Advanced Macroeconomics Lecture 4 Growth facts. Solow-Swan model

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1 Advanced Macroeconomics Lecture 4 Growth facts. Solow-Swan model Marcin Bielecki March 14, 2016

2 GDP per capita and welfare Acemoglu (2009) Introduction to Modern Economic Growth

3 GDP per capita and welfare Acemoglu (2009) Introduction to Modern Economic Growth

4 Growth fact 1 There is enormous variation in per capita income across economies. The poorest countries have per capita incomes that are less than 5 percent of per capita incomes in the richest countries. Income per capita (or GDP per capita) is not the sole measure of what is good, but it s a useful summary statistic. Income per capita ignores distribution of income within a country. Comparing income per capita across countries is not trivial: You have to convert between currencies. Countries have different relative prices for goods.

5 Growth fact 1 GDP per capita GDP per worker LF part. rate Avg. growth Years to Country double United States United Kingdom Japan France Singapore Hong Kong Taiwan South Korea China India Nigeria Uganda Venezuela Haiti Madagascar Zimbabwe Jones and Vollrath (2013) Introduction to Economic Growth

6 Growth fact 1 Acemoglu (2009) Introduction to Modern Economic Growth

7 Growth fact 2 Rates of economic growth vary substantially across countries. Barro and Sala-i-Martin (2004) Economic Growth

8 Growth fact 3 Growth rates are not generally constant over time. For the world as a whole, growth rates were close to zero over most of history but have increased sharply in the twentieth century. For individual countries, growth rates also change over time. The big changes in growth rates over history are from pre-industrial Revolution (close to 0% growth) to modern times (roughly 1.85% growth per year for developed countries). The big changes in growth rates within countries tend to be as they transition from poor to rich (e.g. Japan), after which growth slows down.

9 Growth fact Per capita GDP (1990 dollars, log scale) year Jones and Vollrath (2013) Introduction to Economic Growth

10 Growth fact Japan: Log per capita GDP (1990 Int. GK$) Year

11 Growth fact 3 Vollrath (2016)

12 Growth fact 4 A country s relative position in the world distribution of per capita incomes is not immutable. Countries can go from being poor to being rich, and vice versa. The growth miracles in the table were in 1960 very poor. Now they are among the rich countries. The growth disasters in the table were not that different in 1960 compared to East Asia. Now they are well behind.

13 Growth fact 4 Acemoglu (2009) Introduction to Modern Economic Growth

14 Kaldor s stylized facts In the USA and other developed countries: 1. Per capita output grows over time, and its growth rate does not tend to diminish. 2. Physical capital per worker grows over time. 3. The rate of return to capital is not trending. 4. The ratio of physical capital to output is nearly constant. 5. The shares of labor and physical capital in national income are nearly constant. 6. Real wage grows over time.

15 Kaldor s stylized fact 1 Per capita output grows over time, and its growth rate does not tend to diminish. 35,000 Per capita GDP (1990 dollars, log scale) 20,000 10,000 5,000 2, Year Jones and Vollrath (2013) Introduction to Economic Growth

16 Kaldor s stylized fact 2 Physical capital per worker grows over time USA: capital per worker

17 Kaldor s stylized fact 3 The rate of return to capital is not trending UK Long-Term Nominal Interest Rate US Long-Term Nominal Interest Rate

18 Kaldor s stylized fact 3 The rate of return to capital is not trending. DeLong (2015)

19 Kaldor s stylized fact 3 The rate of return to capital is not trending. Gomme, Ravikumar and Rupert (2015)

20 Kaldor s stylized fact 4 The ratio of physical capital to output is nearly constant USA: capital to GDP ratio

21 Kaldor s stylized fact 5 The shares of labor and physical capital in national income are nearly constant. Acemoglu (2009) Introduction to Modern Economic Growth

22 Kaldor s stylized fact 6 Real wage grows over time USA: Real Compensation Per Hour (Index 2009=100)

23 Explaining growth We want to explain: Why some countries are poor and other rich? Why some countries that were previously poor become rich? Why not all poor countries catch up to rich countries? Why do rich countries still grow?

24 Solow-Swan model Robert Solow (1956) and Trevor Swan (1956) Growth in income per capita comes from two sources: Capital accumulation (endogenous) Improvements in technology (exogenous) But capital accumulation alone cannot sustain growth in the absence of technology improvements Does not explain deep sources of economic growth Departure point for growth theory

25 Simplifications and assumptions Closed economy No government Single, homogenous final good with its price normalized to 1 in each period (all variables are expressed in real terms) Two types of representative agents: Firms Households

26 Production Real GDP is produced according to a neoclassical production function: Y t = F (K t, A t N t ) where Y is real GDP, F is a neoclassical production function, K is capital stock, A is the technology level and N is the number of workers. Technology grows at a constant rate g > 0 and increases productivity of labor (otherwise Kaldor s stylized facts would be violated): A t+1 = (1 + g) A t Very often we use a Cobb-Douglas production function: Y t = K α t (A t N t ) 1 α Like other neoclassical production functions, it exhibits constant returns to scale doubling both inputs K and N doubles the amount produced: (zk t ) α (A t zn t ) 1 α = z α z 1 α Kt α (A t N t ) 1 α = zy t

27 Firms Perfectly competitive firms solve the following profit maximization problem: FOCs: max K t,n t K α t (A t N t ) 1 α r t K t w t N t K t : αkt α 1 (A t N t ) 1 α r t = 0 r t = α Y t K t N t : (1 α) Kt α A 1 α t Nt α w t = 0 w t = (1 α) Y t N t Total factor payments are equal to GDP: r t K t + w t N t = α Y t K t K t + (1 α) Y t N t N t = αy t + (1 α) Y t = Y t

28 Factor shares Calculate the fraction of GDP that is paid to each factor: w t N t Y t = (1 α) Yt N t N t Y t = (1 α) and r t K t Y t = α Yt K t K t Y t = α Cobb-Douglas function implies constant shares of labor and physical capital in income. Confronting with the US data, we can obtain α 1 3 and (1 α) 2 3.

29 Households Own factors of production (capital and labor) and earn income from renting them to firms. Each households supplies one unit of labor and population grows at a constant rate n: N t+1 = (1 + n) N t Capital is accumulated from investment I t and depreciates at rate δ > 0: K t+1 = I t + (1 δ) K t Income of households is consumed or saved (invested): Y t = w t N t + r t K t = C t + S t = C t + I t Do NOT optimize, save a constant fraction s of income I t = sy t and C t = (1 s) Y t

30 GDP per worker Usually we are most interested in GDP per worker (or per capita): Y t N t = K α t (A t N t ) 1 α = A t N t ( Kt A t N t ) α = A t k α t where k is capital per effective unit of labor (AN). Clearly, GDP per worker increases due to improvements in technology and due to capital accumulation. The production function in per worker terms is characterized by diminishing marginal product of capital. GDP per worker increases with k but the size of the increase falls with k. It is also useful to define output per effective unit of labor y: y t = Y t A t N t = k α t

31 Capital accumulation Capital accumulates according to: K t+1 = sy t + (1 δ) K t And capital per effective labor according to: K t+1 = sy t + (1 δ) K t : A t N t K t+1 A t+1 N t+1 A t+1 N t+1 A t N t = s Y t + (1 δ) A t N t k t+1 (1 + g) (1 + n) = sy t + (1 δ) k t k t+1 = skα t + (1 δ) k t (1 + g) (1 + n) k t+1 skα t + (1 δ) k t (1 + n + g) The growth rate of capital per effective labor equals: K t A t N t k t+1 k t = skt α δk t (n + g + ng) k t+1 k t+1 skt α 1 (δ + n + g) k t

32 Balanced growth path (steady state) Variables per effective labor converge to their steady state values. If k t+1 = k t = k then: k (1 + n + g) = s (k ) α + (1 δ) k k (δ + n + g) = s (k ) α (k ) 1 α = k = y = s δ + n + g ( s δ + n + g ( s δ + n + g ) 1 1 α ) α 1 α Variables per worker grow together with increases in technology: Y t = A t y (Y t+1/n t+1 ) = A t+1 = g N t Y t /N t A t And variables like capital and GDP grow at the sum of rates of increase in population and technology.

33 Comparative statics Solow-Swan model predicts that GDP per worker ( Y t = A t N t s δ + n + g ) α 1 α is higher in countries with higher investment share of GDP s and higher technology level A, and lower in countries with higher population growth rate n

34 Investment share of GDP s vs real GDP per worker Y /N Real GDP per worker, 2008, log scale NOR NLD USA AUTAUS BEL GBR FRA CANDNK FIN IRL ISL PRI SWE GRC ITA HKG TTO TWNISR CHEJPN ESP GNQ 50,000 NZL CYP BRB PRT TUR MEX CHL IRN ARG MYS CRI DOM PAN ROM GAB 20,000 URYVEN ZAF BRA COL JAM MUS EGY SLV GTM PER NAMECU DZA SYR FJI 10,000 LKA HND MAR BOL PRY IND IDN PAK PHL NGA PNG 5,000 ZMB NIC COG CMR MRT CIV HTI TCD GMB SEN MLI KEN UGA BENBGD NPL GHA RWA NER COMBFA MDG GIN GNB TZA 2,000 MOZ TGO MWI CAF ETH 1,000 BDI ZAR SGP KOR BWA THA CHN LSO ZWE Investment share of GDP, s, Jones and Vollrath (2013) Introduction to Economic Growth

35 Population growth rate n vs real GDP per worker Y /N Real GDP per worker, 2008, log scale ,000 20,000 10,000 5,000 2,000 1,000 NOR BEL AUTNLD USA SGP AUS FIN DNKFRA ISL GRC CAN HKG IRL JPN SWE ITA GBR TTO CHE PRI TWN ESP ISRGNQ KOR NZL PRT BRB CYP CHL MEX TUR IRN ARG MYS ROM CRI URY DOM BWA GAB JAM MUS ZAF PAN VEN BRA COL SLV PER DZA EGY ECUGTM THA NAM CHN FJI SYR LKA MAR CPV HND IDNIND BOL PRY PHL NGA PNG PAK NIC CMR ZMB COG SEN MRT LSO HTI MLICIV BGD TCDGMB NPL GHA KEN BEN UGA TZA RWA GNB NER COM BFA GIN MWI MOZTGO MDG CAF ETH BDI ZAR JOR ZWE Population growth rate, n, Jones and Vollrath (2013) Introduction to Economic Growth

36 Transition dynamics We are also interested in the determinants of growth rates in GDP per worker. Start with growth rates of GDP per effective labor: ( ) ( y t+1 yt+1 k α ) ( ) ln = ln t+1 kt+1 = α ln α k t+1 y t y t k t k t y t+1 y t k α t α [ sk α 1 t (δ + n + g) ] To obtain growth rate of GDP per worker, add the growth rate of technology g: (Y t+1 /N t+1 ) = α [ skt α 1 (δ + n + g) ] + g Y t /N t = α [ sk α 1 t (δ + n) ] + (1 α) g An increase in s or a decrease in n temporarily increases the growth rate of GDP per worker. Note that even if higher g decreases k it increases the rate of growth of GDP per worker.

37 Investment share of GDP s in growth miracle countries Jones (1998) Introduction to Economic Growth

38 Factor payments once again Using k as capital per effective labor along the BGP, let us revisit factor prices: r = αkt α 1 (A t N t ) 1 α = α (k ) α 1 w = (1 α) Kt α A 1 α t Nt α = (1 α) A t (k ) α The model predicts that along the BGP the interest rates are constant while hourly wages increase at the same rate as GDP per hour: Real mean hourly compensation Real GDP per hour Index 1960= Year

39 Convergence Solow-Swan model predicts that if countries have access to the same technology and share the same steady state, then ones that are poorer should grow faster: Growth rate, JPN GRC PRT FIN ESP NOR SWE ITA IRL CAN AUT DNK FRA CHE GER USA BEL NLD NZL AUS GBR 500 1,000 1,500 2,000 2,500 3,000 3,500 Per capita GDP, 1870 Growth rate, KOR JPN TUR GRC 0.03 IRL PRT ESP FIN ITA 0.03 AUT BEL FRA NOR ISR ISL 0.02 DNK SWE GBR CHL AUS NLD USA 0.01 CAN CHE MEX 0.01 NZL ,000 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 GDP per worker, 1960 Jones and Vollrath (2013) Introduction to Economic Growth

40 No absolute convergence In general it is not true that poorer countries grow faster: 0.06 BWA CHN TWN Growth rate, ROM KOR HKG 0.04 THA SGP MYS LKA JPN IND IDNEGY TUR CYP GRC MAR IRL PRT ESP LSO FINITA PNG PAK PAN TTO AUT GNB TZA MRTCPV MUS DOM ISRPRI FRA BEL NOR 0.02 COG GAB MOZ CHL DNK SWE ISL GBR AUS MWI BFA UGA ETH GHA MLIPHL SYR BRA COLURY BRB NPL ECU GTM IRN BEN TCD BGD GMB CMR PRY HND FJI ARG MEX NAMPER SLV BDI CIV NGA JOR JAM ZAF CRI RWA BOL COM KEN SEN ZMB 0.00 DZA MDG VEN TGO GIN NERHTI CAF NIC ZWE 0.02 NLD USA CAN CHE NZL ZAR 1,000 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 GDP per worker, 1960 Jones and Vollrath (2013) Introduction to Economic Growth

41 Conditional convergence But countries grow faster if they are further from their own steady state: Growth rate of GDP per worker, BWA CHN TWN ROM HKG KOR THA SGP MYS JPN GRC TUR LKA CYP EGYIND MAR IRL IDN ESPPRT FIN LSO ITA TTO BEL PAN AUT ISRPAK PNG FRA GAB DNK COG DOM NOR ISL LUX MUS MRT TZA SWE AUS GBR CHL MWI BRB MOZ BRA NLD COL IRN GTM CAN PHL MLI NPL ECU MEX CHE SYR USA URY UGA ARG GHA NAM CRI PRY SLV BGD GMB ZAF FJI BEN CMR HND BDI JAM PER CIV JOR NZL RWA BOL KEN DZA SEN VEN ZMB TGO NER HTI NIC CAF ZWE Deviation from steady state in 1960, log scale Jones and Vollrath (2013) Introduction to Economic Growth

42 Speed of convergence The model implies a relationship between the distance from steady state and the current rate of growth: ( ) y t+1 yt (1 α) (δ + n + g) ln y t }{{} y Econometric studies both on individual countries and US s states find that β 0.02, meaning that it takes about 35 years to close half of the gap between the current income level and its steady state. Given sensible parameter values α = 0.33, δ = 0.05, n = 0.01 and g = 0.02 the model generates β = 0.053, meaning that it would take about 13 years to close half of the gap, a very unrealistic number. Adding human capital allows the model to assign lower weight to raw labor and be consistent with slow convergence. β

43 Human capital augmented Solow model The production function accounts for human capital: Y t = K α t (A t H t ) 1 α H t = e ψu N t where u is the amount of time spent acquiring human capital (years of schooling) and ψ captures the empirical regularity that schooling boosts individuals wages (estimates point to ψ 0.1, implying that extra year of schooling boosts wages by 10%). Wages contain not only rewards to raw labor (minimum wage level), but also to human capital (actual wage minus the minimum wage). Empirical estimates of the income share of broad capital are consistent with convergence factor β Education generates level effects for GDP per worker: ( Y t = A t e ψu N t s δ + n + g ) α 1 α

44 Years of schooling u vs growth rate in GDP per capita Acemoglu (2009) Introduction to Modern Economic Growth

45 Fit of human capital-augmented Solow model Suggests that poor countries should be richer Predicted steady state value of relative Y/L KOR JPN AUS USA LKA NZLISRCAN HKG BEL CHN JOR IRL JAM BWA ROMMYS ESP TWN CHE CHL BRB GRC DNK SWE FIN ISL FJI FRA ITA AUT NLDSGP DZA ARG CYP LSO THA GHA ECU PER MUS PRT TTO PHL ZAF PAN GAB IRN IDN COL CRI MEX GBR PRY URY NIC HND BOL MAR NAM SLV BRADOM TZABGD IND VEN TUR KEN MRTCOG CMR EGY NPL SEN ZMB PAK UGA BEN GTM GMB PNG SYR NER HTI MLI RWA CIV Relative Y/L Jones and Vollrath (2013) Introduction to Economic Growth

46 Solow residual: accounting for technology differences Relative A SGP TTOGBR AUT NLD BEL DNK FIN FRA GRC HKG ITA IRL ISL TWN SWE AUS USA TUR ESPCHE CAN ISR JPN PRT CYP NZL DOM MEX BRBKOR GTM VENCRI IRN CHL EGY ARG SLV BRA COL PAN URYMYS SYR MUS ZAF GAB BWA ROM NAMPER THA ECU JAM IND DZA PNG MAR HND CHN PAK IDN BOL JOR PRY FJI GMB CIV MLI ZMB PHL LKA COG CMR HTIMRT NIC RWA UGA SEN NER NPL BEN BGD TZA KEN LSO GHA Relative Y/L Jones and Vollrath (2013) Introduction to Economic Growth

47 Takeaway Long run growth stems from improvements in technology Countries can achieve higher steady states if they accumulate more physical and human capital Just as important as accumulation is technology adoption Did not touch on deep causes of growth we treated many choice variables as exogenous parameters: Countries with low s may not protect private ownership properly or have underdeveloped financial system Countries with high n may have high mortality rates incentivizing families to have many children in hopes that at least some survive into adulthood to be able to support their parents once they are old Countries may have high u because they are already rich and children do not have to work there Groups of interest within a country may obstruct technology adoption if they have a monopoly over the old technology