Chapter 4
The Theory of Economic Growth
· The Solow growth model allows us to analyze the determinants of the long-run trend value of the standard of living and its growth rate.
· When the economy is in its balanced-growth equilibrium, the growth rates of output per worker, the capital-labor ratio, and labor efficiency will all be the same. The balanced-growth equilibrium occurs when the capital-output ratio is constant.
· In balanced-growth equilibrium and assuming a Cobb-Douglas production function, the level of output per worker (Y/L) depends upon the saving rate (s), the rate of growth of the labor force (n), the rate of growth of labor efficiency (g), the rate of depreciation of capital (), the parameter of the production function (), and the current level of labor efficiency (E). In balanced-growth equilibrium, the growth rate of labor efficiency alone determines how fast output per worker grows.
Learning Guide
You may find this chapter challenging. There is only one model in this chapter the Solow growth model but it is abstract. Once you see the key point however, the Solow model becomes suddenly simple.
Some professors believe the material covered in Chapter 4 is the most important material of the course. You will especially want to be able to answer the Section C and Section D questions in this chapter and in Chapter 5.
You will need some math skills in this chapter. Math skills covered in Chapter B are indicated with
Short on time?
If you are short on time, you are in a bind. It is almost impossible to study this chapter quickly. At a minimum, learn the equations for K/Y and Y/L. But really: do not try to save time on Chapter 4.
You must understand the Cobb-Douglas production function. The determinants of long-run economic growth and the determinants of balanced-growth equilibrium are very important. Distinguishing between events that change only the level of output per worker and those that also change the growth rate of output per worker is key. You need to acquire both a technical mastery of the topics and a conceptual one.
A. BASIC DEFINITIONS
Before you apply knowledge, you need a basic grasp of the fundamentals. In other words, there are some things you just have to know. Knowing the material in this section won't guarantee a good grade in the course, but not knowing it will guarantee a poor or failing grade.
USE THE WORDS OR PHRASES FROM THE LIST BELOW TO COMPLETE THE SENTENCES. SOME ARE USED MORE THAN ONCE; SOME ARE NOT USED AT ALL.
balanced-growth equilibrium growth rate
capital stock investment
capital-labor ratio Keynesian
capital-output ratio labor force
Cobb-Douglas output per worker
Depreciation saving rate
efficiency of labor Solow
1. ______occurs when the various forces determining economic growth are balanced so that output per worker is increasing from period to period at the same rate as the capital-labor ratio.
2. The ______production function is a particular algebraic form of the general abstract function .
3. The ______has increased due to improvements in technology and in business organization.
4. The ______production function states
.
5. Total saving equals ______spending.
6. ______reduces the capital stock due to wear-and-tear and obsolescence.
7. Output per worker is a function of two inputs: the ______and the ______.
8. The two major factors generating differences between economies' productive potential are the ______and the ______.
9. When the economy is in ______, the growth rate of output per worker equals the growth rate of labor efficiency.
10. The ______is the ratio of the sum of household saving, government saving, and foreign saving to total output.
11. The growth model is also called the ______model, named after the economist who received the Nobel Prize for developing the model.
12. The best proxy we have for the material standard of living is ______.
13. The ______is the total amount of business equipment, machinery and buildings available for producing goods and services.
14. When graphing the production function and the balanced-growth equilibrium line, the ______goes on the horizontal axis and ______goes on the vertical axis.
15. ______refers to construction of buildings and purchases of business equipment and machinery.
16. The ______is the average amount of physical capital available to workers for production.
17. The ______is the share of real output (GDP) that is saved.
18. Real GDP per worker converges to its ______path as the capital-output ratio converges to its equilibrium value.
HINT: In Chapters 4 and 5 we are looking at the long-run economy. In the long run, the economy is always at full employment, always on the production possibilities frontier, always on the aggregate production function. Output and income (real GDP, Y) always equal their potential.
CIRCLE THE CORRECT WORD OR PHRASE IN EACH OF THE FOLLOWING SENTENCES
19. In the very long run, a higher saving rate will / will not increase output per worker and will / will not permanently increase the growth rate of output per worker.
20. Increased obsolescence of capital will increase / decrease the depreciation rate.
21. The economy is / is not in balanced-growth equilibrium when %Δ(Y/L) = %Δ(K/L) = %(E).
22. The economy is / is not in balanced-growth equilibrium when K/Y is constant.
23. The balanced-growth equilibrium line is a straight line / nonlinear curve emanating from the origin.
24. Output per worker increases / decreases when capital per worker increases.
25. Output per worker increases / decreases when technology improves.
26. Output per worker increases / decreases when efficiency of labor increases.
27. The growth rate of the labor force is endogenous / exogenous to the growth model.
28. The growth rate of labor efficiency is endogenous / exogenous to the growth model.
29. The growth rate of output is endogenous / exogenous to the growth model.
30. The growth rate of the capital stock is endogenous / exogenous to the growth model.
31. An increase in government spending, all else constant, causes a(n) increase / decrease in government saving and a(n) increase / decrease in the saving rate.
32. An increase in consumption spending, all else constant, causes a(n) increase / decrease in household saving and a(n) increase / decrease in the saving rate.
33. An increase in imports, all else constant, causes a(n) increase / decrease in foreign saving and a(n) increase / decrease in the saving rate.
______
SELECT THE ONE BEST ANSWER FOR EACH MULTIPLE-CHOICE QUESTION.
34. In balanced-growth equilibrium,
A. the amount of capital per worker is constant.
B. investment per worker is constant.
C. the amount of output per worker is constant.
D. the capital-to-output ratio is constant.
35. The aggregate production function tells us
A. how a firm combines its inputs to produce its output.
B. how the economy's total labor force, capital, and technology can be used to
produce output.
C. the balanced-growth equilibrium.
D. the rate of diminishing returns.
36. To determine the balanced-growth equilibrium value of K/Y, the Solow growth
model requires information about all of the following variables EXCEPT
A. the rate of growth of the labor force.
B. the size of the labor force.
C. the saving rate.
D. the rate of depreciation of capital.
37. Over the last 200 years, the U.S. standard of living has been
A. smoothly increasing from year to year.
B. increasing from decade to decade but not necessarily from year to year.
C. constant.
D. sometimes increasing and sometimes decreasing with no clear trend over time.
TO THE CHALKBOARD
Explaining Figure 4.8
Textbook Figure 4.8 shows how equilibrium output per worker (Y/L) and efficiency of labor (E) grow over time when the economy is in balanced-growth equilibrium. Remember that in that equilibrium, Y/L and K/L will grow at the same rate as labor efficiency: g. A constant rate of growth provides smooth growth but is not graphed as a straight line. A straight line would depict increases that were the same amount each period (such as, $5,000 per month) but would then be a declining rate of growth (percentage change) each period. In equilibrium, the rate of growth (percentage change) is constant from period to period, which means the amount of growth is increasing from period to period. The graphical result is above: a smooth curve whose slope continually increases.If we were to use a logarithmic scale to depict how equilibrium output per worker (Y/L) and efficiency of labor (E) grow over time when the economy is in balanced-growth equilibrium, then we would have a straight line graph. A constant percentage change over time produces a constant slope when log(Y/L) is shown. The graphical result is at the right: a curve with a constant slope. In balanced growth equilibrium, the slopes of the two curves will be equal.
38. The efficiency of labor can increase when
A. workers acquire new and better skills.
B. employers reorganize the work place to increase sales with fewer workers.
C. scientific discoveries make machines more productive.
D. all of the above.
39. Ultimately, the most important factor determining the growth of output per worker
over time is the
A. saving rate.
B. level of output per worker.
C. growth rate of the labor force.
D. growth rate of labor efficiency.
40. A decrease in the growth rate of the labor force, all else constant, will permanently
A. increase the level of Y/L and its growth rate.
B. increase the level of Y/L but have no effect on its growth rate.
C. change neither the level of Y/L nor its growth rate.
D. decrease the level of Y/L but have no effect on its growth rate.
______
TO THE CHALKBOARD
The Key Equations
There are a number of equations in this chapter. Here is a list of the key equations, with a brief description of each. Be sure you learn especially equations [4] and [5].[1] / The general production function. Output per worker depends upon the capital-labor ratio (also known as capital per worker) and the efficiency of labor.
[2] / The Cobb-Douglas production function, which specifies the functional form of the relationship between output per worker, capital per worker, and labor efficiency.
[3] / How capital stock changes from one period to the next. Capital stock at the beginning of period t + 1 equals capital stock at the beginning of the previous period plus the saving rate (s) times last period's output (Yt) minus the depreciation rate () times capital stock at the beginning of period t.
[4] / The balanced-growth equilibrium value of the capital-output ratio equals the saving rate (s) divided by the sum of the labor force growth rate (n), the labor efficiency growth rate (g), and the depreciation rate ().
[5] / The balanced-growth value of output per worker. When the economy is in equilibrium, output per worker is a constant proportion of labor efficiency E: Y/L equals the saving rate (s) divided by the sum of the labor force growth rate (n), the labor efficiency growth rate (g), and the depreciation rate (), all raised to the power ( divided by one minus ), and then multiplied by the value of labor efficiency, E.
B. MANUPULATION OF CONCEPTS AND MODELS
Most instructors expect you to be able to do basic manipulation of the concepts. Being able to do so often means you can earn a C in a course. But if you want a better grade, you'll need to be able to complete this section easily and move on to sections C and D.
NOTE: The distinguishing feature of a Cobb-Douglas production function is that the exponents sum to one (1). For instance, Y = AKβL(1-β) is also Cobb-Douglas because + β (1 - β) = 1. The Cobb-Douglas production function is convenient to use because it has some very nice mathematical properties. A Cobb-Douglas production function exhibits constant returns to scale: if you double all of the inputs, output will also double. Mathematically, the function exhibits constant returns to scale because the exponents sum to 1.
1. The Cobb-Douglas production function is .
A. Suppose labor efficiency, E, is 10,000. Graph the Cobb-Douglas production
function when = 0.2. (You might find it easier to do the graph using a spreadsheet
package such as Quattro Pro or Excel, or using a graphing calculator.)
B. Suppose labor efficiency, E, is 10,000. Graph the Cobb-Douglas production
function when = 0.8.
C. When does a change in the capital to labor ratio generate the larger change in
output per worker, when = 0.2 or when = 0.8? Explain your answer, using the
economic concept of diminishing returns.
Many students are tempted to ask, "Is the Cobb-Douglas production function realistic?" That's the wrong question. The right question is, "Is the Cobb-Douglas production function a good enough approximation of reality to allow us to do reasonable analysis and come up with useful conclusions?" Yes.2. A. Suppose E = 10,000 and = 0.4. Assume
production can be described by the Cobb-Douglas
production function. Compute the value of output
per worker at each level of capital per worker
shown at the right.
B. When K/L doubles, does Y/L double? Why or why not?
In Questions 3, 4, and 5, you will work with the definitions of labor force, labor efficiency, and capital stock, and see how their values change over time.
3. Compute the average value of n, the growth rate of the labor force, by decade, 1950-2000. Consult the Economic Report of the President to locate labor force data.