Solow Growth Model

Michael Bar�

March 4, 2020

Contents

1 Introduction 21.1 Some facts about modern growth . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 The Solow Model 5

3 Qualitative analysis 53.1 Steady state analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Technological improvement . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Higher growth rate of population . . . . . . . . . . . . . . . . . . . . 103.1.3 Higher saving rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.4 Summary of analytical results . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Golden Rule saving rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Quantitative analysis 144.1 Functional forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Assigning numerical value to � . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Equilibrium in the calibrated model with growth . . . . . . . . . . . . . . . . 18

4.3.1 Balanced Growth Path (BGP) . . . . . . . . . . . . . . . . . . . . . . 184.4 Optimal saving rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Conclusions 22

�San Francisco State University, department of economics.

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1 Introduction

The two main branches of macroeconomics are (a) Growth and (b) Business cycles, wheregrowth refers to the general increase in real GDP per capita, while business cycles aredeviations from the growth trend. Usually, macroeconomic courses start with studying ofgrowth for two reasons:

1. The relative importance of growth for the well being of people is much greater thanthat of business cycles. Take for example the recession of 2007 - 2009, with the lowestlevel reached in 2009. There were only 5 years in the entire history prior to 2009, inwhich the real GDP per capita (standard of living) was higher than in 2009 (theseyear are 2004 - 2008). At the lowest point of this recession, the standard of living wasstill higher than all of the previous history of this country, with the exception of only5 years (see the next graph). Therefore, volatility of output around the growth trenddwarfs in comparison to the trend itself.

In addition, consider the di¤erence between 1% growth in standard of living v.s. 2%growth. In the 1% percent case real GDP per capita will double approximately every70 years, while in the 2% case, the doubling time is about 35 years, i.e. after 70 yearsthe standard of living quadruples. In other words, small di¤erences in growth rates,when compounded over a generation or more, have great consequences for standard ofliving.

2. The second reason why we start with growth is that growth models are the mainworkhorse in the study of business cycles, with added stochastic components. So it iseasier to start with growth.

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1.1 Some facts about modern growth

1. Per capita output grows over time, and its growth does not tend to diminish in devel-oped countries.

2. Physical capital per worker grows over time. Capital/output ratio is approximatelyconstant.

3. The rate of return to capital is nearly constant over time.

4. The shares of labor and physical capital in national income are nearly constant overtime.

5. The standard of living di¤ers substantially across countries (see the next �gure).

6. The growth rate of output per worker di¤ers substantially across countries (see thenext �gure).

7. There is no systematic relationship between standards of living and growth rates acrosscountries (poor countries do not tend to grow faster and catch up). In other words,there is no absolute convergence.

Facts 1-4 describe time-series observations. Facts 5-7 describe cross-sectional observa-tions.

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We will later evaluate theories with respect to these facts. For example, we will checkwhether the predictions of our models are consistent with these facts.

1.2 Questions

1. What factors determine which countries prospered?

2. Can we point to speci�c economic policies?

3. Are there speci�c country characteristics that determine economic fate?

4. Is prosperity just a result of luck?

"Once one starts to think about [these questions], it is hard to think about anythingelse". Robert Lucas (1988).

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2 The Solow Model

1. Output is produced with production function Yt = F (Kt; Lt), where Yt is aggregate(real) output, Kt is the stock of physical capital, and Lt is labor services. The func-tion F (�; �) is assumed to exhibit constant returns to scale (CRS), with the followingassumptions:

(a) : F (0; L) = F (K; 0) = 0

(b) : FK (K;L) > 0; FL (K;L) > 0; FKK (K;L) < 0; FLL (K;L) < 0

(c) : limK!0

FK (K;L) = limL!0

FL (K;L) =1, limK!1

FK (K;L) = limL!1

FL (K;L) = 0

The �rst assumption (a) means that both inputs are essential for production. Thesecond assumption (b) means that the marginal products of both inputs are positiveand diminishing, and the last assumption (c) means that the marginal product ofinputs is very high when the input is scarce and very low when it is abundant. Theconditions in (c) are sometimes called Inada conditions, after the Japanese economistKen-Ichi Inada. A production function which is CRS and possesses properties (a) - (c)is called a neoclassical production function.

2. The law of motion of aggregate capital is: Kt+1 = (1� �)Kt + It, where 0 < � < 1 isthe depreciation rate and It is aggregate investment.

3. People save a fraction s of their income. This fraction is exogenous1. Thus, the totalsaving and total investment in this (closed) economy is

St = It = sYt

4. The population of workers grows at a constant rate of n, which is exogenous in thismodel. Thus, the law of motion of population is: Lt+1 = (1 + n)Lt.

5. No government.

3 Qualitative analysis

Now we derive the predictions of the model about output per worker, consumption andinvestment. Our main object of interest is output per worker, denoted by

yt �YtLt:

Another object of interest is consumption per worker:

ct = (1� s) yt1We call a variable endogenous if it is determined within the model and exogenous if it is determined

outside the model. For example, in the model of a market (supply and demand diagram), the price andquantity traded of the good are endogenous variables, while other variables that determine the location ofthe supply and demand curve, such as income and prices of other goods, are assumed exogenous.

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which depends on the consumption rate 1 � s and output per worker. So we start byexamining how output per worker evolves over time in this model. First, we notice thatoutput per worker can be represented as a function of capital per worker:

yt =YtLt=F (Kt; Lt)

Lt= F

�Kt

Lt; 1

�The last equality follows from the CRS assumption. Let kt � Kt

Ltdenote capital per worker

and f (kt) � F�Kt

Lt; 1�be the output per worker as a function of capital per worker2. Thus,

output per worker can be written as

yt = f (kt)

Notice that f (0) = 0, f 0 (k) > 0 and f 00 (k) < 0, from the properties of F (K;L). The keyis therefore to �gure out how capital per worker, kt, evolves over time. To derive the lawof motion of capital per worker, we simply divide the law of motion of aggregate capital byLt+1 and use the de�nition of investment:

Kt+1 = (1� �)Kt + sF (Kt; Lt)

Kt+1

Lt+1=

(1� �)Kt

(1 + n)Lt+sF (Kt; Lt)

(1 + n)Lt

kt+1 =(1� �)(1 + n)

kt +s

(1 + n)f (kt) (1)

The last equation is the law of motion of capital per worker.If the function f (�) does not change over time, i.e., there is no change in productivity

(say due to technological progress), then we can predict the behavior of this economy in theshort run and in the long run pretty easily. Equation (1) gives us the future capital as afunction of current capital kt+1 (kt). This is called a di¤erence equation3. First observe thatkt+1 (0) = 0, i.e. if the economy does not have any physical capital, which is essential inproduction, there is no way to build new one. Next, note that kt+1 (kt) is increasing andstrictly concave:

k0t+1 (kt) =(1� �)(1 + n)

+s

(1 + n)f 0 (kt) > 0

k00t+1 (kt) =s

(1 + n)f 00 (kt) < 0

Moreover,

limkt!0

k0t+1 (kt) =(1� �)(1 + n)

+s

(1 + n)limkt!0

f 0 (kt)| {z }=1

=1

limkt!1

k0t+1 (kt) =(1� �)(1 + n)

+s

(1 + n)limkt!1

f 0 (kt)| {z }=0

=(1� �)(1 + n)

< 1

2This is called intensive form of production function.3A di¤erence equation is a discrete time analog of the di¤erential equation in continuos time.

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Thus, the law of motion of capital per worker starts at the origin, has very a steep slope forsmall kt, the slope decreases as kt becomes large, and eventually the slope becomes less than1. These properties follow from the Inada conditions in assumption (c) above. The graph ofthe law of motion of capital is illustrated in the next �gure.

Figure 1: Law of motion of capital per worker

From the above graph, it is clear that starting from any level of capital per worker k0, itwill converge to the steady state level of kss. The next �gure illustrates the convergence ofcapital per worker to a steady state.

Figure 2: Convergence to steady state

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Suppose the economy starts with k0 < kss capital per worker. The law of motion ofcapital per worker, kt+1 (kt) can be used to obtain k1 (k0). The 450 line helps us to locatek1 on the horizontal axis, so that in period 1 the economy starts with k1. Again, the law ofmotion of capital per worker determines k2 (k1), and the process continues forever. Similarly,Suppose the economy starts above the steady state, k0 > kss. The law of motion helps to�nd k1 (k0), k2 (k1),... Thus far we have proved mathematically, and illustrated graphically,an important proposition about the Solow model.

Proposition 1 If the aggregate production function F (K;L) is CRS, satis�es conditions(a)-(c), and the production function does not change over time (no technological progress),then starting with any positive level of initial capital per worker k0 > 0, the economy willconverge to a steady state. Formally, for any k0 > 0, we have limt!1 kt = kss, such thatkt+1 (kss) = kss.

Since the capital per worker converges to the steady state, output per worker and con-sumption per worker will also converge to steady state levels:

limt!1

yt = yss = f (kss) , and limt!1

ct = (1� s) yss = (1� s) f (kss)

Alternatively, if there was technological progress, then we could write f (kt; t) to indicatethe dependence of the production function on time. In this case technological improvementwould shift the law of motion of capital per worker. The next �gure illustrates a case oftechnological improvement.

Figure 3: Technological improvement

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3.1 Steady state analysis

In the previous section we showed that in the Solow model with no change in produc-tivity, starting with any positive level of capital per worker, the economy will convergeto the steady state. In other words, the prediction of the model is that in the long run,all variables converge to steady state, which is similar to the prediction of the supply anddemand model of a market that the price and quantity will converge to the equilibrium. Soin a sense, the steady state (kss; yss; css) is the model�s prediction about the economy in thelong run. Our next task is to �nd the steady state and investigate how it depends on theexogenous parameters. We start by �nding the steady state level of capital per worker usingthe de�nition of the steady state, that kt+1 (kss) = kss. Thus, substituting kt = kt+1 = kssin the law of motion of capital per worker and rearranging, gives

kt+1 =(1� �)(1 + n)

kt +s

(1 + n)f (kt)

kss =(1� �)(1 + n)

kss +s

(1 + n)f (kss)

kss (1 + n) = (1� �) kss + sf (kss)kss (n+ �) = sf (kss) (2)

Equation (2) characterizes the steady state level of capital and also provides economic intu-ition about the steady state. The left hand side, kss (n+ �), represents the decline in capitalper worker due to depreciation and increasing number of workers ("�ow out"). The righthand side represents the increase in capital per worker due to investment per worker ("�owin"). At the steady state the "�ow out" is exactly o¤set by the "�ow in". We cannot solvefor kss without knowing the function f (�), but we can nevertheless illustrate the steady stategraphically and perform qualitative analysis. The next �gure illustrates the �ows ("�ow in"and "�ow out" in the Solow model).

Figure 4: Steady state with "�ows"

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Figure 4 should not be confused with Figure 1. Both �gures show the steady state, butin a di¤erent way. The curve in Figure 1 is law of motion of capital per worker kt+1 (kt) andthe straight line is 450 which helps us see all the points where kt+1 = kt. The curved linesin Figure 4 on the other hand, are f (kt) and sf (kt) (production and investment), and thestraight line represents the depletion of capital due to depreciation and growth in the numberof workers, kt (n+ �). Figure 4 illustrates not just the steady state capital per worker, butoutput and consumption as well: (kss; yss; css). Thus, Figure 4 is more useful for analyzingthe steady state, while Figure 1�s main purpose is to demonstrate convergence to steadystate (see Figure 2).

3.1.1 Technological improvement

Here we illustrate the e¤ect of a once-and-for-all increase in productivity, so that f (kt)becomes greater for any level of kt. The production function f (kt) has therefore shifted up.As a result, the saving function shifts up as well. All the steady state variables went up:(kss "; yss "; css ").

Figure 5: Technological improvement

The intuition is simple. With higher technology, the return on investment in capital ishigher, so investment and capital go up. As a result, output per worker goes up, and sinceconsumption is a constant fraction of output, (1� s), it also goes up.

3.1.2 Higher growth rate of population

Higher population growth rate, n ", increase the slope of the "�ow out" line. Intuitively,higher growth rate of population results in higher depletion of capital per worker. In a

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sense, growth in the number of workers is similar to depreciation, since with more workerscapital per worker decreases. Thus, the net rate of return on investment in capital is smaller(because it "depreciates" faster), and steady state capital per worker falls.

Figure 6: Higher growth rate of population

As a result of the fall in steady state capital per worker, output per worker falls andconsumption per worker fall as well.

3.1.3 Higher saving rate

Now we increase the saving rate, i.e. s ". The steady state capital per worker goes up,because higher saving rate means that higher fraction of output is saved (=invested) andnot consumed. With higher steady state capital per worker, output per worker goes up aswell. Notice however that when saving rate goes up, the steady state consumption per workerdoes not necessarily go up. In the next �gure it actually goes down. To understand why thisis happening, look carefully at the de�nition of steady state consumption per worker:

css = (1� s)| {z }#

"z }| {f (kss (s))

Notice that steady state capital per worker has increased when saving rate went up, and sodid the output per worker. But at the same time, a smaller fraction of the new output isbeing consumed. Therefore, when s ", consumption per worker does not necessarily goe up.

Figure 7: Higher saving rate

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3.1.4 Summary of analytical results

(1) : Technology " ) kss "; yss "; css "(2) : n " ) kss #; yss #; css #(3) : s " ) kss "; yss "; css?

3.2 Golden Rule saving rate

In the last section we have seen that the steady state depends, among other things, on thesaving (or investment) rate4 s. Now we want to �nd out the optimal saving rate, i.e. thesaving rate that would maximize the steady state level of consumption per worker. Since wedo not have an explicit solution for kss as a function of s, we restrict our attention to �ndingthe optimal level of steady state capital per worker. Formally, we want to solve

maxkss

css = (1� s) f (kss) = f (kss)� sf (kss)s:t:

kss (n+ �) = sf (kss) (the steady state condition)

Substituting the constraint into the objective

maxkss

css = f (kss)� kss (n+ �)

4In closed economies saving rate equals to the investment rate.

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F.O.C.

f 0 (kGR)� (n+ �) = 0

f 0 (kGR) = n+ �

In words, the capital per worker has to be such that the marginal product of capital, i.e.marginal bene�t, is equal to the rate of capital depletion (n+ �), i.e. marginal cost. Intu-itively, suppose that kss > kGR, i.e. the economy saves too much. to see this, notice that ifkss > kGR we have f 0 (kss) < f 0 (kGR) = n + �, so the bene�t from the extra unit of capitalis smaller than its cost - the economy saves too much. Similarly, if kss < kGR, we havef 0 (kss) > (n+ �), so the bene�t from extra unit of capital exceeds its cost, and the economyneeds to save more in order to increase the level of the steady state capital per worker.The golden rule saving rate can be illustrated graphically. Observe that in Figure 7, the

steady state consumption per worker is equal to the vertical distance between the output perworker curve, f (kt), and the "�ow out line", kt (n+ �). In order to �nd the optimal savingrate, we need to maximize the vertical distance between the two lines. The vertical distancein maximized when the slopes of the two lines are the same. When the slopes are di¤erent,this means that the lines are getting closer together or farther away from each other. Thenext �gure illustrates the golden rule saving rate.

Figure 8: Golden rule

The dotted line is parallel to the "�ow out" line, kt (n+ �), so the slope of f (kGR) is thesame as the slope of the "�ow out" line, i.e. f 0 (kGR) = n+ �.

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4 Quantitative analysis

The theoretical model described in the previous section is not suited for quantitative inves-tigation. For example, we want to answer questions such as "if the saving rate goes up by1%, what is the percentage change in the steady state capital per worker, output per worker,and consumption per worker?". In order to answer this and other similar questions, we needto calibrate the model. Calibration consists of two steps:

1. Assigning functional forms to the functions used in the model.

2. Assigning numerical values to the parameters of the model.

What is the best way to calibrate an economic model is an ongoing debate among econo-mists. We will not attempt to resolve the debate, but rather discuss a few possible ways ofcalibrating the model, and mention some of their advantages in disadvantages.

4.1 Functional forms

In the Solow model the only function that we need to calibrate is the production function.Economists usually assume that the production function has the Cobb-Douglas form:

Y = AK�L1��, 0 � � � 1

First, make sure that you know how to check that Cobb-Douglas production function isneoclassical, i.e., it is CRS and satis�es the assumptions (a) - (c). In addition to all of theseproperties, the Cobb-Douglas has the property that the input factor shares are constant(independent of the prices of inputs). Suppose that the �rm is competitive, and solves thefollowing pro�t maximization problem.

maxK;L

� = F (K;L)� rK � wL

where r is the rental rate of capital and w is the real wage rate. The �rst order conditionsfor pro�t maximization are

FK (K;L) = r

FL (K;L) = w

So a competitive �rm pays each factor its marginal product. The total payments to K andL are

rK = FK (K;L)K

wL = FL (K;L)L

Factor shares are the fractions of total output that are paid to each factor. The factor sharesof capital and labor are therefore

Capital share:FK (K;L)K

F (K;L)

Labor share:FL (K;L)L

F (K;L)

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In the Cobb-Douglas case, the factor shares are

Capital share:�AK��1L1��K

AK�L1��= �

Labor share:(1� �)AK�L��L

AK�L1��= (1� �)

Thus, if the production function is Cobb-Douglas, the factor shares are constant (independentof factor prices) and equal to the exponents � and (1� �). In fact, the Cobb-Douglasproduction function is the only production function with this property.So what if the factor shares are constant? If the factor shares in the data are constant,

then the right functional form for the aggregate production function would be the Cobb-Douglas. What we need to check then, is whether the factor shares are indeed constantin the data. We will therefore look at National Income and Product Accounts (NIPA),measure the total income, and decompose it into payments to labor and payments to capital(everything else). We then plot the graphs of these empirical factor shares, and see if theyare indeed constant. The next �gure plots the labor share, 1� �, in the U.S. during 1929 -2006.

We see that indeed the labor share �uctuates mostly between 62% and 70%, with anaverage of 66%. But how do we come up with the numerical values for the factor shares?We provide a detailed answer to this question in the next section.

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4.2 Assigning numerical value to �

We have seen in the previous section that the theory predicts that a fraction � of totaloutput is paid to capital and a fraction 1� � is paid to labor. We now look at the NationalIncome and Product Accounts, and try to decompose the GDP into payments to labor andpayments to capital (everything else). Recall that the GDP can be computed using theincome approach as follows

NI = W + Int+Rent+�p +�B +NFC

GDP = NI +Dep�NFI + Sd

The next two table contain data on the ingredients of National income and its relation toGDP.

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For simplicity, we ignore the Net Factor Income and Statistical Discrepancy, and treatthe di¤erence between GDP and NI as Depreciation (consumption of �xed capital). Thus,we will work with

NI = W + Int+Rent+�p +�B +NFC

GDP = NI +Dep

where Dep = GDP �NINow we need to classify all the above magnitudes into payments to labor and everythingelse. Obviously, compensation of employees is a labor income. But it is a convention totreat proprietors income and nonfactor charges as part labor and part capital income. Inparticular, we assume that a fraction 1� � of �p and NFC is paid to labor, and the rest ispaid to capital. To �nd the labor share then, we have to solve the following equation

(1� �) =W + (1� �) (�p +NFC)

GDP(1� �)GDP = W + (1� �) (�p +NFC)

(1� �) =W

GDP � �p �NFCUsing data from the above tables, for the year 2006. gives: W = 7448:3, GDP = 13194:7,�p = 1006:7, NFC = 993:9. This gives

(1� �) = 7448:3

13194:7� 1006:7� 993:9 = 0:665

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4.3 Equilibrium in the calibrated model with growth

We repeat the description of the model from section 2, but this time the production functionis Cobb-Douglas, and we allow for growth in productivity.

1. Output is produced with production function Yt = AtK�t L

1��t , where Yt is aggregate

(real) output, At is the total factor productivity (TFP), Kt is the stock of physicalcapital, and Lt is labor services. We further assume that

At+1 = (1 + A)At

i.e. the TFP grows at constant, and exogenous, rate A. This assumption is equivalentto At = A0 (1 + A)

t.

2. Capital evolves according to Kt+1 = (1� �)Kt + It, where � is the depreciation rateand It is aggregate investment.

3. People save a fraction s of their income. This fraction is exogenous5. Thus, the totalsaving and total investment in this economy is

St = It = sYt

4. The population of workers grows at a constant rate of n, which is exogenous in thismodel. Thus, Lt+1 = (1 + n)Lt.

5. No government.

In the Cobb-Douglas case, output per worker is given by yt = Atk�t and the law of motionof capital per worker is

kt+1 =(1� �)(1 + n)

kt +s

(1 + n)Atk

�t

The graph of this law of motion, for �xed At, looks like the one in �gure 2. However, whenAt is growing, we cannot plot the law of motion since it will be constantly shifting up (see�gure 3). The good news is that we are able to analyze the dynamics of this model in aspecial case of a balanced growth path.

4.3.1 Balanced Growth Path (BGP)

A balanced growth path is a sequence of endogenous variables such that they all grow atconstant rate (not necessarily the same for all variables). It turns out that if the growth rateof productivity is constant, then the economy converges to the unique BGP.

5We call a variable endogenous if it is determined within the model and exogenous if it is determinedoutside the model. For example, in the model of a market (supply and demand diagram), the price andquantity traded of the good are endogenous variables, while other variables that determine the location ofthe supply and demand curve, such as income and prices of other goods, are assumed exogenous.

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Proposition 2 Along a balanced growth path, output per worker, capital per worker, con-sumption per worker, and saving (=investment) per worker, all grow at the same rate. Inother words, suppose that the economy is on a balanced growth path where yt+1 =

�1 + y

�yt,

kt+1 = (1 + k) kt, ct+1 = (1 + c) ct, st+1 = (1 + s) st. Then the proposition claims that

y = k = c = s

Proof. Divide both sides of the law of motion of capital per worker by kt

kt+1kt|{z}const

=(1� �)(1 + n)| {z }const

+s

(1 + n)

ytkt

This implies that the ratio yt=kt must also be constant, so y = k. Since consumption andsaving (or investment) per worker are proportional to output per worker,

ct = (1� s) ytst = syt

we must have c = s = y

Let denote the common growth rate by . We can �nd this common growth rate by usingthe fact that on a BGP, the ratio yt=kt must be constant. Therefore Atk��1t is constant, and

At+1k��1t+1

Atk��1t

= 1

(1 + A) (1 + )��1 = 1

1 + = (1 + A)1

1��

This means that when TFP grows at 1.2% per year, and capital share is 32%, the growthrate of all the endogenous per worker variables in the Solow model is

1 + = (1 + 0:012)1

1�0:32 = 1:0177

= 1:77%

In the above discussion we analyzed what happens on a BGP, if such BGP exists. Wehave not established yet whether such a path exists.

Proposition 3 Suppose that the TFP grows at a constant rate A. Then the endogenousvariables converge to the unique balanced growth path with the common growth rate =(1 + A)

11�� � 1.

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Proof. De�ne e¢ ciency variables (also called detrended variables) as the original endoge-nous variables divided by the cumulative growth factor:

k�t �kt

(1 + )t, y�t �

yt

(1 + )t, c�t �

ct

(1 + )t

Observe that at time t = 0, the original and detrended variables are the same. Also noticethat when the original variables are on the balanced growth path, the detrended variablesare at a steady state (constant). The idea is to rewrite the law of motion of capital perworker in terms of these detrended variables, and prove that this new law of motion has aunique stable steady state.

kt+1 =(1� �)(1 + n)

kt +s

(1 + n)Atk

�t

k�t+1 (1 + )t+1 =

(1� �)(1 + n)

k�t (1 + )t +

s

(1 + n)A0 (1 + A)

t �k�t (1 + )t��k�t+1 (1 + )

t+1 =(1� �)(1 + n)

k�t (1 + )t +

s

(1 + n)A0�(1 + )t

�1�� �k�t (1 + )

t��k�t+1 (1 + )

t+1 =(1� �)(1 + n)

k�t (1 + )t +

s

(1 + n)A0k

��t (1 + )

t

k�t+1 =(1� �)

(1 + n) (1 + )k�t +

s

(1 + n) (1 + )A0k

��t

The function k�t+1 (k�t ) satis�es the properties of the original model (with no growth) in

section 3, that guarantee a unique steady state as illustrated in �gure 2. In particular,

k�0t+1 (k�t ) =

(1� �)(1 + n) (1 + )

+s

(1 + n) (1 + )�A0k

���1t

Thus,

limk�t!0

k�0t+1 (k�t ) =

(1� �)(1 + n) (1 + )

+s

(1 + n) (1 + )limk�t!0

�A0k���1t| {z }

=1

=1

limk�t!1

k�0t+1 (k�t ) =

(1� �)(1 + n) (1 + )

+s

(1 + n) (1 + )limk�t!1

�A0k���1t| {z }

=0

=(1� �)

(1 + n) (1 + )< 1

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Now we can �nd the steady state in terms for the detrended variables, the way we didfor the original variables in the case of no productivity growth.

k�t+1 =(1� �)

(1 + n) (1 + )k�t +

s

(1 + n) (1 + )A0k

��t

k�ss =(1� �)

(1 + n) (1 + )k�ss +

s

(1 + n) (1 + )A0k

��ss

k�ss (1 + n) (1 + ) = (1� �) k�ss + sA0k��ssk�ss [(1 + n) (1 + )� (1� �)] = sA0k

��ss

[1 + + n+ n � 1 + �] = sA0k���1ss

k�ss =

�sA0

n+ � + + n

� 11�� �=

�sA0

n+ � +

� 11��

y�ss = A0k��ss

c�ss = (1� s)A0k��ss

The last approximation is valid for small values of and n. If the economy starts with initialcapital stock per worker of k0 = k�ss, then the detrended variables are at the steady state,while the original variables are on the balanced growth path.The steady state analysis that we performed in section 3.1 carries through to the steady

state in detrended variables. Everything else equal, if we compare two countries with di¤erentinitial productivity, then the country with higher A0 will have higher k�ss; y

�ss and c

�ss and

thus higher capital per worker, output per worker and consumption per worker on the BGP.Everything else equal, if we compare two countries with di¤erent population growth rates,then the country with higher n should have lower k�ss; y

�ss and c

�ss and thus lower capital per

worker, output per worker and consumption per worker on the BGP. Everything else equal,if we compare two countries with di¤erent investment rates, then the country with higher sshould have higher k�ss and y

�ss, but we can�t tell if c

�ss is higher or lower. Obviously, countries

with higher are growing faster by de�nition of (the BGP growth rate of endogenousvariables).

4.4 Optimal saving rate

In section 3.2 we derived the condition for Golden Rule saving rate, i.e. the saving ratewhich maximizes the steady state consumption per worker:

f 0 (kGR) = n+ � (3)

When a Cobb-Douglas production function is assumed, the output per worker is f (kt) =Atk

�t . Assuming that there is no growth in productivity, At = A 8t, the law of motion of

capital per worker is:

kt+1 =1� �1 + n

kt +sAk�t1 + n

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The steady state of capital per worker is found in the same way as in the last section, byplugging kt = kt+1 = kss into the law of motion of capital per worker:

kss =1� �1 + n

kss +sAk�ss1 + n

kss (1 + n) = (1� �) kss + sAk�sskss (n+ �) = sAk�ss

The left hand side is the "�ow out" of the stock of capital per worker, due to depreciationand growing number of workers. The right hand side is the "�ow in" to the stock of capitalper worker due to investment. The solution is:

kss =

�sA

n+ �

� 11��

Now we �nd the optimal capital stock in the steady state, kGR, which maximizes thesteady state consumption per worker:

maxkss

css = Ak�ss � sAk�ss = Ak�ss � kss (n+ �)

The last step substitutes the steady state condition: "�ow out" = "�ow in". Thus, theoptimal capital per worker must satisfy:

�Ak��1GR = n+ �

Where the left hand side is the marginal product of capital f 0 (kGR) as in (3). Solving forthe golden rule capital per worker, gives:

kGR =

��A

n+ �

� 11��

We see that in order to achieve this level of capital per worker in a steady state, the savingrate must be:

sGR = �

5 Conclusions

We started these notes with some motivating facts and questions. Does the Solow modelhelp us understand the facts and answering some of these questions? The Solow model is thestarting point for looking at the growth experience of a country and comparing cross-countryeconomic performance. The Solow model demonstrates that without growth in productivity,a country cannot experience sustained growth. In other words, growth that is based primarilyon factor accumulation cannot go on forever. So the Solow model tells us that countries thatgrow, do so because of growth in productivity, and the reason why some countries stagnateis the lack of growth of productivity. We can use the Cobb-Douglas production function todecompose a growth of a country into a part coming from factor accumulation v.s. the part

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coming from productivity growth. This is called growth accounting, and is carried out asfollows. Using the production function, we can relate the growth rates of output with thegrowth rates of TFP, capital and labor.

Yt+1Yt

=At+1K

�t+1L

1��t+1

AtK�t L

1��t

1 + Y =�1 + A

��1 + K

�� �1 + L

�1��where hat above the variable denotes the growth rate of the variable:

x � xt+1 � xtxt

Taking ln�s of both sides

ln�1 + Y

�= ln

�1 + A

�+ � ln

�1 + K

�+ (1� �) ln

�1 + L

�For small growth rates, the above is approximately

Y = A+ �K + (1� �) L

So suppose that the growth rate of real GDP is 3%, capital grows at 1% and labor grows at1.5%. Further assume that the capital share is 35%. Thus,

3% = A+ 0:35 � 1% + (1� 0:35) � 1:5%) A = 1:675%

This means that the growth in this country is fueled primarily by growth in productivity.The Solow model can also be used to compare two or more countries at a point in time.

Assuming no growth in productivity, the steady state output per worker is

yss = Ak�ss = A

�sA

n+ �

� �1��

= A1

1��

�s

n+ �

� �1��

We can use this equation to account for cross country di¤erences in GDP per worker, underthe assumption that both countries (i and j) are in the steady state:

yiyj=A

11��i

�sini+�

� �1��

A1

1��j

�sjnj+�

� �1��

This equation can be used to measure how much of the di¤erence in output per workerbetween countries i and j can be accounted for by di¤erences in TFP, investment rate,population growth rate. This cross-country accounting tells us which country speci�c char-acteristics are important for explaining the di¤erence between the two countries. Once thisaccounting is performed for many countries, economists learned that the most importantfactor in accounting for cross-country di¤erences is productivity. The Solow model does notprovide a theory of productivity, but it points out the importance of developing one.

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