I. The Solow model - UAM vs fundamental causes of growth The Solow model helps us understand how...

I. The Solow model

Dynamic Macroeconomic Analysis

Universidad Autonoma de Madrid

Autumn 2014

Dynamic Macroeconomic Analysis (UAM) I. The Solow model Autumn 2014 1 / 38

Objectives

In this first lecture we will study the Solow growth model.

The Solow model is the basis for modern growth theory.

It offers a simplified representation of the mechanics of growth,placing particular emphasis on the role of capital accumulation.

Households own the capital stock and save a fixed proportion of theirincome.

In later themes we will construct a genuine dynamic generalequilibrium models with almost identical predictions.


Proximate vs fundamental causes of growth

The Solow model helps us understand how capital accumulation,population growth and technological change contribute to the growthof living standards.

In the terminology of Acemoglu these are “proximate causes” ofeconomic growth. They fail to explain the persistence of thepronounced differences in living standards.

The study of the “fundamental causes” of these income differences,such as the quality of institutions, is left for future study.

Even so, a thorough understanding of the mechanics of growth isessential.


Main assumptions

A closed economy without a government

Yt = Ct + It

Yt − Ct ≡ St = It

A neo-classical production function (see below)

Yt = F (Kt , Lt , At)

The final good can be consumed or used as capital. Technology, At ,is non-rival and non-excludable.

Households save a fraction s ∈ (0, 1) of their income.

St = It = sYt

while capital depreciates at a constant rate δ so that Dt = δKt


Technology

A neo-classical production function F (K , L, A) with F (0, 0, A) = 0 has thefollowing features:

1 Positive but decreasing marginal products

∂F

∂K= FK > 0 ;

∂F

∂L= FL > 0

∂2F

∂K 2= FKK < 0 ;

∂2F

∂L2= FLL < 0

2 Constant returns to scale in K and L

F (λK , λL, A) = λF (K , L, A)

3 The function F satisfies the Inada conditions

limK→0FK = limL→0FL = ∞; limK→∞FK = limL→∞FL = 0


Production side of the economy

Essentially, there are two alternative interpretations of the production side

Autarky (each household owns its own firm)

A representative firm hires capital and labour on a competitive labourmarket

In the latter case, the representative’s problem is given by

maxK ,L

Π = F (Kt , Lt , At)− rtKt − wtLt

rt = FK (., ., .)wt = FL(., ., .)

F (Kt , Lt , At) = rtKt + wtLt

where the last line follows from CTS.


Continuous time

For convenience we analyze the model in continuous time

The instantaneous change in any variable Xt is denoted by

Xt =dXt

dt

(= lim∆→0

Xt+∆ − Xt

∆

)


Law of motion of the capital stock

The change in the capital stock at t, Kt , is the difference between grossinvestment It , and depreciation, Dt :

Kt = It −Dt

= sYt −Kt

= sF (Kt , Lt , At)− δKt


Units

For future purposes, it is convenient to convert all aggregate variables inper capita quantities. We denote the quantities per capita by lower-caseletters.

For simplicity, let Lt denote both the size of the population and the labourforce. Then

κt =Kt

Lt

yt =Yt

Lt

=1

LtF (Kt , Lt , At) = F (Kt/Lt , 1, At)

= F (κt , 1, At) = f (κt , At)


Benchmark

In our benchmark, we abstract from population growth or technicalprogress. So,

Lt = L

At = A

while

κt =δ

δt

(Kt

L

)=

LKt −Kt0

L2=

Kt

L


The Fundamental Equation of the Solow Model

Combining the expressions for Kt and κt we obtain:

Kt = sF (Kt , L, A)− δKt

Kt

L=

1

LsF (Kt , L, A)− δ

Kt

Lκt = sf (κt)− δκt

Formally, this is a non-linear first-order differential equation.


Example: Cobb-Douglas production function

The Cobb-Douglas production function is a convenient example of aneo-classical production function:

Y = AK αL1−α

y =AK αL1−α

L= AK αL−α

= Aκα = f (κ, A)

The function f (., .) is a strictly concave function of κ with

δf (κ, A)δκ

= f ′(κ, A) = αAκα−1 =αA

κ1−α

limκ→0f ′(κ, A) = ∞limκ→∞f ′(κ, A) = 0


Convergence to steady state


Convergence to steady state

The previous slide has demonstrated that the economy converges(monotonically) to a steady state with κt = κ∗.

The economy replicates itself because each agent (or household) saves aquantity i∗ = sf (κ∗) that exactly compensates for the depreciation of hercapital d∗ = δκ∗.

Formally, for any initial capital stock κ0 ∈ (0, κ∗) the economy willexperience a period of growth in which κt and yt are strictly positive.

Once the economy reaches the steady state, growth comes to an end.

Note, convergence to k∗ will occur for any κ > 0, including the cases inwhich the economy starts out with a κ0 > κ∗ (see below).


What guarantees convergence?

The assumption of a neo-classical production technology is a sufficientcondition to guarantee convergence to a unique steady state:

The Inada condition limK→0FK = ∞ guarantees that the curve sf (κ)is steeper than δκ near the origin;

The diminishing marginal returns to capital guarantee that sf (κ) is astrictly concave function;

The second Inada condition, limK→∞FK = 0, guarantees that thecurve sf (κ) intersects deltaκ for some finite κ.

The intersection is unique because the slope of sf (κ) is a strictlydecreasing function of κ.


Relevant features of the steady state

Our assumptions about technology guarantee that there is a uniquesteady state with a positive capital-labour ratio κ∗.

There is also a trivial steady state with κ∗ equal to zero, but we willignore this unstable steady state.

It is common to talk about steady-state equilibria. But formally theentire trajectory from κ0 to κ∗ should be part of any equilibriumdefinition.

The unique positive levels of κ∗ and y ∗ are strictly increasing in s andA, and decreasing in δ.

By contrast, consumption per capita, c∗, is increasing in A butnon-monotonic in s (lims→0c∗ = lims→1c∗ = 0).


The Golden Rule

We cannot perform a formal welfare analysis as we have not specifiedagents’ preferences over consumption and labour.

Nonetheless, we can ask ourselves whether the steady-state allocationgenerates the maximum feasible level of steady-state consumption.

In any steady state, i∗ = It/L = δκ∗ . Since c∗ = f (κ∗)− i∗ we cantherefore define the golden-rule capital stock as

argmaxκc∗ = f (κ∗)− δκ∗

The FOC that implicitly defines κgold is:

f ′(κgold ) = δ

Let sgold denote the savings rate required to attain a steady state withκgold . There are no forces in the model to guarantee that s = sgold .


Steady-state consumption and the Golden Rule


Dynamic inefficiency

The concept of the Golden Rule is not just useful to identify the allocationwith the highest steady-state consumption level. It also allows us torecognize inefficient allocations.

In particular, any steady state with κ∗ > κgold (or s > sgold) is inefficient.

If the agents were to reduce their savings rate to sgold consumption willconverge to the maximum level cgold . And along the transition to steadystate the agents will enjoy even higher consumption levels.

In other words, if agents value consumption they must be unambiguouslybetter off than under the initial steady state.


Intertemporal tradeoffs

Now suppose the economy is located in a steady state with κ∗ < κgold

(and s < sgold). Once more c∗ < cgold , but this time we cannot makeunambiguous efficiency statements.

To reach the Golden Rule capital stock, the agents have to raise theirsavings rate. This generates an immediate reduction in consumption fromc∗ = (1− s)f (κ∗) to c ′ = (1− sgold )f (κ∗).

Consumption will grow over time and will eventually exceed c∗.

In other words, the agents have to trade off lower consumption todayagainst higher consumption in the future. A formal analysis of the welfareconsequences requires a fully specified model with preferences.


Population growth

Once we allow for population growth, the economy converges to a steadystate in which all aggregate variables grow at the same rate as Lt , butagain there is no growth in per capita variables.

Constant population growth

Lt

Lt= n

κt =δ

δt

(Kt

Lt

)=

LtKt −Kt Lt

L2t

=Kt

Lt− κt

(Lt

Lt

)=

Kt

Lt− κtn


The fundamental equation with population growth

Recall that the evolution of the capital stock is governed by

Kt

Lt=

sF (Kt , Lt , A)Lt

− δKt

Lt

κt =Kt

Lt− κtn

Combining these expressions, we obtain

κt = sf (κt)− (n + δ)κt

In steady state (κt = 0)

sf (κ∗) = (n + δ)κ∗


Steady state with population growth


Characteristics of the steady state

In steady state all per capita variables, such as κt , yt , it or ct , remainconstant over time.

By contrast, all aggregate variables grow at the same rate as thepopulation. For example,

Kt = κ∗Lt

log(Kt) = log(κ∗) + log(Lt)

Taking derivatives with respect to t yields

Kt

Kt= 0 +

Lt

Lt= n

From CRTS, it follows that Yt and hence Ct and It also grow at rate n.


Transitional dynamics

Since yt is stricly increasing in κt , the growth rate of y (yt/t) isproportional to the growth rate of κ

γκ ≡κ

κ= s

f (κ, A)κ− (n + δ)

The concavity of f (., .) implies that the average output per unit of capitalis decreasing in κ.

Hence the growth rate of κ is strictly decreasing in κ with γκ > 0 for allκt < κ∗ and γκ < 0 for κt > κ∗. This combined withlimκ→0(f (κ, A)/κ) = ∞ guarantees existence and uniqueness.


Growth and convergence

Correct interpretation: g=0


Technological progress

In the basic Solow model we can only have sustained growth if there issustained technological progress. Moreover, balanced growth is onlypossible if technological progress is labour-augmenting. Formally, we will

assume

Yt = F (Kt , AtLt)At

At= x

Next, we let Lt = AtLt denote the efficiency units of labour and

κt =Kt

Lt

=Kt

AtLt

yt =F (Kt , Lt)

Lt

= f (κt)


Balanced growth with exogenous technological change

Following the same procedure like before, we arrive at

δ

δt

(Kt

AtLt

)=

Kt

Lt

− κt(x + n)

Kt

Lt

= sf (κt)− δκt

Combing the above equations we arrive at the fundamental equation withtechnological growth

δκt

δt= sf (κt)− (n + δ + x)κt


Interpretation of the balanced growth path

Once the economy reaches the steady state,

All quantities per efficiency unit (y , c and κ) are constant over time

All quantities per capita (y , c and κ) grow at the rate n.

All aggregate variables (Y , C and K ) grow at the rate n + x .

However, it should be reminded that the growth of At is generatedexogenously. Eventually, we would like to understand the drivers behindtechnological progress (investments in human capital, R&D, etc. ).

In order to generate endogenous growth with investment in R&D we needto abandon the assumption of perfect competition.


(Un)conditional convergence

Source: Acemoglu (2008)


(Un)conditional convergence

Source: Acemoglu (2008)


Conditional convergence

The model predicts that the growth rate of capital and income per capitaare falling in the level of capital. Hence, if the level of capital is the ONLYdifference between two countries, then:

The poor country should grow at a faster rate than the rich country

Both countries should eventually converge to the same steady state.

The conditional convergence of income levels is a prediction of the modelthat is verified by the data.

Similarly, nothing prevents the divergence between countries with differentfundamentals.


Decentralized equilibrium

So far we have abstracted from markets, assuming that agentsoperate a backyard technology.

It is straightforward to relax this assumption and assume thathouseholds offer their capital and labour services on competitivemarkets

The representative firm maximizes profits so that

wt = FL(Kt , Lt)Rt = FK (Kt , Lt)

All factor markets clear at all dates


Equilibrium

Consider the version with population growth:

For a given sequence of {L(t), At}∞t=0 and and initial capital stock K0 an

equilibrium path is a sequence of capital stocks, output levels, consumptionlevels, wage rates and rental prices {Kt , Yt , Ct , wt , Rt}∞

t=0 such that

Kt = sF (At , Kt , Lt)− (n + δ)Yt = F (At , Kt , Lt)Ct = (1− s)Yt

Rt = FK () = f ′(κt)wt = FL() = f (κt)− f ′(κt)


Dynamic inefficiency

In this economy, the agents receive steady state net-interest rater ∗ = R∗ − δ = f ′(κ∗)− δ.

The Golden Rule capital stock is defined by f ′(κGOLD) = n + δ

Thus for dynamically inefficient allocations with k∗ > kGOLD , it musttrue that

f ′(κ∗) = r ∗ + n < n + δ

r ∗ < n

In later themes we will see that this condition implies that a PAYGOpension scheme improves welfare


Sustained growth — The AK model

In order to generate sustained growth without exogenous technologicalchange, we need to relax some of the assumptions of the Solow model.

A logical candidate is to consider alternatives to the assumption of aneo-classical production technology

In the Solow model growth vanishes due to the decreasing marginalproduct of capital

We can show that perpetual growth is feasible if output is linear in Kso that

Yt = AKt

This interpretation is acceptable if we use a broad concept of capitalthat includes both human and physical capital.


Sustained growth — The AK model

In the AK -model, per capita output is equal to y = AK /L = Aκ. In otherwords, the fundamental equation remains valid

κ = sy − (n + δ)κ

= sAκ − (n + δ)κ

and so the growth rate of κ can be written as

κ

κ= sA− (n + δ)

which is constant and independent of κ.


Distinctive features of the AK model

1 The model produces sustained growth without sustained growth in anexogenous variable like A

2 The growth rate is increasing in the savings rate

3 No transitional dynamics — the growth rate is constant and equal tosA− (n + δ)

4 No convergence

5 Recessions produce permanent effects on living conditions

6 The economy is never dynamically inefficient


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