing more than 30 times as much as the poorest countries, but are also consuming 30 times as much. Similarly, cross-country differences in health are quite remarkable; while life expectancy at birth is as
Log consumption per capita, 2000
15
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AFG
ALB
DZA
AGO
ATG
ARG
ARM
AUSAUT
AZE
BHS
BHR
BGD
BRB
BLR
BEL
BLZ
BEN
BMU
BTN
BOL BIH
BWA
BRA BRN
BGR
BFA
BDI
KHM
CMR
CAN
CPV
CAF
TCD
CHL
CHN
COL
COM
ZAR
COG
CRI
CIV
HRV
CUB
CYP
CZE
DNK
DJI DMA
DOM
ECU
EGYSLV
GNQ
ERI
EST
ETH
FJI
FIN
FRA
GAB
GMB
GEO
GER
GHA
GRC
GRD
GTM
GIN
GNB
GUYHTI HND
HKG
HUN
ISL
IND
IDN
IRN
IRQ
IRLISR
ITA
JAM
JPN
JOR
KAZ
KEN
KIR
PRK
KOR
KWT
KGZ
LAO
LVA
LBN
LSO
LBR
LBY LTU
LUX
MAC
MKD
MDGMWI
MYS
MDV
MLI
MLT
MRT
MUS
MEX
FSMMDA
MNG
MAR
MOZ
NAM
NPL
NLD
ANT
NZL
NIC
NER NGA
NOR
OMN
PAK
PLWPAN
PNG
PRY
PERPHL
POL
PRT PRI
QAT
ROM RUS
RWA
WSM
STP
SAU
SEN
SCG
SYC
SLE
SGP
SVK
SVN
SLB
SOM
ZAF
ESP
LKA
KNA
LCA
VCT
SDN
SUR
SW Z
SWE
CHE
SYR
TWN
TJK
TZA
THA
TGO
TON
TTO
TUN
TUR
TKM
UGA
UKR
AREGBR
USA
URY
UZB
VUT
VEN
VNM
YEM
ZMB
ZWE
6 7 8 9 10 11 Log GDP per capita, 2000
FIGURE 1.5 The association between income per capita and consumption per capita in 2000. For a definition of the abbreviations used in this and similar figures in the book, see http://unstats.un.org/unsd /methods/m49/m49alpha.htm.
Source: Acemoglu (2008). Introduction to Modern Economic Growth9 / 59
8 . Chapter 1 Economic Growth and Economic Development: The Questions
Life expectancy, 2000 (years)
AFG
AGO
ALB
ANT
ARE
ARG
ARM
AUS AUT
AZE
BDI
BEL
BEN
BFA
BGD
BGR
BHR
BHS
BIH
BLR
BLZ
BOL
BRA
BRB
BRN
BTN
BWA
CAF
CANCHE
CHL
CHN
CIV CMR
COG
COL
COM
CPV
CRICUB
CYP
CZE
DJI
DNK
DOM
DZA
ECU
EGY
ERI
ESP
EST
FIN
FJI
FRA
FSM
GAB
GBR
GEO
GHA
GIN GMB
GNB GNQ
GRC
GTM
GUY
HKG
HND
HRV
HTI
HUN
IDN
IND
IRL
IRN
IRQ
ISL ISRITA
JAM JOR
JPN
KAZ
KEN
KGZ
KHM
KOR
KWT
LAO
LBN
LBR
LBYLCALKA
LSO
LTU
LUX
LVA
MAC
MAR
MDA
MDG
MDV
MEX MKD
MLI
MLT
MNG
MOZ
MRT
MUS
MWI
MYS
NAM
NER NGA
NIC
NLD NOR
NPL
NZL
OMN
PAK
PAN
PERPHL
PNG
POL
PRI
PRK
PRT
PRY
QAT
ROM
RUS
RWA
SAU SCG
SDN SEN
SGP
SLB
SLE
SLV
SOM
STP
SUR
SVK
SVN
SWE
SW Z
SYR
TCD
TGO
THA
TJK
TKM
TON
TTO
TUN
TUR
TZA
UGA
UKR
URY
USA
UZB
VCT
VEN
VNM
VUTWSM
YEM
ZAF
ZMB
ZWE
ETH
GER
30
40
50
60
70
80
6 7 8 9 10 11 Log GDP per capita, 2000
FIGURE 1.6 The association between income per capita and life expectancy at birth in 2000.
high as 80 in the richest countries, it is only between 40 and 50 in many sub-Saharan African nations. These gaps represent huge welfare differences.
Understanding why some countries are so rich while some others are so poor is one of the most important, perhaps the most important, challenges facing social science. It is important both because these income differences have major welfare consequences and because a study of these striking differences will shed light on how the economies of different nations function and how they sometimes fail to function.
The emphasis on income differences across countries implies neither that income per capita can be used as a “sufficient statistic” for the welfare of the average citizen nor that it is the only feature that we should care about. As discussed in detail later, the efficiency properties of the market economy (such as the celebrated First Welfare Theorem or Adam Smith’s invisible hand) do not imply that there is no conflict among individuals or groups in society. Economic growth is generally good for welfare but it often creates winners and losers. Joseph Schumpeter’s famous notion of creative destruction emphasizes precisely this aspect of economic growth; productive relationships, firms, and sometimes individual livelihoods will be destroyed by the process of economic growth, because growth is brought about by the introduction of new technologies and creation of new firms, replacing existing firms and technologies. This process creates a natural social tension, even in a growing society. Another source of social tension related to growth (and development) is that, as emphasized by Simon Kuznets and discussed in detail in Part VII, growth and development are often accompanied by sweeping structural transformations, which can also destroy certain established relationships and create yet other winners and losers in the process. One of the important questions of
Source: Acemoglu (2008). Introduction to Modern Economic Growth10 / 59
Growth Facts
Fact 1. There is enormous variation in incomes percapita across countries. The poorest countries have percapita incomes less than 5% of per capita incomes inthe richest countries. Table
Fact 2. Rates of economic growth vary substantiallyacross countries. Fig
Fact 3. Growth rates are not generally constant overtime. For the world as a whole, growth rates were closeto zero over most of history but have increased sharplyin the 20-th century. Fig
The same applies to individual countries. Countriescan move from being “poor” to being “rich” (e.g.,South Korea), and vice versa (e.g., Argentina). Fig
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Source: Charles Jones. Introduction to Economic Growth.Back
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1.4 Today’s Income Differences and World Economic Growth . 11
Log GDP per capita
United States
United Kingdom
Spain
Singapore
Brazil
South Korea
Botswana
Guatemala
Nigeria
India
7
8
9
10
11
1960 1970 1980 1990 2000
FIGURE 1.8 The evolution of income per capita in the United States, the United Kingdom, Spain, Singapore, Brazil, Guatemala, South Korea, Botswana, Nigeria, and India, 1960–2000.
40 years? Why did Spain grow relatively rapidly for about 20 years but then slow down? Why did Brazil and Guatemala stagnate during the 1980s? What is responsible for the disastrous growth performance of Nigeria?
1.4 Origins of Today’s Income Differences and World Economic Growth
The growth rate differences shown in Figures 1.7 and 1.8 are interesting in their own right and could also be, in principle, responsible for the large differences in income per capita we observe today. But are they? The answer is largely no. Figure 1.8 shows that in 1960 there was already a very large gap between the United States on the one hand and India and Nigeria on the other.
This pattern can be seen more easily in Figure 1.9, which plots log GDP per worker in 2000 versus log GDP per capita in 1960 (in both cases relative to the U.S. value) superimposed over the 45◦ line. Most observations are around the 45◦ line, indicating that the relative ranking of countries has changed little between 1960 and 2000. Thus the origins of the very large income differences across nations are not to be found in the postwar era. There are striking growth differences during the postwar era, but the evidence presented so far suggests that world income distribution has been more or less stable, with a slight tendency toward becoming more unequal.
Source: Acemoglu. Introduction to Modern Economic Growth.Back
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Source: Charles Jones. Introduction to Economic Growth.Back
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The Power of Growth Rates
Time to double (rule of 70): Assume that ytgrows at a constant rate g. Then
yt+1 = (1 + g)yt and yt+k = (1 + g)kyt, k ≥ 0
What is the time needed for yt to double?
We want to solve for k∗ that satisfies2yt = (1 + g)k
∗yt.
Taking natural logs from both sides gives
ln 2 = ln(1 + g)k∗, or k∗ =ln 2
g≈ 0.7
g=
70
100g,
where g is expressed in percentage terms.15 / 59
The Power of Growth Rates
Thus, if g = 0.02 (e.g., U.S.), GDP per capitawill double every 70/2=35 years
if g = 0.06 (e.g., South Korea)≈ every 12years.
If, e.g., the difference in age betweengrandparents and grandchildren is about 48years, Korean (future) grandchildren of thecurrent generation will be about 24 = 16times wealthier than the current generation.
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Kaldor Facts
For the U.S. over the last century:
1 The real interest rate shows no trend, up ordown. (In the classical model related toMPK = α Y
K .)2 The share of labor and capital costs in
income, although fluctuating, have no trend.Fig
3 The average growth rate in output per capitahas been relatively constant over time, i.e.,the U.S. is on a path of sustained growth ofincomes per capita. Fig
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Back
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Fig. 1.4
Source: Charles Jones. Introduction to Economic Growth.19 / 59
Questions About Growth
What determines growth? In particular,–How much capital accumulation accounts forgrowth? (“growth accounting”)–How important is technological progress?(“growth accounting”)
Why do countries grow at different rates?
What causes growth rates to decline?
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Acemoglu. Figure 1.8
“Why is the United States richer in 1960 thanother nations and able to grow at a steadypace thereafter?”
“How did Singapore, South Korea, andBotswana manage to grow at a relativelyrapid pace for 40 years?”
“Why did Spain grow relatively rapidly forabout 20 years but then slow down?”
“Why did Brazil and Guatemala stagnateduring the 1980s?”
“What is responsible for the disastrousgrowth performance of Nigeria?”
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The Lessons of Growth Theory
understand why poor countries are poor
design policies that can help them grow
learn how the growth rates of rich countriesare affected by shocks and the governments’policies
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Acemoglu. Introduction to Modern Economic Growth.Chapter 2.
Our starting point is the so-called Solow-Swanmodel named after Robert (Bob) Solow and TrevorSwan, or simply the Solow model, named after themore famous of the two economists. Theseeconomists published two pathbreaking articles inthe same year, 1956 (Solow, 1956; Swan, 1956)introducing the Solow model. Bob Solow laterdeveloped many implications and applications ofthis model and was awarded the Nobel prize ineconomics for his contributions. This model hasshaped the way we approach not only economicgrowth but also the entire field of macroeconomics.Consequently, a by-product of our analysis of thischapter is a detailed exposition of a workhorsemodel of macroeconomics.
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Solow Growth Model
A major paradigm:–widely used in policy making–benchmark against which most recentgrowth theories are compared
Looks at the determinants of economic growthand the standard of living in the long run
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Solow model
A starting point for more complex models.
Abstracts from modeling heterogeneoushouseholds (in tastes, abilities, etc.),heterogeneous sectors in the economy, andsocial interactions. It is a one-good economywith simplified individual decisions.
We’ll discuss the Solow model both in discreteand continuous time.
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Households and production
Closed economy, with a unique final good.
Time is discrete: t = 0, 1, 2, . . ., ∞(days/weeks/months/years).
A representative household saves a constantfraction s ∈ (0, 1) of its disposable income (abit unrealistic).
A representative firm with the aggregateproduction functionY (t) = F (K(t), L(t), A(t)).
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Production function
Y (t) = F (K(t), L(t), A(t)),
where Y (t) is the total production of the finalgood at time t, K(t)—the capital stock at time t,and A(t) is technology at time t.
Think of L as hours of employment or the numberof employees; K the quantity of “machines” (K isthe same as the final good in the model; e.g., K is“corn”); A is a shifter of the production function(a broad notion of technology: the effects of theorganization of production and of markets on theefficiency of utilization of K and L).
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Assumption on technology
Technology is assumed to be free. That is, it ispublicly available and
Non-rival—consumption or use by othersdoes not preclude an individual’sconsumption or use.
Non-excludable—impossible to preventanother person from using or consuming it.
E.g., society’s knowledge of how to use thewheels.
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Assumptions on production function
Assumption 1: The production function F (K,L,A) is twicedifferentiable in K and L, satisfying:
FK(K,L,A) =∂F (K,L,A)
∂K> 0, FL(K,L,A) =
∂F (K,L,A)
∂L> 0
FKK =∂2F (K,L,A)
∂K2< 0, FLL =
∂2F (K,L,A)
∂L2< 0.
F is constant returns to scale in K and L.
FK > 0, FL > 0: positive marginal products (the level ofproduction increases with the amount of inputs used);FKK < 0, FLL < 0: diminishing marginal products (morelabor, holding other things constant, increases output byless and less).
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Constant returns to scale assumption
F (K,L,A) is constant returns to scale in K andL if λF (K,L,A) = F (λK, λL,A). In this case,F (K,L,A) is also said to be homogeneous ofdegree one.
In general, a function F (K,L,A) is homogeneousof degree m in K and L ifλmF (K,L,A) = F (λK, λL,A).
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Theorem 2.1 (Euler’s theorem)
Let g(x, y, z) be differentiable in x and y and homogeneousof degree m in x and y, with partial first derivatives gx andgy (z can be a vector). Then,
(1) : mg(x, y, z) = gx(x, y, z)x+ gy(x, y, z)y
(2) : gx and gy are homogeneous of degree m− 1 in x and y.
Proof : By definition λmg(x, y, z) = g(λx, λy, z) (2.2).
Part 1: Diff. both sides wrt λ to obtainmλm−1g(x, y, z) = gx(λx, λy, z)x+ gy(λx, λy, z)y, for any λ.Set λ ≡ 1. Thus, mg(x, y, z) = gx(x, y, z)x+ gy(x, y, z)y.
Part 2: Differentiate (2.2) wrt x:λmgx(x, y, z) = λgx(λx, λy, z), orλm−1gx(x, y, z) = gx(λx, λy, z).
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Endowments, market structure and market clearing
Competitive markets.
Households own labor, supplied inelastically:all the endowment of labor L(t) is supplied ata non-negative rental price.
Market clearing condition: Ld(t) = L(t). Rental price oflabor at t is the wage rate at t, w(t).
Households own the capital stock and rent itto firms. R(t) is the rental price of capital att.
Market clearing condition: Kd(t) = Ks(t).Kd
t used in production at t is consistent with households’endowments and saving behavior.
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K(0) ≥ 0—initial capital stock is given.
The price of the final good, P (t), isnormalized to 1 at all t. The final good(within each date) is numeraire.
Capital depreciates at the rate δ ∈ (0, 1)(machines utilized in production lose some oftheir value due to wear and tear). 1 unit now“becomes” 1− δ units next period.
R(t) + (1− δ) = 1 + r(t), where r(t) is thereal interest rate at t.
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Firm optimization and equilibrium
Firms’ objective, at each t, is to maximize profits;firms are competitive in product and rentalmarkets, i.e., take w(t) and R(t) as given.
maxK≥0,L≥0
π = F (Kd, Ld, A(t))−R(t)Kd − w(t)Ld,
where superscript d stands for demand, and π isprofit.
Note that maximization is in terms of aggregate variables(due to the representative firm assumption), and R(t) andw(t) are relative prices of labor and capital in terms of thefinal good.
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Since F is constant returns to scale, there is nowell-defined solution (K∗d, L∗d) (∗ means optimal,or profit-maximizing).If (K∗d ≥ 0, L∗d ≥ 0) and π > 0, then(λK∗d ≥ 0, λL∗d ≥ 0) will bring λπ. Thus, wantto raise K and L as much as possible. A trivialsolution K = L = 0, or multiple values of K andL that give π = 0. When profits are zero, wouldrent L and K so that L = Ls(t) and K = Ks(t).
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Factor prices
w(t) = FL(K(t), L(t), A(t)),
R(t) = FK(K(t), L(t), A(t)).
Using these two equilibrium conditions and Euler’stheorem, we can verify that firms make zero profits in theSolow growth model.
Y (t)︸︷︷︸LHS
= F (K(t), L(t), A(t)) = (by Euler theorem)
FK (K(t), L(t), A(t)) ·K(t) + FL (K(t), L(t), A(t)) · L(t) =
R(t)K(t) + w(t)L(t)︸ ︷︷ ︸RHS
.
Factor payments (RHS) exhaust total revenue (LHS), henceprofits are zero.
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Assumption 2: Inada conditions
We’ll assume that F satisfies Inada conditions:
limL→0
FL(K,L,A) =∞, and limL→∞
FL(K,L,A) = 0 for all K > 0, all A
limK→0
FK(K,L,A) =∞, and limK→∞
FK(K,L,A) = 0 for all L > 0, all A
Capital is essential in production (can be relaxed):
F (0, L,A) = 0 for all L and A.
The conditions say that the “first” units of labor and capitalare highly productive, and that when labor and capital aresufficiently abundant, the use of a marginal unit does not addanything to the existing output.
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MPK, and Inada conditionsIntroduction to Modern Economic Growth
F(K, L, A)
K0
K0
Panel A Panel B
F(K, L, A)
Figure 2.1. Production functions and the marginal product of capital. Theexample in Panel A satisfies the Inada conditions in Assumption 2, while theexample in Panel B does not.
Assumption 2. (Inada Conditions) F satisfies the Inada conditions
limK→0
FK (K,L,A) = ∞ and limK→∞
FK (K,L,A) = 0 for all L > 0 and all A
limL→0
FL (K,L,A) = ∞ and limL→∞
FL (K,L,A) = 0 for all K > 0 and all A.
Moreover, F (0, L,A) = 0 for all L and A.
The role of these conditions–especially in ensuring the existence of interior equilibria–will become later in this chapter. They imply that the “first units” of capital and labor arehighly productive and that when capital or labor are sufficiently abundant, their marginalproducts are close to zero. The condition that F (0, L,A) = 0 for all L and Amakes capital anessential input. This aspect of the assumption can be relaxed without any major implicationsfor the results in this book. Figure 2.1 draws the production function F (K,L,A) as a functionof K, for given L and A, in two different cases; in Panel A, the Inada conditions are satisfied,while in Panel B, they are not.
I refer to Assumptions 1 and 2, which can be thought of as the “neoclassical technology as-sumptions,” throughout much of the book. For this reason, they are numbered independentlyfrom the equations, theorems and proposition in this chapter.
2.2. The Solow Model in Discrete Time
I next present the dynamics of economic growth in the discrete-time Solow model.
2.2.1. Fundamental Law of Motion of the SolowModel. Recall thatK depreciatesexponentially at the rate δ, so that the law of motion of the capital stock is given by
(2.8) K (t+ 1) = (1− δ)K (t) + I (t) ,
where I (t) is investment at time t.
38
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Solow model in discrete timeThe law of motion of K(t) is given by
K(t+ 1) = I(t) + (1− δ)K(t),
where I(t) is investment at time t.
S(t) = sY (t) = I(t) closed economy
C(t) = Y (t)− I(t) = (1− s)Y (t)
Ks(t+ 1) = (1− δ)K(t) + S(t) = (1− δ)K(t) + sY (t) hhld behavior
Ks(t+ 1) = K(t+ 1), L̄(t) = L(t) equilibrium conditions
The fundamental law of motion of the Solow model is:
K(t+1) = sY (t)+(1−δ)K(t) = sF (K(t), L(t), A(t))+(1−δ)K(t).
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Definition of Equilibrium
Equilibrium is defined as an entire path ofallocations, C(t), K(t), Y (t), and prices, w(t) andR(t), given an initial level of capital stock K(0),and given sequences{L(0), L(1), L(2), . . . , L(t), . . .} and{A(0), A(1), A(2), . . . , A(t), . . .}.
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Equilibrium without population growth andtechnological progress
In this case, L̄(t) = L, and A(t) = A.
Example. The Cobb-Douglas productionfunction.
Y (t) = F (K(t), L(t), A(t)) = AK(t)αL1−α, 0 < α < 1.
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Note that F (λK, λL,A) = A(λK(t))α(λL)1−α =Aλαλ1−αK(t)αL1−α = λAK(t)αL1−α = λY (t).
Let λ ≡ 1L .
Then Y (t)L = A
(K(t)L
)α (LL
)1−α= A
(K(t)L
)α11−α = A
(K(t)L
)α.
Define k(t) ≡ K(t)L and y(t) ≡ Y (t)
L . Then y(t) = Ak(t)α.
R(t) = FK = AαK(t)α−1L1−α = Aα(K(t)L
)α−1= Aαk(t)α−1.
w(t) = FL = A(1− α)K(t)αL−α = A(1− α)(K(t)L
)α=
A(1− α)k(t)α.
Note also that w(t)L = Y (t)−R(t)K(t), and so
w(t) = Y (t)L −R(t)k(t) = Ak(t)α −Aαk(t)α−1k(t) =
Ak(t)α −Aαk(t)α = A(1− α)k(t)α = (1− α)y(t).
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Steady-state equilibrium
Recall the law of motion of the capital stock:
K(t+ 1) = (1− δ)K(t) + sF (K(t), L, A),
Divide the equation by L, to obtain,
k(t+ 1) = (1− δ)k(t) + sY (t)
L
= (1− δ)k(t) + sF
(K(t)
L, 1, A
).
Normalize A to one, and define F(K(t)L, 1, 1
)= f(k(t)).
Thus, the law of motion in per worker terms is:
k(t+ 1) = (1− δ)k(t) + sf(k(t)).
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A steady-state equilibrium without populationand technological growth is an equilibrium pathso that k(t) = k∗ for all t.
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Steady-state equilibrium—contd.
For a steady state,
k∗ = sf(k∗) + (1− δ)k∗
sf(k∗) = δk∗
f(k∗)
k∗=δ
s. (2.18)
Also,
y∗ = f(k∗)
c∗ = (1− s)f(k∗).
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Notes
k∗ = K∗
L is constant and L is not growing ⇒K is not growing in the SS;
to keep capital per worker, k, constantinvestment per worker should be equal to theamount of capital per worker that needs to be“replenished” because of depreciation;
k∗ will be a function of s and δ, i.e.,k∗ = k∗(s, δ).
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SS capital-labor ratio; the model without populationand technological growth
Introduction to Modern Economic Growth
k(t+1)
k(t)
45°
sf(k(t))+(1–δ)k(t)k*
k*0
Figure 2.2. Determination of the steady-state capital-labor ratio in theSolow model without population growth and technological change.
Returning to the analysis with the general production function, the per capita represen-tation of the aggregate production function enables us to divide both sides of (2.12) by L toobtain the following simple difference equation for the evolution of the capital-labor ratio:
(2.17) k (t+ 1) = sf (k (t)) + (1− δ) k (t) .
Since this difference equation is derived from (2.12), it also can be referred to as the equi-librium difference equation of the Solow model: it describes the equilibrium behavior of thekey object of the model, the capital-labor ratio. The other equilibrium quantities can all beobtained from the capital-labor ratio k (t).
At this point, let us also define a steady-state equilibrium for this model.
Definition 2.3. A steady-state equilibrium without technological progress and populationgrowth is an equilibrium path in which k (t) = k∗ for all t.
In a steady-state equilibrium the capital-labor ratio remains constant. Since there is nopopulation growth, this implies that the level of the capital stock will also remain constant.Mathematically, a “steady-state equilibrium” corresponds to a “stationary point” of the equi-librium difference equation (2.17). Most of the models in this book will admit a steady-stateequilibrium. This is also the case for this simple model.
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Source: Acemoglu. Introduction to Modern Economic Growth.47 / 59
Investment and consumption in SS
Introduction to Modern Economic Growth
An alternative visual representation shows the steady state as the intersection betweena ray through the origin with slope δ (representing the function δk) and the function sf (k).Figure 2.4, which illustrates this representation, is also useful for two other purposes. First,it depicts the levels of consumption and investment in a single figure. The vertical distancebetween the horizontal axis and the δk line at the steady-state equilibrium gives the amountof investment per capita at the steady-state equilibrium (equal to δk∗), while the verticaldistance between the function f (k) and the δk line at k∗ gives the level of consumptionper capita. Clearly, the sum of these two terms make up f (k∗). Second, Figure 2.4 alsoemphasizes that the steady-state equilibrium in the Solow model essentially sets investment,sf (k), equal to the amount of capital that needs to be “replenished”, δk. This interpretationwill be particularly useful when population growth and technological change are incorporated.
output
k(t)
f(k*)
k*
δk(t)
f(k(t))
sf(k*)sf(k(t))
consumption
investment
0
Figure 2.4. Investment and consumption in the steady-state equilibrium.
This analysis therefore leads to the following proposition (with the convention that theintersection at k = 0 is being ignored even when f (0) = 0):
Proposition 2.2. Consider the basic Solow growth model and suppose that Assumptions1 and 2 hold. Then, there exists a unique steady state equilibrium where the capital-laborratio k∗ ∈ (0,∞) is given by (2.18), per capita output is given by
(2.19) y∗ = f (k∗)
and per capita consumption is given by
(2.20) c∗ = (1− s) f (k∗) .
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Source: Acemoglu. Introduction to Modern Economic Growth.
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Proposition 2.2
Assume that Assumptions 1 and 2 hold. Then there existsa unique equilibrium where k∗ ∈ (0,∞) satisfies (2.18).
Proof. Existence: from Assumption 2, limk→0f(k)k
=∞and limk→∞
f(k)k
= 0, and f(k)k
is continuous. By theintermediate value theorem, there exists k∗ thatsatisfies (2.18). Uniqueness :
∂
∂k
[f(k)
k
]=f ′(k)k − f(k)
k2= −w
k2< 0.
Thus, f(k)k
is everywhere strictly decreasing, and there mustexist one k∗ that satisfies (2.18).
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Non-existence, non-uniqueness of SS
Introduction to Modern Economic Growth
k(t+1)
k(t)
45°
sf(k(t))+(1–δ)k(t)
0
k(t+1)
k(t)
45°
sf(k(t))+(1–δ)k(t)
0
k(t+1)
k(t)
45°
sf(k(t))+(1–δ)k(t)
0Panel A Panel B Panel C
Figure 2.5. Examples of nonexistence and nonuniqueness of interior steadystates when Assumptions 1 and 2 are not satisfied.
Proof. The preceding argument establishes that any k∗ that satisfies (2.18) is a steadystate. To establish existence, note that from Assumption 2 (and from L’Hospital’s rule, seeTheorem A.21 in Appendix A), limk→0 f (k) /k = ∞ and limk→∞ f (k) /k = 0. Moreover,f (k) /k is continuous from Assumption 1, so by the Intermediate Value Theorem (TheoremA.3) there exists k∗ such that (2.18) is satisfied. To see uniqueness, differentiate f (k) /k withrespect to k, which gives
(2.21)∂ (f (k) /k)
∂k=
f 0 (k) k − f (k)
k2= − w
k2< 0,
where the last equality uses (2.15). Since f (k) /k is everywhere (strictly) decreasing, therecan only exist a unique value k∗ that satisfies (2.18).
Equations (2.19) and (2.20) then follow by definition. ¤
Figure 2.5 shows through a series of examples why Assumptions 1 and 2 cannot bedispensed with for establishing the existence and uniqueness results in Proposition 2.2. Inthe first two panels, the failure of Assumption 2 leads to a situation in which there is no steadystate equilibrium with positive activity, while in the third panel, the failure of Assumption 1leads to non-uniqueness of steady states.
So far the model is very parsimonious: it does not have many parameters and abstractsfrom many features of the real world. An understanding of how cross-country differences incertain parameters translate into differences in growth rates or output levels is essential forour focus. This will be done in the next proposition. But before doing so, let us generalizethe production function in one simple way, and assume that
f (k) = Af̃ (k) ,
where A > 0, so that A is a shift parameter, with greater values corresponding to greaterproductivity of factors. This type of productivity is referred to as “Hicks-neutral” (see below).For now, it is simply a convenient way of parameterizing productivity differences acrosscountries. Since f (k) satisfies the regularity conditions imposed above, so does f̃ (k).
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Source: Acemoglu. Introduction to Modern Economic Growth.
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Steady-state predictions
We normalized A to 1 before. Redefine our per workerproduction function as f(k) = Af̃(k). The steady-stateequilibrium will now be obtained from sAf̃(k∗) = δf̃(k∗), andk∗ = k∗(s,A, δ). Thus, sAf̃(k∗(s,A, δ)) = δk∗(s,A, δ).It can be shown that
∂k∗(s,A, δ)
∂A> 0,
∂k∗(s,A, δ)
∂s> 0,
∂k∗(s,A, δ)
∂δ< 0
∂y∗(s,A, δ)
∂A> 0,
∂y∗(s,A, δ)
∂s> 0,
∂y∗(s,A, δ)
∂δ< 0.
Intuition: economies with higher saving rates and bettertechnologies will accumulate higher capitals per worker and willbe richer; higher depreciation rates lead to lower standards ofliving and lower capitals per worker.
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Example
In the steady state:
f̃(k∗)
k∗=
δ
sA,
where k∗ = k∗(A, s, δ).Differentiate both sides wrt s to obtain:
k∗f̃ ′(k∗)∂k∗
∂s− f̃(k∗)∂k
∗
∂s
(k∗)2= − δ
As2.
Rearranging,
∂k∗
∂s=
δ
As2
((k∗)2
f(k∗)− k∗f ′(k∗)
)=
δ
As2(k∗)2
w∗> 0.
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Golden rule steady-state
In steady state,
c∗ = f(k∗(s))− δk∗(s).
Maximizing wrt s, we need to set:
∂c∗
∂s= [f ′(k∗)− δ]∂k
∗
∂s= 0,
or
[f ′(k∗)− δ
] ∂k∗∂s
= 0.
Since ∂k∗
∂s > 0, the golden rule savings rate must yieldf ′(k∗gold) = δ.
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Golden-rule savings rate
Introduction to Modern Economic Growth
the relationship between consumption and the saving rate are plotted in Figure 2.6. Thenext proposition shows that sgold and k∗gold are uniquely defined.
consumption
savings rate
(1–s)f(k*gold)
s*gold 10
Figure 2.6. The “golden rule” level of savings rate, which maximizes steady-state consumption.
Proposition 2.4. In the basic Solow growth model, the highest level of steady-state con-sumption is reached for sgold, with the corresponding steady-state capital level k∗gold such that
(2.23) f 0¡k∗gold
¢= δ.
Proof. By definition ∂c∗ (sgold) /∂s = 0. From Proposition 2.3, ∂k∗/∂s > 0, thus (2.22)can be equal to zero only when f 0 (k∗ (sgold)) = δ. Moreover, when f 0 (k∗ (sgold)) = δ, itcan be verified that ∂2c∗ (sgold) /∂s2 < 0, so f 0 (k∗ (sgold)) = δ indeed corresponds to a localmaximum. That f 0 (k∗ (sgold)) = δ also yields the global maximum is a consequence of thefollowing observations: for all s ∈ [0, 1], we have ∂k∗/∂s > 0 and moreover, when s < sgold,f 0 (k∗ (s))− δ > 0 by the concavity of f , so ∂c∗ (s) /∂s > 0 for all s < sgold. By the converseargument, ∂c∗ (s) /∂s < 0 for all s > sgold. Therefore, only sgold satisfies f 0 (k∗ (s)) = δ andgives the unique global maximum of consumption per capita. ¤
In other words, there exists a unique saving rate, sgold, and also a unique correspondingcapital-labor ratio, k∗gold, given by (2.23), that maximize the level of steady-state consumption.When the economy is below k∗gold, the higher saving rate will increase consumption, whereaswhen the economy is above k∗gold, steady-state consumption can be raised by saving less. In
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Source: Acemoglu. Introduction to Modern Economic Growth.
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Corollary 2.1
1. Let x(t), a and b be scalars. If a < 1, then the unique steadystate of the linear difference equation x(t+ 1) = ax(t) + b isglobally (asymptotically) stable: x(t)→ x∗ = b
1−a .
2. Let g : R→ R be differentiable in the neighborhood of thesteady state x∗, defined as g(x∗) = x∗, and suppose |g′(x∗)| < 1.Then the steady state x∗ of the nonlinear difference equationx(t+ 1) = g(x(t)) is locally (asymptotically) stable. Moreover,if g is continuously differentiable and satisfies |g′(x)| < 1 for allx, then x∗ is globally (asymptotically) stable.
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Global stability
|x(t+ 1)− x∗| = |g(x(t))− g(x∗)|
=
∣∣∣∣∣∫ x(t)
x∗g′(x)dx
∣∣∣∣∣< |x(t)− x∗|.
Thus for any x(0) < x∗, x(t) is an increasing sequence. Thedistance between x(t+ 1) and x∗ becomes smaller than thedistance between x(t) and x∗.
In the Solow model, x(t+ 1) ≡ k(t+ 1), andg(x(t)) ≡ sf(k(t))− δk(t) + k(t). Note thatg′(x) = sf ′(k(t)) + 1− δ is not always less than 1 in absolutevalue. For ex., it tends to a large number when k(t) is small.Need some different arguments.
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Local stability in the Solow model
Since f(k) is strictly concave by assumption:
f(k) > f(0) + f ′(k)(k − 0) = f ′(k)k. (2.29)
In steady state, k∗ = g(k∗) = sf(k∗)− δk∗ + k∗, andg′(k∗) = sf ′(k∗) + 1− δ.
(2.29) implies thatsf(k∗)
k∗= δ︸ ︷︷ ︸
SS
> sf ′(k∗), or −δ < −sf ′(k∗), or
sf ′(k∗)− δ < 0 or sf ′(k∗) + 1− δ < 1—which is required forlocal stability.
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Global Stability in the Solow ModelFor all k(t) ∈ (0, k∗),
k(t+ 1)− k∗ = g(k(t))− g(k∗)
= −∫ k∗
k(t)g′(k)dk < 0,
since g′(k) > 0 for all k. Thus, k(t+ 1) also lies below k∗.Also,
k(t+ 1)− k(t)
k(t)= s
f(k(t))
k(t)− δ
> sf(k∗)
k∗− δ = 0,
where inequality follows from the fact that f(k)k is decreasing in
k. Thus, k(t+ 1) ∈ (k(t), k∗).58 / 59
Transitional dynamics in the basic Solow modelIntroduction to Modern Economic Growth
45°
k*
k*k(0) k’(0)0
k(t+1)
k(t)
Figure 2.7. Transitional dynamics in the basic Solow model.
The analysis has established that the Solow growth model has a number of nice prop-erties; unique steady state, global (asymptotic) stability, and finally, simple and intuitivecomparative statics. Yet, so far, it has no growth. The steady state is the point at whichthere is no growth in the capital-labor ratio, no more capital deepening and no growth inoutput per capita. Consequently, the basic Solow model (without technological progress)can only generate economic growth along the transition path when the economy starts withk (0) < k∗. However, this growth is not sustained : it slows down over time and eventuallycomes to an end. Section 2.7 will show that the Solow model can incorporate economicgrowth by allowing exogenous technological change. Before doing this, it is useful to look atthe relationship between the discrete-time and continuous-time formulations.
2.4. The Solow Model in Continuous Time
2.4.1. From Difference to Differential Equations. Recall that the time periodst = 0, 1, ... could so for refer to days, weeks, months or years. In some sense, the time unit isnot important. This suggests that perhaps it may be more convenient to look at dynamicsby making the time unit as small as possible, that is, by going to continuous time. Whilemuch of modern macroeconomics (outside of growth theory) uses discrete-time models, manygrowth models are formulated in continuous time. The continuous-time setup has a number ofadvantages, since some pathological results of discrete-time models disappear in continuous
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Source: Acemoglu. Introduction to Modern Economic Growth.
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