Cobb-Douglas Production Function -Output Generated by Capital
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Pls provide the Computation of the Total Factor Productivity (TFP) for the years 1993 to 2015 and Conclusion.
Thank you in advance.
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Solution Summary
This solution provides an outline and example of calculating total factor productivity (TFP), which is also known as A, from a 1993-2015 dataset. In this example, GDP represents Y, capital formation represents K, and total employment represents L. The properties of logarithmic functions and the steps to run a regression model in Excel are reviewed to determine the TFP based on estimating the coefficients (i.e. betas) of both capital and labour inputs. The approach in this solution used is one that goes over every step in detail, with the assumption that one has a zero knowledge at the beginning in determining the TFP.
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Calculating Total Factor Productivity (A) in a Cobb-Douglas Production Function (Output Generated by Capital and Labour Inputs)
I used a Cobb-Douglas production function for the productivity calculation because this model demonstrates a relationship between output and two key inputs-capital (K) and labour (L)- which are featured in your example. In addition, this model takes into account total factor productivity, represented as A below.
The typical Cobb-Douglas production model looks like this:
Y=A*(L^Beta 1)*(K^Beta 2)
(Note: Beta 1 can also be known as alpha and Beta 2 referred to as beta; nevertheless, we use Beta 1 and Beta for simplicity)
Essentially, this equation above highlights that Y is impacted by total factor productivity, labour (dependent on its elasticity Beta 1 with respect to Y), and capital (dependent on its elasticity Beta 2 with respect to Y). Using the labels from your example, Y represents GDP, K represents capital formation, and L represents total employment. In order to find out A, we would need to estimate the elasticities for both labour and capital with respect to output (the Beta 1 and Beta 2 values). By using the properties of natural logarithmic functions (ln) to linearize the Cobb-Douglas function, we will use a regression model to provide estimations for Beta 1, Beta 2, and thus find out A.
Steps:
1) Transform the Cobb-Douglas function into a ln function. To do this, we must remember two rules: i) the power rule and ii) the product rule. For the power rule, recall that ln(x^a) = a*ln(x), bringing the a exponent down in front of the ln function. Also, for the product rule, recall that ln(y) = ln (x*y*z) = ln(x) + ln(y) + ln(z), which adds x, y, and z variables in the ln form.
Y=A*(L^Beta 1)*(K^Beta 2)
ln (Y) = ...
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