Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Advanced Macroeconomics Part 3: Economic Growth Univ.‐Prof. Dr. Matthias Neuenkirch WS 2026/2027 Matthias Neuenkirch Advanced Macroeconomics Universität Trier 1/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Outline 1. Economic Growth Facts 2. Malthusian Model of Economic Growth 3. Exogenous Growth: Solow Model 4. Endogenous Growth: Human Capital Accumulation Matthias Neuenkirch Advanced Macroeconomics Universität Trier 2/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Economic Growth Facts ﴾1﴿ Before the Industrial Revolution, standards of living differed little over time and across countries. ﴾2﴿ Since the Industrial Revolution, per capita income growth has be‐ en sustained in the richest countries. ﴾3﴿ There is a positive correlation between the rate of investment and output per worker across countries. ﴾4﴿ There is a negative correlation between the population growth rate and output per worker across countries. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 3/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Natural Logarithm of Per Capita Real GDP Remember: Slope of the natural log of a time series is approxim‐ ately equal to the growth rate of a time series Matthias Neuenkirch Advanced Macroeconomics Universität Trier 4/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Real Income Per Capita vs. Investment Rate Matthias Neuenkirch Advanced Macroeconomics Universität Trier 5/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Real Income Per Capita vs. Population Growth Rate Matthias Neuenkirch Advanced Macroeconomics Universität Trier 6/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Economic Growth Facts ﴾5﴿ Differences in per capita incomes increased dramatically among countries of the world between 1800 and 1950. ﴾6﴿ There is no distinct correlation across countries between the level of output per capita in 1990 and the average rate of growth in output per capita for the years 1990−2022, i.e., there is no strong tendency for rich countries to grow faster than poor countries, and vice versa. ﴾7﴿ Richer countries are much more alike in terms of rates of growth of real per capita income than are poor countries. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 7/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Growth Rate vs Level of Real Income Per Capita Matthias Neuenkirch Advanced Macroeconomics Universität Trier 8/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Outline 1. Economic Growth Facts 2. Malthusian Model of Economic Growth Model Setup Comparative Statics 3. Exogenous Growth: Solow Model 4. Endogenous Growth: Human Capital Accumulation Matthias Neuenkirch Advanced Macroeconomics Universität Trier 9/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Malthusian Model of Economic Growth • Thomas Malthus ﴾1798﴿: “An Essay on the Principle of Populati‐ on.” • Any advances in the technology for producing food inevitably lead to further population growth. • Higher population ultimately reduces the average person to the subsistence level of consumption they had before the advance in technology. • Population and level of aggregate consumption could grow over time, but in the long‐run there is no increase in the standard of living. • His ideas can be easily translated into a formal structure. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 10/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Model Setup • Closed economy two‐period model ﴾current period and future period﴿ without government spending. • Output, i.e., food, is produced from a fixed quantity of land L and labor input ﴾= total population﴿ N. • Constant returns to scale production function: Y = zF(L, N) ﴾1﴿ • Y cannot be stored from one period to the next. • There are no investments and no savings. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 11/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Model Setup • N′ denotes population in the next period. • Population growth rate is an increasing function in consumption per capita: ( ) N′ C =g ﴾2﴿ N N • In equilibrium, consumption equals output: C = zF(L, N) ﴾3﴿ ] [ N′ zF(L, N) =g N N ﴾4﴿ • Substitute ﴾3﴿ into ﴾2﴿: Matthias Neuenkirch Advanced Macroeconomics Universität Trier 12/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Population Growth and Consumption per Worker Matthias Neuenkirch Advanced Macroeconomics Universität Trier 13/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Model Setup • Constant returns to scale property of the production function im‐ plies: xzF(L, N) = zF(xL, xN) for any x > 0. • If x = N1 then: zF(L,N) = zF( NL , 1). N • Rewriting ﴾4﴿ yields future population as a function of the current population: [ ( )] L ′ N = g zF ,1 N ﴾5﴿ N Matthias Neuenkirch Advanced Macroeconomics Universität Trier 14/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Determination of Steady State Population Matthias Neuenkirch Advanced Macroeconomics Universität Trier 15/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Convergence to Steady State • Suppose that N < N∗ : ▶ Relatively large quantity of consumption per worker possible. ▶ Implies that the population increases to N∗ . • Suppose that N > N∗ : ▶ Relatively small quantity of consumption per worker possible. ▶ Implies that the population decreases to N∗ . • Aggregate consumption ﴾output﴿ in equilibrium: C∗ = Y∗ = zF(L, N∗ ) Matthias Neuenkirch Advanced Macroeconomics ﴾6﴿ Universität Trier 16/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Per‐Worker Form • Dividing ﴾1﴿ by N and rearranging yields: NY = zF (L ) N, 1 . • Lower‐case letters denote per‐worker quantities: y ≡ NY , l ≡ NL , C and c ≡ N . • Per‐worker production function: y = zf(l) ﴾7﴿ • Equilibrium condition in per‐worker form: c = zf(l) Matthias Neuenkirch Advanced Macroeconomics ﴾8﴿ Universität Trier 17/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Per‐Worker Production Function Matthias Neuenkirch Advanced Macroeconomics Universität Trier 18/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Steady State Per‐Worker Form • Population growth as function of consumption per worker c: N′ = g(c) N ﴾9﴿ ′ • Steady state: N′ = N = N∗ . It follows NN = 1. • Long‐run standard of living is determined entirely by the function g ﴾and not by z﴿. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 19/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Determination of the Steady State Matthias Neuenkirch Advanced Macroeconomics Universität Trier 20/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Effects of an Increase in z: Comparative Statics • If z increases, this shifts up the per‐worker production function. • In the long‐run, the population increases to the point where per capita consumption c returns to its initial level. • There is no long‐run change in living standards: g(c) is unchan‐ ged. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 21/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Effects of an Increase in z: Comparative Statics ∗ Note: l∗ 1 and l2 are interchanged. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 22/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Effects of an Increase in z: Adjustment to the Steady State Matthias Neuenkirch Advanced Macroeconomics Universität Trier 23/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Population Control in the Malthusian Model • Population control alters the relationship between population growth and per‐capita consumption g(c). • In the long‐run, per capita consumption c increases, and living standards rise. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 24/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Population Control in the Malthusian Model Matthias Neuenkirch Advanced Macroeconomics Universität Trier 25/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics How Useful is the Malthusian Model? • Model provides a good explanation for pre‐1800 growth facts in the world but not afterwards. • Malthus did not understand the role of capital accumulation in growth. • In contrast to land, the capital stock can be increased without any limit. • Malthus did not predict the effects of technological advances on fertility. • The opportunity cost of raising a large family becomes relatively larger in the face of higher market wages. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 26/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Outline 1. Economic Growth Facts 2. Malthusian Model of Economic Growth 3. Exogenous Growth: Solow Model Model Setup Comparative Statics Growth Accounting Convergence 4. Endogenous Growth: Human Capital Accumulation Matthias Neuenkirch Advanced Macroeconomics Universität Trier 27/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Solow Model • Key model and basis for the modern theory of economic growth. • Technological progress is necessary for sustained increases in stan‐ dards of living. • Model does a good job of explaining economic growth facts. • Exogenous growth model: Growth is caused in the model by forces that are not explained by the model itself. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 28/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Consumers • Population is assumed to grow exogenously at a constant rate n > −1: N′ = (1 + n)N ﴾10﴿ • Consumers supply their one unit of time as labor in each period. • Population is identical to the labor force N: no unemployment. • Consumers are assumed to save a constant fraction s of their in‐ come Y ﴾= real output﴿ and to consume the rest C. C = (1 − s)Y Matthias Neuenkirch Advanced Macroeconomics ﴾11﴿ Universität Trier 29/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Representative Firm • Production function: Y = zF(K, N) • Constant returns to scale: Y = zF N ( ) K , 1 ⇔ y = zf(k) N ﴾12﴿ ﴾13﴿ • The slope of the per‐worker production function y = zf(k) is the marginal product of capital MPK . • Evolution of the capital stock: Future capital equals the capital remaining after depreciation at a constant rate d, plus current investment I: K′ = (1 − d)K + I ﴾14﴿ Matthias Neuenkirch Advanced Macroeconomics Universität Trier 30/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Per‐Worker Production Function Matthias Neuenkirch Advanced Macroeconomics Universität Trier 31/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Competitive Equilibrium • Capital market is in equilibrium if S = sY = I. • It follows Y = C + I, i.e., the income‐expenditure identity is the equilibrium condition. • Substituting I = sY = szF(K, N) in ﴾14﴿ yields: K′ = (1 − d)K + szF(K, N) K K′ N′ F(K, N) + (1 − d) = sz ′ NN N N Matthias Neuenkirch Advanced Macroeconomics ﴾15﴿ Universität Trier 32/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Competitive Equilibrium • Rewriting ﴾15﴿ in per‐worker form and rearranging using N′ /N = 1 + n yields the future capital stock per worker k′ as function of the current capital stock per worker k: k′ = Matthias Neuenkirch szf(k) (1 − d)k + 1+n 1+n Advanced Macroeconomics ﴾16﴿ Universität Trier 33/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Determination of Steady State Capital Stock Matthias Neuenkirch Advanced Macroeconomics Universität Trier 34/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Competitive Equilibrium • Substituting the steady state condition k′ = k = k∗ in ﴾16﴿ yields: szf(k∗ ) (1 − d)k∗ + 1+n 1+n szf(k∗ ) = (n + d)k∗ k∗ = Matthias Neuenkirch Advanced Macroeconomics ﴾17﴿ Universität Trier 35/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Determination of Steady State Capital Stock Matthias Neuenkirch Advanced Macroeconomics Universität Trier 36/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Predictions of Solow Model • Model predicts that the quantity of capital per worker converges to a constant k∗ . • It also predicts that the quantity of output per worker converges to a constant, which is y∗ = zf(k∗ ). • If the savings rate s, the labor force growth rate n, and total factor productivity z are constant, then real income and real consump‐ tion per worker cannot grow in the long‐run. ▶ Reason: Marginal product of capital is diminishing. ▶ Output per worker can grow only as long as capital per worker continues to grow. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 37/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Predictions of Solow Model • In the steady state all real aggregate quantities grow at the growth rate in the labor force n. ▶ Aggregate quantity of capital: K = k∗ N. ▶ Because k∗ is a constant and N grows at the rate n, K must also grow at the rate n. ▶ Aggregate real output: Y = y∗ N = zf(k∗ )N. ▶ Aggregate real investment: I = sY = szf(k∗ )N. ▶ Aggregate real consumption: C = (1 − s)zf(k∗ )N. • Exogenous growth model: Growth in key macroeconomic aggre‐ gates is determined by exogenous labor force growth when s, n, and z are constant. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 38/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Comparative Statics • Totally differentiating ﴾17﴿ yields ﴾keeping d fixed﴿: [szf′ (k∗ ) − n − d]dk∗ + zf(k∗ )ds + sf(k∗ )dz − k∗ dn = 0 ﴾18﴿ • Solving for the appropriate derivatives yields: dk∗ ds dk∗ dn dk∗ dz = = = −zf(k∗ ) >0 szf′ (k∗ ) − n − d k∗ <0 szf′ (k∗ ) − n − d −sf(k∗ ) >0 szf′ (k∗ ) − n − d ﴾19﴿ ﴾20﴿ ﴾21﴿ • szf′ (k∗ ) − n − d < 0 since the slope of szf(k∗ ) is smaller than the slope of (n + d)k∗ in the steady state. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 39/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Effects of an Increase in s on k: Comparative Statics Matthias Neuenkirch Advanced Macroeconomics Universität Trier 40/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Effects of an Increase in s: Comparative Statics • The levels of capital per worker and output per worker are higher in the new steady state. • The increase in the savings rate, however, has no effect on the growth rates of aggregate variables. • In the steady state, all real quantities grow at the rate of growth in the labor force n. • In the transition to the new growth path, the rate of growth in Y is higher than n due to the higher rate of capital accumulation. • However, the marginal product of capital diminishes and growth slows down. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 41/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Effects of an Increase in s on y: Adjustment to the New Steady State Matthias Neuenkirch Advanced Macroeconomics Universität Trier 42/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Effects of an Increase in s on c: Comparative Statics • Consumption per worker in the steady state is: c∗ = (1 − s)zf(k∗ ) = zf(k∗ ) − (n + d)k∗ ﴾22﴿ • Response of consumption per worker to changes in the saving rate: ] dk∗ dc∗ dc∗ dk∗ [ ′ ∗ = ∗ = zf (k ) − n − d ﴾23﴿ ds dk ds ds • The sign of zf′ (k∗ ) − n − d is ambiguous, so that the consumption per worker could either increase or decrease with an increase in the savings rate. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 43/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Effects of an Increase in s on c: Comparative Statics Matthias Neuenkirch Advanced Macroeconomics Universität Trier 44/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Golden Rule Quantity of Capital per Worker • The golden rule savings rate is the savings rate sgr that maximi‐ zes consumption per worker in the steady state. • The golden rule steady state quantity of capital per worker solves the problem: max [zf(k∗ ) − (n + d)k∗ ] k∗ Matthias Neuenkirch Advanced Macroeconomics ﴾24﴿ Universität Trier 45/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Golden Rule Quantity of Capital per Worker • First‐order condition: zf′ (k∗gr ) = n + d ﴾25﴿ • The marginal product of capital MPK = zf′ (k∗gr ) equals the popu‐ lation growth rate n plus the depreciation rate d in the golden rule steady state. • The golden rule savings rate sgr is determined by ﴾17﴿: (n + d)k∗gr zf(k∗gr ) ﴾26﴿ Advanced Macroeconomics Universität Trier sgr = Matthias Neuenkirch 46/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Golden Rule Quantity of Capital per Worker Matthias Neuenkirch Advanced Macroeconomics Universität Trier 47/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Effects of an Increase in n on k: Comparative Statics • Recall ﴾20﴿: dk∗ k∗ = < 0. dn szf′ (k∗ ) − n − d • Capital and, as a consequence, output per worker decreases. • Aggregate output, aggregate consumption, and aggregate in‐ vestment grow at the labor force growth rate n in the steady sta‐ te, i.e., growth in all of these variables must also increase. • Higher growth in aggregate income need not be associated with higher income per worker in the long‐run. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 48/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Effects of an Increase in n on k: Comparative Statics Matthias Neuenkirch Advanced Macroeconomics Universität Trier 49/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Effects of an Increase in z on k: Comparative Statics • Recall ﴾21﴿: dk∗ −sf(k∗ ) > 0. = dz szf′ (k∗ ) − n − d • Sustained increases in total factor productivity cause sustained increases in per capita income. • Accepting the Solow model implies that the steady increase in per capita income since the Industrial Revolution was caused by sustained increases in total factor productivity. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 50/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Effects of an Increase in z on k: Comparative Statics Matthias Neuenkirch Advanced Macroeconomics Universität Trier 51/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Real GDP and Linear Trend • Real GDP grew on average at a rate of 2.4% from 1950 to 2023. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 52/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Growth Accounting • Growth accounting is an approach that uses the production function and measurements of aggregate inputs and outputs to attribute economic growth to: ﴾1﴿ Growth in factor inputs K and N. ﴾2﴿ Total factor productivity growth z. • Aggregate Cobb‐Douglas production function: F(K, N) = zKα N1−α , 0 < α < 1 ﴾27﴿ • In an competitive equilibrium, α is the share that capital receives of national income and 1 − α the share that labor receives. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 53/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Growth Accounting • E.g., a labor share in national income of 70% gives: Y = zK0.3 N0.7 ﴾28﴿ • The Solow residual is calculated as: z= Y ﴾29﴿ K0.3 N0.7 • Interpretation: Output that remains to be accounted for after we measure the direct contribution of the capital and labor inputs to output. • Here: Solow residual is taken from the Penn World Table. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 54/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Natural Log of the Solow Residual Matthias Neuenkirch Advanced Macroeconomics Universität Trier 55/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Average Annual Growth Rates of Y, K, N, and z Years Y K N z 1960–1969 1970–1979 1980–1989 1990–1999 2000–2009 2010–2019 2020–2023 4.82 3.25 1.96 2.26 0.79 1.97 0.17 6.87 4.43 2.53 2.54 1.44 1.31 1.22 0.24 0.15 0.45 –0.19 0.47 1.02 0.27 2.48 1.72 0.72 1.57 0.00 0.84 –0.10 Matthias Neuenkirch Advanced Macroeconomics Universität Trier 56/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Reasons for Productivity Slowdown ﴾1﴿ Shift from the production of manufactured goods to the produc‐ tion of services. ▶ Problems associated with measuring real growth in GDP due to changes in the quality of services over time. ▶ There might have been a downward bias in measuring real GDP growth. ﴾2﴿ Increases in the relative price of energy. ▶ Old, energy inefficient, capital equipment became obsolete. ▶ Some of the measured capital stock was not actually productive. ▶ There might have been an upward bias in measuring capital input. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 57/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Reasons for Productivity Slowdown ﴾3﴿ Costs of adopting new technology. ▶ With any dramatically new technology, time is required for workers to learn how to use the new technology. ▶ Productivity growth can be low, because workers are investing so‐ me of their time in learning on the job. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 58/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Convergence in the Solow Growth Model • Prediction of the Solow growth model: Per capita income con‐ verges among countries that are either initially rich or poor but otherwise identical in terms of ▶ total factor productivity z, ▶ labor force growth rate n, ▶ saving rate s. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 59/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Rich and Poor Countries and the Steady State Matthias Neuenkirch Advanced Macroeconomics Universität Trier 60/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Convergence in Income per Worker Matthias Neuenkirch Advanced Macroeconomics Universität Trier 61/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Convergence in Aggregate Output Matthias Neuenkirch Advanced Macroeconomics Universität Trier 62/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Convergence in the Solow Growth Model • However, the poorest countries of the world seem to fall behind the richest ones, rather than catching up. • In reality, countries do not have access to the same technology ﴾and do not have the same institutions﴿. ﴾1﴿ Absorption of new technologies takes some time when these spread from country to country. ﴾2﴿ Barriers to the adoption of new technology ﴾e.g., union power, trade restrictions, government subsidies for otherwise dying sec‐ tors﴿. ﴾3﴿ Differences in efficiency with which factors of production are allo‐ cated across firms in the economy, i.e., differences in institutions ﴾e.g., corruption, monopolies﴿. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 63/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Differences in Total Factor Productivity Matthias Neuenkirch Advanced Macroeconomics Universität Trier 64/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Comparative Statics Growth Accounting Convergence Convergence in the Solow Growth Model • Differences in total factor productivity can explain the disparity in income per worker across countries. • Aggregate output grows at the same rate in the steady state but standards of living are permanently different in the three count‐ ries ﴾with zr , zm , and zp ﴿. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 65/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence Outline 1. Economic Growth Facts 2. Malthusian Model of Economic Growth 3. Exogenous Growth: Solow Model 4. Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence Matthias Neuenkirch Advanced Macroeconomics Universität Trier 66/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence Endogenous Growth: Human Capital Accumulation • Total factor productivity accumulation involves research and de‐ velopment, education, and training on the job. • Total factor productivity accumulation is endogenous. • In this model, the growth rate of per capita income is determined by ▶ the efficiency with which human capital is accumulated and ▶ the fraction of available time devoted to human capital accumu‐ lation. • Model predicts that differences in per capita income can persist forever. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 67/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence Assumptions • Representative consumer allocates time between supplying labor to produce output and accumulating human capital. • The higher the human capital of workers, the more they can pro‐ duce, and the more new human capital can be produced. • Human capital accumulation differs from physical capital accu‐ mulation in that there is no limit to human knowledge or to how productive individuals can become. • Paul Romer: Key feature of knowledge is non‐rivalry. • Lack of diminishing returns to human capital investment leads to unbounded growth. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 68/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence Representative Consumer • One unit of time for work and accumulating human capital, i.e., no leisure. • u is the fraction of time devoted to working in each period. • The consumer’s quantity of human capital Hs is the measure of the productivity of the consumer’s working time. • Hence, the number of efficiency units of labor devoted to work is uHs . • For each efficiency unit of labor supplied, the consumer receives the current real wage w. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 69/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence Representative Consumer • Budget constraint in the current period ﴾assumption: no savings﴿: C = wuHs ﴾30﴿ • Trade‐off between current consumption and future consumption by accumulating human capital. • Time not used for production is used for human capital accumu‐ lation: ′ Hs = b(1 − u)Hs ﴾31﴿ • b > 0 is a parameter that captures the efficiency of the human capital accumulation technology. • Accumulating skills and education is easier, the more skills and education an individual ﴾or society﴿ has. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 70/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence Representative Firm • No physical capital: the representative firm produces output using only efficiency units of labor. • Constant returns to scale production function: Y = zuHd ﴾32﴿ • z > 0 is the marginal product of efficiency units of labor. • uHd is the demand for efficiency units of labor by the represen‐ tative firm. • Firm hires the quantity of efficiency units of labor that maximizes current profits: π = Y − wuHd = zuHd − wuHd = (z − w)uHd Matthias Neuenkirch Advanced Macroeconomics ﴾33﴿ Universität Trier 71/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence Representative Firm • Infeasible cases: ▶ z − w < 0: Firm does not hire. ▶ z − w > 0: Firm wants to hire an infinite quantity of workers. • z−w = 0: Firm is indifferent about the quantity of efficiency units of labor hired. • Firm’s demand curve for efficiency units of labor is infinitely ela‐ stic at w = z. ▶ Real wage w = z is independent of supply curve. ▶ Real wage per hour of work changes in proportion to the quantity of human capital of the representative consumer Hd as wHd = zHd . Matthias Neuenkirch Advanced Macroeconomics Universität Trier 72/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence Determination of the Equilibrium Real Wage Matthias Neuenkirch Advanced Macroeconomics Universität Trier 73/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence Competitive Equilibrium • There is only one market each period ﴾the labor market﴿, which clears at a real wage per efficiency unit of w = z so that Hs = Hd = H. • Thus, we get: C = wuHs = zuH ′ H = b(1 − u)H = b(1 − u)H s ﴾34﴿ ﴾35﴿ • Human capital grows over time without bound if b(1 − u) > 1. • Growth rate of human capital is: H′ − 1 = b(1 − u) − 1 H Matthias Neuenkirch Advanced Macroeconomics ﴾36﴿ Universität Trier 74/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence Human Capital Accumulation Matthias Neuenkirch Advanced Macroeconomics Universität Trier 75/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence Competitive Equilibrium • Growth rate of human capital increases if b ﴾e.g., efficiency in the education sector﴿ increases or if u ﴾i.e., time devoted to produce output﴿ decreases. • ﴾34﴿ also holds in the future period: C′ = zuH′ . • Growth rate of consumption is identical to the growth rate of human capital: C′ zuH′ H′ −1= −1= − 1 = b(1 − u) − 1 C zuH H ﴾37﴿ • Since C = Y, output also grows at the same level. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 76/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence No Convergence • Model economy does not grow because of any exogenous forces ﴾no population growth and no change in production technology as b and z are fixed﴿. • Growth occurs because of endogenous forces, with the growth rate determined by b and u. • Suppose that two countries are identical in all respects ﴾b, z, and u﴿ except that there are differences in the initial level of human capital. • Convergence does not occur as income ﴾consumption﴿ grows at a constant rate b(1 − u) − 1. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 77/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence No Convergence Matthias Neuenkirch Advanced Macroeconomics Universität Trier 78/ 79 Economic Growth Facts Malthusian Model of Economic Growth Exogenous Growth: Solow Model Endogenous Growth: Human Capital Accumulation Model Setup Optimization No Convergence No Convergence • Model is consistent with the fact that there are persistent diffe‐ rences in per capita income among poorer countries and bet‐ ween the poorer and richer countries of the world. • Model appears inconsistent with the fact that per capita incomes have converged among the richer nations. • Human capital externalities in highly developed economies are a possible explanation, i.e., skills are more easily transferred. • In addition, people with high levels of human capital tend to mo‐ ve from the less developed countries to the highly developed countries ﴾“brain drain”﴿. Matthias Neuenkirch Advanced Macroeconomics Universität Trier 79/ 79
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