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Economic Growth I: Capital Accumulation and Population Growth

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What I want you to learn: the closed economy Solow model how a countrys standard of living depends on its saving and population growth rates how to use the Golden Rule to find the optimal saving rate and capital stock

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Why growth matters Data on infant mortality rates: 20% in the poorest 1/5 of all countries 0.4% in the richest 1/5 In Pakistan, 85% of people live on less than $2/day. One-fourth of the poorest countries have had famines during the past 3 decades. Poverty is associated with oppression of women and minorities. Economic growth raises living standards and reduces poverty….

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Income and poverty in the world selected countries, 2000

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links to prepared Gapminder.org notes: circle size is proportional to population size, color of circle indicates continent Income per capita and Life expectancy Infant mortality Malaria deaths per 100,000 Adult literacy Cell phone users per 100,000

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Why growth matters Anything that effects the long-run rate of economic growth – even by a tiny amount – will have huge effects on living standards in the long run. 1,081.4%243.7%85.4% 624.5% 169.2% 64.0% 2.5% 2.0% …100 years …50 years …25 years percentage increase in standard of living after… annual growth rate of income per capita

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Why growth matters If the annual growth rate of U.S. real GDP per capita had been just one-tenth of one percent higher during the 1990s, the U.S. would have generated an additional $496 billion of income during that decade.

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The lessons of growth theory …can make a positive difference in the lives of hundreds of millions of people. These lessons help us understand why poor countries are poor design policies that can help them grow learn how our own growth rate is affected by shocks and our governments policies

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The Solow model due to Robert Solow, won Nobel Prize for contributions to the study of economic growth a major paradigm: widely used in policy making benchmark against which most recent growth theories are compared looks at the determinants of economic growth and the standard of living in the long run

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How Solow model is different from Chapter 3s model 1.K is no longer fixed: investment causes it to grow, depreciation causes it to shrink 2.L is no longer fixed: population growth causes it to grow 3.the consumption function is simpler 4.no G or T (only to simplify presentation; we can still do fiscal policy experiments) 5.cosmetic differences

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The production function In aggregate terms: Y = F (K, L) Define: y = Y/L = output per worker k = K/L = capital per worker Assume constant returns to scale: zY = F (zK, zL ) for any z > 0 Pick z = 1/L. Then Y/L = F (K/L, 1) y = F (k, 1) y = f(k)where f(k) = F(k, 1)

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The production function Output per worker, y Capital per worker, k f(k) Note: this production function exhibits diminishing MPK. 1 MPK = f(k +1) – f(k)

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The national income identity Y = C + I (remember, no G ) In per worker terms: y = c + i where c = C/L and i = I /L

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The consumption function s = the saving rate, the fraction of income that is saved (s is an exogenous parameter) Note: s is the only lowercase variable that is not equal to its uppercase version divided by L Consumption function: c = (1–s)y (per worker)

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Saving and investment saving (per worker) = y – c = y – (1–s)y = sy National income identity is y = c + i Rearrange to get: i = y – c = sy (investment = saving, like in chap. 3!) Using the results above, i = sy = sf(k)

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Output, consumption, and investment Output per worker, y Capital per worker, k f(k) sf(k) k1k1 y1y1 i1i1 c1c1

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Depreciation Depreciation per worker, k Capital per worker, k k = the rate of depreciation = the fraction of the capital stock that wears out each period = the rate of depreciation = the fraction of the capital stock that wears out each period 1

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Capital accumulation Change in capital stock= investment – depreciation k = i – k Since i = sf(k), this becomes: k = s f(k) – k The basic idea: Investment increases the capital stock, depreciation reduces it.

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The equation of motion for k The Solow models central equation Determines behavior of capital over time… …which, in turn, determines behavior of all of the other endogenous variables because they all depend on k. E.g., income per person: y = f(k) consumption per person: c = (1–s) f(k) k = s f(k) – k

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The steady state If investment is just enough to cover depreciation [sf(k) = k ], then capital per worker will remain constant: k = 0. This occurs at one value of k, denoted k *, called the steady state capital stock. k = s f(k) – k

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The steady state Investment and depreciation Capital per worker, k sf(k) k k*k*

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Moving toward the steady state Investment and depreciation Capital per worker, k sf(k) k k*k* k = sf(k) k depreciation k k1k1 investment

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Moving toward the steady state Investment and depreciation Capital per worker, k sf(k) k k*k* k1k1 k = sf(k) k k k2k2

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Moving toward the steady state Investment and depreciation Capital per worker, k sf(k) k k*k* k = sf(k) k k2k2 investment depreciation k

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Moving toward the steady state Investment and depreciation Capital per worker, k sf(k) k k*k* k = sf(k) k k k2k2

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Moving toward the steady state Investment and depreciation Capital per worker, k sf(k) k k*k* k = sf(k) k k2k2 k k3k3

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Moving toward the steady state Investment and depreciation Capital per worker, k sf(k) k k*k* k = sf(k) k k3k3 Summary: As long as k < k *, investment will exceed depreciation, and k will continue to grow toward k *.

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A numerical example Production function (aggregate): To derive the per-worker production function, divide through by L: Then substitute y = Y/L and k = K/L to get

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A numerical example, cont. Assume: s = 0.3 = 0.1 initial value of k = 4.0

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Approaching the steady state: A numerical example Year k y c i δ k Δk Year k y c i δ k Δk … … … …

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NOW YOU TRY: Solve for the Steady State Continue to assume s = 0.3, = 0.1, and y = k 1/2 Use the equation of motion k = s f(k) k to solve for the steady-state values of k, y, and c.

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ANSWERS: Solve for the Steady State

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An increase in the saving rate Investment and depreciation k δkδk s 1 f(k) An increase in the saving rate raises investment… …causing k to grow toward a new steady state: s 2 f(k)

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Prediction: Higher s higher k *. And since y = f(k), higher k * higher y *. Thus, the Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and income per worker in the long run.

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International evidence on investment rates and income per person Income per person in 2003 (log scale) Investment as percentage of output (average )

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The Golden Rule: Introduction Different values of s lead to different steady states. How do we know which is the best steady state? The best steady state has the highest possible consumption per person: c* = (1–s) f(k*). An increase in s leads to higher k* and y*, which raises c* reduces consumptions share of income (1–s), which lowers c*. So, how do we find the s and k* that maximize c*?

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The Golden Rule capital stock the Golden Rule level of capital, the steady state value of k that maximizes consumption. To find it, first express c * in terms of k * : c * = y * i * = f (k * ) i * = f (k * ) k * In the steady state: i * = k * because k = 0.

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Then, graph f(k * ) and k *, look for the point where the gap between them is biggest. The Golden Rule capital stock steady state output and depreciation steady-state capital per worker, k * f(k * ) k *

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The Golden Rule capital stock c * = f(k * ) k * is biggest where the slope of the production function equals the slope of the depreciation line: steady-state capital per worker, k * f(k * ) k * MPK =

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The transition to the Golden Rule steady state The economy does NOT have a tendency to move toward the Golden Rule steady state. Achieving the Golden Rule requires that policymakers adjust s. This adjustment leads to a new steady state with higher consumption. But what happens to consumption during the transition to the Golden Rule?

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Starting with too much capital then increasing c * requires a fall in s. In the transition to the Golden Rule, consumption is higher at all points in time. then increasing c * requires a fall in s. In the transition to the Golden Rule, consumption is higher at all points in time. time t0t0 c i y

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Starting with too little capital then increasing c * requires an increase in s. Future generations enjoy higher consumption, but the current one experiences an initial drop in consumption. then increasing c * requires an increase in s. Future generations enjoy higher consumption, but the current one experiences an initial drop in consumption. time t0t0 c i y

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Population growth Assume the population and labor force grow at rate n (exogenous): EX: Suppose L = 1,000 in year 1 and the population is growing at 2% per year (n = 0.02). Then L = n L = ,000 = 20, so L = 1,020 in year 2.

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Break-even investment ( + n)k = break-even investment, the amount of investment necessary to keep k constant. Break-even investment includes: k to replace capital as it wears out n k to equip new workers with capital (Otherwise, k would fall as the existing capital stock is spread more thinly over a larger population of workers.)

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The equation of motion for k With population growth, the equation of motion for k is: break-even investment actual investment k = s f(k) ( + n) k

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The Solow model diagram Investment, break-even investment Capital per worker, k sf(k) ( + n ) k k*k* k = s f(k) ( +n)k

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The impact of population growth Investment, break-even investment Capital per worker, k sf(k) ( +n 1 ) k k1*k1* ( +n 2 ) k k2*k2* An increase in n causes an increase in break- even investment, leading to a lower steady-state level of k.

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Prediction: Higher n lower k*. And since y = f(k), lower k* lower y*. Thus, the Solow model predicts that countries with higher population growth rates will have lower levels of capital and income per worker in the long run.

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International evidence on population growth and income per person Income per person in 2003 (log scale) Population growth (percent per year, average )

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The Golden Rule with population growth To find the Golden Rule capital stock, express c * in terms of k * : c * = y * i * = f (k * ) ( + n) k * c * is maximized when MPK = + n or equivalently, MPK = n In the Golden Rule steady state, the marginal product of capital net of depreciation equals the population growth rate.

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Alternative perspectives on population growth The Malthusian Model (1798) Predicts population growth will outstrip the Earths ability to produce food, leading to the impoverishment of humanity. Since Malthus, world population has increased sixfold, yet living standards are higher than ever. Malthus neglected the effects of technological progress. Thomas Malthus

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Alternative perspectives on population growth The Kremerian Model (1993) Posits that population growth contributes to economic growth. More people = more geniuses, scientists & engineers, so faster technological progress. Evidence, from very long historical periods: As world pop. growth rate increased, so did rate of growth in living standards Historically, regions with larger populations have enjoyed faster growth. Michael Kremer

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Summary 1. The Solow growth model shows that, in the long run, a countrys standard of living depends: positively on its saving rate negatively on its population growth rate 2. An increase in the saving rate leads to: higher output in the long run faster growth temporarily but not faster steady state growth

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Summary 3. If the economy has more capital than the Golden Rule level, then reducing saving will increase consumption at all points in time, making all generations better off. If the economy has less capital than the Golden Rule level, then increasing saving will increase consumption for future generations, but reduce consumption for the present generation.

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In the simple Solow model, the production technology is held constant. income per capita is constant in the steady state. Neither point is true in the real world: : U.S. real GDP per person grew by a factor of 7.8, or 2.05% per year. examples of technological progress abound Technological progress in the Solow model

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Examples of technological progress From 1950 to 2000, U.S. farm sector productivity nearly tripled. The real price of computer power has fallen an average of 30% per year over the past three decades. Percentage of U.S. households with 1 computers: 8% in 1984, 62% in : 213 computers connected to the Internet 2000: 60 million computers connected to the Internet 2001: iPod capacity = 5gb, 1000 songs. Not capable of playing episodes of True Blood. 2009: iPod capacity = 120gb, 30,000 songs. Can play episodes of True Blood.

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Technological progress in the Solow model A new variable: E = labor efficiency Assume: Technological progress is labor-augmenting: it increases labor efficiency at the exogenous rate g:

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Technological progress in the Solow model We now write the production function as: where L E = the number of effective workers. Increases in labor efficiency have the same effect on output as increases in the labor force.

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Technological progress in the Solow model Notation: y = Y/LE = output per effective worker k = K/LE = capital per effective worker Production function per effective worker: y = f(k) Saving and investment per effective worker: s y = s f(k)

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Technological progress in the Solow model ( + n + g)k = break-even investment: the amount of investment necessary to keep k constant. Consists of: k to replace depreciating capital n k to provide capital for new workers g k to provide capital for the new effective workers created by technological progress

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Technological progress in the Solow model Investment, break-even investment Capital per worker, k sf(k) ( +n +g ) k k*k* k = s f(k) ( +n +g)k

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Steady-state growth rates in the Solow model with tech. progress n + gY = y E LTotal output g(Y/ L) = y EOutput per worker 0y = Y/(L E ) Output per effective worker 0k = K/(L E ) Capital per effective worker Steady-state growth rate SymbolVariable

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The Golden Rule with technological progress To find the Golden Rule capital stock, express c * in terms of k * : c * = y * i * = f (k * ) ( + n + g) k * c * is maximized when MPK = + n + g or equivalently, MPK = n + g In the Golden Rule steady state, the marginal product of capital net of depreciation equals the pop. growth rate plus the rate of tech progress.

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Growth empirics: Balanced growth Solow models steady state exhibits balanced growth - many variables grow at the same rate. Solow model predicts Y/L and K/L grow at the same rate (g), so K/Y should be constant. This is true in the real world. Solow model predicts real wage grows at same rate as Y/L, while real rental price is constant. Also true in the real world.

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Growth empirics: Convergence Solow model predicts that, other things equal, poor countries (with lower Y/L and K/L) should grow faster than rich ones. If true, then the income gap between rich & poor countries would shrink over time, causing living standards to converge. In real world, many poor countries do NOT grow faster than rich ones. Does this mean the Solow model fails?

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Growth empirics: Convergence Solow model predicts that, other things equal, poor countries (with lower Y/L and K/L) should grow faster than rich ones. No, because other things arent equal. In samples of countries with similar savings & pop. growth rates, income gaps shrink about 2% per year. In larger samples, after controlling for differences in saving, pop. growth, and human capital, incomes converge by about 2% per year.

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Growth empirics: Convergence What the Solow model really predicts is conditional convergence - countries converge to their own steady states, which are determined by saving, population growth, and education. This prediction comes true in the real world.

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Growth empirics: Factor accumulation vs. production efficiency Differences in income per capita among countries can be due to differences in: 1. capital – physical or human – per worker 2. the efficiency of production (the height of the production function) Studies: Both factors are important. The two factors are correlated: countries with higher physical or human capital per worker also tend to have higher production efficiency.

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Growth empirics: Factor accumulation vs. production efficiency Possible explanations for the correlation between capital per worker and production efficiency: Production efficiency encourages capital accumulation. Capital accumulation has externalities that raise efficiency. A third, unknown variable causes capital accumulation and efficiency to be higher in some countries than others.

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Average annual growth rates, closedopen Growth empirics: Production efficiency and free trade Since Adam Smith, economists have argued that free trade can increase production efficiency and living standards. Research by Sachs & Warner: 0.7%4.5%developing nations 0.7%2.3%developed nations

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Growth empirics: Production efficiency and free trade To determine causation, Frankel and Romer exploit geographic differences among countries: Some nations trade less because they are farther from other nations, or landlocked. Such geographical differences are correlated with trade but not with other determinants of income. Hence, they can be used to isolate the impact of trade on income. Findings: increasing trade/GDP by 2% causes GDP per capita to rise 1%, other things equal.

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Policy issues Are we saving enough? Too much? What policies might change the saving rate? How should we allocate our investment between privately owned physical capital, public infrastructure, and human capital? How do a countrys institutions affect production efficiency and capital accumulation? What policies might encourage faster technological progress?

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Policy issues: Evaluating the rate of saving Use the Golden Rule to determine whether the U.S. saving rate and capital stock are too high, too low, or about right. If (MPK ) > (n + g ), U.S. is below the Golden Rule steady state and should increase s. If (MPK ) < (n + g ), U.S. economy is above the Golden Rule steady state and should reduce s.

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Policy issues: Evaluating the rate of saving To estimate (MPK ), use three facts about the U.S. economy: 1. k = 2.5 y The capital stock is about 2.5 times one years GDP. 2. k = 0.1 y About 10% of GDP is used to replace depreciating capital. 3. MPK k = 0.3 y Capital income is about 30% of GDP.

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Policy issues: Evaluating the rate of saving 1. k = 2.5 y 2. k = 0.1 y 3. MPK k = 0.3 y To determine, divide 2 by 1:

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Policy issues: Evaluating the rate of saving To determine MPK, divide 3 by 1: Hence, MPK = = k = 2.5 y 2. k = 0.1 y 3. MPK k = 0.3 y

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Policy issues: Evaluating the rate of saving From the last slide: MPK = 0.08 U.S. real GDP grows an average of 3% per year, so n + g = 0.03 Thus, MPK = 0.08 > 0.03 = n + g Conclusion: The U.S. is below the Golden Rule steady state: Increasing the U.S. saving rate would increase consumption per capita in the long run.

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Policy issues: How to increase the saving rate Reduce the government budget deficit (or increase the budget surplus). Increase incentives for private saving: reduce capital gains tax, corporate income tax, estate tax as they discourage saving. replace federal income tax with a consumption tax. expand tax incentives for IRAs (individual retirement accounts) and other retirement savings accounts.

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Policy issues: Allocating the economys investment In the Solow model, theres one type of capital. In the real world, there are many types, which we can divide into three categories: private capital stock public infrastructure human capital: the knowledge and skills that workers acquire through education How should we allocate investment among these types?

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Policy issues: Allocating the economys investment Two viewpoints: 1.Equalize tax treatment of all types of capital in all industries, then let the market allocate investment to the type with the highest marginal product. 2.Industrial policy: Govt should actively encourage investment in capital of certain types or in certain industries, because they may have positive externalities that private investors dont consider.

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Possible problems with industrial policy The govt may not have the ability to pick winners (choose industries with the highest return to capital or biggest externalities). Politics (e.g., campaign contributions) rather than economics may influence which industries get preferential treatment.

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Policy issues: Establishing the right institutions Creating the right institutions is important for ensuring that resources are allocated to their best use. Examples: Legal institutions, to protect property rights. Capital markets, to help financial capital flow to the best investment projects. A corruption-free government, to promote competition, enforce contracts, etc.

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Policy issues: Encouraging tech. progress Patent laws: encourage innovation by granting temporary monopolies to inventors of new products. Tax incentives for R&D Grants to fund basic research at universities Industrial policy: encourages specific industries that are key for rapid tech. progress (subject to the preceding concerns).

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CASE STUDY: The productivity slowdown U.S. U.K. Japan Italy Germany France Canada Growth in output per person (percent per year)

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Possible explanations for the productivity slowdown Measurement problems: Productivity increases not fully measured. But: Why would measurement problems be worse after 1972 than before? Oil prices: Oil shocks occurred about when productivity slowdown began. But: Then why didnt productivity speed up when oil prices fell in the mid-1980s?

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Possible explanations for the productivity slowdown Worker quality: 1970s - large influx of new entrants into labor force (baby boomers, women). New workers tend to be less productive than experienced workers. The depletion of ideas: Perhaps the slow growth of is normal, and the rapid growth during is the anomaly.

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Which of these suspects is the culprit? All of them are plausible, but its difficult to prove that any one of them is guilty.

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CASE STUDY: I.T. and the New Economy U.S. U.K. Japan Italy Germany France Canada Growth in output per person (percent per year)

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CASE STUDY: I.T. and the New Economy Apparently, the computer revolution did not affect aggregate productivity until the mid-1990s. Two reasons: 1.Computer industrys share of GDP much bigger in late 1990s than earlier. 2.Takes time for firms to determine how to utilize new technology most effectively. The big, open question: How long will I.T. remain an engine of growth?

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Endogenous growth theory Solow model: sustained growth in living standards is due to tech progress. the rate of tech progress is exogenous. Endogenous growth theory: a set of models in which the growth rate of productivity and living standards is endogenous.

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A basic model Production function: Y = A K where A is the amount of output for each unit of capital (A is exogenous & constant) Key difference between this model & Solow: MPK is constant here, diminishes in Solow Investment: s Y Depreciation: K Equation of motion for total capital: K = s Y K

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A basic model K = s Y K If s A >, then income will grow forever, and investment is the engine of growth. Here, the permanent growth rate depends on s. In Solow model, it does not. Divide through by K and use Y = A K to get:

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Does capital have diminishing returns or not? Depends on definition of capital. If capital is narrowly defined (only plant & equipment), then yes. Advocates of endogenous growth theory argue that knowledge is a type of capital. If so, then constant returns to capital is more plausible, and this model may be a good description of economic growth.

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A two-sector model Two sectors: manufacturing firms produce goods. research universities produce knowledge that increases labor efficiency in manufacturing. u = fraction of labor in research (u is exogenous) Mfg prod func: Y = F [K, (1-u )E L] Res prod func: E = g (u )E Cap accumulation: K = s Y K

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A two-sector model In the steady state, mfg output per worker and the standard of living grow at rate E/E = g (u ). Key variables: s: affects the level of income, but not its growth rate (same as in Solow model) u: affects level and growth rate of income

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Facts about R&D 1. Much research is done by firms seeking profits. 2. Firms profit from research: Patents create a stream of monopoly profits. Extra profit from being first on the market with a new product. 3. Innovation produces externalities that reduce the cost of subsequent innovation. Much of the new endogenous growth theory attempts to incorporate these facts into models to better understand technological progress.

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Is the private sector doing enough R&D? The existence of positive externalities in the creation of knowledge suggests that the private sector is not doing enough R&D. But, there is much duplication of R&D effort among competing firms. Estimates: Social return to R&D 40% per year. Thus, many believe govt should encourage R&D.

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Economic growth as creative destruction Schumpeter (1942) coined term creative destruction to describe displacements resulting from technological progress: the introduction of a new product is good for consumers, but often bad for incumbent producers, who may be forced out of the market. Examples: Luddites ( ) destroyed machines that displaced skilled knitting workers in England. Walmart displaces many mom and pop stores.

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Chapter Summary 1. Key results from Solow model with tech progress steady state growth rate of income per person depends solely on the exogenous rate of tech progress the U.S. has much less capital than the Golden Rule steady state 2. Ways to increase the saving rate increase public saving (reduce budget deficit) tax incentives for private saving

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Chapter Summary 3. Productivity slowdown & new economy Early 1970s: productivity growth fell in the U.S. and other countries. Mid 1990s: productivity growth increased, probably because of advances in I.T. 4. Empirical studies Solow model explains balanced growth, conditional convergence Cross-country variation in living standards is due to differences in cap. accumulation and in production efficiency

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Chapter Summary 5. Endogenous growth theory: Models that examine the determinants of the rate of tech. progress, which Solow takes as given. explain decisions that determine the creation of knowledge through R&D.

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