Lecture Notes 2 Economic Growth
Jie Chen
1 Introduction
Q: What Economic Growth implies?
o Vast differences in standards of living over time and across countries
o Enormous variations in human welfare
击节碎o Welfare conquences of long-run growth swamp the short-run fluctuations.
o Differences in economic growth are associated with differences in nutrition, literacy,
infant mortality, life expectancy, and other direct measures of well-being.
Q: What is the performance of economic growth?
o Changes in incomes and output, in absolute and relative levels
o Large variations in performance: growth miracles and growth disasters
o Over the whole of modern era, cross-country income differences have widened on
average.
Q: What economic models can tell us about economic growth?
o Through models’ mechanics, our goal is to learn what insights they offer concerning
二级建造师培训学校
worldwide growth and income differences across countries.
o The ultimate objective is to determine whether there are possibilities for raising
overall economic growth or bringing standards of living in poor countries clo to
tho in advanced countries.
2 The Malthusian Model of Economic Growth
Malthus argued: advances in the technology for producing food
o incread population growth
o no increa in the standard of living unless there were some limits on population
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o A dynamic model with many periods
o Confine attention to what happens in the current period and the future period
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Aggregate production function with constant returns to scale:
(2.2.1) Y = z F(L, N)
where Y is current aggregate output
L is current fixed supply of land
N is current labor.
There is no investment and no government spending.
Assume each person is willing to work at any wage and has one unit of labor to supply (a
normalization), so that in Equation (1), N is both the population and the labor input.
Suppo population growth depends on the quantity of consumption per worker
(2.2.2) )('N
C g N N = where N ’ is the population in the future (next) period
g is an increasing function
C is aggregate consumption
so that C/N is current consumption per worker
dg/d(C/N) > 0, mainly due to the fact that higher food consumption per worker reduces death
rates through better nutrition.
All goods produced are consumed, so C = Y .
Hence ,
(2.2.3) C = Y = zF(L,N)
英文电影对白Then u Equation (2.2.3) to substitute for C in Equation (2.2.2):
(2.2.4) )),(('N
N L zF g N N =
The constant-returns-to-scale property of the production function implies that
)1,(),(N
L zF N N L zF N Y == After multiplying each side by N , Equation (4) can be rewritten as
(2.2.5)N N
L zF g N )1,(('=
Population growth depends on consumption per worker in the Malthusian model
• N* is the steady state for the population
• If N < N*, then N’ >N, population increas
• If N > N*, then N’ >N, population decreas
How uful is the Malthusian Model of Economic Growth?
• Before the Industrial Revolution in about 1800: economic growth consistent with the
Malthusian Model
• From the perspective of the 20 and early 21st century: Malthus was proven wrong
Why?
¾ Did not allow for the effect of increas in the capital stock on production, and
¾ Did not account for all the effects of economic forces on population growth
3. Solow Model: Exogenous Growth
3.1. The model tup
The Solow model is the starting point for almost all modern analys of economic growth. Even models that depart fundamentally from Solow’s are often best understood through comparisons with the Solow model.
Four variables: output (Y ), capital (K ), labour (L ), and “knowledge” or the “effectiveness of labour” (A ).leftbehind
let(2.3.1) ))()(),(()(t L t A t K F t Y =, where t denotes time.
Note that, A , and L , enters multiplicatively and AL is referred to as effective labour input . This way of technological progress expression is known as labour-augmenting or Harrod-neutral. If instead, A , and K , enters multiplicatively and AK is referred to as effective capital input . Technological progress is capital-augmenting. In addition, if knowledge enters in the form Y = AF(K, L), technological progress is Hicks-neutral,
Critical assumptions of the Solow model:
The production function is CRS (constant returns to scale) in its two arguments.
(2.3.2) F(cK, cL) = cF(K, L) for all C ≥ 0
Implications of this assumption:
1. Don’t’ take account of the gains from specialization. The economy is big enough.
2. Inputs other than capital, labour and knowledge are relatively unimportant . The availability of nature resources is not a major constraint on growth.
Therefore, (2.3.2) implies that
(2.3.3) ),(1)1,(AL K F AL
AL K F =
Define k = K/AL (the amount of capital per unit of effective labour), y = Y/AL (output per unit of effective labour), and f(k) = F(k,1), we can rewrite (2.3.1) as
(2.3.4) y = f (k )
The intensive-form production function, f(k), is assumed to satisfy that f(0) =0, f’(k) > 0, f’’(k)
< 0. In addition, it is assumed to satisfy the Inada conditions , ∞=→)('lim 0k f k , 0)('lim =∞
→k f k . Otherwi the path of the economy may not converge.
The esnce of growth model concerns how the stocks of capital, labour, and knowledge evolve ove
r time.
The initial levels of capital, labour, and knowledge are taken as given (denoted as L(0), A(0), and K(0)). We assume that labour and knowledge grow at constant rates:
(2.3.5) )()(t nL t L =•
(2.3.6) )()(t gA t A =•
where n and g are exogenous parameters and where a dot over a variable denotes a derivate with respect to time (that is, dt t dX t X /)()(=•
).
The pha the growth rate of X refers to the quantity )(/)(t X t X •. But we also know
that dt t X d t X t X /)(ln )(/)(=•. Thus,
(2.3.7) nt e L t L )0()(=
(2.3.8) gt e A t A )0()(=
That is to say, L and A each grow exponentially.
The final step of the Solow model tup concerns the division of output between consumption and investment. It is assumed that the fraction of output devoted to investment, s , is exogenous and constant. But at the same time existing capital also depreciates at a constant rate δ. Thus:
(2.3.9) )()()(t K t sY t K δ−=•
We now complete the description of the Solow model.
3.2. The dynamics of the Solow model
Since the evolution of labour and knowledge are exogenous in the model, to characterize the dynamics of the economy, we must analyze the development path of capital, K . Since by definition of k(t),
(2.3.10) )
()()()(t L t A t K t k = taking natural log of both sides,
)(ln )(ln )(ln )(ln t L t A t K t k −−=
Differentiate with respect to time t , we have
)
()()()()()()()(t L t L t A t A t K t K t k t k •
•••−−=
Using (2.3.10), )()()()()()()()()()()()(t k t L t L t k t A t A t L t A t K t K t K t k •
•••−−=
Using (2.3.5), (2.3.6) and (2.3.9),
)(n )(g )
()()()()(t k t k t L t A t K t sY t k −−−=•δ )(n )(g )()
()()(Y t k t k t k t L t A t s −−−=δ )()()())((t gk t nk t k t k sf −−−=δ
which reduces to
(2.3.11) )()())(()(t k g n t k sf t k δ++−=•
Equation (2.3.11) is the key equation of the Solow model .
It states that the rate of change of the capital stock per unit of effective labour is the difference between two terms: sf(k(t)), the deepness of capital, which is actual investment per unit of effective labour; (n+g+δ)k , the breakeven investment , the amount of investment that must be done to keep k at its existing level.
First, existing capital stock is depreciating at rate of δ and this capital must be replaced to keep the capital stock from falling: this requires δk amount of investment. Second, the quantity of effective labour is growing at rate n + g , thus the capital stock must grow at rate of n + g to hold k steady.
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3.3. The Steady-State 0)()()())((>⇒++>•t k t k g n t k sf δ, Saving larger than the breakeven investment
0)()()())((<⇒++<•
t k t k g n t k sf δ, Saving smaller than the breakeven investment
0)()()())((=⇒++=•t k t k g n t k sf δ, Saving equal with the breakeven investment, actual investment equals the desired investment
The steady state: 0)(=•t k , *k k =, L grows at n , A grows at g , the capital stock K (= ALk ) grows at n + g , output Y grows at n + g (CRS assumption), capital per worker (K/L ) and output per worker (Y/L ) grows at g .
No matter where k starts, it converges to k * and this leads to the balanced growth path . On the balanced growth path, all inputs grow at a constant and exogenous rate, and the growth rate of output per capita is determined solely by the rate of technology progress, g .
Proof: 0)(')(]/)([]/)([22'<−=−−==k
s k k kf k f s dk k k f d s dk k k sf d F AL
Implication: rich country grows faster than poor country becau they have higher initial capital stock and larger saving rate.s c
3.4. The impact of a change in the saving rate
linakThe Impact on output: in the short run
In short: a permanent change in the saving rate has a level effec t but not a growth effect : it
changes the economy’s balanced growth path, and thus increas the level of Y/L permanently;
