Economic Growth - Theory and Evidence

Economic GrowthTheory and Evidence

Drago [email protected]

Department of EconomicsBI Norwegian Business School

GRA6634 - Advanced Macroeconomics, Fall 2012

DB (BI Norwegian Business School) GRA6634 - Economic Growth Fall 2012 1 / 141

mailto:[email protected]

Readings

Syllabus:These lecture slides!Obstfeld and Rogoff (1996) ch. 7.1-7.2 (not 7.1.2.2-7.1.2.3 and7.2.2.3)Romer (2012) ch. 1.1-1.7, ch. 2.1-2.6, ch. 4.1-4.2Krueger (2007) ch. 3-5, ch. 7

Other relevant material:Barro and Sala-i-Martin (2004) ch. 1.1-1.2, ch. 2.1-2.6Acemoglu (2009) ch. 1-5


Outline

1 Part 1 - Introduction toeconomic growth

MotivationWhat do we want to explainwith growth theory?

2 Part 2 - The Solow modelThe original modelExtension 1:Labor-augmentingtechnological progressExtension 2: Human capital

3 Part 3 - The Neoclassicalgrowth model

Introduction and modelsetupThe competitive equilibriumThe social planner’sproblem and the welfaretheoremsDynamic analysisSteady state, the goldenrule and the modifiedgolden rule

4 Part 4 - Taking stockModern growth theorySummary


Part 1 - Introduction to economic growth Motivation

Outline1 Part 1 - Introduction to economic growth

MotivationWhat do we want to explain with growth theory?

2 Part 2 - The Solow modelThe original modelExtension 1: Labor-augmenting technological progressExtension 2: Human capital

3 Part 3 - The Neoclassical growth modelIntroduction and model setupThe competitive equilibriumThe social planner’s problem and the welfare theoremsDynamic analysisSteady state, the golden rule and the modified golden rule




Why it is important to study economic growth

Is there some action a government of India could take thatwould lead the Indian economy to grow like Indonesia’s orEgypt’s? If so, what, exactly? If not, what is it about the"nature of India" that makes it so? The consequences forhuman welfare involved in questions like these are simplystaggering: Once one starts to think about them, it is hard tothink about anything else.

–Robert E. Lucas, Jr. (1988)



Why it is important to study economic growth (cont.)

Growth in the US:GDP per capita in 1870: $3,340. Average growth rate 1870–2000of 1.8%.Implied GDP per capita in 2000 is $33,330.Would have been $9,450 if average growth rate was 0.8%!Would have been $127,000 if average growth rate was 2.8%!




Figure 1 : Growth differences (Barro and Sala-i-Martin 2004)

Short movie: Hans Rosling – 200 years that changed the world




The difference between economic growth theory and business cyclestheory:

A business cycle is a short term fluctuation in economic growth, atransitory deviation from some long run trend.Economic growth theory is, in contrast to business cyclesanalysis, concerned with the long run trend itself.Wikipedia:

[T]he topic of economic growth is concerned with thelong-run trend in production due to basic causes such asindustrialization. The business cycle moves up and down,creating fluctuations in the long-run trend in economic growth.


Part 1 - Introduction to economic growth What do we want to explain with growth theory?








Stylized facts of economic growth

Any convincing growth model should be able to explain someobservable regularities related to economic growth.Kaldor (1961) offered a set of empirical observations ofindustrialized (and capitalized) countries.

These observations have come to be known as stylized facts abouteconomic growth.The stylized facts are not supposed to hold in the short run – theyare long run statements.



Stylized facts of economic growth (cont.)

Kaldor (1961, pp. 178-179):SF1 Output per worker shows continuing growth "with no

tendency for a falling rate of growth of productivity".SF2 Capital per worker shows continuing growth.SF3 The rate of return on capital is steady.SF4 The capital-output ratio is steady.SF5 Labor and capital receives constant shares of total

income.SF6 There are wide differences in the rate of productivity

growth across countries.



Stylized facts of economic growth (cont.)

Romer (1989, pp. 55):SF7 In the cross-section, the mean growth rate shows no

variation with the level of per capita income.SF8 The rate of growth of factor inputs is not large enough to

explain the rate of growth of output; that is, growthaccounting always finds a residual.

SF9 Growth in the volume of trade is positively correlated withgrowth in output.

SF10 Population growth rates are negatively correlated with thelevel of income.

SF11 Both skilled and unskilled workers tend to migratetowards high-income countries.



Stylized facts of economic growth

This part of the course will be about (standard) models of economicgrowth. We shall. . .

. . . study a few simple, but celebrated growth models.

. . . focus on key equations, both derivations and interpretations.

. . . evaluate their empirical predictions, qualitatively andquantitatively.. . . (try to) make sure you master the basic tools of economicgrowth theory.

Today we will focus on the Solow model.


Part 2 - The Solow model The original model








Introduction

Solow (1956) and Swan (1956) developed a neoclassical growthmodel with some important features:

An aggregate production function with constant returns to scaleand diminishing returns to each input.A classical Keynesian type demand side with a constantsaving-rule.One of the key predictions is that countries with low levels of GDPper capita should converge towards richer countries.



The model

ProductionYt = F (Kt ,Lt ) (1)

where Kt and Lt are capital and labor in period t , respectively.Assumptions:

λYt = F (λKt , λLt ) (for any constant λ > 0)

FK > 0,FL > 0,FKK < 0,FLL < 0,FKL > 0

limK→0

FK = limL→0

FL = +∞, limK→∞

FK = limL→∞

FL = 0

Cobb-Douglas example: Yt = KtαL1−α

t



The model (cont.)

Equilibrium condition in the goods market (aggregate resourceconstraint)

Yt = Ct + It (2)

where Ct and It are consumption and investment, respectively.Savings rate:

St = sYt (3)

where St is aggregate saving and s is the saving rate.Important: Savings are assumed exogenous in the Solow model.



The model (cont.)

Law of motion for capital

Kt+1 = (1− δ) Kt + It (4)

where δ is the depreciation rate of capital from one period to thenext.Equilibrium condition in the financial market: Saving equalsinvestment

St = It (5)

Constant population growth rate

Lt+1 = (1 + n) Lt ⇔ n =Lt+1−Lt

Lt(6)



The model (cont.)

Equations (1)-(6) constitute the Solow model. The system ofequations has 6 variables (Yt ,Kt ,Lt ,Ct , It ,St ) and 3 parameters(s, δ,n).We want to have a model with a steady state, i.e. with variablesthat do not grow in the long run. Need to reformulate equations interms of per capita units.



The model (cont.)

Define output and capital per worker :

yt ≡Yt

Lt=

F (Kt ,Lt )

Lt= F

(Kt

Lt,1)≡ f (kt ) (7)

where kt ≡ KtLt

.Combining the other equations (details in class):

kt+1 − kt =1

1 + n[sf (kt )− (n + δ) kt ] (8)

The model: Two equations (7) and (8), two variables kt and yt .sf (kt ) > (n + δ) kt implies positive capital (per worker) growth,sf (kt ) < (n + δ) kt implies the opposite.



The steady state

DefinitionA steady state is a solution to the model where the variables areconstant over time.

DefinitionA balanced growth path is a solution to the model where the variablesgrow at a constant rate over time.

Our terminology: A steady state is a solution to the model where ratiosbetween variables (e.g. yt ) are constant over time, while the variablesthemselves (e.g. Yt ) evolve along a balanced growth path.



The steady state (cont.)

From (8): kt+1 = kt = . . . ≡ k yields

sf (k) = (n + δ) k , (9)

a fixed point (kt and yt do not change over time).This point is stable because fk > 0 while fkk < 0:

If kt < k , then fk is high, and kt grows towards k .If kt > k , then fk is low, and kt declines towards k .yt moves in the same direction as kt .Taken together, the economy will always converge towards thesteady state.

Is there any other fixed point? Is it stable?




Figure 2 : The Solow model

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Figure 2 : The Solow model

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Problem: No growth in per capita variables

The Solow model in its simplest form predicts the following growthrates:

Capital and output: Kt+1−KtKt

= Yt+1−YtYt

= nCapital and output per worker: kt+1−kt

kt= yt+1−yt

yt= 0

Constant capital-worker and output-worker ratios violate SF1 andSF2.What to do? Introduce technological progress.


Part 2 - The Solow model Extension 1: Labor-augmenting technological progress








The model

ProductionYt = F (Kt ,EtLt ) (10)

where Et is the effectiveness of labor and EtLt are the efficiencyunits of labor (physical labor stock is still Lt ).

Cobb-Douglas example: Yt = Ktα (EtLt )

1−α

Difference between Harrod neutral (labor-augmenting), Solowneutral (capital-augmenting) and Hicks neutral technologicalprogress.

Assume constant technological progress:

Et+1 = (1 + g) Et ⇔ g =Et+1 − Et

Et(11)



The model (cont.)

The remaining equations are as before:Equilibrium in goods market: Yt = Ct + ItSaving: St = sYt

Law of motion for capital: Kt+1 = (1− δ) Kt + ItEquilibrium in financial market: St = ItPopulation growth: Lt+1 = (1 + n) Lt

Equations (2)-(6) and (10)-(11) constitute the Solow model withtechnological progress.



The model (cont.)

Define output and capital per effective worker:

yt ≡Yt

EtLt=

F (Kt ,EtLt )

EtLt= F

(Kt

EtLt,1)≡ f

(kt

)(12)

where kt ≡ KtEt Lt

.Combine equations to get (details in class):

kt+1 − kt =1

1 + n + g

[sf(

kt

)− (n + g + δ) kt

](13)

Key equations as before, except that n is replaced by n + g.



The model (cont.)

Let us look at equation (13) more closely:

kt+1 − kt =1

1 + n + g

[sf(

kt

)− (n + g + δ) kt

]

First term: Gross savings per efficiency worker. yt = f(

kt

)is the

intensive form of the production function.Second term: Period losses in kt = Kt

Et Ltcaused by population

growth, technology growth, and capital depreciation.Both terms multiplied by 1

1+n+g : Net investment in period t mustbe spread out over more efficiency workers in t + 1.



The model (cont.)

Figure 3 : The Solow model with technological progress

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The model (cont.)


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The model (cont.)


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The model (cont.)


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The model (cont.)


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Dynamics and convergence

From figure and equation (13):

kt+1 − kt =1

1 + n + g

[sf(

kt

)− (n + g + δ) kt

]

sf(

kt

)> (n + g + δ) kt implies kt+1 − kt > 0.

sf(

kt

)< (n + g + δ) kt implies the opposite.



Dynamics and convergence (cont.)

Figure 4 : A phase diagram

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Define γk ≡kt+1−kt

ktas the growth rate of the capital to

efficiency-worker ratio and assume that yt = kαt .Rewrite (13) (details in class):

γk ≡kt+1 − kt

kt=

11 + n + g

sf(

kt

)kt

− (n + g + δ)

=

11 + n + g

[skα−1

t − (n + g + δ)]

(14)

This growth rate depends negatively on kt :

∂γk

∂kt=

s1 + n + g

fk t kt − f(

kt

)k2

t

=s

1 + n + g(α− 1) kα−2

t ≤ 0 (15)




Define γy ≡yt+1−yt

ytas the growth rate of the

output-efficiency-worker ratio:

γy ≡yt+1 − yt

yt≈

fk t

(kt+1 − kt

)yt

=fk t kt

ytγk

=α

1 + n + g

[skα−1

t − (n + g + δ)]

(16)

Also this growth rate depends negatively on kt :

∂γy

∂kt= α

∂γk

∂kt

=αs

1 + n + g(α− 1) kα−2

t ≤ 0 (17)




Convergence hypothesis an implication of the Solow model:Countries with low capital intensity (poor countries) should havefaster growth than rich countries.Implies convergence between countries, where initially poorcountries catch up with richer countries.

Absolute convergence: Poor countries should grow faster thanricher countries.Conditional convergence: Poor countries should grow faster thanricher, but otherwise similar countries .




Figure 5 : Absolute convergence (Barro and Sala-i-Martin 2004)




Figure 6 : Conditional convergence (Barro and Sala-i-Martin 2004)



The steady state

Steady state is still defined by kt = kt+1 = . . . ≡ k :

sf(

k)

= (n + g + δ) k (18)

Gross savings just enough to maintain current level of capital perefficiency worker.An analytical solution to the steady state:

Assume Yt = Ktα (EtLt )

1−α.(18) can then be solved for k to yield (details in class):

k ≡ KEL =

(s

n+g+δ

) 11−α

, y ≡ YEL = kα =

(s

n+g+δ

) α1−α

,

c ≡ CEL = (1− s)

(s

n+g+δ

) α1−α

, i ≡ IEL = s

(s

n+g+δ

) α1−α




The model contains a balanced growth path with positive growth incapital and output per capita:

Capital and output:

Kt+1 − Kt

Kt=

Yt+1 − Yt

Yt= n + g

Capital and output per effective worker:

kt+1 − kt

kt=

yt+1 − yt

yt= 0

Capital and output per physical worker:

kt+1 − kt

kt=

yt+1 − yt

yt= g

Positive capital and output per capita growth rates fit SF1 and SF2!DB (BI Norwegian Business School) GRA6634 - Economic Growth Fall 2012 39 / 141


Comparative statics

Figure 7 : A positive shift in the savings rate s

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Comparative statics (cont.)

Figure 8 : Increased savings rate (from Romer (2012) ch. 1)




Figure 9 : A positive shift in g (or in n or δ)

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The golden rule of capital accumulation

Suppose a benevolent policymaker could choose the savings rate s tomaximize consumption?

Using (18) we can write the steady state consumption as afunction of the saving rate c (s):

c (s) = (1− s) f(

k (s))

= f(

k (s))− (n + g + δ) k (s)

First order condition ∂c(s)∂s = 0 delivers consumption maximizing

solution (denoted with star):

fk(

k∗ (s∗))

= (n + g + δ)

Slope of y -curve equal to slope of "break even"-curve.



The golden rule of capital accumulation (cont.)

Figure 10 : The golden rule capital per efficiency worker ratio

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Figure 11 : Consumption and the savings rate

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Saving and efficiencyIf s > s∗: Dynamically inefficient because reduced savings ratewould increase consumption today and consumption in the future.If s < s∗: Dynamically efficient because an increase in the savingsrate in order to increase future consumption would come at thecost of reduced consumption today.

This analysis does not justify normative statements asmicrofoundations are absent!



Lessons about long run levels and growth rates

Countries with. . .high saving rateslow population growthlow technology growthlow depreciation rates

. . . in the long run should have large capital-efficiency-workerratios (they are capital intensive) and largeoutput-efficiency-worker ratios.Changes in saving rates do not lead to changes in long run growthrates, only to changes in long run ratios.



Empirical performance of the Solow model (cont.)

One can rewrite y = kα =(

sn+g+δ

) α1−α to (details in class):

log yt ≈ log E0 + gt +α

1− αlog s (19)

− α

1− αlog (n + g + δ)

where, as before, yt is per capita output.OLS estimates from Mankiw, Romer and Weil (1992). . .

. . . fit the model predictions qualitatively.Higher savings lead to higher steady state output per capita.Higher population growth lead to lower steady state output per capita.

BUT: The point estimate α1−α = 1.42 implies α = 0.59

(unrealistically high), not α = 1/3 as under free competition.


Part 2 - The Solow model Extension 2: Human capital








Why add human capital to the Solow model?

The standard Solow model predicts much higher capital sharethan what we see in data.Why?

Cobb-Douglas technology specification wrong?Mis-measurement of the capital input?

What to do? Mankiw, Romer and Weil (1992) add human capitalto the model.



Why add human capital to the Solow model? (cont.)

Wikipedia: Human capital is the stock of competencies,knowledge, social and personal attributes, including creativity,embodied in the ability to perform labor so as to produceeconomic value. . . .

Standard view is that one can invest in human capital via e.g.education.



The model

Production

Yt = F (Kt ,Ht ,EtLt ) = KtαHt

φ (EtLt )1−α−φ (20)

where Ht is human capital and 0 < α + φ < 1.Equilibrium condition in the goods market (aggregate resourceconstraint)

Yt = Ct + IKt + IHt (21)

where IKt and IHt are investment in physical capital and humancapital, respectively.



The model (cont.)

Savings rates:SKt = sK Yt (22)

SHt = sHYt (23)

where SKt and SHt are aggregate savings in physical and humancapital, and sK and sH are the savings rates satisfying0 ≤ sK + sH ≤ 1.Equilibrium conditions in the financial market: Saving equalsinvestment

SKt = IKt (24)

SHt = IHt (25)



The model (cont.)

Law of motion for physical and human capital

Kt+1 = (1− δ) Kt + IKt (26)

Ht+1 = (1− δ) Ht + IHt (27)

(note: same depreciation rate)Constant population growth rate and technological process

Lt+1 = (1 + n) Lt (28)

Et+1 = (1 + g) Et (29)



The model (cont.)

Equations (20)-(29) constitute the Solow model with humancapital.Again we define output and capital per effective worker:

yt ≡Yt

EtLt=

KtαHt

φ (EtLt )1−α−φ

EtLt= kαt hφt (30)

where kt ≡ KtEt Lt

and ht ≡ HtEt Lt

.



The model (cont.)

Combining equations we end up with two first order differenceequations in kt and ht (details in class):

kt+1 − kt =1

1 + n + g

[sK kαt hφt − (n + g + δ) kt

](31)

ht+1 − ht =1

1 + n + g

[sH kαt hφt − (n + g + δ) ht

](32)

Everything as before, but now also: sH kαt hφt > (n + g + δ) htimplies positive human capital (per effective worker) growth,sH kαt hφt < (n + g + δ) ht implies the opposite.



The steady state

Using (31) and (32), one can solve for steady state k , h, etc.:

k ≡ KEL

=

(sK

1−φsHφ

n + g + δ

) 11−α−φ

h ≡ HEL

=

(sH

1−αsKα

n + g + δ

) 11−α−φ

y = kαhφ =

(1

n + g + δ

) α+φ1−α−φ

sKα

1−α−φ sHφ

1−α−φ



Empirical performance

Rewrite y =(

1n+g+δ

) α+φ1−α−φ sK

α1−α−φ sH

φ1−α−φ to (details in class):

log yt ≈ log E0 + gt +α

1− α− φlog sK (33)

+φ

1− α− φlog sH

− α + φ

1− α− φlog (n + g + δ)

OLS estimates from Mankiw, Romer and Weil (1992). . .. . . are qualitatively and quantitatively consistent with the model.. . . suggest α and φ close to 1/3, in line with what we get from data.


Part 2 - The Solow model Taking stock

Critique of the Solow model

Fits nicely many stylized facts.Drawback: No microfoundations in the model!

The Solow model is a growth model where growth is presentbecause we assume growth...The Solow model offers a partial equilibrium explanation. Whatdetermines the savings rate, the interest rate, etc.?Modern macroeconomics is all about general equilibrium.That is why we will spend the next lectures on the Neoclassicalgrowth model.


Part 2 - The Solow model Taking stock

The Solow model summarized

Any growth model should be able to explain a set of stylized facts.The Solow model fits stylized facts fairly well, and predicts incomeconvergence between (otherwise similar) countries with differentcapital intensities.Harrod neutral technological progress needed to explain positivegrowth in per capita variables.Human capital needed to explain capital and labor shares (ofoutput).Critique: The demand side (households) is treated as purelyexogenous. Unrealistic assumption, modern macro is mostlyabout general equilibrium.


Part 3 - The Neoclassical growth model Introduction and model setup








Introduction

The canonical Solow model (and its close extensions) is, as wesaw earlier, based on the assumption that people always save aconstant fraction of income. That assumption is questionable andmakes the model vulnerable to the Lucas critique.The Lucas critique:

Given that the structure of an econometric modelconsists of optimal decision rules of economic agents,and that optimal decision rules vary systematically withchanges in the structure of series relevant to the decisionmaker, it follows that any change in policy willsystematically alter the structure of econometric models.

–Robert E. Lucas, Jr. (1976)



Introduction (cont.)

The Ramsey-Cass-Koopman model, often called the Neoclassicalgrowth model, is perhaps the single most celebrated model in modernmacroeconomics. The model. . .

. . . is superior to the Solow model in the sense that it explicitlytakes into account the optimal behavior of economic agents (takesinto account the Lucas critique).. . . constitutes the framework for some of the most up to datemacroeconomic models out there, including those used by centralbanks in the industrialized world.. . . is widely used in growth theory, business cycles analysis,monetary policy analysis, public finance, asset pricing andbanking theory.

Most of you will come across the Neoclassical model several timesduring your master, so you should study it well!



Introduction (cont.)

Two types of agents in the model, households and firms. Depending onthe research question, one often wants to add a public sector as well.

Households derive utility from consumption financed by laborincome and asset returns.Firms maximize profits.All markets (market for goods, assets and labor) are perfectlycompetitive.At the heart of the model lies a tradeoff between currentconsumption and investment in future consumption.



Household preferences

Households live forever (broadly speaking) but discount the future.Utility for a household in period t is a function of per capitaconsumption, denoted u (ct ), where ct ≡ Ct

Ltis the per capita

consumption level.Lifetime utility is assumed to be additive:

U (c0, c1, c2, . . . ) = u (c0) + βu (c1) + β2u (c2) + . . .

=∞∑

t=0

βtu (ct ) (34)

β ≡ 11+ρ ∈ (0,1) is a time discount factor (ρ is the discount rate).

Households are impatient to the extent that β < 1.



Household preferences (cont.)

Assumptions:Increasing, but concave utility function:

uct > 0, ucct < 0

Households always prefer more consumption rather than less, butextra consumption adds less utility when the household alreadyconsumes much.Inada type conditions often assumed as well:

limct→0

uct =∞, limct→∞

uct = 0

Isoelastic utility functions are especially popular:

u (ct ) =c1−σ

t1− σ

(35)

where u (ct ) = ln ct if σ = 1.



The budget constraint

Households face a period-by-period budget constraint:

Ct + It = wtLt + rtKt (36)

where wt is the real wage rate and rt is the real rental price oncapital. The left hand side represents total householdexpenditures, the right hand side total household income.The law of motion for capital:

Kt+1 = (1− δ) Kt + It (37)



Reformulation in terms of efficiency units

We want to have a model with a steady state, i.e. with variablesthat do not grow in the long run.At the same time we would like the model to be consistent withlong run growth in e.g. GDP per capita.Need to reformulate equations in terms of efficiency units.



Reformulation in terms of efficiency units (cont.)

Assume the economy is growing because of. . .

. . . constant populationgrowth:

Lt = (1 + n) Lt−1

= (1 + n) (1 + n) Lt−2

= . . . = (1 + n)tL0

= (1 + n)t

. . . constant labor-augmentingtechnology growth:

Et = (1 + g) Et−1

= (1 + g) (1 + g) Et−2

= . . . = (1 + g)tE0

= (1 + g)t




Then define variables in. . .

. . . per capita terms:

ct =Ct

Lt=

Ct

(1 + n)t

kt =Kt

Lt=

Kt

(1 + n)t

yt =Yt

Lt=

Yt

(1 + n)t

. . . per efficiency capita terms:

ct =Ct

EtLt=

ct

(1 + g)t

kt =Kt

EtLt=

kt

(1 + g)t

yt =Yt

EtLt=

yt

(1 + g)t




Rewrite the period utility function (details in class):

u (ct ) = (1 + g)t(1−σ) c1−σt

1− σ= (1 + g)t(1−σ) u (ct ) (38)

Rewrite the lifetime utility function:

U =∞∑

t=0

βtu (ct ) (39)

where β ≡ β (1 + g)1−σ and we assume that β < 1.Finally rewrite the budget constraint to obtain:

ct + (1 + n) (1 + g) kt+1 − (1− δ) kt = wt1Et

+ rt kt (40)


Part 3 - The Neoclassical growth model The competitive equilibrium








The representative household’s problem

The representative household’s problem:

max{ct ,kt+1}∞t=0

∑∞t=0 β

tu (ct )

subject to

ct + (1 + n) (1 + g) kt+1 − (1− δ) kt = wt1Et

+ rt kt ,

ct , kt+1 ≥ 0, k0 > 0, TVC: limt→∞

βtuct kt+1 = 0

(41)

Notice that the representative household still cares about percapita units (ct and kt+1), not per efficiency capita units (ct andkt+1).We will not talk about the transversality condition (TVC). For thoseinterested, it basically states that the discounted present value ofcapital in the infinite future is zero.



The representative household’s problem (cont.)

The Lagrangian:

L =∞∑

t=0

βtu (ct )

−∞∑

t=0

λt

[ct + (1 + n) (1 + g) kt+1 − (1− δ) kt − wt

1Et− rt kt

]

where ct ≡ ct(1+g)t and kt+1 ≡ kt+1

(1+g)t+1 .

Optimality conditions (details in class):

ct : βtuct = λt (42)

ct+1 : βt+1uct+1 = λt+1 (43)kt+1 : λt (1 + n) (1 + g) = λt+1 [rt+1 + (1− δ)] (44)



The Euler equation

Combining the first order conditions we get the Euler equation(details in class):

uct = βuct+1rt+1 + (1− δ)

(1 + n) (1 + g)(45)

Interpretation:LHS shows the utility loss of reducing consumption marginallytoday.RHS shows the discounted utility increase in the next period byreducing consumption marginally today.Euler equation states that an utility maximizing allocation is toequate these two (why?).



The Euler equation (cont.)

Assuming utility is isoelastic, we can rewrite the Euler equation to(details in class):

ct+1 − ct

ct=

[rt+1 + (1− δ)

(1 + ρ) (1 + n) (1 + g)σ

] 1σ

− 1 (46)

Interpretation:(46) shows the optimal consumption growth rate in per efficientcapita terms.Higher interest rate relates to higher consumption growth asmeasured in efficiency units. Why?

rt+1 is the (non-discounted) period price of current consumption interms of future consumption.As rt+1 increases, current consumption becomes more expensive,and the household shifts some of its consumption to the future.



A microfounded savings rate

Critique of Solow model: Savings rate an exogenous parameter.The savings rate in this model however is an endogenous variable(details in class):

st =ityt

=(1 + n) (1 + g) kt+1 − (1− δ) kt

kαt(47)

No people’s marginal propensity to save depends on the state ofthe economy ! This makes the savings rate time varying.Today’s savings rate is. . .

. . . increasing in kt+1 (more future capital requires more savings).

. . . decreasing in kt (more capital today implies less savings neededto meet future capital requirements).. . . increasing in n, g and δ (those parameters discount futurevalues of current capital).



The representative firm’s problem

Think of a single firm that acts competitively.Production:

Yt = Ktα(EtLt )

1−α (48)

Profit maximization problem:

max{Kt ,Lt}∞t=0

Πt = pt [Yt − rtKt − wtLt ]

subject to

Yt = Ktα(EtLt )

1−α,Kt ,Lt ≥ 0

(49)

where the real rate of return on capital and the real wage aregiven by rt ≡ nominal rate of return on capital

ptand wt ≡ nominal wage

pt.



The representative firm’s problem (cont.)

Profit maximizing conditions for firm (details in class):

Kt : rt = αKtα−1(EtLt )

1−α = αkα−1t (50)

Lt : wt = (1− α) Ktα(EtLt )

−αEt = (1− α) kαt (1 + g)t (51)

Real return on capital equal to marginal product of capital, realwage equal to marginal product of labor.



The representative firm’s problem (cont.)

Lessons:Firms, although acting as profit maximizers, earn zero profits(details in class):

Πt = Yt − rtKt − wtLt = 0

This is always the case when production exhibits constant returnsto scale and the firm takes market prices as given.The capital and labor shares of income are both constant:

rtKt

Yt= α

wtLt

Yt= 1− α

This is in line with SF5.



Market clearing

The economy as a whole is subject to a market clearing condition,which states that demand should equal supply:

Ct + It = Yt

orct + it = yt (52)

where, as before, ct ≡ CtEt Lt

, it ≡ ItEt Lt

and yt ≡ YtEt Lt

are the efficiencyunits of consumption, investment and output. Market clearing shouldalso hold in the labor market and the capital market.

We are now ready to define the equilibrium of this economy.



Competitive equilibrium

DefinitionGiven initial capital k0, a competitive equilibrium is allocations for the representativehousehold,

{ct , kt+1

}∞t=0

, allocations for the representative firm,{

kt

}∞t=0

, and (real)

prices {rt ,wt}∞t=0 such that:1 Given {rt ,wt}∞t=0, the household allocation solves the household problem (41).2 Given {rt ,wt}∞t=0, the firm allocation solves the firm problem (49).3 Market clearing in all three markets, i.e. demand equal to supply:

Ct + It = Yt ,

Ldemandt = Lsupply

t ,

K demandt = K supply

t ,



Competitive equilibrium (cont.)

The first order conditions from the household’s problem and thefirm’s problem, together with the prices rt and wt , imply that thecapital market and the labor market clear.Remains to show the goods market clearing condition. From thehousehold’s budget constraint and the firm’s optimality conditions,one can get (details in class)

ct + it = yt ,

which implies market clearing in the goods market.This actually follows from Walras’ law: In a market with xsubmarkets, the last submarket (here the goods market) clears ifthe x − 1 other markets (here capital and labor market) clear.


Part 3 - The Neoclassical growth model The social planner’s problem and the welfare theorems








The social planner’s problem

What we are ultimately after is a good description of the economyand how it grows over time. That is why a characterization of thecompetitive equilibrium was needed.However, in the process we had to. . .

1 . . . solve the household problem.2 . . . solve the firm problem.3 . . . make sure that all markets clear.

Wouldn’t it be great if we were able to do all this stuff in a muchsimpler manner?

It turns out that there is a simpler way to find the competitiveequilibrium, namely to set up and solve the social planner’sproblem.



The social planner’s problem (cont.)

Imagine a social planner that can dictate how all the agents in theeconomy should behave.

Assume the social planner is benevolent, i.e. he would like tomaximize the lifetime utility of households.The planner does not need to care about prices, instead he isconcerned about the resource constraint Ct + It ≤ Yt .Combined with the capital law of motion and the processes forpopulation and technology growth, the resource constraint can bewritten (details in class):

ct + (1 + n) (1 + g) kt+1 − (1− δ) kt = kαt (53)




The social planner’s problem:

max{ct ,kt+1}∞t=0

∑∞t=0 β

tu (ct )

subject to

ct + (1 + n) (1 + g) kt+1 − (1− δ) kt = kαt ,ct , kt+1 ≥ 0, k0 > 0, TVC: lim

t→∞βtuct kt+1 = 0

(54)

Note that the social planner cares about per capita units (ct andkt+1), not per efficiency capita units (ct and kt+1).




The Lagrangian:

L =∞∑

t=0

{βtu (ct )− λt

[ct + (1 + n) (1 + g) kt+1 − (1− δ) kt − kαt

]}(55)

where ct ≡ ct(1+g)t and kt+1 ≡ kt+1

(1+g)t+1 .

Optimality conditions (details in class):

ct : βtuct = λt (56)

ct+1 : βt+1uct+1 = λt+1 (57)

kt+1 : λt (1 + n) (1 + g) = λt+1

[αkα−1

t+1 + (1− δ)]

(58)




Combining equations (details in class):

uct = βuct+1αkα−1

t+1 + (1− δ)

(1 + n) (1 + g)(59)



The first and second welfare theorem

Compare the Euler equation from the social planner’s problem,


t+1 + (1− δ)

(1 + n) (1 + g)

with the Euler equation in competitive equilibrium,

uct = βuct+1rt+1 + (1− δ)

(1 + n) (1 + g).

The two are identical if and only if

rt+1 = αkα−1t+1 .

But this condition must hold in competitive equilibrium, it is theoptimality condition for firms with respect to capital!



The first and second welfare theorem (cont.)

Theorem(First welfare theorem) Suppose we have a competitive equilibriumwith allocations

{ct , kt+1

}∞t=0

. Then the equilibrium is socially optimalin the sense that it solves the social planner’s problem (details inclass).

Theorem

(Second welfare theorem) Suppose the allocations{

ct , kt+1

}∞t=0

solvethe social planner’s problem and hence are socially optimal. Thenthere exists prices {rt ,wt}∞t=0, that together with these allocations forma competitive equilibrium (details in class).



The first and second welfare theorem (cont.)

The two theorems basically states that the competitive equilibriumand the solution to the social planner’s problem are the same.This fact justifies an analysis of the (often much) simpler socialplanner’s economy instead.

The welfare theorems hold because our economy is frictionless.Modern macromodels (e.g. New Keynesian models) often havefrictions which brake down these theorems.


Part 3 - The Neoclassical growth model Dynamic analysis








Key equations of the Neoclassical growth model

The Euler equation and the resource constraint are the two keyequations of the Neoclassical growth model:


t+1 + (1− δ)

(1 + n) (1 + g)

ct + (1 + n) (1 + g) kt+1 − (1− δ) kt = kαt

The Euler equation illustrates the tradeoff between current andfuture consumption.The resource constraint, which is the source of this tradeoff,shows the natural upper limit of economic activity.



Dynamics of consumption

When utility is isoelastic, the Euler equation becomes (details inclass):

1cσt

=1

cσt+1βαkα−1

t+1 + (1− δ)

(1 + n) (1 + g)

or

ct+1 − ct

ct=

[β

(1 + n) (1 + g)

] 1σ [αkα−1

t+1 + (1− δ)] 1

σ − 1 (60)

Consumption per efficiency unit is growing (shrinking) from one

period to the next whenever[

β(1+n)(1+g)

] 1σ[αkα−1

t+1 + (1− δ)] 1

σ> 1

(< 1).



Dynamics of consumption (cont.)

The Euler equation uniquely determines steady state consumption asthe steady state definition ct = ct+1 = . . . = c implies:

1 = βαkα−1 + (1− δ)

(1 + n) (1 + g)

If kt+1 > k :

⇒ βαkα−1

t+1 +(1−δ)(1+n)(1+g) < 1

⇒ ct+1 < ct⇒ Consumption declines

If kt+1 < k :

⇒ βαkα−1

t+1 +(1−δ)(1+n)(1+g) > 1

⇒ ct+1 > ct⇒ Consumption grows



Dynamics of consumption (cont.)

Figure 12 : Consumption dynamics

��

��

∆�� 0



Dynamics of capital

The resource constraint can be rewritten to (details in class):

kt+1 − kt =1

(1 + n) (1 + g)

[f(

kt

)− (n + g + ng + δ) kt − ct

]The resource constraint determines steady state capital as thedefinition kt = kt+1 = . . . = k implies:

c = f(

k)− (n + g + ng + δ) k

When ct > f(

kt

)− (n + g + ng + δ) kt :

⇒ kt+1 < kt⇒ Capital declines

When ct < f(

kt

)− (n + g + ng + δ) kt :

⇒ kt+1 > kt⇒ Capital grows



Dynamics of capital (cont.)

The steady state value of ct is increasing in kt untilfk t = n + g + ng + δ (the golden rule level of capital), and thendecreasing.

If f(

kt

)= (n + g + ng + δ) kt , then ct = 0 (two levels of kt that

satisfy this).



Dynamics of capital (cont.)

Figure 13 : Capital dynamics

��

��

∆�� 0



Dynamics of the economy

Figure 14 : A phase diagram

��

��

∆�� 0

∆�� 0



Dynamics of the economy (cont.)

Figure 15 : The unique equilibrium path

��

��

∆�� 0

��

∆�� 0




Lessons:Any path other than the red line will eventually lead to either ct < 0or kt+1 < 0.These outcomes are not consistent with ct , kt+1 ≥ 0, i.e. therestrictions in both the representative household’s problem andthe the social planner’s problem.The equilibrium path (the red line) therefore covers all possiblecombinations

(ct , kt+1

)that solve these problems, i.e. the

equilibrium path is unique.Movements along the equilibrium path also leads to a uniquesteady state.




Figure 16 : The equilibrium path and the steady state

��

��

∆�� 0

∆�� 0

�

Unique steady state with

�� . . . � and

�� . . . ��



Comparative statics

To facilitate comparative statics, we first write some steady stateidentities from. . . (details in class):

. . . the Euler equation:

1 =αkα−1 + (1− δ)

(1 + ρ) (1 + n) (1 + g)σ

. . . the resource constraint:

c = kα − (n + g + ng + δ) k

The first equation uniquely determines k and, given that k , the secondequation uniquely determines c.




Case 1:Suppose households for some reason care less about the future (theybecome more impatient), reflected by an increase in ρ.

How does this affect the steady state?What about transition dynamics?




By inspecting the Euler equation

1 =αkα−1 + (1− δ)

(1 + ρ) (1 + n) (1 + g)σ,

we conclude that an increase in ρ must be accompanied by adecrease in k (remember α < 1).By inspecting the resource constraint

c = kα − (n + g + ng + δ) k ,

we observe that ρ is not part of this equation, i.e. it remainsunchanged.




Figure 17 : The effects of increased impatience (ρ ↑)

��

��

∆�� 0

∆�� 0

��

��

��




We see from the figure that consumption (in efficiency units)jumps up to a higher level, and then declines gradually until itreaches a lower steady state level.Capital on the other hand declines monotonically towards the newsteady state.Intuitively households who care less about the future shiftconsumption towards the present at the cost of lower futureconsumption.




Case 2:Suppose there is an exogenous increase in the growth rate oftechnology, reflected by an increase in g.

How does this affect the steady state?What about transition dynamics?




By inspecting the Euler equation

1 =αkα−1 + (1− δ)

(1 + ρ) (1 + n) (1 + g)σ,

we conclude that an increase in g must be accompanied by adecrease in k .By inspecting the resource constraint

c = kα − (n + g + ng + δ) k ,

we observe that higher g, for any given level of k , implies adownward shift in c.




Figure 18 : The effects of increased technology growth (g ↑)

��

��

∆�� 0

∆�� 0

��

��

��




The initial jump in consumption can be positive or negative(positive in figure), but it must be followed by a decline towards alower steady state level.Capital declines gradually until it reaches the new steady steadystate level.This is qualitatively the same long run effect as in the Solowmodel.



Some lessons from the Neoclassical growth model

Households face a tradeoff between current and futureconsumption.

More consumption today comes at the cost of less capitalinvestment, and thus less consumption in the future.The household chooses an allocation such that one is indifferentbetween marginally more consumption or investment today.

Firms choose an allocation such that factor prices are equal tomarginal products.The competitive (or decentralized) equilibrium is socially efficientin the sense that a benevolent social planner would choose thesame allocation.There exists a unique and stable equilibrium path in which theeconomy moves along. Consumption is a jump variable whilecapital is a state variable along this path.


Part 3 - The Neoclassical growth model Steady state, the golden rule and the modified golden rule








The steady state

Steady state capital per efficient worker can be derived as (detailsin class):

k =

[α

(1 + ρ) (1 + n) (1 + g)σ − (1− δ)

] 11−α

k depends negatively on ρ,n,g, δ and σ, and positively on α.Steady state output, consumption and investment, all in efficiencyunits, follow:

y = kα

c = kα − (n + g + ng + δ) k

i = (n + g + ng + δ) k




Once we know k , y , c and i , we also know the steady state values ofkt , yt , ct and it :

kt = kEt = k (1 + g)t

yt = yEt = y (1 + g)t

ct = cEt = c (1 + g)t

it = iEt = i (1 + g)t

It is straight forward to show that the per capita variables kt , yt , ctand it all grow along a balanced growth path at the rate g, in linewith SF1 and SF2.It is also straight forward to show that most ratios, such as thecapital-output ratio Kt

Yt, are constant in the steady state. This fits

with SF4.




The steady state savings rate where β < 1 (details in class):General case:

s = α(1 + n) (1 + g)− (1− δ)

(1+n)(1+g)β

− (1− δ)

Special case with δ = 1:s = αβ

Lessons:If households care much about the future (β close to 1), theirsteady state savings rate is large.If the marginal productivity of capital is high (reflected by high α),then future capital is valuable and the steady state savings rate islarge.




What about steady state prices?The real interest rate (details in class):

r = (1 + ρ) (1 + n) (1 + g)σ − (1− δ)

The real wage:wt = (1− α) kα (1 + g)t

Lesson: The real rate is constant, in line with SF3, while the real wagegrows along a balanced growth path at the rate g.



The golden rule and the modified golden rule

Do you remember the golden rule level of capital from the Solowmodel? That was the level of capital (per efficiency unit) thatmaximizes long run consumption (per efficiency unit).It is time to ask what is the level of capital that maximizes steadystate consumption in the Neoclassical model.



The golden rule and the modified golden rule (cont.)

From the steady state resource constraint

c = kα − (n + g + ng + δ) k

we get the golden rule capital level kg (details in class):

kg =

[α

(1 + n) (1 + g)− (1− δ)

] 11−α

(61)

This is the steady state capital level that will yield the highestpossible steady state consumption level.




From the steady state Euler equation

1 = βαkα−1 + (1− δ)

(1 + n) (1 + g)

we get the modified golden rule capital level k (details in class):

k =

α(1+n)(1+g)

β− (1− δ)

11−α

< kg (62)

where the inequality requires β ≡ β (1 + g)1−σ ≡ (1+g)1−σ

1+ρ < 1.The modified golden rule capital level is lower than the golden rulecapital level.




Further insight is gained when we compare saving rates acrossregimes:

One can find the golden rule savings rate sg as (details in class):

sg = α

The savings rate under the modified golden rule is just the steadystate savings rate we found before:

General case:

s = α(1 + n) (1 + g)− (1− δ)

(1+n)(1+g)β

− (1− δ)< sg

Special case with δ = 1:

s = αβ < sg

Evidently people do not save enough to maximize consumption!DB (BI Norwegian Business School) GRA6634 - Economic Growth Fall 2012 123 / 141



The vertical kg-line must cross the (inverted U-shaped)(∆kt+1 = 0)-curve when the latter is horizontal (consumption ismaximized exactly at that point).As k < kg , we see from the figure that c < cg .

Figure 19 : Consumption under the golden rule and the modified golden rule

��

��

∆�� 0

∆�� 0

��

� ��

Golden rule capital level




Why do households in the competitive equilibrium, i.e. the sociallyoptimal solution, choose a steady state consumption level lower thanthe highest consumption level possible? After all, households only getutility from consumption!

Answer:The representative household’s objective is to maximize lifetime(per capita) utility, not lifetime consumption.The representative household is impatient to the extent that β < 1.This impatience should be taken into account by letting thehousehold consume a little more today, at the expense of lowerconsumption in the future.


Part 3 - The Neoclassical growth model Taking stock

Model predictions and empirical performance

The Neoclassical growth model is similar in many aspects to theSolow model.It fits with many stylized facts (in particular SF1-SF5), but not all.In addition to predictions it share with the Solow model, itsuggests:

Expected real interest rates correlate positively with consumptiongrowth rates. Alternatively: Real returns correlate negatively withper capita consumption levels.Capital intensity correlates positively with consumption intensity(think about phase diagram).Impatient countries should consume more and save less thanpatient countries in the long run (think about modified golden rule).



Critique of the Neoclassical growth model

The main critique is about the driver of economic growth, anexogenous process:

Model predicts economies converge to a steady state path alongwhich it grows with a constant rate (growth rate g for per capitavariables). This critical growth rate is determined outside themodel, and independent of production technology, preferences, orgovernmental policy behavior.A growth model where growth is present entirely because weassume growth...Either same growth rate in all economies or different growth ratesabout which model has nothing to say. E.g. evidence that long rungrowth rates are positively correlated with investment share ofoutput (Romer 1989, Lucas 1988).This critique is of course equally relevant for the Solow model.



Critique of the Neoclassical growth model (cont.)

Secondary critique is about the implications for cross-country capitalflows:

A model equation is rt+1 = αkα−1t+1 (countries with low capital

intensities should have high real interest rates).But subject to one world interest rate, capital should flow from richto poor countries (if financial markets are complete).An implication is that all countries should have the same capitalintensity even outside the steady state!Puzzle: This is not reflected well in the data.This critique is more relevant for the Neoclassical growth modelthan for the Solow model.



Summary

The model (in a nutshell). . .. . . is superior to the Solow model in the sense that it has a propermicrofoundation.. . . fits with many of the stylized facts, generally the same as theSolow model.. . . is critized heavily on the basis that its growth engine is nothingmore than an exogenous technological process (totallyindependent of preferences and policy behavior).. . . still serve as a framework for state-of-the-art models in manyfields within modern macroeconomics.



Central aspects to remember

1 How to set up and solve the optimal growth problem (the socialplanner’s problem). The solution mirrors the competitiveequilibrium and is used for drawing the phase diagram, studyingthe steady state, etc.

2 The Euler equation because it determines. . .. . . the slope of optimal consumption growth between periods.. . . the steady state capital level, and therefore also the steady stateoutput, consumption and investment levels (all in efficiency units).

3 How to establish the phase diagram. It is used for studyingdynamics outside the steady state.

4 How to derive steady state expressions. They are used to studychanges to the steady state.

5 The key lesson from the modified golden rule: Long runconsumption is lower because people care less about the future– they consume more today at the cost of future consumption.


Part 4 - Taking stock Modern growth theory








The main problem to address

Both the Solow model and the Neoclassical model are growthmodels where the growth rate itself is assumed exogenous, andtherefore left unexplained.There is a newer strand of economic growth literature (born in thelate 80s), often referred to as endogenous economic growththeory, which aims to address the sources of the long run growthrate.We will only mention this literature very briefly.



The externality model

Example from McCallum (1996):Suppose per capita production is Cobb-Douglas. We assumen = g = 0 to fix things:

yt = kαt kηtwhere kt is economy-wide average capital per capita.Each firm is small and views kt as given. The Euler equation is(assuming σ = δ = 1):

ct+1

ct= αβkα−1

t+1 kηt+1 = αβkα+η−1t+1

where the last equality follows from the assumption that all firmsare identical, implying kt+1 = kt+1.If, by chance, α + η = 1 (the AK-model), then it is possible with anever-growing economy where the growth rate is determined bydeep parameters. Also possible with multiple equilibria.



Other determinants of economic growth

Other model classes are expanding varieties models, models ofSchumpeterian growth, models of directed technological change.Important determinants for economic growth are also political andother institutions. Numerous models are devoted to these aspects(see Acemoglu (2009) for a review).

Economic growth theory seems to have gone through a significantresurgence since the early 90s, and is now one of the main fields inmacroeconomics.


Part 4 - Taking stock Summary








The models you have learned

1 The Solow modelThe original versionThe version with technological progressThe model with human capital

2 The Neoclassical growth model (often referred to as theRamsey-Cass-Koopman model)

Notice:Continuous time versions of these two models are covered extensivelyin the Romer book, discrete time versions are discussed briefly in theObstfeld and Rogoff book (see reading list). The best treatment (by far)of the Neoclassical growth model is in the chapters by Krueger.



Main lessonsSome stylized facts by Kaldor and Romer:

SF1 Output per worker showscontinuing growth "with notendency for a falling rate ofgrowth of productivity".

SF2 Capital per worker showscontinuing growth.

SF3 The rate of return on capitalis steady.

SF4 The capital-output ratio issteady.

SF5 Labor and capital receivesconstant shares of totalincome.

SF6 There are wide differences inthe rate of productivitygrowth across countries.

SF7 In the cross-section, themean growth rate shows no

variation with the level of percapita income.

SF8 The rate of growth of factorinputs is not large enough toexplain the rate of growth ofoutput; that is, growthaccounting always finds aresidual.

SF9 Growth in the volume oftrade is positively correlatedwith growth in output.

SF10 Population growth rates arenegatively correlated with thelevel of income.

SF11 Both skilled and unskilledworkers tend to migratetowards high-incomecountries.



Main lessons (cont.)

The Solow model:Technological progress needed to have steady state per capitagrowth.Key equation (you must know how to derive and interpret it):

kt+1 − kt =1

1 + n + g

[skαt − (n + g + δ) kt

]You should be able to draw and use the figure we studied in class.The convergence hypothesis: Poor countries (those with lowcapital intensity) should grow faster than rich countries.Steady state expressions, you should train on deriving them.Human capital in the model helps targeting certain parameters.




The Neoclassical growth model:The model is, in contrast to the Solow model, microfounded.The representative household’s utility function and budgetconstraint.The representative household’s problem and the firm’s problem incompetitive equilibrium.The Euler equation and the firm’s optimality conditions.The optimal growth problem (social planner’s problem) and itssolution.Key equations (you must know how to derive and interpret them):


t+1 + (1− δ)

(1 + n) (1 + g)

kt+1 − kt =1

(1 + n) (1 + g)

[kαt − (n + g + ng + δ) kt − ct

]DB (BI Norwegian Business School) GRA6634 - Economic Growth Fall 2012 139 / 141



The Neoclassical growth model (cont.):The phase diagram, you should be able to draw the diagram andexplain its foundations (using the two key equations).Steady state expressions, derive them until you know them.The golden rule and the modified golden rule. You should be ableto derive steady state expressions under both rules, and to tellwhy people save less than under the golden rule.As a minimum the Neoclassical model fits SF1-SF5.The main objection is that the key statistic, i.e. the long run growthrate, is determined outside the model and therefore leftunexplained.



Good luck on the midterm and the exam!


Economic Growth - Theory and Evidence

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