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Suppose that the production function is given by Y=K1/2L1/2

a. Derive the steady state levels of capital per worker and output per worker in terms of the saving rate, s, and the depreciation rate, δ.

b. Suppose δ = 0.05 and s = 0.2. Find out the steady state output per worker.

c. Suppose δ = 0.05 but s increases to 0.5. Find out the steady state output per worker and compare your result with your answer in part b. Explain the intuition behind your results.

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The Facts of Growth Chapter 10 Chapter 10 Outline The Facts of Growth 10-1 10-2 10-3 10-4 Measuring the Standard of Living Growth in Rich Countries since 1950 A Broader Look across Time and Space Thinking about Growth: A Primer The Facts of Growth • Growth is the steady increase in aggregate output over time. • We now shift our focus from economic fluctuations and the determination of output in the short and medium run to growth and the determination of output in the long run. Figure 10-1 U.S. GDP since 1890. U.S. GDP per person since 1890 Panel A Panel A shows the enormous increase in U.S. output since 1890, by a factor of 46. Figure 10-1 U.S. GDP since 1890. U.S. GDP per person since 1890 (cont’d) Panel B Panel B shows that the increase in output is not simply the result of the large increase in U.S. population from 63 million to more than 300 million over this period. Output per person has risen by a factor of 9. 10-1 Measuring the Standard of Living • We care about growth because we care about the standard of living. • Output per person, rather than output itself, is the variable we compare over time or across countries. • We need to correct for variations in exchange rates and systematic differences in prices across countries. • When comparing the standard of living across countries, we use purchasing power parity (PPP) numbers which adjust the numbers for the purchasing power of different countries. • The right measure on the production side is output per worker or output per hour worked. FOCUS: The Construction of PPP Numbers • Consider this example: – United States: Each year, people buy a new car for $10,000, and spend another $10,000 on food. – Russia: People spend 20,000 rubles on cars (each lasts for 15 years) a year, and 40,000 rubles on food. • If the exchange rate is $1 = 30 rubles, consumption per person in Russia is only 10% of U.S. consumption per person. • If we use U.S. prices for both countries and assume people spending all money on food, then consumption per person is $20,000 ($10,000+$10,000) in the U.S., but $10,700 [(1/15)x$10,000)=(1x$10,000)] in Russia, so Russian consumption per person is 53.5% of U.S. consumption per person. 10-2 Growth in Rich Countries since 1950 Table 10-1 The Evolution of Output per Person in Four Rich Countries since 1950 • There has been a large increase in output per person due in part to the force of compounding. • There has been a convergence of output per person across countries. FOCUS: Does Money Lead to Happiness? Figure 1 Life Satisfaction and Income per Person • Easterlin paradox: Once basic needs are satisfied, higher income per person does not increase happiness, and the level of income relative to others, rather than the absolute level of income, matters 10-2 Growth in Rich Countries since 1950 Figure 10-2 Growth Rate of GDP Per Person since 1950 versus GDP per Person in 1950; OECD Countries Countries with lower levels of output per person in 1950 have typically grown faster. 10-3 A Broader Look across Time and Space • From the end of the Roman Empire to roughly year 1500, Europe was in a Malthusian trap or Malthusian era with stagnation of output per person because most workers were in agriculture with little technological progress. • After 1500, growth of output per person turned positive but still small. • Between 1820 and 1950, U.S. growth was still 1.5% per year. • Sustained growth was high since 1950. 10-3 A Broader Look across Time and Space Figure 10-3 Growth Rate of GDP per Person since 1960, versus GDP Per Person in 1960 (2005 dollars); 85 Countries There is no clear relation between the growth rate of output since 1960 and the level of output per person in 1960. 10-3 A Broader Look across Time and Space • For the OECD countries, there is clear evidence of convergence. • Convergence is also visible for many Asian countries, especially for those with high growth rates, called the four tigers—Singapore, Taiwan, Hong Kong, and South Korea. • Most African countries were very poor in 1960, and some of them had negative growth of output per person between 1960 and 2011 due in part to internal or external conflicts. 10-4 Thinking About Growth: A Primer • Aggregate production function F: where Y is output, K is capital, and N is labor. • The function F depends on the state of technology. • Constant returns to scale: • Decreasing returns to capital: Increases in capital lead to smaller and smaller increases in output. • Decreasing returns to labor: Increases in labor lead to smaller and smaller increases in output. 10-4 Thinking About Growth: A Primer • The production function and constant returns to scale imply a simple relation between output per worker (Y/N) and capital per worker (K/N): • Increases in capital per worker: Movements along the production function. • Improvements in the state of technology: Shifts (up) of the production function. • Growth comes from capital accumulation (a higher saving rate) and technological progress (the improvement in the state of technology). 10-4 Thinking About Growth: A Primer Figure 10-4 Output and Capital per Worker Decreasing returns to capital: Increases in capital per worker lead to smaller and smaller increases in output per worker. 10-4 Thinking About Growth: A Primer Figure 10-5 The Effects of an Improvement in the State of Technology An improvement in technology shifts the production function up, leading to an increase in output per worker for a given level of capital per worker. Saving, Capital Accumulation, and Output Chapter 11 Chapter 11 Outline Saving, Capital Accumulation, and Output 11-1 11-2 Interactions between Output and Capital The Implications of Alternative Saving Rates 11-3 Getting a Sense of Magnitudes 11-4 Physical versus Human Capital APPENDIX The Cobb-Douglas Production Function and the Steady State Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-2 Saving, Capital Accumulation, and Output • Since 1970, the U.S. saving ratio—the ratio of saving to gross domestic product—has averaged only 17%, compared to 22% in Germany and 30% in Japan. • Even if a lower saving rate does not permanently affect the growth rate, it does affect the level of output and the standard of living. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-3 11-1 Interactions between Output and Capital • Output in the long run depends on two relations: – The amount of capital determines the amount of output – The amount of output being produced determines the amount of saving, which in turn determines the amount of capital being accumulated over time Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-4 11-1 Interactions between Output and Capital Figure 11-1 Capital, Output, and Saving/Investment Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-5 11-1 Interactions between Output and Capital • Recall Chapter 10: or • Assume that N is constant, and there is no technological progress, so f does not change over time: • Higher capital per worker leads to higher output per worker. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-6 11-1 Interactions between Output and Capital • Assume: – The economy is closed: I = S + (T − G) – Public saving (T − G) is 0: I = S – Private saving is proportional to income: S = sY • So the relation between output and investment: It = sYt • Investment is proportional to output. • The higher (lower) output is, the higher (lower) is saving and so the higher (lower) is investment. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-7 11-1 Interactions between Output and Capital • The evolution of the capital stock is: • Replace investment by the above expression and divide both sides by N: or • The change in the capital stock per worker is equal to saving per worker minus depreciation. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-8 11-2 The Implications of Alternative Saving Rates • Combining equations (11.1) and (11.2): • If investment per worker exceeds (is less than) depreciation per worker, the change in capital per worker is positive (negative). Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-9 11-2 The Implications of Alternative Saving Rates Figure 11-2 Capital and Output Dynamics When capital and output are low, investment exceeds depreciation and capital increases. When capital and output are high, investment is less than depreciation and capital decreases. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-10 11-2 The Implications of Alternative Saving Rates • The state in which output per worker and capital per worker are no longer changing is called the steady state of the economy. • The steady-state value of capital per worker is such that the amount of saving per worker is sufficient to cover depreciation of the capital stock per worker. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-11 Focus: Capital Accumulation and Growth in France in the Aftermath of World War II • France suffered heavy losses in capital when World War II ended in 1945. • The growth model predicts that France would experience high capital accumulation and output growth for some time. • From 1946 to 1950, French real GDP indeed grew at 9.6% per year. Table 1 Proportion of the French Capital Stock Destroyed by the End of World War II Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-12 11-2 The Implications of Alternative Saving Rates • The saving rate has no effect on the long-run growth rate of output per worker, which is equal to zero. • The saving rate determines the level of output per worker in the long run. • An increase in the saving rate will lead to higher growth of output per worker for some time, but not forever. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-13 11-2 The Implications of Alternative Saving Rates Figure 11-3 The Effects of Different Saving Rates A country with a higher saving rate achieves a higher steady-state level of output per worker. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-14 11-2 The Implications of Alternative Saving Rates Figure 11-4 The Effects of an Increase in the Saving Rate on Output per Worker in an Economy Without Technological Progress An increase in the saving rate leads to a period of higher growth until output reaches its new higher steady-state level. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-15 11-2 The Implications of Alternative Saving Rates Figure 11-5 The Effects of an Increase in the Saving Rate on Output per Worker in an Economy with Technological Progress An increase in the saving rate leads to a period of higher growth until output reaches its new higher steady-state level. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-16 11-2 The Implications of Alternative Saving Rates • What matters to people is not how many is produced, but how much they consume. • Governments can affect the saving rate by: – changing public saving (budget surplus) – using taxes to affect private saving • Golden-rule level of capital: The level of capital associated with the saving rate that yields the highest level of consumption in steady state. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-17 11-2 The Implications of Alternative Saving Rates Figure 11-6 The Effects of the Saving Rate on Steady-State Consumption per Worker An increase in the saving rate leads to an increase, then to a decrease in steady-state consumption per worker. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-18 11-2 The Implications of Alternative Saving Rates • For a saving rate between zero and the golden-rule level, a higher saving rate leads to higher capital per worker, higher output per worker and higher consumption per worker. • For a saving rate greater than the golden-rule level, a higher saving rate still leads to higher capital per worker and output per worker, but lower consumption per worker. • An increase in the saving rate leads to lower consumption for some time but higher consumption later. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-19 11-3 Getting a Sense of Magnitudes • Assume the production function f: • so that equation (11.3) becomes: • which describes the evolution of capital over time. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-20 FOCUS: Social Security, Saving, and Capital Accumulation in the United States • Social Security, introduced in 1935, has led to a lower U.S. saving rate and thus lower capital accumulation and lower output per person in the long run. • Social Security is a pay-as-you-can system that taxes workers and redistributes the tax contributions as benefits to current retirees, resulting in lower private saving as workers anticipate receiving benefits when they retire. • An alternative is a fully-funded system that pays back the principal plus interest to the workers when they retire, resulting in lower private saving but higher public saving as the System invests their contributions in financial assets. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-21 11-3 Getting a Sense of Magnitudes • Equation (11.7) implies that capital per worker in the steady state (K*/N) becomes: • Combining equations (11.6) and (11.8) gives the steady state output per worker: • In the long run, output per worker doubles when the saving rate doubles. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-22 11-3 Getting a Sense of Magnitudes Figure 11-7(a) The Dynamic Effects of an Increase in the Saving Rate from 10% to 20% on the Level and the Growth Rate of Output per Worker It takes a long time for output to adjust to its new higher level after an increase in the saving rate. Put another way, an increase in the saving rate leads to a long period of higher growth. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-23 11-3 Getting a Sense of Magnitudes Figure 11-7(b) The Dynamic Effects of an Increase in the Saving Rate from 10% to 20% on the Level and the Growth Rate of Output per Worker It takes a long time for output to adjust to its new higher level after an increase in the saving rate. Put another way, an increase in the saving rate leads to a long period of higher growth. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-24 11-3 Getting a Sense of Magnitudes • In the steady state, consumption per worker is: • Given equations (11.8) and (11.9), the steadystate consumption per worker is: • Table 11-1 gives the steady-state values of capital per worker, output per worker and consumption per worker for different saving rates (given δ=10%) Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-25 11-3 Getting a Sense of Magnitudes Table 11-1 The Saving Rate and the Steady-State Levels of Capital, Output, and Consumption per Worker Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-26 11-4 Physical versus Human Capital • Human capital (H): The set of skills of the workers in the economy built through education and on-the-job training. • The production function with human capital: • As for physical capital (K) accumulation, countries that save more or spend more on education can achieve higher steady-state levels of output per worker. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-27 11-4 Physical versus Human Capital • Models of endogenous growth: Steady-state growth in outpour per worker depends on variables such as the saving rate and the rate of spending on education, even without technological progress. • However, the current consensus is that given the rate of technological progress, higher rates of saving or spending on education do not lead to a permanently higher growth rate. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-28 APPENDIX: The Cobb-Douglas Production Function and the Steady State • The Cobb-Douglas production function: which gives a good description of the relation between output, physical capital, and labor in the United States from 1899 to 1922. • In steady state, saving per worker must be equal to depreciation per worker, implying that: s(K*/N)α = δ(K*/N) where K* is the steady-state level of capital. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-29 APPENDIX: The Cobb-Douglas Production Function and the Steady State • The preceding expression can be rewritten as: s = δ(K*/N) 1-α • The steady-state level of capital per worker becomes: (K*/N) = (s/δ) α/(1-α) • If α = 0.5, then: K*/N = s/δ which implies that a doubling of the saving rate leads to a doubling in steady-state output per worker. Copyright © 2017 Pearson Education, Inc. All rights reserved. 11-30 Question 3 Suppose that the production function is given by Y=K1/2L1/2 a. Derive the steady state levels of capital per worker and output per worker in terms of the saving rate, s, and the depreciation rate, 8. b. Suppose 8 = 0.05 and s = 0.2. Find out the steady state output per worker. c. Suppose 8 = 0.05 but s increases to 0.5. Find out the steady state output per worker and compare your result with your answer in part b. Explain the intuition behind your results.
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