# Ken Jackson. January 31st, 2013

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1 Wilfrid Laurier University January 31st, 2013

2 Recap of the technology models Do the models match historical data? growth accounting Estimating technology change through history A revised model of technology change

3 A One-Country Model of Technology Driven The pace of technological change ŷ = γ A µ L γ A is the share of the population in research µ is the cost of invention, L is the size of the labour force Economic growth increases with population size Should large countries grow faster than small countries? Increasing the fraction of the population in research increases economic growth Is this true historically?

4 Two-Country Model of Technology Driven Differentiate between inventors and imitators It should be cheaper to imitate - follower countries should have an easier time Very different from conditional convergence Leader and follower countries will both grow at the same pace If lead country increases research inputs - growth increases What are the key assumptions driving this result?

5 Models of Technology - A Summary The number of researchers matter, and the cost of invention matters There are two implicit assumptions here: There are constant returns to scale in research - double the number of researchers doubles technology growth The cost of research is constant - regardless of what has been discovered How do these fit the historical evidence?

7 Review of How do we account for changes in income? accounting is introduced in chapter 7 of the textbook Start with an equation for income Y = AK α (hl) 1 α Or in per capita terms: y = Ak α h 1 α Income per capita = productivity(a) * factors of production(k,h)

8 Calculating Rates Defining a growth rate for variable, X Take a natural logarithm - ln X Derivative with respect to time: ln X t = X t X Derivative of the natural logarithm is the growth rate

9 - Factors of Production Define X = k α h 1 α To find the growth rate of X, first take the natural logarithm: ln X = ln k α h 1 α = α ln k + (1 α) ln h Then differentiate both sides with respect to time Note that: X t X ln X = t = Ẋ X Ẋ = α k ḣ + (1 α) X k h ˆX = αˆk + (1 α)ĥ

10 Continued Now we can rewrite the original equation as: y = AX Taking logarithms: lny = lna + lnx (1) Taking derivatives with respect to time: ŷ = Â + ˆX = Â + αˆk + (1 α)ĥ

11 Continued rates ŷ = Â + αˆk + (1 α)ĥ (income) = growth rate(productivity) + growth(factors of production) Or rearranging: Â = ŷ αˆk (1 α)ĥ rate of technology = growth rate of income - growth rate of factors of production

12 Using for the Solow Model Measure growth rates of income Measure growth rates of physical and human capital Estimate the share α of income that is paid to capital The results are the growth rate of technology

13 productivity growth CAF NIC NER ZMB VEN MOZ TGO TZA ZWE JPN BEL ITA AUT ISR CHL NOR FRA ESP FIN SYR KEN GHA AUS IND DNK GRC MWI BRA USA URYGBR PAK ECU NLD SWE DOM CAN COL ZAF PAN BOL GTM DZA ARG PER PRY LKA CHE TTOUGA BGD MEX JAM SEN PHL MLI CRI CMR PNG HND NPL SLV JOR HKG IRL MUS THA MYS IDN KOR BWA rate of income,

14 How did technology change before 1800? The growth accounting based on the Solow model is difficult No measure of capital formation - can we ignore capital formation? Labour was not the only factor of production Land matters far more than capital in an agricultural society Landowners received a significant share of total income

15 A Modified Solow Model Add land to the Solow model, but remove capital Y = AX β L 1 β We have maintained the constant returns to scale idea - but not in reproducible factors As before - all income is paid to land-owners or labour (calculate marginal products) Dividing both sides by the size of the population we get: y = A ( ) β X L

16 in the Modified Solow Model Taking logs of this equation and calculating growth rates gives us ŷ = Â + β ˆX βˆl (income) = growth(technology) + growth(land) - growth(labour) rate of labour has direct consequences on growth rate of income Alternatively: growth(land) - growth(labour) = growth(land per capita)

17 Fixed Factors Does it matter that land is in the equation Land is fixed - increases in population reduce the effectiveness of labour In the Solow model this happens, but investment in capital compensates rate of income = growth rate of productivity + growth rate of physical and human capital per person Here - no compensating factor The fixed factor implies a much stronger negative effect of fertility growth Similar effects to the Malthusian model (without the feedback effect)

18 As before: ŷ = Â + β ˆX βˆl rate of land is zero Rearranging the last equation Â = ŷ + βˆl (technology) = growth(income) + growth(labour force) Historical measure of β 1 3 Calculate growth rates over 2 historical time periods

20 Industrial - period of rapid technological change Roughly in Britain Key technological changes Textiles - productivity improves by 200x Energy - production of coal rises 10x for use in steam engines Transportation - steam engines reduce the cost of water and rail transport Metallurgy - use of coal increases iron production 20x from

23 Industrial Up to rapid technological change Income growth driven by productivity growth model should explain why productivity started to increase After investment in factors of production is important Introduce mass production Solow growth model becomes a relevant approach

24 What Made the Industrial Special Prior to many important inventions The wheel BC Writing BC Clocks Movable type But earlier inventions didn t build on each other In the Ind.Rev. - the change in power acts as a General Purpose Technology GPT complements many additional inventions

25 Post-Industrial Productivity and capital investment both key factors - at different times Capital investment Productivity improvements through the World Wars Capital investment again in the 1970 s and 1980 s

27 and the Models Productivity growth has not continued to rise with population Productivity is not only technology - might be missing something else Relationship between research effort and research output may be different

28 The Cost of Invention How do we model technology growth that becomes harder as new ideas are developed? Model the cost of each invention as µa φ Assume that 0 < φ < 1 This represents the fishing out effect

29 Decreasing Returns to Scale Difficult to assume that there are constant returns to scale in research More people in research tend to chase after the same ideas Assume that the production of technology increases with L λ A The overall technology production function is now: Â = Lλ AA φ µ How does this affect the model developed previously?

30 The Scaled Model The growth rate of technology is defined as: Can we find a steady-state? Â = Lλ AA φ µ In a steady-state economy - growth rate of technology is constant Â = g A Implies that L λ AA φ is constant

31 Calculating the Rate As usual - take logs and differentiate this equation with respect to time Rearranging this equation: µg A = L λ AA φ 0 = λ ˆ L A φâ Â = λ φ ˆ L A In the steady-state, the share of the population in research (γ A ) must be constant rate of researchers must equal the population growth rate (n) Rearranging this equation now results in: Â = λn φ

32 The New We have reduced the scale effect considerably Compare the new growth rate with the old growth rate: Â = λn φ Â = L A µ No longer implies that large countries should grow faster Population growth generates technological improvements - no diminishing return In the Solow model - capital is divided between people (rivalry matters)

33 So, Is Good or Bad? Population growth drives technology growth But population growth reduces the amount of capital/land available for each worker A key difference - ideas flow across countries far easier than capital does High fertility in a country may generate more technology - this benefits everyone in the world But the country with high fertility will itself be poor Particularly true in agricultural countries with a key fixed factor (land)

34 Where Does Population Occur? Where is population growth occurring? The assumption in the technology model is that increases in population increase the share of the population in research This is only true if population growth occurs in countries where people do research If technology development is limited by capital access - population growth in poor countries would have a limited effect Population growth is almost entirely in countries that have very low incomes And a significant part is in countries with an agricultural base

35 total fertility rate NER AGO BFA TCD MWI UGA MLI ZAR YEM BDI COG RWA SLE GNB ETH MRT GNQ ERI MDG BEN SAU TZA BTN NGA ZMBDJI GIN MOZ SEN TGO GMB LAO NAM CIV CMR CAFPAK SDN GTM KEN STP LSO HTI PNG SWZ NPL COM OMN GHA GAB KHM BWA HND BOL PRY ZWE JOR CPV SYR NIC PHL EGY DZA GRD TJK BGD SLV BLZ BHR INDECU MYS MAR PER ISR DOM FJI VEN ZAF MNG KGZ UZB IDN COL IRN MEX JAM CRI PAN ARG GUY TKM LBNTUR VNM ALB BRA CHL BHS LKA VCT URY KNA TUN SYC AZE LCAKAZ MUS NZL CHN THA PRI CYP MKD TTO MLT BRB ATG CUB PRT MDA HRV POL KOR ARM ROM BGR LTU BLR GRC ESTHUN SVK UKR LVA RUS CZE SVN ESP GEO KWT ISL FRA IRLNOR GBR FINLD AUSDNK SWE BEL CAN CHE SGP AUT JPN ITA MAC HKG USA LUX real gdp per capita 2000

36 total fertility rate AGO YEM COG GNQ MRT DJI GIN ZMB NGA SEN MOZ CIV PAK GTM KEN STP SWZ LSO PNG HTI GAB BWA HND PRYBOL ZWE JOR CPV SYR PHL GRD DZA EGY SLV TJK BLZ BGD ECUMYS IND ZAF VEN MAR PER DOM FJI MEX ARG CRI COL IDN IRN JAM PAN SUR LBN TUR TKM KNA URYBRA VNM SYC VCT CHL MUS TUN LKA LCAKAZ AZE FRA CHN LUX NOR THA DNK FIN ATG BRB MKD GBR TTO BEL CUB SGP PRT POL KOR HRV MDA SVK EST BGR BLR ITA LTU ROM SVN ESP ARM CZE LVA RUS UKR NER BFA TCD MWI UGA MLI RWA SLE MDG BEN BTN TZA TGO GMB CMR SDN NPL COM GHA KHM NIC MNGUZB KGZ GEO BDI ETH LAO CAF ALB GNB agriculture, value added/gdp 2000

37 total fertility rate NER AGO BFA MWI MLI UGA YEM ETH MRT BEN ERI MDG SAU TZA BTN DJI GIN NGA ZMB MOZSEN LAOGMB TGONAM CMR CIV CAF PAK SDN GTM KEN NPL COM OMN GHA GAB KHM HND PRY BOL BWA ZWE SYR JOR NIC PHL DZA EGY GRD BGD SLV BLZ BHR ARE MMR IND ECU MYS MAR PER VEN ZAF ISR FJI MNG COL IRN BRN KWT MEX QAT IDN PAN JAM ARG CRI GUYTUR LBN VNM ALB BRA CHL URY KNA LKA TUN VCT SYC MUS LCA CHN THA CYP BRB MLT YUG TTO CUB MDA POLHRV KOR ARM ROMBGR LTU GRCHUN SVK UKR RUS EST ITA CZE LVA ESP MAC NZL FRA IRL GBR BEL PRT AUT JPN SVN HKG ISL NLD FIN CAN DNK LUXAUS NOR SGP SWE CHE USA computers per 1,000 people 2000

38 Prospects for World from Technology Worldwide technology grows in response to growth in research Increasing growth rates: Increases in the labour force - population and participation Fraction of the labour force in research and development Countries advance to the cutting edge

39 Sectors of Technology Substitutes vs. complements Productivity improvements of a product with a necessary complement Pushes labour into the complementary product Productivity improvements of a product with many substitutes Creative destruction - labour is moved into the product that is developing

40 Goods vs. Services Productivity advances have been focused in goods production Consumption is shifting to services - goods and services are complements Can productivity in services develop at a similar rate?

41 Productivity is more than technology An attempt to differentiate productivity and technology What other factors affect productivity? Forms of inefficiency

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