Economic Growth I: Capital Accumulation And Population Growth PDF

CHAPTER 8 Economic Growth I: Capital Accumulation and Population Growth © 2016 Worth Publishers, all rights reserved  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...

CHAPTER 8 Economic Growth I: Capital Accumulation and Population Growth © 2016 Worth Publishers, all rights reserved  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 2  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…. CHAPTER 8 Economic Growth I 3  Income and poverty in the world selected countries, 2010 100 Zambia % of population living on $2/day or less Nigeria 75 Senegal 50 Indonesia Kyrgyz Republic Georgia Peru Panama 25 Uruguay Mexico Poland 0 $0 $3.500 $7.000 $10.500 $14.000 Income per capita in U.S. dollars  links to prepared graphs @ Gapminder.org notes: circle size is proportional to population size, color of circle indicates continent, press “play” on bottom to see the cross section graph evolve over time Income per capita and ▪ Life expectancy ▪ Infant mortality ▪ Malaria deaths per 100,000 ▪ Cell phone users per 100 people CHAPTER 8 Economic Growth I 5  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 increase in standard rate of income of living after… 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% CHAPTER 8 Economic Growth I 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 CHAPTER 8 Economic Growth I 9  How Solow model is different from Chapter 3’s 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 CHAPTER 8 Economic Growth I 10  How Solow model is different from Chapter 3’s model 4. no G or T (only to simplify presentation; we can still do fiscal policy experiments) 5. cosmetic differences CHAPTER 8 Economic Growth I 11  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) CHAPTER 8 Economic Growth I 12  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 CHAPTER 8 Economic Growth I 13  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 CHAPTER 8 Economic Growth I 14  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) CHAPTER 8 Economic Growth I 15  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) CHAPTER 8 Economic Growth I 16  Output, consumption, and investment Output per f(k) worker, y c1 y1 sf(k) i1 k1 Capital per worker, k CHAPTER 8 Economic Growth I 17  Depreciation Depreciation δ = the rate of depreciation per worker, δk = the fraction of the capital stock that wears out each period δk δ 1 Capital per worker, k CHAPTER 8 Economic Growth I 18  Capital accumulation The basic idea: Investment increases the capital stock, depreciation reduces it. Change in capital stock = investment – depreciation Δk = i – δk Since i = sf(k) , this becomes: Δk = s f(k) – δk CHAPTER 8 Economic Growth I 19  The equation of motion for k Δk = s f(k) – δk ▪ 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) CHAPTER 8 Economic Growth I 20  The steady state Δk = s f(k) – δk If investment is just enough to cover depreciation [sf(k) = δk ], then capital per worker will remain constant: Δk = 0. This occurs at one value of k, denoted k*, called the steady state capital stock. CHAPTER 8 Economic Growth I 21  The steady state Investment and δk depreciation sf(k) k* Capital per worker, k CHAPTER 8 Economic Growth I 22  Moving toward the steady state Δk = sf(k) − δk Investment and δk depreciation sf(k) Δk investment depreciation k1 k* Capital per worker, k CHAPTER 8 Economic Growth I 23  Moving toward the steady state Δk = sf(k) − δk Investment and δk depreciation sf(k) Δk k1 k2 k* Capital per worker, k CHAPTER 8 Economic Growth I 24  Moving toward the steady state Δk = sf(k) − δk Investment and δk depreciation sf(k) Δk investment depreciation k2 k* Capital per worker, k CHAPTER 8 Economic Growth I 25  Moving toward the steady state Δk = sf(k) − δk Investment and δk depreciation sf(k) Δk k2 k3 k* Capital per worker, k CHAPTER 8 Economic Growth I 26  Moving toward the steady state Δk = sf(k) − δk Investment and δk depreciation Summary: sf(k) As long as k < k*, investment will exceed depreciation, and k will continue to grow toward k*. k3 k* Capital per worker, k CHAPTER 8 Economic Growth I 27 NOW YOU TRY Approaching k* from above 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. Show what happens to k over time. Does k move toward the steady state or away from it? 28  A numerical example Production function (aggregate): 1/2 1/2 Y = F (K , L ) = K × L = K L To derive the per-worker production function, divide through by L: 1/2 1/2 1/2 Y K L ⎛K ⎞ = =⎜ ⎟ L L ⎝L ⎠ Then substitute y = Y/L and k = K/L to get 1/2 y = f (k ) = k CHAPTER 8 Economic Growth I 29  A numerical example, cont. Assume: ▪ s = 0.3 ▪ δ = 0.1 ▪ initial value of k = 4.0 CHAPTER 8 Economic Growth I 30  Approaching the steady state: A numerical example Year k y c i δk Δk 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 2.096 1.467 0.629 0.440 0.189 4 4.584 2.141 1.499 0.642 0.458 0.184 … 10 5.602 2.367 1.657 0.710 0.560 0.150 … 25 7.351 2.706 1.894 0.812 0.732 0.080 … 100 8.962 2.994 2.096 0.898 0.896 0.002 … ∞ CHAPTER 8 9.000 3.000 2.100 Economic Growth I 0.900 0.900 0.000 31 NOW YOU TRY Solve for the steady state Continue to assume s = 0.3, δ = 0.1, and y = k 1/2 Use the equation of motion Δk = s f(k) − δk to solve for the steady-state values of k, y, and c. 32 ANSWERS Solve for the steady state s f (k *) = δ k * eq'n of motion with Δk = 0 0.3 k * = 0.1k * using assumed values k * 3= = k * k * Solve to get: k * = 9 and y * = k * = 3 Finally, c * = (1 − s )y * = 0.7 × 3 = 2.1 33  An increase in the saving rate An increase in the saving rate raises investment… …causing k to grow toward a new steady state: Investment δk and depreciation s2 f(k) s1 f(k) * k k 1 k * 2 CHAPTER 8 Economic Growth I 34  Prediction: ▪ 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. ▪ Are the data consistent with this prediction? CHAPTER 8 Economic Growth I 35  International evidence on investment rates and income per person Income per 100.000 person in 2010 (log scale) 10.000 1.000 100 0 9 18 28 37 46 55 Investment as percentage of output (average 1960-2010)  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*? CHAPTER 8 Economic Growth I 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*) − i* In the steady state: = f (k*) − δk* i* = δk* because Δk = 0. CHAPTER 8 Economic Growth I 38  The Golden Rule capital stock steady state output and depreciation δ k* Then, graph f(k*) and δk*, f(k*) look for the point where the gap between c * gold them is biggest. * * i gold = δ k gold * * y gold = f (k gold ) * k gold steady-state capital per worker, k* CHAPTER 8 Economic Growth I 39  The Golden Rule capital stock c* = f(k*) − δk* δ k* is biggest where the slope of the f(k*) production function equals the slope of the * depreciation line: c gold MPK = δ * k gold steady-state capital per worker, k* CHAPTER 8 Economic Growth I 40  Math details CHAPTER 8 Economic Growth I 41  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? CHAPTER 8 Economic Growth I 42  Starting with too much capital If k * > k gold * then increasing c* y requires a fall in s. In the transition to c the Golden Rule, consumption is i higher at all points in time. t0 time CHAPTER 8 Economic Growth I 43  Starting with too little capital * * If k < k gold then increasing c* requires an y increase in s. c Future generations enjoy higher consumption, but the current i one experiences an initial drop in consumption. t0 time CHAPTER 8 Economic Growth I 44  Population growth ▪ Assume the population and labor force grow at rate n (exogenous): ΔL = n L ▪ EX: Suppose L = 1,000 in year 1 and the population is growing at 2% per year (n = 0.02). ▪ Then ΔL = n L = 0.02 ×1,000 = 20, so L = 1,020 in year 2. CHAPTER 8 Economic Growth I 45  Break-even investment ▪ (δ + n)k = break-even investment, the amount of investment necessary to keep k constant. ▪ Break-even investment includes: ▪ δ k to replace capital as it wears out ▪ nk 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.) CHAPTER 8 Economic Growth I 46  The equation of motion for k ▪ With population growth, the equation of motion for k is: Δk = s f(k) − (δ + n) k actual break-even investment investment CHAPTER 8 Economic Growth I 47  The Solow model diagram Investment, Δk = s f(k) − (δ+n)k break-even investment (δ + n ) k sf(k) k* Capital per worker, k CHAPTER 8 Economic Growth I 48  The impact of population growth Investment, break-even (δ +n2) k investment (δ +n1) k An increase in n causes an increase sf(k) in break-even investment, leading to a lower steady-state level of k. k2* k1* Capital per worker, k CHAPTER 8 Economic Growth I 49  Prediction: ▪ 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. ▪ Are the data consistent with this prediction? CHAPTER 8 Economic Growth I 50  International evidence on population growth and income per person Income per 100.000 person in 2010 (log scale) 10.000 1.000 100 0 1 2 3 4 5 Population growth (percent per year, average 1961-2010)  The Golden Rule with population growth To find the Golden Rule capital stock, express c* in terms of k*: c* = y* − i* = f (k* ) − (δ + n) k* In the Golden c* is maximized when Rule steady state, MPK = δ + n the marginal product of capital net of or equivalently, depreciation equals MPK − δ = n the population growth rate. CHAPTER 8 Economic Growth I 52  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 neglected the effects of technological progress. CHAPTER 8 Economic Growth I 53  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. CHAPTER 8 Economic Growth I 54  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 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 55  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. 56