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Total Factor Productivity
Analytical Exercises
Simple vs. Compound Interest Rate
• If you have a time deposit and receive a
simple interest rate, then, after 1 year, your
account will increase by the interest rate.
Bt  (1  r ) Bt 1
Bt  (1  r ) B0
t
Bt  Bt 1
r
Bt 1
• If you receive a compound interest rate, rr(n),
you may receive your extra income in
increments which appear n times per year.
rr n
Bt  (1  ) Bt 1
n
rr nt
Bt  (1  ) B0
n
• Define the compound interest rate with
continuous compounding ρ = rr(n→∞), we
calculate growth with the anti-log, e,
Bt  e  Bt 1
Bt  e  t B0
ln Bt  ln B0
 ln Bt  ln Bt 1   
t
• Refer to the log-difference as the
continuous growth rate,
Natural Logarithmic Function
•
Natural Log Function has a number of useful
(for the study of productivity) properties
1. Log of a Product is the sum of the logs
ln( X  Y ) ln( X )  ln(Y )
2. Log of exponent
ln( X  )   ln( X )
3. Derivative of logarithm is the inverse
d ln( X ) 1

dX
X
Growth Accounting
• To study growth, take derivative of production
function with respect to time. By the chain rule,
Notation for
dX
X
dt
CRTS implies
βt = 1-αt,
Price taking
implies 1-αt is
labor’s share
of income.
dY F F dK F dL



dt
t K dt L dt
F
Y t

Y
Y
F
 K
F
K L

Y K
Y
K
L
L
L
Y TFP
K
L

  t  t
Y TFP
K
L
Continuous Rate of Change
• The time derivative of the logarithm of time
dependent variable is the variables’ continuous
rate of change. d ln( X t ) X
dt

X
• Most economic series are observed only
intermittently. We approximate the continuous
rate of change with the first difference of the
natural log:
d ln X
 ln X t  ln X t  ln X t 1 
t
dt
Cobb-Douglas
• Cobb-Douglas function is log-linear
Yt  ( Kt

)
1


 ( At Lt ) 
ln Yt   ln K t  (1   ) ln At  (1   ) ln Lt
• Easy to do Growth Accounting because factor
intensities are constant. TFP growth is
proportional to technology growth.
Y TFP
K
L

   (1   )
Y TFP
K
L
TFP
A
 (1   )
TFP
A
TFP Growth
• TFP is log linear
1 at
 Yt 
TFPt   
 Lt 
[ at ]
 Yt 
 
 Kt 
Yt
Yt
 ln TFPt  (1  at ) ln  at ln
Lt
Kt
ln TFPt  (1  at ) ln Yt  ln Lt   (at ) ln Yt  ln Kt 
ln TFPt  ln Yt  (1  at ) ln Lt  (at ) ln Kt
• TFP growth rate is the gap between GDP
growth rate and the weighted average of the
growth rate of the factors of production.
ln TFPt  ln TFPt 1
 ln Yt  ln Yt 1  (1  at )[ln Lt  ln Lt 1 ]  at [ln K t  ln K t 1 ]
tTFP  tY   at tL   (1  at ) tK 
Total Factor Productivity
• Total factor productivity measures the total
effectiveness of an economy in applying all of its
factors of production.
• TFP is a geometrically weighted average of
capital and labor productivity with factor
intensity, at and 1-at = WL PY used as weights.
[1 at ]
 Yt 
TFPt   
 Lt 
 Yt 
 
 Kt 
at
Wt Lt
[1  at ] 
PY
t t
Cobb-Douglas TFP
• Since
TFP
A
 (1   )
TFP
A
1
t
we can write TFPt  A
• Under Cobb-Douglas, the level of TFP is a
geometrically weighted function of capital
productivity and labor productivity.
Yt   K t  ( Lt At )1  Yt  Yt1

1
 Yt 
 
 Lt 

 Yt 
1

(
A
)
 TFPt
 
t
 Kt 
Growth Accounting Exercise: Celtic
Tiger
• Economy of Ireland has been one of the most
successful in Europe in recent years
transforming from one of the poorer countries
of Europe to one of the richest over a period of
about 25 years.
• Growth accounting is a technique that
economists use as a first step in explaining the
sources of growth.
Growth Accounting
•
The growth accounting equation divides the
sources of growth into 3 parts
1. Growth due to labor (i.e. growth in labor weighted
by labor intensity, βt)
2. Growth due to capital (i.e. growth in capital
weighted by capital intensity, αt)
3. Total Factor Productivity Growth (i.e. change in
the production function itself).
Economic Growth Rates
Average Continuous Growth Rate
GDP Growth 1980-1984
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
Ireland
Europe
Data
• Question: What part of Irish growth is derived
from labor growth, capital growth, or TFP
growth?
• Irish data is from " Marcel P. Timmer, Gerard
Ypma and Bart van Ark (2003), IT in the
European Union: Driving Productivity
Divergence?, GGDC Research Memorandum
GD-67 (October 2003), University of
Groningen, Appendix Tables
Use approximation of growth
accounting function
• For every year, assume CRTS and write the
approximate equation
 ln Yt   ln TFPt  t  ln Kt  (1  t ) ln Lt
• Approximate labor intensity with the average
of labor share of income in the previous year
and the current year.


Wt Lt Wt 1Lt 1
1  t  

 2
Pt 1Yt 1 
t t
 PY
GDP
Irish
Data
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2000 prices
(millions of Euros)
32,509
33,565
34,295
33,750
35,175
36,260
36,015
37,882
40,108
42,661
46,536
47,353
48,938
50,236
53,265
58,716
63,786
71,028
77,554
86,526
95,398
101,131
107,334
111,255
116,729
Gross fixed
capital stock
2000 prices
(millions of Euros)
55,335
59,156
62,980
65,906
67,943
69,383
70,473
71,589
72,984
74,877
77,609
80,397
82,537
84,429
86,318
88,692
92,067
96,841
103,247
111,183
119,886
128,007
135,394
142,310
149,178
Total hours
(in millions)
2,230
2,191
2,174
2,146
2,097
2,084
2,114
2,114
2,111
2,120
2,212
2,170
2,130
2,151
2,226
2,334
2,422
2,465
2,558
2,677
2,806
2,871
2,902
2,855
2,903
1-α
0.766
0.767
0.742
0.733
0.730
0.709
0.695
0.690
0.681
0.662
0.650
0.654
0.663
0.667
0.658
0.637
0.612
0.589
0.574
0.559
0.546
0.542
0.529
0.522
0.526
Growth
Accounting
ΔlnY
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
Average
Δ lnK
Δ lnL
1-α
0.032
0.022
-0.016
0.041
0.030
-0.007
0.051
0.057
0.062
0.087
0.017
0.033
0.026
0.059
0.097
0.083
0.108
0.088
0.109
0.098
0.058
0.060
0.036
0.048
0.067
0.063
0.045
0.030
0.021
0.016
0.016
0.019
0.026
0.036
0.035
0.026
0.023
0.022
0.027
0.037
0.051
0.064
0.074
0.075
0.066
0.056
0.050
0.047
-0.017
-0.008
-0.013
-0.023
-0.006
0.014
0.000
-0.002
0.004
0.043
-0.019
-0.019
0.010
0.034
0.047
0.037
0.018
0.037
0.046
0.047
0.023
0.011
-0.016
0.017
0.766
0.754
0.738
0.732
0.720
0.702
0.692
0.685
0.672
0.656
0.652
0.659
0.665
0.663
0.647
0.624
0.600
0.581
0.566
0.553
0.544
0.536
0.526
0.524
0.053
0.041
0.011
0.644
Contribution
to Growth
αΔlnK
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
Average
(1-α)Δ lnL
Δ lnTFP
0.016
0.015
0.012
0.008
0.006
0.005
0.005
0.006
0.008
0.012
0.012
0.009
0.008
0.007
0.010
0.014
0.020
0.027
0.032
0.034
0.030
0.026
0.024
0.022
-0.013
-0.006
-0.010
-0.017
-0.004
0.010
0.000
-0.001
0.003
0.028
-0.013
-0.012
0.007
0.023
0.031
0.023
0.011
0.021
0.026
0.026
0.013
0.006
-0.009
0.009
0.030
0.012
-0.018
0.050
0.029
-0.021
0.046
0.052
0.051
0.047
0.018
0.036
0.012
0.028
0.057
0.046
0.077
0.040
0.052
0.038
0.016
0.028
0.021
0.017
0.015
0.006
0.032
Growth Accounting Results
• Growth Accounting suggests that
approximately 30% of the growth in Irish GDP
is due to growth in the capital stock about 10%
is due to growth in the labor input and about
60% is due to TFP growth.
• Of course, this brings up the question, why did
TFP grow so much.
– Note, we also did not adjust labor for quality
improvements.
Cobb-Douglass
• Easier to calculate if we assume constant TFP growth.
Previous slide shows average (1-α) ≈2/3. Assume
Cobb-Douglas production function and use this value.
• Note average of ΔlnX from period 1 to T is
ln( X T )  ln( X 0 )
T
So we only need to know start and stop values to
calculate Growth Accounting. Results, not much
different.
Average TFP Growth Cobb Douglas
Q
K
L
1980
32,509
55,335
2,230
2001
116,729
149,178
2,903
Average Continuous Growth Rate
1-α = 2/3
ΔlnQ
0.053
ΔlnK
0.041
αΔlnK
0.014
ΔlnL
0.011
(1-α)ΔlnL
0.007
ΔlnTFP
0.032
Item for Discussion
East Asian Miracle
• Another set of miracle economies over the last
30 years have been the economies of East and
Southeast Asia.
• What is the source of growth in these
economies?
• What are the implications for future growth.