ECON 581. The Solow Growth Model, Continued. Instructor: Dmytro Hryshko

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1 ECON 581. The Solow Growth Model, Continued Instructor: Dmytro Hryshko 1 / 38

2 The Solow model in continuous time Consider the following (difference) equation x(t + 1) x(t) = g(x(t)), where g( ) is some function. In terms of our law of motion, e.g., x(t) = k(t), and g(x(t)) sf(k(t)) δk(t). The equation specifies the change in x between two discrete time points. What happens inside of the interval [t, t + 1] is not known. Assume that we can partition time very finely. If t and t + 1 are close enough, the following approximation is reasonable x(t + t) x(t) tg(x(t)). 2 / 38

3 Dividing both sides by t and taking the limits gives x(t + t) x(t) lim t 0 t = ẋ(t) g(x(t)), where ẋ(t) = dx(t) the time derivative of x(t). dt In terms of our law of motion, k(t) = sf(k(t)) (n + δ)k(t) 3 / 38

4 The law of motion of capital in continuous time In continuous time, we refer to w(t) and R(t) as instantaneous rental rates (the flows received at instant t). Assume that the labor force grows at a constant rate n: L(t) = exp(nt)l(0) Define k(t) = K(t) L(t). log L(t) = nt + log(l(0)) L(t) L(t) = n log(k(t)) = log(k(t)) log(l(t)) k(t) k(t) = K(t) K(t) L(t) L(t) = sf (K(t), L(t), A) = K(t) = sf(k(t)) k(t) K(t) sf (K(t), L(t), A) δk(t) n = n K(t) K(t) (n + δ) = sf (K(t), L(t), A)/L(t) K(t)/L(t) (n + δ) k(t) = sf(k(t)) (n + δ)k(t). (n + δ) Again, the steady-state occurs when k(t) = 0, and sf(k ) = (n + δ)k. 4 / 38

5 Add-on to the steady-state predictions Since we introduced population growth into the model, k = k (s, A, δ, n), and y = y (s, A, δ, n). We can show that k (s, A, δ, n) n < 0, y (s, A, δ, n) n < 0. Economies with higher population growth rates have lower capital-per-worker ratios and lower incomes per worker. 5 / 38

6 Factor shares in total income Consider again the Cobb-Douglas production function Y = AK α L 1 α, 0 < α < 1. The share of capital costs in total income is defined as R(t)K(t) Y (t). In competitive factor markets R(t) = F K (K(t), L(t), A(t)) and w(t) = F L (K(t), L(t), A(t)). Thus, R(t)K(t) Y (t) = AαK(t)α 1 L(t) 1 α K(t) Y (t) =Y (t) {}}{ = α AK(t) α L(t) 1 α = α. Y (t) From Euler theorem, Y (t) = w(t)l(t) + R(t)K(t). Thus, w(t)l(t) Y (t) = 1 α. For the Cobb-Douglas function, the shares of capital and labor costs in total income are constant! 6 / 38

7 For this function, y(t) = Y (t) L(t) = AK(t)α L(t) 1 α L(t) = A ( ) K(t) α = Ak(t) α. L(t) Thus, k(t) = sak(t) α (n + δ)k(t), and k occurs when sa(k ) α = (n + δ)k. Hence, k = ( ) 1 sa 1 α. n + δ 7 / 38

8 Transitional dynamics Given k(0) > 0 need to solve for k(t) from k(t) = sak α (t) (n + δ)k(t). Let x(t) k(t) 1 α. Thus, ẋ(t) = (1 α)k(t) α k(t). ẋ(t) = (1 α)sa (1 α)(n + δ) x(t). }{{}}{{} b a Define y(t) = x(t) + b/a. Then ẏ(t) = ẋ(t) = b + ax(t) = b + a(y(t) b/a) = ay(t). The solution is y(t) = c exp(at). Thus, x(t) = b/a + c exp(at), x(0) = b/a + c, and x(t) = b/a + (x(0) + b/a) exp(at). So, [ [ k(t) = x(t) 1 α 1 sa = n + δ + k(0) 1 α sa ] ] 1 1 α exp( (1 α)(n + δ)t). n + δ 8 / 38

9 Solow model with technological progress Introduce sustained growth by allowing technological progress in the form of changes in A(t). How to model technological growth and its impact on Y (t)? Introduce technological progress so that the resulting allocations (Y (t), K(t), and C(t)) are consistent with balanced growth, as defined by the Kaldor facts constant K Y, stable r(t) and factor income shares, constant growth in Y/L. 9 / 38

10 Types of technological progress Define different types of neutral technological progress. Hicks-neutral: F (K(t), L(t), A(t)) = A(t)F (K(t), L(t)). Solow-neutral (capital-augmenting): F (K(t), L(t), A(t)) = F (A(t)K(t), L(t)). Harrod-neutral (labor-augmenting): F (K(t), L(t), A(t)) = F (K(t), A(t)L(t)). Balanced growth is possible in the long-run only if technological progress is Harrod-neutral or labor-augmenting. 10 / 38

11 Theorem 2.6 (Uzawa s Theorem I) Consider a growth model with aggregate production function Y (t) = F (K(t), L(t), Ã(t)), CRS in K and L (Ã can be a vector). The aggregate resource constraint K(t) = Y (t) C(t) δk(t); population grows at a constant rate, L(t) = exp(nt)l(0), and there exists T < such that for all t T, Ẏ (t)/y (t) = g Y > 0, K(t)/K(t) = g K > 0, and Ċ(t)/C(t) = g C > 0. Then 1 g Y = g K = g C. 2 For any t T, there exists a function F : R+ 2 R + homogeneous of degree 1 in two arguments, such that the aggregate production function can be represented as Y (t) = F (K(t), A(t)L(t)), where A(t) R + and A(t)/A(t) = g = g Y n. 11 / 38

12 Example. can be represented as Y (t) = (A K (t)k(t)) α (A L (t)l(t)) 1 α where A(t) = A K (t) α 1 α AL (t). Y (t) = K(t) α (A(t)L(t)) 1 α, 12 / 38

13 The Solow model with technological progress: continuous time Let Y (t) = F (K(t), A(t)L(t)) be constant returns to scale in K(t) and L(t), and Ȧ(t) A(t) = g > 0. Define the aggregates as ratios to effective labor, A(t)L(t). Thus, k(t) = is capital per effective labor, and ŷ(t) = Y (t) A(t)L(t) K(t) A(t)L(t) is output per effective labor. Note that k(t) k(t) = K(t) K(t) g n; ŷ(t) = y(t) = Y (t) L(t) Y (t) A(t)L(t) = F (K(t),A(t)L(t)) A(t)L(t) = F = ŷ(t)a(t) = A(t)f(k(t)). ( K(t) A(t)L(t), 1 ) = f(k(t)), and 13 / 38

14 k(t) sf (K(t), A(t)L(t)) δk(t) = (n + g) k(t) K(t) sf (K(t), A(t)L(t))/(A(t)L(t)) = (δ + n + g) K(t)/(A(t)L(t)) = sf(k(t)) (n + g + δ). k(t) An equilibrium is defined as before, now in terms of the constancy of k(t) = K(t) A(t)L(t). The growth rate of K(t) L(t), however, will be equal to g. 14 / 38

15 Equilibrium with technological progress Consider the Solow growth model with labor-augmenting technological progress at the rate g and population growth at the rate n. Then there exists a unique balanced growth path where the effective capital-labor ratio is constant and given by sf(k ) = (n + g + δ)k, where k = K AL. Output per worker, capital per worker and consumption per worker will grow at the rate g. 15 / 38

16 Proposition 2.12 Let A(0) be the initial level of technology. Denote the BGP level of effective capital-labor ratio by k (A(0), s, n, g, δ) and the level of output per capita by y (A(0), s, δ, g, n, t). Then k (A(0), s, δ, n, g) A(0) k (A(0), s, δ, n, g) n = 0, < 0, k (A(0), s, δ, n, g) s k (A(0), s, δ, n, g) δ > 0 < 0 y (A(0), s, δ, n, g) A(0) y (A(0), s, δ, n, g) n > 0, < 0, y (A(0), s, δ, n, g) s y (A(0), s, δ, n, g) δ > 0 < / 38

17 What difference does technological progress bring? The model now generates the growth in output per capita, and so can be mapped to the data better. The disadvantage is that this growth is driven exogenously. 17 / 38

18 The Solow Model and the Data. Growth Accounting Production function: Y (t) = F (K(t), L(t), A(t)); competitive factor markets. Differentiate wrt t: Thus, Ẏ = F K K + F L L + FA A = F K K K K + F LL L L + F A AA A. Ẏ Y }{{} g = F KK Y }{{} ε K K }{{} K g K + F L L L }{{ Y }}{{} L ε L g L + F A A A, } Y{{ A} x where x is the contribution of technological growth (total factor productivity) to economic growth. 18 / 38

19 In competitive markets, w = F L and R = F K, and so ε l = F LL Y = wl Y = α L, and ε k = α K, the income shares of labor and capital. Thus, the contribution of TFP to growth can be calculated as: x = g α K g K α L g L. An estimate of TFP growth at t is ˆx(t) = g(t) α K (t)g K (t) α L (t)g L (t). 19 / 38

20 Solow (1957): developed the growth accounting framework and applied to U.S. data for assessment of the sources of growth during the early 20th century. Conclusion: a large part of of the growth was due to technological progress. 20 / 38

21 Table 8-3 Accounting for Economic Growth in Canada Mankiw and Scarth: Macroeconomics, Canadian Fifth Edition Copyright 2014 by Worth Publishers 21 / 38

22 Possible biases in estimation of x: 1 Inflated ˆx if g L or g K underestimated (e.g., how to account for changes in human capital, changes in the prices of machines?). 2 Mismeasurement in g (e.g., due to changes in relative prices and the quality of products). 22 / 38

23 The Solow model and regression analysis Equilibrium is described by y(t) = A(t)f(k(t)) (3.6) k(t) k(t) = sf(k(t)) (n + g + δ). k(t) (3.7) Take natural logs from (3.6) and differentiate wrt t: ẏ(t) y(t) = A(t) A(t) + f (k(t)) f(k(t)) k(t) = g + f (k(t))k(t) k(t) f(k(t)) k(t). (3.8) }{{} =ε k (k(t)) 23 / 38

24 Note that (3.7) can be written as k(t) sf(exp log k(t)) = (n + g + δ). k(t) exp log k(t) Taking a first-order Taylor expansion around log k, we obtain k(t) k(t) sf(k ) k n δ g + }{{} =0 since SS sf (k )k k sf(k )k (k ) 2 (log k(t) log(k )) ( f (k )k ) sf(k ) = f(k 1 ) k (log k(t) log k ) = (ε k (k ) 1)(n + g + δ) (log k(t) log k ). 24 / 38

25 Using (3.8), ẏ(t) y(t) g ε k(k )(1 ε k (k ))(n + g + δ) (log k(t) log k ). We can write (3.6) as log y(t) = log A(t) + log f(exp log k(t)). Approximating around log k, and so log y(t) = log y + f (k )k f(k (log k(t) log k ), ) }{{} =ɛ k (k ) ẏ(t) y(t) g (1 ε k(k ))(n + g + δ) (log y(t) log y ). (3.10) }{{} convergence 25 / 38

26 Cobb-Douglas example: Y (t) = A(t)K(t) α L(t) 1 α ẏ(t) y(t) g (1 α)(n + g + δ) (log y(t) log y ) (3.11) ( = ) ( ) (log y(t) log y ) 3 = (log y(t) log y ) The implied rate of convergence is too high: the gap in incomes between similar countries should be halved in little more than 10 years. g i,t,t 1 = b 0 + b 1 log y i,t 1 + ɛ i,t (3.12) OECD countries: ˆb 1 is negative. Fig The whole world: no such evidence. Fig / 38

27 Introduction to Modern Economic Growth Average Growth Rate of GDP TUR JPN PRT GRC ESP IRL ITA FIN BEL FRA GBR ISL NOR DNK AUS SWE Log GDP per Worker in 1960 NLD CAN LUX USA CHE Source: Figure Acemoglu Annual (2009). growth rate of GDP per worker between 1960 and 2000 versus log GDP per worker in 1960 for core OECD countries. Back 27 / 38

28 sperous over the postwar period. Average Growth Rate of GDP CHN GNQ TWN KOR HKG THA MYS ROM JPN SGP IRL LKA LUX PRT GHA LSO ESP PAK AUT IND GRC IDN EGY CPV ITA MUS BEL MAR TUR ISR FIN FRA NOR PAN SYR DOM ISL GBR MWI NPL USA BRA DNK ETH GAB CHL AUS NLD PRY GNB CIV IRN SWE TZA CAN BFA BEN PHL TTO COL BRB CHE CMRZWE ECU MEX URY GMB MOZ MLI COG GTM ZAF UGA CRI ARG DZA NZL HND BOL SLV BDI PER KEN NGA RWA TGO ZMB JAM COM SEN GIN VEN TCD JOR NER MDG NIC Log GDP per Worker in 1960 Figure Source: Acemoglu Annual(2009). growth rate of GDP per worker between 1960 and 2000 versus log GDP per worker in 1960 for the entire world. 28 / 38

29 More realistically, y differs across countries and depends on their characteristics (institutions, investment behavior). This motivates to run g i,t,t 1 = b i 0 + b 1 log y i,t 1 + ɛ i,t (3.13) Likely, cov(b i 0, log y i,t 1) > 0 so that ˆβ 1 is biased upward. Conditional convergence: g i,t,t 1 = X i,tβ + b 1 log y i,t 1 + ɛ i,t, (3.14) where X i,t is a column vector (constant, schooling rates, fertility rate, investment rate, openness, rule of law and democracy, etc.) These regressions tend to deliver a negative estimate of b 1. Do they deliver causal effects of X s on growth as well? 29 / 38

30 Problems with (3.14): g i,t,t 1 = X i,t β + b1 log y i,t 1 + ɛ i,t 1 X s and y i,t 1 are endogenous. ˆβ s will not measure causal effects and ˆb 1 is inconsistent. 2 If X s include openness and schooling/investment rates, β on the openness will not measure its full effect on growth since it will also likely affect schooling/investment rates. 3 Regressions assume closed Solow economies, or noninteracting islands. 30 / 38

31 The Solow model with human capital Y (t) = K(t) α H(t) β (A(t)L(t)) 1 α β, (3.21) 0 < α, β < 1, α + β < 1. Output per effective unit of labor is ŷ t = k(t) α h(t) β. Laws of motion of physical and human capital per effective labor are: k(t) = s k f (k(t), h(t)) (δ k + g + n)k(t) ḣ(t) = s h f (k(t), h(t)) (δ h + g + n)h(t). 31 / 38

32 The implied steady-state value of ŷ is ( ) α ŷ s ( k 1 α β = n + g + δ k s h n + g + δ h s k (s h ) is more important if α (β) is larger. ) β 1 α β. (3.23) 32 / 38

33 A wold of augmented Solow economies. Mankiw, Romer and Weil (1992) Y j (t) = K j (t) α H j (t) β (A j (t)l j (t)) 1 α β, j = 1,..., J. Output per worker is then ( yj = A j (t) s k,j n j + g j + δ k ) α 1 α β ( s h,j n j + g j + δ h ) β 1 α β. (3.24) Assume A j (t) = Āj exp(gt). Then, the BGP income for country j in logs is ( ) log yj α (t) = log Āj + gt + 1 α β log s k,j n j + g + δ k ( ) β + 1 α β log s h,j. (3.25) n j + g + δ h 33 / 38

34 MRW estimated: log y j (t) = const + α 1 α log(s k,j) α 1 α log (n j + g + δ k ) + ɛ j Table 3.1, column (1): ˆα 2/3 too large. Table 3.1 log y j (t) = const + + α 1 α β log(s k,j) β 1 α β log(s h,j) α 1 α β log (n j + g + δ k ) β 1 α β log (n j + g + δ h ) + ɛ j. Table 3.2, column (1): ˆα 1/3 reasonable. Table 3.2 Adj. R human and physical investments explain more than 3/4 of the cross-country differences in per capita incomes. 34 / 38

35 Table 3.1 Estimates of the Basic Solow Model MRW Updated data log(s k ) (.14) (.11) (.13) log(n + g + δ) (.56) (.55) (.36) Adj R Implied α No. of observations Source: Acemoglu (2009). Back 35 / 38

36 imilar, though the Adjusted R 2 is somewhat lower. Table 3.2 Estimates of the Augmented Solow Model MRW Updated data log(s k ) (.13) (.11) (.13) log(n + g + δ) (.41) (.45) (.33) log(s h ) (.07) (.07) (.13) Adj R Implied α Implied β No. of observations To the Source: extent Acemoglu that these (2009). Back regression results are reliable, they give a big boost to th 36 / 38

37 Calibrating productivity differences Start with the production function: Y j = K α j (A j H j ) 1 α, estimate K j, and H j. For any year, construct predicted series Ŷ j based on the estimated A US and compare with observed Y j. Ŷ j = K 1/3 j (A US H j ) 2/3, 37 / 38

38 Introduction to Modern Economic Growth 45 Predicted Log GDP per Worker UGA MWI NZL CHE NOR DNK AUS ISR SWE JPNFIN AUT CAN NLD USA BEL GBR GRC ISL FRA GUY KOR BRBIRL ARG ESP ITA PER HKG JAM ECU FJI PAN VEN ZMB SGP THA ZWE PHL CHL ZAF URY COG MUS TTO MYS CRI BRA PRT MEX BOL IRN CHN LKA GHA TUR DOM COL KEN NICTUN PRY IND SYR HND SLV PAK BWA BGD JOR SEN LSO IDN PNG GTM NER NPL BEN TGO MLI CAF MOZGMB RWA SLE CMR EGY HUN CYP 45 Predicted Log GDP per Worker NOR JPNCHE KOR NZL SWE GER CAN FIN AUS DNK USA AUT HUN GRC ISR ISL FRA NLD HKG BEL GBR PAN ESP ARG ITA PERTHA MYS ECU CHL IRL JAMPHL MEX LSO VEN JOR BRB PRT CRI ZAF URY ZMB CHN DZA BRA TTO ZWE TUR NIC LKA IDN PRY IRN HND BOL COL COG SYR DOM TUN MUS IND KEN SLV TGO GHANPL CMR PAK EGY MWI BGD GTM MLI NER RWA UGA MOZ BEN GMB SEN Log GDP per Worker Log GDP per Worker 2000 Figure 3.2. Predicted and actual log GDP per worker across countries, 1980 and (2) However, differently from the regression analysis, this exercise shows that there are significant technology (productivity) differences. There are often large gaps between predicted and actual incomes, showing the importance of technology differences across countries. This can be most easily seen in Figure 3.2, where practically all observations are above the 45, which implies that the neoclassical model is overpredicting the income level of countries that are poorer than the United States. (3) The same pattern is visible in Figure 3.3, which plots the estimates of the technology differences, Aj/AUS, against log GDP per capita. These differences are often substantial. More importantly, these differences are also strongly correlated with income per capita; richer countries appear to have better technologies. (4) Also interesting is the pattern that the empirical fit of the Solow growth model seems to deteriorate over time. In Figure 3.2, the observations are further above the Source: Acemoglu (2009). Differences in physical and human capital still matter a lot. There are significant productivity differences: the Solow model overpredicts the incomes in poor countries. The empirical fit of the Solow model deteriorates with time. 38 / 38

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