2.     A decline in productivity growth is the primary reason for the slowdown in output growth in the United States since 1973. Productivity growth may have declined because of deterioration in the legal and human environment, reduced rates of technological innovation, and the effects of high oil prices. To some extent the apparent decline in productivity may be due to measurement difficulties.

3.     The rise in productivity growth in the 1990s occurred because of the revolution in information and communications technologies (ICT). Not only were there improvements in ICT, but also government regulations did not rein in the growth of productivity in the United States, as they did in other countries, such as those in Europe. In addition, intangible investment (research and development, reorganization of firms, and worker training) allowed the ICT improvements to boost productivity.

4.     A steady state is a situation in which the economy’s output per worker, consumption per worker,
and capital stock per worker are constant. 

5.     If there is no productivity growth, then output per worker, consumption per worker, and capital per worker will all be constant in the long run. This represents a steady state for the economy.

1.     (a)  The destruction of some of a country’s capital stock in a war would have no effect on the steady state, because there has been no change in s, f, n, or d. Instead, k is reduced temporarily, but equilibrium forces eventually drive k to the same steady-state value as before.

(b)  Immigration raises n from n1 to n2 in Figure 6.3. The rise in n lowers steady-state k, leading to a lower steady-state consumption per worker.

Figure 6.3

(c)  The rise in energy prices reduces the productivity of capital per worker. This causes sf(k) to shift down from sf1(k) to sf2(k) in Figure 6.4. The result is a decline in steady-state k. Steady-state consumption per worker falls for two reasons: (1) Each unit of capital has a lower productivity, and (2) steady-state k is reduced.

Figure 6.4

(d)  A temporary rise in s has no effect on the steady-state equilibrium.

(e)  The increase in the size of the labor force does not affect the growth rate of the labor force, so there is no impact on the steady-state capital-labor ratio or on consumption per worker. However, because a larger fraction of the population is working, consumption per person increases.

2.     (a)  Solow model

      The rise in capital depreciation shifts up the (n + d)k line from (n + d1)k to (n + d2)k, as shown in Figure 6.5. The equilibrium steady-state capital-labor ratio declines. With a lower capital-labor ratio, output per worker is lower, so consumption per worker is lower (using the assumption that the capital-labor ratio is not so high that an increase in k will reduce consumption per worker). There is no effect on the long-run growth rate of the total capital stock, because in the long run the capital stock must grow at the same rate (n) as the labor force grows, so that the capital-labor ratio is constant.

Figure 6.5

(b)  Endogenous growth model

      In an endogenous growth model, the growth rate of output is DY/Y = sA - d, so the rise in the deprecia-tion rate reduces the economy’s growth rate. Similarly, the growth rate of capital equals DK/K = sA - d, which also declines when the depreciation rate rises. Since consumption is a constant fraction of output, its growth rate declines as well. So the increase in the depreciation rate reduces the long-run growth rate of the capital stock, as well as long-run capital, output, and consumption per worker.

5.     The initial level of the capital-labor ratio is irrelevant for the steady state. Two economies that are identical except for their initial capital-labor ratios will have exactly the same steady state.

        Since the two economies must have the same growth rate at the steady state, and since the economy with the higher current capital-labor ratio has higher current output per worker, then the country with the lower current capital-labor ratio must grow faster.

        The answer holds true regardless of which country is in a steady state. If the country with a higher initial capital-labor ratio is in a steady state at capital-labor ratio k*, then the other country’s capital-labor ratio will rise until it too equals k*. So the country with the lower capital-labor ratio grows faster than the one with the higher capital-labor ratio.

        If the country with the lower initial capital-labor ratio is in a steady state at capital-labor ratio k*, then the other country’s capital-labor ratio is too high and it will decline until it equals k*. So the country with the higher capital-labor ratio must grow more slowly than the country with the lower capital-labor ratio. If the two countries are allowed to trade with each other, then their convergence to the same capital-labor ratio and output per worker will occur even faster.

2.    

 

20 Years Ago

Today

Percent Change

Y

1000

1300

30%

K

2500

3250

30%

N

  500

  575

15%

(a)  DA/A = DY/Y - aK DK/K - aN DN/N

                      = 30% - (0.3 ´ 30%) - 0.7 ´ 15%

                      = 30% - 9% - 10.5%

                      = 10.5%

      Capital growth contributed 9% (aK DK/K), labor growth contributed 10.5% (aN DN/N), productivity growth was 10.5%.

(b)  DA/A = 30% - (0.5 ´ 30%) - (0.5 ´ 15%)

            = 30% - 15% - 7.5%

            = 7.5%

      Capital growth contributed 15% (aK DK/K), labor growth contributed 7.5% (aN DN/N), productivity growth was 7.5%.

3.     (a) 

Year

K

N

Y

K/N

Y/N

1

200

1000

617

0.20

0.617

2

250

1000

660

0.25

0.660

3

250

1250

771

0.20

0.617

4

300

1200

792

0.25

0.660

      This production function can be written in per-worker form since Y/N = K.3N.7/N = K.3/N.3 =
(K/N).3. Note that K/N is the same in years 1 and 3, and so is Y/N. Also, K/N is the same in years 2 and 4, and so is Y/N.

(b) 

Year

K

N

Y

K/N

Y/N

1

200

1000

1231

0.20

1.231

2

250

1000

1316

0.25

1.316

3

250

1250

1574

0.20

1.259

4

300

1200

1609

0.25

1.341

      This production function can’t be written in per-worker form since Y/N = K.3N.8/N = K.3/N.2. Note that K/N is the same in years 1 and 3, but Y/N is not the same in these years. The same is true for years 2 and 4.

4.     To answer this problem, an approximate solution can be found by finding the ratio GDP (2006)/GDP (1950), taking the natural logarithm of that ratio and dividing by 56. This is the answer given in the table below.

 

Real GDP Per Capital

Growth

1950

2006

 

      Ratio

Rate

Australia

7,412

24,343

 

  3.28

2.1%

Canada

7,291

24,951

 

  3.42

2.2%

France

5,186

21,809

 

  4.21

2.6%

Germany

3,881

19,993

 

  5.15

2.9%

Japan

1,921

22,462

 

11.69

4.4%

Sweden

6,739

24,204

 

  3.59

2.2%

United Kingdom

6,939

23,013

 

  3.32

2.1%

United States

9,561

31,049

 

  3.25

2.1%

        Germany and Japan had the highest growth rates because damage from World War II caused capital per worker to be lower than its steady-state level, and thus output per worker was temporarily low.

5.     (a)  sf(k) = (n + d)k

      0.3 ´ 3k.5 = (0.05 + 0.1)k

      0.9k.5 = 0.15k

      0.9/0.15 = k/k.5

      6 = k.5

      k = 62 = 36

      y = 3k.5 = 3 ´ 6 = 18

      c = y - (n + d)k = 18 - (0.15 ´ 36) = 12.6

(b)  sf(k) = (n + d)k

      0.4 ´ 3k.5 = (0.05 + 0.1)k

      1.2k.5 = 0.15k

      1.2/0.15 = k/k.5

      8 = k.5

      k = 82 = 64

      y = 3k.5 = 3 ´ 8 = 24

      c = y - (n + d)k = 24 - (0.15 ´ 64) = 14.4

(c)  sf(k) = (n + d)k

      0.3 ´ 3k.5 = (0.08 + 0.1)k

      0.9k.5 = 0.18k

      0.9/0.18 = k/k.5

      5 = k.5

      k = 52 = 25

      y = 3k.5 = 3 ´ 5 = 15

      c = y – (n + d)k = 15 - (0.18 ´ 25) = 10.5

(d)  sf(k) = (n + d)k

      0.3 ´ 4k.5 = (0.05 + 0.1)k

      1.2k.5 = 0.15k

      1.2/0.15 = k/k.5

      8 = k.5

      k = 82 = 64

      y = 4k.5 = 4 ´ 8 = 32

      c = y – (n + d)k = 32 - (0.15 ´ 64) = 22.4