SolowLectureNotes
1Math preparation
In this chapter we deal with the movement of the macroeconomic variables,such as GDP,over time.There are two ways of dealing with time in macroeconomics:discrete time and continuous time.Discrete-time models treat the time as integers:t=0,1,2,....Continuous-time models treat the time as real numbers.Here we will consider a continuous-time model.
In a continuous-time model,we view a particular macroeconomic variable as a function of time.For example,when we consider GDP,instead of calling it as one number Y,we consider it as a function of time t,Y(t).In this way,we can capture the movement of GDP over time.For example,sometimes we want to see how much GDP changes per unit of time.Mathematically, the change of GDP can be viewed as the slope of Y(t)with respect to t.And we know that the slope can be expressed mathematically as a derivative.Therefore,
the change of GDP per unit of time,measured at time t=dY(t) dt
.
Since we use this expression a lot,we use the short-cut expression by de?ning˙Y(t)≡dY(t)/dt. In usual mathematics,the use of′is more common as a shortcut for a derivative(i.e.Y′(t)≡dY(t)/dt),but a convention in growth economics is to use the˙instead.For example,if the GDP is a linear function of t,that is,Y(t)=at(where a is a constant number),then the change of GDP per unit of time is equal to˙Y(t)=dY(t)/dt=a.If GDP is an exponential function of t,that is,Y(t)=exp(bt)(where b is a constant),then the change of GDP per unit of time is equal to˙Y(t)=dY(t)/dt=b exp(bt).
Note that there is a economic meaning of˙Y(t).When˙Y(t)=0,it means that the GDP is not changing over time.When˙Y(t)>0,it means that the GDP is increasing over time.When ˙Y(t)<0,it means that the GDP is decreasing over time.
Since we analyze economic growth in this chapter,we will have many occasions where we look at the growth rate.The growth rate of a variable is de?ned as the change of the variable divided by the level of the variable.For example,the growth rate of Y(t),g Y,is de?ned by
g Y≡change of Y(t)
level of Y(t)
=
˙Y(t)
Y(t)
.
For example,when Y(t)=at,g Y=˙Y(t)/Y(t)=a/(at)=1/t.When Y(t)=exp(bt), g Y=˙Y(t)/Y(t)=b exp(bt)/exp(bt)=b.Therefore,when Y(t)is an exponential function of t,the growth rate of Y(t)is constant.
There is another,very convenient,way of looking at the growth rate.Before explaining it, let’s refresh our memory of calculus a little bit.
?Chain rule:
Suppose that a variable y is a function of a variable x:y=g(x).Also suppose that a variable z is a function of y:z=f(y).Therefore,z is eventually a function of x: z=f(g(x)).The right hand side,f(g(x)),is called a composite function.There is a rule called chain rule in calculus that allows us to di?erentiate z with respect to x,that is,to take a derivative of a composite function.The formula is:
d dx f(g(x))=
d f(y)
dy
·dg(x)
dx
,where y=g(x).
?Properties of the natural log:
In this chapter,a natural log of a variable x is written as log(x).The formula for a derivative of a natural log function is:
d dx log(x)=
1
x
.
A natural log has the following properties:
log(ab)=log(a)+log(b),
log (a
b
)
=log(a)?log(b),
and
log(xα)=αlog(x).
Equipped with this knowledge,now let’s take a natural log of a variable,for example,Y(t), and di?erentiate it.
d dt log(Y(t))=
d log(y)
dy
·dY(t)dt,where y=Y(t) =
1
y
·˙Y(t),where y=Y(t)
=
1
Y(t)
·˙Y(t)
=
˙Y(t)
Y(t)
=g Y.
That is,when we take a natural log of a variable and take a derivative(i.e.calculate the slope of log(Y(t))),we obtain the growth rate of the variable.We will use this technique again and again.This property is also useful when we want to visualize the growth of a variable—if we
plot log(Y(t))against t,the slope is the growth rate of Y(t).If the graph looks like a straight line,it means that Y(t)is growing at a constant rate.
We can also apply the properties of the natural log and derive the following.(Let us denote the growth rate of a variable X(t)as g X.)
?When Z(t)=X(t)Y(t),
g Z=d
dt
log(Z(t))
=d
dt
log(X(t)Y(t))
=d
dt
(log(X(t))+log(Y(t)))
=d
dt
log(X(t))+
d
dt
log(Y(t))
=g X+g Y.?When Z(t)=X(t)/Y(t),
g Z=d
dt
log(Z(t))
=d
dt
log(X(t)/Y(t))
=d
dt
(log(X(t))?log(Y(t)))
=d
dt
log(X(t))?
d
dt
log(Y(t))
=g X?g Y.?When Z(t)=[X(t)]α,
g Z=d
dt
log(Z(t))
=d
dt
log([X(t)]α)
=d
dt
(αlog(X(t)))
=αd
dt
log(X(t))
=αg X.
Now let’s look at the examples that we have seen before.When Y(t)=at,log(Y(t))= log(at)=log(a)+log(t).Therefore,g Y=d log(Y(t))/dt=0+d log(t)/dt=1/t,which is
the same as the one we obtained before.When Y(t)=exp(bt),log(Y(t))=log(exp(bt))=bt. Therefore,g Y=d log(Y(t))/dt=d(bt)/dt=b,which is also the same as what we derived before.
2Solow model
In this section,we will go over the Solow model,developed by Robert Solow.The Solow model is sometimes called the Solow-Swan model or the Neoclassical Growth Model.It should be noted,however,when people say“the Neoclassical Growth Model,”it can mean a di?erent (but related)model called the Ramsey model.The Solow model occupies an important place in the modern macroeconomics.It is not only an important framework for thinking about economic growth,but also a foundation of almost all the macroeconomic models that are used in the research frontier during the past30years.
2.1Setting up the model
The Solow model has two main ingredients:(i)the(Neoclassical)production function and(ii) the equation that governs the evolution of capital stock.We will explain them in turn.
The production function,sometimes called the Neoclassical production function or the ag-gregate production function,describes how the output is produced from the capital stock and the labor input.In a general form,it can be written as
Y(t)=F(K(t),A(t)L(t)).(1)
The function F(·,·)is the production function.It has two inputs:K(t)and A(t)L(t).K(t)is the capital stock at time t,and A(t)L(t)is the labor input at time t.The labor input at time t consists of two parts:A(t)and L(t).L(t)is the number of workers at time t.For simplicity, we will represent L(t)by the population.In reality,of course,there are some people who are not working,but this is a good approximation when we think about the long-run growth.A(t) is the e?ciency of labor.When A(t)is higher,each worker is more productive.Note that here we are making an implicit assumption that“hiring two workers”and“hiring one worker who is twice more productive”results in the same labor input.The labor input A(t)L(t)is often called the labor input measured by the e?ciency unit of labor.It is also called the amount of e?ective worker.In general,A(t)is interpreted as the technology level of the economy.
Below,we will use the Cobb-Douglas form for the production function:
Y(t)=[K(t)]α[A(t)L(t)]1?α,(2)
where0<α<1.When the production function is Cobb-Douglas form,the production function can also be rewritten as
Y(t)=z(t)[K(t)]α[L(t)]1?α(3)
by de?ning z(t)≡[A(t)]1?α.Note that this transformation may not be possible for a more general production function(1).
The second ingredient is an equation that describes how the capital stock evolves over time. It embodies two ideas:(i)the capital stock can be increased by investment and(ii)without investment,capital stock depreciates(loses a part of its value)over time.The equation looks like the following:
˙K(t)=I(t)?δK(t),(4)
whereδ>0.The left hand side is“the change in capital stock.”The?rst term in the right hand side,I(t),is the investment at time t.The second term represents the depreciation:the valueδK(t)of capital stock is lost by wear and tear per each unit of time.δis called the depreciation rate.
To proceed,we make two additional assumptions.The?rst is that people save a constant fraction of their income.In equation,this is expressed as
S(t)=sY(t),(5)
where0 Recall that from the expenditure approach to GDP,Y(t)=C(t)+I(t)(there is no govern-ment spending,export,or import in this economy)holds.From the income approach,Y(t)is also a national income,which is either consumed or saved.Therefore,Y(t)=C(t)+S(t).From these equations,we can see that I(t)=S(t)has to hold.From(5),this implies I(t)=sY(t). Using this,(4)can be rewritten as ˙K(t)=sY(t)?δK(t). From the production function(2),this can be rewritten as ˙K(t)=s[K(t)]α[A(t)L(t)]1?α?δK(t).(6) The second additional assumption is that the population(that is,L(t))and technology (that is,A(t))both grow at a constant rate.In particular,we assume that ˙L(t) L(t) =n(7) and ˙A(t) A(t) =x.(8) 2.2Analyzing the model At this point,it seems that we are dealing with many variables,such as K(t),L(t),A(t), and Y(t),all of which change over time.It seems like a complicated task to keep track of all these time-varying variables.It turns out,however,we have to analyze only one variable to characterize this model.In particular,we will keep track of a variable k(t)that is de?ned by k(t)= K(t) A(t)L(t) . k(t)can be interpreted as the amount of capital stock per e?ective worker.To see how k(t) moves over time,let’s take a natural log of both sides of this equation: log(k(t))=log ( K(t) A(t)L(t) ) =log(K(t))?log(A(t)L(t)) =log(K(t))?log(A(t))?log(L(t)). Taking derivatives of both sides with respect to t: d dt log(k(t))= d dt log(K(t))? d dt log(A(t))? d dt log(L(t)). Therefore, ˙k(t) k(t)= ˙K(t) K(t) ? ˙A(t) A(t) ? ˙L(t) L(t) . Now,let’s replace˙K(t),˙A(t)/A(t),and˙L(t)/L(t)here using(6),(7),and(8): ˙k(t) k(t)= s[K(t)]α[A(t)L(t)]1?α?δK(t) K(t) ?x?n =s[K(t)]α?1[A(t)L(t)]1?α?δ?x?n =s [ K(t) A(t)L(t) ]α?1 ?δ?x?n =s[k(t)]α?1?δ?x?n. Multiplying k(t)on both sides,we obtain ˙k(t)=s[k(t)]α?(δ+x+n)k(t).(9) The equation(9)describes the change of k(t)as a function of only the level of k(t).In this sense,to analyze the movement of k(t)using(9),we only need to keep track of k(t). The equation(9)can easily be analyzed using a diagram.In Figure1,we draw the?rst term of the right hand side(that is,s[k(t)]α)and the second term of the right hand side(that is,(δ+x+n)k(t))as functions of k(t).The equation(9)tells us that the vertical distance between these two is the change of k(t)at time t,˙k(t). Notice that there is a value of k(t),depicted as k?in Figure1,where s[k(t)]αis equal to (δ+x+n)k(t).At this point,˙k(t)=0holds,which means that once k(t)reaches there,k(t) does not move—it will stay at k?forever.This value k?is called the steady state value of k(t). We can also see that on the left side of k?,s[k(t)]α>(δ+x+n)k(t)holds,which means that ˙k(t)>0.That is,when k(t)is smaller than k?,k(t)increases over time.On the opposite side, if k(t)is larger than k?,s[k(t)]α<(δ+x+n)k(t)holds,so that˙k(t)<0and k(t)decreases over time. As a result,when the initial value of k(t),k(0)(value of k(t)at time0),is smaller than k?, k(t)keeps decreasing over time,and gradually approaches to k?.When k(0)is larger than k?, k(t)keeps increasing over time and gradually approaches to k?.See Figures2and3. Therefore,k?is the value of k(t)that any economy approaches to in the long run.It turns out that we can solve for k?analytically.Since s[k?]α=(δ+x+n)k? holds,dividing both sides by k?yields s[k?]α?1=(δ+x+n). Dividing both sides by s,we obtain [k?]α?1=δ+x+n s . Therefore, k?=( δ+x+n s )1 α?1 or k?=( s δ+x+n )1 1?α .(10) We can see from here that k?is increasing in s,and decreasing inδ,x,and n. 2.3Long-run behavior of the economy Now we will map the behavior of k(t)into the behavior of the macroeconomic variables that we are interested in.Note that we already know how L(t)and A(t)behave over time—they are given,by assumption,by(7)and(8).In other words,their behavior is given exogenously from outside.One variable that we are always interested in macroeconomics is GDP,Y(t). The production function(2)indicates that Y(t)is a?ected by three things:K(t),A(t),and L(t).Since A(t)and L(t)are given exogenously,the only variable that is determined inside the model is K(t).In other words,K(t)is an endogenous variable.So it is the key to understand how K(t)moves over time. Let’s proceed.First,let’s analyze the long-run behavior of the economy.In the long run, we know that k(t)will be constant at k?.From the de?nition of k(t),it means that k?= K(t) A(t)L(t) . Therefore, K(t)=k?A(t)L(t) and ˙K(t) K(t)=0+ ˙A(t) A(t) + ˙L(t) L(t) =x+n, and K(t)grows at the rate of x+n. From(2),the growth rate of the GDP is ˙Y(t) Y(t)=α ˙K(t) K(t) +(1?α) ( ˙A(t) A(t) + ˙L(t) L(t) ) =α(x+n)+(1?α)(x+n)=x+n. This is a striking result:the long-run growth rate of the GDP is determined only by x and n, that is,the rate of technological change and the population growth rate.But how about the saving rate?Isn’t it the case that if we save more,the economy grows faster?Here we have shown that the answer is“no”in the long run.In the next section,we will see that the growth rate of the GDP increases with the saving rate in the short run.It also turns out that even in the long run,the saving rate a?ects the level of GDP,not the growth rate.This can be seen from the production function: Y(t)=[K(t)]α[A(t)L(t)]1?α=[ K(t) A(t)L(t) ]α A(t)L(t)=[k?]αA(t)L(t) in the long run.Since A(t)and L(t)are given exogenously,the level of Y(t)is determined by the level of k?.Sinceα>0,Y(t)is increasing in k?.From(10),k?is increasing in the saving rate s.Therefore,increasing the saving rate increases the level of GDP in the long run,but not the growth rate.Forcing people to save,for example,cannot sustain the economic growth in the long run. Another,perhaps more important,macroeconomic variable is per capita GDP,?y(t)= Y(t)/L(t).From the above equation,it is easy to see that ?y(t)=[k?]αA(t) and since k?is constant,it grows at the same rate as A(t).Therefore the growth rate of per capita GDP is x.So,the important conclusion here is that if we want to keep increasing the per capita GDP(that is,each person’s income)over time,we have to rely on the technological progress.Again,encouraging saving wouldn’t help the growth of per capita GDP in the long run. One consequence of this result is that if there are two countries share the same rate of technological progress x,even if the other parameters(such as s,δ,α,and n)are di?erent,the growth rate of per capita income will eventually be equal across these countries.If,in addition, the population growth rate n is also common,then the growth rate of GDP will also be the same in the long run. Another,quite strong,prediction of this theory is that if A(t)is common across countries and s,δ,α,and n are the same across two economies,these economies end up at the same level (and of course,the same growth rate)of?y(t)in the long run,no matter how di?erent initial conditions are across these counties.So,if we believe that these factors are the same across di?erent countries,even if their per capita income levels are currently di?erent right now,their per capita income levels will eventually become equal. 2.4Short-run behavior of the economy Next,we look at the behavior of the economy in the short run.To proceed,let’s introduce a new variable y(t),which is de?ned as y(t)≡ Y(t) A(t)L(t) . y(t)can be interpreted as output per e?ective worker.From the production function, y(t)=[K(t)]α[A(t)L(t)]1?α A(t)L(t) = [ K(t) A(t)L(t) ]α =[k(t)]α. Sinceα>0,y(t)moves together with k(t)—when k(t)increases y(t)increases,and when k(t) decreases y(t)decreases.In fact,since ˙y(t) y(t)=α ˙k(t) k(t) , y(t)grows faster when k(t)grows faster,and y(t)grows slower when k(t)grows slower.From (9), ˙k(t) k(t)=α s[k(t)]α?(δ+x+n)k(t) k(t) =αs[k(t)]α?1?α(δ+x+n).(11) The?nal expression is decreasing in k(t)—k(t)grows faster when the level of k(t)is small. This,in turn,means that y(t)grows faster when y(t)is small and y(t)grows slower when y(t) is large.It can also be seen that the growth rate of k(t)is increasing in s for a given k(t).This means that the saving rate actually in?uences the growth in the short run. Now let’s look at the GDP.From the de?nition of y(t), Y(t)=y(t)A(t)L(t).(12) Therefore, ˙Y(t) Y(t)= ˙y(t) y(t) +x+n=α ˙k(t) k(t) +x+n.(13) We know that˙k(t)is negative when k(t)>k?and˙k(t)is positive when k(t) We can repeat the same analysis for per capita GDP?y(t)=Y(t)/L(t).Since ?y(t)=y(t)A(t) and ˙?y(t)?y(t)= ˙y(t) y(t) +x=α ˙k(t) k(t) +x, an economy with small?y(t)grows faster in the short run.See Figure4. 2.5Empirical predictions We have already mentioned some predictions of the Solow model.One strong prediction is that, if A(t),s,δ,α,and n are common across countries,initially poor countries grow faster and initially rich countries grow slower,and eventually all countries will have the same per capita income?y(t)=[k?]αA(t),where k?is given as(10).This prediction is called as convergence. Many researchers tried to see if there actually is convergence across countries.It turns out that the convergence actually happens across similar countries,such as within OECD countries or within EU countries.The convergence is also observed across di?erent states and regions within one country.For example,a state whose per capita income was lower at1880tended to have grown faster during the period of1880–2000.See Figure5.However,if we look at the whole world,there is no such a tendency—there are many countries who started poor and did not grow.See Figure6.Many researchers believe that it is because the factors A(t),s,δ,α,and n are not common across these di?erent countries.In particular,the di?erence in A(t)(and consequently the di?erence in x)seems quite enormous.So,the important research question is:why is A(t)so di?erent across countries? 3Measurement:growth accounting One issue that is closely related to the Solow model is how we can measure the technology level A(t).In this section,we use the production function(3)instead of(2)for the ease of exposition.So,now the question is how to measure z(t).(As we noted,we can always convert z(t)back to A(t)by the relationship z(t)≡[A(t)]1?α.)z(t)is sometimes called the total factor productivity. Writing(3)again: Y(t)=z(t)[K(t)]α[L(t)]1?α. We can measure Y(t)(GDP),K(t)(capital stock),and L(t)(population)in the data.The idea is to use this relationship to“back out”the variable z(t)which we cannot see directly in the data. First,let’s take a natural log of both sides of the above equation: log(Y(t))=log(z(t))+αlog(K(t))+(1?α)log(L(t)). By di?erentiating both sides: ˙Y(t) Y(t)= ˙z(t) z(t) +α ˙K(t) K(t) +(1?α) ˙L(t) L(t) .(14) Rearranging the terms: ˙z(t) z(t)= ˙Y(t) Y(t) ?α ˙K(t) K(t) ?(1?α) ˙L(t) L(t) . It is known thatαis around1/3in reality.Therefore,everything in the right hand side can be measured in the data.Thus we can“back out”˙z(t)/z(t)from this equation.˙z(t)/z(t)is sometimes called the Solow residual(since it is measured as a“residual”). What can we tell by measuring˙z(t)/z(t)?The equation(14)tells us there are only three ways to increase the GDP:(i)increase z(t)(ii)increase K(t),or(iii)increase L(t).By knowing ˙z(t)/z(t),we can tell exactly how much each components are contributing to the GDP growth. See Table1. Suppose,for example,that you found that a country’s GDP is growing because K(t)is growing,while z(t)(therefore A(t))is not growing.This is worrisome for this country,because the Solow model taught us that in the long run,the economic growth will stop unless z(t) (A(t))grows.A country can grow fast by increasing saving rate and force K(t)to grow in the short run,but this kind of growth is not sustainable.See Table2and Krugman’s article about the East Asian growth.