Econ 320 Lecture 09 Practice Problems

What is necessary for steady-state growth in the Solow model? - endogenous technological change - MPK = δ + n + g - a higher saving rate - technological progress

- technological progress

If Y is output, K is capital, u is the fraction of the labor force in universities, L is labor, E is the stock of knowledge, and the production Y = F (K, (1 - u) EL) exhibits constant returns to scale, then output (Y) will double if: K is doubled. K and u are doubled. K and E are doubled. L is doubled.

K and E are doubled.

Population Growth and Technological Progress — Quick Quiz Problem The purpose of (exogenous/endogenous) growth theory is to explain technological progress. Some of these models do so by questioning the Solow model's assumption of (constant/diminishing) returns to capital.

endogenous diminishing

According to the Kremerian model, large populations improve living standards because: crowded conditions put more pressure on people to work hard. there are more people who can make discoveries and contribute to innovation. more people have the opportunity for leisure and recreation. most people prefer to live with many other people.

there are more people who can make discoveries and contribute to innovation.

Which of these statements is NOT true about the steady state in the Solow Model with population and technological progress: capital per effective worker and output per effective worker are constant. capital per worker and output per worker grow at the rate of technological progress. total capital stock and total output grow at the rate of population growth. saving per effective worker and consumption per effective worker are constant.

total capital stock and total output grow at the rate of population growth.

Introducing labor-augmenting technological progress into the production function requires that the production function Y = f(K, L) be rewritten as: - Y = E × f(K, L). - Y = f(K, E × L). - E × Y = f(K, L). - Y = f(E × K, E × L).

- Y = f(K, E × L). Labor-augmenting technological progress makes each unit of labor more effective, as if there were an actual increase in labor and expresses the labor input in efficiency units.

In the Solow model with population growth, a rise in the growth rate of the labor force will: - cause capital per worker to fall. - cause capital per worker to rise. - lower the rate of depreciation. - cause an increase in investment.

- cause capital per worker to fall.

Sustained growth of output per worker: is the main prediction of the Solow model. is not explained by capital accumulation in the Solow model. is linked to temporary negative growth of capital per worker. results in a higher saving rate.

is not explained by capital accumulation in the Solow model.

In the Solow model with technological progress, the steady-state growth rate of total output is: 0. g. n. n + g.

n + g

Who were the Luddites? - consumers of woven fabrics who resisted the replacement of artisanal production by mass-production - factory owners who introduced new weaving machines into their factories, threatening the jobs of artisan weavers - developers of new technologies that made it possible for low-skilled workers to operate weaving machines - skilled artisans in England who opposed the introduction of weaving machines

- skilled artisans in England who opposed the introduction of weaving machines

Suppose an economy has 100 units of capital, 100 units of labor, and the efficiency of each worker is equal to 2. The effective number of workers for this economy is _____ and the capital per effective worker is _____. 50; 2 200; 1/2 50; 1/2 200; 2

200; 1/2

If the per-worker production function is given by y = k^1/2, the saving rate is s = 0.25, the population growth rate is n = 0.01, the depreciation rate is δ = 0.02, and the rate of labor-augmenting technological progress is g = 0.02, then the steady-state level of capital per effective worker is: 2.0. 20. 25. 30.

25

If the per-worker production function is given by y = k^1/2, the saving rate is s = 0.25, the population growth rate is n = 0.01, and depreciation is δ = 0.04, then the steady-state level of capital per worker is: 2. 3. 5. 25.

25 In the steady state, sf(k*) = (δ + n)k*. In the steady state, sf(k*) = (δ + n)k*, so with the data given, 0.25 × (k*)^1/2 = (0.04 + 0.01) × k*.

Suppose an economy is described by the Solow model. The rate of population growth is 1 percent, the rate of technological progress is 3 percent, the depreciation rate is 5 percent, and the saving rate is 10 percent. In steady state, output per worker grows at a rate of ________ percent. 1 3 2 4

3%

An economy has the production function y = k1/2 and has a saving rate of s = 0.20, a depreciation rate of δ = 0.03, a population growth rate of n = 0.05, and a rate of labor-augmenting technological progress of g = 0.02. This implies that in the steady state, total output grows at _____ percent. 2 4 7 12

7 Y = n + g

The growth rate of a labor force is 0.03, and in 2020, 80 million people are working. At this rate, in 2021, _____ million people will be working one year later. 77.6 80 82.4 104

82.4

When u — the fraction of the labor force in universities — increases, the growth rate of the stock of knowledge E _____, and break-even investment ____. increases; increases increases; decreases decreases; increases decreases; decreases

increases; increases

In the Solow model with technological progress and population growth, _____ reflects society's knowledge of production methods. the efficiency of labor the labor-force participation rate productivity endogenous technological growth

the efficiency of labor

The rate of labor-augmenting technological progress (g) is the growth rate of: labor. the efficiency of labor. capital. output.

the efficiency of labor.

In the Solow model, _____ is output per worker. - n - y - g - k

y

When u increases, the immediate impact is that more of the labor force is employed in research activities, and, initially, output in the manufacturing sector tends to: - remain the same. - fluctuate. - increase. - decrease.

- decrease Given a constant L, if u rises, what must be the immediate impact on 1 − u?

Suppose that an economy has reached the Golden Rule steady state and then, because of favorable economic policies, the rate of technological progress increases. In order to return to the Golden Rule steady state, the saving rate would need to: - increase. - do nothing; the saving rate is irrelevant to the Golden Rule steady state. - remain the same. - decrease.

- decrease. At the Golden Rule level of capital per effective worker, MPK = δ + n + g, so if g increases, then the MPK would need to increase to return the economy to a Golden Rule steady state; does this imply that the saving rate would have to decrease or increase?

In the Solow model, technological progress is determined outside the model. However, a different set of growth models, called ____ growth theory, explain technological progress within the model. - endocardial - endogamous - exogenous - endogenous

- endogenous

In the two-sector model, break-even investment must perform all of these functions EXCEPT: - give firms permanent monopolies on new products so they can recoup their investments. - provide capital for a growing population. - provide capital for the greater stock of knowledge created by universities. - compensate for depreciation.

- give firms permanent monopolies on new products so they can recoup their investments.

In the Solow model, with labor-augmenting technological progress and population growth, if capital per effective worker, k = K / (E × L), is constant, then capital per worker, K / L, will: - decrease. - increase. - remain the same. - fluctuate.

- increase. Technological progress does not cause the actual number of workers to change; however, because each worker in effect comes with more units of labor over time, technological progress changes the effective number of workers.

Regarding the social return to research, the textbook suggests it: - is much larger than the returns to physical capital. - is about the same as the returns to physical capital. - is not as large as the returns to physical capital. - cannot be measured.

- is much larger than the returns to physical capital. Although theory alone cannot determine whether research effort is greater or less than optimal, the empirical work on this topic is usually less ambiguous. Many studies have suggested that the "standing on shoulders" externality is important and that, as a result, the social return to research is large — often more than 40 percent per year. This rate of return is impressive, especially when compared with the return to physical capital, which in Chapter 10 we will show to be about 6 percent per year. In the judgment of some economists, this finding justifies substantial government subsidies to research.6

In the Solow model with population growth and technological progress, the break-even amount of investment: - keeps the rate of labor-augmenting technological progress constant. - equals the rate of population growth. - keeps the stock of capital per worker constant. - equals the depreciation rate, δ.

- keeps the stock of capital per worker constant. An economy is in a steady state if capital per worker k is unchanging.

Investment in the two-sector model is determined by the saving rate and total output. Break-even investment is determined by [δ + n + g(u)]k. The intersection of the investment curve and the break-even investment curve determines the: - steady-state level of capital per effective worker. - Golden Rule level of capital per worker. - steady-state level of capital per worker. - Golden Rule level of capital per effective worker.

- steady-state level of capital per effective worker.

Private firms have NO incentive to engage in research and development if: - the results create a less competitive marketplace. - there are positive externalities from research and development. - they cannot realize a positive return on their investment. - innovations give firms temporary monopolies.

- they cannot realize a positive return on their investment. If one thinks about the process of research and development for even a moment, three facts become apparent. First, although knowledge is largely a public good (that is, a good nonrival in use and freely available to everyone), much research is done by firms driven by the profit motive. Second, research is profitable because innovations give firms temporary monopolies, either due to the patent system or due to the advantage of being the first firm on the market with a new product. Third, when one firm innovates, other firms build on that innovation to produce the next generation of innovations. These (essentially microeconomic) facts are not easily connected with the (essentially macroeconomic) growth models we have discussed so far.

Malthus felt anti-poverty programs would be counterproductive because: - the poor were poor because they were lazy. - lifting people out of poverty would cause problems for employers. - they merely enable the poor to have more children. - they would be run by charities and governments.

- they merely enable the poor to have more children.

In the Solow model with technological progress, the steady-state growth rate of capital per effective worker is: 0. g. n. n + g.

0

The effective number of workers is the product of E and L. When E grows at a rate of g = 0.05, and L grows at a rate of n = 0.01, the effective number of workers grows at a rate of: −0.04. 0.04. 0.05. 0.06.

0.06.

Population Growth and Technological Progress — Work It Out In the nation of Winknam, the capital share of GDP is 35 percent, the average growth in output is 4.0 percent per year, the depreciation rate is 6.0 percent per year, and the capital-output ratio is 3.5. Suppose that the production function is Cobb-Douglas and that Winknam has been in a steady state. Round answers to two places after the decimal when necessary. b. In the initial steady state, what is the marginal product of capital (MPK)?

0.1

A country's labor force is 60 million people in year 1. The labor force is 62.4 million people two years later. What is the growth rate of the labor force? 1 percent per year 2 percent per year 2.4 percent per year 62.4

2 percent per year Solve the future value formula, FV = PV(1 + r)t, for r.

A country's labor force is 80 million people in year 1 and 82 million in year 2. What is the labor force growth rate? 2.5 percent 2 percent 3 percent 2 million

2.5% The formula for a percentage change is (V2 − V1)/V1 × 100, where V1 is the initial value and V2 is the current value.

Population Growth and Technological Progress — End of Chapter Problem Suppose an economy is described by the Solow model. The rate of population growth is 1%, the rate of technological progress is 3%, the depreciation rate is 5%, and the saving rate is 10%. In a steady state, output per person grows at a rate of = %

3% output per person = rate of tech progress

Population Growth and Technological Progress — Work It Out In the nation of Winknam, the capital share of GDP is 35 percent, the average growth in output is 4.0 percent per year, the depreciation rate is 6.0 percent per year, and the capital-output ratio is 3.5. Suppose that the production function is Cobb-Douglas and that Winknam has been in a steady state. Round answers to two places after the decimal when necessary. a. In the initial steady state, what is the savings rate (𝑠)?

35%

If the per-worker production function is given by y = k1/2, the saving rate is s = 0.16, the population growth rate is n = 0.02, and depreciation is δ = 0.06, then the steady-state level of capital per worker is: 4. 6. 8. 16.

4

If the per-worker production function is given by y = k1/2, the saving rate is s = 0.16, the population growth rate is n = 0.03, the depreciation rate is δ = 0.03, and the rate of labor-augmenting technological progress is g = 0.02, then the steady-state level of capital per effective worker is: 4. 8. 16. 32.

4

The production function of an economy is y = k1/2, in which k is capital per effective worker. When K = 1,200, E = 1.5, and L = 50, then y equals: 4. 6. 9. 24.

4

Which of these statements is NOT true about the creation of knowledge and the process of research and development? Knowledge is a private good, that is, rival and excludable. Much of the research and development is done by firms driven by the profit motive. Patents provide firms with monopoly power that makes research profitable. Most innovations build on previous innovations.

Knowledge is a private good, that is, rival and excludable.

_____ predicted that the pressures of an increasing population would cause widespread poverty. Michael Kremer Thomas Piketty Thomas Malthus Paul Samuelson

Thomas Malthus

Population Growth and Technological Progress — Work It Out An economy has a Cobb-Douglas production function: 𝑌=𝐾𝛼(𝐿𝐸)1−α The economy has a capital share of 0.25, a saving rate of 46 percent, a depreciation rate of 3.25 percent, a rate of population growth of 5.25 percent, and a rate of labor-augmenting technological change of 3.5 percent. It is in steady state. At what rates do total output and output per worker grow? Total output growth rate: % Output per worker growth rate: % Output per effective worker? increases in the steady state. is constant in the steady state and does not change. declines in the steady state.

Total output growth rate: 8.75% Output per worker growth rate: 3.5% is constant in the steady state and does not change.

In the two-sector model, what would result in a constant growth rate in the efficiency of labor? a constant fraction of the labor force in universities increasing the fraction of the labor force in universities to offset diminishing returns to labor a lower rate of depreciation a saving rate that results in an amount of investment that offsets depreciation

a constant fraction of the labor force in universities

In the two-sector model, what would result in a constant growth rate in the efficiency of labor? a constant fraction of the labor force in universities increasing the fraction of the labor force in universities to offset diminishing returns to labor a saving rate that results in an amount of investment that offsets depreciation a lower rate of depreciation

a constant fraction of the labor force in universities

Population Growth and Technological Progress — End of Chapter Problem Assume the economy is in a steady-state for all of the following questions. a. In a steady-state, we know that 𝑠𝑓(𝑘)=(𝛿+𝑛+𝑔)𝑘. By rearranging this equation, we can prove that 𝑓=𝛿+𝑛+𝑔/𝑠; so the production function is constant and does not exhibit diminishing returns. 𝑘/𝑦=𝑠/𝛿+𝑛+𝑔; meaning the capital-output ratio is constant. 𝑠𝑓=𝛿+𝑛+𝑔; so saving is a function of depreciation and the growth rates of population and technological progress. b. In the Solow model with technological progress, capital's share of income is constant*, and labor's share of income is constant*. c. In the Solow model with technological progress, total capital income and total labor income grow the rate: n+g* d. In the Solow model with technological progress, the real rental price of capital grows at a rate of zero*, and the real wage grows at a rate of g*. (Hint: The real rental price of capital equals total capital income divided by the total capital stock, and the real wage equals total labor income divided by the labor force.)

a. 𝑘/𝑦=𝑠/𝛿+𝑛+𝑔; meaning the capital-output ratio is constant.

In the Solow growth model with population growth but no technological progress, the steady-state amount of investment can be thought of as a break-even amount of investment because the quantity of investment just equals the amount of: output needed to achieve the maximum level of consumption per worker. capital needed to replace depreciated capital and to equip new workers. saving needed to achieve the maximum level of output per worker. output needed to make the capital per worker ratio equal to the marginal product of capital.

capital needed to replace depreciated capital and to equip new workers.

The ratio k = K / (E × L) is: capital per effective worker. capital gains. labor-augmenting technological progress. the labor theory of value.

capital per effective worker.

In the two-sector endogenous growth model, income growth persists because the: production function shifts exogenously. saving rate exceeds the rate of depreciation. creation of knowledge in universities never slows down. fraction of the labor force in universities is large.

creation of knowledge in universities never slows down

When u increases, the immediate impact is that more of the labor force is employed in research activities, and, initially, output in the manufacturing sector tends to: remain the same. fluctuate. increase. decrease.

decrease

An increase in u initially will _____ output and consumption; in the long run, output will grow more quickly because of the faster growth in _____. increase; E increase; K decrease; K decrease; E

decrease; E

In the Solow model with population growth and no technological progress, an increase in the population growth rate leads to a(n) _____ in the effective investment rate leading to a(n) _____ in the steady-state income per worker. increase; increase increase; decrease decrease; increase decrease; decrease

decrease; decrease

When u decreases, break-even investment _____, and the growth rate of the stock of knowledge E ____. decreases; decreases increases; decreases decreases; increases increases; increases

decreases; decreases

The efficiency of labor: is the marginal product of labor. is the rate of growth of the labor force. depends on the knowledge, health, and skills of labor. equals output per worker.

depends on the knowledge, health, and skills of labor.

In the Solow growth model of an economy with population growth but no technological change, the break-even level of investment must do all of these EXCEPT: offset the depreciation of existing capital. provide capital for new workers. equal the marginal productivity of capital (MPK). keep the level of capital per worker constant.

equal the marginal productivity of capital (MPK).

Analysis of population growth around the world concludes that countries with high population growth tend to: have high income per worker. have a lower level of income per worker than countries with low population growth. have the same standard of living as other parts of the world. be the high-income-producing nations of the world.

have a lower level of income per worker than countries with low population growth.

Population Growth and Technological Progress — Quick Quiz Problem Models of Schumpeterian creative destruction aim to explain why seeming technological progress can reduce average incomes. how entrepreneurs with new products displace incumbent producers. why economies grow quickly after suffering the ravages of war. how old capital is best retired and replaced with new capital.

how entrepreneurs with new products displace incumbent producers.

Population Growth and Technological Progress — Work It Out An economy has a Cobb-Douglas production function: 𝑌=𝐾𝛼(𝐿𝐸)1−α The economy has a capital share of 0.25, a saving rate of 46 percent, a depreciation rate of 3.25 percent, a rate of population growth of 5.25 percent, and a rate of labor-augmenting technological change of 3.5 percent. It is in steady state. b. Solve for capital per effective worker (𝑘∗), output per effective worker (𝑦∗), and the marginal product of capital.

k* = 6 y* = 1.57 MPK = 0.065

Population Growth and Technological Progress — End of Chapter Problem Suppose an economy described by the Solow model has the following production function: 𝑌=𝐾12(𝐿𝐸)12 a. For this economy, what is 𝑓(𝑘)? 𝑓(𝑘) = b. Use your answer in part a to solve for the steady-state value of y as a function of s, n, g, and 𝛿. 𝑦∗ = Suppose two neighboring economies have the above production function, but they have different parameter values. Atlantis has a saving rate of 28% per year and a population growth rate of 1% per year. Xanadu has a saving rate of 10% per year and a population growth rate of 4% per year. In both countries, 𝑔=0.02 and 𝛿=0.04. Solve for the steady-state value of y for each country. c. 𝑦∗𝐴𝑡𝑙𝑎𝑛𝑡𝑖𝑠y = d. 𝑦∗𝑋𝑎𝑛𝑎𝑑𝑢y =

k^1/2 y= s/(𝛿+n+g) c. 𝑦∗𝐴𝑡𝑙𝑎𝑛𝑡𝑖𝑠y = 4 d. 𝑦∗𝑋𝑎𝑛𝑎𝑑𝑢y = 1

Thomas Malthus believed that higher population growth allows economies to take advantage of economies of scale. larger populations experience greater innovation because they have more scientists and inventors. higher population growth depresses the steady-state amount of capital per worker. larger populations put a strain on an economy's food-producing capacity.

larger populations put a strain on an economy's food-producing capacity.

Population Growth and Technological Progress — Work It Out An economy has a Cobb-Douglas production function: 𝑌=𝐾𝛼(𝐿𝐸)1−α The economy has a capital share of 0.25, a saving rate of 46 percent, a depreciation rate of 3.25 percent, a rate of population growth of 5.25 percent, and a rate of labor-augmenting technological change of 3.5 percent. It is in steady state. c. The economy has (less/more/the same amount) capital than at the Golden Rule steady state. To achieve the Golden Rule steady state, the saving rate needs to (decrease/increase/stay the same). d. Suppose the change in the saving rate you described in part c occurs. During the transition to the Golden Rule steady state, the growth rate of output per worker will be (greater than/less than/the same as) the rate you derived in part a. After the economy reaches its new steady state, the growth rate of output per worker will be ?

more decrease less than 3.5%

Population Growth and Technological Progress — Quick Quiz Problem In the Solow model with population growth and technological progress, at the Golden Rule steady state, the marginal product of capital MPK equals n + g g n + g + δ n

n + g + δ

Schumpeter's thesis of "creative destruction" is an explanation of economic progress resulting from: using up scarce natural resources to create new products. breaking down barriers to trade and development. new product producers driving incumbent producers out of business. creating new methods to destroy the environment.

new product producers driving incumbent producers out of business.

If MPK > δ + n in the steady state, then the slope of the _____ is larger than the slope of the break-even investment line, and increasing steady-state capital per worker will _____ consumption per worker. production function; increase production function; decrease consumption function; increase consumption function; decrease

production function; increase

In a steady-state economy with a saving rate s, population growth n, depreciation rate 𝛿, and labor-augmenting technological progress g, the steady-state ratio of capital per effective worker (k*), in terms of output per effective worker (f (k*)), is sf (k) / (𝛿 + n + g). s / ((f (k))(𝛿 + n + g)). f (k) / ((s)(𝛿 + n + g)). (s - f (k)) / (𝛿 + n + g).

sf (k) / (𝛿 + n + g).

The efficiency of labor reflects: society's knowledge of production methods. how much workers produce per time period. the percentage of time workers engage in production as opposed to non-productive activities. the percentage of people who could be in the labor force who are actually in the labor force.

society's knowledge of production methods.

Investment in the two-sector model is determined by the saving rate and total output. Break-even investment is determined by [δ + n + g(u)]k. The intersection of the investment curve and the break-even investment curve determines the: steady-state level of capital per worker. steady-state level of capital per effective worker. Golden Rule level of capital per worker. Golden Rule level of capital per effective worker.

steady-state level of capital per effective worker.

Endogenous growth theory rejects the assumption of exogenous: production functions. rates of depreciation. population growth rates. technological change.

technological change.

According to the Solow model, persistently rising living standards can only be explained by: population growth. capital accumulation. saving rates. technological progress.

technological progress

In the Solow model with population growth and technological progress, what explains persistently rising living standards? technological progress depreciation saving capital accumulation

technological progress

In the Solow growth model with population growth but no technological progress, when the economy finds itself at the Golden Rule steady state, the marginal product of capital minus the rate of depreciation will equal: 0. the population growth rate. the saving rate. output per worker.

the population growth rate.

In a steady state with population growth and technological progress: the real rental price of capital is constant and the real wage grows at the rate of technological progress. the real rental price of capital grows at the rate of technological progress and the real wage is constant. both the real rental price of capital and the real wage grow at the rate of technological progress. both the real rental price of capital and the real wage are constant.

the real rental price of capital is constant and the real wage grows at the rate of technological progress.

In the Solow model, with labor-augmenting technological progress and population growth, if the production function is y = k1/2, s = 0.5, δ = 0.02, n = 0.01, and g = 0.02, then the steady-state level of capital per worker is _____ the Golden Rule level. less than more than the same as exactly positively correlated to

the same as

If the production function is y = k1/2, the steady-state value of y in the Solow model with population growth and technological progress is: y = ((s + g) / (𝛿 + n))1/2. y = (s + g) / (𝛿 + n). y = (2 / (𝛿 + n + g))1/2. y = s / (𝛿 + n + g).

y = s / (𝛿 + n + g)

Population Growth and Technological Progress — Work It Out In the nation of Winknam, the capital share of GDP is 35 percent, the average growth in output is 4.0 percent per year, the depreciation rate is 6.0 percent per year, and the capital-output ratio is 3.5. Suppose that the production function is Cobb-Douglas and that Winknam has been in a steady state. Round answers to two places after the decimal when necessary. c. Suppose that public policy alters the saving rate so that the economy reaches the Golden Rule level of capital. What will the marginal product of capital be at the Golden Rule steady state (𝑀𝑃𝐾∗𝑔𝑜𝑙𝑑)? 𝑀𝑃𝐾∗𝑔𝑜𝑙𝑑= d. What will the capital-output ratio be at the Golden Rule steady state (𝐾/𝑌∗𝑔𝑜𝑙𝑑)? 𝐾/𝑌∗𝑔𝑜𝑙𝑑= e. What must the saving rate be to reach the Golden Rule steady state (𝑠𝑔𝑜𝑙𝑑)? 𝑠𝑔𝑜𝑙𝑑= %

𝑀𝑃𝐾∗𝑔𝑜𝑙𝑑= 0.1 𝐾/𝑌∗𝑔𝑜𝑙𝑑= 3.5 𝑠𝑔𝑜𝑙𝑑= 35%