Problem Set 2: Working with the Solow Growth Model

http://www.j-bradford-delong.net/2008_html/20080203_ps2.html

1. What does it mean for an economy to have constant returns to scale?

2. How can you tell whether or not an economy in steady-state growth is at the "Golden Rule" level of saving and the capital stock?

3. Why might you think that the "Golden Rule" level of saving and the capital stock is a good place for an economy to be?

4. What do you think were the principal causes of the productivity slowdown of the 1970s?

5. Suppose that an economy's production function is Y = K^0.5(EL)^0.5 ; suppose further that the savings rate is 40% of GDP, that the depreciation rate d is 4% per year, the population growth rate n is 0% per year, and the rate of growth g of the efficiency of the labor force is 2% per year. What is the steady-state capital-output ratio? How fast does output per worker grow along the steady-state growth path? How fast does total output grow along the steady-state growth path?

6. Suppose that all variables are the same as in problem 5 save the production function, which instead is: K^0.8(EL)^0.2 ; how would your answers be different? Why would your answers are different?

7. In which of the two previous problems--5 and 6--is the steady-state growth path a better guide to how the economy will actually behave? Wht?

8. Japan has had a very high savings rate and a high growth rate of output per worker over the past half century, starting from an initial post-WWII very low level of capital per worker. What does the analysis of chapter 4 suggest about Japan's ability to sustain a higher growth rate than other industrial countries?

9. For what reasons might it make sense for the government to adopt an "industrial policy", and subsidize investment?

10. For what reasons might it make sense for the government to adopt a "technology policy", and subsidize research and development?

11. What are the qualitative effects of each the following changes on a growing economy with positive technological progress. Sketch the time paths of Y, K, C; Y/L, K/L; C/L; Y/(EL), K/(EL), C/(EL) (assume the economy is in steady-state balanced-growth equilibrium before the change hits): (a) The rate of depreciation falls (b) The savings rate falls and the rate of population growth rises (c) An earthquake destroys half of the capital stock (but no people are harmed). (d) The rate of technological progress increases (e) The labor supply suddenly increases, and at the same time, population growth falls to zero.

12. Suppose that you had a model with resource depletion—that is, with the production function: Y=K^α(EL)^{(1−α−β)}R^β; but with the growth rate g_R of natural resources R being negative. Briefly sketch out if, and how, analysis of this model would differ from the analysis of the resource-based models considered on Monday.

13. Consider an economy with the production function: Y=K^α(EL)^(1−α). (a) Suppose α = 1/2, E=1, L=100, and K=64; what is output per worker Y/L? (b) Suppose α = 1/2, E=3, L=196, and K=49; what is output per worker Y/L? (c) If both capital K and labor L triple, what happens to total output Y? (d) Holding E=1, suppose that capital per worker increases from 4 to 8 and then from 8 to 12. What happens to output per worker?

14. Suppose that environmental regulations lead to a diversion of investment spending from investments that boost the capital stock to investments that replace polluting with less-polluting capital. In our standard growth model, what would be the consequences of such a diversion for the economy's capital-output ratio and for its balanced-growth path? Would it make sense to say that these environmental regulations diminished the economy's wealth?

15. Suppose we have our standard growth model with s =15 percent, n = 1 percent, g = 1 percent, and α = 3 percent. Suppose also that the current level of the efficiency of labor E is $5,000 per year and the current level of capital per worker is $25,000. Suppose further that the parameter α in the production function: Y=K^α(EL)^(1−α); is equal to 1: α = 1. Then: (a) What can you say about output per worker in this economy? What would you project output per worker to be at some point in the distant future? (b) Suppose the savings rate s is not 15 percent but 17.5 percent. How would that change your projections of the future growth of output per worker? (c) Why aren’t the normal Solow model tools of analysis and rules of thumb much use when α = 1?

Comments

[Anela Chan]

in question 13:
(c) If both capital K and labor L triple, what happens to total output Y? (d) Holding E=1, suppose that capital per worker increases from 4 to 8 and then from 8 to 12. What happens to output per worker?

Are you referring to alpha = .5, or a general case w/ any alpha?

[Coleman Maher]

when is this due?