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.

• If you receive a compound interest rate, rr(n), you may receive your extra income in increments which appear n times per year.

11 0

1

(1 ) (1 )t t tt t t

t

B BB r B B r B r

B

1 0(1 ) (1 )n ntt t t

rr rrB B B B

n n

• Define the compound interest rate with continuous compounding ρ = rr(n→∞), we calculate growth with the anti-log, e,

• Refer to the log-difference as the continuous growth rate,

1 0

01

ln lnln ln

tt t t

tt t

B e B B e B

B BB B

t

• 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

2. Log of exponent

3. Derivative of logarithm is the inverse

Natural Logarithmic Function

ln( ) ln( ) ln( )X Y X Y

ln( ) ln( )X X

ln( ) 1d X

dX X

Growth Accounting

• To study growth, take derivative of production function with respect to time. By the chain rule,

dY F F dK F dL

dt t K dt L dt

Notation for dX

Xdt

F F FY K Lt K L

Y Y YY K LK L

t t

Y TFP K L

Y TFP K L

CRTS implies βt = 1-αt, Price taking implies 1-αt is labor’s share of income.

Continuous Rate of Change

• The time derivative of the logarithm of time dependent variable is the variables’ continuous rate of change.

• Most economic series are observed only intermittently. We approximate the continuous rate of change with the first difference of the natural log:

ln( )td X X

dt X

1

lnln ln ln t

t t t

d XX X X

dt

Cobb-Douglas• Cobb-Douglas function is log-linear

• Easy to do Growth Accounting because factor intensities are constant. TFP growth is proportional to technology growth.

1

ln ln (1 ) ln (1 ) ln

( ) ( )

t t t t

t t t t

Y K A L

Y K AL

(1 ) (1 )Y TFP K L TFP A

Y TFP K L TFP A

TFP Growth• TFP is log linear

1 [ ]

ln (1 ) ln ln

ln (

ln ln (1 ) ln ( ) l

1 ) ln l

n

n ( ) ln ln

t t

t

t

a a

t t t tt t t t

t t t t

t t t t

t t t t t t

t

Y Y Y YTFP TFP a a

L K L K

TFP

TFP Y a L a

L K

K

a Y a Y

• TFP growth rate is the gap between GDP growth rate and the weighted average of the growth rate of the factors of production.

1

1 1 1

ln ln

ln ln (1 )[ln l

(1

n ] [ln ln

)

]

TFP Y L

t t

t t t t t t

Kt t t t t

t t

t

TFP TFP

Y Y a L L a K

a a

K

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 = used as weights.

[1 ]

[1 ]t ta a

t t t tt t

t t t t

Y Y W LTFP a

L K PY

WLPY

• Since we can write

• Under Cobb-Douglas, the level of TFP is a geometrically weighted function of capital productivity and labor productivity.

Cobb-Douglas TFP

(1 )TFP A

TFP A

1t tTFP A

1 1

1

1

( )

( )

t t t t t t

t tt t

t t

Y K L A Y Y

Y YA TFP

L K

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

GDP Growth 1980-1984

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

Ireland Europe

Ave

rag

e C

on

tin

uo

us

Gro

wth

Rat

e

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

• Approximate labor intensity with the average of labor share of income in the previous year and the current year.

ln ln ln (1 ) lnt t t t t tY TFP K L

1 1

1 1

1 2t t t tt

t t t t

W L W L

PY P Y

Irish Data

GDP Gross fixed Total hours capital stock

2000 prices 2000 prices(millions of Euros) (millions of Euros) (in millions) 1-α

1980 32,509 55,335 2,230 0.7661981 33,565 59,156 2,191 0.7671982 34,295 62,980 2,174 0.7421983 33,750 65,906 2,146 0.7331984 35,175 67,943 2,097 0.7301985 36,260 69,383 2,084 0.7091986 36,015 70,473 2,114 0.6951987 37,882 71,589 2,114 0.6901988 40,108 72,984 2,111 0.6811989 42,661 74,877 2,120 0.6621990 46,536 77,609 2,212 0.6501991 47,353 80,397 2,170 0.6541992 48,938 82,537 2,130 0.6631993 50,236 84,429 2,151 0.6671994 53,265 86,318 2,226 0.6581995 58,716 88,692 2,334 0.6371996 63,786 92,067 2,422 0.6121997 71,028 96,841 2,465 0.5891998 77,554 103,247 2,558 0.5741999 86,526 111,183 2,677 0.5592000 95,398 119,886 2,806 0.5462001 101,131 128,007 2,871 0.5422002 107,334 135,394 2,902 0.5292003 111,255 142,310 2,855 0.5222004 116,729 149,178 2,903 0.526

GrowthAccounting

ΔlnY ΔlnK ΔlnL 1-α

1981 0.032 0.067 -0.017 0.7661982 0.022 0.063 -0.008 0.7541983 -0.016 0.045 -0.013 0.7381984 0.041 0.030 -0.023 0.7321985 0.030 0.021 -0.006 0.7201986 -0.007 0.016 0.014 0.7021987 0.051 0.016 0.000 0.6921988 0.057 0.019 -0.002 0.6851989 0.062 0.026 0.004 0.6721990 0.087 0.036 0.043 0.6561991 0.017 0.035 -0.019 0.6521992 0.033 0.026 -0.019 0.6591993 0.026 0.023 0.010 0.6651994 0.059 0.022 0.034 0.6631995 0.097 0.027 0.047 0.6471996 0.083 0.037 0.037 0.6241997 0.108 0.051 0.018 0.6001998 0.088 0.064 0.037 0.5811999 0.109 0.074 0.046 0.5662000 0.098 0.075 0.047 0.5532001 0.058 0.066 0.023 0.5442002 0.060 0.056 0.011 0.5362003 0.036 0.050 -0.016 0.5262004 0.048 0.047 0.017 0.524

Average 0.053 0.041 0.011 0.644

Contributionto Growth

αΔlnK (1-α)ΔlnL ΔlnTFP

1981 0.016 -0.013 0.0301982 0.015 -0.006 0.0121983 0.012 -0.010 -0.0181984 0.008 -0.017 0.0501985 0.006 -0.004 0.0291986 0.005 0.010 -0.0211987 0.005 0.000 0.0461988 0.006 -0.001 0.0521989 0.008 0.003 0.0511990 0.012 0.028 0.0471991 0.012 -0.013 0.0181992 0.009 -0.012 0.0361993 0.008 0.007 0.0121994 0.007 0.023 0.0281995 0.010 0.031 0.0571996 0.014 0.023 0.0461997 0.020 0.011 0.0771998 0.027 0.021 0.0401999 0.032 0.026 0.0522000 0.034 0.026 0.0382001 0.030 0.013 0.0162002 0.026 0.006 0.0282003 0.024 -0.009 0.0212004 0.022 0.009 0.017

Average 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

So we only need to know start and stop values to calculate Growth Accounting. Results, not much different.

0ln( ) ln( )TX X

T

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 ΔlnK ΔlnL

0.053 0.041 0.011

αΔlnK (1-α)ΔlnL ΔlnTFP

0.014 0.007 0.032

Item for DiscussionEast 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.