Due Date: Thursday, September 8Th (At the Beginning of Class) s1

Problem Set 2 FE411 Spring 2007 Rahman

Due Date: Thursday, February 8th (at the beginning of class)

INSTRUCTIONS: Use your own paper to answer the questions. Please turn in your problem sets with your name clearly marked on the front page and all pages stapled together. You are encouraged to work together, but you must hand in your own work. You must show your work for credit and answer in complete sentences when appropriate (such as when the question asks you to “describe” or “explain”).

1) The Solow Growth Model

Assume that production is a function of capital and labor, and that the rate of savings and depreciation are both constant, as described in Chapter 3’s version of the Solow Model. Further, assume that the production per worker can be described by the function:

y  f (k)  k (1/ 3) where k is capital per effective worker.

a. Using the functional form for y, write the functions for both investment and depreciation in the Solow Model.

i = γk0.33 d = δk

b. Using the functional form above, write the Law of Motion of Capital (Δk = …). What condition for the Law of Motion of Capital must hold for the economy to be in a steady state? Show the steady state condition on a graph with the investment function and the depreciation of capital per worker.

Δk = γk0.33 – δk

For steady state, this expression must equal zero. [Graph not shown, see book & notes].

c. Solve for the steady-state value of y as a function of γ and δ.

Ans: y =( γ/δ)0.5

of 6 Problem Set 2 FE411 Spring 2007 Rahman d. A developed country has a saving rate of 28%. A less-developed country has a saving rate of 10%. In both countries δ = 0.04. Find the steady-state value of y for each country. How much richer is the developed country compared to the less- developed one?

Plug into equation above. Developed country: y = 2.65 Developing country: y = 1.58

2.65/1.58 ≈ 1.67. The developed country is 67% richer than the developing country. e. What policies might the less-developed country pursue to raise its level of income?

Induce savings: tax incentives, balancing budget, etc. f. Explain why the savings rate in an economy is so important for the steady state. If the less-developed country increases its savings rate, show on the graph from b what happens to the investment function and the steady state level of capital. What is the cost today of increasing the savings rate? What are the benefits of doing so?

Savings translates into permanent increases of living standards, through raising capital stocks per person. On the graph [not shown], you would draw a new investment line, and illustrate a higher kss and yss.

The cost of raising γ is a lower consumption per person today. The benefits come in the form of higher output per person. Whether or not this also translates into higher long-run consumption per person is uncertain. g. Finally, redo part d, but now assume that production is described by:

y  f (k)  k (2 / 3)

Now, finding the new steady-state values for each country, how much richer is the developed country compared to the developing one? Why is your answer different from your answer in part d?

y =( γ/δ)2

Plug into equation above. Developed country: y = 49

of 6 Problem Set 2 FE411 Spring 2007 Rahman Developing country: y = 6.25

49/6.25 ≈ 7.84. The developed country now is almost 8 times richer than the developing country. With α = 2/3, capital has become more important in production, which means investment becomes more important. Differences in investment then will magnify differences in steady-state output.

2) The Possibility of Poverty Traps

Suppose that there are no investment flows among countries, so that the fraction of output invested in a country is the same as the fraction of output saved. Saving in an economy is determined as follows: There is a subsistence level of consumption per worker, c*, people will consume all of their income. All income per worker in excess of c* will be split between consumption and investment, with a fraction γ going to investment and the rest going to consumption. Use a diagram like Figure 3.4 from the text to analyze the steady states of this economy.

First, define k* as the level of capital per worker necessary to achieve the consumption level c*. If the initial level of k is above k*, savings will be positive, and if k is below k*, savings will be zero. Let’s also define another variable, kthreshhold. This is the level of capital at which depreciation is equal to savings and distinct from the steady-state level of capital, if it exists. We are now ready to begin our analysis. There are two cases.

Case 1: Depreciation is always greater than savings: δk > γf(k), for all k.

In the figure below, at any initial level of k, depreciation is always greater than savings. The level of capital falls over time, as does the level of income per worker. Consequently, the economy will continue to stagnate until the level of income and the capital stock are zero.

of 6 Problem Set 2 FE411 Spring 2007 Rahman

Production y

Depreciation

Investment

Case 2: Depreciation is not always greater than savings: δk ≤ γf(k), for some k.

The figure below shows two possible scenarios. If the initial level of capital k is equal to or below k*, then savings in the economy will be zero. The level of capital in the economy remains zero and we achieve the same result as in the first case. On the other hand, if k > kthreshhold, then the level of savings exceeds the level of depreciation and the capital stock rises over time. The capital stock will reach a steady-state value, as will income. Finally, if k* < k < kthreshhold, then the amount of savings does not exceed the amount of depreciation. The level of capital stock begins to fall and we are in the first case where both income and capital go to zero. In the end, the ultimate determinant of where the economy rests is determined by the initial level of capital. Here history matters – you need to start “rich enough” to stay rich.

Depreciation

Investment

k* kthreshhold kss k

3) Divergence

Country X and Country Y have the same level of output per worker. They also have the same values for the rate of depreciation, δ, and the measure of productivity, A. In Country X output per worker is growing, while in Country Y it is falling. What can you say about the two countries’ rates of investment?

It must be the case that Country X has a higher investment rate than Country Y, and that both countries are out of their steady-states. In the long run, when both countries are in their steady states, Country X will be permanently richer than Country Y.

4) The Golden Rule of Capital Accumulation

The production function per worker is, as described in Chapter 3:

y  f (k)  Ak 

Note that, in our simple closed economy case, any output that is not saved is consumed. In other words:

c  Ak   Ak  where c is consumption per worker, and γ is the savings/investment rate.

Find the value of γ that will maximize the steady-state level of consumption per worker.

The key to this problem is realizing that in steady-state, investment equals depreciation. Thus, we can write:

 c ss  Ak ss  k ss

In this form, we are able to take the derivative to find the necessary condition that will guarantee consumption maximization. Taking the derivative of above with respect to k and setting equal to zero, we get:

 1 Ak ss  

That is, the marginal product of capital must equal the rate of depreciation. Combining this consumption maximization condition with the steady-state condition:

 1  saving  y  f (k)  k  Ak ss k  Ak ss  f (y)  y

Thus it is easy to see that γ must equal α by the above string of equalities. In any steady-state level of consumption per worker, the investment rate must equal the value α to maximize this long-run consumption.

5) Convergence vs. Divergence

Read Chapter 3 of the Easterly book. The economist William Baumol argued that there has been a general tendency toward convergence of national incomes. The economist Lant Pritchett, on the other hand, argued the opposite – that the gap between rich and poor countries has only grown over the past century.

a. In a sense they are both correct. In two or three sentences, explain how.

See pages 60-65 of Easterly for full discussion.

b. The Solow model illustrated in Weil’s Chapter 3 can illustrate 2 two countries that are converging, or two countries diverging, or even two countries whose differences in incomes are permanent. In a sentence or two, how is the Solow model able to achieve such variety in results (no equations necessary, just intuition)?

Convergence or divergence can occur, depending on differences on initial conditions and differences in investment rates. See pages 67-69 of Weil for further discussion.