Econ 320 Lecture 08 Practice Problems

Consider an economy with y = f(k) = k2/3, s = 0.25, and δ = 0.05. In this economy, the steady-state capital stock per worker equals: 5. 25. 100. 125.

125 Using the information given, sf(k*) = δk* becomes 0.25 × k*2/3 = 0.05 × k*, or 0.25 / 0.05 = k*1/3; solve for k*.

If y = k1/2, there is no population growth or technological progress, 5 percent of capital depreciates each year, and a country saves 20 percent of output each year, then the steady-state level of capital per worker is: 2. 4. 8. 16.

16

When the capital stock per worker is lower than the steady-state capital stock per worker, the capital stock per worker will: shrink because investment exceeds depreciation. shrink because depreciation exceeds investment. increase because investment exceeds depreciation. increase because depreciation exceeds investment.

increase because investment exceeds depreciation. When k < k*, investment will exceed depreciation.

Suppose that the per-worker production function y = f(k) = k0.5, that the saving rate is s = 0.2, and that the depreciation rate is δ = 0.02. If k = 49, then the capital stock per worker will: increase. decrease. remain the same. be impossible to compute given then information available.

increase.

If the production function is y = 2k1/2, δ = 0.1, and k = 10, then the saving rate is _____ the Golden Rule level. the same as erratic compared to more than less than

less than

If the production function is y = 3k1/3, δ = 0.2, and k = 10, then the saving rate is ____ the Golden Rule level. erratic compared to the same as more than less than

less than

If the production function is y = 5k1/5, δ = 0.5, and k = 1, then the saving rate is _____ the Golden Rule level. less than more than the same as erratic compared to

less than

If the production function is y = k1/2, s = 0.4, and δ = 0.1, then the steady-state level of capital per worker is _____ the Golden Rule level. less than more than the same as irrelevant to

less than

If the production function is y = k^1/4, δ = 0.1, and k = 2, then the saving rate is _____ the Golden Rule level. greater than the same as less than erratic compared to

less than MPK = 1 / 4k^3/4; is the net marginal product (MPK − δ) greater than, equal to, or less than zero? What does this imply about k and the current saving rate?

Ch8 Quick Quiz In the steady state of the Solow model, higher population growth leads to a ___lower/higher)____ level of income per worker and ___lower/higher)____ growth in total income.

lower/higher

The consumption function in the Solow model assumes that society saves a: constant proportion of income. smaller proportion of income as it becomes richer. larger proportion of income as it becomes richer. larger proportion of income when the interest rate is higher.

constant proportion of income.

If an economy has achieved the Golden Rule steady state, then: consumption per worker cannot be increased. output matches depreciation. investment is larger than depreciation. consumption per worker can be increased by lowering the saving rate.

consumption per worker cannot be increased.

The Golden Rule level of capital is the steady-state value of k that: maximizes saving. maximizes growth. maximizes output. maximizes consumption.

maximizes consumption.

If the production function is y = k1/4, δ = 0.05, and k = 5, then the saving rate is _____ the Golden Rule level. less than erratic compared to greater than the same as

less than

If the production function is y = 2k1/2 and δ = 0.2, then the Golden Rule level is reached when k is: 7. 25. 50. 14.

25

Consider an economy with y = f(k) = k1/2, s = 0.25, and δ = 0.05. If the capital stock per worker in year 1 is at k = 16, then the capital stock in year 2 equals: - 15.8. - 16. - 16.2. - 17.

- 16.2.

Suppose that the per-worker production function y = f(k) = k1/3 and that the saving rate is s = 0.2. If k = 125,000, investment per worker is i equals: 10. 25. 50. 100.

10

If a war destroys a large portion of a country's capital stock but the saving rate is unchanged, the Solow model predicts that output will grow and that the new steady state will approach: a higher level of output per person than before. the same level of output per person as before. a lower level of output per person than before. the Golden Rule level of output per person.

the same level of output per person as before.

If a country had a saving rate of 100 percent, then: there would be no consumption. there would be infinite growth. there would be no steady state. well-being would be maximized.

there would be no consumption.

If a country had a saving rate of 100 percent, then: well-being would be maximized. there would be no steady state. there would be infinite growth. there would be no consumption.

there would be no consumption.

The Golden Rule level of steady-state consumption per worker is:

AB

In the steady state, depreciation equals investment. Therefore, the equation sf(k*) = δk* can be solved for k*, the steady-state level of capital per worker. For f(k) = k1/2, s = 0.1, and δ = 0.05, the steady-state level of capital k* equals: 0.2. 0.4. 2. 4.

4

Two countries, Anastasia and Beersheba, have identical production functions y = f(k), but Anastasia has a higher saving rate than Beersheba. This implies that, for identical levels of capital per worker, Anastasia has: - lower output per worker than Beersheba. - higher consumption per worker than Beersheba. - lower consumption per worker than Beersheba. - lower investment per worker than Beersheba.

- lower consumption per worker than Beersheba. c = (1 − s)y = (1 − s)f(k), so with identical k, what would the higher s of Anastasia imply regarding consumption per worker in the two countries?

In this graph, starting from capital-labor ratio k1, the capital-labor ratio will: decrease. remain constant. increase. first decrease and then remain constant.

increase

Two countries, Argo and Beaurepaire, have identical production functions. If Argo has a higher saving rate than Beaurepaire, and Argo has a higher rate of depreciation than Beaurepaire, then: - Argo has a higher steady-state capital stock per worker than Beaurepaire. - Beaurepaire has a higher steady-state capital stock per worker than Argo. - it is indeterminate whether Argo or Beaurepaire has a higher steady-state capital stock per worker. - steady-state capital stocks per worker in Argo and Beaurepaire will be equal.

- it is indeterminate whether Argo or Beaurepaire has a higher steady-state capital stock per worker. The steady-state capital stock per worker increases as the saving rate increases and decreases as the depreciation rate increases.

Two countries, Agora and Bensalem, have identical production functions f(k) and saving rates s, but Agora has a higher capital-labor ratio k than Bensalem. This implies all of these EXCEPT that: - per capita investment in Agora is higher than in Bensalem. - output is higher in Bensalem than in Agora. - the marginal product of capital in Agora is lower than in Bensalem. - per capita output in Agora is higher than in Bensalem.

- output is higher in Bensalem than in Agora.

When the economy is in the steady state and the saving rate increases, the growth rate of output per worker: - may be positive, neutral, or negative for all subsequent periods. - will be positive until the economy reaches a new steady state. - will decline until the economy reaches a new steady state. - will be positive for all subsequent periods.

- will be positive until the economy reaches a new steady state.

The steady-state level of capital occurs when the change in the capital stock per worker (Δk) equals: 0. the saving rate. the depreciation rate. the population growth rate.

0

Consider the production function y = k^0.5. The marginal product of capital MPK = f(k + 1) − f(k) for k = 25 is: 0.1. 0.5. 1. 25.

0.1 MPK = 26^0.5 − 25^0.5

Assume that the per-worker production function y = f(k) = k0.5 and that the saving rate is s = 0.2. If k = 49, then investment per worker is: 0.7. 1.4. 7. 14.

1.4

Suppose that the per-worker production function y = f(k) = k^1/3 and that the saving rate is s = 0.2. If k = 125,000, investment per worker is i equals: 10. 25. 50. 100.

10 y = 125,000^1/3 = 50, and so i = sy = 0.2 × 50.

_____ cause(s) the capital stock to rise, while _____ cause(s) the capital stock to fall. Inflation; deflation Interest rates; the discount rate Investment; depreciation International trade; depressions

Investment; depreciation

An economy will _____ move to a steady state but _____ to the Golden Rule steady state. always; never always; not necessarily sometimes; not necessarily rarely; never

always; not necessarily

Suppose the economy is originally at a steady state where the marginal product of capital is less than the depreciation rate. If the saving rate of the economy changes to a rate consistent with the golden rule level of capital, then at the new steady state capital per worker will be higher compared to the original steady state. output per worker will be higher compared to the original steady state. investment per worker will be higher compared to the original steady state. consumption per worker will be higher compared to the original steady state.

consumption per worker will be higher compared to the original steady state.

Assume that a war reduces a country's labor force but does not directly affect its capital stock. If the economy was in a steady state before the war and the saving rate does not change after the war, then, over time, capital per worker will _____, and output per worker will _____ as it returns to the steady state. decline; increase increase; increase decline; decrease increase; decrease

decline; decrease

If a production function has constant returns to scale, then: the law of diminishing returns is violated. the rate of profit remains constant as output expands. doubling the amounts of capital and labor will double output. doubling the amounts of capital and labor will triple output.

doubling the amounts of capital and labor will double output.

If the production function is y = 5k^1/5, δ = 0.1, and k = 50, then the saving rate is _____ the Golden Rule level. the same as more than less than erratic compared to

more than MPK = 1 / k^4/5; is the net marginal product (MPK − δ) greater than, equal to, or less than zero? What does this imply about k and the current saving rate?

Two countries, Agora and Bensalem, have identical production functions f(k) and saving rates s, but Agora has a higher capital-labor ratio k than Bensalem. This implies all of these EXCEPT that: output is higher in Bensalem than in Agora. per capita investment in Agora is higher than in Bensalem. per capita output in Agora is higher than in Bensalem. the marginal product of capital in Agora is lower than in Bensalem.

output is higher in Bensalem than in Agora.

The production function in Figure 8-1, y = f(k), implies that as k increases: output Y decreases at a decreasing rate. output Y increases at a constant rate. output per worker increases at a constant rate. output per worker increases at a decreasing rate.

output per worker increases at a decreasing rate. The Production Function The production function shows how the amount of capital per worker k determines the amount of output per worker y = f(k). The slope of the production function is the marginal product of capital: If k increases by 1 unit, y increases by MPK units. The production function becomes flatter as k increases, indicating diminishing marginal product of capital.

In the Solow growth model, the assumption of constant returns to scale means that: all economies have the same amount of capital per worker. the steady-state level of output is constant, regardless of the number of workers. the saving rate equals the constant rate of depreciation. the number of workers in an economy does not affect the relationship between output per worker and capital per worker.

the number of workers in an economy does not affect the relationship between output per worker and capital per worker.

Which per-worker production function does NOT exhibit diminishing returns to capital? y = k y = k0.5 y = k0.7 y = k0.8

y = k

What are the two ways to determine the Golden Rule level of the capital stock? - Find the capital stock at which steady-state consumption is maximized, and find the capital stock at which the the depreciation rate equals the saving rate. - Find the capital stock at which capital per worker is maximized, and find the capital stock at which the marginal product of capital equals the saving rate. - Find the capital stock at which steady-state consumption is maximized, and find the capital stock at which the net marginal product of capital equals zero. - Find the capital stock at the saving rate is maximized, and find the capital stock at which the net marginal product of capital equals zero.

- Find the capital stock at which steady-state consumption is maximized, and find the capital stock at which the net marginal product of capital equals zero.

An economy begins with a level of steady-state capital per worker that is less than the Golden Rule level of capital per worker, and policymakers increase the saving rate to sgold. When the economy reaches the steady state again, consumption per worker will be greater than its initial level, investment per worker will be _____ than its initial level, and the MPK will be _____ than its initial level. - less; less - greater; less - less; greater - greater; greater

- greater; less Beginning in any steady state, an increased saving rate will move the economy to a new steady state with a higher stock of capital per worker and a higher output per worker.

In the steady state, depreciation equals investment. Therefore, the equation sf(k*) = δk* can be solved for k*, the steady-state level of capital per worker. For y = f(k) = k^1/2, s = 0.25, and δ = 0.05, the steady-state level of capital k* equals: 2.5. 5. 12.5. 25.

25 k*/f(k*) = s/d = k*/√k* = .25/.05 = 5^2 = 25

Ch8 Quick Quiz An economy without population growth or technological progress has the production function 𝑦=20𝑘1/2. The current capital stock is 100, the depreciation rate is 10%, and the population growth rate is 2%. For income per person to grow, what rate must the saving rate exceed?

6%

The formula for steady-state consumption per worker (c*) as a function of output per worker and investment per worker is: c* = f (k*) - 𝛿k*. c* = f (k*) + 𝛿k*. c* = f (k*) ÷ 𝛿k*. c* = k* - 𝛿f (k)*.

c* = f (k*) - 𝛿k*.

If the production function is y = k^1/2, s = 0.25, and δ = 0.05, then the steady-state level of capital per worker is _____ the Golden Rule level. less than more than the same as irrelevant to

Less than find the golden state savings rate and compare it to the actual savings rate If y = k^1/2, s = 0.25, and δ = 0.05, the steady state occurs when 0.25 × k^1/2 = 0.05 × k, or at k* = 25

Which of these statements is NOT true about the steady state of the basic Solow model? The capital per worker and output per worker are constant. The investment per worker is always equal to the depreciation per worker. The marginal product of capital always is equal to the depreciation rate. The saving and consumption per worker are constant.

The marginal product of capital always is equal to the depreciation rate.

Capital Accumulation as a Source of Growth — End of Chapter Problem Consider Swan Island, an economy described by the Solow model. There is no population growth or technological progress. The production function is 𝑦=20𝑘13. The initial capital stock per worker is 125. According to the national income accounts, investment equals 18 percent of national income, and depreciation equals 12.5 percent of national income. Calculate the following: a. National income y = b. Consumption c = c. Saving rate s = d. Depreciation 𝛿𝑘= e. Depreciation rate 𝛿 = f. Change in the capital stock in the next period Δ𝑘 = g. Steady-state capital stock k* = h. Steady-state income y* =

a. National income y = 100 b. Consumption c = 82 c. Saving rate s = .18 d. Depreciation 𝛿𝑘= 12.5 e. Depreciation rate 𝛿 = .1 f. Change in the capital stock in the next period Δ𝑘 = 5.5 g. Steady-state capital stock k* = 216 h. Steady-state income y* = 120

An increase in the saving rate will have: neither a level effect nor a growth effect. a level effect and a growth effect. a level effect but no growth effect. a growth effect but no level effect.

a level effect but no growth effect. While income per person may change from one steady state to the next (level effect), the growth rate of income per person returns to zero each time the economy returns to the steady state (growth effect); therefore, would an increase in the saving rate have a growth effect?

Capital Accumulation as a Source of Growth — End of Chapter Problem In the discussion of German and Japanese postwar growth, the text describes what happens when part of the capital stock is destroyed in a war. By contrast, suppose that a war does not directly affect the capital stock but that casualties reduce the labor force. Assume that the economy was in a steady state before the war, that the saving rate is unchanged, and that the rate of population growth is the same as before the war. a. What is the immediate impact on total output? b. What is the immediate impact on output per person? c. What happens subsequently to output per person in the postwar economy? d. What happens to the growth rate of output per worker after the war but before the economy reaches a new steady state?

a. It decreases b. It increases c. It declines d. It is less than zero The decline in L means that capital per worker, 𝑘=𝐾𝐿, rises. What happens to output per person when each worker is equipped with more capital?

Capital Accumulation as a Source of Growth — End of Chapter Problem Suppose the steady-state capital stock is initially below the Golden Rule level. Use the Solow growth model to assess the following claim: "Devoting a larger share of national output to investment would help restore rapid productivity growth and rising living standards." a. Productivity growth will initially ____(rise/remain the same/grow)___(rise/remain unchanged/fall)__ and ____(return to its initial level/fall to a lower level/rise to a higher level)____ as the economy achieves a new steady state. b. Living standards will initially ___(rise/remain unchanged/fall)____ and ___(return to its initial level/fall to a lower level/rise to a higher level)____ as the economy achieves a new steady state.

a. rise/return to its initial level b. fall/rise to a higher level Higher investment causes growth of the capital stock to accelerate. Given increased growth of the capital stock, what would happen to output and output per worker? Assuming the economy begins with an initial steady-state capital stock below the Golden Rule level, the immediate effect of devoting a larger share of national output to investment is that the economy devotes a smaller share to consumption; that is, "living standards," as measured by consumption, fall. The higher investment rate means that the capital stock increases more quickly, so the growth rates of output and output per worker rise. The productivity of workers is the average amount produced by each worker or output per worker. So productivity growth rises. Hence, the immediate effect is that living standards fall but productivity growth rises. In the new steady state, output grows at rate n, while output per worker grows at rate zero. This means that in the steady state, productivity growth is independent of the rate of investment. Since we begin with an initial steady-state capital stock below the Golden Rule level, the higher investment rate means that the new steady state has a higher level of consumption, so living standards are higher. An increase in the investment rate increases the productivity growth rate in the short run but has no effect in the long run. Living standards, on the other hand, fall immediately and only rise over time. That is, the quotation emphasizes growth but not the sacrifice required to achieve it.

The supply of goods and services can grow for all of these reasons EXCEPT: an increase in the labor force. an increase in the interest rate. technological improvement. an increase in the capital stock.

an increase in the interest rate.

Ch8 Quick Quiz If the economy has more capital than in the Golden Rule steady state, reducing the saving rate will _____(increase/decrease)________ steady-state income and _______(increase/decrease)________ steady-state consumption.

decrease/increase A decline in saving entails a decline in investment and thus in the capital stock. If the capital stock is above the Golden Rule level, the decline in the capital stock k will decrease output f(k), but it will decrease depreciation δk (the amount of output needed to replace worn-out capital) even more. Thus, consumption—the difference between output and depreciation, f(k) - δk—will rise.

If the per worker production function for an economy is given by y = k^1/2, the saving rate is 0.3, the depreciation rate is 10%, and the economy starts off with 25 units of capital per worker, then the capital per worker will _____ and output per worker will _____ as the economy approaches the steady state. rise; rise rise; fall fall; rise fall; fall

fall; fall

In the Solow model, if the economy starts with more capital per worker than the steady-state level of capital per worker, then the capital per worker will _____ and the output per worker will _____ as the economy approaches steady state. fall; fall fall; rise rise; fall rise; rise

fall; fall

Suppose an economy is at its steady-state equilibrium and there is a permanent reduction in the saving rate of the economy. In this case, as the economy approaches its new steady state, capital per worker will _____ and output per worker will _____. rise; rise rise; fall fall; rise fall; fall

fall; fall

Ch8 Quick Quiz According to the Solow model, if an economy increases its saving rate, then in the new steady state, compared with the old one, the marginal product of capital will be _____(lower/the same/higher)_______ , and the growth rate of income per person will be ____(lower/the same/higher)_______ .

higher/the same An increase in saving increases investment and thus output. Recall one of the most basis assumptions of economic analysis—one concerning the marginal product of an input.

If the production function is y = 5k1/5, δ = 0.5, and k = 50, then the saving rate is _____ the Golden Rule level. the same as more than erratic compared to less than

more than

Suppose that an economy is in its steady state and the capital stock is above the Golden Rule level. Assuming that there are no population growth or technological change, if the saving rate falls: output, consumption, investment, and depreciation will all decrease. output and investment will decrease, and consumption and depreciation will increase. output and investment will decrease, and consumption and depreciation will increase and then decrease but finally approach levels above their initial state. output, investment, and depreciation will decrease, and consumption will increase and then decrease but finally approach a level above its initial state.

output, investment, and depreciation will decrease, and consumption will increase and then decrease but finally approach a level above its initial state.

When an economy's capital is below the Golden Rule level, reaching the Golden Rule level: produces lower consumption at all times in the future. requires higher consumption levels at all times. requires initially reducing consumption to increase consumption in the future. requires initially increasing consumption to decrease consumption in the future.

requires initially reducing consumption to increase consumption in the future.

When an economy begins above the Golden Rule level, reaching the Golden Rule level: results in lower consumption at all times in the future. results in higher consumption at all times in the future. requires initially reducing consumption to increase consumption in the future. requires initially increasing consumption to decrease consumption in the future.

results in higher consumption at all times in the future.

The Solow model shows that a key determinant of the steady-state ratio of capital to labor is the: level of output. labor force. saving rate. capital elasticity in the production function.

saving rate.

An economy will _____ move to the Golden Rule steady state. sometimes always naturally never

sometimes In the Solow model, the economy always moves to the steady state but not necessarily to the Golden Rule steady state.

If an economy is in a steady state with no population growth or technological change and the marginal product of capital is less than the depreciation rate: the economy is following the Golden Rule. steady-state consumption per worker would be higher in a steady state with a lower saving rate. steady-state consumption per worker would be higher in a steady state with a higher saving rate. the depreciation rate should be decreased to achieve the Golden Rule level of consumption per worker.

steady-state consumption per worker would be higher in a steady state with a lower saving rate.

Ch8 Quick Quiz In the basic Solow model, at the Golden Rule steady state, the marginal product of capital equals the depreciation rate. output per worker. the saving rate. consumption per worker.

the depreciation rate.

The Cobb-Douglas production function Y = K1/2 L1/2 can be rearranged into the output per worker function: y = k1/2. y = k1/2 / L. y = k1/2 × K1/2. yL = k1/2.

y = k1/2.

The Cobb-Douglas production function Y = K2/3 L1/3 can be rearranged into the output per worker function: y = k3. y = k2. y = k2/3. y = k1/3.

y = k2/3.

Ch8 Quick Quiz Which of the following production functions has constant returns to scale? 𝑌=𝐾1/3𝐿1/3 𝑌=𝐾2+𝐿 𝑌=𝐾2𝐿 𝑌=𝐾+𝐿

𝑌=𝐾+𝐿 A production function 𝑌=𝐹(𝐾,𝐿) has constant returns to scale if 𝑧𝑌=𝐹(𝑧𝐾,𝑧𝐿) > 1. In the case of 𝑌=𝐾+𝐿: 𝑧𝑌=𝑧𝐾+𝑧𝐿=𝑧(𝐾+𝐿) Thus, the function exhibits constant returns to scale.