Chapter I- Introduction

Origin of the Report

We were assigned this topic to our group by our honorable course teacher to learn more about
production function for partially fulfilling the objectives of the course Microeconomics. And, this
research is a purely academic research on production function. In economics, a production
function relates physical output of a production process to physical inputs or factors of
production. The production function is one of the key concepts of mainstream neoclassical
theories, used to define marginal product and to distinguish allocative efficiency, the defining
focus of economics. The primary purpose of the production function is to address allocative
efficiency in the use of factor inputs in production and the resulting distribution of income to
those factors, while abstracting away from the technological problems of achieving technical
efficiency, as an engineer or professional manager might understand it. Production function
denotes an efficient combination of inputs and outputs.

In macroeconomics, aggregate production functions are estimated to create a framework in

which to distinguish how much of economic growth to attribute to changes in factor allocation
(e.g. the accumulation of capital) and how much to attribute to advancing technology. Some
non-mainstream economists, however, reject the very concept of an aggregate production
function.

Objectives

The paper has been made on the basis of several objectives.

 To fulfill the course objectives.

 To know more about production function.

 To learn to estimate Cobb-Douglas production function using OLS method.

 Finally, to enhance knowledge.

Scopes

The paper is of immense importance. We got to know the usage of OLS method to estimate
Cobb-Douglas production function. We estimated Cobb-Douglas production function for
Bangladesh and the USA considering GDP as the output. It will help us to have a better
understanding of the economy of these countries.

Limitations
Like every other report, we have some limitations of our study. As we did not get a huge span of
time, there might be some issues we could not cover due to shortage of time. All the data and
information we have provided here are secondary. We did not collect data from primary sources
and conducted the research on the basis the secondary data we collected.

Chapter II- Literature Review

Fitting of Cobb-Douglas Production Functions: Revisited by Prajneshu
The set of Cobb-Douglas production functions is usually fitted by first linearizing the models
through logarithmic transformation and then applying method of least squares. However, this
procedure is valid only when the underlying assumption of multiplicative error-terms is justified.
Unfortunately, this assumption is rarely satisfied in practice and accordingly, the results
obtained are of doubtful nature. Further, nonlinear estimation procedures generally yield
parameter estimates exhibiting extremely high correlations, implying thereby that the
parameters are not estimated independently. In this paper, use of expected-value parameters
has been highlighted and the advantages of their use have also been discussed. Finally, the
developed methodology has been illustrated by applying it to the wheat yield time-series data
of Punjab.

The Cobb-Douglas Production Function: An Antipodean Defence? by Iain Fraser

In this paper a re-examination of the original time-series data sets used by Professor Douglas
and associated researchers to establish the existence of an aggregate production function is
undertaken. Particular attention is paid to the issue of whether the data provide deductive
support for the ‘Laws of Production’ as claimed by Douglas (1948). Various statistical methods
are used to analyse the data to see if the claims of Douglas are justified.

Arguing a Case for Cobb-Douglas Production Function by K. V. Bhanu Murthy

The paper argues that Cobb-Douglas (CD) production function merits use for analysing the
production process, not because it should be looked upon as a simple tool which can be
handled easily or as a crude remedy for estimation, but because of the advantages it possesses.
These advantages are due to the fact that it can handle multiple inputs in its generalised form.
Even in the face of imperfections in the market, it does not introduce distortions of its own.
Unconstrained CD-function further increases its potentialities to handle different scales of
production. Various econometric estimation problems, such as serial correlation,
hetroscidasticity and multicolinearity can be handled adequately and easily. It is argued that
most of its criticism is focussed on its inflexibility and admits that except for one obvious
assumptions all other assumptions can be relaxed. It is further argued that it facilitates
computations and has the properties of explicit representability, uniformity, parsimony and
flexibility. Even the problem of simultaneity can be overcome. The paper argues that aggregate
technology can also be well represented by it.
Specification and Estimation of Cobb-Douglas Production Function Models by A.
Zellner, J. Kmenta and J. Dreze
In this paper, they took up the problem of specifying and estimating a model of a profit
maximizing firm operating with a Cobb-Douglas production function. Their model differs from
the traditional production model considered in the literature, in that they assume that: (a) the
production process is neither instantaneous nor deterministic; and (b) entrepreneurs are aware
of the stochastic nature of production in their profit maximizing endeavors.

Cobb-Douglas Production Function: The Case of a Converging Economy by Dana

Hájková and Jaromír Hurník
The Cobb-Douglas production function is often used to analyse the supply-side performance
and measurement of a country’s productive potential. This functional form, however, includes
the assumption of a constant share of labor in output, which may be too restrictive for a
converging country. For example, labor share in the Czech Republic gradually increased over the
last decade. In this paper, they tested whether this fact renders the application of the Cobb-
Douglas production function unreliable for the Czech economy. They applied a more general
form of production function and allow labor share to develop according to the empirical data.
For the period 1995–2005, they did not find significant difference between the calculation of
the supply side of the Czech economy by the Cobb- -Douglas production function and a more
general production function.
Chapter III- Data Sources & Descriptive Statistics
Bangladesh USA
Labor ( Capital ( Capital (
Year GDP ( Pi ) Li ) Ki ) GDP ( Pi ) Labor ( Li ) Ki )
1980 2.3153E+10 36271231 2.97E+09 5.93355E+12 114426623 1.18E+12
1981 2.4828E+10 37314068 3.65E+09 6.0875E+12 116086918 1.2E+12
1982 2.5358E+10 38414011 3.96E+09 5.97118E+12 117622651 1.13E+12
1983 2.6342E+10 39572437 4.14E+09 6.24778E+12 119002680 1.22E+12
1984 2.7607E+10 40789459 4.55E+09 6.70132E+12 120901964 1.4E+12
1985 2.853E+10 42066363 4.82E+09 6.98537E+12 123155364 1.5E+12
1986 2.9721E+10 43402785 5.09E+09 7.23067E+12 125286805 1.54E+12
1987 3.0842E+10 44803318 5.61E+09 7.48098E+12 127413592 1.57E+12
1988 3.1587E+10 46279964 5.7E+09 7.79547E+12 129518612 1.61E+12
1989 3.2483E+10 47848276 5.98E+09 8.08239E+12 132018718 1.66E+12
1990 3.4309E+10 45674580 6.31E+09 8.23752E+12 131153889.4 1.65E+12
1991 3.5505E+10 46998550 6.47E+09 8.23142E+12 131437031.8 1.59E+12
1992 3.7438E+10 48066630 6.72E+09 8.52408E+12 133025228.5 1.65E+12
1993 3.9201E+10 49155610 7.48E+09 8.75813E+12 134067234.6 1.74E+12
1994 4.0726E+10 50268400 8.17E+09 9.11176E+12 136192353.6 1.84E+12
1995 4.2812E+10 51408980 8.95E+09 9.35951E+12 137994512.5 1.94E+12
1996 4.4749E+10 52535690 1E+10 9.71478E+12 139993681.6 2.09E+12
1997 4.6758E+10 53693860 1.07E+10 1.01507E+13 142528147.4 2.24E+12
1998 4.9178E+10 54876770 1.16E+10 1.06024E+13 144665086 2.44E+12
1999 5.1475E+10 56077320 1.26E+10 1.10991E+13 146800039.9 2.64E+12
2000 5.42E+10 57288130 1.35E+10 1.15533E+13 148722663.9 2.81E+12
2001 5.6952E+10 58857570 1.46E+10 1.16661E+13 149966888.2 2.8E+12
2002 5.9135E+10 60458430 1.57E+10 1.18744E+13 151082493.3 2.75E+12
2003 6.1937E+10 62066180 1.69E+10 1.22077E+13 152029648.2 2.85E+12
2004 6.5183E+10 63655210 1.83E+10 1.26699E+13 153225855.9 3.02E+12
2005 6.9443E+10 65211610 2.01E+10 1.30937E+13 155191428.9 3.19E+12
2006 7.4076E+10 66634470 2.21E+10 1.34429E+13 156990287.3 3.26E+12
2007 7.9305E+10 68031030 2.36E+10 1.3682E+13 158167857.5 3.22E+12
2008 8.4074E+10 69420510 2.6E+10 1.36421E+13 159730505.9 3.06E+12
2009 8.8316E+10 70827370 2.79E+10 1.32634E+13 159692048.1 2.66E+12
2010 9.3236E+10 72274490 3.03E+10 1.35993E+13 159299853.3 2.69E+12
 2011 9.9264E+10 73948480 3.31E+10 1.3817E+13 160635343.6 2.79E+12
2012 1.0574E+11 75642880 3.67E+10 1.41377E+13 161916912.1 2.97E+12
2013 1.121E+11 77357730 3.86E+10 1.44515E+13 163165995.3 3.06E+12
2014 1.1895E+11 79094810 4.24E+10 1.47966E+13 164403409.2 3.19E+12
We know,

Mean, x =
∑x , where ∑x = Summation of elements, n = Number of elements.
n
th th
n n
Median = ()
2 ( )
term +
2
+1 term

2
Mode = The most frequently occurring element.

Bangladesh USA
GDP Labor Capital GDP Labor Capital
55608205.
Mean 54985930414 8 14721309157 1.02915E+13 1.42E+08 2.23E+12
Median 46757686560 53693860 10667505962 1.01507E+13 1.43E+08 2.24E+12
Mode N/A N/A N/A N/A N/A N/A

Comment: For both Bangladesh and the USA, the mean value of GDP is greater than the
median. That means, it is positively skewed and the performance is not so good.
Again,
n

Variance, σ2 =
∑ ( x i−x )2
i=1
n−1

√
n

Standard Deviation, σ = ∑ (x i−x)2

i=1
n−1
σ
Coefficient of Variation, CV = X 100
x

Bangladesh USA
GDP Labor Capital GDP Labor Capital
Variance 7.62596E+20 1.58E+14 1.26E+20 8.28235E+24 2.43E+14 5.14E+23
StDev 27615133354 12582327 1.12E+10 2.87791E+12 15580480 7.17E+11
Mean 54985930414 55608206 1.47E+10 1.02915E+13 1.42E+08 2.23E+12
CV 50.22218074 22.62675 76.29342 27.96385552 10.99981 32.12147

Comment: The coefficients of variation for GDP, labor and capital of Bangladesh are greater than
those of the USA. That mean, the economic situation of Bangladesh is very inconsistent and
risky.
Chapter IV- Methodology & Findings
Cobb-Douglas Production Function
In economics, the Cobb-Douglas functional form of production functions is widely used to
represent the relationship of an output to inputs. It was proposed by Knut Wicksell (1851 -
1926), and tested against statistical evidence by Charles Cobb and Paul Douglas in 1928.
In 1928, Charles Cobb and Paul Douglas published a study in which they modeled the growth of
the American economy during the period 1899 - 1922. They considered a simplified view of the
economy in which production output is determined by the amount of labor involved and the
amount of capital invested. While there are many other factors affecting economic
performance, their model proved to be remarkably accurate.
Let us consider the following Cobb-Douglas production function,
Pi = A 0 L∝i K βi
...................................................................... (1)
where, P = level of output variable, L = quantity of input variable labor, K = quantity of input
variable capital and A 0 is constant. α and β are two parameters, α is called the
output elasticity with respect to input variable labor, β is called the output elasticity of input
variable capital. To estimate α and β , we have to take logarithms on both sides to make it
a linear equation. Taking logarithms on both sides of the equation (1), we have,
ln Pi = ln ( A 0 L∝i K βi )
= ln ( A 0 ) + ln ( L∝i ) + ln ( K iβ )
= ln ( A 0 ) + α ln ( Li ) + β ln ( K i )
This model is known as the double-log transformation.

Let us define, Y i = ln Pi , α 0 = ln A0 , X 1 i = ln ( Li ) and X 2 i = ln ( K i ). So,

the equation will be,
Y i = α 0 + α X 1i + β X 2i

This is a linear equation and we can apply the OLS method to estimate α 0 , α and β .
Ordinary Least Squares (OLS) Method
In statistics, ordinary least squares (OLS) or linear least squares is a method for estimating the
unknown parameters in a linear regression model, with the goal of minimizing the sum of the
squares of the differences between the observed responses (values of the variable being
predicted) in the given dataset and those predicted by a linear function of a set of explanatory
variables. Visually this is seen as the sum of the squared vertical distances between each data
point in the set and the corresponding point on the regression line – the smaller the differences,
the better the model fits the data. The resulting estimator can be expressed by a simple
formula, especially in the case of a single regressor on the right-hand side.
The OLS estimator is consistent when the regressors are exogenous, and optimal in the class of
linear unbiased estimators when the errors are homoscedastic and serially uncorrelated. Under
these conditions, the method of OLS provides minimum-variance mean-unbiased estimation
when the errors have finite variances. Under the additional assumption that the errors
are normally distributed, OLS is the maximum likelihood estimator.
OLS is used in fields as diverse as economics (econometrics), political
science, psychology and electrical engineering (control theory and signal processing).
Y i = α 0 + α X 1i + β X 2i

The equation stated above is a linear equation and we can apply the Ordinary Least Squares
(OLS) method to estimate α 0 , α and β which are given by,
SP ( X 1 , Y ) SS ( X 2 )−SP ( X 2 ,Y ) SP(X 1 , X 2 )
α^ = 2
SS ( X 1) SS ( X 2 ) −[ SP ( X 1 , X 2) ]

SP ( X 2 , Y ) SS ( X 1 )−SP ( X 1 ,Y ) SP(X 1 , X 2 )
^β = 2
SS ( X 1) SS ( X 2 ) −[ SP ( X 1 , X 2) ]

and
α^0 = Ý - α^ X́ 1 - ^β X́ 2

So, the estimated equation will be,

 Y^i = α^0 + α^ X 1i + ^β X 2i

Note:
¿ X1i
¿
SP( X 1 ,y) = n - X́ 1 )( y i - ý )
∑¿
i=1

¿ X2i
¿
SP( X 2 ,y) = n - X́ 2 )( y i - ý )
∑¿
i=1

n
SS( X 1 ) = ∑ ( X 1 i− X́ 1)2
i=1

n
SS( X 2 ) = ∑ ( X 2 i− X́ 2)2
i=1

¿ X1i
¿
SP( X 1 , X 2 ) = n - X́ 1 )( X 2 i - X́ 2 )
∑¿
i=1

For Bangladesh,

Table: Calculation of Y i = ln Pi , X 1 i = ln ( Li ) and X 2 i = ln ( K i ).

X1i X2i
Labor ( Capital ( Y i = ln = ln ( = ln (
Year Country GDP ( Pi ) Li ) Ki ) Pi Li ) Ki )
2315314306 3627123 23.8653963 17.4065 21.8110
1980 Bangladesh 8 1 2967761199 8 4 7
2482802840 3731406 23.9352390 17.4348 22.0169
1981 Bangladesh 2 8 3646082672 3 8 2
2535793992 3841401 23.9563577 17.4639 22.0993
1982 Bangladesh 3 1 3959284556 3 3 3
2634209333 3957243 17.4936 22.1434
1983 Bangladesh 7 7 4137933752 23.994434 4 6
2760738574 4078945 24.0413491 17.5239 22.2383
1984 Bangladesh 5 9 4549907915 7 3 7
2853002862 4206636 24.0742230 17.5547 22.2959
1985 Bangladesh 2 3 4819434295 1 6 2
2972069586 4340278 24.1151094 17.5860 22.3505
1986 Bangladesh 1 5 5089987214 7 3 4
3084187994 4480331 24.1521393 17.6177 22.4484
1987 Bangladesh 2 8 5613753008 4 9 9
1988 Bangladesh 3158709898 4627996 5699753665 24.1760146 17.6502 22.4636
 0 4 1 2 9
3248309298 4784827 24.2039855 17.6835 22.5117
1989 Bangladesh 5 6 5980501589 8 5 7
3430937633 4567458 24.2586845 17.6370 22.5654
1990 Bangladesh 2 0 6310350747 2 5 6
3550513625 4699855 24.2929432 17.6656 22.5897
1991 Bangladesh 9 0 6465535056 1 3 5
3743756918 4806663 24.3459405 22.6290
1992 Bangladesh 0 0 6724640892 6 17.6881 4
3920146336 4915561 24.3919799 22.7348
1993 Bangladesh 0 0 7475379976 1 17.7105 8
4072644985 5026840 24.4301435 17.7328 22.8232
1994 Bangladesh 1 0 8165921452 9 9 4
4281216452 5140898 24.4800881 17.7553 22.9149
1995 Bangladesh 5 0 8950516468 2 2 8
4474852414 5253569 1002025317 23.0278
1996 Bangladesh 2 0 9 24.5243243 17.777 7
4675768656 5369386 1066750596 17.7988 23.0904
1997 Bangladesh 0 0 2 24.5682445 1 7
4917834455 5487677 1163965323 24.6187192 23.1776
1998 Bangladesh 9 0 1 1 17.8206 8
5147505014 5607732 1263910388 24.6643630 17.8422 23.2600
1999 Bangladesh 9 0 1 6 4 6
5419977626 5728813 1353871616 24.7159426 23.3288
2000 Bangladesh 0 0 0 2 17.8636 2
5695165487 5885757 1461297562 24.7654685 17.8906 23.4051
2001 Bangladesh 5 0 3 8 3 8
5913468239 6045843 1569727489 24.8030834 17.9174 23.4767
2002 Bangladesh 2 0 4 3 7 5
6193741052 6206618 1689927080 17.9437 23.5505
2003 Bangladesh 0 0 1 24.8493902 1 4
6518264152 6365521 1828646093 24.9004590 17.9689 23.6294
2004 Bangladesh 8 0 6 4 9 3
6944294308 6521161 2007188513 24.9637712 17.9931 23.7225
2005 Bangladesh 9 0 2 9 5 9
7407608477 6663447 2205658377 25.0283585 18.0147 23.8168
2006 Bangladesh 2 0 5 7 3 8
7930484611 6803103 2363293858 25.0965650 18.0354 23.8859
2007 Bangladesh 2 0 7 7 7 1
8407407282 6942051 2595253654 25.1549640 18.0556 23.9795
2008 Bangladesh 6 0 3 7 9 4
8831571471 7082737 2786994748 18.0757 24.0508
2009 Bangladesh 9 0 3 25.2041839 6 1
9323649172 7227449 3025690310 25.2584050 18.0959 24.1329
2010 Bangladesh 0 0 5 2 8 9
9926365646 7394848 3314974961 25.3210453 18.1188
2011 Bangladesh 2 0 4 4 8 24.2243
7564288 3665292550 25.3842213 18.1415 24.3247
2012 Bangladesh 1.05737E+11 0 0 9 3 6
7735773 3861922365 25.4426185 18.1639 24.3770
2013 Bangladesh 1.12096E+11 0 9 6 5 2
 7909481 4242516797 25.5019833 18.1861 24.4710
2014 Bangladesh 1.18952E+11 0 3 2 6 1
861.480139 623.309 809.569
7 1 5

n n n

Here, Ý =
∑Yi = 24.61371828, X́ 1 = ∑ X1i = 17.80883, X́ 2 = ∑ X2i =
i=1 i=1 i =1
n n n
23.13056.

Table: Calculation of y i = Y i - Ý , x 1i = X1i - X́ 1 and x 2i = X2i - X́ 2 .

x 1i x 2i
Yi = X1i X2i yi = = =
ln = ln ( = ln ( Yi - X1i X2i
Year Country Pi Li ) Ki ) Ý - X́ 1 - X́ 2
- -
Banglade 23.865 17.406 21.811 0.7483 1.3194
1980 sh 4 54 07 2 -0.4023 8
- - -
Banglade 23.935 17.434 22.016 0.6784 0.3739 1.1136
1981 sh 24 88 92 8 5 4
- -
Banglade 23.956 17.463 22.099 0.6573 1.0312
1982 sh 36 93 33 6 -0.3449 3
- - -
Banglade 23.994 17.493 22.143 0.6192 0.3151 0.9870
1983 sh 43 64 46 8 9 9
- -
Banglade 24.041 17.523 22.238 0.5723 0.8921
1984 sh 35 93 37 7 -0.2849 8
- -
Banglade 24.074 17.554 22.295 0.2540 0.8346
1985 sh 22 76 92 -0.5395 7 3
- -
Banglade 24.115 17.586 22.350 0.4986 0.7800
1986 sh 11 03 54 1 -0.2228 2
- - -
Banglade 24.152 17.617 22.448 0.4615 0.1910 0.6820
1987 sh 14 79 49 8 4 7
- -
Banglade 24.176 17.650 22.463 0.1586 0.6668
1988 sh 01 22 69 -0.4377 1 7
 - - -
Banglade 24.203 17.683 22.511 0.4097 0.1252 0.6187
1989 sh 99 55 77 3 9 9
- -
Banglade 24.258 17.637 22.565 0.3550 0.1717
1990 sh 68 05 46 3 8 -0.5651
- - -
Banglade 24.292 17.665 22.589 0.3207 0.1432 0.5408
1991 sh 94 63 75 8 1 1
- - -
Banglade 24.345 17.688 22.629 0.2677 0.1207 0.5015
1992 sh 94 1 04 8 3 1
- - -
Banglade 24.391 17.710 22.734 0.2217 0.0983 0.3956
1993 sh 98 5 88 4 3 8
- - -
Banglade 24.430 17.732 22.823 0.1835 0.0759 0.3073
1994 sh 14 89 24 7 5 2
- - -
Banglade 24.480 17.755 22.914 0.1336 0.0535 0.2155
1995 sh 09 32 98 3 1 8
- - -
Banglade 24.524 23.027 0.0893 0.0318 0.1026
1996 sh 32 17.777 87 9 3 8
- - -
Banglade 24.568 17.798 23.090 0.0454 0.0100 0.0400
1997 sh 24 81 47 7 2 9
Banglade 24.618 17.820 23.177 0.0050 0.0117 0.0471
1998 sh 72 6 68 01 68 26
Banglade 24.664 17.842 23.260 0.0506 0.0334 0.1295
1999 sh 36 24 06 45 1 04
Banglade 24.715 17.863 23.328 0.1022 0.0547 0.1982
2000 sh 94 6 82 24 72 62
Banglade 24.765 17.890 23.405 0.1517 0.0817 0.2746
2001 sh 47 63 18 5 99 18
Banglade 24.803 17.917 23.476 0.1893 0.1086 0.3461
2002 sh 08 47 75 65 34 96
Banglade 24.849 17.943 23.550 0.2356 0.1348 0.4199
2003 sh 39 71 54 72 79 79
Banglade 24.900 17.968 23.629 0.2867 0.1601 0.4988
2004 sh 46 99 43 41 59 7
Banglade 24.963 17.993 23.722 0.3500 0.1843 0.5920
2005 sh 77 15 59 53 16 29
 Banglade 25.028 18.014 23.816 0.4146 0.6863
2006 sh 36 73 88 4 0.2059 2
Banglade 25.096 18.035 23.885 0.4828 0.2266 0.7553
2007 sh 57 47 91 47 42 5
Banglade 25.154 18.055 23.979 0.5412 0.2468 0.8489
2008 sh 96 69 54 46 6 78
Banglade 25.204 18.075 24.050 0.5904 0.2669 0.9202
2009 sh 18 76 81 66 24 58
Banglade 25.258 18.095 24.132 0.6446 0.2871 1.0024
2010 sh 41 98 99 87 49 33
Banglade 25.321 18.118 24.224 0.7073 0.3100 1.0937
2011 sh 05 88 3 27 47 44
Banglade 25.384 18.141 24.324 0.7705 0.3327 1.1942
2012 sh 22 53 76 03 01 02
Banglade 25.442 18.163 24.377 0.3551 1.2464
2013 sh 62 95 02 0.8289 19 59
Banglade 25.501 18.186 24.471 0.8882 0.3773 1.3404
2014 sh 98 16 01 65 25 5

Table: Calculation of SP( X 1 ,Y), SP( X 2 , Y), SS( X 1 ), SS( X 2 ) and SP( X 1 , X 2 ).

Year Country yi x1 i yi x2 i x 1i2 x 2i2 x 1i x 2i

Banglades 0.30104 0.98739 0.16184 1.74103 0.53082
1980 h 8 8 3 7 4
Banglades 0.25371 0.41644
1981 h 8 0.75558 0.13984 1.24019 7
Banglades 0.22672 0.67788 0.11895 1.06343
1982 h 3 9 6 1 0.35567
Banglades 0.19519 0.61129 0.09934 0.97435 0.31112
1983 h 2 2 4 6 2
Banglades 0.16306 0.51065 0.08116 0.79599 0.25418
1984 h 7 9 7 3 2
Banglades 0.13707 0.45028 0.06455 0.69661 0.21205
1985 h 1 2 3 5 9
Banglades 0.11108 0.38892 0.04963 0.60842 0.17378
1986 h 9 3 9 5 6
Banglades 0.03649 0.46522 0.13030
1987 h 0.08818 0.31483 6 2 3
Banglades 0.06942 0.29189 0.02515 0.44471 0.10577
1988 h 5 1 8 4 4
Banglades 0.05133 0.25353 0.01569 0.38289 0.07752
1989 h 4 7 7 7 6
1990 Banglades 0.06098 0.20063 0.02950 0.31933 0.09707
 h 8 8 8 3
Banglades 0.04593 0.17347 0.02050 0.29247 0.07744
1991 h 7 7 8 1 6
Banglades 0.13429 0.01457 0.25151
1992 h 0.03233 4 7 5 0.06055
Banglades 0.02180 0.08773 0.00966 0.03890
1993 h 4 7 9 0.15656 7
Banglades 0.01394 0.05641 0.00576 0.09444
1994 h 2 7 8 7 0.02334
Banglades 0.02880 0.00286 0.04647 0.01153
1995 h 0.00715 8 3 5 5
Banglades 0.00284 0.00917 0.00101 0.01054 0.00326
1996 h 5 9 3 4 8
Banglades 0.00045 0.00182 0.00160 0.00040
1997 h 6 3 0.0001 7 2
Banglades 0.00023 0.00013 0.00222 0.00055
1998 h 5.89E-05 6 8 1 5
Banglades 0.00169 0.00655 0.00111 0.01677 0.00432
1999 h 2 9 6 1 7
Banglades 0.00559 0.02026 0.03930 0.01085
2000 h 9 7 0.003 8 9
Banglades 0.01241 0.04167 0.00669 0.07541 0.02246
2001 h 3 3 1 5 3
Banglades 0.02057 0.06555 0.01180 0.11985 0.03760
2002 h 2 7 1 1 9
Banglades 0.03178 0.09897 0.01819 0.17638 0.05664
2003 h 7 7 2 2 6
Banglades 0.04592 0.14304 0.02565 0.24887 0.07989
2004 h 4 6 1 1 9
Banglades 0.20724 0.03397 0.35049
2005 h 0.06452 1 2 8 0.10912
Banglades 0.08537 0.28457 0.04239 0.47103 0.14131
2006 h 4 6 5 5 3
Banglades 0.10943 0.36471 0.05136 0.57055 0.17119
2007 h 3 8 7 4 4
Banglades 0.13361 0.45950 0.72076 0.20957
2008 h 2 6 0.06094 4 9
Banglades 0.15760 0.07124 0.84687 0.24563
2009 h 9 0.54338 8 4 8
Banglades 0.18512 0.64625 0.08245 1.00487 0.28784
2010 h 1 5 5 2 8
Banglades 0.21930 0.77363 0.09612 1.19627 0.33911
2011 h 4 5 9 5 2
 Banglades 0.25634 0.92013 1.42611 0.39731
2012 h 7 6 0.11069 8 3
Banglades 0.29435 0.12610 1.55365 0.44264
2013 h 8 1.03319 9 9 1
Banglades 0.33516 1.19067 0.14237 1.79680 0.50578
2014 h 5 5 4 7 6
12.7342 1.76096 20.2021 5.94211
3.74119 7 9 1 4

Putting the values of all the terms in equations, we have,

3.74119 X 20.20211−12.73427 X 5.942114
α^ = = -0.33221
1.760969 X 20.20211−(5.942114)2
12.73427 X 1.760969−3.74119 X 5.942114
^β = 2 = 0.728058
1.760969 X 20.20211−(5.942114 )
α^0 = 24.61371828 – (-0.33221 X 17.80883) – (0.728058 X 23.13056) = 13.6896

Therefore, the estimated equation is,

Y^i = 13.6896 - 0.33221 X 1 i + 0.728058 X 2 i

^
A 0 = anti log(13.6896) = 881696.4987

Thus, we have,
^
Pi = 881696.4987 L−0.33221
i K 0.728058
i

Since α^ cannot be negative for an underdeveloped country like Bangladesh, OLS method is
not applicable for this country. After applying AR (Auto Regressive) model, we have,
α^ = 0.243714, ^β = 0.5189 and α^0 = 8.3050

Therefore, the estimated equation is,

Y^i = 8.3050 + 0.243714 X 1 i + 0.5189 X 2 i ; R2 = 0.99710058

^
A 0 = anti log(8.3050) = 4044.042138

Thus, we have,
^
Pi = 4044.042138 L0.243714
i
0.5189
Ki

Comment: From the estimated result, it is found that the output elasticity with respect to labor
is 0.243714 which means that for increasing 100% input labor the output will be increased by
24.3714% while the input capital is constant, and the output elasticity of capital is 0.5189 which
means that for increasing 100% input capital the output will be increased by 51.89% while the
input labor is constant. Since, 0.243714 + 0.5189 is less than 1, the production function is said
to be decreasing return to scale. From the estimated value of R2 , it can be said that 99.7% of
the total variation in the dependent variable GDP is explained by the fitted regression equation
and 0.3% is explained by the random factors. Thus, the fit is very good.

For the USA,

Table: Calculation of Y i = ln Pi , X 1 i = ln ( Li ) and X 2 i = ln ( K i ).

X1i = X2i =
GDP ( Labor ( Capital ( Y i = ln ln ( Li ln ( K i
Year Country Pi ) Li ) Ki ) Pi ) )
5.93E+1 1.14E+0 1.18E+1 29.4116443 18.5554 27.7928
1980 USA 2 8 2 6 4 8
6.09E+1 1.16E+0 18.5698 27.8115
1981 USA 2 8 1.2E+12 29.4372592 5 9
5.97E+1 1.18E+0 1.13E+1 29.4179650 18.5829 27.7563
1982 USA 2 8 2 4 9 1
6.25E+1 1.19E+0 1.22E+1 29.4632477 18.5946 27.8269
1983 USA 2 8 2 3 6 1
1.21E+0 29.5333253 18.6104 27.9670
1984 USA 6.7E+12 8 1.4E+12 2 9 5
6.99E+1 1.23E+0 29.5748385 18.6289 28.0347
1985 USA 2 8 1.5E+12 2 6 6
7.23E+1 1.25E+0 1.54E+1 29.6093527 18.6461 28.0653
1986 USA 2 8 2 1 2 8
7.48E+1 1.27E+0 1.57E+1 29.6433844 18.6629 28.0830
1987 USA 2 8 2 3 5 9
1.61E+1 29.6845642 18.6793
1988 USA 7.8E+12 1.3E+08 2 3 4 28.1064
8.08E+1 1.32E+0 1.66E+1 29.7207088 18.6984 28.1353
1989 USA 2 8 2 4 5 4
8.24E+1 1.31E+0 1.65E+1 29.7397202 18.6918 28.1343
1990 USA 2 8 2 6 8 4
1991 USA 8.23E+1 1.31E+0 1.59E+1 29.7389796 18.6940 28.0920
 2 8 2 2 4 8
8.52E+1 1.33E+0 1.65E+1 29.7739159 18.7060 28.1326
1992 USA 2 8 2 8 5 8
8.76E+1 1.34E+0 1.74E+1 29.8010036 18.7138
1993 USA 2 8 2 4 5 28.1825
9.11E+1 1.36E+0 1.84E+1 29.8405868 18.7295
1994 USA 2 8 2 5 8 28.2418
9.36E+1 1.38E+0 1.94E+1 29.8674139 18.7427 28.2922
1995 USA 2 8 2 1 2 9
9.71E+1 2.09E+1 29.9046698 18.7571 28.3663
1996 USA 2 1.4E+08 2 4 1 1
1.02E+1 1.43E+0 2.24E+1 29.9485622 18.7750 28.4376
1997 USA 3 8 2 1 5 8
1.06E+1 1.45E+0 2.44E+1 29.9920996 18.7899 28.5229
1998 USA 3 8 2 6 3 2
1.11E+1 1.47E+0 2.64E+1 30.0378872 18.8045 28.6030
1999 USA 3 8 2 2 8 8
1.16E+1 1.49E+0 2.81E+1 30.0779938 18.8175
2000 USA 3 8 2 5 9 28.6643
1.17E+1 30.0877063 18.8259 28.6588
2001 USA 3 1.5E+08 2.8E+12 5 3 9
1.19E+1 1.51E+0 2.75E+1 30.1054099 18.8333
2002 USA 3 8 2 9 4 28.6411
1.22E+1 1.52E+0 2.85E+1 30.1330910 18.8395 28.6795
2003 USA 3 8 2 6 9 8
1.27E+1 1.53E+0 3.02E+1 30.1702494 18.8474 28.7359
2004 USA 3 8 2 9 2 8
1.31E+1 1.55E+0 3.19E+1 18.8601 28.7904
2005 USA 3 8 2 30.2031543 7 9
1.34E+1 1.57E+0 3.26E+1 30.2294712 18.8716 28.8120
2006 USA 3 8 2 1 9 4
1.37E+1 1.58E+0 3.22E+1 18.8791 28.7995
2007 USA 3 8 2 30.2471006 7 7
1.36E+1 3.06E+1 30.2441801 28.7499
2008 USA 3 1.6E+08 2 2 18.889 2
1.33E+1 2.66E+1 30.2160323 18.8887 28.6098
2009 USA 3 1.6E+08 2 7 6 5
1.36E+1 1.59E+0 2.69E+1 30.2410363 28.6209
2010 USA 3 8 2 6 18.8863 7
1.38E+1 1.61E+0 2.79E+1 30.2569240 18.8946 28.6572
2011 USA 3 8 2 2 5 7
1.41E+1 1.62E+0 2.97E+1 30.2798695 18.9025 28.7181
2012 USA 3 8 2 9 9 2
 1.45E+1 1.63E+0 3.06E+1 30.3018199 18.9102 28.7481
2013 USA 3 8 2 9 8 6
1.48E+1 1.64E+0 3.19E+1 30.3254215 18.9178 28.7896
2014 USA 3 8 2 4 3 6
656.698 993.261
1047.26059 3 3
n n n

Here, Ý =
∑Yi = 29.92173115, X́ 1 = ∑ X1i = 18.76281, X́ 2 = ∑ X2i =
i=1 i=1 i =1
n n n
28.37889.

Table: Calculation of y i = Y i - Ý , x 1i = X1i - X́ 1 and x 2i = X2i - X́ 2 .

X1i = X2i = yi = x 1i = x 2i =
Y i = ln ln ( Li ln ( Yi - X1i - X2i -
Year Country Pi ) Ki ) Ý X́ 1 X́ 2
29.4116443 18.5554 27.7928
1980 USA 6 4 8 -0.510087 -0.20737 -0.58602
18.5698 27.8115
1981 USA 29.4372592 5 9 -0.484472 -0.19296 -0.5673
29.4179650 18.5829 27.7563
1982 USA 4 9 1 -0.503766 -0.17982 -0.62259
29.4632477 18.5946 27.8269
1983 USA 3 6 1 -0.458483 -0.16815 -0.55198
29.5333253 18.6104 27.9670
1984 USA 2 9 5 -0.388406 -0.15232 -0.41185
29.5748385 18.6289 28.0347
1985 USA 2 6 6 -0.346893 -0.13385 -0.34413
29.6093527 18.6461 28.0653
1986 USA 1 2 8 -0.312378 -0.11669 -0.31352
29.6433844 18.6629 28.0830
1987 USA 3 5 9 -0.278347 -0.09986 -0.2958
29.6845642 18.6793
1988 USA 3 4 28.1064 -0.237167 -0.08347 -0.2725
29.7207088 18.6984 28.1353
1989 USA 4 5 4 -0.201022 -0.06436 -0.24355
29.7397202 18.6918 28.1343
1990 USA 6 8 4 -0.182011 -0.07093 -0.24456
29.7389796 18.6940 28.0920
1991 USA 2 4 8 -0.182752 -0.06877 -0.28681
29.7739159 18.7060 28.1326
1992 USA 8 5 8 -0.147815 -0.05676 -0.24621
29.8010036 18.7138
1993 USA 4 5 28.1825 -0.120728 -0.04896 -0.19639
29.8405868 18.7295
1994 USA 5 8 28.2418 -0.081144 -0.03323 -0.13709
 29.8674139 18.7427 28.2922
1995 USA 1 2 9 -0.054317 -0.02009 -0.0866
29.9046698 18.7571 28.3663
1996 USA 4 1 1 -0.017061 -0.0057 -0.01259
29.9485622 18.7750 28.4376 0.026831 0.05878
1997 USA 1 5 8 1 0.01224 3
29.9920996 18.7899 28.5229 0.070368 0.02712 0.14402
1998 USA 6 3 2 5 2 1
30.0378872 18.8045 28.6030 0.116156 0.04177 0.22418
1999 USA 2 8 8 1 2 3
30.0779938 18.8175 0.156262 0.05478
2000 USA 5 9 28.6643 7 4 0.28541
30.0877063 18.8259 28.6588 0.165975 0.06311 0.27999
2001 USA 5 3 9 2 5 8
30.1054099 18.8333 0.183678 0.07052 0.26220
2002 USA 9 4 28.6411 8 7 3
30.1330910 18.8395 28.6795 0.211359 0.07677 0.30068
2003 USA 6 9 8 9 6 1
30.1702494 18.8474 28.7359 0.248518 0.08461 0.35708
2004 USA 9 2 8 3 4 9
18.8601 28.7904 0.281423 0.41159
2005 USA 30.2031543 7 9 1 0.09736 8
30.2294712 18.8716 28.8120 0.307740 0.10888 0.43314
2006 USA 1 9 4 1 5 8
18.8791 28.7995 0.325369 0.11635 0.42067
2007 USA 30.2471006 7 7 4 7 2
30.2441801 28.7499 0.12618 0.37102
2008 USA 2 18.889 2 0.322449 9 8
30.2160323 18.8887 28.6098 0.294301 0.12594 0.23095
2009 USA 7 6 5 2 8 9
30.2410363 28.6209 0.319305 0.12348 0.24207
2010 USA 6 18.8863 7 2 9 9
30.2569240 18.8946 28.6572 0.335192 0.13183 0.27837
2011 USA 2 5 7 9 7 7
30.2798695 18.9025 28.7181 0.358138 0.13978
2012 USA 9 9 2 4 4 0.33923
30.3018199 18.9102 28.7481 0.380088 0.14746
2013 USA 9 8 6 8 9 0.36927
30.3254215 18.9178 28.7896 0.403690 0.15502 0.41076
2014 USA 4 3 6 4 4 9

Table: Calculation of SP( X 1 ,Y), SP( X 2 , Y), SS( X 1 ), SS( X 2 ) and SP( X 1 , X 2 ).
2 2
Year Country yi x1 i yi x2 i x 1i x 2i x 1i x 2i
0.10577 0.34341
1980 USA 4 0.29892 0.043 7 0.12152
 0.09348 0.27484 0.03723 0.32183 0.10946
1981 USA 4 3 4 3 7
0.09058 0.31363 0.03233 0.38761 0.11195
1982 USA 6 8 4 4 2
0.07709 0.25307 0.02827 0.30468 0.09281
1983 USA 6 5 6 6 8
0.05916 0.15996 0.02320 0.16961 0.06273
1984 USA 2 4 1 9 2
0.04643 0.11937 0.01791 0.11842 0.04606
1985 USA 3 8 7 8 3
0.03645 0.09793 0.01361 0.09829 0.03658
1986 USA 3 6 7 3 6
0.02779 0.08233 0.00997 0.02953
1987 USA 6 6 2 0.0875 9
0.01979 0.06462 0.00696 0.07425 0.02274
1988 USA 7 7 8 4 7
0.01293 0.00414 0.05931 0.01567
1989 USA 7 0.04896 2 9 4
0.04451 0.00503 0.05980 0.01734
1990 USA 0.01291 2 1 8 6
0.01256 0.05241 0.01972
1991 USA 8 5 0.00473 0.08226 4
0.03639 0.00322 0.06062 0.01397
1992 USA 0.00839 4 2 1 5
0.00591 0.00239 0.03857 0.00961
1993 USA 1 0.02371 7 1 5
0.00269 0.01112 0.00110 0.01879 0.00455
1994 USA 7 4 4 4 6
0.00109 0.00470 0.00040 0.00173
1995 USA 1 4 3 0.0075 9
0.00021 0.00015
1996 USA 9.73E-05 5 3.25E-05 8 7.18E-05
0.00032 0.00157 0.00345
1997 USA 8 7 0.00015 5 0.00072
0.00190 0.01013 0.00073 0.02074 0.00390
1998 USA 9 5 6 2 6
0.00485 0.00174 0.05025 0.00936
1999 USA 2 0.02604 5 8 5
0.00856 0.04459 0.00300 0.08145 0.01563
2000 USA 1 9 1 9 6
0.01047 0.04647 0.00398 0.07839 0.01767
2001 USA 6 3 4 9 2
2002 USA 0.01295 0.04816 0.00497 0.06875 0.01849
 4 1 4 2
0.01622 0.06355 0.00589 0.09040 0.02308
2003 USA 7 2 5 9 5
0.02102 0.08874 0.00715 0.12751 0.03021
2004 USA 8 3 9 2 5
0.02739 0.11583 0.00947 0.16941 0.04007
2005 USA 9 3 9 3 3
0.03350 0.13329 0.01185 0.18761 0.04716
2006 USA 8 7 6 8 3
0.03785 0.13687 0.01353 0.17696 0.04894
2007 USA 9 4 9 5 8
0.04068 0.11963 0.01592 0.13766 0.04681
2008 USA 9 7 4 1 9
0.03706 0.06797 0.01586 0.05334 0.02908
2009 USA 7 2 3 2 9
0.03943 0.07729 0.05860 0.02989
2010 USA 1 7 0.01525 2 4
0.04419 0.01738 0.07749 0.03670
2011 USA 1 0.09331 1 4 1
0.05006 0.12149 0.11507 0.04741
2012 USA 2 1 0.01954 7 9
0.05605 0.14035 0.02174 0.05445
2013 USA 1 5 7 0.13636 6
0.06258 0.16582 0.02403 0.16873 0.06367
2014 USA 2 4 2 1 9
1.11835 3.38792 0.42583 4.03492 1.27945
4 1 3 4 7

Putting the values of all the terms in equations, we have,

1.118354 X 4.034924−3.387921 X 1.279457
α^ = = 2.189478
0.425833 X 4.034924−( 1.279457)2
3.387921 X 0.425833−1.118354 X 1.279457
^β = = 0.145375
0.425833 X 4.034924−(1.279457)2
α^0 = 29.92173115 – (2.189478 X 18.76281) – (0.145375 X 28.37889) = -15.2846

Therefore, the estimated equation is,

Y^i = -15.2846 + 2.189478 X 1 i + 0.145375 X 2 i ; R2 = 0.998158

^
A 0 = anti log(-15.2846) = 2.30132 X 10−7
Thus, we have,
^
Pi = 2.30132 X 10−7 L2.189478
i
0.145375
Ki

Comment: From the estimated result, it is found that the output elasticity with respect to labor
is 2.189478 which means that for increasing 100% input labor the output will be increased by
218.94% while the input capital is constant, and the output elasticity of capital is 0.145375
which means that for increasing 100% input capital the output will be increased by 14.53%
while the input labor is constant. Since, 2.189478 + 0.145375 is greater than 1, the production
function is said to be increasing return to scale. From the estimated value of R2 , it can be
said that 99.8% of the total variation in the dependent variable GDP is explained by the fitted
regression equation and 0.2% is explained by the random factors. Thus, the fit is very good.