Given the production function Y = A and fixed values for the saving rate and depreciation, if productivity is growing at an average rate of three percent, and the labor input grows at two percent, there is a unique growth rate of capital that is
sustainable. That is, if the growth rate of capital is either higher or lower than this steady-state value, then it must eventually change, even if nothing else in the economy changes. Calculate this steady-state growth rate of capital, and explain why it alone is a sustainable rate. [Hint: Use the fact that the growth rates of output and capital per worker are 43% higher than the growth rate of productivity.]
Since productivity is growing at three percent, capital per worker is growing at 4.3%
( = 0.03 × 1.43). Since labor grows at two percent, capital must grow at 4.3% + 2% = 6.3%. Putting this into the growth accounting equation: = 0.03 + 0.3 × 0.063 + 0.7 * 0.02 = 0.063. Thus, capital and output grow at the same rate. If capital were to grow at a faster rate, say eight percent, the growth of output could not keep up: 0.03 + 0.3 × 0.08 + 0.7 × 0.02 = 0.068. A growing capital stock requires enough saving to compensate for depreciation and capital dilution, but saving can increase only as fast as output. If the growth rate of capital were too low, say five percent, output would be growing faster than capital, so saving a constant fraction of output would result in investment that is increasingly large relative to depreciation and capital dilution. The investment needed to keep capital growing at a constant rate can occur only when output grows at the same rate as the capital stock.
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