the empirical performance of the solow modeldejong/solow model, empirical performance.pdf · 2010....

The Empirical Performance of the Solow Model

David N. DeJongUniversity of Pittsburgh

Econ. 1540, Spring 2010

DND () Empirical Performance Econ. 1540, Spring 2010 1 / 31

Conditional Convergence

A sharp empirical prediction of the Solow Model is that poor countries,relative to their long-run steady states, should grow relatively fast.


Conditional Convergence, cont.

0

0.05

0.1

0.15

0.2

0.25

0.3

1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4

n+ δ+ g versus s � ekα�1,s = α = 0.33, n+ δ+ g = 0.14, ek� = 3.6.



Note that this prediction does not rule out the coexistence of RICH andPOOR countries, since di¤erences in steady states can arise due todi¤erences in savings rates, fertility rates, etc.:

ek� = � sn+ δ+ g

� 11�α

.



Evidence favoring the prediction of conditional convergence is prevalent:

Japan, Germany, etc., post-WWII

Asian Tigers (S. Korea, Thailand, etc., beginning in the 1960s)

Ireland, 1980s (Emerald Tiger)

China, 1990s-today, big time


Accounting for Di¤erences in Income Levels

What does the Solow Model predict in accounting for di¤erences inincome levels?Recall the expression for steady state income (per e¤ective unit of labor)for hypothetical country i :

ey �i =

�Yi ,tAi ,tLi ,t

��=

�si

ni + δi + gi

� αi1�αi

� θαi1�αii .

Note that this is NOT a function of time t. Thus in per captia terms,

y �i ,t =

�Yi ,tLi ,t

��= θ

αi1�αii Ai ,t .


Accounting for Income Di¤erences, cont.

De�ne the per capita income of country i relative to the U.S. as

byi ,t = yi ,tyus ,t

.

The Solow Model predicts this ratio to be

byi ,t = θαi1�αii Ai ,t

θαus1�αusus Aus ,t

.

We will confront this prediction with data under alternative scenarios.



Under all scenarios, we will assume

αi = 0.33, δi + gi = 0.075 8i .

The assumption for α implies

αi1� αi

=1/32/3

=12.

Further, note that

bθi =θiθus

=si

ni + δi + gi/

susnus + δus + gus

=si/sus

(ni + δi + gi ) / (nus + δus + gus )

=bsi

\(ni + 0.075).



Therefore the Solow model carries the following prediction for relativeincomes:

byi ,t = bθi � Ai ,tAus ,t= bθi � bAi ,t=

bsi\(ni + 0.075)

!0.5� bAi ,t ,

where\(ni + 0.075) =

(ni + 0.075)(nus + 0.075)


Scenario 1: Technology and Education are Equal AcrossCountries

Under this scenario,

Ai ,tAus ,t

= 1 8i , byi ,t = bsi

\(ni + 0.075)

!0.5.

For, say, France,

bsi =0.2450.204

= 1.2

\(ni + 0.075) =0.0049+ 0.0750.0096+ 0.075

= 0.944.

Soθiθus

= 1.2706,�θiθus

�0.5= 1.127.


Scenario 1: Technology and Education are Equal AcrossCountries

Bottom Line: France should be 12.7% richer than the U.S.

But in the data, France is only 78.3% as rich as the U.S.


Scenario 1, cont.

This is a general problem for the model:

Under Scenario 1, predicted relative incomes are far greater thanactual incomes.

It turns out that a large aspect of this problem is due to systematicdi¤erences in educational attainment across countries. Since theversions of the model we have studied abstract from attainment, we willdigress to extend the model in this dimension, and �nd a vastimprovement in empirical performance.


Extending the Solow Model to Include EducationalAttainment

Yt = K αt (AtHt )

1�α (1)

Yt = Ct + It (2)

Ct = (1� s)Yt (3)�K t = It � δKt (4)�LtLt

= n (5)

�AtAt

= g (6)

Ht = eψuLt (7)


Extending the Model, cont.

Endogenous: Yt , Ct , It , Kt , t = 1, 2, 3, ...Exogenous: K0, Lt , At , u, t = 0, 1, 2, 3, ...Parameters: α, s, δ, n, g ,ψ

ψ : returns to �schooling�u : time spent �studying� (exogenous for now, endogenous later)


Extending the Model, cont.

Note the role played by ψ : this indicates the impact on H of increasingstudy time u :

∂H∂u

=∂

∂ueψuLt =

∂ψu∂u

eψuLt

= ψH,

thus∆HH= ψ.

I.e., a given change in u results in a percentage change in H of ψ � ∆u.

Empirically, ψ is estimated around 0.1.


Collapsing the Model

If we normalize variables as

eeyt = Yt/AtHt ,etc., the model once again collapses as

eeyt = eekα

t (8)eey t = eec t +eei t (9)eec t = (1� s) eey t (10)�eek t = eei t � (n+ δ+ g)eek t . (11)

Exercise: verify.


Steady State Behavior

The condensed model is once again

�eek t = seekα

t � (n+ δ+ g)eek t ,thus in the steady state:

eek� =

�KtAtHt

��=

�Kt

AteψuLt

��=

�KtAthLt

��=

�s

n+ δ+ g

� 11�α

.

Likewise for output:

eey � =

�s

n+ δ+ g

� α1�α

= θα1�α


Steady State Behavior, cont.

Bottom Line: The introduction of eduacational attainment in the SolowModel has level a¤ects, but not growth a¤ects. That is, di¤erences ineducational attainment is predicted to help account for di¤erences inLEVELS of per capita output, etc., but not the GROWTH RATE of percapita output, etc.


Scenario 2: Educational Attainment Varies AcrossCountries

Given eey � = θα1�α ,

then

y �t = eey � � h � At= θ

α1�α � h � At .

Assuming At is equalized across countries,

byi ,t =

�θiθus

�0.5� bhi ,

bhi = eψuj/eψuus

= eψ(uj�uus )


Scenario 2, cont.

For France,

bhi = e0.1(7.42�11.89)

= 0.64.

Thus

byi ,t =

�θiθus

�0.5� bhi

= 1.127 � 0.64= 0.72,

compared with 0.783 in the data (1997).

Bottom line: accounting for di¤erences in education results in a dramaticimprovement in the model�s performance. The model does well forrelatively rich countries, but not for poor countries.


Scenario 3: Accounting for Di¤erences in TechnologyAcross Countries

If we relax the restriction that technology is identical across countries,then the Solow Model�s prediction for relative incomes is

byi ,t =

�θiθus

�0.5� bhi � Ai ,tAus ,t

=

�θiθus

�0.5� bhi � bAi ,t .

This raises the question: how do we compute Ai ,t ?


Measuring Technological Progress

We can measure Ai ,t indirectly using the Cobb-Douglas productionfunction and observations on (y , k, h) . To see how, start with

Yt = K αt (AthLt )

1�α ,

and divide by Lt :yt = kα

t (Ath)1�α .

Next divide by kαt : �

ytkt

�α

y1�αt = (Ath)

1�α .


Measuring Technological Progress, cont.

Next raise both sides to the power 11�α :�

ytkt

� α1�α

yt = Ath.

Finally, divide by h = eψu :

At =�ytkt

� α1�α

yte�ψu

(This is the equation on p. 60 of the text.) In relative terms:

bAi ,t = byi ,tbki ,t

! α1�α byi ,te�ψ(ui�uus ).


Measuring Technological Progress, cont.

One last problem: data on ki ,t is not available in the Penn World Dataset.These must be constructed using data on ii ,t .To accomplish, begin with the law of motion for capital:

�K t = It � δKt .

Using�K t = Kt+1 �Kt ,

re-write as

Kt+1 = It + (1� δ)Kt= It + 0.94Kt .

Given K1950 and fItg2009t=1950 , we can construct fKtg2009t=1951 .


Measuring Capital, cont.

Pinning down K0 : two approaches.

Approach 1: Assume

gK =Kt+1 �Kt

Kt= gY ,

calculate gY as the sample average of the growth rate of real GDP fordates near time 0, then using

Kt+1 �KtKt

= gY

=It � δKtKt

,

solve for K0 to obtain

K0 =I0

gY + δ.



Approach 2: Use steady state expressions from the Solow Model.Speci�cally,

eek� =

�s

n+ δ+ g

� 11�α

eey � =

�s

n+ δ+ g

� α1�α

,

so eek�eey � =

�s

n+ δ+ g

� 11�α

��

sn+ δ+ g

�� α1�α

=

�s

n+ δ+ g

� 11�α�

α1�α

=

�s

n+ δ+ g

�| {z }2.41 in U.S.

.



Thus

K0 =�

sn+ δ+ g

�Y0


Scenario 3, cont.

Returning to the Solow Model�s prediction for relative incomes:

byi ,t =

�θiθus

�0.5� bhi � Ai ,tAus ,t

=

�θiθus

�0.5� bhi � bAi ,t ,

where

bAi ,t = byi ,tbki ,t

! α1�α byi ,te�ψ(ui�uus ).


Scenario 3, cont.

For France, cirica 1990, we have

byi ,t = 0.83,bki ,t = 1.026

e�ψ(ui�uus ) = 1.56,

thus

bAi ,t =

�0.831.026

�0.5� 0.83 � 1.56

= 1.16


Scenario 3, cont.

Thus regarding the prediction for relative income:

byi ,t = � θiθus

�0.5| {z }

1.127

� bhi|{z}0.64| {z }

0.72

� Ai ,tAus ,t| {z }1.16

| {z }0.835

Compared with the actual value of 0.83, the model�s prediction isremarkably close.


Bottom Line for Deriving Predicted Relative Incomes

byi ,t = �bθi�0.5 � bhi � bAi ,t ,where

bθi =bsi

\(ni + 0.075),

bhi = e�ψ(ui�uus ),

bAi ,t =

byi ,tbki ,t! α

1�α byi ,te�ψ(ui�uus ),

k0 =

�s

n+ δ+ g

�y0,

kt+1 = it + 0.94kt .


the empirical performance of the solow modeldejong/solow model, empirical performance.pdf · 2010....

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