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,

λ(K,N):4 Indeed, FK(Kt, Nt) = FK(λKt, λNt) and FN (Kt, Nt) = FN (λKt, λNt); hence,

λYt = λFK,tKt + λFN,tNt

This does also imply that, for given input prices (taken as invariant with respect to the pro-

duction scale) CRTS technologies can accommodate any scale of production at the same unit

(and marginal) cost; implying there is no efficient–minimum unit cost– scale. Also, any pro-

duction scale yields zero profit. If prices are, (ν, w) = (FK , FN ), which it is going to be at an

4Graphically, in a plane (K,N), CRTS technologies have technical rates of substitutions, FK/FN which are

constant across isoquants if measured along any ray from the origin. Fixing any K′, N ′ and expanding them

by λ > 0 generate a ray originated in (K′, N ′) of constant slope N ′/K′. Along a ray, input and output expand

proportionally (as λ increases), but since FK/FN stays constant (by homog. of degree zero), each point in which

the ray crosses an isoquant λF (k′, N ′) corresponds to the same technical rate of substitution.

4

interior firm optimum, total production costs change proportionally with (Kt, Nt) and revenues

Yt = F (Kt, Nt).5

In the special case of a Cobb-Douglas technology,

Yt = αYtKtKt + (1− α)

YtNtNt

implies that YtKt

:= κ and YtNt

:= η remain constant as we change (Kt, Nt) in the same proportion.

To summarize, under CRTS the efficient, as well as the competitive equilibrium, number

of firms/plants and their individual production levels are undetermined. This is essentially,

without loss of generality, one can assume a representative firm.

3. The Solow-Swan growth model

Solow (1956) and Swan (1956) growth model is a simplified version of RCK, in which con-

sumers’ behavior is not represented endogenously: households inelastically supply labor and

have a constant average saving rate s (i.e. also a constant average propensity to consume,

c = 1− s). Hence, aggregate consumption is,

Ct = (1− s)Yt

More precisely, a Solow-Swan economy E is represented by its fundamentals

E = (F, s, n, µ, δ,K0, N0, A0)

respectively, a technology, an average saving rate, a rate of growth of the population of workers,

a rate of technological innovation (or change), a rate of capital depreciation, an initial capital

stock and population of workers, an initial productivity level. Time t evolves over an infinite

horizon, in T = {0, 1, 2, ...}. In every period t, capital and labor are demanded by a represen-

tative firm so as to maximize its profits. Labor and capital rental markets are competitive.

The law of capital accumulation is,

Kt+1 = It +Kt(1− δ)

The economy income accounting is,

Ct + It = Yt

In this economy, the latest two conditions define the following feasibility constrain,

Kt+1 = F (Kt, Nt, At) +Kt(1− δ)− Ct

Definition 1 (Solow-Swan Competitive Equilibrium). In an economy E a Solow-Swan Com-

petitive Equilibrium is a sequence of allocations (Y ,C, I,K,N ,A) and prices (w,ν) such that,

at all t in T ,

• Yt = F (Kt, Nt, At)

• Ct = (1− s)Yt, It = sYt

• Kt+1 = It +Kt(1− δ)5For any given (Kt, Nt), profits are,

F (Kt, Nt)− νtKt − wtNt = FK,tKt + FN,tNt − νtKt − wtNt = 0

5

• νt = FK(Kt, Nt, At), wt = FN (Kt, Nt, At)

• Nt = (1 + n)tN0, At = (1 + µ)tA0

In this economy markets are competitive, implying that firms demand inputs up to the point

at which their marginal productivities equate prices. As we said earlier, CRTS implies zero

profits (see the discussion in the remark 2.1): at all dates t in T

Πt = F (Kt, Nt, At)− νtKt − wtNt = 0

Since in the Solow economy, labor supply is inelastic and corresponds to the whole labor

force, the labor market equilibrium is attained at full employment, as the real wage rate adjusts

so that the labor demand equals the whole labor force. Thus, the population (or workforce)

dynamics fully translates into labor input dynamics, Nt = (1 + n)tN0.

Using the law-of-motion of capital, one finds that the equilibrium rate of growth of capital

is,

(1)∆Kt+1

Kt= s

YtKt− δ

An equilibrium with balanced growth is one at which (Y,C,K) grow at the same constant

rate g. Later we say that an equilibrium is a steady-state if g is zero.

To study the dynamics and establish the existence (or compute) balance growth, it is often

useful to transform the original variables expressing them in per-capita terms or, when there

is growth in labor input productivity, in efficiency units. We are going to show that such a

transformation allows to easily compute the steady-state of the transformed economy and that

this corresponds to the balanced growth equilibrium of the original economy. Indeed, let us

consider the second case and, 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:

yt :=YtNt

=Kαt (AtNt)

1−α

AtNt=

(Kt

AtNt

)α= kαt

In this economy, in efficiency units, there is no balanced growth, except for the steady state

(zero growth). Indeed, since yt = kαt ,

yt+1

yt=

(kt+1

kt

)αand a balanced growth g would require, 1 + g = (1 + g)α, which holds if and only if g = 0.

Going back to the original economy, we are going to show that, if it exists, a balanced growth

is one with the economy growing at the (instantaneous) rate η = µ + n. Indeed, consider the

definition of the rate of growth of capital in efficiency units; by simple transformations,

∆kt+1

kt=Kt+1/Nt+1

Kt/Nt− 1(2)

=Kt+1

Kt

Nt

Nt+1− 1

=

(∆Kt+1

Kt+ 1

)Nt

Nt+1− 1

6

Notice that,

Nt = AtNt = [

(1+η)︷ ︸︸ ︷(1 + µ)(1 + n)]tN0 = (1 + η)tN0

1 + η := 1 + n+ µ+ nµ ≈ 1 + n+ µ

as nµ ≈ 0.6 Hence, the rate of growth of N is constantly equal to 1 + n + µ. Using this into

(2),

∆kt+1

kt=

(∆Kt+1

Kt+ 1

)1

1 + η− 1

=1

1 + η

(∆Kt+1

Kt+ 1− 1− η

)yielding,

(3) (1 + η)∆kt+1

kt=

∆Kt+1

Kt− η

At a steady state of the model with variables in efficiency units, ∆kt+1

kt= 0 implies,

∆Kt+1

Kt= η =: g > 0

One can also simplify (8), using ∆kt+1

ktη ≈ 0 (i.e. the products of the growth rate of capital in

efficiency units with n and µ are zero), so as to directly consider the following is approximation,

(4)∆kt+1

kt=

∆Kt+1

Kt− η

We are left to check that all activity variables growth at the per-period rate g (or η), so that

we have a balanced growth equilibrium. By (1), g = s(Yt/Kt)−δ. Rearranging, we obtain that

the capital-output ratio is constant,

(5)Kt

Yt=

s

g + δ

Denote the fraction on the right hand side z, and consider Kt = zYt at two consecutive periods.

Subtracting, yields

∆Kt+1 = z∆Yt+1

and, dividing through by Kt = zYt,

g :=∆Kt+1

Kt=

∆Yt+1

Yt:= gY

We are left to check that consumption also grows at the same rate. Since, Ct = (1− s)Yt,

gC :=Ct+1 − Ct

Ct=

(1− s)∆Yt+1

Ct=

(1− s)∆Yt+1

(1− s)Yt= gY

Thus, we have used the steady-state of the economy in efficiency units to compute the balanced

growth path of the original economy gY = gC = g = n+ µ.

6These type of approximations, taking the product of growth rates to be zero, tend to be exact as we reduce

the time-interval and, in the limit, gives back the characterization obtained in continuous time models.

7

We end this section computing the equilibrium of the economy in efficiency units and drawing

some considerations. Recalling (8), and using (1) and (5) with g = η,

(6)∆kt+1

kt= s

ytkt− δ − η = skα−1

t − δ − η

At steady-state, the latest yields,

(7) k∗ =

(s

δ + η

) 11−α

As for the other activity variables, y∗ = (k∗)α, c∗ = (1− s)(k∗)α, (I/N)∗ = s(k∗)α.

Finally, we derive equilibrium prices. Given profits,

Πt = AtNt[f(kt)− νtkt]− wtNt

at the firm optimum, each factor demand satisfies marginal factor productivity equal factor

price. For capital,

νt = f ′(kt) = αkα−1t = α

ytkt

= αYtKt

Therefore, ν∗ is constant at the balance growth. Turning to labor demand, using zero-profit

(CRTS),

wt = At[f(kt)− νtkt] = At[f(kt)− f ′(kt)kt] = (1− α)YtNt

Implying that at balanced growth the wage rate increases at the rate of growth of labor pro-

ductivity µ, so as to keep the labor demand constant. Indeed, at the steady-state capital

level,

wt = Atkα∗ (1− α)

wt+1

wt=At+1

At= 1 + µ

A few general comments follow.

• At a balanced growth, total savings sy∗ is used to replace capital depreciation δk∗, to

provide newly born agents with the same amount of capital in efficiency units ηk∗. This

is so by, (6) and ∆kt+1/kt = 0, implying sy∗ = δk∗ + ηk∗.

• A change in the saving rate s does only have temporary effects on the growth rate. This

is again by (6) and ∆kt+1/kt = 0 and it occurs exactly because savings are carried out

only to keep the capital in efficiency units constant.

• A change in the saving rate s has permanent effects on the steady-state level of capital

k∗, by (7).

• A country has an higher saving rate s if and only if it has a higher capital-income ratio

k∗/y∗. (6) and ∆kt+1/kt = 0 yield,

k∗

y∗=

s

δ + n+ µ

• Two countries with same fundamentals and different saving rates have the same growth

rate and different steady-state capital levels of activities (capital, income, consumption).

8

• At balanced growth, real gross-interest rate ν∗ is constant (i.e. it has a zero long-run

growth),

ν∗ = αy∗

k∗= α

δ + n+ µ

sincreasing in the rate of technological progress, in the productivity of capital and de-

creasing in s.

• In the Cobb-Douglas economy, α = νK/Y and 1 − α = wN/Y measure the constant

shares of GDP that, respectively, go to capitalists and workers.

Most of the empirical evidence is drawn in per-capita terms, rather than in efficiency units.

It is easy to check that, if a balanced growth exists it is one with the economy with the per-

capita variables growing at the constant rate µ, measuring the rate of growth of technological

progress. To see this, just start out again from equation (2) and re-iterate the derivation until

equation (8) using kt := Kt/Nt rather than Kt.

3.1. Steady-state properties. We are going to prove that in the Solow-Swan economy a

steady state exists, it is unique and stable.

3.1.1. Existence. Define the rate of growth function, g : R→ R such that

g(kt) = sf(kt)

kt− (δ + η)

g(·) has the following properties (start looking at figure 1):

• Since f(0) = 0 and Inada, (using de l’Hopital) limk→0+ g(k) = limk→0+ sf′(k)−(δ+η) =

+∞ > 0.

• Again by Inada, limk→+∞ g(k) = limk→+∞ sf′(k)− (δ + η) = −(δ + η) < 0

• By continuity of g(·), there exist a k sufficiently closed to zero and a k sufficiently large,

such that g(k) > 0 and g(k) < 0.

Therefore, we can apply the Intermediate Value Theorem (IVT) and conclude that there exists

a k < k∗ < k such that g(k∗) = 0; thus k∗ > 0 is a steady state.7

3.1.2. Uniqueness. To prove uniqueness of the steady state, it suffices to show that g(·) is

strictly monotone.

g′(k) = s

[f ′(k)k − f(k)

k2

]=s

k

[f ′(k)− f(k)

k

]We claim that this is negative for all k > 0, if f is strictly concave. To prove our claim we shall

exploit the strict concavity of f (along with f(0) = 0) and the following simple lemma.

Lemma 1 (Concavity). Let f : R→ R be any C1, concave function with f(0) ≥ 0. Then, for

all k > 0,

f ′(k) ≤ f(k)

k

with strict inequality if f(·) is strictly concave.

7The IVT says that if F (·) is a real-valued, continuous function on the interval (a, b), and F (a) < c < F (b),

then there is a z ∈ (a, b) such that F (z) = c.

9

To see why, use concavity to attain,

f(0)− f(k) ≤ f ′(k)(0− k)

holding with strict inequality if f is strictly concave. Then, use f(0) ≥ 0 and k > 0 to prove

lemma.

3.1.3. Stability. We are left to show that the steady state k∗ is (asymptotically) stable. From

the definition of g(·) above,

∆kt+1 = g(kt)kt or kt+1 = [1 + g(kt)]kt

Local stability, in a neighborhood of k∗, can be shown using a linear approximation of the last

(non-linear) difference equation; namely, kt+1 = [1 + g(k∗)]k∗+ {g′(k∗)k∗+ [1 + g(k∗)]}(kt−k∗)and using g(k∗) = 0,

(8) kt+1 − k∗ = [1 + g′(k∗)k∗](kt − k∗)

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,

the results leads to an upward biased estimate.

16