note7_n6 - Econ303 Solow Growth Model Part Three Two...

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Econ303 Solow Growth Model: Part Three 1 Two puzzles 1. Growth of capital and labor can not fully account for the growth of GDP in the US. Based our model of GDP determination, the output at time t is determined by capital and labor input and the technology that is used in production. Technology is well presented by a Cobb-Douglas production function: Y t = A t K α t L 1 - α t The total factor productivity A t is measured empirically by the following method A t = Y t K α t L 1 - α t A t is also referred as Solow residual . Solow model I and II assumes that technology doesn’t improve over time. For the Cobb- Douglas production function, it means that the total factor productivity A t stays the same over time. Under these models, GDP in an economy grows over time only because input, capital and labor, grows over time. Let Δ Y denote the change of output, Δ K the change of capital stock, and Δ L the change of labor. The change of capital times the marginal product of capital, MP K , is the contribution of capital on the change of output. The change of labor times the marginal product of labor, MP L , is the contribution of labor. Without technology progress, the change of output can be expressed by the sum of the contributions from capital and labor Δ Y = MP K · Δ K + MP L · Δ L Divide the above by GDP, Y , we have, Δ Y Y = MP K · Δ K Y + MP L · Δ L Y To express everything in terms of growth rates we can do the following Δ Y Y = MP K · K Y · Δ K K + MP L · L Y · Δ L L At the equilibrium MP K is equal to the rental rate of capital, and MP L is equal to the wage rate. So MP K · K Y is the capital share and MP L · L Y the labor share. Without technology progress, growth of GDP can be accounted by the following equa- tion: Δ Y Y = Capital share · Δ K K + Labor share · Δ L L In the US, during the period of 1950-1985, the growth rate of GDP, Δ Y Y is 3.2%, the contribution from capital accumulation, Capital share · Δ K K , is 1.1%, and the
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Econ303 Solow Growth Model: Part Three 2 contribution from labor input increase, Labor share · Δ L L , is 0.9%. The growth of capital and labor is not enough to account for the growth of GDP. The remaining 1.2% is attributed to productivity increases during this period. Let Δ A denote the change of total factor productivity. Now the growth of GDP can be decomposed into three parts Δ Y Y = Capital share · Δ K K + Labor share · Δ L L + Δ A A Data can be collected on the growth rate of GDP, capital and labor. The growth of productivity is computed as the residue. Such exercise is called the Growth Ac- counting . 2. The difference in capital per capita between the US and India can not fully explain the difference in GDP per capita between the two countries, if we assume that the two countries have the same technology.
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