# Macroeconomics II. Growth

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1 Macroeconomics II Growth Growth Possibilities We previously referred to the aggregate production function Y = A K α L 1- α. The growth rate of real GDP, Y, is generated by the contributions of A, K and L. Today: Understand when and how capital accumulation takes place. Relationship between capital accumulation and growth. Use one of most famous models in Economics: Solow Model. 1

2 The Solow Model Given 0 < α < 1, the production function is: Y = A K α L 1- α. What happens as we increase the amounts of labor or capital? Output goes up, but at a decreasing rate. Let s look at some examples. Cobb-Douglas example (cont d) Output = (Capital Stock) (1000) Output Capital Stock 2

3 Cobb-Douglas example (cont d) Previous examples illustrate the fact that there are decreasing marginal returns to inputs: Increasing the quantity of an input, while holding TFP and the other input fixed, raises output but by progressively smaller amounts. This feature of the production function is the the main force limiting growth through capital accumulation: As the capital stock grows larger, increments in capital will have smaller and smaller impact on output. The Solow Model (cont d) Let us examine the role of capital further. What is capital? Total value of the machines and buildings used to produce output. How does capital evolve over time? Investment: adding to the existing stock. Capital depreciates (wears out) with usage. 3

4 The Solow Model (cont d) Assume constant depreciation per unit of time, δ. Further, depreciation rate proportional to the capital stock: δk. Recall I was used to denote gross investment in capital goods. Given depreciation δk and investment I, how does capital next period relate to capital today? (Ignore 1-period lag). Simple: K t+1 = K t - δ K t + I t = (1- δ) K t + I t. National Income Accounting Detour GDP: Market value of the final goods and services produced in a given country in a given period of time. GDP is distributed as income to people involved in production: Wages paid to workers Profits paid to capital owners, Income is spent: GDP = C + I + G + (X-M) Three alternative but equivalent ways of computing GDP. 4

5 The Solow Model (cont d) From expenditure approach to measuring GDP: Y = C + I +G + (X-M). For now: Ignore government, G = 0; Assume closed economy: X-M = 0. Left with Y = C + I. From the income approach, we know that GDP will also be distributed to agents as income. Income, in turn, is either consumed or saved: Y = C + S. The Solow Model (cont d) Putting together Y = C + I and Y = C + S, it follows that saving equals investment: S = I. What determines saving? For now, assume households save a constant fraction s of their income: S = sy. We only need an extra step before starting to see the implications of the model for growth. 5

6 The Solow Model (cont d) For now, let us assume labor is constant: L t = L. To simplify and plot, express variables in per capita terms: Define: I Y K i = y = k = L L L Then: α 1 α α sak L K α i = sa = sak L L α 1 α AK L K α y = A = Ak L L α 1 α Kt sakt Lt k t+ 1 = 1 δ + = kt 1 δ + sak L L α ( ) ( ) α t Output, Saving, and Investment y=af(k) i=s Af(k) Capital per worker - k 6

7 Output, Saving, and Investment (cont d) Output Real GDP pc C Investment Investment (20% of GDP) Capital per person Steady State Increase in capital per worker: k t+ 1 kt = kt 1 Steady State: k α α ( δ ) + sakt kt = sakt δkt α 1 kt = sakt δk = 0 t+ t 7

8 Steady State (cont d) Investment = Depreciation Output Real GDP pc Depreciation = δ x Capital Stock Investment Capital per worker Steady-State (cont d) Investment lower than depreciation: capital declines. Output Real GDP pc Depreciation Investment K Low K ss K High Capital per worker 8

9 Steady-State (cont d) Investment greater than depreciation: capital increases. Output Real GDP pc Depreciation Investment K Low K ss K High Capital per worker Growth Rate Capital pe r worke r - k 9

10 Implications Countries with lower capital stocks should be growing the fastest: High investment rates and output growth rates, in transition. Countries with very high capital stocks should be eating some of that capital. Convergence Theory: Countries would converge to unique steady-state. Only transient differences in growth rates depending on a country s initial conditions. Examining Convergence Investing to Grow Average Annual Growth Rate Argentina Canada US France South Korea Japan 1 Chile Italy Investment Rate 10

11 Examining Convergence (cont d) 11 Real GPD per Capita Log US Dollars United States Japan Canada United Kingdom Italy France Examining Convergence (cont d) Growth Rate of Real GDP per Capita, KOR TUR PRT Convergence among OECD countries JPN GRC ESP MEX IRL AUT ITA FRA FIN NOR BEL NLDDNK SWE GBR LUX AUS CAN NZL USA Real GDP per Capita, 1955 CHE 11

12 Examining Convergence? Growth Rate of Real GDP per Capita, TWN KOR THA JPN CYP CHN MYS PRT GRCESP AUT BRA ISR IRL ITA DOM IRN FRA NOR LUX EGY FINBEL INDMAR JOR TTO PAK ZWE PAN TUR DNK CHL ISL LKA MUS NLD COL GBR AUS CAN ECU PRY SWE MEX KEN PHL JAM GTM MWI NZL CRI ZAF UGA ARG GHAHND PER URY GUY SLV ETH NGA BOL ZMB VEN ZAR (Lack of) convergence in the world NIC USA CHE Real GDP per Capita, 1955 Examining Convergence (cont d) Data reject (Unconditional) Convergence Theory. However, it appears that countries/regions with similar characteristics converge. Conditional Convergence Theory: A country will grow faster the further away it is from its own steady-state. Countries/regions within homogenous regions (OECD countries, individual states within USA) would be predicted to converge. What determines steady-state? Public policies (infrastructure, tax law, property rights): INSTITUTIONS. 12

13 Comparative Statics Under the model, what policies are conducive to growth? What happens if: Saving rate changes? Consider population growth? Saving Rate Higher saving rate higher steady-state capital stock. Output Real GDP pc Depreciation Investment (30% of GDP) Investment (20% of GDP) K 20% K 30% Capital per worker 13

14 Saving Rate (cont d) IMPORTANT: At steady-state, growth rate of capital (and output) are identical (zero?) across countries whose initial conditions were different. Steady-state growth rates are therefore independent of the saving rate. Then why bother with the saving rate? The level of output per person (GDP pc) is NOT independent of the saving rate. Saving Rate (cont d) Higher saving rate associated with higher living standards, at the steady-state. Log of Real GDP per Capita, Investment and GDP per Capita LUX USA CANDNKAUSHKG MAC IRL BEL CHE GBR GER NLD FRA AUT FIN ISL JPN NOR SWE ITA NZL ATG BRB ESP SVN PRT KOR ISR MUS KNA CZE GRC TTO CHL ARG EST HUN SVK URY SYC MYS GAB HRV MEX BLR POL KAZ LVA LTU DMA RUS VCT ZAF BLZ TUR BRA BGR COL CRI IRN GRD GEO DOM LBN LCA TUN PAN VEN THA MKD DZASWZ EGY SLV PRY UKR PER GTM SYR CHN CPV GNQALB IDN MAR JOR ROM ECU JAM KGZ LKA PHL GINARM BOL AZE IND ZWE CMR MDA CIV PAKHND BGD NIC COG SEN COM LSO GMBBEN GHA TJK NPL KEN UGA MOZ MLI BFA MDG RWA YEMTGO NER TCD ZMB NGA MWI ETH GNB BDI TZA Average Investment Rate,

15 Saving Rate (cont d) Given previous results, should governments try to promote massive saving? What are the implications of saving today? Postponing consumption. How much should a country optimally save? Golden Rule. The Golden Rule Depreciation = δ x Capital Stock δ is a technological parameter NIPA 3% for structures 8% for equipment New Capital = Investment δ x K Steady State Because of diminishing returns, will reach point where increases in capital stock don t pay off. Investment = δ *K 15

16 The Golden Rule (cont d) Make consumption as high as possible. Real GDP pc C I Steady State Investment = δ x K Maximize this Output K s Capital per person Choose value of s that delivers highest consumption. The Golden Rule (cont d) NOTE: C is getting smaller as K increases to K 1. Steady State Investment = δ x K Real GDP pc C I Output K 0 K 1 Capital per person 16

17 The Golden Rule (cont d) NOTE: C is getting smaller as K decreases to K 2. Steady State Investment = δ x K Real GDP pc C I Output K 2 K 0 Capital per person Golden Rule (cont d) Capital stock is too high Output is used to maintain an overly-large capital stock Consumption is low Capital stock is too low Output is used to support consumption Capital is too low to produce sufficient output 17

18 Golden Rule (cont d) δ = MPK Real GDP pc K s C I Steady State Investment = δ x K Output Capital per person Golden Rule: Marginal Product of Capital = Rate of Depreciation. Golden Rule (cont d) Questions: If saving/investment is so important, why do some countries not save? Some may be too poor to save (poverty traps). Can saving rates be permanently influenced by policy? Possible, not obvious, someone must pay the bill. Is increasing the saving rate always beneficial? Those who save now are reducing their consumption. There may be benefits for future generations but they come at a cost for the current cohorts. Transition issues not considered in our static analysis of the Golden Rule. 18

19 Does Aid Work? It could be the case that some countries are too poor to save: If income is below some subsistence level, they cannot afford to save. Poverty trap. Aid programs: A big inflow of investment is required for the country to get past the poverty trap and start to grow. Does Aid Work? (cont d) y=af(k) δk i=s * Af(k) Capital pe r worke r - k 19

20 Does Aid Work? (cont d) Historically, aid has not met with much success. In the very least, for aid to work we need good institutions. Does Aid Work? (cont d) Live Aid Pledge (July 2 nd 2005) To the 8 most powerful leaders in the world: 50,000 people are dying, needlessly, every day of extreme poverty. At this year's G8 summit meeting, it is within your power to put an end to this tragedy. It is an extraordinary opportunity which it would be shameful to ignore. We urge you to take these 3 steps to make extreme poverty history... double the aid sent to the world's poorest countries, fully cancel their debts, change the trade laws so that they can build their own future. 20

21 Does Aid Work? (cont d) From World Vision online: WE RECOGNIZE that a pact including such measures as fair trade, debt relief, fighting corruption and directing additional support for basic needs education, health, clean water, food, and care for orphans would transform the futures and hopes of an entire generation in the poorest countries, at a cost equal to just one percent more of the US budget. Application: The Asian Miracle Why did Asian economies grow so fast after 1950? Can this experience be repeated elsewhere? Let s do some growth accounting. 21

22 The Asian Miracle (cont d) The Asian Miracle (cont d) Evidence suggests that Southeast Asia achieved rapid growth as a result of capital accumulation and, to a lesser extent, employment. As such, these rapid growth phenomena can be thought of as a case study in capital accumulation. As the model indicates, growth from accumulation of capital is not sustained. Developed economies grow mostly out of TFP. Miracle put into perspective. Cost: high investment hurt old at expense of young. 22

23 Population Growth Remember law of motion for capital per worker? k + i t = (1 d) kt 1 Transform it to accommodate population growth rate, n. k t= k (1 t 1 d n) + i Intuition: more capital needed to keep up with growing number of people using it. t t Population Growth (cont d) Population growth reduces output per capita in equilibrium y=af(k) (δ+n)k δk i=s Af(k)

24 Population Growth (cont d) Log Real GDP per Capita, Population Growth and GDP per Capita LUX USA DNK NORCHE JPN HKG BEL AUT CAN AUS SGP GBR ITA FIN SWEFRANLD ISL IRL BRB CYP NZL ESP ISR KOR TWN PRTGRC ARG MUS URY TTOPRI SYC CHL GAB MYS BRA THA ZAF MEX TUR VEN COL BWA FJI PAN ROM CRI IDN DOMECU DZA PRY IRN JAM SLVPER NAM EGY MAR GTM SYR GUY CHN LKACPV PNG PHL GINBOL ZWE IND CMR HND HTI PAK COG NIC AGO BGDCOM GNQ LSO SEN NPL MRTGHA KEN BEN GMB GNB MOZ CAF BFA TCD NGA MLI RWA MDG TGO ZMB UGA MWI NER BDI ETH TZA ZAR CIV Average Growth Rate of Population, JOR Comparative Statics Should we conclude that countries that save more and have fewer children become rich? Or that countries that become rich save more and have fewer kids? Saving and fertility rates are clearly endogenous variables: Influenced by the level of pc income. Better interpretation: Growth comes from a common driving force to GDP pc, saving, fertility, 24

25 Demographic Transition Model Model of demographic transition based on experience of Western Europe. Summary Decreasing marginal returns to capital imply that growth exclusively based on capital accumulation will cease. In addition, redistributive effects across cohorts (young/old). Need to consider other sources of growth. 25

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