Savings, Investment and Economic Growth · Chapter 2 Savings, Investment and ... consumption and...

Chapter 2

Savings, Investment andEconomic Growth

In this chapter we begin our investigation of the determinants of economicgrowth. We focus primarily on the relationship between savings, investment,physical capital accumulation and economic growth.

The starting point for the analysis of this process is the Solow [1956]model. This model is based on a neoclassical production function and theassumption of a constant exogenous savings rate. Given that in a closedeconomy savings are equal to investment, the process of capital accumulationdepends on the savings rate which determines the investment rate.1

In this model, capital accumulation per worker continues until savingsper employee are equated with depreciation and the additional investmentrequired to maintain a constant ratio of capital to labor.

In the case where technical progress raises labor productivity continu-ously, at a constant exogenous rate, then capital accumulation per e�ciencyunit of labor continues until savings per e�ciency unit of labor are equatedto depreciation plus the additional investment required to provide for pop-ulation growth and technical progress.

In the long-run equilibrium of this model, alternatively referred to as thesteady state or the balanced growth path, economic growth is exogenous andequal to the rate of population growth plus the rate of technical progress.Essentially, on the balanced growth path, per capita output grows at theexogenous rate of technical progress.

During the adjustment process towards the balanced growth path, an

1This model is often referred to as the Solow-Swan model, as a similar analysis waspublished in the same year by Swan [1956].

35

36 Ch. 2 Savings, Investment and Economic Growth

economy that has an initial capital stock lower than its steady state value,experiences a growth rate which is higher than its long-run growth rate.Capital accumulates at a rate which exceeds the sum of the rate of growthof population and the rate of technical progress, and so does output. For aneconomy that has an initial capital stock which is higher than its steady statevalue, the growth rate during the transition is below the long-run growthrate, as capital accumulates at a rate that falls short of the sum of the rateof growth of population and the rate of technical progress.

This model predicts that economies converge to a unique balanced growthpath. A “poor” economy, it terms of its initial capital stock, and a “rich”economy, in terms of its initial capital stock, converge to the same balancedgrowth path, provided that they are characterized by the same savings rateand the same technological and demographic parameters.

However, if two economies have di↵erent savings rates, di↵erent totalfactor productivity, di↵erent initial labor e�ciency, di↵erent rates of pop-ulation growth or a di↵erent depreciation rate of capital, they will con-verge to di↵erent balanced growth paths. Convergence in this model is thusconditional. The conditions are related to the structural characteristics ofdi↵erent economies, such as their savings and investment rates, total fac-tor productivity, the rate of population growth and the rate of technicalprogress.

This model predicts that a higher savings (and investment) rate results inhigher steady state capital and output per worker. Furthermore, it predictsa positive impact on capital and output per worker from higher total factorproductivity and initial labor e�ciency, and a negative impact from the rateof population growth, the rate of technical progress and the depreciation rateof capital.

The Solow model is a key model and an important reference point inthe theory of economic growth. Although its roots lie in older models, andalthough it has a number of theoretical and empirical weaknesses, this modelprovides a very useful, simple and flexible framework for the analysis of thegrowth process and has stood the test of time well.

However, the accumulation of physical capital, which is the main en-gine of economic growth in the Solow model, cannot fully explain either thelong-term growth of output per worker and per capita income that has beenobserved in developed economies, or the large di↵erences in labor produc-tivity and living standards per head between developed and less developedeconomies.

Only a small part of these di↵erences can be explained by the accu-mulation of physical capital. Most of it is accounted for by di↵erences in

George Alogoskoufis, Dynamic Macroeconomics 37

total factor productivity and technical progress, which in the Solow modelare considered exogenous parameters. In this sense, the Solow model, likeall models that rely on similar assumptions about technology and technicalprogress, shows us how to overcome its weaknesses, introduce human as wellas physical capital, and to try to explain technical progress endogenously.

2.1 The Solow Growth Model

In order to account for the process of economic growth, the Solow model fo-cuses on three main aggregate endogenous variables. Total output (Y ), thetotal physical stock (K) and aggregate consumption (C). Two additionalendogenous variables, the real wage w and the real interest rate r, are de-termined if one assumes competitive markets for factors of production. Thenumber of employees (L) is assumed proportional to an exogenously evolv-ing total population, and the e�ciency of labor (h) is assumed to evolveexogenously as well, growing at a constant rate of technical progress.

Thus, the rate of growth in the number of employees is equal to thepopulation growth rate (n) and is considered exogenous. The rate of growthin the e�ciency of labor is equal to the rate of exogenous technical progress(g).

The model explains the level and rate of growth of output and physicalcapital as functions of these exogenous factors (n and g), the saving rate(s), which is also considered exogenous, total factor productivity and theexogenous rate of depreciation of capital (�). Once the capital stock, output,consumption and investment are determined, one can derive the real interestrate r (renumeration of capital) and real wages w (remuneration of labor), asthese depend on the ratio of capital to total labor e�ciency. Obviously, allendogenous variables are determined simultaneously in a dynamic generalequilibrium.

2.1.1 The Neoclassical Production Function

A key assumption that di↵erentiates the Solow model from previous modelsof economic growth is the neoclassical production function. This assumes apositive but finite elasticity of substitution between factors of production.Thus, at each point in time t, the economy is assumed to possess a stockof capital, number of employees and labor e�ciency, which are combinedto produce output, through this neoclassical production function. This isassumed to take the form,


Y (t) = F (K(t), h(t)L(t)) (2.1)

It is worth noting the following characteristics of the neoclassical pro-duction function:

First, time t enters the production function solely through the factors ofproduction K(t) and h(t)L(t). Output can change over time only throughchanges in the quantity or e�ciency of factors of production.

Second, technical progress is assumed to increase only the e�ciency oflabor h(t). This is called labor augmenting technical progress, or Harrodneutral technical progress.

Third, the production function is characterized by constant returns toscale. Multiplying all factors of production by any non negative real number,multiplies the scale of production by the same non negative real number.Thus, (2.1) satisfies,

�Y (t) = F (�K(t),�h(t)L(t))

for any [� � 0.Because of the assumption of constant returns to scale, the production

function can be re-written as,

y(t) = f (k(t)) (2.2)

where,y = Y/hL is output per e�ciency unit of labork = K/hL is capital per e�ciency unit of laborf(k) = F (k, 1) is the production function per e�ciency unit of labor(2.2) is often referred to as the production function in intensive form.

The intensity of production (output per e�ciency unit of labor) depends onthe intensity of capital (capital per e�ciency unit of labor).

Fourth, it is assumed that the production function satisfies the followingproperties:

f(0) = 0, f 0 =@f

@k> 0, f 00 =

@2f

@k2< 0

The marginal product of capital per e�ciency unit of labor is positive butdeclining. The production function in intensive form, with these additionalassumptions is depicted in Figure 2.1.

Finally, it is assumed that,


Figure 2.1: The Production Function in Intensive Form

limk!0

f 0(k) = 1, limk!1

f 0(k) = 0

These final assumptions are called the Inada conditions. The Inada con-ditions ensure that the marginal product of capital is very high when capitalintensity is low, and very low when capital intensity is high. These condi-tions are necessary and su�cient in order to prove the global uniqueness ofthe balanced growth path.2

2.1.2 The Cobb-Douglas Production Function

A particular production function which is often used in the theory of growth,but also more generally in macroeconomics, is the Cobb Douglas productionfunction. This takes the form,3

2See Inada [1964].3This production function was first put forward by Cobb and Douglas [1928], to fit em-

pirical equations for production, employment and the capital stock in U.S. manufacturing.


F (K(t), h(t)L(t)) = AK(t)↵(h(t)L(t))1�↵ (2.3)

where, A > 0 and 0 < ↵ < 1. A is defined as total factor productivity,and ↵ as the exponent (share) of capital in total production. 1 � ↵ is thecorresponding exponent (share) of labor.

The Cobb Douglas production function in intensive form is given by,

y(t) = f(k(t)) = Ak(t)↵ (2.4)

One can easily confirm that the Cobb Douglas production function (2.3)satisfies all the assumptions we have made about the neoclassical produc-tion function. The marginal product of capital and labor are positive anddeclining, and the Inada conditions are satisfied.

In addition, for the Cobb Douglas production function, labor augmentingtechnical progress (Harrod neutral) does not di↵er from capital augmentingtechnical progress, or technical progress that augments both factors (Hicksneutral). The reason is that in the Cobb Douglas production function thefactors of production enter multiplicatively, and thus, it does not matterwhich of the factors of production is multiplied by technical progress.4

2.1.3 Population Growth and Technical Progress

We shall analyze the Solow model in continuous time, assuming that t is acontinuous variable, where t 2 [0,1).5

We shall assume that the number of employees is a constant fraction oftotal population, and grows continuously at a constant, exogenous rate ofpopulation growth n.

L(t) = L(0)ent (2.5)

where L(0) is the number of employees at time 0, and e is the base ofnatural logarithms.

We shall also assume that the e�ciency of labor also grows continuouslyat a constant exogenous rate of technical progress g.

It is thus known as the Cobb Douglas production function, although this functional formwas alluded to earlier by economists such as Clark, Wicksell and Wicksteed.

4In Appendix A we introduce a more general production function, the CES (constantelasticity of substitution) production function, put forward by Arrow et al. [1961]. TheCES production function encompasses the Cobb Douglas, the Leontief and the linear pro-duction function as special cases, but it does not necessarily satisfy the Inada conditions.

5In section 2.6 we also analyze the Solow model in discrete time, where t = 0, 1, 2, ...is an integer, that refers to discrete time periods, like years, months, weeks, days etc.


h(t) = h(0)egt (2.6)

where h(0) is the e�ciency of labor at time 0.From (2.5) and (2.6) it follows that,

L(t) =dL(t)

dt= nL(t) (2.7)

h(t) =dh(t)

dt= gh(t) (2.8)

A dot on top of a variable denotes its first derivative with respect totime, i.e its change over time.6

2.1.4 Savings, Capital Accumulation and Economic Growth

The output produced accrues to households as income, which is either con-sumed or saved. In the Solow model, the share of income which is saved isassumed exogenous, and denoted by s.

Aggregate consumption C, is thus given by,

C(t) = (1� s)Y (t) (2.9)

The demand for total output consists of consumption plus gross invest-ment.

Y (t) = C(t) + I(t) (2.10)

where I(t) is gross investment.(2.10) is an equilibrium condition in the product market, stating that

total production (output supply) is equal to the demand for output.Gross investment consists of additions to the capital stock, plus replace-

ment investment, and is given by,

I(t) = K(t) + �K(t) (2.11)

where 0 < � < 1 is the constant exogenous rate of depreciation of thecapital stock.

Substituting the consumption function (2.9) and the definition of grossinvestment (2.11), in the equilibrium condition (2.10), we get,

6Technically, (2.7) and (2.8) are first order linear di↵erential equations, whose solu-tion is given by (2.5) and (2.6) respectively. For an introduction to ordinary di↵erentialequations see Mathematical Annex 1.


Y (t) = (1� s)Y (t) + K(t) + �K(t) (2.12)

Solving (2.12) for the change in the capital stock, we get,

K(t) = sY (t)� �K(t) (2.13)

From (2.13), the accumulation of capital is determined by the di↵erencebetween savings and replacement investment. To the extent that savings ishigher than replacement investment, the capital stock grows over time. Ifsavings is lower than replacement investment, the capital stock is reducedover time.

Dividing (2.13) through by hL, taking into account that L grows at arate n, and h grows at a rate g, we get,

k(t) = sy(t)� (n+ g + �)k(t) (2.14)

(2.14) is the capital accumulation equation expressed in e�ciency unitsof labor. To the extent that savings per e�ciency unit of labor is higher thanthe investment required to keep capital per e�ciency unit of labor constant,capital per e�ciency unit of labor grows over time. In the opposite case, itdeclines over time.

Using the production function in intensive form, (2.2), to replace for yin (2.14), we get,

k(t) = sf(k(t))� (n+ g + �)k(t) (2.15)

The non linear di↵erential equation (2.15) is the key equation of theSolow model. It suggests that the change over time in capital per e�ciencyunit of labor is determined by the di↵erence of two terms that both dependon the level of capital per e�ciency unit of labor. The first term is currentsavings and investment per e�ciency unit of labor, and the second term issteady state investment per e�ciency unit of labor. Steady state, or long runequilibrium, investment is defined as the investment rate that keeps capitalper e�ciency unit of labor constant.

2.1.5 The Balanced Growth Path and the Convergence Pro-cess

Given that the aggregate labor force, in e�ciency terms, is increasing atan exogenous rate n + g, and that a fraction � of the capital stock needsto be replaced at every moment, due to depreciation, the investment that


is required to keep the capital stock per e�ciency unit of labor constant isgiven by (n+ g + �)k. This we shall denote as steady state investment.

Steady state capital per e�ciency unit of labor is thus determined by,

k(t) = 0,) sf(k(t)) = (n+ g + �)k(t) (2.16)

We shall denote the capital stock per e�ciency unit of labor k that satis-fies (2.16) as k⇤. One can easily deduce that k⇤ is constant and independentof time. k⇤ defines the so called balanced growth path or steady state of themodel, as all other steady state variables in this model depend on k⇤.

On the balanced growth path, the steady state capital stock, output,consumption and investment per e�ciency unit of labor are constant. Theper capita steady state capital stock, output, consumption and investmentare all growing at the exogenous rate of technical progress g. The aggregatesteady state capital stock, output, consumption and investment are growingat the rate g + n, which is the sum of population growth and the rate oftechnical progress.

The determination of k⇤, and the dynamic adjustment of k towards k⇤

are depicted in Figure 2.2. The straight line depicts steady state investment(n+ g+ �)k. The curved line sf(k) depicts current savings and investment.At the point k⇤, current savings and investment are equal to steady statesavings and investment.

We have also depicted the production function in intensive form f(k)which relates output e�ciency unit of labor to capital per e�ciency unit oflabor.

To the left of k⇤ current investment is higher than steady state invest-ment, and k is increasing over time. To the right of k⇤ current investmentis lower than steady state investment, and k is declining over time.

The equilibrium at k⇤ is unique and globally stable. Irrespective of initialconditions, the economy converges to k⇤, which is the steady state capitalstock per e�ciency unit of labor.

2.1.6 The Rate of Growth of Capital and Output

To examine the behavior of the growth rate of capital per e�ciency unit oflabor in the Solow model, we can divide both sides of equation (2.15) byk(t). We then get that the growth rate of capital per e�ciency unit of laboris given by,

�k

(t) =k(t)

k(t)= s

f(k(t))

k(t)� (n+ g + �)


Figure 2.2: Equilibrium in the Solow Model

The growth rate of capital (per e�ciency unit of labor) in the Solowmodel depends on the di↵erence of the average product of capital, multipliedby the savings rate, from the sum of population growth, the rate of technicalprogress and the depreciation rate. If average savings exceed average steadystate investment per unit of capital, then capital per e�ciency unit of labordisplays positive growth, as the capital stock increases at a rate higher thann + g. In the opposite case, capital per e�ciency unit of labor displaysnegative growth, as the capital stock either falls, or grows at a rate lowerthan n+g. In the steady state, capital per e�ciency unit of labor is constant,as the capital stock rises at a rate equal to n+ g. The determination of thegrowth rate is depicted in Figure 2.3.

The downward sloping curve depicts savings per unit of capital. Ascapital per e�ciency unit of labor increases, savings per unit of capital fall,since the marginal and average product of capital is decreasing. This curveapproaches infinity as capital tends to zero, and zero as capital tends toinfinity, because of the Inada conditions. The straight line n+ g+ � depicts


Figure 2.3: The Determination of the Growth Rate

steady state investment per unit of capital. It is the investment requiredto keep capital per e�ciency unit of labor constant. Because of the Inadaconditions, the two curves intersect at a positive capital stock per e�ciencyunit of labor, which is the steady state capital stock k⇤. To the left of k⇤ thegrowth rate of k is positive and declining, and to the right of k⇤ the growthrate of k is positive and increasing. At k⇤, the balanced growth path, thegrowth rate of k is equal to zero, and the capital stock itself grows at theexogenous rate n+ g.

From the production function, the growth rate of output per e�ciencyunit of labor is given by,

�y

(t) =y(t)

y(t)= f 0(k(t))

k(t)

k(t)

Since the marginal product of capital is positive, the growth rate ofoutput per e�ciency unit of labor has the same sign as the growth rate ofcapital per e�ciency unit of labor.


To the left of k⇤, output per e�ciency unit of labor grows at a positiveand declining rate. In fact, the growth rate of output declines at a faster ratethan the growth rate of capital, because of the declining marginal productof capital.

To the right of k⇤, output per e�ciency unit of labor grows at a negativeand increasing rate. In fact, the growth rate of output increases at a fasterrate than the growth rate of capital, because of the increasing marginalproduct of capital.

At k⇤, the growth rate of output per e�ciency unit of labor is zero, andtotal output increases at the exogenous rate n+ g.

Since consumption is a constant fraction of output in this model, thegrowth rate of consumption is equal to the growth rate of output �

y

.

From the above analysis it follows that the Solow growth model doesnot explain the rate of long-run economic growth, i.e the rate of economicgrowth on the balanced growth path, as this is equal to the sum of twoexogenous parameters, g and n. It does not explain the growth rate of percapita income and consumption along the balanced growth path either, asthis is equal to the rate of exogenous technical progress g.

What the Solow model does explain is the level of the per capita capitalstock and per capita output and income, the level of per capita consumptionand real wages and the real interest rate, on the balanced growth path.These depend on all the parameters of the model, as we shall shortly see.

In addition, the Solow growth model explains the process of convergencetowards the balanced growth path. The process of convergence predictedby the model is the result of the accumulation of physical capital. Thegrowth rate of output, or output per capita, in the convergence processdi↵ers from the long run growth rate g + n or g, to the extent that, duringthe convergence process, the economy accumulates capital at a di↵erent ratethan g + n. Note that the convergence process is asymptotic, in the sensethat the steady state (or balanced growth path) is the limit as time goes toinfinity.

2.1.7 The Significance of the Inada Conditions

One can use (2.16) to show that if the Inada conditions are not satisfied, asteady state may not exist.

Assume that as capital per e�ciency unit of labor tends to infinity, themarginal product of capital remains positive, and the average product ofcapital converges not to zero, but to a positive value, say !, where,


! >n+ g + �

s

Thus, from (2.16) the growth rate of capital per e�ciency unit of labordoes not converge to zero either, and k⇤ does not exist. In such a case,capital per e�ciency unit of labor grows continuously, and the Solow modelbecomes an endogenous growth model, with the long run growth rate ofcapital per e�ciency unit of labor being equal to,

�k

= s! � (n+ g + �) > 0

Assume in the opposite case that as capital per e�ciency unit of labortends to zero, the marginal product of capital does not tend to infinity, asrequired by the Inada conditions, but to a level that makes the averageproduct of capital equal to �, where,

� <n+ g + �

s

From (2.16), and the properties of the production function, k will bedriven to zero and a steady state will not exist.

As is shown in Appendix A, the CES production function does not nec-essarily satisfy the Inada conditions, and thus may not be compatible withthe existence of a steady state capital stock and output per e�ciency unitof labor.

2.2 Competitive Markets, the Real Interest Rateand Real Wages

As we have presented it so far, the Solow model assumes a single domesticfirm and one national household which owns this firm. However, due tothe constant returns to scale hypothesis, all the properties of this model gothrough, when one assumes competitive markets, with many identical firmsand households.

Suppose there is a large number of households owning capital and sup-plying one unit of labor per member. Firms rent capital and labor in com-petitive capital and labor markets. The interest rate is r(t) and the realwage (per e�ciency unit of labor) is w(t). Each firm uses capital and laborand produces according to a production function which, in intensive form, isgiven by (2.2). Each firm pays the return on capital to households holdingits shares, and real wages to its workers.


The conditions for profit maximization on the part of firms are that themarginal product of capital equals the user cost of capital (the real interestrate plus the depreciation rate), and that the marginal product of laborequals the real wage. Therefore it holds that,

f 0(k(t)) = r(t) + � (2.17)

f(k(t))� k(t)f 0(k(t) = w(t) (2.18)

One can easily conclude that, when (2.17) and (2.18), are satisfied, firmshave zero profits and factor payments exhaust real output. This is a conse-quence of constant returns to scale.

The total household income per e�ciency unit of labor is equal to grossoutput and is given by,

(r(t) + �) k(t) + w(t)

The condition equating savings and investment per e�ciency unit oflabor is given by,

k(t) = s ((r(t) + �) k(t) + w(t))� (n+ g + �)k(t) (2.19)

Substituting (2.17) and (2.18) in (2.19), we have the basic accumulationequation of the Solow model.

k(t) = sf(k(t))� (n+ g + �)k(t)

Consequently, the Solow model is compatible with the existence of com-petitive markets for goods, labor and capital.

In the process of adjustment towards the balanced growth path from theleft, i.e when the initial capital per e�ciency unit of labor is less than itssteady state value, real wages are rising and real interest rates are falling,reflecting the evolution of the falling marginal product of capital and therising marginal product of labor.

On the balanced growth path the real wage (per e�ciency unit of labor)remains constant and the same happens with the real interest rate. However,the real wage per employee, along with all other per capita variables, isgrowing at a rate g, the exogenous rate of technical progress.


2.3 The Savings Rate and the Golden Rule

One can prove that a rise in the savings rate results in an increase in steadystate capital and output. It can also be shown that the rate of growth of theper capita capital stock and per capita output and income rise temporarilyabove the rate of long-run economic growth g + n. The relevant analysis ispresented in Figure 2.4.

Figure 2.4: Implications of a Rise in the Savings Rate

2.3.1 The Savings Rate and the Balanced Growth Path

We assume that the initial balanced growth path is at (y⇤, k⇤) in Figure2.4. A rise in the savings rate from s to s0 leads to an increase in savingsand investment that initiates a process of capital accumulation, which grad-ually causes an increase in output and income per e�ciency unit of labor.The economy starts converging to a new balanced growth path (y⇤⇤, k⇤⇤)which is characterized by both higher capital and higher income. During the


adjustment process, savings and investment exceed equilibrium investment,and the rate of growth exceeds the long-run growth rate g + n.

The process of convergence towards the new balanced growth path overtime is depicted in Figure 2.5. Figure 2.5 depicts the so called impulseresponse function of the Solow model.7

The rise in the savings rate leads to capital accumulation that exceedsthe level required to maintain the capital stock per e↵ective unit of laborat its initial steady state level k⇤. Capital starts accumulating at a fasterrate, leading to parallel rise in output and income per e↵ective unit of labor,and the process continues until the economy gradually converges to the newbalanced growth path k⇤⇤. This process of convergence is asymptotic.8

2.3.2 The Savings Rate, the Golden Rule and Dynamic In-e�ciency

Capital and income increase definitely and unequivocally following a rise inthe savings rate. What happens to consumption is more uncertain, as arise in the savings rate reduces consumption for any given level of income.Initially, as capital, output and income are given, a rise in the savings ratecauses a temporary fall in consumption. Gradually capital output and in-come rise and so does consumption. Whether consumption per capita on thenew balanced growth path will be higher or lower than on the original bal-anced growth path, depends on the di↵erence between the marginal productof capital and n+ g + �. The latter is the marginal increase in steady stateinvestment. Consumption will be higher in the new balanced growth pathif the marginal product of capital is higher than n + g + �, and it will belower in the opposite case.

To see this, recall that steady state consumption is given by,

c⇤ = f(k⇤)� (n+ g + �)k⇤ (2.20)

It follows that the change in steady state consumption following a risein the savings rate is given by,

@c⇤

@s=�f 0(k⇤)� (n+ g + �)

� @k⇤

@s(2.21)

7An impulse response refers to the reaction of any dynamic system to some externalchange. The impulse response function (IRF) describes the reaction of the system asa function of time (or possibly as a function of some other independent variable thatparameterizes the dynamic behavior of the system).

8The term asymptotic means that a variable approaches its steady state value arbi-trarily closely as t tends to infinity.


Figure 2.5: Implications of a Rise in the Savings Rate

Since the last term in the right hand side of (2.21) has been shown tobe positive, the impact of the change in the savings rate on steady stateconsumption per e↵ective unit of labor depends on the di↵erence betweenthe marginal product of capital f 0(k⇤) from the equilibrium investment raten+ g + �.

Another way to express this is to say that the change in steady stateconsumption depends on the di↵erence between the net (of depreciation)marginal product of capital f 0(k⇤)� � and the long-run growth rate g + n.

If the net marginal product of capital is smaller than the long-run growthrate, then the extra product from the accumulation of capital will not besu�cient to fund the higher equilibrium investment rate, and consumptionwill have to go down. If the net marginal product of capital is higher thanthe long-run growth rate, then the extra product from the accumulation ofcapital will be more than su�cient to fund the higher equilibrium investmentrate, and consumption will also increase.

In the special case where the net marginal product of capital on the


original balanced growth path is exactly equal to the long-run growth rate,equilibrium consumption will remain unchanged following an infinitesimallysmall rise in the savings rate.

In the latter case, equilibrium consumption is at its highest possible level,and the value of k⇤ that corresponds to this case is referred to as the goldenrule capital stock.9

The golden rule savings rate is defined as the savings rate that impliesa steady state capital stock (per e↵ective unit of labor) that maximizessteady state consumption (per e↵ective unit of labor). Since the welfare ofhouseholds is usually assumed to depend on consumption, the maximizationof steady state consumption per e↵ective unit of labor is a proxy for themaximization of steady state consumer welfare.

From (2.20), the first order conditions for the maximization of consump-tion require,

f 0(k⇤) = n+ g + � ) f 0(k⇤)� � = n+ g (2.22)

From (2.22), the steady state capital stock that maximizes steady stateconsumption is the one that results in a net marginal product of capitalequal to the long-run growth rate. This is what determines the golden rulecapital stock. Since in a competitive equilibrium the net marginal productof capital is equal to the real interest rate, this implies that the golden rulecapital stock is the one for which the real interest rate is equal to the steadystate growth rate.

r = g + n

If the savings rate is such that the steady state capital stock is higherthan the golden rule capital stock, then the economy is said to be character-ized by dynamic ine�ciency. Steady state consumption and, presumably,consumer welfare could be increased by reducing the savings rate, consumingmore and accumulating less capital. By reducing the savings rate, house-holds will be able to enjoy higher steady state consumption, despite thereduction in steady state capital and output.

Thus, an increase in the savings rate is not necessarily always desirablein the Solow model, despite the fact that it results in an increase in steadystate capital and income. It is only desirable as long as the savings rateis below the golden rule rate, i.e. the rate that maximizes steady state

9The significance of the golden rule for the Solow growth model and growth theorywas first highlighted by Phelps [1961], although the concept was previously alluded to byJohn von Neumann and Maurice Allais.


consumption per e�ciency unit of labor. In such a case, an increase in thesavings rate increases steady consumption.

A major problem with the Solow model is that, since the savings rate isexogenous, there is nothing in the model that can help exclude the posibillityof dynamic ine�ciency, that is a suboptimally high savings rate.

2.3.3 The Elasticity of Steady State Output with Respect tothe Savings Rate

One can show that in the Solow model, the long run elasticity of outputwith respect to the savings rate is equal to the ratio of the share of capitalto the share of labor in total output.

To prove this, we start from the change in steady state output followinga change in the savings rate. This is equal to,

@y⇤

@s= f 0(k⇤)

@k⇤

@s(2.23)

k⇤ is defined from,

sf(k⇤) = (n+ g + �)k⇤ (2.24)

Di↵erentiating (2.24) with respect to s, we get,

@k⇤

@s=

f(k⇤)

(n+ g + �)� sf 0(k⇤)(2.25)

Substituting (2.25) in (2.23), we get,

@y⇤

@s=

f 0(k⇤)f(k⇤)

(n+ g + �)� sf 0(k⇤)(2.26)

From (2.26) and (2.24), the long-run elasticity of output with respect tothe savings rate is given by,

s

y⇤@y⇤

@s=

s

f(k⇤)

f 0(k⇤)f(k⇤)

(n+ g + �)� sf 0(k⇤)

=(n+ g + �)k⇤f 0(k⇤)

f(k⇤)(n+ g + �)[1� k⇤f 0(k⇤)/f(k⇤)]

=k⇤f 0(k⇤)

f(k⇤)[1� k⇤f 0(k⇤)/f(k⇤)]

(2.27)

(2.27) can be re-written as,


s

y⇤@y⇤

@s=

k⇤f 0(k⇤)/f(k⇤)

1� [k⇤f 0(k⇤)/f(k⇤)]=

↵K

(k⇤)

1� ↵K

(k⇤)(2.28)

where ↵K

(k⇤) is the elasticity of total output with respect to capital,at the steady state. With competitive markets factor incomes are equal totheir marginal products. In such a case, the elasticity of total output withrespect to capital is equal to the ratio of the share of capital to the share oflabor in total output.

A commonly accepted estimate of the share of capital in total output is1/3. Using this estimate, the long run elasticity of total output with respectto the savings rate is equal to 1/2.

2.4 The Speed of Convergence towards the Bal-anced Growth Path

In the neighborhood of the balanced growth path, the speed of convergenceof k towards k⇤ depends on their di↵erence. On the basis of widely acceptedvalues for the parameters of the model, one can show that the speed ofconvergence in the Solow model is about 4% per annum. As a result, theSolow model predicts that it should take slightly more than 17 years to closehalf of any given gap between k andk⇤.

In order to derive the speed of convergence we start from the basic ac-cumulation equation of the Solow model.

k(t) = sf(k(t))� (n+ g + �)k(t) (2.29)

Steady state capital (per e↵ective unit of labor) k⇤ is determined from(2.29) for k(t) = 0.

In order to determine the speed at which k(t) approaches k⇤, we linearize(2.29) around k⇤. From the linear Taylor approximation of the non-lineardi↵erential equation (2.29) around k⇤, we get,

k(t) ' @k

@k

!

k=k

⇤

(k(t)� k⇤) (2.30)

where the first derivative is taken from (2.29).(2.30) can be written as,

k(t) ' �� (k(t)� k⇤) (2.31)


where � = �⇣@k

@k

⌘

k=k

⇤= �[sf 0(k⇤)� (n+ g + �)].

(2.31) implies that around the steady state k⇤, k approaches k⇤ with aspeed that depends on its distance from k⇤. The rate at which k(t)� k⇤ isreduced is approximately constant and equal to �. We shall refer to � asthe speed of convergence.

Solving the linear first order di↵erential equation (2.31), we get that,

k(t) ' k⇤ + e��t (k(0)� k⇤) (2.32)

where k(0) is the initial value of k.In order to calculate the speed of convergence � in terms of the structural

parameters of the model, we rearrange the definition of � in (2.31) as,

� = (n+ g + �)� sf 0(k⇤)

= (n+ g + �)[1� k⇤f 0(k⇤)

f(k⇤)]

= (n+ g + �)[1� ↵K

(k⇤)](2.33)

where ↵K

(k⇤) is the share of capital in total income at the steady state.In order to proceed from the initial expression in (2.33) we used the fact thatin the steady state sf(k⇤) = (n + g + �)k⇤ in order to eliminate s. To getto the final expression in terms of the share of capital we used the fact thatin a competitive equilibrium the return on capital is equal to its marginalproduct.

Widely accepted annual estimates of n+g+ � determine it at about 6%.For example, this would be the result with n = 1%, g = 2% and � = 3%.With the share of capital estimated at about 1/3, (2.33) implies an annualspeed of convergence of about 4%.

Thus, on the basis of these estimates, the Solow model implies that eachyear roughly 4% of the gap between the current capital stock (and income)and the steady state capital stock (and income) is closed through the processof capital accumulation.

From (2.32) we can estimate how many years it will take with this speedof convergence to close a particular percentage of the gap between k(0) andk⇤ .

In order to calculate the number of years required to cover half of theinitial di↵erence, we need to calculate the time span t that satisfies,

e��t = 0.5


for � = 4%. This suggests that t = �ln(0.5)/� = 0.69/� = 0.69/0.04 =17.3.

Thus, for an annual speed of convergence of 4%, it would take 17.3 yearsto cover half of any initial di↵erence between the initial capital stock (andreal income) and its steady state value. This is often referred to as the halflife of the convergence process.

In order to calculate the number of years required to cover two thirds ofthe initial di↵erence, we need to calculate the time span t that satisfies,

e��t = 0.333

for � = 4%. This suggests that t = �ln(0.333)/� = 2.1/� = 2.1/0.04 =27.5.

It would take 27.5 years to cover two thirds of any di↵erence betweenthe initial capital stock (and real income) and its steady state value.

Econometric evidence from, among others, Mankiw, Romer and Weil(1992), suggests that the speed of convergence for a large number of economiesin the post war period was on average about 2% per annum. Thus, the speedof convergence predicted by the Solow model, based on the parameter esti-mates we used, is on the high side compared with the econometric evidence.We shall return to this issue in Chapter 7.

2.5 The Process of Economic Growth and the SolowModel

The Solow model, like any other economic model, is based on relativelysimple and, many would claim, largely unrealistic assumptions. However, itconstitutes a significant improvement over previous models which did notrely on the neoclassical production function. Such were for example themodels of Harrod (1939) and Domar (1946), which were based on Leontie↵(1941) production functions with constant coe�cients and a zero elasticityof substitution between capital and labor.

The question is whether the model of Solow (and all models based onsimilar assumptions about the technology of production) can o↵er an ade-quate and satisfactory account of the process of economic growth in the realworld. To answer this question, we must delve a little deeper into the mainfeatures of long run growth.


2.5.1 The Kaldor Stylized Facts of Economic Growth

An important early codification of the main empirical features pertaining tolong run growth, is due to Kaldor [1961], who based them on the long rungrowth experience of Great Britain and the USA. According to Kaldor, agrowth theory should be consistent with the following six (6) stylized factsabout long-run growth:

1. Per capita GDP is growing over time, and the growth rate is notdeclining.

2. Physical capital per worker is growing over time.3. The long run rate of return on capital is roughly constant.4. The long run capital-output ratio is roughly constant.5. The shares of labor and capital in Gross Domestic Product do not

display a long-term trend.6. The growth rate of labor productivity varies substantially between

countries.These stylized facts remain roughly in force today, with the addition of

some newer ones.10

The Solow model is at a general level consistent with all of these basicempirical characteristics. However, the process of physical capital accumu-lation, which is the main engine of economic growth in the Solow model, isnot su�cient as an explanation of either the long run growth of output perworker that has been observed historically in almost all developed economiesof the world, or the large di↵erences in output per worker between developedand less developed economies.

Only a small part of these phenomena can be explained by the accumu-lation of physical capital. The largest part appears to be due to technicalprogress and to di↵erences in total factor productivity and the e�ciency oflabor, which in the Solow model are considered exogenous.11

The Solow model identifies three sources of di↵erences in output perworker between countries or between periods: First, di↵erences in capital perworker, secondly, di↵erences in total factor productivity and labor e�ciency,

10Jones and Romer [2010] have recently codified a number of additional stylized factsthat a theory of economic growth must be able to account for. We shall examine theseadditional stylized facts in Chapter 7.

11It is worth mentioning that Kaldor, who was quite critical of neoclassical theory,considered the Solow model to be incompatible with at least some of the stylized factsthat he identified, mainly stylized facts 1,2 and 6. The reason he was critical is that theSolow model is compatible with these facts only when one assumes exogenous technicalprogress, which drives the e�ciency of labor, per capita income, per capita consumptionand real wages along the balanced growth path.


and thirdly, di↵erences in initial conditions.To simplify the analysis of the impact of each of these di↵erences, we

will use the Solow model, assuming a Cobb Douglas production function.

2.5.2 Di↵erences in Economic Growth between Economies

In the Solow model, based on the Cobb Douglas production function, capitalper e�ciency unit of labor on the balanced growth path is defined by thecondition,

sA(k⇤)↵ = (n+ g + �)k⇤ (2.34)

From (2.34) if follows that the steady state capital stock, per e�ciencyunit of labor, is given by,

k⇤ =

✓sA

n+ g + �

◆ 11�↵

(2.35)

From (2.35) and the production function (2.4), output per e�ciency unitof labor is given by,

y⇤ = A(k⇤)↵ = A

✓sA

n+ g + �

◆ ↵

1�↵

(2.36)

The per capita product on the balanced growth path is thus given by,

y⇤(t) =Y ⇤(t)

L(t)= y⇤h(t) = A(k⇤)↵h(0)egt (2.37)

where the hat over a variable denotes the per capita magnitude.Based on (2.37), di↵erences in capital per worker, for realistic estimates

of the parameters of the model, cannot explain the di↵erences in output perworker that we observe in the real world.

For example, let us assume that we want to explain a ratio x in outputper worker between two economies, economy 1 (a developed economy) andeconomy 2 (a less developed economy). From (2.37), assuming that allother parameters except for the capital stock are the same between the twoeconomies, we must have that,

y1(t)

y2(t)=

Ak1(t)↵(h(0)egt)1�↵

Ak2(t)↵(h(0)egt)1�↵

=

k1(t)

k2(t)

!↵

= x (2.38)


To explain this ratio, capital per worker should di↵er by x to the power1/↵, where ↵ is the share of capital in domestic income. Since ↵ is ofthe order of 1/3, to explain that GDP per worker in developed countriesis currently 20 times higher than in less developed countries, capital perworker should be 8,000 times (20 raised to the 3rd power) higher. But thisis not the case. In developed economies capital per worker is only 20-30times higher than in less developed economies. Thus, we cannot account fordi↵erences in per capita output and income solely on the basis of di↵erencesin the per capita capital stock.

We can certify this indirectly as well. If the di↵erences in output perworker were due only to di↵erences in physical capital per worker, thenwe should observe huge di↵erences in the rate of return to capital betweenperiods and between countries. However, such huge di↵erences do not exist.

To explain the large di↵erences between developed and less developedcountries on the balanced growth path, we should allow for di↵erences intotal factor productivity and the e�ciency of labor. Allowing for such dif-ferences in (2.38), we have that,

y1(t)

y2(t)=

A1k1(t)↵(h1(0)egt)1�↵

A2k2(t)↵(h2(0)egt)1�↵

=A1k1(t)↵(h1(0))1�↵

A2k2(t)↵(h2(0))1�↵

= x (2.39)

Di↵erences in total factor productivity and the initial e�ciency of labor,along with di↵erences in physical capital per worker, can explain almost alldi↵erences in output per worker that we observe in the real world. For exam-ple, if the developed countries have capital per worker 30 times higher thanthe less developed countries, a total factor productivity which is three timesthat of the less developed countries (A1 = 3A2) and three times the initiale�ciency of labor (h1(0) = 3h2(0)), then (2.39) predicts that, along thebalanced growth path, their output per worker and their per capita incomewill be about 20 times higher than those of the less developed countries.

However, total factor productivity and the e�ciency of labor are notexplained by the Solow model, but considered exogenous. Therefore, onecould say that this model does not explain the process of long run growth,but only assumes it.12

12Mankiw et al. [1992] have generalized the Solow model, attributing di↵erences in thee�ciency of labor to investment in human capital (education of the labor force). However,they retained the assumption that total factor productivity increases at an exogenous rateg. The generalized Solow model which they put forward is analyzed in Chapter 7, andseems to explain the growth experience of 98 non-oil producing countries after 1960 fairly


This is why this model, like all models based on similar assumptionsabout the technology of production and technical progress, is often referredto as exogenous growth model. It assumes that total factor productivity A,the initial e�ciency of labor h(0), and the rate of technical progress g, areall exogenous parameters.

2.5.3 Conditional Convergence

Our analysis in the previous section makes clear that the process of con-vergence predicted by the Solow model does not entail convergence to thesame per capita income for all economies. The per capita income to whichan economy converges is determined by (2.36) and (2.37) as,

y⇤(t) = A(k⇤)↵h(0)egt = A

✓sA

n+ g + �

◆ ↵

1�↵

h(0)egt (2.40)

To the extent that parameters such as the rate of savings and investments, total factor productivity A, the population growth rate n, the depreciationrate � and the initial labor e�ciency h(0) di↵er between two economies, theseeconomies will converge towards di↵erent levels of per capita income, evenif along the balanced growth path, per capita income is growing at the samerate of technological progress g.

Convergence towards di↵erent levels of per capita income, which de-pend on the parameters characterizing the structure of di↵erent economies,is called conditional convergence. The per capita income towards whicheconomies converge in the Solow model, and the other exogenous growthmodels which we shall analyze in the next few chapters, depends on theirspecific characteristics. Not all economies converge to the same per capitaincome. Each economy converges to the per capita income which is de-termined by its own technological, demographic and savings (investment)parameters.

2.5.4 Convergence with a Cobb Douglas Production Func-tion

The Solow model with a Cobb Douglas production function, can be solvedanalytically not only for the steady state variables, as we have done so far,but also for their evolution during the process of convergence. This is be-cause the first order di↵erential equation that characterizes the convergence

well. See also Jones [2002] and Chapter 7 for other generalized models for economic growthwhich rely on investment in both physical and human capital.


process in this case has the form of a Bernoulli equation, which can be con-verted to a linear di↵erential equation in the capital-output ratio, and thussolved analytically.

With a Cobb Douglas production function, output per e�ciency unit oflabor is given by,

y(t) = Ak(t)↵

Therefore, the adjustment of capital per e�ciency unit of labor is givenby,

k(t) = sAk(t)↵ � (n+ g + �)k(t) (2.41)

This is a Bernoulli equation, which can be converted to a linear di↵er-ential equation if we define a new variable z, as,13

z(t) =k(t)

y(t)=

1

Ak(t)1�↵ (2.42)

This variable is none other than the capital-output ratio.From (2.42) it follows that,

z(t) =@z(t)

@k(t)k(t) =

1� ↵

Ak(t)�↵k(t) (2.43)

By substituting (2.41) in (2.43) we get,

z(t) = (1� ↵)s� �z(t) (2.44)

where � = (1 � ↵)(n + g + �). The parameter � is just the speed ofconvergence.

(2.44) is a first order linear di↵erential equation in the variable z (thecapital output ratio), and can be solved as,

z(t) =s

n+ g + �+

✓z0 �

s

n+ g + �

◆e��t (2.45)

Substituting from the definition of the capital-output ratio with a CobbDouglas production function, the convergence process of capital and outputper e�ciency unit of labor is given by,

13This solution method was analyzed in Chiang [1974] and is also discussed in Jones[2002] and Barro and Sala-i Martin [2004].


k(t) =

As

n+ g + �

⇣1� e��t

⌘+ k0

1�↵e��t

� 11�↵

(2.46)

y(t) = A

As

n+ g + �

⇣1� e��t

⌘+⇣y0A

⌘ 1�↵

↵

e��t

� ↵

1�↵

(2.47)

The limit of (2.46) and (2.47), as time tends towards infinity, is thebalanced growth path, as determined by (2.35) and (2.36).

2.6 Dynamic Simulations of a Calibrated SolowModel

In order to further investigate the process of dynamic adjustment that char-acterizes the Solow model, we can simulate, for calibrated numerical valuesof the parameters, the transition from a balanced growth path to another,when there is an exogenous permanent change in particular parameters,such as the savings rate or total factor productivity, or the rate of growthof population.

To simulate the model numerically we shall convert it from a continuous-time model to a discrete-time model.

2.6.1 The Solow Model in Discrete Time

Instead of assuming that time is a continuous variable, time is now measuredas successive time periods, where t = 0, 1, 2, .... The variable x

t

, indicatesthe variable x in period t.

Population and the e�ciency of labor grow at constant exogenous ratesn and g per period respectively. Thus, we have,

Lt

= L0(1 + n)t (2.48)

ht

= h0(1 + g)t (2.49)

The production function is given by,

Yt

= F (Kt

, ht

Lt

) (2.50)

and is characterized by constant returns to scale and diminishing returnsof individual factors.


We assume, as in the case of continuous time, that the consumptionfunction is characterized by a fixed savings rate s.

Ct

= (1� s)Yt

= (1� s)F (Kt

, ht

Lt

) (2.51)

The accumulation of capital is determined by,

Kt+1 �K

t

= F (Kt

, ht

Lt

)� Ct

� �Kt

= sF (Kt

, ht

Lt

)� �Kt

(2.52)

Thus, in discrete time, the accumulation of capital per e�ciency unit oflabor is given by,

kt+1 =

1

(1 + n)(1 + g)(sf(k

t

) + (1� �)kt

) (2.53)

It can easily be shown diagrammatically (see Figure 2.6), that the dif-ference equation (2.53) converges to a unique equilibrium.

In Figure 2.6 the curve depicts the accumulation equation (2.53) and the450 straight line the steady state equilibrium condition k

t

= kt+1) = k⇤. The

process of convergence is determined by (2.53) and the equilibrium towardswhich the economy converges determines the balanced growth path.14

2.6.2 The Calibrated Solow Model

For the purposes of the simulation we shall used a calibrated version of themodel, assuming that the production function is Cobb Douglas,

yt

= f(kt

) = Ak↵t

(2.54)

where A > 0 is total factor productivity, and 0 < ↵ < 1 the exponent(share) of capital in the production function. 1�↵ is the exponent of labor.

Substituting (2.54) in (2.53), the capital accumulation equation is givenby,

kt+1 =

1

(1 + n)(1 + g)(sAk↵

t

+ (1� �)kt

) (2.55)

From (2.55), the steady state capital stock, per e�ciency unit of labor,is given by,

14Note that (2.53) is a non-linear first order di↵erence equation which can be analyzeddiagrammatically.


Figure 2.6: Convergence in the Solow Model in Discrete Time

k⇤ =

✓sA

(1 + n)(1 + g)� (1� �)

◆ 11�↵

(2.56)

The remaining variables are all functions of k and their steady statevalues a function of k⇤.

Output is given by (2.54), and steady state output is given by,

y⇤ = A

✓sA

(1 + n)(1 + g)� (1� �)

◆ ↵

1�↵

(2.57)

Consumption is given by,

ct

= (1� s)Ak↵t

(2.58)

Finally, the real interest rate and the real wage are given by,

rt

= ↵Ak↵�1t

� � (2.59)


wt

= (1� ↵)Ak↵t

(2.60)

Simulating (2.55) numerically, for specific parameter values, we can cal-culate the dynamic adjustment of capital towards the balanced growth path.The dynamic adjustment of the other variables can be calculated then from(2.54), (2.58), (2.59) and (2.60).

The calibrated values of the initial parameters in the simulation are asfollows: A = 1, ↵ = 0.333, s = 0.30, n = 0.01, g = 0.02, � = 0.03. These arethe same as the values used to calculate the speed of convergence in section2.4.

2.6.3 Dynamic Simulations of the Calibrated Solow Model

In Figure 2.7 we present the dynamic adjustment of the calibrated Solowmodel following a permanent increase in the saving rate s by 5%.

Figure 2.7: Dynamic Simulation of the Solow Model Following a PermanentIncrease in the Savings Rate by 5%


In the simulation of Figure 2.7, the economy is on the initial balancedgrowth path, and in period 1, the savings rate increases permanently, andunexpectedly, by 5%, from 0.30 to 0.315. The increase in the saving rateleads to a decrease in consumption, gradual accumulation of capital, a grad-ual increase in production, a gradual increase in real wages and a gradualfall in real interest rates. The reason behind increasing real wages is thegradual increase in the marginal product of labor caused by the accumula-tion of capital. The reason behind the gradual reduction in the real interestrate is the gradual reduction of the marginal product of capital caused bythe accumulation of capital. The economy gradually converges towards anew balanced growth path. In this new balanced growth path, capital pere�ciency unit of labor is higher by approximately 7.6%, output and realwages by 2.5%, consumption by 0.3% (due to the increase in the savingrate), while the real interest rate has fallen by 0.3 percentage points.

In Figure 2.8 we present the dynamic adjustment of the model followingan increase in total factor productivity A by 5%.

In the simulation of Figure 2.8 the economy is on the initial balancedgrowth path, and in period 1, total factor productivity A increases perma-nently and unexpectedly by 5%, from 1 to 1.05. This increase leads immedi-ately to an increase in output, consumption, savings, the marginal productof labor (real wage) and the marginal product of capital (real interest rate).The increase in savings causes a gradual accumulation of capital, which leadsto a further gradual increase in output and consumption, a further gradualincrease in real wages, but a gradual fall in real interest rates. The reasonfor the gradual decrease of the real interest rate is the gradual reductionof the marginal product of capital caused by the accumulation of capital.The economy gradually converges to a new balanced growth path. In this,capital per e�ciency unit of labor is increased by about 7.6%, output, con-sumption and real wages also increased by 7.6%, while the real interest rate,after the initial increase, has returned to its original equilibrium. The equi-librium real interest rate, because the production function is assumed CobbDouglas, is independent of total factor productivity A. The reason why anincrease in productivity by 5% leads to an increase in real income by 7.6%,i.e more than 5%, is that the increase in productivity causes an increasein savings and capital accumulation, which in turn causes additional sec-ondary increases in real incomes and consumption. This can be confirmedfrom (2.57), where the elasticity of steady state output with respect to totalfactor productivity A is equal to 1/(1� ↵) > 1.

Finally, in Figure 2.9 we present the dynamic adjustment of the modelfollowing an increase in the rate of population growth n by 5%.


Figure 2.8: Dynamic Simulation of the Solow Model Following a PermanentIncrease in Total Factor Productivity by 5%

In the simulation of Figure 2.9 the economy is on the initial balancedgrowth path, and in period 1, the rate of growth of population n increasespermanently and unexpectedly by 5%, from 1% to 1.05%. This increaseleads to a gradual decumulation of capital and a gradual decline in output,consumption and the real wage (the marginal product of labor). The realinterest rate (marginal product of capital) gradually increases. The economygradually converges to a new balanced growth path. In this, capital pere�ciency unit of labor is lower by about 1.3%, output, consumption andreal wages are lower by 0.4%, while the real interest rate has increasedfrom 3.68% to 3.74%. The e↵ects of an increase in population growth onsteady state per capita income and consumption in the Solow model are notinsignificant, but they are not huge either.


Figure 2.9: Dynamic Simulation of the Solow Model Following a PermanentIncrease in the Rate of Population Growth by 5%

2.7 Conclusions

The Solow model is a key model in the theory of economic growth. Althoughit is rooted in older models, and although it has theoretical and empiricalweaknesses, this model provides an extremely useful, relatively simple, andflexible framework for the analysis of the process of economic growth.

However, the process of physical capital accumulation, which is the mainengine of economic growth in the Solow model, cannot fully explain eitherthe long-run growth of output per worker that has been observed in de-veloped economies, or the large di↵erences in output per worker betweendeveloped and less developed economies. In fact, only a small part of thesephenomena can be explained by the accumulation of physical capital. To ex-plain the rest, one has to rely on total factor productivity and the e�ciencyof labor (technological progress), which for the Solow model are consideredexogenous.


In this sense, the Solow model, and all the models that make similarassumptions about technology and technical progress, shows us how to over-come its weaknesses and to try to explain total factor productivity, labore�ciency and technical progress. Extensions of the Solow model to addressthese weaknesses are examined in Chapter 7.

Another theoretical weakness of the Solow model is the assumption thatthe savings rate is exogenous. Although at the time that the Solow modelfirst appeared this was a widely accepted assumption, in the context of key-nesian macroeconomics, the assumption is not satisfactory as it does not takeinto account the underlying determinants of household savings behavior.

In the next two chapters we examine two alternative classes of dynamicgeneral equilibrium models of savings behavior, where savings are the resultof rational intertemporal behavior on the part of households that have accessto competitive capital markets. These two classes of models, which areessential building blocks of modern intertemporal macroeconomics, are therepresentative household and the overlapping generations class of models.

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