Simple Solow model
1. Analyzing economic growth with the Solow Model Suppose that the aggregate production function for a closed private economy is Y= Kos that the saving rate (a) is 20% and the depreciation rate (5) is 5%. The economy is in long-run, steady-state equilibrium in 2007. This means that in macroeconomic equilibrium Y=C+I=C+S. The economy has constant supplies of labor and natural resources, so the only variable factor of production is capital. Both capital (K) and output (Y) are measured in trillions of dollars.
Remember that in the macro-economy, output equals income, and growth (rate) is measured by (percentage) increases in Y.
1) Draw a curve (identify by the label fk) representing the aggregate production function for this economy. Make your graph precise with axes that are scaled and labeled. To find hypothetical values of Y easily, use values of K such as 1, 4, 9, 16, 25, ...
2) Draw a curve (identify by the label afk ) representing the saving function (S- cY).
3) Draw a line showing the amount of new capital necessary to replace the capital lost through depreciation. This is the steady-state investment function (I=6K). Note: You can graph questions 1, 2 and 3 together in one graph.
4) What is the steady-state equilibrium dollar value of the capital stock in 2007? (Hint: Set S=I and solve for K)
5) What is the steady-state equilibrium dollar value of total output in 2007?
(Hint= Enter the value of K that you found in (4) into the production function and solve for Y)
6) What is the dollar amount of saving for 2007 in this economy? (Hint: multiply the Y you found in (5) by the saving rate (sometimes called the average propensity to save).
7) What is the annual dollar amount of depreciation in this economy? (Hint: This is the amount of the capital stock that wears out each year. In the steady state, it also equal the amount of investment)
8) How much is consumed in 2007? (Hint: Y=C+I where C=consumption)
9) What is the rate of economic growth in steady-state equilibrium?
10) If the saving rate rose to 25%, what would happen to this economy? There are two answers to this question:
A. In the short run before investment increases the capital stock, Y would remain at its 2007 level. Explain what would happen to C and I. Show in graph
B. In the long run, the economy would move to a new equilibrium. Calculate the new steady state values for K, Y, C, S, and I. Sketch the appropriate new curves in your diagram.
11) How much economic growth occurred as a result of the increase in the propensity to save? Express your answer in dollars and in terms of percentage growth. Can this continue indefinitely?
12) Instead of an increase in the saving rate, suppose that technical progress raised the average productivity of capital (i.e. Y/K). This could be represented by a new aggregate production function (Y=106). Assume that the saving rate remains at 20% and the depreciation rate at 5%. Answer questions 4, 5, 6 and 8 again with respect to 2007. Sketch another new curve in the diagram to show the new steady state equilibrium. How much growth occurred? Could it continue indefinitely?
13) If this economy opens to international trade, would the dynamic effects be more like those you calculated for question 10 or for 12?
4. The following tables show data on investment rates and output per worker for three pairs of countries. For each country pair, calculate a) the ratio of GDP per worker in steady state that is predicted by the Solow model, assuming that all countries have the same values of A and 6 and that the value of a is 1/3. Then.
b) Calculate the ratio of GDP per worker for each pair of countries. For which pair of countries does the Solow model do a good job of predicting relative income? For which pairs does the Solow model do a poor job?
a.)
|
|
Investment rate
|
(Average 1975-2009)
Output per Worker in 2009
|
|
Country
|
|
Thailand
|
35.2%
|
$13,279
|
|
Bolivia
|
12.6%
|
$8,202
|
b.)
|
|
Investment rate
(Average 1975-2009)
|
Output per Worker in 2009
|
|
Country
|
|
|
|
Nigeria
|
6.4%
|
$6,064
|
|
Turkey
|
16.3%
|
$29,699
|
c)
| |
Investment rate |
Output per Worker in 2009 |
| (Average 1975-2009) |
| Country |
|
|
| Japan |
29.90% |
$57,929 |
| New Zealand |
18.60% |
$49,837 |
5. Country X and Country Y have the same level of output per worker. They also have the same values for the rate of depreciation, 6, and the measure of productivity, A. In country X output per worker is growing, whereas in Country Y it is falling. What can you say about the two countries' rates of investment?
6. In a country the production function is y - k112. The fraction of output invested, sty, is 0.25. The deprecation rate, 8., is 0.05.
a. What are the steady-state levels of capital per worker, k, and output per worker. y?
b. In year 1, the level of capital per worker is 16. In a table such as the following one, show how capital and output change over time (the beginning if filled in as a demonstration). Continue this table up to year 8.
| Year |
Capital |
Output |
Investment
|
Depreciation |
Change in |
| k |
y = k1/2 |
σy |
δk |
Capital Stock |
| |
|
|
|
σy - δk |
| 1 |
16 |
4 |
1 |
0.8 |
0.2 |
| 2 |
16.2 |
|
|
|
|
C. Calculate the growth rate of output between years 1 and 2.
d. Calculate the growth rate of output between years 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?