Hint: To refresh your memory of derivatives, read Calculus Tricks #1 while working through this assignment. The last page (p. 22) has a detailed example of how to derive the marginal product of labor from the production function.
In its introduction to the Solow Model without technological progress, Lecture 4 contains a derivation of the marginal product of capital.
ANSWER: The marginal product of labor is the derivative of the production function with respect to labor:
So to find the marginal product of labor, we must take the derivative of the production function with respect to labor:
Because the exponent on labor, , is negative, we can move it to the denominator:
which highlights that the marginal product of labor is a function of the capital-labor ratio.
|MPL0 (K = 1)
|MPL1 (K = 2)
ANSWER: When the capital stock increases, the marginal product of labor is higher at every amount of labor.
In Lecture 2, you learned that a firm hires labor up to the point where the wage equals the price times the marginal product of labor (MPL), i.e. , where labor is supplied at wage rate, , and the labor demand is given by . Since we're now discussing economy-wide aggregates, it's convenient to normalize the price level to .
ANSWER: The Solow Model assumes that labor is fully-employed, so – at a given point in time – labor supply is constant in the Solow Model. The labor force grows over time, but it is constant at a given point in time.
In 2003, Pres. George W. Bush convinced Congress to reduce the maximum tax rate that shareholders pay on dividends from 38.6 percent to 15 percent. In lobbying for this measure, he argued that cutting the tax would encourage people to invest more – i.e. increase the economy's saving rate.
Opponents of the policy argued that cutting the tax on dividends was a giveaway to Pres. Bush's rich friends and that it would not benefit workers.
Answer the following questions using the Solow Model without technological progress. Throughout the problem, assume that the U.S. economy was in steady state when Pres. Bush announced his dividend tax plan. Until part e., assume that Pres. Bush's tax policy would increase the saving rate.
ANSWER: Because we are assuming that Pres. Bush's tax policy would increase the saving rate, such a tax policy would increase steady-state consumption per worker if the previous saving rate was lower than the golden-rule level.
ANSWER: Because the economy would converge to a higher steady-state level of capital per worker, the marginal product of labor will increase as capital per worker increases.
ANSWER: Because the Solow Model assumes that the labor supply is constant at a given point in time, the higher marginal product of labor will increase equilibrium wages.
ANSWER: No. If the change in tax policy increases the saving rate, workers will benefit from higher wages.
ANSWER: Yes. If the change in tax policy does not increase the saving rate, then it's a giveaway to the rich without any benefit for workers.
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