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# 01 Solow model.pdf

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University of Maryland

Economics

ECON 325

Sadik Arouba

Fall

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The Solow Growth Model
Reading:
Romer, Chapter 1;
Robert E. Lucas Jr., “Why Doesn’t Capital Flow from Rich to Poor Countries?” AER
Papers and Proceedings, 92-96, 1990;
Martin Feldstein and Charles Horioka, “Domestic Savings and International Capital
Flows,” Economic Journal, 314-329, 1980.
Mankiw, Romer and Weil. “A Contribution to the Empirics of Economic Growth,”
Quarterly Journal of Economics 107 (2), 1992, 407-437.
Stylized Facts of Economic Growth
Little growth in per-capita output before the Industrial Revolution: technological progress
and productivity gains translated into population growth rather than output growth; most
people attained little more than subsistence income throughout most of human history.
Average growth in GDP per capita: 0% in Western Europe and India during the first
millenium, 0.14% in Western Europe and 0.02% in India between 1000 and 1820.
Population growth was similarly 0% during the first millenium and 0.2% in Western
Europe and 0.13% in India between 1000 and 1820 [Maddison, 2001].
World population grew on average less than 0.1% per year between 1 and 1750 [Livi-Baci,
A Concise History of World Population, Blackwell, 1997].
Rapid growth in output and standard of living since the Industrial Revolution: per-capita
incomes increased 50-300 fold in Western Europe and the US during the past 200 years.
The Great Divergence: little difference in per capita incomes across countries until the
Industrial Revolution, on-going divergence since then. The ratio between per capita
incomes in the richest regions (US, Canada and New Zealand) and the poorest ones
(Africa) was 3 in 1820; this ratio has increased to 20 by 1998 [Maddison, 2001].
By the end of the first millenium, China and India were both richer and more
technologically advanced then Europe. By 1700, the core areas of Europe and Asia had
similar consumption levels and overall wellbeing. Subsequently, Industrial Revolution took
place in Europe whose growth has far outpaced that of China and India.
Different explanations for divergence and reversal or roles:
Landes (The Wealth and Poverty of Nations: Why Some Are So Rich and Some Are So
Poor, 1999): European nations embraced insitutions that encouraged entrepreneurship,
invention and technological progress. Competition between nations created a strong
incentive to utilize technological progress to gain a competitive (and military)
advantage. The latter was not the case in China, India or the Ottoman empire which
were subject to political hegemony and (initially) free from an outside threat.
Pomeranz (The Great Divergence: China, Europe, and the Making of the Modern
World Economy, 2001): Europe benefited from having convenient sources of coal that replaced timber at the eve of the Industrial Revolution. Because of trade and
colonization of the New World, Europe gained a new source of primary products. This
allowed increased population growth in Europe and, by releasing labor from
aggriculture, facilitated its specialiazation in manufacturing .
Growth often varies considerably over time: the same country may experience growth
accelerations and slowdowns: Mexico, Soviet Union/Russia, Japan.
Growth miracles – South Korea, Taiwan, Singapore, Hong Kong, and more recently China
and Botswana – and growth disasters – Argentina (one of the richest countries in 1900) and
Sub-saharan Africa.
Cross-country differences in growth rates and output levels are correlated with differences
in other measures of well-being: nutrition, literacy, infant mortality, life expectancy, etc.
Lucas (1988): Once one starts to think about economic growth, it is hard to think about
anything else.
Potential sources of growth:
factor accumulation: physical capital, labor and human capital;
technological progress;
institutions.
Neoclassical economics initially focused on factor accumulation (Solow and Ramsey
models of growth), then on technological progress (endogenous growth models). More
recently, institutional economics emphasises the importance of institutions. The Solow Growth Model
Robert Solow (1956), T.W. Swan (1956).
Assumptions
Savings and investment decisions are exogenous (no individual optimization). Factor
accumulation and technological growth are also exogenous.
Production function, with physical capital K, labor L and knowledge or technology A:
Y F K t
Time affects output only through K, L and A. Technology is labor-augmenting: AL is
effective labor. Land and natural resources are ignored (not considered among factors of
production).
Constant returns to scale (CRTS): FcK,cAL cF K,A for ny c ≥ 0.
Example: Cobb-Douglas function
1−
Y K AL
displays CRTS:
1−
cK cAL
c K c 1−AL 1−
cK A 1−
1
Because of CRTS, we can express Y t intensive form,taking c AL :
K 1
F AL ,1 AL FK,AL
K Y
or, denoting k AL , y AL and f F ,1 ,we can relate output per unit of effective
labor as a function of capital per unit of effective labor:
y f .
The intensive-form production function is assumed to have the following properties:
f 0
f 0
f 0
and Inada conditions:
lim
k→0
lim k 0
k→
Example: Cobb-Douglas function:
Y K A 1− which translates into
y k .
Diminishing marginal product of capital:
−1
f k 0
f" − 1− k −2 0
∂Y
Note that f ∂K:
∂Y −1 1−
∂K K AL
−1
K
AL −1
−1
k
Evolution of Effective Labor
Labor and knowledge grow at constant exogenous rates n and g:
L nL t
A gA t
Note that the growth rate of a variable equals the rate of change of its log:
dlnL dlnL t dL 1 L
dt dL dt L L L n
Using this, and taking the initial value of t as given:
dlnL
n
dt
dlnL ndt
lnL lnL 0
Therefore,
lnL lnL 0
lnL lnL 1 lnL 0 2
...
lnL lnL 0 nt
Similarly,
lnA lnA 0 t
Hence, for given initial levels of L and A, this implies that labor and knowledge grow
exponentially:
L L 0 e nt gt
A A 0
Dynamics of Capital
Assume constant and exogenous savings rate, s, (i.e. not a result of individual optimization
decision) and constant depreciation rate of capital, :
̇
K sY t K t
Dynamics of capital per unit of effective labor, k AL :
̇
k K − K A t
A 2
K − K L A
A A L A
sY K t − k g
A
sf − n g k
The first term, sf is the actual investment in physical capital per unit of effective
labor. The second term, n g k t he effective depreciation of capital per unit of
effective labor.
Steady-state (equilibrium) occurs at such value of capital per effective labor, k ,where
k 0:
∗ ∗
sf − n g k 0.
At k ∗, investment equals effective depreciation and k remains constant over time. Because
of the Inada conditions, there is only a single value of k 0.
Behavior of aggregate variables in the steady state: effective ∗abor, AL∗ grows at rate
n g Capital grows at the same rate (note that K ALk , with k constant). Because of
CRTS, aggregate output grows at rate n .
Output per unit of effective labor, y, is constant. Capital per worker, K , and output per
Y L
worker, L,growatrate g.
Balanced growth path
In the steady state, all variables grow at constant rates:
∗
Capital per unit of effective labor, k : constant;
Labor and technology grow at rates n and g, respectively;
Capital, K ALk grows at rate n g;
Because of CRTS, output, Y,alsogrowsatrate n g
Capital per worker, L , and output per worker, L ,growatrate g.
Hence, the equilibrium (steady state) rate of growth of output per capita is determined by
the rate of technological progress only. Comparative Statics: Change in the Savings Rate
Recall: in the steady state:
s k n g k ∗
The savings rate, s, is a key parameter of the Solow model. An increase in s implies higher
actual investment; k grows until it reaches its new (higher) steady-state value. In the
transition to the new steady state, the rate of growth of output per worker accelerates.
Once the new steady state is attained, all variables grow again at the same rates as before;
output per worker again grows at the rate of growth of technological progress, g,whichis
independent of s. An increase in the savings rate only leads to a temporary increase in the
growth rate of output per worker (but a permanent rise in the level of capital per worker and
output per worker).
In the Solow model, only changes in technological progress have permanent growth effects,
all other changes have level effects only.
Effect on Consumption
Household welfare depends on consumption rather than output.
Fraction of output that is consumed is 1− s sothat c 1 − f k n s increases, c
initially falls but then rises gradually as output per worker, , increases. Eventually, it
may be greater or smaller than before the change in s.
In the steady state (using sf n g k ∗)
∗ ∗ ∗
c f − sf
f − n g k ∗
How do changes in the savings rate affect consumption? Noting that k depends on s as
well as n, g,and :
∗
∂c∗ f s,n,g, − n g ∂k s,n,g, .
∂s ∂s
∂k ,n,g
The last term, ∂s , is always positive: an increase in the savings rate always translates
into a higher k .
The impact of changes in the savings rate on consumption depends on the sign of
k∗ − n g i.e. whether the marginal product of capital, f′ k exceeds
n g Because of the properties that we assumed to hold for the production function,
f k exceeds n g fr small values of k and falls short of it for large values of k.
Golden-rule level of capital: The highest possible level of consumption is attained at the
level of k such that
∗
∂c 0
∂s
or f n g
Note: in the Solow model, households/firms do not make any optimization decisions; the
savings rate is exogenous. Therefore, there is no reason to expect the golden-rule level of
k to prevail rather than some other level of k .∗
Effect on Output
∗
Effect on steady-state output, y :
∂y ∗ ∗ ∂k s,n,g,
f′
∂s ∂s
In the steady state k 0, therefore we can use the following property
∗ ∗
sfk s,n,g, n g k n,g, .
Differentiating both sides w.r.t. s:
∗ ∗
sf′k∗ ∂k fk∗ n g ∂k
∂s ∂s
∗ ∗ ∂k∗
f n g− sf′ ∂s
∂k∗
We can solve for ∂s:
∗ ∗
∂k fk .
∂s n g sf′
∂y∗
and substitute this back into the expression for ∂s above. The effect of changes in the
savings rate on output then is
∗ ∗
∂y f′ fk .
∂s n g sf′
Elasticity of output s.r.t. the savings rate can be obtained by multiplying both sides of the
above equation by y :
∗ ∗ ∗
∂y s s f .
∂s y∗ fk n g− sf′k ∗
∗ ∗ ∗ ∗
Note that in the steady state, s k n g k and s n g k/f k :
∂y ∗ n g ∗f′k∗
s∗ ∗ ∗ ∗ ∗
∂s y k n g− n g k f k
k ∗f
∗ ∗ ∗ ∗
k 1 − k f′k
k∗f′ ∗
∗ ∗ ∗
1 − k f′
k f′ k∗ /f k∗ is the elasticity of output w.r.t. capital at k k Denoting this k :
K
∂y∗ s K ∗
∂s y∗ 1 − K . To evaluate this, note that under competitive markers and in absence of externalities,
capital earns its marginal product
∂Y ∂ALf ∂k ALf′ 1 f′ .
∂K ∂k ∂K AL
The share of output earned by capital in the steady state is
∗ ∗
k f ∗
y K
1 ∂y∗ s 1
Given that this is approximately 3 in most countries, ∂s y∗ ≈ 2 . Thus, 1% increase in
the savings rate increases the steady-state level of output per effective labor by
approximately 0.5% – a modest effect. Relevance of the Solow Model
Two sources of variation in output per worker according to the Solow model:
Differences in capital per worker, K (these, in turn, depend on differences in the
L
savings rate and population growth);
Differences in knowledge, A.
The notion of knowledge is very vague in the Solow model: the very variable that drives
growth in the steady state is not analyzed by the model at all!
Differences in capital per worker can only account for a fraction of the observed
differences over time or across countries. Assuming a Cobb-Douglas production function,
y k with 1 , a 10-fold difference in output per worker would require a 1000-fold
3
difference in capital per worker:
Y Ak 3
L
Y 1
10 A 000k 3
L
Given the differences in capital per worker between rich and poor countries, the marginal
return to capital should be much higher in the latter and we should observe large flows of
capital from rich to poor countries. Neither is the case, however (Lucas, 1990, AER Papers
and Proceedings).
Assume all countries share the same CRTS production function: Y F K,AL , an have
access to the same technology. If we observe differences in output per worker, these must
be due to differences in capital per worker. Then, the marginal product of capital (and the
return to capital) should be correspondingly higher in the poorer country.
If movement of capital across countries is unrestricted, capital should flow from rich to
poor countries until capital per worker is equalized across countries (so that returns to
capital and to labor are equalized).
Assume standard Cobb-Douglas production function, Y K AL 1−,or y k in

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