# Lecture 12 The Solow Model and Convergence. Noah Williams

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1 Lecture 12 The Solow Model and Convergence Noah Williams University of Wisconsin - Madison Economics 312 Spring 2010

2 Recall: Balanced Growth Path All per-capita variables grow at rate g. All level variables grow at rate g + n. k = s k α (n + g + d) k ( s k(t) = A(t) ) 1 1 α n + g + d y(t) = ( s A(t) n + g + d ( s K(t) = N (t)a(t) n + g + d ( s Y (t) = N (t)a(t) n + g + d ) α 1 α ) 1 1 α ) α 1 α

4 Growth Facts Stylized facts, originally due to Kaldor. Empirical regularities of the growth process for the US and for most other industrialized countries 1 Output (real GDP) per worker y = Y N and capital per worker k = K N grow over time at relatively constant and positive rate. 2 They grow at similar rates, so that the ratio between capital and output, K Y is relatively constant over time 3 The real return to capital r (and the real interest rate r d) is relatively constant over time. 4 The capital and labor shares are roughly constant over time.

5 Capital and Output Growth US post-wwii data 300 K/N Y/N Capital and Output, 1948=

6 Capital/Output Ratio 3.6 Capital/Output Ratio

7 Interest Rate and Depreciation d r Implied Depreciation and Interest Rates Percent

8 Evaluation of the Model: Growth Facts The Solow model matches the stylized facts. Output and capital per worker grow at the same constant, positive rate in BGP of model. In long run model reaches BGP. Capital-output ratio K Y constant along BGP Interest rate constant in balanced growth path Capital share equals α, labor share equals 1 α in the model (always, not only along BGP) Success of the model along these dimensions, but source of growth, technological progress, is left unexplained.

9 Some Development Facts Stylized facts for cross-country comparisons. 1 Enormous variation of per worker income across countries. 2 Enormous variation in growth rates of per worker income across countries. 3 Growth rates are not constant over time for a given country. 4 Countries change their relative position in the international income distribution.

10 40 35 Distribution of Relative Per Worker Income Number of Countries Income Per Worker Relative to US

11 Output per Capita as a Share of US Level 100% 75% Canada USA Japan 50% France Germany (W) 25% Italy Britain 0% Years

12 25 Distribution of Average Growth Rates (Real GDP) Between 1960 and Number of Countries Average Growth Rate

13 Evaluation of the Model: Development Facts Differences in income levels across countries explained in the model by differences in s, n and d. Variation in growth rates: in model permanent differences can only be due to differences in rate of technological progress g. Temporary differences can be explained by transition dynamics. That growth rates are not constant over time for a given country can be explained by transition dynamics and/or shocks to n, s and d. Changes in relative position: in the model countries whose s moves up, relative to other countries, move up in income distribution. Reverse with n.

14 The Convergence Hypothesis Fact: Enormous variation in incomes per worker across countries Question: Do poor countries eventually catch up? Convergence hypothesis: They do, in the right sense. Main prediction of convergence hypothesis: Poor countries should grow faster (per capita) than rich countries. Why? Recall: k k = skα 1 (n + d), and: ẏ y = α k k

15 The Solow Model and Convergence Analyze countries with same s, n, d, α, g Eventually same growth rate of output per worker and same level of output per worker (absolute convergence). Countries starting further below the balanced growth path (poorer countries) should grow faster than countries closer to balanced growth path. Seems to be the case for the sample of now industrialized countries. World capital markets should speed this process. Capital should flow from rich (high K low MPK) to poor countries (low K, high MPK).

16 Figure 1.a: Growth Rate Versus Initial Per Capita GDP 3 JPN Growth Rate of Per Capita GDP, NOR FIN ITL CAN DNK GER SWE AUT FRA BEL NLD USA GBR NZL AUS Per Capita GDP, 1885

17 Figure 1.b: Growth Rate Versus Initial Per Capita GDP 5 JPN Growth Rate of Per Capita GDP, TUR POR GRC ESP IRL ITL FRA AUT BEL FIN GER NOR GBR DNK SWE NLD CAN CHE AUS USA NZL Per Worker GDP, 1960 x 10 4

18 Figure 1.c: Growth Rate Versus Initial Per Capita GDP Growth Rate of Per Capita GDP, KOR HKG OAN SGP JPN SYC CYP LSO THA PRT GRC ESP MYS IDN ITA JOR EGY TUR SYRIRL ISR AUT FRA FIN BEL CHN YUG BOL ECU PRY BRA GER LUX CMR GIN NAM TUN COL DZA ISL NOR NLD CAN GAB PNG MUS MEX GBR BGD CSK PAN DNK CHE ZAF FJI NGA GTM CIV HND LKA DOM SWEAUS GNBPHL COM CRI SLV CHL MAR URY IND COG CIV JAM NZL CAF ZMB SEN ZWE PER IRN KEN TTO GMB BEN TGOGHA MOZ TCD MLI RWANIC VEN UGA MRT MLI MDG CAF MWI BFA BDI BFALSO MOZ PAK GUY USA Per Worker GDP, 1960 x 10 4

19

20 Conditional Convergence Countries with same g but potentially differing s, n, d, α. Countries have different balanced growth path. Countries that start further below their balanced growth path (countries that are poor relative to their BGP) should grow faster than rich countries (relative to their BGP). This is called conditional convergence. Data for full sample lend support to conditional convergence. Industrialized countries as of 1885: similar savings rates, population growth rates. US States: Strong evidence of convergence across states. Again similar technology, saving, population. Baro and Sala-i-Martin (1992)

21

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23 Conclusion: The Solow Growth Model Offers a simple and elegant account of a number of growth facts. Leaves unexplained factors that make countries leave (or not attain) their BGP. Leaves unexplained why certain countries have higher s, n than others. Leaves unexplained technological progress, the source of growth.

24 Endogenous Growth Models Now briefly discuss some models which try to explain sources of growth endogenous growth models. An active research topic in late 1980s-1990s, not as much lately. Romer (1986, 1990), Lucas (1988) most influential: models of R & D, human capital. More recently Acemoglu et al: role of institutions in growth.

25 What s in TFP? Institutions & Geography Aside from innovations (which we ll turn to next), infrastructure, institutions, and geography are also important. Interesting comparison: experiences of former colonies. Acemoglu, Johnson and Robinson (2001). Small initial differences in income. Differences in settlers mortality influenced whether colony was run for extraction or whether colonists developed institutions. Those colonies where institutions took hold developed faster. Large differences in outcomes still today!

26

27 What s in TFP? Ideas and Human Capital Relatively new branch of economic theory: endogenous growth theory seeks to explain how technical change happens. Simple endogenous growth model (AK model): aggregate production function Y = AK. (Ignore labor and population growth, could think of this as per capita production.) Not subject to diminishing returns: MPK is constant F K = Y K = A. Idea: Aggregate capital K captures not just increases in physical capital but changes in the makeup of that capital.

28 Human Capital as a Source of Growth Human capital: knowledge, skills, and training of individuals. As economies become richer they invest in human capital in the same proportion, offsetting the diminishing marginal product of physical capital alone. Explicitly: production depends on human capital H, physical capital K: Y = zh θ K 1 θ Say H = hk, so that human capital is constant fraction of physical, then letting A = zh θ : [ Y = z(hk) θ K 1 θ = zh θ] K = AK

29 Other Interpretations Research and development programs are part of capital investment. They increase the stock of knowledge, which offsets diminishing marginal products of capital accumulation. Learning by doing: as economies produce more they learn better how to produce.

30 Implications of the Endogenous Growth Model Again savings constant fraction s of output. So: K = sak dk Since Y = AK, Ẏ Y = K K = sa d Growth of output depends on the saving rate, even in the long run. No steady state. Higher savings more human capital, R&D, learning by doing. So higher savings leads to productivity improvements and higher growth. Important implication, some evidence that measured TFP does depend on savings, human capital.

31 TFP growth rate, (Labor share=0.65, no returns to education) IDN IRL PRT MUS TUN CHL BWACOG MYS ZWE ITA ISR IND KEN COL URY BRA BEL SWESP NLD AUS GRC AUT DNK PAK PRY MAR USAGBR FRA ECU EGY GHA GTMLKA UGA BGD DOM SYRTUR CAN TTO ZAF DZA PAN MLI PHL ARG PER JAMEX BFA BOL NZL CHE ETH SLV CIV NPL HND MWI VEN SEN JOR CRI RWA BDI TZA BENCMR PNG MDG CAF MRT TGO NGA NER ZMB AGO MOZ ZAR NIC HKG KOR THA JPN FIN NOR Saving rate, Figure 1: Relation of TFP growth to saving rate SGP TFP growth rate, (Labor share=0.65, no returns to education) THA IDN PRT NOR TUN MUS ZWE BWA ITAMYS CHL COG JPNISR KEN BRAIND COL URYGRCAUT SWE PAK MAR PRY ECU GBR FRA ESP BEL FIN AUSDNK NLD USA GTM UGA BGD GHA DOM TUR LKA SYR EGY CAN DZA PAN ZAF ARGMEX PER TTO BFA MLI PHL JAM MWI CIV NPL HND BOL CHE NZL BDI TZA ETH SLV VEN SEN CRI RWA BEN PNG CMR MDG CAF MRT NGA NER ZMB TGO AGO MOZ ZAR NIC Human capital investment rate, Figure 2: Relation of TFP growth to schooling rate HKG SGP KOR IRL JOR 45

32 TFP growth rate, (Labor share=0.65, no returns to education) PRT NOR IRL THA IDN ITA JPN MUS CHL COG ISR TUN MYS ZWE BWA BEL FIN URY AUT GRC SWE DNK GBR FRA ESPNLD AUS IND BRA COL KEN USA MAR PAK ECU PRY CAN LKA EGY ARG CHE NZL TTO BGD GHA GTM TUR PAN DOM SYR JAM ZAF UGA PER BFA MLI PHL MEX DZA BOL NPL MWI HND SLV VENCIV ETH SEN CRI BDI TZA RWA PNG CMRBEN MDG CAF MRT NGA TGO AGO NER ZMB MOZ HKG KOR Labor force growth rate, Figure 3: Relation of TFP growth to labor force growth rate SGP ZAR NIC JOR TFP growth rate, (Labor share=0.65, 7% return to education) IRL IDN COG PRT MUSCHL ITA IND KEN TUN MYS COL URY BWA BRA BEL ZWEAUS DNK AUT ISR GTMLKA PAK PRY USAGBR SWESPGRC NLD FRA ECU UGA GHA BGDDOM CAN MLI EGY BOL TTO ZAF SYRTUR DZA ARGJAM PAN MWIPHL PERNZL MEX CHE SEN SLV NPL HND TZA CRI VEN RWA BENCMR PNG JOR MOZ CAF TGO NER ZAR NIC ZMB HKG KOR THA Saving rate, Figure 4: Relation of TFP growth to saving rate JPN FIN NOR SGP 46

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