Solutions to the Problems in the Textbook Conceptual Problems:

1.

The producti production on function function provides provides a quantitative quantitative link link between between inputs and output. output. For example, example, the Cobb-Douglas production function mentioned mentioned in the text is of the form: θ

θ

Y = F(N,K) = AN 1- K , where Y represents the level of output. (1 - θ ) and θ are weights equal to the shares of labor (N) and capital (K) in production, while A is often used as a measure for the level of technology. It can be easily shown that labor and capital each contribute to economic growth by an amount that is equal to their individual growth rates multiplied by their respective share in income.

2. The Solow model model predicts predicts converge convergence, nce, that is, is, countries countries with the the same production production function function,, savings savings rate, and population growth will eventually reach the same level of income per capita. In other words, a poor country may eventually catch up to a richer one by saving at the same rate and making technological innovations. However, if these countries have different savings rates, they will reach different levels of income per capita, even though their long-term growth rates will be the same.

3. A production production functio function n that omits omits the stock of natural natural resources resources cannot cannot adequately adequately predic predictt the impact of a significant change in the existing stock of natural resources on the economic performance of a country. For example, the discovery of new oil reserves or an entirely new resource would have a significant effect on the level of output that could not be predicted by such a production function.

4. Interp Interpret reting ing the Solow Solow residu residual al purely purely as techno technolog logica icall progre progress ss would would ignor ignore, e, for example example,, the impact that human capital has on the level of output. In other words, this residual not only captures the effect of technological progress but also the effect of changes in human capital (H) on the growth rate of output. To eliminate this problem we can explicitly include human capital in the production function, such that Y = F(K,N,H) = AN aK bHc

with a + b + c = 1.

Then the growth rate of output can be calculated as ∆Y/Y

= ∆A/A + a( ∆ N/N) + b(∆K/K) + c( ∆H/H).

5. The savings savings function function sy = sf(k) assumes assumes that a constant constant fraction fraction of output output is saved. saved. The investmen investmentt requirement, that is, the (n + d)k-line, represents the amount of investment needed to maintain a constant capital-labor ratio (k). A steady-state equilibrium is reached when saving is equal to the

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investment requirement, that is, when sy = (n + d)k. At this point the capital-labor ratio k = K/N is not changing, so capital (K), labor (N), and output (Y) all must be growing at the same rate, that is, the rate of population growth n = ( ∆ N/N).

6. In the the long long run, run, the rate rate of popul populati ation on growt growth h n = ( ∆ N/N) determines the growth rate of the steadystate output per capita. In the short run, however, the savings rate, technological progress, and the rate of depreciation can all affect the growth rate. 7. Labor Labor product productivi ivity ty is defined defined as Y/N, that that is, the ratio ratio of output output (Y) to labor labor input input (N). A surge surge in labor productivity therefore occurs if output grows at a faster rate than labor input. In the U.S. we have experienced such a surge in labor productivity since the mid-1990s due to the enormous growth in GDP. This surge can be explained from the introduction of new technologies and more efficient use of existing technologies. Many claim that the increased investment in and use of computer technology has stimulated economic growth. Furthermore, increased global competition has forced many firms to cut costs by reorganizing production and eliminating some jobs. Thus, with large increases in output and a slower rate of job creation we should expect labor productivity to increase. (One should also note that a higher-skilled labor force also can contribute to an increase in labor productivity, productivity, since the same number of workers can produce more output if workers are more highly skilled.)

Technical Problems:

1.a. According to Equation (2), the growth of output output is equal to the growth in labor times the labor share plus the growth of capital capital times the capital share plus the rate of technical progress, that that is, ∆Y/Y

= (1 - θ )(∆ N/N) + θ (∆K/K) + ∆A/A, where

1 - θ is the share of labor (N) and θ is the share of capital (K). Thus if we assume that the rate of technological progress ( ∆A/A) is zero, then output grows at an annual rate of 3.6 percent, since ∆Y/Y

= (0.6)(2%) + (0.4)(6%) + 0% = 1.2% + 2.4% = + 3.6%,

1.b.The so-called "Rule of 70" suggests that the length of time it takes for output to double can be calculated by dividing 70 by the growth rate of output. Since 70/3.6 = 19.44, it will take just under 20 years for output to double at an annual growth rate of 3.6%, 1.c. Now that ∆A/A = 2%, we can calculate economic growth as ∆Y/Y

= (0.6)(2%) + (0.4)(6%) + 2% = 1.2% + 2.4% + 2% = + 5.6%.

Thus it will take 70/5.6 = 12.5 years for output to double at this new growth rate of 5.6%. 5. 6%. 2.a. According to Equation (2), the growth of output output is equal to the growth in labor times the labor share plus the growth of capital times the capital share plus the growth rate of total factor productivity (TFP), that is,

27

∆Y/Y

= (1 - θ )(∆ N/N) + θ (∆K/K) + ∆A/A, where

1 - θ is the share of labor (N) and θ is the share of capital (K). In this example θ = 0.3; therefore, if output grows at 3% and labor and capital grow at 1% each, then we can calculate the change in TFP in the following way 3% = (0.3)(1%) + (0.7)(1%) +

∆A/A

==> ∆A/A = 3% - 1% = 2%,

that is, the growth rate of total factor productivity is 2%. 2.b.If 2.b. If both labor and the capital capital stock are fixed and output grows at 3%, then all this growth growth has to be contributed to the growth in factor productivity, that is, ∆A/A = 3%.

3.a. If the capital capital stock grows grows by ∆K/K = 10%, the effect on output would be an additional a dditional growth rate of ∆Y/Y = (.3)(10%) = 3%. 3.b. If labor grows grows by ∆ N/N = 10%, the effect on output output would be an additional additional growth rate of ∆Y/Y

= (.7)(10%) = 7%.

3.c. If output output grows at ∆Y/Y = 7% due to an increase in labor by ∆ N/N = 10%, and this increase in labor is entirely due to population growth, then per capita income would decrease and people’s welfare would decrease, since ∆y/y

= ∆Y/Y - ∆ N/N = 7% - 10% = - 3%.

3.d. If this increase in labor is due to an influx of women into the labor force, the overall population does not increase and income per capita would increase by ∆y/y = 7%. Therefore people's welfare would increase.

4.

Figure 3-4 shows output output per head as a function of the capital-labor capital-labor ratio, that is, y = f(k). The The savings functi function on is sy = sf(k), sf(k), and it inter intersec sects ts the straig straight ht (n + d)k-li d)k-line, ne, repres represent enting ing the invest investme ment nt requirement. At this intersection, the economy is in a steady-state equilibrium. Now let us assume that the economy is in a steady-state equilibrium before the earthquake hits, that is, the steady-state capital-labor ratio is currently k *. Assume further, for simplicity, that the earthquake does not affect people's savings behavior. behavior. If the earthquake destroys one quarter of the capital stock but less than one quarter of the labor force, then the capital-labor ratio falls from k * to k 1 and per-capita output falls from y * to y1. Now saving is greater than the investment requirement, that is, sy 1 > (d + n)k 1, and the capital stock and the level of output per capita will grow until the steady state at k * is reached again. However, if the earthquake destroys one quarter of the capital stock but more than one quarter of the labor force, then the capital-labor ratio increases from k * to k 2. Saving now will be less than the investment requirement and thus the capital-labor ratio and the level of output per capita will fall until the steady state at k * is reached again.

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If exactly one quarter of both the capital stock and the labor stock are destroyed, then the steady state is maintained, that is, the capital-labor ratio and the output per capita do not change. If the severity of the earthquake has an effect on peoples’ savings behavior, then the savings function sy = sf(k) will move either up or down, depending on whether the savings rate (s) increases (if people save more, so more can be invested in an effort to rebuild) or decreases (if people save less, since they decide that life is too short not to live it up). y y = f(k) y2 y* y1

(n+d)k sy

0 k *

k 1

k 2

k

5.a. An increase in the population growth rate (n) affects the investment requirement, and the (n + d)kline gets steeper. As the population grows, more saving must be used to equip new workers with the same amount of capital that the existing workers already have. Therefore output per capita (y) will decrease as will the new optimal capital-labor ratio, which is determined by the intersection of the sycurve and the (n1 + d)k-line. Since per-capita output will will fall, we will have a negative growth rate in the short run. However, the steady-state growth rate of output will increase in the long run, since it will be determined by the new and higher rate of population growth. y

(n1 + d)k y = f(k)

yo

(no + d)k

y1

sy

0 k 1

k o

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k

5.b. Starting from an initial steady-state equilibrium at a level of per-capita output y *, the increase in the population growth rate (n) will cause the capital-labor ratio to decline from k * to k 1. Output per capita will also decline, a process that will continue at a diminishing rate until a new steady-state level is reached at y1. The growth rate of output will gradually adjust to the new and higher level n 1. y y*

y1

to

t1

t

to

t1

t

k k *

k 1

6.a. Assume the production function function is of the form Y = F(K, N, Z) = AK a N bZc ==> ∆Y/Y

= ∆A/A + a( ∆K/K) + b(∆ N/N) + c(∆Z/Z), with a + b + c = 1.

Now assume that there is no technological progress, that is, ∆A/A = 0, and that capital and labor grow at the same rate, that is, ∆K/K = ∆ N/N = n. If we also assume that all natural resources available are fixed, such that ∆Z/Z = 0, then the rate of output growth will be ∆Y/Y

= an + bn = (a + b)n.

In other words, output will grow at a rate less than n since a + b < 1. Therefore output per worker will fall. 6.b. If there is technological progress, progress, that is, ∆Y/Y

∆A/A

> 0, then output will grow faster than before, namely

= ∆A/A + (a + b)n.

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If ∆A/A > c, then output will grow at a rate larger than n, in which case output per worker will increase. 6.c. If the supply of natural resources is fixed, then output can only grow at a rate that is smaller than the rate of population growth and we should expect limits to growth as we run out of natural resources. However, if the rate of technological progress is sufficiently large, then output can grow at a rate faster than population, even if we have a fixed supply of natural resources.

7.a. If the production function is of the form Y = K 1/2(AN)1/2, and A is normalized to 1, then we have Y = K 1/2 N1/2 . In this case capital's and labor's shares of income are both 50%. 7.b. This is a Cobb-Douglas production function. function. 7.c. A steady-state equilibrium is reached when sy = (n + d)k. From Y = K 1/2 N1/2 ==> Y/N = K 1/2 N-1/2 ==> y = k 1/2 ==> sk 1/2 = (n + d)k ==> k -1/2 = (n + d)/s d)/s = (0.07 + 0.03)/(.2) 0.03)/(.2) = 1/2 ==> k 1/2 = 2 = y ==> k = 4 .

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8.a. If technological progress occurs, then the level of output per capita for any given capital-labor ratio increases. increases. The function y = f(k) increases increases to y = g(k), and thus the savings savings function increases increases from sf(k) to sg(k).

y

g(k)

y2

f(k) (n +d)k sg(k)

y1 sf(k)

0 k 1

k 2

k

8.b. Since g(k) > f(k), it follows that sg(k) > sf(k) for each level of k. Therefore the intersection of the sg(k)-curve with the (n + d)k-line is at a higher level of k. The new steady-state equilibrium will now be at a higher level of saving and output per capita, and at a higher capital-labor ratio. 8.c. After the technological progress occurs, the level of saving and investment will increase until a new and higher optimal capital-labor ratio is reached. The ratio of investment to capital will also increase in the transition period, since more has to be invested to reach the higher optimal capital-labor ratio. k k 2

k 1 0 t1

t2

t

9. The Cobb-D Cobb-Dougl ouglas as produc production tion function function is define defined d as

32

θ

θ

Y = F(N,K) = AN 1- K . The marginal product of labor can then be derived as θ

θ

θ

θ

MPN = (∆Y)/(∆ N) = (1 - θ )AN- K = (1 - θ )AN1- K /N = = (1 - θ )(Y/N) ==> labor's share share of income = [MPN*(N)]/Y [MPN*(N)]/Y = (1 - θ )(Y/N)*[(N)/(Y)] = (1 - θ ) Additional Problems 1. Assume Assume labor's share share of income is 80%, 80%, capital's capital's share share of income is 20%, 20%, and the level of technology is fixed. Give a Cobb-Douglas aggregate production function with only two inputs, labor (N) and capital (K), that will represent such an economy. If labor grows by 2.4% and capital grows by 1.2%, by how much will output in this economy grow?

The Cobb-Douglas production function is of the general form: θ

θ

Y = F(N,K) = AN1- K , where Y represents the level of output. (1 - θ ) and θ are weights equal to the shares of labor (N) and capital (K) in production, while A is often used as a measure for the level of technology. In this case, this production function function has the following form: form: 0.8 0.2 Y = N = N K The growth of output can be calculated by using the following formula ∆Y/Y

= (1 - θ )(∆ N/N) + θ (∆K/K) + ∆A/A = (0.8)(2.4%) + (0.2)(1.2%) + 0% = 1.92% + 0.24% = + 2.16%,

2. Assume a Cobb-Douglas Cobb-Douglas aggregate aggregate production function function in which labor's share of income is 80% and capital's share of income is 20%. By how much would real GDP in this economy grow if labor grows at 3%, the capital stock grows at 2% and total factor productivity increases by 2.4%?

The growth of output can be calculated as the sum of the growth in labor times the labor share plus the growth of capital times the capital share plus the change in total factor productivity, productivity, that is, ∆Y/Y

= (1 - θ )(∆ N/N) + θ (∆K/K) + ∆A/A, where

1 - θ is the share of labor (N), θ is the share of capital (K), and productivity. productivity. Therefore, we can calculate total output growth as ∆Y/Y

∆A/A

= (0.8)(3%) + (0.2)(2%) + 2.4% = 2.4% + 0.4% + 2.4% = 5.2%.

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is the change in total factor

3. Assume a Cobb-Douglas Cobb-Douglas aggregate aggregate production function function in which labor's share of income is 80% and capital's share of income is 20%. By how much will real GDP in this economy grow if labor grows at 2.5%, the capital stock grows at 1.5%, and total factor productivity increases by 2.1%?

The growth of output can be calculated as the sum of the growth in labor times the labor share plus the growth of capital times the capital share plus the change in total factor productivity, productivity, that is, ∆Y/Y

= (1 - θ )(∆ N/N) + θ (∆K/K) + ∆A/A, where

1 - θ is the share of labor (N) and θ is the share of capital (K), and productivity. productivity. Therefore, we can calculate total output growth as ∆Y/Y

∆A/A

is the change in total factor

= (0.8)(2.5%) + (0.2)(1.5%) + 2.1% = 2.0% + 0.3% + 2.1% = 4.4%.

4. Assume that labor's share of income is is three times as large as capital's capital's share, and that capital, labor, and output all grow at the same rate (n). Use a Cobb-Douglas aggregate production function that has constant returns to scale to show that labor will be a larger source of growth than capital if there is no technological progress.

With the share of labor being 1 aggregate production function: θ

θ

= 0.75 and the share of capital being

θ

= 0.25, we get the following

θ

Y = AN1- K = AN3/4K 1/4 We can get the growth accounting equation by taking the total derivative of this equation and then dividing both sides of the equation by Y, that is ∆Y/Y

= ∆A/A + (1 - θ )(∆ N/N) + θ (∆K/K) = 0 + (0.75)n + (0.25)n = n.

Therefore we see that labor contributes three times as much to growth as capital in the absence of technological progress, that is, if ∆A/A = 0.

5. Assume an economy economy in which the level level of technology and the stock of capital capital are fixed. If there is a sudden increase in the labor force due to immigration, how would the standard of living be affected? Would it make a difference if the increase in the labor force were instead caused by a higher labor force participation rate of women?

If the increase in the labor force were entirely due to an influx of foreigners, then the population would increase and living standards would decrease, since per-capita income would decrease. However, if the increase in the labor force were due to an increase in the labor force participation of women, then income per capita would increase since the overall population would not increase. Therefore there would be an increase in living standards.

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6.

Briefly Briefly distinguish distinguish between between these two concepts: concepts: decreasing decreasing returns returns to scale and the law of diminishing marginal returns.

Decreas Decreasing ing return returnss to scale scale occur occur when when a propor proportio tional nal increa increase se in all inputs inputs leads leads to a less-t less-than han- proportional increase in output. The law of diminishing marginal returns states that the marginal product of each unit of input will decline as the amount of that input is increased, while all other inputs constant are held constant. In other words, if we only increase one of several inputs, output will increase but at a decreasing rate.

7.

Comment on the following statement: "Technological advances increase output in the short run but have no impact on long-run growth in output per capita, since this is determined entirely by the rate of population growth."

When technologic technological al advances advances occur, occur, then each input input becomes becomes more productive productive and the total factor productivity and growth of output rise. This implies that technological advances can have very profound implications for output growth. Total output grows at a rate equal to the sum of the growth rate of the population and the the rate of technological progress. progress.

8. "The Solow growth model model predicts that a country with a low savings savings rate can eventually eventually achieve the same living standard as a country with a high savings rate as long as both countries have the same same rate rate of popula populatio tion n growth growth and access access to the same same tec techno hnolog logy." y." Commen Commentt on this this statement.

The Solow model predicts that two countries that have the same production function, savings rate, and population growth will eventually reach the same level of income per capita. In other words, a poor country may eventually catch up to a richer one by saving at an equivalent rate to that of the richer country and by implementing technological innovations. However, if the two countries have different savings rates, then they will reach different levels of income per capita, even though their growth rates ultimately will be the same. Therefore the country with the lower savings rate will have a lower standard of living.

9.

"The neoclassica neoclassicall growth model model predicts that a nation's nation's per-capita per-capita GDP can only grow grow if technological progress occurs." Comment on this statement.

The neoclassical growth model generally assumes a production function with constant returns to scale. In the absence of technological progress, this model reaches a steady-state equilibrium at an optimal capitaloutput ratio k *. At this point output grows at the same rate as the population. This implies that output per capita remains constant. In the short-run, however, the capital-output ratio k may be below the optimal level k *. If so, the level of saving will exceed the level of investment needed to keep the capital-labor ratio constant. In this case, the capital-labor ratio will increase (which implies that output per head will increase) until the optimal level k * is reached. At k * the capital-labor ratio and output per head are both constant again.

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10. "A higher populati population on growth growth is always desirab desirable le since it will lead lead to higher living living standards standards." ." Comment on this statement with the help of a diagram derived from the neoclassical growth model.

If the rate of population growth (n) increases, the capital stock will grow at a lower rate than population. Since the country has to feed its people, not enough will be saved and invested to keep the capital-labor ratio (k) at its original level. Therefore, the capital-labor ratio (k) decreases until a new steady state is reached. In other words, the investment requirement, that is, the (n+d)k-line gets, steeper as population growth (n) increases. The (n+d)k-line will now intersect the sy-curve at a lower steady-state capital-labor ratio (k). This implies a lower level of output per capita (y), and therefore a lower living standard. Thus the statement is wrong.

y (n1 +d)k y

y1 y2

(n + d)k sy

k 2

k 1

k

11. "If the U.S. can increase increase its national national savings rate, rate, it can achieve a higher long-term long-term growth growth rate." Comment on this statement with the help of a diagram derived from the neoclassical growth model.

If the savings rate increases, then the sy-curve shifts up and we get a new intersection with the (n+d)k line at a higher steady state capital-labor ratio and a higher level of output per capita. In other words, as the level of savings and investment increases, the labor force will be supplied with more capital until a new and higher optimal capital-labor ratio (k) and output per capita level (y) is reached. Thus, the growth rate of real output will increase only temporarily. But in the long run, while the level of output per capita is higher, the growth rate of real r eal output will again be equal to the population growth (n).

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y y2

f(k)

y1

(n + d)k s’f(k) sf(k)

k k 1

k 2

12. "An increase increase in the depreciation depreciation rate reduces reduces the level of output per capita capita and the capitallabor ratio." Comment on this statement

An increasing depreciation rate (d) indicates that the capital stock is wearing out at a faster rate than previously. previously. This leads to a reduction in the optimal capital-labor ratio and in the level of output. In other words, the (n + d)k-line becomes steeper as the depreciation rate increases. The savings function sy = sf(k) now intersects the new (n + d')k-line at a lower capital-labor ratio (k), which implies a lower level of output per capita (y).

13. True True or false? false? Why? Why? "The neoclassical growth model predicts that a decrease in the savings rate permanently affects the growth rate of real output."

False. As the savings rate decreases, a smaller proportion of output is saved at each level of the capitallabor ratio (k). The savings curve shifts from sy = sf(k) to s 1y = s1f(k). Now savings is less than the investment requirement (n + d)k at the original capital-labor ratio k *, so the capital stock begins to grow more slowly. During the transition to the new equilibrium (at a lower steady-state capital-labor ratio and a lower output per capita), there is a temporary decline in the growth of output, and the level of output per capita decreases. However, in the long run the growth rate of real output will not be affected. y yo

y = f(k) (n+d)k

y1 sy

37

s1y

0 k o

k 1

k

14. Comment Comment on the following following stateme statement: nt: "A nation nation with with a dec declin lining ing popula populatio tion n growth growth will will experi experienc encee a fall fall in living living standa standards rds;; therefore governments should design policies to increase population growth."

If a nation's population growth declines, then the current level of saving will be higher than the level of investmen investmentt required required to maintain maintain a constant constant capital-labor capital-labor ratio. Therefore Therefore the capital-la capital-labor bor ratio will increase until a new steady-state equilibrium is reached. The new optimal capital-labor ratio is determined by the intersection of the saving-curve (sy) with the new (n 1 + d)k-line, that is, the new investment requirement line. The new and larger optimal capital-labor ratio (k *) implies a larger level of income per capita (y*) and therefore therefore a higher higher standard standard of living living for the country's country's population population.. Therefore, Therefore, there is no need to design policies to increase population growth. As a matter of fact, as the preceding analysis suggests, an increase in population growth would decrease living standards.

y y

y = f(k)

1

(no +d)k yo (n1 +d)k sy

0 k o

k 1

k

15. "A high level of public spending crowds out investment spending. Therefore, the government needs to cut its expenditures to free resources for investment." Comment on this statement.

By decreasing the level of public spending and reducing the budget deficit, the government can increase the level of national saving. Therefore, more funds will be available for investment. If the rate of capital accumulation increases, then the growth rate of output can increase at least temporarily, and this will

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increase the level of output per capita. However, the neoclassical growth model predicts that the longterm growth rate will not be affected since this rate is solely determined by population growth and technological progress. In addition, a decrease in government spending may, in the short run, lead to a recession, negatively affecting the level of investment spending.

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