14.02 Principles of Macroeconomics

Problem Set #6

Posted: Wednesday, April 18, 2001

Due Date: Wednesday, April 25, 2001

Part I: True or False? Briefly explain your answer. Note that all of the points in this section are for the explanations, not for just stating "true" or "false." (30 points, 6 each)

1. If s is lower than the savings rate associated with the golden rule, there will be no cost in increasing consumption forever.
2. Growth in the long run is driven entirely by how much capital a country accumulates (saving rates less depreciation).
3. Solow's model can't give an explanation for the high growth rate experienced by Europe in the two decades after World War II.
4. "Convergence" refers to the near universal tendency of poorer countries to grow more rapidly than richer countries.
5. A country's saving rate will not affect the long run equilibrium level and growth rate of capital and output.

Part II: Long Question on Growth (70 points, 10 each)

1. Generate the growth equation for steady state capital per capita as a function of s, f(k) and d, where s is the exogenous savings rate, f(k) is output per capita (as a function of only k) and d is the depreciation rate. Graph the 2 components that determine this dynamic system, and find the long run equilibrium for k.
2. Deduce and graph the growth rate equation for capital per capita (i.e.(dk/dt)*(1/k)) as in the previous question. What can you say about the growth rate of countries with a small amount of capital compared to an identical country with more capital?
3. Solve for consumption per capita in the long run. For what level of savings rate does consumption per capita in long run equilibrium reach its maximum (i.e. the Golden Rule level)? (Do not find the level of savings rate, but find the condition that must be satisfied in the long run). Graph this condition in a graph similar to the one in question 1.
4. Redo question 1 assuming that population grows at a rate 'n'. (Hint:(dK/dt)*(1/L) is not equal to dk/dt but the former can be written in terms of the latter and n*k.) Graph. What is the long run growth rate of k, output per capita and consumption per capita?
5. Redo question 1 assuming that there is technological growth in the form F(K,A*L), where A grows at a rate z. For this question, change the variable of interest from k to k'= k/A. What is the long run growth rate of k’, output per capita, and consumption per capita?
6. Go back to question 1. Once the country reaches equilibrium, output per capita growth is zero. In order to have growth in the long run, someone proposes to increase the savings rate each time we approach equilibrium. Will this policy work? What might be the problem(s) with it?
7. If F(K,L)= K + L and L grows at a rate n, at what rate will k grow? Graph your result.