Lecture04_note - Macroeconomics II Lecture 4 Growth theory:...

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Unformatted text preview: Macroeconomics II Lecture 4 Growth theory: the Solow Model Macroeconomics Instructor: Dang Vu University of Economics and Business - VNU In this chapter, you will learn… the closed economy Solow model how a country’s 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 LECTURE 4 Economic Growth: Solow model slide 1 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…. LECTURE 4 Economic Growth: Solow model slide 2 1 Macroeconomics II Income and poverty in the world selected countries, 2000 100 Madagascar % of popu ulation living on $2 per day or less 90 India Nepal Bangladesh 80 70 60 Botswana Kenya 50 China 40 Peru 30 Mexico Thailand 20 Brazil 10 0 $0 Russian Chile Federation $5,000 S. Korea $10,000 $15,000 $20,000 Income per capita in dollars 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. annual growth rate of income per capita …25 years …50 years …100 years 2.0% 64.0% 169.2% 624.5% 2.5% 85.4% 243.7% 1,081.4% LECTURE 4 percentage increase i t i in standard of living after… Economic Growth: Solow model slide 4 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. LECTURE 4 Economic Growth: Solow model slide 5 2 Macroeconomics II 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 government’s policies LECTURE 4 Economic Growth: Solow model slide 6 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 LECTURE 4 Economic Growth: Solow model slide 7 How Solow model is different from Lecture 3’s model 1. K is no longer fixed: investment causes it to grow, depreciation causes it to shrink 2. 2 L is no longer fixed: population growth causes it to grow 3. the consumption function is simpler LECTURE 4 Economic Growth: Solow model slide 8 3 Macroeconomics II How Solow model is different from Lecture 3’s model 4. no G or T (only to simplify presentation; we can still do fiscal policy experiments) 5. 5 cosmetic differences LECTURE 4 Economic Growth: Solow model slide 9 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) LECTURE 4 where f(k) = F(k, 1) Economic Growth: Solow model slide 10 The production function Output per worker, y f(k) MPK = f(k +1) – f(k) 1 Note: this production function exhibits diminishing MPK. Capital per worker, k LECTURE 4 Economic Growth: Solow model slide 11 4 Macroeconomics II 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 LECTURE 4 Economic Growth: Solow model slide 12 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 y that is not equal to its uppercase version divided by L Consumption function: c = (1–s)y (per worker) LECTURE 4 Economic Growth: Solow model slide 13 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 lecture 3!) Using the results above, i = sy = sf(k) LECTURE 4 Economic Growth: Solow model slide 14 5 Macroeconomics II Output, consumption, and investment Output per worker, y f(k) c1 sf(k) y1 i1 Capital per worker, k k1 LECTURE 4 Economic Growth: Solow model slide 15 Depreciation Depreciation per worker, k = the rate of depreciation = the fraction of the capital stock that wears out each period k 1 Capital per worker, k LECTURE 4 Economic Growth: Solow model slide 16 Capital accumulation The basic idea: Investment increases the capital stock, depreciation reduces it. Change in capital stock k = investment – depreciation i – k = Since i = sf(k) , this becomes: k = s f(k) – k LECTURE 4 Economic Growth: Solow model slide 17 6 Macroeconomics II The equation of motion for k k = s f(k) – k The Solow model’s 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) LECTURE 4 Economic Growth: Solow model slide 18 The steady state k = s f(k) – k 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. LECTURE 4 Economic Growth: Solow model slide 19 The steady state Investment and depreciation k sf(k) k* LECTURE 4 Economic Growth: Solow model Capital per worker, k slide 20 7 Macroeconomics II Moving toward the steady state k = sf(k) k Investment and depreciation k sf(k) k investment depreciation k1 LECTURE 4 k* Capital per worker, k Economic Growth: Solow model slide 21 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) k k1 LECTURE 4 k* Capital per worker, k Economic Growth: Solow model slide 22 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) k k1 k2 LECTURE 4 k* Economic Growth: Solow model Capital per worker, k slide 23 8 Macroeconomics II Moving toward the steady state k = sf(k) k Investment and depreciation k sf(k) k investment depreciation k2 LECTURE 4 k* Capital per worker, k Economic Growth: Solow model slide 24 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) k k2 LECTURE 4 k* Capital per worker, k Economic Growth: Solow model slide 25 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) k k2 k3 k* LECTURE 4 Economic Growth: Solow model Capital per worker, k slide 26 9 Macroeconomics II Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) Summary: As long as k < k*, investment will exceed depreciation, and k will continue to grow toward k*. k3 k* LECTURE 4 Capital per worker, k Economic Growth: Solow model slide 27 Now you try: Draw the Solow model diagram, labeling the steady state k*. On the horizontal axis, pick a value greater than k* for the economy s initial capital stock. Label it k1. economy’s stock Show what happens to k over time. Does k move toward the steady state or away from it? LECTURE 4 Economic Growth: Solow model slide 28 A numerical example Production function (aggregate): Y F (K , L ) K L K 1 / 2L1 / 2 To derive the per-worker production function, divide through by L: LECTURE 4 Economic Growth: Solow model slide 29 10 Macroeconomics II A numerical example, cont. Assume: s = 0.3 = 0.1 initial value of k = 4.0 LECTURE 4 Economic Growth: Solow model slide 30 Approaching the steady state: A numerical example Assumptions: y k ; s 0.3; 0.1; initial k 4.0 k Year k y c i dk 1 4.000 2.000 1.400 0.600 0.400 0.200 2 4.200 2.049 1.435 0.615 0.420 0.195 3 4.395 4 395 2.096 2 096 1.467 1 467 0.629 0 629 0.440 0 440 0.189 0 189 4 4.584 2.141 1.499 … 10 5.602 2.367 1.657 … 25 7.351 2.706 1.894 … 100 8.962 2.994 2.096 … ¥ 3.000 2.100 CHAPTER 9.000 7 Economic Growth I 0.642 0.458 0.184 0.710 0.560 0.150 0.812 0.732 0.080 0.898 0.896 0.002 0.900 0.900 0.000 31 slide Exercise: Solve for the steady state Continue to assume s = 0.3, = 0.1, and y = k 1/2 Use the equation of motion q k = s f(k) k to solve for the steady-state values of k, y, and c. LECTURE 4 Economic Growth: Solow model slide 32 11 Macroeconomics II An increase in the saving rate An increase in the saving rate raises investment… …causing k to grow toward a new steady state: Investment and depreciation k s2 f(k) s1 f(k) LECTURE 4 k* k 2* 1 Economic Growth: Solow model k slide 34 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. LECTURE 4 Economic Growth: Solow model slide 35 International evidence on investment rates and income per person Income per 100,000 person in 2000 (log scale) 10,000 1,000 100 0 5 10 15 20 25 30 35 Investment as percentage of output (average 1960-2000) LECTURE 4 Economic Growth: Solow model slide 36 12 Macroeconomics II 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 consumption’s share of income (1–s), which lowers c*. So, how do we find the s and k* that maximize c*? LECTURE 4 Economic Growth: Solow model slide 37 The Golden Rule capital stock * k gold 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*) = f (k*) LECTURE 4 i* k* In the steady state: i* = k* because k = 0. Economic Growth: Solow model slide 38 The Golden Rule capital stock steady state output and depreciation Then, graph f(k*) and k*, look for the point where the gap between them is biggest. * * y gold f (k gold ) LECTURE 4 k* f(k*) * c gold * * i gold k gold * k gold Economic Growth: Solow model steady-state capital per worker, k* slide 39 13 Macroeconomics II The Golden Rule capital stock c* = f(k*) k* is biggest where the slope of the production function equals l the slope of the depreciation line: k* f(k*) * c gold MPK = * k gold LECTURE 4 steady-state capital per worker, k* Economic Growth: Solow model slide 40 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 s. This adjustment leads to a new steady state with higher consumption. But what happens to consumption during the transition to the Golden Rule? LECTURE 4 Economic Growth: Solow model slide 41 Starting with too much capital * If k * k gold then increasing c* requires a fall in s. In the transition to the Golden Rule, consumption is higher at all points in time. y c i t0 LECTURE 4 Economic Growth: Solow model time slide 42 14 Macroeconomics II Starting with too little capital * If k * k gold then increasing c* requires an increase in s. Future generations enjoy higher consumption, but the current one experiences an initial drop in consumption. LECTURE 4 y c i t0 Economic Growth: Solow model time slide 43 Population growth Assume that the population (and labor force) grow at rate n. (n is exogenous.) L n L EX: Suppose L = 1,000 in year 1 and the population is growing at 2% per year (n = 0.02). Then L = n L = 0.02 1,000 = 20, so L = 1,020 in year 2. LECTURE 4 Economic Growth: Solow model slide 44 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 nk to equip new workers with capital (Otherwise, k would fall as the existing capital stock would be spread more thinly over a larger population of workers.) LECTURE 4 Economic Growth: Solow model slide 45 15 Macroeconomics II The equation of motion for k With population growth, the equation of motion for k is k = s f(k) ( + n) k actual investment LECTURE 4 break-even investment Economic Growth: Solow model slide 46 The Solow model diagram Investment, break-even investment k = s f(k) ( +n)k ( + n ) k sf(k) k* LECTURE 4 Capital per worker, k Economic Growth: Solow model slide 47 The impact of population growth Investment, break-even investment ( +n2) k ( +n1) k An increase in n causes an increase in breakeven investment, leading to a lower steady-state level of k. sf(k) k2* LECTURE 4 Economic Growth: Solow model k1* Capital per worker, k slide 48 16 Macroeconomics II 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. LECTURE 4 Economic Growth: Solow model slide 49 International evidence on population growth and income per person Income 100,000 per Person in 2000 (log scale) 10,000 1,000 100 0 1 2 3 4 5 Population Growth (percent per year; average 1960-2000) LECTURE 4 Economic Growth: Solow model slide 50 The Golden Rule with population growth To find the Golden Rule capital stock, express c* in terms of k*: c* = y* = f (k* ) i* ( + n) k* c* is maximized when MPK = + n or equivalently, MPK = n LECTURE 4 In the Golden Rule steady state, the marginal product of capital net of depreciation equals the population growth rate. Economic Growth: Solow model slide 51 17 Macroeconomics II Alternative perspectives on population growth The Malthusian Model (1798) Predicts population growth will outstrip the Earth’s 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 omitted the effects of technological progress. LECTURE 4 Economic Growth: Solow model slide 52 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. LECTURE 4 Economic Growth: Solow model slide 53 Chapter Summary 1. The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate negatively on its p p g y population g 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. LECTURE 4 Economic Growth: Solow model slide 54 18 Macroeconomics II Chapter 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. LECTURE 4 Economic Growth: Solow model slide 55 19 ...
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