Economic Growth: Malthus and Solow

Economic Growth: Malthus and Solow Economics 3307 - Intermediate Macroeconomics Aaron Hedlund Baylor University Fall 2013 Econ 3307 (Baylor University) Malthus and Solow Fall 2013 1 / 35

Introduction Two questions: 1 What explains differences in growth within a country over time? 2 What explains differences in growth between countries? Econ 3307 (Baylor University) Malthus and Solow Fall 2013 2 / 35

Introduction Differences across time: Japanese boy born in 1880 had a life expectancy of 35 years, today 81 years. An American worked 61 hours per week in 1870, today 34. Differences across countries: Average American income is 5 times larger than Mexican s, 14 times larger than Indian s, and 35 times larger than African s (using PPP). Out of 6.4 billion people, 0.8 do not have access to enough food, 1 to safe drinking water, and 2.4 to sanitation. Life expectancy in rich countries is 77 years, 67 years in middle income countries, and 53 years in poor countries. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 3 / 35

Introduction The evolution of GDP per capita across countries, 1820-2000: Econ 3307 (Baylor University) Malthus and Solow Fall 2013 4 / 35

Introduction The evolution of GDP per capita across countries, 1000-2000: Econ 3307 (Baylor University) Malthus and Solow Fall 2013 5 / 35

Introduction North Korea vs. South Korea: Econ 3307 (Baylor University) Malthus and Solow Fall 2013 6 / 35

Growth Facts Kaldor s stylized facts of economic growth: 1 Real GDP per worker y = Y N and capital per worker k = K N time at relatively constant and positive rates. grow over 2 They grow at similar rates, so the capital-output ratio K Y is approximately constant over time. 3 The real return to capital and the real interest rate are relatively constant over time. 4 The capital and labor shares are roughly constant over time. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 7 / 35

Growth Facts More growth facts: 1 Pre-1800: constant per capita income across time and countries. 2 Post-1800: sustained growth in rich countries (2% in U.S. since 1900). 3 Across countries, high investment high standard of living and high population growth low standard of living. 4 Divergence of per capita incomes from 1800-1950. 5 From 1960-2000, no relationship between output levels and output growth across countries. 6 Richer countries more alike in growth rates than are poor countries. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 8 / 35

Can Money Buy Happiness? Econ 3307 (Baylor University) Malthus and Solow Fall 2013 9 / 35

Economic Growth and Welfare The consequences for human welfare involved in questions [of economic growth] are simply staggering: Once one starts to think about them, it is hard to think about anything else. - Robert Lucas Econ 3307 (Baylor University) Malthus and Solow Fall 2013 10 / 35

Economic Growth and Welfare Growth is far more important to welfare than business cycles because of compounding. Suppose constant growth rate of g: 1 + g = How long does it take for GDP to double? 2y = (1 + g) t y t = yt y t 1 y t = (1 + g) t y 0. ln 2 70/(g 100%) ln(1 + g) Rule of 70. Example: t = 35 years with 2% growth, t = 23 years with 3% growth, and t = 70 years with 1% growth. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 11 / 35

Malthusian Growth Robert Malthus, An Essay on the Principle of Population, 1798. Idea: technological advances for producing food higher population growth lower average consumption until only subsistence. No long run increase in standard of living without population growth limits. A good description of economic history prior to the Industrial Revolution. The English Economy 1275-1800 Econ 3307 (Baylor University) Malthus and Solow Fall 2013 12 / 35

Malthusian Growth: Ingredients of the Model Output Y t = zf (L t, ) where L t is land and is labor. Interpret Yt as (perishable) food. No savings or investment. Fixed supply of land: L t = L for all t. No government spending: Gt = 0. Inelastic labor supply: Nt = total labor = population. +1 = + Births t Deaths t = (1 + birth rate t death rate t ) Assume birth rate is an increasing function of Ct decreasing function of Ct. Thus, +1 = g ( Ct ) with g > 0 and death rate is a Econ 3307 (Baylor University) Malthus and Solow Fall 2013 13 / 35

Equilibrium and Steady State of the Malthus Model Goods market clearing: C t = Y t = zf (L, ) +1 ( ) L, 1 Nt CRTS F zf (L,Nt) = zf +1 = g ( zf Unique steady state where N = g ( zf ( L N, 1 )) N. = g ( zf (L,Nt) ( L Nt, 1 )). ). When < N, population increases: +1 >. When > N, population decreases: +1 <. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 14 / 35

Determinants of Living Standards in the Malthus Model Let y t Yt, l t L, and c t Ct. Then y t = zf (l t ) is the per-worker production function. Equilibrium: c t = zf (l t ) and +1 = g(c t ). In steady state, +1 = 1 g(c ) = 1 and c = zf (l ). Consumption per-worker determined solely by g, implying that technological change has no impact on long run living standards. Malthus solution: state-mandated population control. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 15 / 35

The Effects of Technological Progress The effects of an increase in z in the short run and long run: Econ 3307 (Baylor University) Malthus and Solow Fall 2013 16 / 35

The Effects of Population Control The effects of introducing population control: Econ 3307 (Baylor University) Malthus and Solow Fall 2013 17 / 35

Long Run Growth and the Solow Model Malthus accurate prior to 1800 because of agricultural economy. Main reasons for stagnation in the Malthus model: no accumulation of production inputs other than labor. 1 No accumulation of other production inputs always cursed by diminishing returns in the long run. 2 Assumes population growth increases in consumption per worker. In reality, death rates decrease but so do birth rates. Introducing capital raises prospects for long run growth because more capital more output more capital. Capital begets capital. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 18 / 35

Ingredients of the Solow Model The Representative Firm: zf (Kt,Nt) CRTS production Yt = zf (K t, ) Yt = Firm profits are π t = zf (K t, ) w t r tk t. = zf ( ) K t, 1. Let y t = Yt, k t = Kt, and f (k t ) = F ( ) K t, 1. Then y t = zf (k t ). Households: Exogenous population growth +1 = (1 + n) with inelastic household labor supply nt s = 1 Nt s =. Households own the capital and rent it to firms. Exogenous savings rate s c t = (1 s)(w t + r t kt s + πt ) and i t = s(w t + r t kt s + πt ), where w t + r t kt s + πt is household income. Capital accumulation: K s t+1 = (1 d) k s t + i t = (1 d)k s t + I t. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 19 / 35

Competitive Equilibrium of the Solow Model A competitive equilibrium is prices {w t, r t } t=0 and allocations {Nt d, Kt d } t=0 and (ks 0, {c t, i t } t=0 ) s.t. 1 Consumers satisfy their budget constraint: c t + i t = w t + r t kt s + πt. 2 N d t and K d t maximize firm profits π t = zf (K t, ) w t r t K t. 3 The labor market clears: N d t =. 4 The capital market clears: K d t = K s t = k s t. 5 The goods market clears: c }{{} t + i t = zf (K }{{} t d, Nt d ). C t I t Taking prices as given, households choose how much to consume c t and how much to invest i t in new capital. Taking prices as given, firms choose N d t and K d t to maximize profits. The prices adjust to clear each of the markets. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 20 / 35

Competitive Equilibrium, Cont d There is no consumer optimization because we did not specify preferences. However, the budget constraint must be satisfied. Walras law: (1) + (2) + (3) + (4) (5). From (1), c t + i t = w t + r t k s t + π t c t + i t = C t + I t = w t + r t K s t + π t ( w t + r t k s t + π t From (2), π t = zf (Kt d, Nt d ) w t Nt d r t Kt d ( ) C t + I t = w t + r t Kt s + zf (Kt d, Nt d ) w t Nt d r t Kt d From (3) and (4), = N d t and K s t = K d t K t C t + I t = w t + r t K t + zf (K t, ) w t r t K t C t + I t = zf (K t, ) ) Econ 3307 (Baylor University) Malthus and Solow Fall 2013 21 / 35

Solving the Model Investment i t = s(w t + r t k s t + πt ) i t = s(w t +r t K t +π t ) = szf (K t, ) K t+1 (1 d)k t }{{} =i t=i t = szf (K t, ). The dynamics of K t and are therefore K t+1 = (1 d)k t + szf (K t, ) +1 = (1 + n) In per-worker terms, k t+1 = 1 d 1 + n k t + szf (k t) 1 + n Steady state szf (k ) = (n + d)k. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 22 / 35

Analyzing the Steady State An increase in s causes an increase in k and y but not always c. The golden rule savings rate s gr maximizes steady state consumption c = (1 s gr )zf (k gr ) = zf (k gr ) (n + d)k gr. Optimality condition: ( zf (k gr ) (n + d) ) dk gr ds = 0 MP K = n + d. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 23 / 35

Analyzing the Steady State An increase in n causes a decrease in k, y, and c. An increase in z causes an increase in k, y, and c. No limit to long run economic growth as long as z keeps rising. Fluctuations in z can also be used to study business cycles. 2008-2009 recession: credit disruptions reflected in TFP decrease. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 24 / 35

Simulated Effects of a Temporary TFP Drop Calibrate a version of the Solow model and simulate a 5% drop in z. zf (K, N) = zk 0.36 N 0.64 zf (k) = zk 0.36, s = 0.14, d = 0.1, n = 0.01. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 25 / 35

The Solow Model with Continual Technological Progress Production Y t = F (K t, A t ) with labor augmenting technological progress. Growth rates 1 + g = A t+1 A t and 1 + n = +1. Let x t = Xt A t = xt A t for X t = C t, K t, Y t. {}}{ Rewrite K t+1 = (1 d)k t + sf (K t, A t ) as k t+1 (1 + g)(1 + n) = (1 d) k t + f ( k t ) Dividing by (1 + g)(1 + n) gives k t+1 = I t 1 d (1 + g)(1 + n) k s t + (1 + g)(1 + n) f ( k t ) Steady state k implies long run balanced growth path. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 26 / 35

The Solow Model with Continual Technological Progress Compute growth rate of per capita X t as g x = X t+1/+1 X t/ = x t+1 x t. Steady state growth rates g y = g k = g c = g. To see this, note that k t = Kt k t+1 k t = At+1 A t k t+1 k t (1 + g) k k A t = kt A t k t = A t k t. Therefore = 1 + g in the steady state. Wage growth w t+1 w t = F N(K t+1,a t+1 +1 )A t+1 F N (K t,a t)a t = F N(K t+1 /(A t+1 +1 ),1)A t+1 F N (K t/(a t),1)a t = F N( k t+1,1)a t+1 F N ( k (1 + g) F N( k,1) = 1 + g. t,1)a t F N ( k,1) Rental rate growth r t+1 r t = F K (K t+1,a t+1 +1 ) F N (K t,a t) F K ( k,1) F K ( k,1) = 1. Mathematical note: if F (ak, an) = af (K, N) for all a, then F K (ak, an) = F K (K, N) and F N (ak, an) = F N (K, N) for all a. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 27 / 35

Evaluating the Solow Model Two difficulties: 1 Difficulty evaluating cross-country measurements. 2 Limited time span of data. Are economies in steady state? Two predictions of the Solow model: 1 Higher savings rates s = It Y t lead to higher living standards. 2 Higher population growth n = Nt+1 leads to lower living standards. The data show a positive correlation between It Y t negative correlation between +1 and Yt. The Solow model matches the Kaldor facts well. and Yt and a Econ 3307 (Baylor University) Malthus and Solow Fall 2013 28 / 35

Evaluating the Solow Model Limitations of the Solow model: 1 Savings and population growth rates are not exogenous. 2 No steady state growth unless z is continually increasing. 3 Technological progress not exogenous. 4 The model cannot account for the magnitude of development differences between countries. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 29 / 35

From Malthus to Solow Pre-Industrial Revolution: land-intensive production with fixed supply of land and decreasing return to labor. Post-Industrial Revolution: constant returns to scale production with labor and capital inputs. What accounts for the transition from stagnation to steady growth? Econ 3307 (Baylor University) Malthus and Solow Fall 2013 30 / 35

From Malthus to Solow Malthus to Solow (Hansen and Prescott, 2002): Two technologies: Malthus technology and Solow technology. Y Mt = A Mt K φ Mt Nµ Mt L1 φ µ Mt and Y St = A St K θ St N1 θ St where L Mt is in fixed supply and θ > φ. Along the equilibrium growth path, only Malthus technology is used in early stages of development because the Solow technology is unprofitable. As TFP grows, firms adopt the Solow technology. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 31 / 35

From Malthus to Solow Econ 3307 (Baylor University) Malthus and Solow Fall 2013 32 / 35

Growth Accounting Growth accounting decomposes economic growth into growth of factor inputs and TFP, where Y = zf (K, N). Cobb-Douglas a good fit for U.S. data: Y = zk 0.36 N 0.64 Generate Solow residuals as follows: ẑ t = Ŷ t ˆK 0.36 t ˆN 0.64 t Econ 3307 (Baylor University) Malthus and Solow Fall 2013 33 / 35

Solow Residuals and the Productivity Slowdown/Recovery Average Annual Growth Rates in the Solow Residual Years Average Annual Growth Rate 1950 1960 1.42 1960 1970 1.61 1970 1980 0.50 1980 1990 1.05 1990 2000 1.36 2000 2007 0.76 Three common reasons for the slowdown: 1 Measurement problems due to change in quality of goods/services during shift from manufacturing to services. 2 Increases in relative price of energy. Old capital not energy efficient, became obsolete. 3 Disruption arising from costs of adopting new technology. Beginning of IT revolution. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 34 / 35

A Growth Accounting Exercise Average Annual Growth Rates Years Ŷ ˆK ˆN ẑ 1950 1960 3.48 3.68 1.11 1.42 1960 1970 4.19 3.86 1.80 1.61 1970 1980 3.19 3.24 2.36 0.50 1980 1990 3.26 2.85 1.81 1.05 1990 2000 3.28 2.72 1.43 1.36 2000 2007 2.32 2.64 0.93 0.76 ( ) 1 Growth rate for X between years m and n is gmn X = Xn n m X m 1. Cobb-Douglas Y m = z m KmN α m 1 α, Y n = z n Kn α Nn 1 α ( ) α ( ) 1 α = zn Kn Nn z m K m N m 1 + gmn y = (1 + gnm)(1 z + gmn) K α (1 + gmn) N 1 α Yn Y m Approximation g y mn g z mn + αg K mn + (1 α)g N mn. Econ 3307 (Baylor University) Malthus and Solow Fall 2013 35 / 35