jones growth model chapter2

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T H S O L O W M O O L 2

Charles JonesIntroduction To Economic Growth nd Edition

ChapterThe Solow Model

Al l theory depends on assumptions which are not quitetrue. That is what makes it theory. The art of success-ful theorizing is to make the inevitable simplifyingassumptio ns in such a way that the final results arenot very sensitive. -ROBERT SOLOW 1956), P 65.

n 1956, Robert Sol ow published a seminal paper on economicgrowth and development titled A Contribution to the Theory of Eco-nomic Growth. For this work and for his subsequent contributions toour understanding of eco nomic growth, Solow was awarded the NobelPrize in economics in 19 87. In this chapter, we develop the mod el pro-posed by Solow and explore its ability to explain the stylized facts ofgrowth and development discussed in Chapter 1 As we will see, thismodel provides an i mport ant cornerstone for understanding wh y somecountries flourish whil e others are impoverished.

Following the advic e of Solow in the quotation above, we will makeseveral assumptions t hat m ay seem to be heroic. Nevertheless, we hopethat these are simplifying assumptions in that, for the purposes at hand ,they do not terribly distort the picture of the world we create. For ex-ample, the world we consider in this chapter will consist of countriesthat produce and c onsume only a single, homogeneous good (output).Conceptually, as well as for testing the model using empirical data, it isconvenient to think of this output as units of a country's gross domes-tic product, or GDP. One implication of this simplifying assumption is

that there is no international trade in the model because there is onlya single good: I 'll give you a 1941 Joe DiMaggio autograph in exchangefo r. . yo ur 1941 Joe DiMaggio autograph? Another assumption of th emodel is that technology is exogenous- that is, the technology avail-able to firms in this simple world is unaffected by the actions of thefirms, including research and development (R&D).These are assump-tions that we will relax later on, but for the mome nt, and for Solow, theyserve well. Much progress in economics has been made by creating avery simple world and then seeing how it behaves and misbehaves.

Before presenting the Solow model, it is worth stepping back to con-sider exactly what a model is and wha t it is for. In modern economics, amodel i s a mathematical representation of some aspect of the economy.It is easiest to thi nk of models as toy economies populated by robots. Wespecify exactly how the robots behave, which is typically to maximizetheir o wn utility. We also specify the constraints t he robots face in seek-ing to maximize their utility. For exampl e, the robots that populate oureconomy may want to consume as much out put as possible, but they arelimited in h ow much outp ut they can produce by the techniques at theirdisposal. The best models are often very simpl e but convey enormousinsight into how the world works. Consider the supply and demandframework in microeconomics. This basic tool is remarkably effectiveat predicting ho w the prices and qu antities of goods as diverse as healthcare, computers, and nuclear weapons will respond to changes in theeconomic environment.

With this understanding of how and why economists develop mod-els, we pause to highlight one of the important assumptions we willmake until the final chapters of this book. Instead of writing down util-ity functions that the robots in our economy maximize, we will sum-marize the results of utility maximization with elementary rules thatthe robots obey. For example, a common problem in economics is foran individual to decide how much to consume today and how much tosave for consumptio n in the future. Another is for individuals to deci dehow much time to spend going to school to accumulate skills and howmuch time t o spend working in th e labor market. Instead of writingthese problems down formally, we will assume that individuals save aconstant fraction of their income a nd sp end a constant fraction of theirtime accumulating skills. These are extremely useful simplifications;without th em, the models are difficult to solve without more advanced


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T H E S O L O W M O D E L T H E B S I C S O L O W M O O E L 3

mathematical techniques. For many purposes, these are fine assump-tions to make in our first pass at understanding economic growth. Restassured, however, that we will relax these assumptions in Chapter 7.

2 1 T H E B A SI C S O L O W M O D E LThe Solow model is built around two equations, a production functionand a capital accum ulation equation. The production function describeshow inputs such as bu lldozers, semiconductors, engineers, and steel-workers combine to produce output. To simplify the model, we groupthese inputs into two categories, capital, K, and labor, L, and denote out-put as Y. The production function is assumed to have the Cobb-Douglasform and is given by

where a is some number between and 1.l Notice that this produc-tion function exh ibits constant returns to scale: if all of the inpu ts aredoubled, output will exactly d o ~ b l e . ~

Firms in this economy pay workers a wage, w, for each unit oflabor and pay r in o rder to rent a unit of cap ital for one period. Weassume there are a large number of firms in the economy so that perfectcompetition prevails and the firms are price-takers.3 Normalizing theprice of ou tput i n our economy to unity, profit-maximizing firms solvethe following problem:

max F(K,L) K wL.K L

According to the first-order cond itions for this problem, firms will hirelabor until the marginal product of labor is equal to the wage and will'Charles Cobb and Paul Douglas 1928) proposed this functi onal form in their analysis ofU S manufacturing. Interestingly, they argued that this pro duction function, with a valuefor a of 1/4 it the d ata very well without allowing for technological progress.2Recall that if F(aK, aL) aY for any number a > 1 hen we say that the productionfunction exhibits co nstan t returns to scale. If F(aK, aL)> aY, then the production func-tion exhibits increasi ng returns to scale, and if the inequality is reversed the productionfunction exhibits decrea sing returns to scale.3 Y o ~ ay recall from microeconomics tha t with constant returns to scale the number offirms is indeterminate--i.e., not pinn ed down by the model.

rent cap ital until the marginal product of capital is equal to the rentalprice:

Notice that wL rK = Y That is, payments to the inputs ( factorpayments ) completely exhaust the value of output produced so thatthere are no economic profits to be earned. This important result is ageneral property of production functions with constan t returns to scale.Notice also that the share of ou tput paid to labor is wL/Y = a andthe share paid to capital is rK/Y a These factor shares are thereforeconstant over time, consistent with Fact 5 from Chapter 1

Recall from Chapter that the stylized facts we are typically in-terested in explaining involve output per worker or per capita o utput.With this interest in mind, we can rewrite the production fu nction inequation (2.1) in terms of outp ut per worker, y = Y/L, and cap ital perworker, k = K/L:

y = k . (2.2)This production function is graphed in Figure 2.1. With more capitalper worker, firms produce more output per worker. However, there arediminishingretur ns to capital per worker: each additional unit of cap italwe give to a single worker increases the output of that worker by lessand less.

The second key equation of the Solow model is an equation tha tdescribes how capital accumulates. The capital accumulation equationis given by

K = S Y - d ~ . (2.3)This kind of equation will be used throughout this book and is veryimportant, so let's pause a moment to explain carefully what this equa-tion says. According to this equa tion, the change in the capital stock,I;(% equal to the amount of gross investment, sY, less the amo unt ofdepreciation that occurs during th e production process, dK We'll nowdiscuss these three terms in more detail. I


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T H E B S I C SO L O W M O D E L 54 2 T H E S O L O W M O O E Lr A C O H H D O U G LA S PR O O U C TIO N

F U N C T I O N

The term on the left-hand si de of equation (2.3) is the continuoustime version of Kt l Kt,hat is, the change in the capital stock perperiod. We use the dot notation4 to denote a derivative with respect

to time:

The second term of equat ion (2.3) represents gross investment. Fol-lowing Solow, we assume that workers/consumers save a constant frac-tion, s of their combined wage and renta l income, = w L rK.Theeconomy is closed, so that saving equals investment, and the only useof investment i n this economy is to accumulate capital. The consumersthen rent this capital to firms for use in production, as discussed above.

The third term of equation (2.3) reflects the depreciation of the cap-ital stock that occurs during production. The standard functional formused here implies that a constant fraction, d, of the capital stock depre-ciates every period (regardless of how much output is produced). For

4Appendix A discusse s the meaning of this notation in more detail.

example, we often assume d = .05 o that percent of the machinesand factories in our model economy wear out e ach year.

To study the evolution of output per person i n this economy, werewrite the capital accumulation equation in terms of capital per person.Then the production function in equation (2.2) will tell us the amountof ou tput per person produced for whatever capital stock per personis present in the economy. This rewriting is most easily accomplishedby using a simple mathematical trick that is often used in the study ofgrowth. The mathematical trick is to take logs and then derivatives(see Append ix A for further discussion). Two examples of this trick aregiven below.xample 1 :

xample :y = ka *logy= alogk

Y==3 - a .kApplying Example o equation (2.3) will allow us to rewrite the

capital accumulation equation in terms of capi tal per worker. But beforewe proceed, let's first consider the growth rate of the labor force,L/L.An important assumption that will be maintained throughout most ofthis book is that the labor force participation rate is constant and thatthe population growth rate is given by the parameter n.=This impliesthat the labor force growth rate,L/L s also given by n. If n = .01 henthe population and th e labor force are growing at one percent per year.This exponential growth can be seen from the relationship

Take ogs and differentiate this equation, and what do you get?.

Often, it is convenient in describing he model to assume that the labor force participationrate is unity-i.e., every member of the population is also a worker.


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6 2 THE SOLOW MOUE THE B S I C S O L O W M O O L 7Now we are ready to combine Example 1 and equation (2.3):

This now yields the capital accumulation equation in per worker terms:

This equation says that the change in capital per worker each period isdetermined by three terms. Two of the terms are analogous to the origi-nal capita l accumulation equation. Investment per worker, sy. increasesk, while depreciation per worker, dk. reduces k. The term that is newin this equation is a reduction in k because of population growth, thenk term. Each period, there are nL new workers around who were notthere during the last period. If there were no new investment and nodepreciation, capital per worker would decline because of the increasein the labor force. The amou nt by which it would decline is exactly nk,as can be seen by setting to zero in Example 1~ S O LV IN G TH E B A S I C S OLO W M O D E LWe hav e now laid out the basic elements of the Solow model and it istime to begin solving the model. What does it mean to solve a model?To answer this question we need to explain exactly what a model is andto define some concepts.

In general, a model consis ts of several equations that describe the re-lation ships among a collection of endogenous variables-that is, amongvariables whose values are determined within the model itself. For ex-ample . equation (2.1) shows how output is produced from capital andlabor, and equation (2.3) shows how capital is accumulated over time.Output, Y nd capital, K are endogenous variables, as are the respec-tive per worker versions of these variables, y and k.

Notice that the equations describing the relationships among en-dogenous variables also involve parameters and exogenous variables.Parameters are terms such as a s, ko, and n that stand in for singlenumbers. Exogenous variables are terms that may vary over time butwhose values are dete rmi ned outside of the model-i.e., exogenously.

The number of workers in this economy, L is an example of an exoge-nous variable.

With these concepts explained, we are ready to tackle the questionof what it means to salve model. Solving a model means obtainingthe values of each endogenous variable when given values for the ex-ogenous variables and parameters. Ideally, one would like to be able toexpress each end ogenou s variable as a function only of exogenous vari-ables and paramete rs. Sometimes this is possible; other times a diagramcan provide insights into the nature of the solution but a computer isneeded for exact values.

For this purpose, it is helpful to think of the economist as a labora-tory scientist. The economist sets up a model and has control over theparameters and exogenous variables. The experiment is the modelitself. Once the model is setup, the economist starts the experimentand watches to see how the endogenous variables evolve over time.The economist is free to vary the parameters and exogenous variablesin different experiments to see how this changes the evoluti on of theendogenous variables.

In the case of the Solow model, our solution will proceed in severalsteps. We begin with several diagrams that provide insight into thesolution. Then, in Section 2.1.4, we provide an analytic solution for thelong-run values of th e key endogenous variables. A full solution of themodel at every point in time is possible analytically, but this derivationis somewhat difficult and is relegated to the appendix of this chapter.

Tbd T H E S O L O W D I A R A MAt the beginning of this s ection we derived the two key equations ofthe Solow model in terms of output per worker and capi tal per worker.These equations are

and

-Now we are ready to ask fund amen tal questions of our model. For exam-ple, an economy star ts out wit h a given stock of capita l per worker, ko,and a given population growth rate, depreciation rate, and investment


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28 T H E S O L O W M O D E L T H E B S I C S O L O W M O O E L 9

rate. How does output per worker evolve over time in thi s economyi.e., how does the economy grow? How does output per worker comparein the long run between two economies that have different investmentrates?

These questions are most easily analyzed in a Solow diagram, asshow n in Figure 2.2. The Solow diagram consists of two curves, plottedas functions of the capital-labor ratio, k. The first curve is the am ountof investment per person, sy sk . This curve has the same shapeas the production function plotted in Figure 2.1, but it is translateddown by the factor s The second curve is the line (n d)k, whichrepresents the amount of new investment per person required to keepthe amount of capital per worker constant -both deprecia tion and thegrowing workforce tend to reduc e the amount of capital per personin the economy. By no coincidence , the difference between these twocurves is the change in the amount of capital per worker. When thischange is positive and the economy is increasing its capital per worker,we say that capital de epening is occurring. When this per worker changeis zero but the actual capital s tock K is growing (because of populationgrowth), we say that only capit al widening is occurring.

To consider a specific example, suppose an economy has capitalequal to the amount kotoday, as dra wn i n Figure 2.2. What happens over

2 2 TH E B SI C SO LO W D I G R M

time? At kc the amount of investment per worker exceeds the amountneeded to keep capital per worker constant, so that capital deepeningoccurs-that is, k increases over time. This capital deepening willcontinue until k k , t which point sy (n d)k, so that k 0. Atthis point , the am ount of capital per worker remains constant, and wecall such a point a ste ady state.

What would happen if instead the economy began with a capitalstock per worker larger than k ? At points to the right of k in Figure 2.2,the am ount of investment per worker provided by the economy is lessthan the amount needed to keep the capital-labor ratio constant. Theterm k is negative, and therefore the a mount of capital per worker beginsto decline in this economy. This decline occurs until the amount ofcapital per worker falls to k .

Notice that the Solow diagram determines the steady-state valueof capi tal per worker. The production function of equatio n (2.4) the ndetermines the steady-state value of out put per worker, y*, as a functionof k*. t is sometimes convenient to inclu de the production function inthe Solow diagram itself to make this point clearly. This is done in

2.b TH E SO LO W D I G R M N D TH E PR O D U C TI O NFU N C TI O N

onsumption


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3 T H E S O L O W M O O E L T H E B S I C S O L O W M O O E L 31

Figure 2.3. Notice that steady-state consumption per worker is thengiven by the differencebetween steady-state output per worker, y*,andsteady-state investment per worker, sy*.; 3 COMPARATIVE STATICSComparative statics are used to examine the response of the model tochanges in the values of various parameters. In this sect ion, we willconsider what happens to per capita income in an economy that beginsin steady state but then experiences a shock. The shocks we willconsider are an increase in the investment rate, s, and an increase inthe population growth rate, n.AN INCREASE N THE lNVESTM ENT RATE Consider an economy that hasarrived at its steady-state value of outp ut per worker. Now suppo se thatthe consumers in that economy decide to increase the investment ratepermanently from s to some value s'. What happ ens to k and y in thiseconomy?

k t b U R E 2.4 A N I N C R E A S E I N T H E I N V E S T M E N TRATE

n+ d k

The answer 1s found in Figurc 2.4. The increase in the investmentrate shifts the s\: curve upward to s'y. At the current value of the capi-tal stock, A , investnlent per worker now exceeds the amount requiredto keep capital per worker constant, and therefore the economy be-gins capital deepening again This capital deepening continues untils'y = n d ) L and the capital stock per worker reaches a higher value,indicated by the point k '. Froin the produ ction function, we know thatthis higher level of capital per worker will be associated with higherper c apita out put; the economy is now richer th an it was before.A N l N C R E A S E IN THE POPULATlON GROWTH RATE Now consider an alterna-tive exercise. Suppo se an economy has reached its steady state, but th enbecause of immigratio n, for example, the po pula tion growth rate of theeconomy rises from n to n'. What happens to k and y in this economy?

Figure 2.5 computes the answer graphically. The n d k curverotates up an d to the left to the new curve (n' d k.At the current valueof the capital stock, k , investment per worker is now no longer highenough to keep the capital-labor ratio constan t in th e face of the rising

A N I N C R EA SE I N PO PU LA TIO NGROWTH


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32 T H E S O L O W M O O E L T H E B S I C S O L O W M O O E L 33

population. Therefore the capital-labor ratio begins to fall. t continuesto fall until the point at which sy (n' d ) k , indicated by k' inFigure 2.5. At this point, the econom y has less capital per worker tha nit began with and is therefore poorer: per capita output is ultimatelylower after the increase in populatio n growth in this example. Why?6 - rr PROPER TIES OF THE STE DY ST TE

By definition, the steady-state quantity of capital per worker is deter-mined by the condition that k 0 quations ( 2 . 4 )and (2.5) allow us touse thi s condition to solve for the steady-state quantities of capital perworker an d output per worker. Substituting from (2.4) into ( 2 . 5 ) ,

deepening more difficult, and these econonlies tend to accumulate lesscapital per worker.

How well do these predictions of the Solow model hold up empiri-cally? Figures 2.6 and 2.7 plot GDP per worker against gross investmentas a share of GDP and against population growth rates, respectively.Broadly speaking, the predictions of the Solow model are borne out bythe empirical evidence. Countries with high investment rates tend to bericher on average than countries with low investment rates, and coun-tries with high population growth rates tend to be poorer on average.At this level, then, the general predictions of the Solow model seem tobe supported by the data.

k sk (n d) k ,and setting this equation equal to zero yields . t : z b G D P PER WO RK ER VER SU S TH E I N VESTME N T R TE

Substituting this int o the pro ductio n function reveals the steady-statequantity of output per worker, y * :

Notice that the endogenous variable y is now written in terms of theparameters of the model. Thus , we have a solution for the model, atleast in th e steady state.

REALGOP PERWORKER,40,000 USA

IRL NORN E L Cl s

r r AFRA CHE

SGP

E W TZL FINl*N JPNKOR

This equation reveals the Solow model's answer to the question 20,000 T T ~ mRCPRT MYSCHLWhy are we so rich and they so poor? Countries that have highsavings/investment rates will tend to be richer, ceteris paribm6 Such URYcountries accumulate more capital per worker, and countries with more MUS wFJ C O @ ~ ~ ~ZAcapital per worker have more out put per worker. Countries that have 1 0,000 TUNNAM ZAF P @ ~ J ~ ~ ~UN BGR-high popul ation growth rates, in contrast, will te nd to be poorer, accord- LKA PRY WSM l0 US King to th e Solow model. A higher fraction of savings in these economiesmust go simp ly to keep the capital-labor ratio constant in the face of agrowing population. This capital- widening requirement makes capital 1,000

INVESTMENT SHARE OFGOP. S 1980-97Ceteris paribus is Latin for all other things being equal.


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2 GD P P E R WOR K E R V E R S U S P OP U L T ION GR OWT H R T E SREALGDP PERWORKER,1997

40,000

30,000

20,000

1 0,000

1,000

USA

SGPT o Cfi&BELITAF E

ZL HK ISRISPAN

ML+N KORBHR

PRT GRC n oA ~ ~ M E XE Y ~ ~URY

ARE

U JOR

ROM J V ~ ~B

SAU

-0.01 0.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07POPULATION GROWTHRATE, 1980-97

E ONOMI G R O W T H IN THE SIMPLE MODELWhat does economic growth look like in the steady state of this simpleversion of the Solow model? The answer is that there is n o per capitagrowth in this version of the model Outp ut per worker (and thereforeper person, since we've assumed the labor force participation rate isconstant) is constant in the steady state. Output itself, Y is growing, ofcourse, but only at the rate of population growth.7

This version of the model fits several of the stylized facts discussedin Chapter 1 It generates differences in per capita income across coun-tries. It generates a constant capital-output ratio (because both k and y7This can be seen easily by applying the take logs and differentiate rick to y Y/L

T H E B S I C S O LO W M O D E L 5

are constant, implying that K /Y is constant). It generates a constantinterest rate , the marginal product of capital. However, it fails to pre-dict a very important stylized fact: that economies exhibit sustained percapita income growth. In this model, economies may grow for a while ,but not forever. For example, an economy that begins with a stock ofcapital per worker below its steady-state value will experience growthin k and y along the tr nsition p th to the steady state. Over time, how-ever, growth slows down as the economy approaches its steady state,and eventually growth s tops altogether.

To see that growth slows down along the transition path, notice twothings. First, from the capital accumulation equation (equation (2.5)),one can divide both sid es by k to get

Because is less than one, as k rises, the growth rate of k graduallydeclines. Second, from Example 2, the growth rate of y is proportionalto the growth rate of k, so that the same statement holds true for outputper worker.

T R N S I T I O N D Y N M I C S


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T H S O L O W M O O LThe transition dynamics implied by equation (2 .6) are plotted in

Figure 2.8.' The first term on the right-hand side of the equation iss k ' which is equal to s y / k . The higher the level of capital per worker,the lower the average product of capi tal, y / k , because of diminishingreturns to capital accumulation ( a is less than one). Therefore, thiscurve slopes downward. The second term on the right-hand side ofequation (2.6) is n d, which doesn't depend on k , so it is plotted asa horizontal line. The difference between the two lines in Figure 2.8is the growth rate of the capital stock, or k / k . Thus, the figure clearlyindicates that the further an economy is below i ts steady-state value ofk , he faster the economy grows. Also, the further a n economy is abovejts steady-state value of k , he faster k declines.

+ TECHNOLOGY ND THE SOLOW MO DELTo generate sustained growth in per capita income in this model, wemust follow Solow and introduce technological progress to the model.This is accomplished by adding a technology variable, A, to the pro-duction function:

Y F(K,AL) = K (AL)'- . (2.7)Entered this way, the technology variable is said to be labor-augmenting or Ha rrod- ne~t ral. ~echnological progress occurs when

increases over time unit of labor, for example, is more productivewhen the level of technology is higher.

An important assumption of the Solow model is that technologicalprogress is exogenous: in a common phrase, technology is like mannafrom heaven, in that it descends upon the economy automatically andregardless of whatever else is going on in the economy. Instead of mod-eling carefully where technology comes from, we simply recognize forthe moment that there is technological progress and make the assump-his alterna tive version of the Solow diagram makes the growth implications of the

Solow model much more transparent. Xavier Sala-i-Martin 1 9 9 0 )emphasizes this point.'The other possibilities are F [ A K ,L) , which is know n as capital-augmenting or Solow-neutral technology, and A F K , L) , which is known as Hicks-neutral technology. Withthe Cobb-Douglas functional form assumed here, this distinction is less important.

tion that A is growing t a constant rate:

where g is a parameter representing the growth rate of technology. Ofcourse, this assumption about technology is unrealistic, and explaininghow to relax this assumption is one of the major accomplishments ofthe new growth theory that we will explore in later chapters.

The capital accumulation equation in the Solow model with tech-nology i s the same as before. Rewriting it slightly, we get

To see the growth implicat ions of the model wit h technology, firstrewrite the production function (2.7) in terms of output per worker:

Then take logs and differentiate:

Finally, notice from the capital accumulation equation (2.8) that thegrowth rate of K will be constant if and only if Y / K is constant. Fur-thermore, if Y / K is constant, y / k is also constant, and most important,y and k will be growing at the same rate. A situation in which capital,output, consumption, and population are growing at constant rates iscalled a balanced growth path. Partly because of its empirical appeal ,this is a situation that we often wish to analyze in our models. Forexample, according to Fact in Chapter 1 this situation describes theU.S. economy.

Let's use the notation g, to denote the growth rate of some variablealong a balanced growth path. Then, along a balanced growth p$th,gy = gk according to the argument above. Substituting thi s relationship

into equation (2.9)and recalling that A / A = g,

That is, along a balanced growth path in the Solow model, output perworker and capital per worker both grow at the rate of exogenous tech-


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8 2 T H E S O L O W M O O E L T E C H N O L O E Y ND T H E S O L O W M O O E L 9nological change, g. Notice that in the model of Section 2.1, there wasno technological progress, and therefore there was no long-run growthin output per worker or capital per worker; g,, k g 0 The modelwith technology reveals that technological progress is th e source of sus -tained per capita growth. In this chapter, this result is little more thanan assumption; in later chapters, we will explore the result in muchmore detail and come to the same conclusion.

THE SOLOW DI GR M WITH TECHNOLOGYThe analysis of the Solow model with technological progress proceedsvery much like the analysis in Section 2.1: we set up a differentialequation and analyze it in a Solow diagram to find the steady state. Theonly important difference is tha t the variable k is no longer constant inthe long run, so we have to write our differential equation in terms ofanother variable. The ne w state variable will be k K/AL. Notice thatthis is equivalent to k / A an d is obviously constant along the balancedgrowth path because gk g~ g The variable therefore representsthe ratio of capital per worker to technology. We will refer to this asthe capital-technology ratio (keeping in mind that the numerator iscapital per worker rather than the total level of capital).

Rewriting the production function in terms of k we get

where = Y/AL y /A . Following the terminology above, we willrefer to as the output-tech nology ratio. 1

Rewriting the capital accum ulation equation in terms of is accom-plish ed by following exactly th e methodology used in Secti on 2.1. First,note that

Combining this with the capital accumulation equation reveals that

''The variables 7 and are sometimes referred to as output per effective unit of labor ,and capital per effective unit of labor. This labeling is motivated by the fact that tech-nologlcal progress is labor-augmenting. AL S then the effective amount of labor usedIn production.

* : , T H E S O L O W D I G R M W I T HT E C H N O L O G I C L P R O G R E S S

The similarity of equations (2.11) and (2.12) to their counterparts inSection 2.1 should be obvious.

The Solow diagram with technological progress is presented in Fig-ure 2.9. The analysis of this diagram is very similar to the analysis whenthere is no technological progress, but the interpretation is slightlydiffe rent . If the economy begins with a capital-technology ratio tha tis below its steadystate level, say at a point such as , the capital-technology ratio will rise gradually over time. W hy? Because the amountof investment being undertaken exceeds the amount needed to keep thecapital-technology ratio constant. This will be true until s j (n+g+ dlat the point k t which point the economy is i n steady state and growsalong a balanced growth path.

SOLV ING FOR THE STE DY ST TEThe steady-state output-technology ratio is determined by the produc-tion function and the condition that O Solving for I; , we find


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T H E S O L O W M O O E Lthat

Substituting this into the production fullction yields

To see what this implies about output per worker, rewrite the equationas

where we explicitly note the dependence of y and A on time. From equa-tion (2.131, we see that outpu t per worker along the balanced growthpath is determined by technology. the investment rate, and the popu-lation growth rate. For the special case of g = and A. = -i.e.. ofno technological progress-this result is identical to that derived inSection 2.1.

An interesting result is apparent from equation (2.13) and i s dis-cussed in more detail in Exercise at the end of this chapter. Thatis, changes in the investment rate or the population growth rate affectthe long-run level of ou tpu t per worker but do not affect the long-rungrowth rate of output per worker. To see this more clearly, let s considera simple example.

Suppose an economy begins in steady state with investment rate sand then permanently increases its investment rate to S (for example,because of a permanent subsidy to investmen t). The Solow diagram forthis policy change is drawn in Figure 2.10, and the results are broadlysimilar to the case with no technological prog ess. At the initial capital- 1technology ratio investment exceeds the amount needed to keep thecapital-technology ratio constant, so begins to rise. 1

To see the effects on growth, rewrite equation 2.12) as

T E C H NO L O G Y A N T H E SO L O W M O O E L 4

I A N I N C R E A S E I N T H EI N V E S T M E N T R A T E

and note that j?/ is equal to k- . igure 2.11 illustrates the transitiondynamics implied by this equation. As the diagram shows, the increasein the investment rate to s raises the growth rate temporarily as theeconomy transits to the new steady state,k .ince g is constant, fastergrowth in along the transition path implies that output per workerincreases more rapidly than technology: y y > g. The behavior of thegrowth rate of output per worker over time is disp layed in Figure 2.12.

Figure 2.13 cumulates the effects on growth to show what happensto the (log) level of output per worker over time. Prior to the policychange, output per worker is growing at the constant rate g, so that thelog of outpu t per worker rises linearly. At the time of the policy change.t , output per worker begins to grow more rapidly. This more rapidgrowth continues temporarily until the ou tput-technology ratio reachesits new steady state. At this po int, growth has retu rned to its long-runlevel of g .

This exercise illustrates two important points. First, policy changesin the Sdlow model increase growth rates, but only temporarily alongthe transition to the new steady state. That is, policy changes have nolong-run growth effect. Second, policy changes can have level effects.


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4 T H E S O L O W MODEL

A N I N C R E A S E I N T H E I N V E S T M E N TRATE TRANSIT ION DYNA MICS

E V L U T I N G T H E S O L O W M O D E L 43

THE EFFECT OF AN INCREASE I NINVESTMENT ON y

t TIM

That is, a permanen t policy change can permanent ly raise or lower)the level of per capita output.

E 2 12 T H E E F F E C T O F AN I N C R E A S E I NI N V E S T M E N T O N G R O W T H

t TIM

3 EVALUATING THE SOLOW MODE LHow does the Solow model answer the key questions of growth anddevelopment? First, the Solow model appeals to differences in invest-ment rates and population growth rates and perhaps) to exogenousdifferences in technology to explain differences in per capita incomes.Why are we so rich an d they so poor? According to the Solow model,it is because we invest more and have lower population growth rates,both of which allow us to accumulate more capital per worker an d thusincrease labor productivity. In the next chapter, we will explore thishypothesis more carefully and see that it is firmly supported by dataacross the countr ies of the world.

Second, why do economies exhibit sustained growth in the Solowmodel? The answer is technological progress. As we saw earlier, with-out technological progress, per capita growth will eventually cease asdiminishing returns to capital set in. Technological progress, however,can offset the tendency for the marginal p roduc t of capital to fall, and


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T H E S O L O W M O O E L G R O W T H A C C O U N T I N G T H E P A O O UC T l V l T Y S L O W O O W N A N 0 T H E N E W E C O N O M Y 5

in the long run, countries exll11)it per ( ap it , ~ ~o wt h t the rate of t ech-nological progress.

How, then, does the Solow model account for differences in growthrates across countries? At first glance, i t may seem fi at the Solow modelcannot do s o, except by appealing to differences in (unmodeled) tech-nological progress. A more subtle explanation, however, can be foundby appealing to transition dynamic:^. We have seen several examplesof how transition dyna mics can allow countri es to grow at rates differ-ent from their long-run growth rates. For example, an economy with acapital-technology ratio below its long-run level will grow rapidly untilthe capital-technology ratio reaches its steady-state level. This reason-ing may help explain why countries such as Japan and Germany, whichhad their capital stocks wiped out by World War 11 have grown morerapidly than the United States over the last fifty years. Or it may explainwhy an economy that increases its investment rate will grow rapidly as

I N V E S T M E N T R A TE S I N S O M E N E W L Y I N D U S T R I A L I Z IN GE C O N O M I E S

RATEOINVESTMENT 45AS APERCENTAGE

O GDP)

I1950 1960 1970 1980 1990 2000YEAR

t makcs the transition to a higher output-tccllnology ratio. This expla-nation may work wel l for countries such as South Korea, Singapore, andTaiwan. Their investment rates have increased dramatically since 1950,as shown in Figure 2.14. The explanation may work less well, however,for an economy such as Hong Kong's. This kind of reasoning raises aninteresting question: can countries permanently grow at different rates?'This question will be discussed in more detail in later chapters.

GROWTH ACCOUNTING THE PRODUCTIVITY SLOWDOWNAND THE NEW ECONOMYWe have seen in the Solow model that sustained growth occurs only i nthe presence of technological progress. Without technological progress,capital accumulation runs into diminishing returns. With technologicalprogress, however, improvements in technology continually offset thediminishing returns to capital accumulation. Labor productivity growsas a result, both direc tly because of the improvements i n technologyand indirectly because of the additional capital accumulation theseimprovements make possible.

In 1957, Solow published a second article, Technical Change andthe Aggregate Production Function, in which he performed a simpleaccounting exercise to break dow n growth in outp ut into growth i n cap-ital, growth in labor, and growth in technological change. This growth-accounting exercise begins by postulating a production function suchas

where B is a Hicks-neutral productivity term.'' Taking logs and dif-ferentiating this production function, one derives the key formula ofgrowth accounting:

In i ac:~. his growth accounting can he done with a much more general productionI unction such as B t ) F K , ), and the results are very similar.


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6 T H E S O L O W M O D E L G R O W T H A C C O U N T I N G T H E P R O D U C T I V I T Y S L O W D O W N A N 0 T H E N E W E C O N O M Y 7

This equation s~iyshat output growth is cqual to a weightcd averageof capital and labor growth plus the growth rate of B This last term,B/B s comnlonly referred to as total factor productivity growth orrzzultifactor productik ity growtlz Solow, as well as economists such asEdward Denison and Dale Jorgenson who followed Solow's appr oach,have used this equation to understand the sources of growth in output.

Since we are primarily interested here in the growth rate of outputper worker instead of total output, it is helpful to rewrite equation (2.14)by subtracting L L from both sides:

That is, the growth rate of output per worker is decomposed into thecontribution of physical capital per worker and the contribu tion frommultifactor productivity growth.

The U.S. Bureau of Labor Statistics (BLS) provides a detailed ac-counting of U.S. growth using a generalization of equati on (2.15) . Itsmost recent numbers are reported in Table 2.1. They generalize thisequation in a cou ple of ways. First , the BLS measures labor by calculat -

Labor composition 0 2 0 2 0 0 0 3 0I Multifactorproductivity 1 4 2 1 0 6 0 5 0 6 1 4

ureau of abor Statistics 2000).

ing total hours worked rather than just the number of workers. Second,the BLS includes an add itional term in equation (2.15) to adjust forthe changing composition of the labor force-to recognize, for example,that the labor force is more educated today than it was forty years ago.

As can be seen from the table, output per hour in the private busi-ness sector for the Unite d States grew at an average annual rate of 2.5percent between 1948 and 1998. The contribution from capital per hourworked was 0.8 percentage points, and the changing composition of thelabor force contributed another 0.2 percentage points. Multifactor pro-ductivity growth accounts for the remaining 1.4 percentage points, bydefinition. The implication is that about one-hal f of U.S. growth wasdue to factor accumulation and one-half was d ue to the improvementin the productivity of these factors over this pe riod. Because of the wayin whi ch it is calculated, economists have referred to this 1.4 percentas the residual or even as a measure of our ignorance. One inter-pretation of the multif actor productivity growth t erm is that it is due totechnological change; notice that in terms of the production function inequation (2.7),B A1-a. his interpretation will be explored in laterchapters.

Table 2.1 also reveals how GDP growth and its sources have changedover time in the Un ited States. One of the important stylized facts re-vealed i n the table is the productivity growth slowdown that occurredin the 1970s. The top row shows that growth in outpu t per hour (alsoknown as labor productivity) slowed dramatically after 1973; growthbetween 1973 and 1995 was nearly 2 percentage points slower thangrowth between 1948 and 1973. What was the s ource of this slow-down? The next few rows show that the changes in the contributionsfrom capital per worker and labor composition are relatively minor. Theprimary culprit of the productivity slowdown is a substantial decline inthe growth rate of multifactor productivity. For some reason, the resid-ual was much lower after 1973 han before: the bulk of the productivi tyslowdown is accounted for by the measure of ou r ignorance.':A similarproductivity slowdown occurred throughout th e advanced countries ofthe world.

Various explanations for the productivity slowdown have been ad-vance.d. For example, perhaps the sharp rise in energy prices in 1973and 1979 contributed to the slowdown. One problem with this expla-nation is that in real terms energy prices were lower in the late 1980s


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8 T H E S O L O W M O O E L G R O W T H A C C O U N T I N G T H E P R O OU C T l Vl T Y S L O W O O W N A N 0 T H E N E W E C O N O M Y 9than th :v worc befolc. thc oil shocks. Anothcr explan,li~onmay involvcthe cl ~l ng ing omposliion of the labor f orcr or the scc-toral shift in theeconolny away from manufacturing (urhich tends t o have high laborproductivity) toward services (many of which have low labor produc-tivity). This explanation receives some support from rcicent evidencethat productivity growth recovered substantially in thc 1980s in manu-facturing. It is possible that a slowdown in resources spent on researchand development (R&D) n the late 1960s contributed to the slowdownas well. Or, perhaps it is not the 1970s and 1980s that need to be ex-plained but rather the 1950s and 1960s: growth may simply have beenartificially and temporarily high i n the years following World War I1 be-cause of the applic ation to the priva te sector of new technologies createdfor the war. Nevertheless, careful work on the productivity slowdownhas failed to provide a complete explanation.12

The flip side of the productivity slowdown after 1973 is the rise inproductivi ty growth in the 1995-98 period, sometimes labeled the NewEconomy. Growth in output per hour and in multifactor productivityrose substan tially in this period , returning about 50 percent of the wayback to the growth rates exhibited before 1973. As shown in Table 2.1,the increase in growth rates is partially associated with an increasein the use of information technology. Before 1973, this component ofcapital accumulation contributed on ly 0.1 percentage points of growth,but by the late 1990s, this contribution had risen to 0.8 percentagepoints. In addit ion, evidence suggests that as much as half of the risein multifactor productivity growth in recent years is due to increases inefficiency of the production of inf ormat ion technology.

Recently, a number of economists have suggested that the informa-tion-technology revolution associated wi th the widespread adoption ofcomputers might explain both the productivity slowdown after 1973as well as the recent rise in producti vity growth. According to this hy-pothesis, growth slowed temporarily while the economy adapted itsfactories to the new prod uction techniques associated with informationtechnology an d as workers learn ed to take advantage of the new tech -nology. The recent upsurge in productivity growth, then, reflects the

lZTh e fall 988 issue of the ]ournc;l of Economic Perspectives contains wveral papersciiscussing potential explanations of the productivity slowdown.

successful widespread adoption of this new technology. The recent up-surge in productivity growth, then, reflects the successful widespreadadoption of this new technology.' Whether or not this view is correctremains to be seen.

Growth accounting has also been used to analyze economic growthin countries other than the United States. One of the more interestingapplications is to the NICs of South Korea, Hong Kong, Singapore, andTaiwan. Kecall from Chapter that average annua l growth rates haveexceeded 5 percent in these economies since 1960. Alwyn Young (1995)shows that a large part of this growth is the result of factor accumulation:increases in investment i n physical capital and education, increases inlabor force participation, and a shift from agriculture into manufactur-ing. Support for Young's result is provided in Figure 2.15. The verticalaxis measures growth in output per worker, while th e horizontal axismeasures growth in Harrod-neutral (i.e., labor-augmenting) total fac-tor productivity. That i s, instead of focusing on growth in B, whereB Alp , we focus on the growth of A. (Notice tha t with i hegrowth rate of A is sim ply 1.5 times the growth rate of B. This changeof variables is often convenient because along a steady-state balancedgrowth path, g g ~ .ountries growing along a balanced growth path,then, should lie on the 45-degree line i n the figure.

Two features of Figu re 2.15 stand out. First , while th e growth ratesof ou tput per worker in the East Asian countri es are clearly remark-able, their rates of growth in total factor productivity (TFP) are less so.A number of other countri es such as Italy, Brazil, and Chile have alsoexperienced rapid TFP growth. Total factor productivity growth, whiletypically higher than in th e United States, was not exceptional in theEast Asian economies. Second, the East Asian countries are far abovethe 45-degree line. This shift means that growth in output per workeris much higher than TFP growth would suggest. Singapore is an ex-treme example, with slightly n eg a t i ve TFP growth. Its rapid growth ofoutput per worker is entirely attributable to growth in capital an d ed-ucation. More generally, a key source of the rapid growth performance

' See Paul David (1990) an d Jeremy Greenwood an d Mehmet Yorukoglu (1997). Moregenerally, a nice collection of papers on'the New Economy can be found in the Fall2000 issue of the ]ournu1 of Eco nomic Perspectives


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2 T H E S O L O W M O O E L Exercises 51

GROWTHRATEOF 0.05

r r 6 8 t r I L GROWTH ACCOUNTINGJPN

DEUFAAKOR OAN ITA

0 03 SGP

ARG

7 ' COL-0.01 I I I

-0.01 0 0.01 0.02 0.03 0.04 0.05GRO WM RATE OFAHARRODHEUTRAL)SOURCE Author s calculations using the data collection reported in Table 10.8

of Barro and Sala-i- Martin (1998).Note: The years over which growth rates are calculated vary across countries:

1960-90 for OE D members, 194 0-80 for Latin America, and 1966-90 for EastAsia.

of these countr ies is factor accumulati on. Therefore, Young concludes,the framework of the Solow model ( and the extension of the m odel inChapter 3) can explain a substantial amount of the rapid growth of theEast Asian economies.

APPENDIX: CLOSED FORM SOLUTIONOF THE SOLOW MODELIt is possible to solve analytically for output per worker y( t) at each pointin time in th e Solow model. The derivation of this solution is beyondthe scope of thi s book. One derivation can be found i n the appendix t o

Chapter 1 of Barro and Sala-i-Martin (1998). Another can be found in"A Note on the Closed-Form Solu tion of the Solow Model," which canbe downloaded from my Web page at http://emlab.berkeley.edu/users/chad/papers.html#closed form. The key%sight is to recognize that thedifferential equation for the capital-output ratio in th e Solow model islinear and can be solved using stan dard techniques.

Although the method of solution is beyond t he scope of this book,the exact solution is still of interest. It illustrates nicely what it meansto "solve" a model:

In this expression, we have defined a new parameter: = 1 a ) (ng d).Notice that output per worker at any time t is written as a functionof the paramete rs of the model as well as of the exogenous variable A(t).

To interpret this expression, notice that at t = 0, output per workeris simply equal to yo, which i n turn is given by the parameters of themodel; recall that yo = &A:- . That's a good thing: our solution saysthat o utput per worker starts at the level given by the production func-tion At the other extreme, consider wha t happens as t gets very large,in the limit going off to inhi ty . In this case, e-" goes to zero, so weare left with an expression that is exactly that given by equation (2.13):output per worker reaches its steady-state value.

In between t = 0 and t = 03 output per worker is some kind ofweighted average of its initial value an d its steady-state value. As timegoes on, all that changes are the weights.

The interested reader will find it very useful to go back and rein-terpret the Solow diagram and the various comparative static exerciseswith this solution in mind.

EXERCISES1 A decrease in the investment rate. Suppose the U.S. Congress en-

acts legislation that discourages saving and investment, such as theelimination of the investment tax credit that occurred in 1990. As aresult, suppose the investment rate falls permanentl y from s' to s".


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5 T H E S O L O W M O E L E x e r c i s e s 53

Examine this policy changc in tlic Solow n~odel ith technologicalprogress, assuming that th e economv I)cy ns in steady state. Sketcha graph of how (the natural log of) output p : r worker evolves overtime with and without the policy change Mako a similar graph forthe growth rate of output per worker. Does the po li c ~hange perma-nently reduce the level or the grow111 ute of output per worker?2 An illcrease in the labor force. Shocks to a n economy, such as wars,

famines, or the unification of two economies, often generate largeone-time flows of workers across borders. What are the short-run andlong-run effects on an eco nomy of a one-time permanent increase inthe stock of labor? Examine this question in the context of the Solowmodel with g and n > 0.3. An income tax. Suppos e the IJ.S. Congress decides to levy an inc ometax on both wage income and capital income. Instead of receivingwL rK Y, consumers receive 1 r)wL 1 r)rK 1 )Y.Trace the consequences of this tax for output per worker in th e shortan d long runs, starting from steady state.4. Manna falls faster. Supp ose that there is a permanent increase in therate of technological progress, so that g rises to g . Sketch a gr aph ofthe growth rate of outpu t pe r worker over time. Be sure to pay closeattention to the transition dynamics.5. Can we save too much? Consumption is equal to output minu s in-vestment: c 1 s ) ~ .n the context of the Solow model withno technological progress, what is the savings rate that maximizessteady-state consumption per worker? What is the marginal productof capital in this steady state? Show this point in a Solow diagram.Be sure to draw the production function on the diagram, and sho wconsumption and saving and a line indicating the marginal productof capital. Can we save too much?6. Solow 19513)versus Solow 1957). In the Solow model with tech-nological progress, consider an economy that begins in steady statewit h a rate of technological progress, g of 2 percent. Suppose g risespermanently to 3 percent. Assume a 1/3.

b) Using equation (2.15), perform t he growth accounting exercisefor this economy, both before the cha nge and after the economyhas reached its new balanced growth path. (Hint: recall that B =A I How muc h of the increase in the growth rate of output perworker is d ue to a change in the growth rate of capital per worker,and how much is due to a change in multifactor productivitygrowth?

(c) In what sense does the growth accounting result in part (b) pro-duce a misleading picture of this experiment?

a) What is the growth rate of output per worker before the change,and what happens to this growth rate in the long run?

jones growth model chapter2

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