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Growth_Model_Part3_SolowSwan_Model - Disclaimer These notes...

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Disclaimer: These notes were prepared based on lectures of Prof Sala-i- Martin’s 2008 Fall Course of Intermediate Macro-W3213. Contents of these notes might not match completely with the current teachings in class. An updated version would be available later in the semester. 2.13 Measuring Productivity: The method of the Solow Residual. Imagine that the production function is Cobb Douglas 𝑌 = ?? 𝛼 ? 1 −α 2. A2.1 . We have learnt that technology is the source of growth in the long run. Now we want to MEASURE the growth rate of technology (or the rate of technological progress). Unfortunately, we cannot do it directly since “technology” cannot be measured. Solow came up with a trick to measure the growth rate of A: ? ? = 𝛥? ? indirectly. Take logarithms of the production function 𝑌 = ?? 𝛼 ? 1 −α . : ??? 𝑌 = ??? ? + ??? ( ? 𝛼 ) + ??? ( ? 1 −𝛼 ) Using the laws of logarithms we can bring the exponents down and write the equation as: ??? 𝑌 = ??? ? + 𝛼??? ? + (1 − 𝛼 ) ??? ? . Now remember that the derivative of log x is : 𝛥 log ? = 𝛥? ? . Use this rule to take derivatives of both sides of our equation and get: 𝛥𝑌 𝑌 = 𝛥? ? + 𝛼 𝛥? ? + 1 − 𝛼 𝛥? ? 2.10 The key now is to realize that WE CAN MEASURE the following components of the above equation: 𝛥𝑌 𝑌 is the growth rate of aggregate GDP (this is measured in the national accounts), 𝛥? ? is the growth rate of the capital stock (this is also measured in the N.A.), 𝛥? ? is the growth rate of labor (again, this is measured), (1- α) is the fraction of total national income that accrues to workers (=wage bill divided by national income i.e. ?? 𝑌 ). α is the fraction of national income that accrues to capital. Hence, we can measure everything in the above equation except for 𝛥? ? . As a result, we can measure 𝛥? ? indirectly by rearranging the equation:
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𝛥? ? = 𝛥𝑌 𝑌 𝛼 𝛥? ? 1 − 𝛼 𝛥? ? 2.11, i.e., the growth rate of productivity can be measured as the difference (or the residual) between the growth rate of GDP 𝛥𝑌 𝑌 and the growth rate of the two inputs ( 𝛥? ? and 𝛥? ? ) each of which is weighted by its share or contribution to the production of GDP. Some Results In class we presented some results for a bunch of countries (taken from Chapter 10 of the book “ECONOMIC GROWTH” by Barro and Sala -i-Martin.) The main message of the table is that productivity growth in MIRACLE COUNTRIES (Hong Kong, Singapore, Taiwan, and South Korea) is NOT really spectacular in the sense that it is comparable to the growth rates for Mexico or Brazil. Hence, the miracle is NOT why they have so much productivity growth but HOW they achieved enormous savings and investment rates (which led to large increases in the capital stock and the stock of labor).
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