Answers

**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.

**Problem 1**

In its introduction to the Solow Model without technological progress, Lecture 4 contains a derivation of the marginal product of capital.

- If the production function is given by:
$Y={K}^{\alpha}\xb7{L}^{(1-\alpha )}$
,
what is the marginal product of
__labor__?

**ANSWER:** The marginal product of labor is the derivative
of the production function with respect to labor:

$$MPL\equiv \frac{dY}{dL}$$

So to find the marginal product of labor, we must take the derivative of the production function with respect to labor:

$$\frac{dY}{dL}=(1-\alpha )\cdot {K}^{\alpha}\cdot {L}^{-\alpha}$$

Because the exponent on labor, $L$, is negative, we can move it to the denominator:

$$MPL=(1-\alpha )\cdot {\left(\frac{K}{L}\right)}^{\alpha}$$

which highlights that the marginal product of labor is a function of the capital-labor ratio.

- Assuming that $\alpha =0.5$ and that $K=1$, calculate the marginal product of labor from one unit of labor input to five units. Hint: Use a calculator!
- On a graph, plot the marginal product of labor using the values you just calculated.
- Assuming that $\alpha =0.5$ and that $K=2$, calculate the marginal product of labor from one unit of labor input to five units.
- On the same graph, plot the marginal product of labor using the values you just calculated.

**ANSWERS:**

Labor | MPL_{0} (K = 1) |
MPL_{1} (K = 2) |
---|---|---|

1 | 0.50 | 0.71 |

2 | 0.35 | 0.50 |

3 | 0.29 | 0.41 |

4 | 0.25 | 0.35 |

5 | 0.22 | 0.32 |

- What happens to the marginal product of labor when the economy's stock of capital increases?

**ANSWER:** When the capital stock increases, the marginal product of labor
is higher at every amount of labor.

**Problem 2**

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. $w=p\xb7MPL$ , where labor is supplied at wage rate, $w$, and the labor demand is given by $p\xb7MPL$. Since we're now discussing economy-wide aggregates, it's convenient to normalize the price level to $p=1$.

- If we assume that the wage rate,
$w$,
is constant at a given point in time, then how will the quantity
of labor that the economy demands respond to a sudden increase in the capital stock?
**ANSWER:**At the new equilibrium, the quantity of labor demanded will be higher.

- If we assume that the quantity of labor supplied,
$L$,
is constant at a given point in time, then how
will the wage rate respond to a sudden increase in the capital stock?
**ANSWER:**At the new equilibrium, the wage rate will be higher.

- Which assumption does the Solow Model make?

**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.

**Problem 3**

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.

- Under what condition would Pres. Bush's tax policy increase steady-state consumption per worker? Under what condition would it decrease steady-state consumption per worker?

**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.

- How would the marginal product of labor differ between the initial steady state and the one to which the economy will converge to after reduction of the tax on dividends?

**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.

- How would Pres. Bush's tax policy affect wages, $w$? Hint: remember that: $w=p\xb7MPL$

**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.

- Given your answers to the previous three questions, was Pres. Bush's tax policy a giveaway to the rich without any benefit for workers?

**ANSWER:** No. If the change in tax policy increases the saving rate,
workers will benefit from higher wages.

- Now assume that Pres. Bush's tax policy would not increase the saving rate. Under this assumption, was the tax policy a giveaway to the rich without any benefit for workers?

**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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