Lecture notes on modern growth theories Part 1 Mario Tirelli Very preliminary material. Not to be circulated without permission of the author. October 21, 2019 Contents 1. Introduction 1 2. Preliminary considerations 3 3. The Solow-Swan growth model 5 3.1. Steady-state properties 9 3.2. Conclusions 11 4. The Solow model predictions and the empirical evidence 12 4.1. The main source of growth: Solow’s residuals 13 4.2. The Solow model and cross-country income variations 14
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Lecture notes on modern growth theories
Part 1
Mario Tirelli
Very preliminary material.
Not to be circulated without permission of the author.
October 21, 2019
Contents
1. Introduction 1
2. Preliminary considerations 3
3. The Solow-Swan growth model 5
3.1. Steady-state properties 9
3.2. Conclusions 11
4. The Solow model predictions and the empirical evidence 12
4.1. The main source of growth: Solow’s residuals 13
4.2. The Solow model and cross-country income variations 14
1. Introduction
Kaldor (1957) and Solow (1957) first highlighted the following empirical regularities on
growth, which are not totally undisputed today:
(1) Real output roughly grows at a constant rate, g.
(2) Real capital has roughly the same constant rate of growth of output, gK ≈ g.
(3) Labor input has a constant growth rate that is higher than that of real capital and
output, gN > g.
(4) The ratio of profits on capital Π/K and the real interest rate r are both roughly
constant.
(5) Economies with a high profit/income ratio Π/Y tend to have a high investment/output
ratio I/Y .
(6) Cross-country comparisons reveal a high variance of output per-capita and growth rates.
2) implies that the ratio of capital to output stays constant over time. Indeed, 2) says that
Kt+1
Kt− 1 =
Yt+1
Yt− 1, at all t
which trivially implies, Kt+1/Yt+1 = Kt/Yt, at all dates t.
Next, 2) plus 4) imply that the income distribution between capital owners and workers is
roughly constant over time. Indeed, they do imply that the share of income that goes to the
capital YK/Y is constant (i.e. it has a steady, zero growth rate): given that Y K , essentially,
equals the sum of the (real) rental income, rK, going to capital owners and (real) corporate
profits, Π, going to those who invest capital into production activities,
Y Kt
Yt= rt
Kt
Yt+
Πt
Yt
= rtKt
Yt+
Πt
Kt
Kt
Yt
=Kt
Yt
(rt +
Πt
Kt
)
Since the terms in parenthesis are roughly constant, by 2) and 4), we have the result.
If we assume capital market-clearing, that is I = sY , then 5) can be restated by saying that
the average saving rate s is higher in countries with higher return from investing Π/Y .
Moreover, if we assume that capital depreciates at a constant rate δ, the simple capital-
accumulation accounting is,
Kt+1 = It + (1− δ)Kt
By 1) and 2) the latest implies that the average saving rate s is constant.1 Therefore, the first
four facts (and constant depreciation δ) imply that the economy experiences a balanced growth,
1To see this, rewrite the accumulation equation as, It/Kt = g+δ and use the fact that It = stYt. This yields,
(s) st =g + δ
(Yt/Kt)
where the denominator is constant by 2).
1
with the main economic variables (Y,K, I, C, where C = (1− s)Y ) growing at a constant rate
g. Thus, at the balanced growth, the scale of the economy (the output level Y ) changes over
time, but the ratios of the variables to output tend to be constant.
It remains to consider fact 6). This is a very important stylized fact, raising many relevant
questions, which turn to be central in economics: Why are some countries so much richer than
others? How much of these income differences are explained by differences in growth rates?
Do countries with different per-capita income display convergence over time or not? Finally,
which are the determinants, or fundamental causes, of growth and growth differentials?
Some answers to the first questions are well exposed in Acemoglu (2009) (chapter 1). To
summarize, growth differentials are relevant in explaining cross-country income differentials
only if we take a sufficiently long time perspective; looking at post-war date does not suffice,
essentially because by the II world war time the variance of per-capita income was already
very high. Maddison’s studies indicate that, in its large proportions, this originated with
the industrial revolution in the XIX century. Moreover, there is not evidence of per-capita
income convergence across the world. Convergence is observed for those countries with similar
socio-economic characteristics or fundamentals. For example, in the post-war period, OECD
countries tend to show convergence: relative to the US, lower income countries have grown
faster, catching up or reducing their initial poorer condition. Thus, we can conclude that data
signal conditional convergence, as opposed to absolute convergence.
Equipped with all this empirical evidence we still have to understand which are the fun-
damental causes of growth, cross-country income and growth differentials, conditional conver-
gence. We need a theory. A model that could capture the growth stylized facts and explain
those fundamental causes.
In these notes we shall focus on modern neoclassical growth. To this end, we shall begin
with Solow’s growth model and then extend the analysis to Ramsey-Cass-Koopmans’.2
There are three fundamental reasons for studying modern neoclassic theory and its workhorse
model due to Ramsey, Cass and Koompmans (RCK). First, this theory addresses both growth
and business cycle as two integrated phenomena, with a unique economic model and set of
analytic tools.3 Roughly speaking, the long-run behavior of the economy (namely, its balanced
growth path) determines the variables’ secular trends. Instead, short-run dynamics around
these trends, describe the variables’ cyclical fluctuations. Some of these fluctuations character-
ize business cycles.
A second reason for studying RCK is that, unlike previous research, it builds up on general
equilibrium theory, with all the advantages that this implies in term of economic analysis and
policy. RCK allows both to analyze equilibria occurring at different policy regimes (e.g. tax and
2Cass, D.: ”Optimum Growth in an Aggregative Model of Capital Accumulation,” Review of Economic
Studies, Vol. 32 (1965). Koopmans, T,. C.: ”On the Concept of Optimal Economic Growth,” Semaine D’Etudes
sur le Role de L’Analyse Econometrique dans la Formulation de Plans de Developpement, Rome: Pontificia
Academia Scientiarum, 1965. Ramsey, F. P.: ”A Mathematical Theory of Saving,” Economic Journal, Vol. 38
(1928).3This idea is not new in economics and emerges, for example, in the works of Hicks and Goodwin in the
1950s, as well as in the structural macro-econometric models of Tinbergen, Klein and Modigliani in the 1960s.
2
fiscal policy schemes) and their welfare properties. Hence, welfare analysis, based on efficiency
and distributional considerations, can be used to guide policy decisions.
A third reason for studying RCK is that the theory passed a few important empirical tests,
being able to match many important statistics characterizing economic growth and business
cycles; early models were simple and still able to provide a good representation of complex
phenomena.
Finally, the neoclassical general equilibrium approach has been adopted by the New-Keynesian
school and its models exploited by most of the international institutions involved in economic
analysis (central banks, IMF, public authorities, etc.). Modern general equilibrium models
(computable general equilibrium models - CGE) are often sophisticated, incorporating more
realistic descriptions of the economy, including financial market frictions and other market fail-
ures (e.g. externalities, public goods, imperfect competition and asymmetric information), but
they are all deep-rooted in the RCK model.
2. Preliminary considerations
Assumption 1. The technology is represented by a production function, F : R3+ → R+,
F (Kt, Nt, At)
that satisfies the following properties.
• It is twice continuously differentiable in K,N and it is strictly increasing in K,N ,
FK , FN > 0, concave in K,N , FKK , FNN ≤ 0, and F (0, ·, ·) = 0.
• It has constant return to scale in the input factors K,N (CRTS).
• It satisfies Inada conditions:
limx→0
Fx = +∞, limx→∞
Fx = 0, for x ∈ {K,N}
The CRTS assumption, formally, says that the production function is homothetic in the
input factors (implying homogeneous of degree one in K,N). Hence, by Euler’s Theorem,
F (Kt, Nt, At) = FK(Kt, Nt, At)Kt + FN (Kt, Nt, At)Nt, for all (Kt, Nt, At)
and FK , FL are homogeneous of degree zero in K,N .
Observe that At represents the technological progress. Three typical specifications are, the
Hicks-neutral, the Harrod-neutral and the Solow-neutral technological progress. In the first,
At shifts up and down isoquants, G(Kt, Nt), that is F (Kt, Nt, At) = AtG(Kt, Nt). In the other
two, At enters as the multiplicand of Nt and Kt, respectively. For example, for a Cobb-Douglas
function,
F (Kt, Nt, At) = AtKαt N
1−αt , F (Kt, Nt, At) = Kα
t (AtNt)1−α, F (Kt, Nt, At) = (AtKt)
αN1−αt
We say that the technological progress is exogenous (i.e. part of the economic fundamentals)
when, as we do here, we assume that its dynamics is governed by a given, exogenous, model.
Typically, one assumes that (the deterministic part) of this model has a constant per-period,
growth rate, µ,
µ :=At+1 −At
At, at all t ∈ T
3
Hence, the dynamics of technological progress, at a given initial level, say, A0 = 1, is,
A1 = (1 + µ)A0 = 1 + µ
A2 = (1 + µ)A1 = (1 + µ)2
...
At = (1 + µ)t
A similar assumption is often used for demographics. The population, or work-force, grows
at a constant, per-period rate n, according to the deterministic model,
Nt = (1 + n)tN0, t ∈ T
Next, for any variable Xt, let Nt := AtNt and xt := Xt/Nt be that variable in efficiency
units. In the case of a Harrod-neutral, Cobb-Douglas technology F (Kt, Nt) = Kαt N
1−αt ,
yt :=YtNt
=Kαt (Nt)
1−α
Nt=
(Kt
Nt
)α= kαt
Define the function f : R+ → R to be such that
f(kt) = F (kt, 1)
Clearly, under the above assumptions, f is a differentiable functions, which is strictly increasing
and strictly concave. Moreover, it satisfies Inada conditions in k. You should verify such
properties as an exercise.
Remark 2.1 (Production under CRTS). CRTS technologies have the property that the productive-
efficient level of output of a single plant/firm is not determined. This is just the consequence
of the fact that a technology is homogeneous of degree one in inputs. So that, proportionally
increasing inputs one attains an disproportional increase in output; hence, the scale of produc-
tion does not affect the amount of input required to produce a unit of output. More formally, if
F is CRTS, FK and FN are homogeneous of degree zero and, by Euler’s theorem,
Yt = FK,tKt + FN,tNt
Thus, any production scale λY , λ > 0, can be achieved by proportionally expanding inputs,
k∗ is (locally) asymptotically stable if and only if |1 + g′(k∗)k∗| < 1. Since we have shown that
g′(k) < 0 for all k > 0, we immediately have that 1 + g′(k∗)k∗ < 1. So, we only have to show
that 1 + g′(k∗)k∗ > −1, that is g′(k∗)k∗ > −2. Observe that, by definition,
g′(k∗)k∗ = s
[f ′(k∗)−
f(k∗)
k∗
]> −sf(k∗)
k∗= −(δ + η)
Hence, a sufficient condition is δ + η < 2, which is typically true (annual data say that,
δ ≈ 3− 4%, n ≈ 1− 2%, µ ≈ 1− 2%, overall about 5− 8%).
Finally, letting kt be initial capital (if you like, relabel kt as k0), one can observe that two
countries with same fundamentals but different levels of accumulated capital, say k1t < k2
t < k∗,
are characterized by the poorer 1 growing faster than the richer 2 (i.e., k1t+1 − k∗ > k2
t+1 − k∗,given that the two countries have the same steady state k∗ and function g(·)).
Figure 1. k∗ is the unique, stable steady state
10
Since g is differentiably strictly convex, it is natural to guess that the steady state is also
globally stable. The proof is simple and can be found in Acemoglu (2009) (proposition 2.5, p.
45).
3.1.4. Speed of convergence. Equation (8) can also be used to determined the speed at which
the economy converges as a function of the initial distance from the steady state. Indeed,
kt+1 − k∗ = [1 + g′(k∗)k∗](kt − k∗)
= [1− (1− α)(δ + η)](kt − k∗)
The latest can be rewritten as,
(9) kt+1 − kt = −[(1− α)(δ + η)](kt − k∗)
This says that the higher is the distance from steady state at t, in the sense of (kt − k∗) < 0,
the higher will be the capital accumulation from t to t + 1, ∆kt+1. The speed of convergence
is by how much ∆kt+1 changes with such distance; which is measured by [(1 − α)(δ + η)]. In
problem 1 below, you are asked to show that the speed of convergence of income is α times
that of capital.
3.2. Conclusions. To summarize, the Solow model allows to, at least, capture some basic,
qualitative, empirical facts.
• Variables such as income consumption and capital tend to grow at the same rate and
the ratio of capital stock to output is roughly constant.
• The level of balanced growth g depends on demographics, the rate of growth of popu-
lation (or labor force) and on the rate of technological progress.
• Two economies (or countries) with same fundamentals, but different saving rates would
have the same rate of growth; that is, contrary to the empirical evidence, saving rates do
not covariate with the growth rate in the medium- and short-run too. A change in the
saving rate does only affect the steady-state variables in levels and the capital-output
ratio.
• Two economies with the same fundamentals, except for their initial capital stock in
per-capita terms, are characterized by the fact that the poorest will grow faster, and
catch-up (conditional convergence).
Problem 1. Show that the speed of convergence of y (approximately) is α[(1−α)(δ+η)].[Hint.
Use the fact that ∆kt+1/kt ≈ log kt+1 − log kt, to rewrite equation (9) as,
log kt+1 − log kt = [(1− α)(δ + η)](log k∗ − log kt). Then, use the production function. ]
Problem 2. Consider the Solow model, for simplicity, without technological progress.
Let the per-capita production function be f(k) = 3√k, the depreciation rate δ is 10%, and the
population growth rate n is 5%. Further, assume that individuals save 30% of their income.
(1) What are the steady state, per-capita values of capital, output and consumption (kt, yt, ct)?
(2) What happens when the saving rate is s = .4? and when n = .06 (and s = .3)?
11
(3) Suppose the production function is of the form
F (K,N) =KN
.25N + .5K
Give conditions under which a steady state exists.[Hint. this depends on the fact that
k > 0 and implies that the interest rate is ”high enough”. This condition was un-
necessary in the first item because the Inada conditions were satisfied; here they are
not!?]
Problem 3. Do some comparative statics, also using the following graphical representation.
(1) Argue, also graphically, that a permanent increase in s has temporary effects on the
growth rate and permanent ones on the steady-state levels of k∗ and y∗.
(2) Explain the effects a of a permanent increase in s on investments and consumption.
(3) Explain the effects of a permanent increase in the rate of growth of technological progress
µ on (K,C, I).
sf(k)
(∆+g)k
k*
10 20 30 40 50 60
-1
1
2
3
4
5
6
4. The Solow model predictions and the empirical evidence
Section 3.2 already provides a partial answer to the question of how good is the Solow model
in explaining the main key facts on long-run macro dynamics. Yet there is more to be assessed.
Indeed, so far, our discussion has only been qualitative; we have not said anything on the
potentials of the model to actually capture and predict economic figures. Can we use the
Solow model to do quantitative analysis, at least, for what concern the interpretation of past
dynamics? The answer to this question is important since it also provides some key elements
and thoughts on the directions of subsequent research and theory development, which we shall
examine later in the course.
12
4.1. The main source of growth: Solow’s residuals. Solow, using US data from 1901 to
1949, estimated that changes in productivity accounted for the 87.5% of the growth of output
per-worker, while only about 12.5% was accounted for by increased capital per worker (i.e.
to capital accumulation). Revising his analysis with more recent data one finds that the two
components, respectively, account for the 2/3 and 1/3 of the growth of output per-worker.
Instead the contribution of variations in labor input is roughly zero (recall that the changes of
the average hours of work, per-worker, has no trend).
Let us summarize Solow’s analysis. Assume that output is represented by an aggregate
production function, whose arguments are labor (N), capital (K) and an index of technological
productivity (A), say the total factor productivity (TFP),
Yt = F (Kt, Nt, At)
Take the logarithmic transformation, lnYt = lnF (Kt, Nt, At), and totally differentiate it with
respect to time. Then, use the definition of instantaneous rate of change,
∂(lnYt)/∂t = (∂Yt/∂t)/Yt =: Y /Y
Y
Y=
1
Y
(FKK + FN N + FAA
)=
1
Y
(FKK
K
K+ FN N
N
N+ FAA
A
A
)
=
εK︷ ︸︸ ︷FK
K
Y
K
K+
εN︷ ︸︸ ︷FN
N
Y
N
N+ FA
A
Y
A
A
= εKK
K+ εN
N
N+ FA
A
Y
A
A︸ ︷︷ ︸u
where εK and εN denote the elasticity of output with respect to K and N , respectively.
u := FAA
Y
A
A
denotes the contribution of technological progress (TFP change) to growth. Solow estimates u
as the residuals (thereafter called the Solow residuals) of the following regression,
g = γ1gK + γ2gN + u
where (g, gK , gN ) are the growth rates of output, capital and labor input. The result is a low
R-squared, denoting that most of the long-run growth is not explained by (gK , gN ) and that
γ1 ≈ 1/3 and γ2 ≈ 0.8
8A similar analysis has been carried out to estimate the contribution of inputs variation in explaining real
business cycles. Results typically reveal that both total factor productivity and labor productivity are procyclical
and slightly lead the cycle. The contribution of variations in labor input is about 2/3 and that of changes of
TFP is about 1/3; while, capital per-worker is about constant over the business cycle. See, for example, chapter
1 in the Handbooks of Macroeconomics vol. 1. Stock J. and M. Watson, 1999. ’Business Cycle Fluctuations in
US Macroeconomic Time Series’.
13
Finally, notice that adopting the Cobb-Douglas specification, α = εK ≈ γ1 = 1/3 explains
why in most empirical analysis this assumption is made.
4.2. The Solow model and cross-country income variations. Mankiw, Romer and Weil
(1992) (hereafter, MRW) argued that,9
the empirical predictions of the Solow model are, to a first approximation, con-
sistent with the evidence. Examining recently available data for a large set of
countries, we find that saving and population growth affect income in the di-
rections that Solow predicted. Moreover, more than half of the cross-country
variation in income per capita can be explained by these two variables alone.[p.
407]
Thus, countries with higher income per capita tend to have higher saving rate s and lower pop-
ulation growth n. Yet, observe that MRW only talk about quality predictions. Quantitatively,
the effect of population growth and of the saving rates are too large. Let us briefly see this.
As above, we treat productivity (the A term) as due to level of technology and assume that
technology is a labor-augmenting Cobb-Douglas, Yt = Kαt (AtNt)
1−α. Recall that the economy
growth rate is g = η = n+ µ and that the steady-state capital (in efficiency units) is,
k∗ =
(s
δ + n+ µ
) 11−α
In per capita, the production function is,
YtNt
= At
(Kt
AtNt
)α= Atk
αt
Taking logs on both sides and evaluating at the steady state (kt = k∗),
ln
(YtNt
)= lnAt + α ln k∗
= lnA0 + t ln(1 + µ) +α
1− αln s− α
1− αln(δ + n+ µ)
Since, n and s vary across countries, the capital stock (per capita) will also vary explaining,
explaining different levels of income per capita. t ln(µ+ 1) is a linear trend with slope equal to
the (gross-) rate of growth of labor productivity (or technological progress) A.
The empirical (Solow) model is, for each country j in the cross-section,
ln
(YjNj
)= a0 + b1 ln sj + b2 ln(δ + µ+ nj) + εj
where a0 = lnA0 is the intercept capturing, not only technology, but resource endowments,
climate, institutions of country in the base year (1960, while data refer to 1985);10 ε is an error
term (also interpretable as the stochastic component of technological progress) which is assumed
to be independently distributed across countries and independent on the regressors. Moreover,
9Mankiw, N. G., Romer, D., Weil, D. N. (1992). A Contribution to the Empirics of Economic Growth. The
Quarterly Journal of Economics, 107(2), 407-437.10MRW take investments and population growth as the averages of the sample variables for the period
1960-1985; they also assume that µ+ δ is 0.05. See the sample description below.
14
one can measure the average saving rate using Say’s law, sYt = It, as the investment-income
ratio. In the regression, one can test or restrict coefficient to satisfy,
b1 = −b2 =α
1− α
Figure 2. MRW table I - OLS regressions.
Coefficients on investment (capturing the saving) rate (ln(I/Y )) and population growth have
the predicted signs and are significant in most samples.11 Equality restriction on coefficients b
can’t be rejected. Regressions explain large fraction of the cross-country variation in income
per capita: the adjusted R-squared in the intermediate sample is 0.59. This is in contrast with
the common wisdom and previous empirical findings, which says that the Solow model explains
most of the cross-country income variation, based on differences in labor productivity (again,
the A term). Nonetheless, estimates on different country groups reveal that the impacts of
saving and labor force growth are much larger than model predicts: the value of α implied
by the coefficients tends to exceed the capital-income share of 1/3 (it is more than 1/2). If
11The first (non-oil) sample includes 98 countries; oil countries are excluded because most of their GDP comes
from oil extraction, as opposed to value added. The ‘intermediate sample’ is obtained from the first, dropping
23 countries, mostly with a small population. The OECD sample contains 22 countries with population greater
than one million. The sample years are 1960 and 1985. In the table g is the rate of growth of technological
progress, corresponding to our µ.
15
one had to constraint the coefficient to be 1/3, then the constrained regression would see the
R-squared drop from 0.59 to 0.28.12
Therefore, quantitatively, MRW conclude that the effect of population growth and of the
saving rates are too large. For this reason they suggest to go beyond the Solow model textbook
form, toward one including a broader definition and specification of ‘capital’. Economists agree
that reducing accumulation to savings in physical capital is fallacious. In particular they have
long stress the importance of human capital accumulation to explain economic growth; where,
they normally summarize with human capital accumulation activities such as work training,
schooling and others (e.g. including health care), which enhance labor productivity through
a costly and timely investment. Just to grasp the concept, a simple representation of human
capital into a standard Cobb-Douglas technology is,
Yt = AtKαt (htNt)
1−α
where ht is the quantity of labor supplied by each of the Nt workers. Intuitively, everything
else equal, ht increases with education (e.g. years and quality of schooling), with work training,
with health-care accessibility (e.g. the extension and quality of the public health-care system),
and the latest are all increasing in per-capita income.
The practical reason why introducing human capital into the Solow model might improve
its empirical predictions is twofold.
(1) For any saving rate into human capital, higher s or lower n leads to higher Y/N and this
increases human capital accumulation, activating a further indirect effect that boosts
up Y/N ; hence, one can now explain higher Y/N with more plausible levels of s and n;
or, for actual values of s, n, one attains lower estimates of α.
(2) Human capital accumulation may be correlated with saving rates s and population
growth n: the population spends more time in the education system in countries with
a better system; this decision is costly, implying both that part of the family income is
saved and invested into education (this lowers saving that goes into physical capital s)
and that people enter in the labor force later in life (something that contributes to lower
n). Therefore, omitting human capital biases upward the estimates of the coefficients
attached to s and n.13
Since including human capital into the Solow model implies a change in the theory, we
postpone its analysis.
12Estimates based on more recent data confirm these results. See ‘Is growth exogenous? taking Mankiw,
Romer, and Weil seriously’, in Ben S. Bernanke & Kenneth Rogoff, 2002. NBER Macroeconomics Annual 2001,
Vol. 16, NBER Books, National Bureau of Economic Research, June.13Not accounting for human capital accumulation and fixing α to (say) the true value, implies that one can
explain the higher Y/N of a country with higher investment in human capital only with a saving rate s higher
than the true rate. Analogously, if for this country one uses the true s and estimates α with the Solow regression,