ECON 805: Solow Growth Model

What Predictions Does This Model Deliver?

1. Predictions about growth rates.

2. Predictions about the cross-sectional distribution of income.

Key Concepts:

Steady-State: A steady-state is said to exist when the endogenous variables donot change. In our model, the key variable will be the capital stock (defined ineither per capita terms or in effective labor units). In the basic Solow model, theeconomy reaches a steady-state, the two forces that impact capital accumulation(investment and depreciation exactly offset). At this point, we will also showthat consumption and investment (both defined in either per capita terms or ineffective labor units) will also attain a steady-state.

Balanced Growth Path: A balanced growth occurs when all endogenousvariables in grow at a constant rate. Typically, the rate is determine by thegrowth rate of the of the exogenous variables.As a reminder, and endogenous variable is a variable that is determined “inside”the economics model and exogenous variable is a variable that is determinedoutside of the model. In the context of the basic Solow model, we assume thatpopulation growth (and the growth rate of technology) are exogenous. Theendogenous variables include the capital stock, consumption, investment.

Solow Model: Introduction

The Solow model is a simple dynamic, general equilibrium model which is usefulas a starting point to explain differences in income across countries and differ-ences in growth rates over time. We start with some simplifying assumptions–probably too simple. In particular, we assume the following:

• individuals save (or consume) a constant fraction of their income,

• the growth rate of the population is exogenous,

• (later) we assume that there is an underlying growth rate of technologythat also is exogenously given.

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All of these assumptions are extreme, in that, we think that rational agents makeoptimal choices when deciding how much to consume today and how much toconsume in the future; that is, savings and the savings rate is not literallyexogenous. We also think that agents are rational in the sense that they makechoices about the number of children they would like to have; that is, populationgrowth is not exogenous. Finally, we think that technological progress doesnot grow exogenously. Real resources—time and goods—must be allocated to’enhance’ the productivity of inputs used in the production process. Finally,there are other aspects of the real world that are not incorporated in this model.For example, the model does not include actions by the government (taxing andspending), the model also does not incorporate money or uncertainty. As thecourse progresses, we will relax all of these assumptions to construct a moregeneral model. For now these, assumptions allow us to get a first look at whysome countries are so rich and some countries are so poor.The basic Solow model presumes differences in income across countries are dueto the amount of physical capital available for production. Intuitively, all elseheld equal, we would expect countries that have more capital would also producemore output. To see if this indeed is the case, we plot the amount of (naturallog) of the per capita capita stock on the x-axis and natural log of output percapita on the y-axis. We see that there is a strong relationship between thetwo variables–the correlation coefficient is 0.97.

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Figure 1: Per Capita Capital Stock and Per Capita Real GDP

The question we seek to answer in these notes is to what extent can differencesin capital stocks and savings (investment) rates account for differences in in-come. Savings rates are important part of this explanation because a society’sdecision to save implies that it is foregoing current consumption in favor of fu-ture consumption. When and individuals save, the savings must be channeledsomewhere. As we show below, in a closed economy, what is not consumed (thatis what is saved) must be allocated to investment– and capital accumulation.Let us now be more clear what we mean by capital and investment. Capital (asstated by Weil) has five key characteristics. They are:

• Capital is productive,

• Capital is produced,

• Capital can earn a return,

• Capital’s use is limited, and

• Capital wears out.

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Let us take each of these in turn. Capital is productive, in that, the more capitalper worker employed in producing output the more output produced. For a givenamount of labor, the more capital there is the more goods and services can beproduced. Clearly, I could produce more output with a computer, some software,and a printer than I could if I just had a typewriter and a hand calculator. Wealso think that for a fixed amount of workers the additional benefits of addingadditional unit of capital declines as more capital is added; that is, we say thatcapital has a positive but diminishing marginal product. We define the marginal

product of capital as the additional amount of capital that can be produced byadding one more unit of capital–holding all other inputs constant. We will saymore about this below.We also say that capital is produced ; that is capital is built or created. Thefact that real resources must be allocated to produce capital implies that some-thing must be given up to produce more capital. If we think of a simple worldwhere the output produced has two uses–consumption and investment, any ad-ditions to the capital stock that occur through investment imply that currentconsumption must be sacrificed. Weil states that capital has to be produced soit is different than something like land (or a natural resource). While this istrue in some sense it is also not quite true literally. If we want to use land, thensome resources must be allocated to the land to make it usable.Capital earns a return. Because some current consumption must be sacrificed,there must be an incentive for individuals to forego current consumption. Thecompensation for the sacrificed consumption is the return to capital. The returnmay occur as dividends that are paid out to stockholders of a company. Theymay occur to entrepreneurs who buy capital and use it to produce output; inthis case the return to capital may show up as part of their wages; that isthere returns may not be as high had they not bought the capital. In this case,capital’s return shows up as part of wages.Capital depreciates. When capital is used in production it will lose some of itsvalue through normal wear and tear. For now, we will assume that the capitalstock depreciates at a constant rate. Later we will allow the depreciation ofcapital to depend on the its usage rate; that is, when capacity utilization is highthe capital stock will wear out faster.

Set-up of the Basic Solow Model

1. One good is produced: The good is consumed or saved (Denote Yt as theaggregate amount of output produced.)

2. Labor force grows at a constant rate n (n > �1) ;where Lt = L0(1 + n)

t,) Lt+1

Lt = 1 + n,) Lt+1�Lt

Lt = n

3. Aggregate household consumption is a constant fraction of output: Ct =

(1� s)Yt s 2 (0, 1),

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4. Investment: Capital accumulates in the standard fashion:

Kt+1 = It + Kt � �Kt � 2 (0, 1)

which can be rearranged to read as

Kt+1 = It + (1� �)Kt

which just simply implies that tomorrow’s capital stock (Kt+1) is equal tothe capital stock that does not depreciate from use today ((1� �) Kt) plusthe amount of current output that gets allocated to capital accumulation(It) . Note if the amount of output that gets allocated to capital accu-mulation exceeds the amount of capital that depreciates (It > �Kt) thenthe aggregate capital stock will grow (Kt+1 > Kt) . Given the standardproperties of the production function, if capital grows output must growas well.

5. Households accumulate assets according to:

At+1 = (1 + rt)At + wtNt � Ct

orAt+1 �At = rtAt + wtNt � Ct

where we define At to be assets held at time t.

6. Assets come in the form of either bond holdings (Bt) or in the form ofcapital (Kt) . Individuals issue and hold bonds. In this set-up, there areno government issued bonds. Firms do not issue bonds directly. Sincethere is no financial intermediaries household will issue there own bondsto borrow and hold bonds in order to save. Therefore, the only holdersof bonds are households. In the aggregate net bond holding will be zero.Because for every borrower there is a saver; so net bond holdings must bezero. Keep in mind, in equilibrium, net bond holding will be zero and theonly asset held is the capital stock. We will show this must be true moreformally below.

7. The production function is given by:

Yt = ZtF (K

dt , L

dt )

For now, we assume that Zt = 1. We assume that the production functionexhibits constant returns to scale in capital and labor; that is, if we writethe production function as and if we double the inputs of capital andlabor output would exactly double. Generally speaking, if we increase theamount of capital and labor by any factor, say �, then output will increaseby �. That is �Yt = F (�K

dt ,

��L

dt

�) . If we let � = 1/L

dt we get

Yt

Lt= F (

Kt

Lt, 1)

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we can define yt =

YtNt

as output per-worker and kt =

KtNt

as capital perworker then we can write the production function as

yt = F (kt, 1)

In this case, since the second argument in the production function doesnot vary, we can write the production function as

yt = f(kt)

This is the form we will use below. Note again, this simply states thatoutput per effective labor unit is a function of capital per effective laborunits.An example that we will use frequently in this course is the Cobb-Douglasproduction function. We can write this as

Yt = K

↵t (Lt)

1�↵

which can be written in output per capita as a function of the per capitacapital stock

yt = k

at

For the Cobb-Douglas production function we define the marginal product

of capital as@Yt

@Kt= ↵

⇣(Kt)

↵�1(Lt)

1�↵⌘

= ↵k

1�↵t

If we were to graph the marginal product of capital it would look like thediagram drawn belowNote that this production function exhibits positive but diminishing marginalproduct. Another property that we will use below is that in a world wherethere is perfect competition and inputs are paid their marginal product wecan define capital’s share of output as the fraction of output (or income)that is paid to capital; that is, capital’s share = (capital income)/output.

If capital gets paid its marginal product then the payment to capital isMPK⇤K = ↵

⇣(Kt)

↵�1(Lt)

1�↵⌘⇤Kt = ↵Yt., so capital’s share of output

is given by

capital

0s share =

↵

⇣(Kt)

↵�1(XtNt)

1�↵⌘

Kt

Yt=

↵Yt

Yt= ↵

A similar calculation would show that labor’s share of output is equal to(1� ↵). The basic properties of the production function are:

(a) FL > 0, FK > 0, (Positive marginal product of capital and labor)(b) FLL > 0, FKK > 0, (Diminishing marginal products of capital and

labor)

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(c) INADA conditions: F (K, 0) = F (0, L) = 0, and limK!0 FK =

1 and limL!0FL =1Technical Note: A balanced growth path implies that the economy, ata certain point in time, will reach a point where all the endogenousvariables grow at a constant rate. In this model, the endogenousvariables (capital, output, and consumption) all grow at a constantrate once we are on the balanced growth path. At what rate will thevariables grow? The aggregate variables will grow at the rate samerate as the exogenous variables. In this model, population grows atconstant rate so the aggregate endogenous variables will grow at therate of population growth; that is, the gross growth rate of output,capital and consumption will grow at a rate of (1 + n) per unit oftime. A special growth path is what is termed a steady state. Whenan economy reaches a steady state, the endogenous variables do notgrow at all. In the basic Solow model, per capita output, capital,consumption and investment will converge to a steady state. We willuse the concept of a steady state throughout the class. There areobvious reasons why we would want to do so. One reason is that itis easy to depict steady-states in two dimensions. The second reasonis that steady states look a lot like standard economic equilibriumanalysis; that is, the steady state occurs where two curves cross. Thisshould become clearer below,

8. The Economy’s resource constraint is satisfied: Yt = Ct + It or Yt =

Ct + Kt+1 � (1� �)Kt.

Goal:

1. To use these tools to see to what extent differences in the capital stock(savings rate) can explain differences in income.

2. To understand why some countries grow faster than others.

Household and Firm Behavior

Households

We define all of our endogenous variables in per capita terms; that is, we netout the growth rate of the variable that grows exogenously (population). Bynormalizing all variables by the size of the population at time t (Lt), we removethe exogenous growth components from all the variables.We already know that per capita income in effective labor units is given by:yt = f(kt)

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To get per capita consumption and the per capita capital stock, we simply dividethe aggregate variable by the population; that is

ct =

Ct

Ltand kt =

Kt

Lt

Recall the asset accumulation equation for the household is given by

At+1 �At = rtAt + WtLt � Ct

Dividing through by the population at time t, yields

At+1

Lt� At

Lt= rt(

At

Lt) + Wt �

Ct

Lt

which equalsAt+1

Lt� at = rtat + Wt � ct

Note that we did not write At+1/Lt = at; the reason for this is because the timesubscripts on assets at time t and the population at time t do not align. Thereis a simple fix for this problem: We rewrite

At+1

Lt=

At+1

Lt+1

✓Lt+1

Lt

◆= at+1(1 + n)

so that the asset accumulation equation is given by

at+1(1 + n)� at = rtat + Wt � ct

orat+1(1 + n) = (1 + rt)at + Wt � ct

or by subtracting (1 + n)at from both sides of the above equation we have

(at+1 � at)(1 + n) = (rt � n)at + Wt � ct

It may be easier to think of this budget equation in terms of the sources of funds

equal uses of funds. The sources of funds are the incomes received Wt + rtat.The uses of funds are what the individuals do with their income. They use theirincome to accumulate assets and to buy the consumption goods. We saw abovethat then change in assets is given by (at+1 � at) + at+1n and consumption isgiven by ct. Thus, the economy’ uses of funds are (at+1 � at) + at+1n + ct.

Therefore, the budget constant is given by

Wt + (rt � n)at = (at+1 � at)(1 + n) + ct

In this model, household’s are not choosing consumption and saving optimallysince, by assumption, the household consumes a constant fraction of income;that is ct = (1� s) [Wt + rtat]

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More on assets: we stated above that households can hold two different typesof assets: bonds or capital. Total asset holdings are given by

At = Bt + Kt

The returns to each of these assets will determine the how much of their wealththey hold in bonds and capita. The return on bond holding is denoted by r

bt ,

and if an individual purchases a bond at time t (Bt), the return is (1 + r

bt )Bt

(1 + r

bt )Bt �Bt =

✓(1 + r

bt )� 1

◆Bt

Alternatively, can obtain capital and rent it to the firm. If they do so they get agross return of RtKt; however, when the firm uses the capital stock, the capitalstock depreciates by �Kt. The net return from holding wealth in capital andrenting it to the firm

(1 + ⇢

kt � �)K �Kt =

✓(1 + ⇢

kt � �)� 1

◆Kt

Note then the asset equation is given by

At+1 = Bt+1 + Kt+1 = (1 + r

bt )Bt + (1 + ⇢

kt � �)Kt + WtLt � Ct

or

Bt+1 + Kt+1 = (1 + r

bt )(Bt + Kt)+

(⇢

kt � �)� r

bt

�Kt + WtLt � Ct

orAt+1 = (1 + r

bt )At+

(⇢

kt � �)� r

bt

�Kt + WtLt � Ct

The key term is the term in brackets. Note

• if(⇢

kt ��)�r

bt

�> 0 then the return on capital is holding capital is greater

than the return on holding bonds. In this case, everyone would want toborrow. However, not everyone can borrow! This must drive the returnon bonds up!

• if(⇢

kt ��)�r

bt

�< 0 the return on holding bonds is greater than the return

to capital. In this case, everyone would want to hold bonds. However,this cannot be an equilibrium since we will show below R = FK andlimK!0FK =1.

This implies that (⇢

kt � �) = r

bt . We simply state this as (⇢

kt � �) = r

bt = rt or

⇢

kt = rt + �

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Firms

Firms hire workers and rent capital to produce output. We think of firms asrenting capital from households to produce output. None of the results wouldchange if we assumed that the firms owned the capital stock and the householdsowned the firms. We define Rt as the rental price of capital and so that paymentsto the households will be proportional to the capital stock. If we define � as thedepreciation rate per unit of capital the net return of a renting a unit of capitalto the firm is R � �. In the absence of uncertainty, the return from holding aone-period bond and the return from renting a unit of capital to the firm mustbe the same; that is R� � = r or rt + � = R. So we can write the firms problemas

⇧ = max

K,LF (K

d, L

d)� ⇢ ⇤K

d �WL

d

I have dropped the time subscripts here because the problem is a static problem;that is, the firm decides on how many workers and how much capital to hireeach period. How much labor and capital to employ next period does not enterinto the firm’s decision set. The maximization problem can be written in percapita terms as:

⇧ = max

K,LL

d⇥f

�k

d�� ⇢k

d⇤�WL

d

and the first-order conditions are

@⇧

@k

= L

d⇥f

0 �k

d�� ⇢

⇤= 0

@⇧

@L

=

⇥f(k

d)� ⇢k

d⇤�W�

⇢k

d⇥f

0 �k

d�� ⇢

⇤�= 0

The term in curly brackets is zero because of the first-order condition withrespect to k. These first-order conditions imply

f

0 �k

dt

�= ⇢t

Wt = f(k

dt )� f

0(k

dt )k

dt

Equilibrium

Now we can list the conditions and the corresponding equations that we use tosolve the model:

1. Households:

(a) Households budget constraint is satisfied (at+1 � at)(1 + n) = (rt �n)at + Wt � ct

(b) Agents spend a constant fraction of income on consumption; that isct = (1� s)yt = (1� s)f(kt)

2. Firms: Firms maximize profits: f

0(kt) = rt +� and Wt = f(kt)�f

0(kt)kt

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3. No arbitrage condition: In equilibrium, the only asset held by thehousehold is the capital stock at = kt (bond holding in the aggregatemust be zero bt = 0), and the return on bonds is equal to the net returnon capital.

4. Market Clearing: Markets are in equilibrium (We will often state thisas markets clear.)

(a) Factor markets clear: Lt = L

dt and Kt = K

dt

(b) Goods market clearing: yt = ct + it,

We should be a little more careful about the goods market clearing condition.Recall from before that we could write market clearing as

Yt = Ct + Kt+1 � (1� �)Kt

Dividing this equation through by Lt yields:

yt = ct +

Kt+1

Lt� (1� �)kt

As with the asset accumulation equation, the time period and so we do as beforewe rewrite the equation as

yt = ct +

Kt+1

Lt+1

✓Lt+1

Lt

◆� (1� �)kt

yt = ct + kt+1(1 + n)� (1� �)kt

which can written as

f(kt) = ct + (1 + n)kt+1 � (1 + n)kt � [(1� �)� (1 + n)]kt

f(kt) = ct + (1 + n)(kt+1 � kt)� [(1� �)� (1 + n)]kt

or(kt+1 � kt) =

f(kt)� ct � (n + �)kt

1 + n

Now if we substitute for ct = (1� s)f (kt), we obtain

(kt+1 � kt) =

sf(kt)� (n + �)kt

1 + n

The resource constraint informs that the path of the capital stock is given by

(kt+1 � kt) =

sf(kt)� (n + �)kt

1 + n

(1)

What happens if we start from the (aggregated) households’ budget constraint(written in per capita terms)? We know that

at+1(1 + n) = (1 + rt)at + Wt � ct

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In equilibrium, kt = at and bt = 0, so that

kt+1(1 + n) = (1 + rt)kt + Wt � ct

Next recall that in equilibrium rt = ⇢t � � so that

kt+1(1 + n) = (1 + ⇢t � �)kt + Wt � ct

Also in equilibrium, markets clear Lt = L

dt and k

dt = kt. Also recall that profit

maximization impliesWt = f(kt)� f

0(kt)kt

and⇢t = f

0(k)

substituting this into the household’s budget constraint yields

kt+1(1 + n) = (f

0(k)t + 1� �)kt + f(kt)� f

0(kt)kt � ct

and after some rearranging

kt+1(1 + n) = f(kt) + (1� �)kt � ct

Now subtract (1 + n)kt from both sides to obtain

(1 + n)(kt+1 � kt) = f(kt)+

✓(1� �)� (1 + n)

◆kt � ct

(1 + n)(kt+1 � kt) = f(kt)� ct � (n + �)kt

or(kt+1 � kt) =

sf(kt)� (n + �)kt

1 + n

(2)

Note this is the same as equation (1) . It should not be much of a surprise(recall Walras’s law): If N-1 markets are in equilibrium, the Nth market will beas well. We just showed that if the factor markets are in equilibrium (and giventhe factor payments), the households budget constraint must equal the resourceconstraint.

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Characterization of Steady-State

Figure 2: Per Capita Output and Saving

The above figure depicts the graph of the production and the saving function.Where do we end up?Ask ourselves is it possible that kt+1 � kt = 0? If so, it means that the capitalstock stops growing and output per worker stops growing. Define i

⇤= (n + �) k .

This means the level of investment is just sufficient to keep up with the growthrate of the population, and replace the capital that gets worn out in usage.Stated differently, i

⇤ means we are investing enough to replace depreciated cap-ital, and give every new entrant into the labor force kt units of capital. We cancombine this graph with the savings to determine how the capital stock evolves.Putting the i* line together with the savings function we get

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Figure 3: Saving and i* Line

Alternatively, we can depict the relationship between savings and investment asdepicted below

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Graph what happens to output, the capital stock, consumption and investmentwhen1. the savings rate increases.2. population growth increases.3. exogenous technological growth increases.

The model predicts that countries with higher saving rates will tend to havehigher levels of GDP per capita. A simple look at the data show that the datais consistent with this prediction.

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Figure 4: Investment Share and Per Capita GDP

The model predicts that countries with higher population growth rates will tendto have lower levels of GDP per capita. A simple look at the data show thatthe data is consistent with this prediction.

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Figure 5: Population Growth and Real Per Capita GDP

Comparative Steady-State Statics:

Steady state income is characterized by the following conditions:

yss = f(kss)

sf(kss) = (n + �)kss

We are interested in how the steady-state level of income will change if one ofthe exogenous variables change.

Long-Run Implications: Change in the Savings Rate:

We totally differentiate the system of equations with respect to the two en-dogenous variables {yss, kss} as the endogenous variable s changes. To this

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end,

dyss = f

0(kss)dkss

{sf 0(kss)� (n + �)}dkss = �f(kss)ds

Then rearranging the second condition yields:

dk

ds

=

�f(kss)

sf

0(k)� (n + �)

> 0

and substituting into the first-condition

dyss

ds

s

yss=

�sf

0(kss)

sf

0(k)� (n + �)

Next noting that sf(kss) = (n+�)kss and rewriting this expression as (n + �) =

sf(kss)/kss and substituting into the above expression for (n + �) yields

dyss

ds

s

yss=

�sf

0(kss)

sf

0(k)� sf(kss)/kss

Finally canceling out the s term and multiplying through by kss/f(kss) yields

dyss

ds

s

yss=

�f

0(kss)kss

f 0(k)kss

f(kss) � 1

Denoting capital’s share of income as ↵(kss) = f

0(kss)kss/f(kss) then implies

dyss

ds

s

yss=

↵(kss)

1� ↵(kss)

Cross-country estimates of ↵(kss) place it in the range of .33 to 0.40. Assuming↵(kss) = 1/3, then the elasticity of output with respect to a change in thesavings rate is is

dyss

yss

s

ds

= 1/2.

So for example, the United States income in 2000 was $31843.15 and theirsavings rate was 19% Sierra Leone had a savings rate of 9.6 percent and theirincome level was 421.95. The question we ask is to what extent these differencesin income are explained by the differences in the savings rate. Stated differently,suppose Sierra Leone increased their savings rate to be equal to that of theUnited States savings rate what by how much would their income increase.ds/s = (.19 � .096)/.096 = 0.97917 ==>

dyss

yss=

12

dss ⇡ 0.50. Sierra Leone’s

income per person would increase to $641.72. Thus, the Solow model gets thingsright qualitatively it seems to miss quantitatively.You should repeat these exercises for a change in n; that is compute dy

yndn .

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An Example with a Specific Functional Form of the Pro-duction Function:

Suppose production was given by y = k

↵. In this case, the steady-state level of

output can be solved as following steady-state savings is equal (syss) is equalto steady-state investment ((n + �)kss) or

syss = (n + �)kss ) kss =

syss

(n + �)

and substitute this expression into the production function to get

yss =

✓syss

(n + �)

◆↵

and solving for yss yields

yss =

✓s

(n + �)

◆ ↵1�↵

Investment is given by

savingsss = s

✓s

(n + �)

◆ ↵1�↵

You should verify that steady-state savings is equal to steady-state investment;that is

s

✓s

(n + �)

◆ ↵1�↵

= (n + �)

✓s

(n + �)

◆ 11�↵

The steady-state level of capital is given by

kt+1 � kt = 0 ) sf(kss)� (n + �)kss

which impliessk

↵= (n + �)kss

kss =

✓s

n + �

◆ 11�↵

With this functional form, we can infer information about factor prices (W andr) in the steady state. Recall that

W = f(k)� f

0(k)k = (1� ↵)k

↵

In the steady-state, we have

Wss = (1� ↵)

✓s

n + �

◆ ↵1�↵

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The return to capital (⇢) is given by

⇢ss = ↵k

↵�1ss

and substituting for kss =

✓s

n+�

◆ 11�↵

yields

⇢ss =

↵(n + �)

s

so thatrss = ⇢ss � � =

↵(n + �)

s

� �

Cross Country Income Comparisons

Assuming similar world technology growth and similar depreciation rates, wecan then compare differences in steady-state output between two countries saythe US and country j as follows:

y

USss =

✓s

US

(n

US+ �)

◆ ↵1�↵

(3)

y

jss =

✓s

j

(n

US+ �)

◆ ↵1�↵

(4)

y

jss

y

USss

=

0

@ s

j/s

US

(nJ+�)(nUS+�)

1

A

↵1�↵

(5)

Another nice property of the Cobb-Douglas production function is that thesteady-state level of output is linear in logs. Recall

yss =

✓s

n + �

◆ ↵1�↵

and taking the logs we have

ln(yss) =

↵

1� ↵

ln(s)� ↵

1� ↵

ln(n + �)

Take-Aways

We constructed a simple model of that can account for differences in (steady-state) income per capita that shows how saving, investment, and populationgrowth help determine the (steady-state) level of income per capita. We canuse this simple model to highlight the importance of savings and investment isfor long run growth. We showed that

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• the (steady-state) level of output increases as the saving rate increases

• the (steady-state) level of output is decreasing in the rate of populationgrowth

• the model with a Cobb-Douglas production function allows for simpleanalytic solutions that is consistent with

– labor’s share of output being (roughly) constant over time– capital’s share of output being (roughly) constant over time– the stylized fact that countries that save more (invest more) tend to

have higher levels of output per capita

• the Cobb-Douglas production function also delivers a simple expressionthat is linear in logs.

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