Lecture notes on modern growth theories

Lecture notes on modern growth theories Part 1 Mario Tirelli Very preliminary material. Not to be circulated without permission of the author. January 18, 2019 Contents 1. Introduction 1 2. Preliminary considerations 3 3. The Solow-Swan growth model 5 3.1. Steady-state properties 9 3.2. Conclusions 11 4. The Solow model predictions and the empirical evidence 12 4.1. The main source of growth: Solow s residuals 13 4.2. The Solow model and cross-country income variations 14

1. Introduction Kaldor (1957) and Solow (1957) first highlighted the following empirical regularities on growth, which are not totally undisputed today: (1) Real output roughly grows at a constant rate, g. (2) Real capital has roughly the same constant rate of growth of output, g K g. (3) Labor input has a constant growth rate that is higher than that of real capital and output, g N > g. (4) The ratio of profits on capital Π/K and the real interest rate r are both roughly constant. (5) Economies with a high profit/income ratio Π/Y tend to have a high investment/output ratio I/Y. (6) Cross-country comparisons reveal a high variance of output per-capita and growth rates. 2) implies that the ratio of capital to output stays constant over time. Indeed, 2) says that +1 1 = Y t+1 Y t 1, at all t which trivially implies, +1 /Y t+1 = /Y t, at all dates t. Next, 2) plus 4) imply that the income distribution between capital owners and workers is roughly constant over time. Indeed, they do imply that the share of income that goes to the capital Y K /Y is constant (i.e. it has a steady, zero growth rate): given that Y K, essentially, equals the sum of the (real) rental income, rk, going to capital owners and (real) corporate profits, Π, going to those who invest capital into production activities, Y K t Y t = r t Y t + Π t Y t = r t + Π t Y t Y t = K ( t r t + Π ) t Y t Since the terms in parenthesis are roughly constant, by 2) and 4), we have the result. If we assume capital market-clearing, that is I = sy, then 5) can be restated by saying that the average saving rate s is higher in countries with higher return from investing Π/Y. Moreover, if we assume that capital depreciates at a constant rate δ, the simple capitalaccumulation accounting is, +1 = I t + (1 δ) By 1) and 2) the latest implies that the average saving rate s is constant. 1 Therefore, the first four facts (and constant depreciation δ) imply that the economy experiences a balanced growth, 1 To see this, rewrite the accumulation equation as, It / = g + δ and use the fact that I t = s t Y t. This yields, (s) s t = g + δ (Y t / ) where the denominator is constant by 2). 1

with the main economic variables (Y, K, I, C, where C = (1 s)y ) growing at a constant rate g. Thus, at the balanced growth, the scale of the economy (the output level Y ) changes over time, but the ratios of the variables to output tend to be constant. It remains to consider fact 6). This is a very important stylized fact, raising many relevant questions, which turn to be central in economics: Why are some countries so much richer than others? How much of these income differences are explained by differences in growth rates? Do countries with different per-capita income display convergence over time or not? Finally, which are the determinants, or fundamental causes, of growth and growth differentials? Some answers to the first questions are well exposed in Acemoglu (2009) (chapter 1). To summarize, growth differentials are relevant in explaining cross-country income differentials only if we take a sufficiently long time perspective; looking at post-war date does not suffice, essentially because by the II world war time the variance of per-capita income was already very high. Maddison s studies indicate that, in its large proportions, this originated with the industrial revolution in the XIX century. Moreover, there is not evidence of per-capita income convergence across the world. Convergence is observed for those countries with similar socio-economic characteristics or fundamentals. For example, in the post-war period, OECD countries tend to show convergence: relative to the US, lower income countries have grown faster, catching up or reducing their initial poorer condition. Thus, we can conclude that data signal conditional convergence, as opposed to absolute convergence. Equipped with all this empirical evidence we still have to understand which are the fundamental causes of growth, cross-country income and growth differentials, conditional convergence. We need a theory. A model that could capture the growth stylized facts and explain those fundamental causes. In these notes we shall focus on modern neoclassical growth. To this end, we shall begin with Solow s growth model and then extend the analysis to Ramsey-Cass-Koopmans. 2 There are three fundamental reasons for studying modern neoclassic theory and its workhorse model due to Ramsey, Cass and Koompmans (RCK). First, this theory addresses both growth and business cycle as two integrated phenomena, with a unique economic model and set of analytic tools. 3 Roughly speaking, the long-run behavior of the economy (namely, its balanced growth path) determines the variables secular trends. Instead, short-run dynamics around these trends, describe the variables cyclical fluctuations. Some of these fluctuations characterize business cycles. A second reason for studying RCK is that, unlike previous research, it builds up on general equilibrium theory, with all the advantages that this implies in term of economic analysis and policy. RCK allows both to analyze equilibria occurring at different policy regimes (e.g. tax and 2 Cass, D.: Optimum Growth in an Aggregative Model of Capital Accumulation, Review of Economic Studies, Vol. 32 (1965). Koopmans, T,. C.: On the Concept of Optimal Economic Growth, Semaine D Etudes sur le Role de L Analyse Econometrique dans la Formulation de Plans de Developpement, Rome: Pontificia Academia Scientiarum, 1965. Ramsey, F. P.: A Mathematical Theory of Saving, Economic Journal, Vol. 38 (1928). 3 This idea is not new in economics and emerges, for example, in the works of Hicks and Goodwin in the 1950s, as well as in the structural macro-econometric models of Tinbergen, Klein and Modigliani in the 1960s. 2

fiscal policy schemes) and their welfare properties. Hence, welfare analysis, based on efficiency and distributional considerations, can be used to guide policy decisions. A third reason for studying RCK is that the theory passed a few important empirical tests, being able to match many important statistics characterizing economic growth and business cycles; early models were simple and still able to provide a good representation of complex phenomena. Finally, the neoclassical general equilibrium approach has been adopted by the New-Keynesian school and its models exploited by most of the international institutions involved in economic analysis (central banks, IMF, public authorities, etc.). Modern general equilibrium models (computable general equilibrium models - CGE) are often sophisticated, incorporating more realistic descriptions of the economy, including financial market frictions and other market failures (e.g. externalities, public goods, imperfect competition and asymmetric information), but they are all deep-rooted in the RCK model. 2. Preliminary considerations Assumption 1. The technology is represented by a production function, F : R 3 + R +, that satisfies the following properties. F (, N t, A t ) It is twice continuously differentiable in K, N and it is strictly increasing in K, N, F K, F N > 0, concave in K, N, F KK, F NN 0, and F (0,, ) = 0. It has constant return to scale in the input factors K, N (CRTS). It satisfies Inada conditions: lim F x = +, x 0 lim F x = 0, for x {K, N} x The CRTS assumption, formally, says that the production function is homothetic in the input factors (implying homogeneous of degree one in K, N). Hence, by Euler s Theorem, F (, N t, A t ) = F K (, N t, A t ) + F N (, N t, A t )N t, for all (, N t, A t ) and F K, F L are homogeneous of degree zero in K, N. Observe that A t represents the technological progress. Three typical specifications are, the Hicks-neutral, the Harrod-neutral and the Solow-neutral technological progress. In the first, A t shifts up and down isoquants, G(, N t ), that is F (, N t, A t ) = A t G(, N t ). In the other two, A t enters as the multiplicand of N t and, respectively. For example, for a Cobb-Douglas function, F (, N t, A t ) = A t Kt α Nt 1 α, F (, N t, A t ) = Kt α (A t N t ) 1 α, F (, N t, A t ) = (A t ) α Nt 1 α We say that the technological progress is exogenous (i.e. part of the economic fundamentals) when, as we do here, we assume that its dynamics is governed by a given, exogenous, model. Typically, one assumes that (the deterministic part) of this model has a constant per-period, growth rate, µ, µ := A t+1 A t A t, at all t T 3

Hence, the dynamics of technological progress, at a given initial level, say, A 0 = 1, is, A 1 = (1 + µ)a 0 = 1 + µ A 2 = (1 + µ)a 1 = (1 + µ) 2. A t = (1 + µ) t A similar assumption is often used for demographics. The population, or work-force, grows at a constant, per-period rate n, according to the deterministic model, N t = (1 + n) t N 0, t T Next, for any variable X t, let N t := A t N t and x t := X t / N t be that variable in efficiency units. In the case of a Harrod-neutral, Cobb-Douglas technology F (, N t ) = Kt α N t 1 α, y t := Y t N t = Kα t ( N t ) 1 α N t = Define the function f : R + R to be such that f(k t ) = F (k t, 1) ( Kt N t ) α = k α t Clearly, under the above assumptions, f is a differentiable functions, which is strictly increasing and strictly concave. Moreover, it satisfies Inada conditions in k. You should verify such properties as an exercise. Remark 2.1 (Production under CRTS). CRTS technologies have the property that the productiveefficient level of output of a single plant/firm is not determined. This is just the consequence of the fact that a technology is homogeneous of degree one in inputs. So that, proportionally increasing inputs one attains an disproportional increase in output; hence, the scale of production does not affect the amount of input required to produce a unit of output. More formally, if F is CRTS, F K and F N are homogeneous of degree zero and, by Euler s theorem, Y t = F K,t + F N,t N t Thus, any production scale λy, λ > 0, can be achieved by proportionally expanding inputs, λ(k, N): 4 Indeed, F K (, N t ) = F K (λ, λn t ) and F N (, N t ) = F N (λ, λn t ); hence, λy t = λf K,t + λf N,t N t This does also imply that, for given input prices (taken as invariant with respect to the production scale) CRTS technologies can accommodate any scale of production at the same unit (and marginal) cost; implying there is no efficient minimum unit cost scale. Also, any production scale yields zero profit. If prices are, (ν, w) = (F K, F N ), which it is going to be at an 4 Graphically, in a plane (K, N), CRTS technologies have technical rates of substitutions, FK/F N which are constant across isoquants if measured along any ray from the origin. Fixing any K, N and expanding them by λ > 0 generate a ray originated in (K, N ) of constant slope N /K. Along a ray, input and output expand proportionally (as λ increases), but since F K/F N stays constant (by homog. of degree zero), each point in which the ray crosses an isoquant λf (k, N ) corresponds to the same technical rate of substitution. 4

interior firm optimum, total production costs change proportionally with (, N t ) and revenues Y t = F (, N t ). 5 In the special case of a Cobb-Douglas technology, implies that Yt := κ and Yt N t Y t = α Y t + (1 α) Y t N t N t := η remain constant as we change (, N t ) in the same proportion. To summarize, under CRTS the efficient, as well as the competitive equilibrium, number of firms/plants and their individual production levels are undetermined. This is essentially, without loss of generality, one can assume a representative firm. 3. The Solow-Swan growth model Solow (1956) and Swan (1956) growth model is a simplified version of RCK, in which consumers behavior is not represented endogenously: households inelastically supply labor and have a constant average saving rate s (i.e. also a constant average propensity to consume, c = 1 s). Hence, aggregate consumption is, C t = (1 s)y t More precisely, a Solow-Swan economy E is represented by its fundamentals E = (F, s, n, µ, δ, K 0, N 0, A 0 ) respectively, a technology, an average saving rate, a rate of growth of the population of workers, a rate of technological innovation (or change), a rate of capital depreciation, an initial capital stock and population of workers, an initial productivity level. Time t evolves over an infinite horizon, in T = {0, 1, 2,...}. In every period t, capital and labor are demanded by a representative firm so as to maximize its profits. Labor and capital rental markets are competitive. The law of capital accumulation is, The economy income accounting is, +1 = I t + (1 δ) C t + I t = Y t In this economy, the latest two conditions define the following feasibility constrain, +1 = F (, N t, A t ) + (1 δ) C t Definition 1 (Solow-Swan Competitive Equilibrium). In an economy E a Solow-Swan Competitive Equilibrium is a sequence of allocations (Y, C, I, K, N, A) and prices (w, ν) such that, at all t in T, Y t = F (, N t, A t ) C t = (1 s)y t, I t = sy t +1 = I t + (1 δ) 5 For any given (Kt, N t), profits are, F (, N t ) ν t w t N t = F K,t + F N,t N t ν t w t N t = 0 5

ν t = F K (, N t, A t ), w t = F N (, N t, A t ) N t = (1 + n) t N 0, A t = (1 + µ) t A 0 In this economy markets are competitive, implying that firms demand inputs up to the point at which their marginal productivities equate prices. As we said earlier, CRTS implies zero profits (see the discussion in the remark 2.1): at all dates t in T Π t = F (, N t, A t ) ν t w t N t = 0 Since in the Solow economy, labor supply is inelastic and corresponds to the whole labor force, the labor market equilibrium is attained at full employment, as the real wage rate adjusts so that the labor demand equals the whole labor force. Thus, the population (or workforce) dynamics fully translates into labor input dynamics, N t = (1 + n) t N 0. Using the law-of-motion of capital, one finds that the equilibrium rate of growth of capital is, (1) +1 = s Y t δ An equilibrium with balanced growth is one at which (Y, C, K) grow at the same constant rate g. Later we say that an equilibrium is a steady-state if g is zero. To study the dynamics and establish the existence (or compute) balance growth, it is often useful to transform the original variables expressing them in per-capita terms or, when there is growth in labor input productivity, in efficiency units. We are going to show that such a transformation allows to easily compute the steady-state of the transformed economy and that this corresponds to the balanced growth equilibrium of the original economy. Indeed, let us consider the second case and, for any variable X t, let N t := A t N t and x t := X t / N t be that variable in efficiency units. In the case of a Harrod-neutral, Cobb-Douglas technology: y t := Y t = Kα t (A t N t ) 1 α ( ) α Kt = = kt N α t A t N t A t N t In this economy, in efficiency units, there is no balanced growth, except for the steady state (zero growth). Indeed, since y t = kt α, y t+1 y t = ( ) α kt+1 and a balanced growth ḡ would require, 1 + ḡ = (1 + ḡ) α, which holds if and only if ḡ = 0. Going back to the original economy, we are going to show that, if it exists, a balanced growth is one with the economy growing at the (instantaneous) rate η = µ + n. Indeed, consider the definition of the rate of growth of capital in efficiency units; by simple transformations, k t (2) k t+1 k t = +1/ N t+1 / N t 1 = +1 Nt 1 N t+1 ( ) Kt+1 Nt = + 1 1 N t+1 6

Notice that, (1+η) {}}{ N t = A t N t = [(1 + µ)(1 + n)] t N 0 = (1 + η) t N 0 1 + η := 1 + n + µ + nµ 1 + n + µ as nµ 0. 6 Hence, the rate of growth of N is constantly equal to 1 + n + µ. Using this into (2), ( ) k t+1 Kt+1 1 = + 1 k t 1 + η 1 = 1 ( ) Kt+1 + 1 1 η 1 + η yielding, (3) (1 + η) k t+1 = +1 η k t At a steady state of the model with variables in efficiency units, k t+1 k t = 0 implies, +1 = η =: g > 0 One can also simplify (8), using k t+1 k t η 0 (i.e. the products of the growth rate of capital in efficiency units with n and µ are zero), so as to directly consider the following is approximation, (4) k t+1 = +1 η k t We are left to check that all activity variables growth at the per-period rate g (or η), so that we have a balanced growth equilibrium. By (1), g = s(y t / ) δ. Rearranging, we obtain that the capital-output ratio is constant, (5) = s Y t g + δ Denote the fraction on the right hand side z, and consider = zy t at two consecutive periods. Subtracting, yields and, dividing through by = zy t, +1 = z Y t+1 g := +1 = Y t+1 Y t := g Y We are left to check that consumption also grows at the same rate. Since, C t = (1 s)y t, g C := C t+1 C t C t = (1 s) Y t+1 C t = (1 s) Y t+1 (1 s)y t = g Y Thus, we have used the steady-state of the economy in efficiency units to compute the balanced growth path of the original economy g Y = g C = g = n + µ. 6 These type of approximations, taking the product of growth rates to be zero, tend to be exact as we reduce the time-interval and, in the limit, gives back the characterization obtained in continuous time models. 7

We end this section computing the equilibrium of the economy in efficiency units and drawing some considerations. Recalling (8), and using (1) and (5) with g = η, (6) k t+1 k t At steady-state, the latest yields, = s y t δ η = skt α 1 δ η k t (7) k = ( s ) 1 1 α δ + η As for the other activity variables, y = (k ) α, c = (1 s)(k ) α, (I/ N) = s(k ) α. Finally, we derive equilibrium prices. Given profits, Π t = A t N t [f(k t ) ν t k t ] w t N t at the firm optimum, each factor demand satisfies marginal factor productivity equal factor price. For capital, ν t = f (k t ) = αk α 1 t = α y t k t = α Y t Therefore, ν is constant at the balance growth. Turning to labor demand, using zero-profit (CRTS), w t = A t [f(k t ) ν t k t ] = A t [f(k t ) f (k t )k t ] = (1 α) Y t N t Implying that at balanced growth the wage rate increases at the rate of growth of labor productivity µ, so as to keep the labor demand constant. Indeed, at the steady-state capital level, A few general comments follow. w t = A t k α (1 α) w t+1 w t = A t+1 A t = 1 + µ At a balanced growth, total savings sy is used to replace capital depreciation δk, to provide newly born agents with the same amount of capital in efficiency units ηk. This is so by, (6) and k t+1 /k t = 0, implying sy = δk + ηk. A change in the saving rate s does only have temporary effects on the growth rate. This is again by (6) and k t+1 /k t = 0 and it occurs exactly because savings are carried out only to keep the capital in efficiency units constant. A change in the saving rate s has permanent effects on the steady-state level of capital k, by (7). A country has an higher saving rate s if and only if it has a higher capital-income ratio k /y. (6) and k t+1 /k t = 0 yield, k y = s δ + n + µ Two countries with same fundamentals and different saving rates have the same growth rate and different steady-state capital levels of activities (capital, income, consumption). 8

At balanced growth, real gross-interest rate ν is constant (i.e. it has a zero long-run growth), ν = α y k = αδ + n + µ s increasing in the rate of technological progress, in the productivity of capital and decreasing in s. In the Cobb-Douglas economy, α = νk/y and 1 α = wn/y measure the constant shares of GDP that, respectively, go to capitalists and workers. Most of the empirical evidence is drawn in per-capita terms, rather than in efficiency units. It is easy to check that, if a balanced growth exists it is one with the economy with the percapita variables growing at the constant rate µ, measuring the rate of growth of technological progress. To see this, just start out again from equation (2) and re-iterate the derivation until equation (8) using k t := /N t rather than. 3.1. Steady-state properties. We are going to prove that in the Solow-Swan economy a steady state exists, it is unique and stable. 3.1.1. Existence. Define the rate of growth function, g : R R such that g(k t ) = s f(k t) k t (δ + η) g( ) has the following properties (start looking at figure 1): Since f(0) = 0 and Inada, (using de l Hopital) lim k 0 + g(k) = lim k 0 + sf (k) (δ+η) = + > 0. Again by Inada, lim k + g(k) = lim k + sf (k) (δ + η) = (δ + η) < 0 By continuity of g( ), there exist a k sufficiently closed to zero and a k sufficiently large, such that g(k) > 0 and g( k) < 0. Therefore, we can apply the Intermediate Value Theorem (IVT) and conclude that there exists a k < k < k such that g(k ) = 0; thus k > 0 is a steady state. 7 3.1.2. Uniqueness. To prove uniqueness of the steady state, it suffices to show that g( ) is strictly monotone. [ f g ] (k)k f(k) (k) = s k 2 = s [ f (k) f(k) ] k k We claim that this is negative for all k > 0, if f is strictly concave. To prove our claim we shall exploit the strict concavity of f (along with f(0) = 0) and the following simple lemma. Lemma 1 (Concavity). Let f : R R be any C 1, concave function with f(0) 0. Then, for all k > 0, f (k) f(k) k with strict inequality if f( ) is strictly concave. 7 The IVT says that if F ( ) is a real-valued, continuous function on the interval (a, b), and F (a) < c < F (b), then there is a z (a, b) such that F (z) = c. 9

To see why, use concavity to attain, f(0) f(k) f (k)(0 k) holding with strict inequality if f is strictly concave. Then, use f(0) 0 and k > 0 to prove lemma. 3.1.3. Stability. We are left to show that the steady state k is (asymptotically) stable. From the definition of g( ) above, k t+1 = g(k t )k t or k t+1 = [1 + g(k t )]k t Local stability, in a neighborhood of k, can be shown using a linear approximation of the last (non-linear) difference equation; namely, k t+1 = [1 + g(k )]k + {g (k )k + [1 + g(k )]}(k t k ) and using g(k ) = 0, (8) k t+1 k = [1 + g (k )k ](k t k ) k is (locally) asymptotically stable if and only if 1 + g (k )k < 1. Since we have shown that g (k) < 0 for all k > 0, we immediately have that 1 + g (k )k < 1. So, we only have to show that 1 + g (k )k > 1, that is g (k )k > 2. Observe that, by definition, [ g (k )k = s f (k ) f(k ] ) > s f(k ) = (δ + η) k k Hence, a sufficient condition is δ + η < 2, which is typically true (annual data say that, δ 3 4%, n 1 2%, µ 1 2%, overall about 5 8%). Finally, letting k t be initial capital (if you like, relabel k t as k 0 ), one can observe that two countries with same fundamentals but different levels of accumulated capital, say k 1 t < k 2 t < k, are characterized by the poorer 1 growing faster than the richer 2 (i.e., k 1 t+1 k > k 2 t+1 k, given that the two countries have the same steady state k and function g( )). Figure 1. k is the unique, stable steady state 10

Since g is differentiably strictly convex, it is natural to guess that the steady state is also globally stable. The proof is simple and can be found in Acemoglu (2009) (proposition 2.5, p. 45). 3.1.4. Speed of convergence. Equation (8) can also be used to determined the speed at which the economy converges as a function of the initial distance from the steady state. Indeed, The latest can be rewritten as, k t+1 k = [1 + g (k )k ](k t k ) = [1 (1 α)(δ + η)](k t k ) (9) k t+1 k t = [(1 α)(δ + η)](k t k ) This says that the higher is the distance from steady state at t, in the sense of (k t k ) < 0, the higher will be the capital accumulation from t to t + 1, k t+1. The speed of convergence is by how much k t+1 changes with such distance; which is measured by [(1 α)(δ + η)]. In problem 1 below, you are asked to show that the speed of convergence of income is α times that of capital. 3.2. Conclusions. To summarize, the Solow model allows to, at least, capture some basic, qualitative, empirical facts. Variables such as income consumption and capital tend to grow at the same rate and the ratio of capital stock to output is roughly constant. The level of balanced growth g depends on demographics, the rate of growth of population (or labor force) and on the rate of technological progress. Two economies (or countries) with same fundamentals, but different saving rates would have the same rate of growth; that is, contrary to the empirical evidence, saving rates do not covariate with the growth rate in the medium- and short-run too. A change in the saving rate does only affect the steady-state variables in levels and the capital-output ratio. Two economies with the same fundamentals, except for their initial capital stock in per-capita terms, are characterized by the fact that the poorest will grow faster, and catch-up (conditional convergence). Problem 1. Show that the speed of convergence of y (approximately) is α[(1 α)(δ +η)].[hint. Use the fact that k t+1 /k t log k t+1 log k t, to rewrite equation (9) as, log k t+1 log k t = [(1 α)(δ + η)](log k log k t ). Then, use the production function. ] Problem 2. Consider the Solow model, for simplicity, without technological progress. Let the per-capita production function be f(k) = 3 k, the depreciation rate δ is 10%, and the population growth rate n is 5%. Further, assume that individuals save 30% of their income. (1) What are the steady state, per-capita values of capital, output and consumption (k t, y t, c t )? (2) What happens when the saving rate is s =.4? and when n =.06 (and s =.3)? 11

(3) Suppose the production function is of the form F (K, N) = KN.25N +.5K Give conditions under which a steady state exists.[hint. this depends on the fact that k > 0 and implies that the interest rate is high enough. This condition was unnecessary in the first item because the Inada conditions were satisfied; here they are not!?] Problem 3. Do some comparative statics, also using the following graphical representation. (1) Argue, also graphically, that a permanent increase in s has temporary effects on the growth rate and permanent ones on the steady-state levels of k and y. (2) Explain the effects a of a permanent increase in s on investments and consumption. (3) Explain the effects of a permanent increase in the rate of growth of technological progress µ on (K, C, I). 6 5 ( +g)k 4 3 2 sf(k) 1 1 10 20 30 k 40 50 60 4. The Solow model predictions and the empirical evidence Section 3.2 already provides a partial answer to the question of how good is the Solow model in explaining the main key facts on long-run macro dynamics. Yet there is more to be assessed. Indeed, so far, our discussion has only been qualitative; we have not said anything on the potentials of the model to actually capture and predict economic figures. Can we use the Solow model to do quantitative analysis, at least, for what concern the interpretation of past dynamics? The answer to this question is important since it also provides some key elements and thoughts on the directions of subsequent research and theory development, which we shall examine later in the course. 12

4.1. The main source of growth: Solow s residuals. Solow, using US data from 1901 to 1949, estimated that changes in productivity accounted for the 87.5% of the growth of output per-worker, while only about 12.5% was accounted for by increased capital per worker (i.e. to capital accumulation). Revising his analysis with more recent data one finds that the two components, respectively, account for the 2/3 and 1/3 of the growth of output per-worker. Instead the contribution of variations in labor input is roughly zero (recall that the changes of the average hours of work, per-worker, has no trend). Let us summarize Solow s analysis. Assume that output is represented by an aggregate production function, whose arguments are labor (N), capital (K) and an index of technological productivity (A), say the total factor productivity (TFP), Y t = F (, N t, A t ) Take the logarithmic transformation, ln Y t = ln F (, N t, A t ), and totally differentiate it with respect to time. Then, use the definition of instantaneous rate of change, (ln Y t )/ t = ( Y t / t)/y t =: Ẏ /Y Ẏ Y = 1 ( ) F K K + F N Ṅ + F A Ȧ Y = 1 ( F K K K Y K + F NṄ N ) N + FAȦA A = ϵ K {}}{ F K K Y = ϵ K K K + ϵ N ϵ N {}}{ K K + N Ṅ F N Y N + F A A Y Ṅ N + F A Ȧ A Y A }{{} u Ȧ A where ϵ K and ϵ N denote the elasticity of output with respect to K and N, respectively. A Ȧ u := F A Y A denotes the contribution of technological progress (TFP change) to growth. Solow estimates u as the residuals (thereafter called the Solow residuals) of the following regression, g = γ 1 g K + γ 2 g N + u where (g, g K, g N ) are the growth rates of output, capital and labor input. The result is a low R-squared, denoting that most of the long-run growth is not explained by (g K, g N ) and that γ 1 1/3 and γ 2 0. 8 8 A similar analysis has been carried out to estimate the contribution of inputs variation in explaining real business cycles. Results typically reveal that both total factor productivity and labor productivity are procyclical and slightly lead the cycle. The contribution of variations in labor input is about 2/3 and that of changes of TFP is about 1/3; while, capital per-worker is about constant over the business cycle. See, for example, chapter 1 in the Handbooks of Macroeconomics vol. 1. Stock J. and M. Watson, 1999. Business Cycle Fluctuations in US Macroeconomic Time Series. 13

Finally, notice that adopting the Cobb-Douglas specification, α = ϵ K ˆγ 1 = 1/3 explains why in most empirical analysis this assumption is made. 4.2. The Solow model and cross-country income variations. Mankiw, Romer and Weil (1992) (hereafter, MRW) argued that, 9 the empirical predictions of the Solow model are, to a first approximation, consistent with the evidence. Examining recently available data for a large set of countries, we find that saving and population growth affect income in the directions that Solow predicted. Moreover, more than half of the cross-country variation in income per capita can be explained by these two variables alone.[p. 407] Thus, countries with higher income per capita tend to have higher saving rate s and lower population growth n. Yet, observe that MRW only talk about quality predictions. Quantitatively, the effect of population growth and of the saving rates are too large. Let us briefly see this. As above, we treat productivity (the A term) as due to level of technology and assume that technology is a labor-augmenting Cobb-Douglas, Y t = Kt α (A t N t ) 1 α. Recall that the economy growth rate is g = η = n + µ and that the steady-state capital (in efficiency units) is, ( k = In per capita, the production function is, Y t N t = A t s δ + n + µ ( Kt A t N t ) 1 1 α ) α = A t k α t Taking logs on both sides and evaluating at the steady state (k t = k ), ( ) Yt ln = ln A t + α ln k N t = ln A 0 + t ln(1 + µ) + α 1 α ln s α ln(δ + n + µ) 1 α Since, n and s vary across countries, the capital stock (per capita) will also vary explaining, explaining different levels of income per capita. t ln(µ + 1) is a linear trend with slope equal to the (gross-) rate of growth of labor productivity (or technological progress) A. The empirical (Solow) model is, ( ) Yt ln = a 0 + b 1 ln s + b 2 ln(δ + µ + n) + ϵ N t where a 0 = ln A 0 is the intercept capturing, not only technology, but resource endowments, climate, institutions of country in the base year (1960, t being year 1985); 10 ϵ is an error term (also interpretable as the stochastic component of technological progress) which is assumed to be independently distributed across countries and independent on the regressors. Moreover, 9 Mankiw, N. G., Romer, D., Weil, D. N. (1992). A Contribution to the Empirics of Economic Growth. The Quarterly Journal of Economics, 107(2), 407-437. 10 MRW take investments and population growth as the averages of the sample variables for the period 1960-1985; they also assume that µ + δ is 0.05. See the sample description below. 14

one can measure the average saving rate using Say s law, sy t = I t, as the investment-income ratio. In the regression, one can test or restrict coefficient to satisfy, b 1 = b 2 = α 1 α Figure 2. MRW table I - OLS regressions. Coefficients on investment (capturing the saving) rate (ln(i/y )) and population growth have the predicted signs and are significant in most samples. 11 Equality restriction on coefficients b can t be rejected. Regressions explain large fraction of the cross-country variation in income per capita: the adjusted R-squared in the intermediate sample is 0.59. This is in contrast with the common wisdom and previous empirical findings, which says that the Solow model explains most of the cross-country income variation, based on differences in labor productivity (again, the A term). Nonetheless, estimates on different country groups reveal that the impacts of saving and labor force growth are much larger than model predicts: the value of α implied by the coefficients tends to exceed the capital-income share of 1/3 (it is more than 1/2). If 11 The first (non-oil) sample includes 98 countries; oil countries are excluded because most of their GDP comes from oil extraction, as opposed to value added. The intermediate sample is obtained from the first, dropping 23 countries, mostly with a small population. The OECD sample contains 22 countries with population greater than one million. The sample years are 1960 and 1985. In the table g is the rate of growth of technological progress, corresponding to our µ. 15

one had to constraint the coefficient to be 1/3, then the constrained regression would see the R-squared drop from 0.59 to 0.28. 12 Therefore, quantitatively, MRW conclude that the effect of population growth and of the saving rates are too large. For this reason they suggest to go beyond the Solow model textbook form, toward one including a broader definition and specification of capital. Economists agree that reducing accumulation to savings in physical capital is fallacious. In particular they have long stress the importance of human capital accumulation to explain economic growth; where, they normally summarize with human capital accumulation activities such as work training, schooling and others (e.g. including health care), which enhance labor productivity through a costly and timely investment. Just to grasp the concept, a simple representation of human capital into a standard Cobb-Douglas technology is, Y t = A t K α t (h t N t ) 1 α where h t is the quantity of labor supplied by each of the N t workers. Intuitively, everything else equal, h t increases with education (e.g. years and quality of schooling), with work training, with health-care accessibility (e.g. the extension and quality of the public health-care system), and the latest are all increasing in per-capita income. The practical reason why introducing human capital into the Solow model might improve its empirical predictions is twofold. (1) For any saving rate into human capital, higher s or lower n leads to higher Y/N and this increases human capital accumulation, activating a further indirect effect that boosts up Y/N; hence, one can now explain higher Y/N with more plausible levels of s and n. (2) Human capital accumulation may be correlated with saving rates s and population growth n: the population spends more time in the education system in countries with a better system; this decision is costly, implying both that part of the family income is saved and invested into education (this lowers saving that goes into physical capital s) and that people enter in the labor force later in life (something that contributes to lower n). Therefore, omitting human capital biases upward the estimates of the coefficients attached to s and n. Since including human capital into the Solow model implies a change in the theory, we postpone its analysis. 12 Estimates based on more recent data confirm these results. See Is growth exogenous? taking Mankiw, Romer, and Weil seriously, in Ben S. Bernanke & Kenneth Rogoff, 2002. NBER Macroeconomics Annual 2001, Vol. 16, NBER Books, National Bureau of Economic Research, June. 16