M03_WEIL7590_02_ISM_C03

14WeilEconomic Growth, Second EditionChapter 3Physical Capital13

Chapter 3Physical Capital

Note:Special icons in the margin identify problems requiring a computer or calculator and those requiring calculus .

(Solutions to Problems

1.Explain whether or not the following is physical capital.

a) a delivery truck

b) milk

c) farmland

d) the Pythagorean theorem

ANSWER The five key characteristics of physical capital are that it is productive, it is produced, its use is limited, it can earn a return, and it wears out.

a.A delivery truck is physical capital because it is productive, i.e., by allowing a delivery man to drive instead of walk it increases his output (deliveries made); it has been produced itself; it is rival in its use, i.e., only one person can drive it to use it at a time; it can earn a return if rented out; and it suffers wear and tear (depreciation) for the period of its use.

b.Milk is not physical capital because it is not productive. (To be slightly technical, milk is used in making other thingsfor example, cheese. But in this case milk is a raw material rather than a factor of production. In economic terms, cheese production is the value added resulting from using capital and labor to turn milk into cheese.)

c.Farmland is not physical capital because even though it allows a worker to produce more output, it has not been produced itself.

d.The Pythagorean Theorem is not physical capital because it is non-rival in its use. That is, an unlimited amount of people can use it at the same time.

2.Use the Solow model to describe a country as follows:

Production function: y = k1/2

k=400

The fraction of output invested is 50%. The depreciation rate is 5%.

Is the country at its steady state level of income? If not, is it above or below?ANSWER (NO CALCULATOR NEEDED) To find the steady-state value of output in the country, we refer to Equation (3.3) on page 63

Plugging in values: A 1, ( 0.5, ( 0.5, and ( 0.05, we get:

Simplifying the above equation, we get

To find the current output per worker, we substitute in k 400 into the production function to get:

That is, the level of current output is 20 whereas the steady-state output level is 10. Therefore, we conclude thatso the country is above its steady-state level of output per worker.

4.The investment rate is 5% in country 1 and 20% in country 2. The two countries have the same level of productivity A and the same depreciation rate . Assuming that the level of is 1/3, calculate the steady state value of the ratio between output per worker in country 1 and output per worker in country 2. What would it be the ratio if were 1/2?ANSWER Denoting each variable by the appropriate country subscript i, we write Equation (3.3) from page 63 in ratio form. That is,

Since productivity, A, and depreciation, (, are the same, we can cancel them and rewrite the previous ratio with the appropriate values: and setting

Forwe get,

EMBED Equation.DSMT4 Therefore, when the ratio is 0.5 or 1 to 2 and when the ratio is 0.25 or 1 to 4. For bigger values of , the marginal productivity of capital in both countries is higher. This disproportionately favors the country with the higher propensity to save and invest.5.Here are the data on investment rates and output per worker for three pairs of countries. Investment rate (1970-2005)Output per worker in 2005 ($)

Thailand30.3%14260

Bolivia9.9%6912

Nigeria7.5%3468

Turkey14.6%17491

Japan31.3%48389

New Zealand20.7%43360

For each country pair, compute the ratio of GDP per worker in steady state that is predicted by the Solow model, assuming that all countries have the same A and and that the value of is 1/3. Then calculate the actual ratio of GDP per worker for each pair of countries. For which pair of countries does the Solow model do a good job in predicting relative income? For which countries does it do a poor job?

ANSWER (CALCULATOR NEEDED) Since we know productivity, A, and depreciation, (, are the same, we know that they will cancel out in our steady state ratio analysis. Therefore, with ( 1/3, our equation of interest boils down to

EMBED Equation.DSMT4 for all three pairs of countries.Using a subscript T for Thailand and a subscript B for Bolivia, we rewrite the previous equation for Thailand and Bolivia as

Substituting in and we get the steady state ratio to be:

The actual ratio is,

Therefore, with a prediction error of about 15%, the Solow Model does a good job in predicting relative income for Thailand and Bolivia.Then, using a subscript N for Nigeria and a subscript T for Turkey, we rewrite the previous equation, with and to get,

The actual ratio is,

Therefore, with a prediction error of 243%, the Solow Model does a poor job in predicting relative income for Nigeria and Turkey.

Using a subscript J for Japan and a subscript N for New Zealand, we rewrite the previous equation, with and to get,

The actual ratio is

Therefore, with a prediction error of some 9.8%, the Solow Model does a good job in predicting relative income for Japan and New Zealand.

6.Country X and Y: same level of output per worker, same depreciation rates and same productivity. Yet output per worker is growing in country X and falling in country Y. What can you say about the two countries investment rates?ANSWER If output per worker is rising in Country X and output per worker is falling in Country Y, we can be assured that both countries are not in their respective steady states. Instead, they are converging to their respective steady states. In addition, for Country X and Country Y, we are given information that depreciation, productivity, and output per worker are identical. By the process of elimination, the only difference between the countries can and must be the level of capital stocks. Capital stock levels follow the process:

As such, we can conclude that differences in investment rates are responsible for the divergence in output per worker. Specifically, a rise in output per worker for Country X and a fall in output per worker for Country Y imply that the ratio of steady-state output per worker is,

Consequently, we can determine that Country X has a higher investment rate than that of Country Y, and these differences account for the falling and rising levels of output per worker.

7. In a country, the production function is: y=k1/2, the fraction of output invested is 0.25 and the depreciation rate is 0.05.

(a) Compute the steady state levels of k and y

(b) In year 1, the level of k is 16. In a table with Year, capital per worker k, output per worker y=k1/2, investment (y, depreciation (k, change in capital stock ((y - (k) as columns, show how capital and output change over time. Continue this table until year 8.(c) Calculate the growth rate of output between year 1 and 2(d) Calculate the growth rate of output between year 7 and 8

(e) Comparing your answers from parts (c) and (d), what can you conclude about the speed of output growth as a country approaches its steady state?

ANSWER (CALCULATOR NEEDED)

(a) First we find the steady-state level of capital per worker. Using the values for investment, ( 0.25, depreciation, ( 0.05, productivity, A 1, and ( 0.5, we get,

That is, the steady-state level of capital per worker is 25. Plugging in into the production function we get the steady-state level of output per worker to be:

That is, the steady-state level of output per worker is 5.

(b) For year 2, using 16.2 as the value for capital per worker, calculate output, y, followed by investment (y, depreciation (k, and then change in capital stock. Add the value for change in capital stock to 16.2, the value for capital per worker in year 2, to get capital per worker for year 3. Use year 3 capital to obtain all the values for year 3 and continue up to year 8. The filled in table is below.

YearCapitalOutputInvestmentDepreciationChange in Capital Stock

116.004.001.000.800.20

216.204.021.010.810.20

316.404.051.010.820.19

416.594.071.020.830.19

516.784.101.020.840.19

616.964.121.030.850.18

717.144.141.040.860.18

817.324.161.040.870.17

(c) The growth rate of output between years 1 and 2 is given by:

That is, output per worker grew at a rate of 0.5 percent between years 1 and 2. (Using exact values, the growth rate is approximately 0.62 percent for years 1 and 2.)

(d) The growth rate of output between years 7 and 8 is given by:

That is, output per worker grew at a rate of 0.48 percent between years 7 and 8. (Using exact values, the growth rate is approximately 0.52 percent for years 7 and 8.)

(e) The speed of growth has changed from 0.50 percent to 0.48 percent implying that growth has slowed down at a rate of 4 percent. Thus, as a country reaches their steady-state value, the rate of growth slows.

8.(DIFFICULT PROBLEM) Suppose that there are no investment flows among countries, so that the fraction of output invested is the same as the fraction of output saved. Saving is determined as follows. There is a subsistence level of consumption per worker c*. If income per worker is equal to c*, then 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 (going to saving and investment, and the rest going to consumption. Use a diagram like Figure 3.4 to analyze the steady state of this economy.

ANSWER Before beginning the analysis, we define two new variables. The level of capital per worker necessary to achieve consumption level c* is denoted k*. (Technically, k* is given by). Therefore, if the initial level of capital, ki is above k*, savings will be positive, and if ki is below k*, savings will be 0. The second variable I is denoted (refer to second figure). It 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 Saving;

In the figure below, at any initial level of depreciation is always greater than savings. This is because the economy is very poor and income is not high enough to satisfy the basic consumption needs of people and then leave some resources for saving and investment as well. 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. The economy implodes to zero. People will be taken to starvation (unless some foreign aid saves them).

Case 2.Depreciation is not always greater than Saving; for some k

The figure below shows two possible scenarios. If the initial level of capital, is equal to or below k*, then savings in the economy will be zero. The level of capital in the economy falls due to depreciation and we achieve the same result as in the first case. On the other hand, if then the level of savings exceeds or will exceed the level of depreciation and the capital stock rises over time. The capital stock will reach a state-state value as will income. If 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 levels. In the end, the ultimate determinant of where the economy rests is determined by the initial level of capital, commensurate with the initial level of income. This is the multiple steady state equilibrium, with two possible steady states. Which steady state is eventually reached depends on the initial level of capital.

9. The golden rule of investment. Calculate the investment rate that maximizes the steady state level of consumption per worker in an economy where the production is y=k.ANSWER First, in a steady-state level that maximizes consumption per worker, the change in capital stock will be zero. That is,

Rearranging the previous equation, we know that investment must equal depreciation.

Second, given that any output not saved is consumed, we can write an equation for consumption as,

In the last part of the previous equation, we replace savings with depreciation and write output in functional form. In this form, we are able to take the derivative to find the necessary condition that will guarantee consumption maximization. Taking the derivative with respect to k and rearranging,

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

Therefore, it is easy to see the ( must equal ( by the above string of equalities. In any steady-state level of consumption per worker, the investment/saving rate must equal the value (.

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