Theory of economic growth
Slides made by: K˚are Bævre, Department of Economics, University of Oslo
Slight adjustment: Miroslav Hlouˇsek, ESF MU
4 The Solow-model and growth econometrics
Required reading: Mankiw (1995), Mankiw, Romer and Weil (1992) (sections:
I, II.A, III.A), BSiM: 1.2.10-1.2.11, 10.1-10.2,10.5
4.1 The text-book model and stylized facts
• We now try to let the text-book Solow-model explain the main patterns
of how income and growth differs between countries. We assume that
all countries share the same production function.
• The parameter α plays a crucial role in the formulaes for the quantitative
implications we derived in section 3.2.3. Note that
α =
f (k)k
y
=
FKK
Y
=
RK
Y
= Capital’s share of income
• It is a fairly general finding that capital’s share of income is approximately
1/3. Hence we can use the estimate α = 1/3 to calibrate and
get the predicted size of the effects described above.
• Most formulaes we derive for the special case of a CD-production function,
also hold (at least as approximations) with a general production
function. One important exception is relationship between y and R
where elasticity of substitution between capital and labor appears.
4.1.1 The magnitude of international differences
• α = 1/3 implies that a four times higher savings rate only implies a
twice as high level of production per capita. But we need a model that
is able to explain that income levels can vary by a factor of 10 (at least).
The differences in s and n needed to account for such differences are
far to high.
• α/(1 − α) must be higher ⇒ we need a larger α!
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• Alternatively: The results in Mankiw, Romer and Weil (1992), section
I, show that the estimated effects of saving and population growth are
too strong to fit with the model. Besides: the empirical model does
not explain too much of the data (low R2
).
4.1.2 The rate of convergence
• With α = 1/3, and a n of 1 per cent, a x of 2 per cent and a δ of 3 per
cent (yearly rates), we get a rate of convergence β = 4 per cent.
• But the observed rate of convergence is roughly 2 per cent ⇒ we need
a larger α!
4.1.3 The rates of return
• With α = 1/3, a poor country where income is only 1/10 of that in a
rich country would have rates of returns that where 100 times higher
than in the rich country.
• This must be moderated if σ > 1, which seems plausible. See the
discussion in Mankiw (1995) page 287–288.
• But still, we do not observe anything close to this, and the flow of
capital from rich to poor countries is very modest.
• 1−α
ασ
is too large ⇒ we need a larger α!
4.2 The augmented Solow model (human capital)
4.2.1 A reassessment of capital
• There is more to capital than only physical capital.
• Levels of human capital have risen considerably.
• A reinterpretation of the Solow model where K is a broader measure
of capital will increase the elasticity α.
• Note that if we interpret human capital into K we must take into
consideration that an important share of the wages we observe is a
remuneration of the human capital of the workers, and hence should
be included in income accruing to K.
• Increasing α is the solution to all the three problems raised by Mankiw
(1995).
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• To see this somewhat more formally, we augment the production function
to include human capital
Y = Kα
Hη
(TL)(1−α−η)
⇒ ˆy = ˆkαˆhη
(1)
• We preserve the assumption of constant returns to scale (the exponents
sum to 1).
• We assume that α + η < 1, so there is (still) decreasing returns to the
accumulated factors.
• Consumption, physical capital and human capital are prodcued by the
same production function, i.e. we produce skills very much like cars
and computers.
• We will later return (topic 13) to the plausibility of the assumption of
single sector production.
• Savings can now be used to invest in both new physical capital (K)
and human capital (H).
• For simplicity we assume that both types of capital depreciates at the
same rate δ. Then the fundamental equation becomes
˙ˆk +
˙ˆh = sˆkαˆhη
− (n + x + δ) · (ˆk + ˆh)
• Equality of rates of return to physical and human capital requires that
α
ˆy
ˆk
− δ = η
ˆy
ˆh
− δ ⇒ ˆh =
η
α
ˆk (2)
i.e. it will require that there is a fixed relationship between ˆk and ˆh.
• Note that we by assumption immediately readjust any combination of
K and H to achieve this ratio. (Turn K into H or vice versa). Is this
plausible?
• Using (2), we can rewrite the fundamental equation to
˙ˆk = sAˆkα+η
− (n + x + δ)ˆk
where A = ηηα1−η
α+η
is a constant.
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• Thus we are back to a simple fundamental equation for ˆk just like
the one we had in the case with only physical capital. The only, but
important, difference is that α is replaced by α + η. I.e. it is as if we
have a larger α in the text-book model.
• Note that we can characterize the full system by a single equation for
ˆk, because movements in ˆh will always follow the movements in in ˆk
due to (2).
• Why does inclusion of human capital improve our predictions?
1. Differences in savings rates affect how much we have of the accumulated
input. The role of the accumulated input is now larger
(both physical capital (elasticity α), and new human capital
(elasticity η)), and hence translates in larger differences in y.
2. Convergence is slower. Informally: there is more inertia, because
we have a broader base for capital. Formally: diminishing returns
sets in more slowly because the production function is less concave
in the accumulated inputs (α+η > α), hence we get to the steadystate
more slowly.
3. Given differences in y translates to smaller differences in rates
of return because the marginal return to the accumulated factor
declines more slowly.
4.2.2 An alternative formulation, Mankiw-Romer-Weil
• Above we assumed that production were distributed trough investments
on the two types of capital so that rates of return were equated.
• In the long run, equality of returns seems reasonable. But the ability
to substitute freely between H and K is perhaps not always plausible.
• For this reason it is worth also considering an alternative formulation.
This formulation is particularly important because it is employed in a
very influential study by Mankiw, Romer and Weil (1992), from now
on MRW.
• We now assume instead that an exogenous and fixed share, sk of income
is invested in physical capital, and a share sh in human capital. That
is:
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˙ˆk = sk
ˆkαˆhη
− (n + x + δ)ˆk (3)
˙ˆh = sh
ˆkαˆhη
− (n + x + δ)ˆh (4)
• This system is basically the same as before, but since we now have two
dynamic equations the details become somewhat more complicated.
The problem is to ensure that there exists a steady-state.
• Consider a diagram in (ˆk, ˆh)-space. Draw the line characterizing the
values of ˆk and ˆh for which
˙ˆk = 0, and a similar curve for the case
where
˙ˆh = 0. Show that you end up in the (unique) point where the
two curves intersect, i.e. the steady state.
• The steady state (
˙ˆk =
˙ˆh = 0) gives:
ˆk∗
=
s1−η
k sη
h
n + x + δ
1/(1−α−η)
(5)
ˆh∗
=
sα
k s1−α
h
n + x + δ
1/(1−α−η)
(6)
(7)
• Plugging this back in the production function we find that the income
per capita in the steady-state can be written as
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ln((Y (t)/L(t))∗
) = ln(T(t))+
α
1 − α − η
ln(sk)+
η
1 − α − η
ln(sh)−
α + η
1 − α − η
ln(n+x+δ)
(8)
which is the equivalent of equation (9) in lecture note 1 for the model
without human capital.
• This log-linear formulation is very convenient for empirical analysis,
because it can be implemented in a familiar linear regression framework.
References
[1] Mankiw, N. Gregory, The Growth of Nations, Brookings Papers on
Economic Activity, 1995, 275-310.
[2] Mankiw, N. Gregory, Romed David and Weil, N. David, A
Contribution to the Empirics of Economic Growth, Quarterly Journal
of Economics, May 1992, 107 (2), 407-437.
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