the empirical performance of the solow modeldejong/solow model, empirical performance.pdf · 2010....
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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.
DND () Empirical Performance Econ. 1540, Spring 2010 2 / 31
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.
DND () Empirical Performance Econ. 1540, Spring 2010 3 / 31
Conditional Convergence, cont.
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
.
DND () Empirical Performance Econ. 1540, Spring 2010 4 / 31
Conditional Convergence, cont.
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
DND () Empirical Performance Econ. 1540, Spring 2010 5 / 31
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 .
DND () Empirical Performance Econ. 1540, Spring 2010 6 / 31
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.
DND () Empirical Performance Econ. 1540, Spring 2010 7 / 31
Accounting for Income Di¤erences, cont.
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).
DND () Empirical Performance Econ. 1540, Spring 2010 8 / 31
Accounting for Income Di¤erences, cont.
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)
DND () Empirical Performance Econ. 1540, Spring 2010 9 / 31
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.
DND () Empirical Performance Econ. 1540, Spring 2010 10 / 31
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.
DND () Empirical Performance Econ. 1540, Spring 2010 11 / 31
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.
DND () Empirical Performance Econ. 1540, Spring 2010 12 / 31
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)
DND () Empirical Performance Econ. 1540, Spring 2010 13 / 31
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)
DND () Empirical Performance Econ. 1540, Spring 2010 14 / 31
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.
DND () Empirical Performance Econ. 1540, Spring 2010 15 / 31
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.
DND () Empirical Performance Econ. 1540, Spring 2010 16 / 31
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
DND () Empirical Performance Econ. 1540, Spring 2010 17 / 31
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.
DND () Empirical Performance Econ. 1540, Spring 2010 18 / 31
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 )
DND () Empirical Performance Econ. 1540, Spring 2010 19 / 31
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.
DND () Empirical Performance Econ. 1540, Spring 2010 20 / 31
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 ?
DND () Empirical Performance Econ. 1540, Spring 2010 21 / 31
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 .
DND () Empirical Performance Econ. 1540, Spring 2010 22 / 31
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 ).
DND () Empirical Performance Econ. 1540, Spring 2010 23 / 31
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 .
DND () Empirical Performance Econ. 1540, Spring 2010 24 / 31
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.
DND () Empirical Performance Econ. 1540, Spring 2010 25 / 31
Measuring Capital, cont.
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.
.
DND () Empirical Performance Econ. 1540, Spring 2010 26 / 31
Measuring Capital, cont.
Thus
K0 =�
sn+ δ+ g
�Y0
DND () Empirical Performance Econ. 1540, Spring 2010 27 / 31
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 ).
DND () Empirical Performance Econ. 1540, Spring 2010 28 / 31
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
DND () Empirical Performance Econ. 1540, Spring 2010 29 / 31
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.
DND () Empirical Performance Econ. 1540, Spring 2010 30 / 31
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 .
DND () Empirical Performance Econ. 1540, Spring 2010 31 / 31