Economic Growth Theory. Vahagn Jerbashian. Lecture notes

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1 Economic Growth Theory Vahagn Jerbashian Lecture notes This version: February 2, 206 Contents Empirical observations; Kaldor stylized facts of growth; Neoclassical production function; The Solow-Swan model 4 Growth matters over long time periods 4 Kaldor stylized facts of growth 6 Neoclassical production function 6 Solow-Swan model - (Solow, 956; Swan, 956) 7 Market equilibrium 8 Solow model and Kaldor stylized facts 3 Additional: Harrod (939)-Domar (946) model 5 Additional: Poverty traps 7 Some proofs and additional results 8 Continuous time optimal control - the basics and applications 20 Necessary conditions 2 Suffi cient conditions 22 Infinite horizon discounted problem 24 Many states and controls 24 Continuous time Bellman equation (Hamilton-Jacobi-Bellman equation) 25 These notes may contain typos/mistakes and are subject to changes/updates during our course Please keep track if there are any

2 The Ramsey-Cass-Koopmans model - (Ramsey, 928; Cass, 965; Koopmans, 965) 27 Main assumptions 27 Market equilibrium 28 Transition dynamics 3 The behavior of the savings rate 32 Evaluation of the model 34 Extensions 35 Government in the Ramsey model 35 Productive public expenditures in the Ramsey model 37 The AK model - Spillovers à la Romer (986) 40 Main assumptions 40 Market equilibrium 4 Social Planner s problem 45 Human capital accumulation: The Lucas (988) model 48 Preliminary remarks 48 Main assumptions 49 Market equilibrium 50 Transition dynamics 53 Kaldor stylized facts and first models of endogenous growth 56 Externalities in the production of goods and human capital 57 R&D based models of growth: The Romer (990) model - horizontal product innovation 64 Main assumptions 64 Market equilibrium 65 The Social Planner s problem 7 Additional - Erosion of monopoly power 72 R&D based models of growth: The Jones (995) model - horizontal product innovation, (semi-endogenous) growth without scale effects 75 Motivation 75 Main assumptions 76 Market equilibrium 77 R&D based models of growth: The Smulders and van de Klundert (995) model - vertical product innovation and market structures 83 Motivation 83 Main assumptions 83 Market equilibrium 85 2

3 R&D based models of growth: The Grossman and Helpman (99) model - vertical product innovation with creative destruction 96 Main Assumptions 96 Market equilibrium 97 Further comments 0 Scale effects 02 Innovation by market leader 02 The Social Planner s problem 04 Technology Diffusion and Growth - Barro and Sala-i-Martin (997) 07 Main assumptions 07 Market equilibrium 09 Directed Technical Change: Acemoglu (2002) - horizontal product innovation 4 Main assumptions 5 Market equilibrium 7 Further applications and discussion 23 Directed Technical Change: Acemoglu (998) - vertical product innovation 25 Main assumptions 25 Market equilibrium 26 3

4 Empirical observations; Kaldor stylized facts of growth; Neoclassical production function; The Solow-Swan model Growth matters over long time periods For the welfare of a single country example: postwar growth experience of the US GDP per capita was,233 $US in 950 This increased steadily to 36,098 $US by 2004, hence increasing by a factor of around 3 in 50 years This is due to an average annual growth of 22% If it had grown at % only, then GDP per capita would have increased by a factor of 7 by 2004 If it had grown at 3%, then GDP per capita would have increased by a factor of 5 by 2004 For the welfare across countries explore the properties of the distribution of PPP-adjusted GDP per capita across the available set of countries The distribution has shifted to the right due to growth in per capita income in most countries The dispersion has increased There is indication of stratification of countries/ "twin peaks" which can indicate that there is no convergence between high income and low income countries 4

5 The differences in real GDP per capita remain rather persistent over time Important exception: East Asian countries 5

6 Kaldor stylized facts of growth Per capita output grows over time and its growth rate does not tend to decline 2 Physical capital per worker grows over time 3 The physical capital to output ratio is nearly constant 4 Rate of return on capital is nearly constant 5 The shares of capital and labor in national income are nearly constant 6 The growth rate of output per worker within a country may exhibit short periods of acceleration Across countries - the growth rates of output per worker differ Neoclassical production function Y = F (K, L), where K is the capital input, and L is the labor input Constant Returns to Scale (CRTS): F (λk, λl) = λy ; λ > 0 F is increasing and concave function That is F K, F L > 0 and F KK, F LL < 0,where F X = F/ X for X F K, F L > 0 and F KK, F LL < 0 diminishing returns to inputs Inada conditions: lim K 0 F K = lim L 0 F L = +, lim F K = K + lim F L = 0 L + Essentiality of all inputs: F (0, L) = F (K, 0) = 0 (this follows from F < 0 and CRTS) 6

7 Solow-Swan model - (Solow, 956; Swan, 956) Main assumptions Neoclassical production function One sector model of growth Y = C + I (resource constraint), where Y is the aggregate output, C is the aggregate consumption, and I is the aggregate investment the investment I covers the depreciated amount of capital and generates new capital, ie, I = K + δk, where K = dk/dt is the new capital, δ is the depreciation rate, and δk is the depreciated amount of capital Closed economy Closed economy insures that I = S, where S is are the aggregate savings Exogenous savings rate S = I = sy ; s (0, ) Further assumptions Exogenous rate of population growth, L(t) = L(0)e nt L/L = n, where L/L is the growth rate of population (To keep it simple) No technological progress From Y = C + I and I = K + δk it follows that K = Y C δk This is the standard law of motion of capital Further, from Y = C + I and I = sy it follows that C = ( s)y Therefore, the law of motion of capital can be rewritten as K = sy δk The further analysis we will perform in per-capita terms Denote the macroeconomic aggregates in per-capita terms by small letters, k : = K L ; c := C L, y : = Y L = F ( K L, ) = f(k) 7

8 From these definitions it follows that f (k) > 0; f (k) < 0, ( ) K k = d /dt = K L L nk, where f (k) = df(k)/dk; f (k) = d 2 f(k)/dk 2 Therefore, the law of motion of capital in per-capita terms is given by k = sf (k) (n + δ) k Market equilibrium Household (HH) - representative HH offers labor L and holds assets W The marginal unit of labor earns income w and the assets earn returns r Evolution of asset holdings, W : Ẇ = rw + wl C In per-capita terms: ϖ = (r n)ϖ + w c The means of savings is lending to other HH and/ or owning capital There is no optimization problem given that the savings rate is exogenously fixed Firm - representative Perfect competition in input and output markets It chooses labor, L, and capital, K, to maximize its profits within every period: π = F (K, L) RK wl, given the wage, w, and the rental rate, R, for the services of a unit of capital First order conditions (optimal rules) are: [K] : [L] : π K = 0 F K = f (k) = R F (K, L) F (K/L, ) = = f (k) K K/L π L = 0 F L = f(k) kf (k) = w F (K, L) LF (K/L, ) = = f(k) kf (k) L L 8

9 General equilibrium results Capital market clearing condition The net rate of return from a unit of capital for a HH is R δ given its depreciation within a period No arbitrage condition requires that R δ = r or r = f (k) δ and W = K Given that W = K, F K = R, and F L = w the constraint Y = C + K + δk Ẇ = rw + wl C reduces to the resource Steady-state/Balanced growth path Steady-state of the economy - where all variables grow at constant rates (balanced growth path - BGP) Denote From the law of motion of capital Ż Z = g Z K = sy δk follows that g K = K K = s Y K δ Given that s, δ = const it must be the case that the aggregate output Y and the total capital stock K grow at the same rate when the growth rate of total capital stock g K is constant, ie, g K = g Y From the resource constraint C = ( s) Y Therefore, the aggregate consumption and output grow at the same rate, ie, g C = g Y = g K Y K = F (, L ) K, therefore, since Y K = const the ratio L K should be constant This implies that g Y = g K = g C = n or, equivalently, g c = g k = g y = g K n = 0 Neoclassical production function implies Uniqueness of the steady-state, ie, k = 0 sf (k ) = (n + δ) k k = k (s, n, δ) () The steady-state is unique since ϕ (k) := sf (k) (n + δ)k has the following properties: ϕ (0) = 0, lim k 0 ϕ (k) = +, lim k + ϕ (k) = (n + δ) and ϕ (k) < 0 9

10 On a graph the steady-state value of capital is determined from the intersection of sf(k ) curve and (n + δ)k line, ie, From () it follows that k s (n + δ) < 0, ϕ(k,s) s = f (k ) and k s > 0: dϕ (k, s) = ϕ(k,s),s) s ϕ(k,s) = ϕ(k k k > 0 dk + ϕ s ds = 0; ϕ(k,s) k = sf (k ) During the transition the model gradually converges to this unique steady-state The transition can be easily derived from the law of motion of capital written in per-capita terms d k dk = sf (k) (n + δ) In the neighborhood of steady-state, given that f < 0 and s f(k ) ( ) k = (n+δ), d k dk < 0 Therefore, the transition drawn in k, k is given by the following phase diagram For example when Y = K α L α the steady state per-capita capital stock and output are y = f (k) = k α k = ( s n+δ ) α y = ( s n+δ ) α α No growth in per-capita terms in the steady-state (in the absence of exogenous technological progress) Per-capita endogenous growth takes place only due to capital accumulation during the transition to the steady-state 0

11 Convergence g k = s f(k) (n + δ), k (2) g k k = (k)k f(k) sf k 2 < 0 (3) Absolute convergence means that countries with lower capital per worker would exhibit higher growth, irrespective of any other characteristics of the economies Conditional convergence means that each country converges to its own steady-state In other words, there is convergence of capital/ output per capita for a set of countries with similar savings, depreciation and population growth rates Solow model predicts conditional convergence The data do not support "absolute convergence", ie, the data suggest that the poor do not grow faster than the rich and gradually catch up in terms of welfare independent of the specific characteristics of the countries The data provide some support of "conditional convergence", ie the data suggest that every country grows faster the further away it is from its own steady-state The latter is a function of the special characteristics of each country

12 Illustration of conditional convergence From the law of motion of capital written in per-capita terms it follows that k s = (n + δ) f(k ), (4) [ ] f(k)/k g k = (n + δ) f(k )/k Given the diminishing returns to capital, we know that f(k)/k decreases as k increases As a result, as k increases towards its steady-state value k, g k falls to zero (its steady-state value) The model only predicts that we grow faster the further away we are from the steady-state Exogenous technological change and per-capita growth Redefine L in the production function F () in the following way L = A L, where L is the labor and A is the technology it uses for producing the output Therefore, L is the effective labor Let L grow at rate n and A grow at rate g A Denote the aggregates in effective labor units as ỹ = Y A L, k = Y A L, c = Y A L 2

13 Repeating the analysis above for these variables it is straight forward to show that gỹ = g k = g c = g K g A L = 0 Therefore, g y = g k = g c = g K n = g A This implies that there can be per-capita growth in Solow-Swan model in case there is exogenous technological progress, g A > 0 Solow model and Kaldor stylized facts Per capita output grows over time and its growth rate does not tend to decline 2 Physical capital per worker grows over time In this model both and 2 require exogenous labor augmenting technology which grows at constant rate g A, so that g k = g y = g A 3 Rate of return on capital is nearly constant: In this model, in the steady-state r = f (k ) δ 4 The physical capital to output ratio is nearly constant In steady-state this model predicts that g K = g Y Y K = const 5 The shares of capital and labor in national income are nearly constant: r k f(k ) = f (k )k f(k ) w = const; f(k ) = f (k )k f(k ) = const 6 The growth rate of output per worker within a country may exhibit short periods of fast growth: Explained through transition dynamics Across countries the growth rates of output per worker differ: assumption on exogenous technological change Solow model fails given the The (other) weaknesses of Solow model The savings rate is exogenous It cannot explain differences in y in terms of s or n It cannot explain differences in y in terms of differences in k since the income levels differ more than capital levels Differences in y need to be explained by differences in (labor augmenting) technology used which is exogenous 3

14 Endogenously generated per-capita growth rates can differ only along the transition path Persistent differences cannot be explained It can only explain short episodes of growth Solution: Different production function, ie, drop the decreasing returns to scale and/or the Inada conditions Golden rule level of capital and inter-generational issues In Solow-Swan model the golden rule level of capital is the amount of capital that maximizes consumption in the steady state In order to find it use the steady state conditions and write c = ( s) f (k ) = f (k ) (n + δ) k k G f (kg) = n + δ; s G = (n + δ) f(kg ) If k < kg k since s > 0 the saving rate should be increased in order to arrive at k G The dynamics of the system will be the following in such a case If k > kg k since s > 0 the saving rate should be decreased in order to arrive at k G The dynamics of the system will be the following in such case 4

15 In case we have overlapping generations increasing the savings rate s can imply lower consumption for the current generation and higher consumption for upcoming generations Additional: Harrod (939)-Domar (946) model Secular stagnation was very popular about a century ago (it seems to get some attention now too) (Harrod, 939; Domar, 946) present a model and offer a related discussion Suppose that production function takes Leontief form Y = F (K, L) = min (AK, BL), where A, B > 0 (Notice that this is the limiting case of constant elasticity of substitution production function were elasticity of substitution is 0, ie, inputs are complements) function implies that F (K, L) = { AK if K B A L, BL if K > B A L Therefore, if K B A L some part of labor force is idle and there is unemployment circumstance the unemployment rate is L A B K L Such a production In turn, when K > B A L some part of the capital is idle In percentage terms, idle capital is K B A L K Consider now production function in per-capita terms It is given by f (k) = min (Ak, B) In such a Therefore, f (k) = { Ak if k B A, B if k > B A This following figure offers a plot of this function 5

16 Notice also that the ratio of output and capital have the following form f (k) k = { A if k B A, B k if k > B A Therefore, the fundamental equation of Solow-Swan model is { k k = sa (n + δ) if k B A, s B k (n + δ) if k > B A This following figure plots this relationship for sa < n + δ and sa > n + δ Notice that the rate of growth of capital is negative when sa < n + δ in both cases () k B A and (2) k > B A since s B k < sa Therefore, in such a case over time per capita capital, output, and consumption decline to 0 Moreover, the steady-state features permanently growing pool of unemployed people In turn, when sa > n + δ there exists steady-state level of per-capita capital k, k = sb n + δ 6

17 Apparently, k > B A The economy converges to this level of capital if it starts above or below it At the steady-state there is a growing pool of idle capital given by K B A L K = k B A k The only way to have a full employment of capital and labor in this model AK = BL is for the parameter values sa = n + δ, which is unlikely to hold There are couple of problems, however, with this model First, the average product of capital K in this model would usually depend on K and adjust to satisfy equality s f (k) k = n + δ in the steady-state as in Solow-Swan model Second, the rate of savings could adjust to satisfy this condition In particular, when agents maximize their discounted life-time utility selecting the amount of their savings they would not save (at constant rate) when the marginal product of capital is zero Additional: Poverty traps One popular notion in development economics concerns poverty traps Definition An economy is in a poverty trap if there are multiple stable equilibria and the economy appears to be in an equilibrium which does not deliver the highest level of income (wealth) In order to generate a poverty trap in a model economy consider the following setup Let an economy have an access to two types of technologies: primitive (A) and modern (B) Primitive and modern goods are produced according to Y A = AK 2 L 2, Y B = BK 2 L 2, where A < B In order to use this better technology, the country has to pay managerial cost in each and every point in time which is proportional to labor force bl, where b > 0 This cost needs to be financed by the government which levies a tax on households to finance its expenditures In per-capita terms, net of these costs, production functions are given by y A = Ak 2, y B = Bk 2 b, and are drawn together below 7

18 If the government pays managerial costs than the firms use the modern technology Meanwhile, they use the primitive technology if the government does not pay managerial costs It is sensible ( ) to pay the costs if net benefits are positive, ie, y B b > y A or k > b 2 ( ) 2 B A Denote k = b B A and notice that at this point y B = y A Assume that the government pays the costs if k > k and does not do so for lower values of k The economy is still governed by the fundamental equation of the Solow-Swan model with a slight modification k = { since at k it has to be that y B = y A sak 2 (n + δ) k if k k, sbk 2 b (n + δ) k if k > k, Consider the case when there exist k A and k B such that sak 2 (n + δ) k and sbk 2 b (n + δ) k are zero There are then 3 steady-states in this economy k A = k B; = k B;2 = ( ) sa 2, n + δ sb (sb) 2 4 (n + δ) b 2 (n + δ) sb + (sb) 2 4 (n + δ) b 2 (n + δ) 2 2, It can be easily shown that kb; is not stable in the sense that it is not an attractor Whereas, k A and kb;2 are stable and k A is a poverty trap Some proofs and additional results That g k k < 0 in (3) follows from f (x) < 0 Intuitively, if f < 0 then f function grows at a lower rate at each point than a linear function that is tangent to it at that point To prove it use f (tx + ( t) x 2 ) > tf (x ) + ( t) f (x 2 ) 8

19 which holds for t [0, ] Rearranging one gets f (tx + ( t) x 2 ) f (x 2 ) t > f (x ) f (x 2 ) The limit of the left-hand side when t 0 is equal to f (x 2 + t (x x 2 )) f (x 2 ) (x x 2 ) lim = (x x 2 ) f (x 2 ) t 0 t (x x 2 ) Therefore, (x x 2 ) f (x 2 ) > f (x ) f (x 2 ) Now substitute x = 0 and x 2 = k kf (k) < f (k) Meanwhile, in case of Cobb-Douglas production function it is possible to obtain the closed form solution for k (t) From (2) or the fundamental equation Denote v = k α then The solution of this differential equation is v (t) = v + (n + δ)v = s α s + v (0) exp [ ( α) (n + δ) t] n + δ Therefore, k α (t) = s n + δ + k α (0) exp [ ( α) (n + δ) t] 9

20 Continuous time optimal control - the basics and applications The general continuous time constrained optimal control problem can be written in the following form max {u t x t} t t=t0 F = t t 0 f (t, x t, u t ) dt st dx t dt =: ẋ t = g (t, x t, u t ) (6) x (t 0 ) = x 0, x (t ) is free (7) For simplicity in the reminder of the text assume that f and g are continuously differentiable functions of time and u is piecewise continuous function of time (5) The constraint (6) is the law of motion of the state variable x t which is predetermined at the beginning of each "period" (eg, capital) Meanwhile u t is the control variable (eg, the amount of consumption) which, given the value of x t and its law of motion (6), we choose in order to maximize F (eg, the lifetime utility of household) The solution of this problem is the optimal path of state and control variables, (x t, u t ) This path should be feasible In other words, it should satisfy the dynamic constraint (6), ie, the law of motion, and the initial condition (7) Moreover, given the definition of control and state variables u t = u t (x t ) The values of x (t ) and u (t ) satisfy maximization problem (ie, these values are a choice) Digression: When we consider a household s intertemporal problem we - usually - have f (t, x t, u t ) = f (u t (x t )) exp ( ρt), where e ρt is the discounting function and ρ is the discount rate, f is the instantaneous (one period) utility from consumption c t ( u t ) - erroneously we use the letter u for f The optimal consumption c t, in turn, is function of capital k t ( x t ) Meanwhile, the constraint (6) represents the accumulation rule of assets/capital - in such case we basically solve the optimal consumption and saving paths, where the latter determines the optimal path of capital The rigorous approach to solving the problem is through Lagrangian Let q t be the Lagrange multiplier of the constraint (6) The optimal problem written in terms of Lagrangian is the following max {u t x t} t s=t0 L = Integrate the last term by parts t t 0 t t 0 q t ẋ t dt = {f (t, x t, u t ) + q t [g (t, x t, u t ) ẋ t ]} dt t t 0 q t dx t = q t x t + q t0 x t t t 0 x t q t dt

21 and rewrite the L t max {u t x t} t s=t0 L = [f (t, x t, u t ) + q t g (t, x t, u t ) + q t x t ] dt q (t ) x (t ) + q (t 0 ) x (t 0 ) t 0 Necessary conditions Let u t be the optimal control function Construct a family of "comparison" controls u t +αh t, where h t is some function and α is a real number Denote y (t, α) the path of the state variable generated by the control u t + αh t Assume that y (t, α) is differentiable in arguments and y (t 0, α) = x (t 0 ) for any α [ie, the optimal path x t and y (t, α) start from the same point] Notice that y (t, 0) = x t With this comparison controls the value of the Lagrangian L is L (α) = t [f (t, y (t, α), u t + αh t ) + q t g (t, y (t, α), u t + αh t ) + q t y (t, α)] dt t 0 q (t ) y (t, α) + q (t 0 ) x 0 Further, for simplicity let L (α) have one and interior maximum and let it be differentiable Consider the following first order condition with slight abuse of previous notation 0 = dl (α) dα L α (0) = t t 0 q (t ) y α (t, 0) := L α (0) (8) α=0 ( f x y α + q t g xy α + q t y α + f uh t + q t g uh t ) dt Apparently, the exact value of the RHS of this expression depends on q t It depends also on h t and the way h t influences the path of the state variable y (t, α) Meanwhile, the condition (8) should hold for any h t (thus any y α) Therefore, one should select q t (and q t ) so that it eliminates the influence of h t - note that we are basically deriving the envelope condition which states that the gradient of the maximand at the optimal point is orthogonal Select q t = [ f x (t, x, u ) + q t g x (t, x, u ) ] (9) q (t ) = 0 (0) Under such a choice, 0 = L (0) () = t [ f u (t, x, u ) + q t g u (t, x, u ) ] h t dt, t 0 2

22 which should hold for any h t Therefore, it should hold also for h t = f u (t, x, u ) + q t g u (t, x, u ), which means that t [ f u (t, x, u ) + q t g u (t, x, u ) ] 2 dt = 0 (2) t 0 This in turn implies that f u (t, x, u ) + q t g u (t, x, u ) = 0 (3) The equations (9), (0), and (3) are the necessary conditions for optimality Together with (6) and (7) they determine the optimal path of control and state variables (x t, u t ) A simple way for deriving the necessary conditions Form a Hamiltonian H (t, x t, u t, q t ) f (t, x t, u t ) + q t g (t, x t, u t ), where q t is the costate variable and is part of the solution to the optimal problem The necessary conditions are obtained as: H u H x H q = 0, (4) = q, (5) = ẋ (6) Notice that (4) is the same as (3), (5) is the same as (9), and (6) is (6) In addition, one gets an obvious condition x (t 0 ) = x 0 and q (t ) = 0 The latter plays the role of transversality condition (TVC) in terms of finite time problem Digression: The TVC requires that in a dynamically optimal path the choices are made in a way that ensures that at the end of the time horizon the state variable (eg, capital) has no value and therefore the constraint is not binding In terms of economics, one wants the value of capital in terms of utility to be zero at the planning horizon If its value is positive then at the end of the time the choice leaves a positive value of capital that gives no utility, which is against the optimality In terms of economics, the costate variable measures the shadow value of the associated state variable Hence, it captures the gains (value) in the optimal control problem that stem from marginally increasing the state variable Suffi cient conditions In order the necessary conditions to be also suffi cient we need further conditions the functions f and g are concave in both arguments the optimal trajectories of x, u, and q satisfy the necessary conditions 22

23 x t and q t are continuous functions with q t 0 for all t and if g is nonlinear in x or u, or both In order to prove the suffi ciency define f := f (t, x, u ) and g := g (t, x, u ) and t D := (f f) dt t 0 Given that we are solving for a maximum we need to show that D 0 Since f is concave f f f x (x x) + f u (u u) Therefore, D = t t 0 t t 0 [ f x (x x) + f u (u u) ] dt (7) [(x x) ( qg x q) + (u u) ( qg u)] dt Notice that t t 0 q (x x) dt = = t t 0 t t 0 (x x) dq = (x x) q t t0 + (g g) qdt t t 0 (g g) qdt since x (t 0 ) = x (t 0 ) and q (t ) = 0 Therefore, (7) can be written as t D [(g g) gx (x x) gu (u u)] qdt 0 t 0 The latter integral is greater or equal to zero since q 0 and g is a concave function of x and u This shows that the necessary conditions together with concavity of f and g and non-negativity of q are also suffi cient conditions 23

24 Infinite horizon discounted problem A usual economic problem is written as max {u t x t} + t=0 st U = + 0 f (x t, u t ) exp ( ρt) dt (8) ẋ t = g (t, x t, u t ) (9) x (0) = x 0 > 0 (20) Notice that while f - the instantaneous utility - is at time t the costate involves the value of changing the state from x t incrementally over time, ie, to t + dt Thus the costate (and the Hamiltonian) has to take this into account The present value Hamiltonian (discount factor = e ρt ) is H P = f (x t, u t ) e ρt + q P t g (t, x t, u t ) While, the current value Hamiltonian (discount factor = ) is H C = e ρt H P = f (x t, u t ) + q C t g (t, x t, u t ), q P t = q C t exp ( ρt) The necessary conditions for optimality for present value Hamiltonian are Hu P = f u (x t, u t ) e ρt + qt P g u (t, x t, u t ) = 0, [ ] q t P = Hx P = fx (x t, u t ) e ρt + qt P g x (t, x t, u t ), lim t + qp t x t = 0 For current value Hamiltonian using the the definition of q C t ( q P t = q C t e ρt ρq C t e ρt ) the necessary conditions are Hu C = f u (x t, u t ) + qt C g u (t, x t, u t ) = 0, [ ] q t C = ρqt C Hx C = ρqt C fx (x t, u t ) + qt C g x (t, x t, u t ), lim t + qc t x t exp ( ρt) = 0 The last conditions are the TVCs for infinite horizon optimal problem They states that the value of state variable in terms of utility should be zero in the limit when t approaches + Many states and controls There could be many state and control variables - the numbers do not need to coincide more than one state simply one adds extra costate variables (multiplying the RHS of the dynamic 24 For

25 constraints) to the Hamiltonian condition for each control variable For more than one control, one needs to derive one optimal Continuous time Bellman equation (Hamilton-Jacobi-Bellman equation) This section is for those who are familiar with recursive dynamic programming in discrete time It illustrates the analogy between continuous time necessary conditions and the conditions derived for discrete time Here I consider only the discounted problem, though all the logic can be applied for the more general case With a slight abuse of notation define the maximized value of the objective function as a function of the initial state x t and initial time t [it s suffi cient since u t = u (x t )] V (t, x t ) = max {u s:ẋ s=g(x s,u s) x s} + s=t + This can be rewritten in recursive form in the following way: for any t V (t, x t ) = max {u s:ẋ s=g(x s,u s) x s} t+ t s=t t+ t Subtract from both sides V (t, x t ) and divide by t 0 = max t t f (x s, u s ) exp [ ρ (s t)] ds f (x s, u s ) exp [ ρ (s t)] ds + V (t + t, x t+ t ) exp ( ρ t), {u s:ẋ s=g(x s,u s) x s} t+ t s=t t t+ t + V (t + t, x t+ t) exp ( ρ t) V (t, x t ) t t f (x s, u s ) exp [ ρ (s t)] ds Take the limit t 0 (ie, continuous time) To evaluate the first term in cuerly barckets use the L Hopital s rule: } lim t 0 = lim t 0 t+ t t t (t+ t) t f (x s, u s ) exp [ ρ (s t)] ds f (x t+ t, u t+ t ) exp ( ρ t) t t = f (x t, u t ) 25

26 Meanwhile, apply the definition of differential in order to get V (t + t, x t+ t ) exp ( ρ t) V (t, x t ) lim t 0 t { [exp ( ρ t) ] V (t + t, xt+ t ) = lim + V (t + t, x t+ t) V (t, x t+ t ) t 0 t t + V (t, x } t+ t) V (t, x t ) t = ρv (t, x t ) + V (t, x t ) + V x (t, x t ) ẋ t In sum this means that { ρv (t, x t ) = max f (xt, u t ) + V x (t, x t ) g (x t, u t ) + V } (t, x t ), (2) u t x t which is the Hamilton-Jacobi-Bellman equation The second term in RHS captures the value gains from marginal change in the state variable, while the third term stands for the gains over time The maximization gives the FOC: f u + V xg u = 0, which is the necessary condition for optimality, H C u = 0, where V x = q C t This shows how the costate captures the effect of the change of the state on the objective function in current value terms It also shows that qt C depends on dynamic decisions The envelope condition is ρv x = f x + V xg x + V xxg + V x This is the necessary condition which describes the dynamics of the costate variable q C t ẋ t = g (x t, u t ) given that 26

27 The Ramsey-Cass-Koopmans model - (Ramsey, 928; Cass, 965; Koopmans, 965) The Solow-Swan model assumes exogenous and constant savings rate, when the savings are the source of capital accumulation and are a decision variable for the savers (households) The Ramsey- Cass-Koopmans (in short Ramsey model) model endogenizes the savings rate We will see that in the steady-state the saving rate in the Ramsey model is constant, similar to Solow-Swan model Therefore, basically we will simply re-examine the results of the Solow-Swan model, while relaxing the assumption of exogeneity of the savings Main assumptions Neoclassical production function: Y = F (K, AL), where A is labor augmenting technology One-sector model of growth ie, both capital and consumption goods are produced with the same technology From the consumption-side A continuum of infinitely lived and identical households (HHs) of mass L The representative household (HH) is endowed with a unit of labor and chooses its consumption c, labor supply (and the evolution of assets ϖ) to maximize the lifetime utility U, where U = + 0 u(c t )L exp ( ρt) dt u(c) is the instantaneous utility from consumption of amount c of final good in per-capita terms The instantaneous utility function is increasing and concave in c [ie, u > 0, u < 0] and satisfies the Inada conditions [ie, limu (c) = +, c 0 lim c + u (c) = 0] The concavity implies that HH prefers to smooth consumption over time The pure rate of time preference is ρ > 0 The budget constraint of HH is ϖ = (r n)ϖ + w c Since the utility here should be percieved in cardinal sense, the HH maximizes simply its utility multiplied by the size of the representative HH Further assumptions Technology grows at exogenous rate Ȧ A = g A, A (0) > 0 - given Population grows at exogenous rate L L = n, L (0) > 0 - given 27

28 Market equilibrium The firm side is similar to Solow-Swan model Formally, setting the final goods as numeraire the representative firm s optimization problem is π = max {F (K, AL) RK wl} K,L Therefore, the first order conditions (optimal rules) are [K] : [L] : π K = 0 F K = R, (22) π L = 0 F L = w (23) The representative HH chooses consumption path to maximize its lifetime utility Its means of savings is accumulation of capital Formally, the HH s problem is max c st + 0 u(c t ) exp [ (ρ n) t] dt, ϖ = (r n)ϖ + w c, ϖ(0) > 0 - given If written in terms of current value Hamiltonian the HH s problem is max {u(c) + q [(r n)ϖ + w c]}, c st ϖ(0) > 0 - given, where the q is the shadow price of a unit of assets Therefore, the optimal rules are Denote H = u(c) + q [(r n)ϖ + w c] [c] : H c = 0 u (c) = q, (24) [ϖ] : q = q (ρ n) H = q (ρ r), (25) [T V C] : lim τ + From the first optimal rule it follows that ϖ ϖ(τ)q(τ) exp [ (ρ n) τ] = 0 q = ċu (c) Therefore, r ρ = q q = du /dt u cu (c) = ċ c u (c), 28

29 or the optimal consumption path is ċ c = u (c) u (r ρ) (c)c The TVC states that the value of the current asset holdings in infinity is zero Formally, this is part of the open boundary problem given by the maximization of H Note the following ċ c > 0 if r ρ > 0 The sensitivity of the growth of consumption to r ρ is higher, the lower is u (c) u (c)c, which is the intertemporal elasticity of substitution This elasticity is a measure of the responsiveness of consumption to changes in the marginal utility, ie, it measures the willingness to deviate from consumption smoothing In a special case of constant intertemporal elasticity of substitution (CIES) utility function the growth rate of consumption is given by u(c) = c θ θ ; θ > 0 ċ c = (r ρ) θ since u (c) u (c)c = c θ θc θ = θ The CIES assumption is preserved in what follows The equilibrium in the asset market delivers again R = r + δ, W = K This gives the law of motion for capital K = Y C δk, given that Y = F (K, AL) = RK + wl The last equation is implied by the homogeneity of degree one assumption and states that the final good producer earns zero profit (note that there is perfect competition in final good market) Balanced growth path All variables of the model need to grow at constant rates (BGP) On a BGP g c = ċ c = const We have CIES utility function and ρ = const, therefore, g c = (r ρ) θ 29

30 On the BGP, therefore the interest rate should be constant, r = const In our setup, constant interest rate then will imply that savings rate is constant The intuition behind is that on the BGP there should be no shifts in the shares of aggregates (notice that C + S = Y ) Use the constant returns to scale assumption and write r + δ = F (K, AL) K = F ( K AL, ) ( ) K K = f (26) AL AL Given that f < 0 the ratio K AL should be constant on the BGP in order to have r = const Given that the ratio K AL that is constant from the constant returns to scale assumption it follows ( Y K = F, AL ) = const K Given that Y K it follows that the ratio C K is constant on the balanced growth path from the law of motion of capital also should be constant, C K = Y K δ g K Therefore, g K = g Y = g C Moreover, from Y AL K = F (, K ) it follows that on the balanced growth path g K = g Y = g C = g L + g A = n + g A In order to derive the steady-state and to characterize the transition dynamics, redefine the model in units of effective labor, ie, AL Let ỹ := Y AL, k := K AL, and c := C AL Also, for Y = AL F ( K AL, ) := AL f( k) From these definitions it follows that R = F K = f ( k), w/a = F L /A = f( k) f ( k) k In the steady-state gỹ = g k = g c = 0 The steady-state and the transition dynamics of the model can be summarized by the following system of equations c c = [ ( k ) ] f δ ρ θg A, (27) θ ( k) k f k = c k (δ + n + g A ) (28) k k (0) > 0 given lim k (τ) q (τ) exp [ (ρ n g A ) τ] = 0 (29) τ + The first equation follows from the optimal path of consumption given that c c = ċ c g A, 30

31 and ( k ) r = f δ The second equation follows from the law of motion of capital given that k k = k k g A = K K (n + g A) k In the steady-state k c c = = 0 we can solve for the steady-state values of c and k from (27) and (28) Let F (K, AL) = K α (AL) α f ( k ) = δ + ρ + θg A ( ) k α α = f ( k ) ( α = δ + ρ + θg A δ + ρ + θg A c = f ( k ) (δ + n + g A ) k ( ) α c α [ α = α (δ + n + g A) δ + ρ + θg A δ + ρ + θg A ) α α, (30) ] (3) Transition dynamics ) The transition dynamics of the model in ( k, c space is characterized by the Jacobian of the system of equations (28) and (27) evaluated in the neighborhood of the steady-state J = = k k c k k c c c ) ( k f (δ + n + g A ) θ f ( k) c θ ] [f ( k) δ ρ θg A Notice that at the steady-state c = 0, therefore, J SS = ( f ( kss ) (δ + n + g A ) θ f ( k SS ) c SS 0 ) = θ f ( k SS ) c SS < 0 Since det J = µ µ 2, where µ,2 are the eigenvalues of the matrix J, we have that µ and µ 2 have different signs This means that we have saddle path with one stable arm and one unstable arm The stable arm corresponds to negative eigenvalue, while the unstable arm corresponds to the positive eigenvalue 2 The phase diagram of the system is as follows 2 For 2x2 matrices eigenvalues µ,2 can be found from det M = µ µ 2 and trace(m) = µ + µ 2 3

32 Starting at any level of capital k (0) > 0 the economy makes a discrete jump to the saddle-path k which passes through the intersection of k c c = 0 and = 0 and moves along that path toward the steady-state (This saddle-path is illustrated in the above figure) In other words, given k (0) > 0 consumers select c (0) so that the economy is on the saddle-path Any other c (0) violates either Euler equation or the TVC If c (0) is higher, then the economy transits toward k = 0 and hits the vertical axis in finite time At the point where k = 0 consumption c jumps to 0 which violates Euler equation (Notice also that at that point if θ the instantaneous utility u is ) If c (0) is lower, then the economy transits toward c = 0 and hits the horizontal axis in finite time In order to see what happens during this transition rewrite the transversality condition (29) using (25) and (26) in the following manner { τ q (τ) = q (0) exp lim k (τ) exp τ + { 0 τ 0 [ ( k )] } ρ + δ f dv [ ( k ) ] f δ n g A dv } = 0 ) At the steady-state in order TVC to hold it has to be the case that f ( k (δ + n + g A ) > 0 At the ) same time at the steady-state f ( k) (δ + ρ + θg A ) = 0 Therefore, k that satisfies f ( k = δ + n + g A (denote it by k ) is higher than steady-state level of k since f < 0 At k (for fixed k) consumption is c = k [ ] f( k ) (δ + n + g k A ) > 0 Moreover, at k consumption declines and k increases However, as k ) increases f declines therefore f ( k (δ + n + g A ) declines below zero and TVC does not hold The behavior of the savings rate The savings rate is given by s = f( k) c = c It endogenously changes along the transition f( k) f( k) path It can increase or decrease during transition During the transition there are two forces that 32

33 affect the saving rate: income effect and substitution effect If the the economy starts with low level of capital then as capital increases the interest rate declines The inter-temporal substitution effect then reduces willingness to save However, higher income tends to increase willingness to save Consider the simple case of Cobb-Douglas production function, ỹ = k α In such a case in the steady-state the saving rate can be pinned down from the steady-state values of capital and consumption per (effective) labor, (28) and (27), s = ( k c ) α = α (δ + n + g A) δ + ρ + θg A Note that the TVC condition (29) implies that ρ r (ρ n g A ) < 0 Therefore, at the steady-state from (28) it follows that n + ( θ) g A < ρ This parameter restriction allows to have bounded utility and well-defined optimal problem implies that s < In order to assess the behavior of the savings rate on the transition path consider the average propensity to consume x = c ỹ = s and assume that The growth rate of x is x x = c c ỹ ỹ = c c α Meanwhile, the transition dynamics of x can be described by the expressions (27) and (28), x x = k k = { [ ] α k α (δ + ρ + θg A ) θ [ α (δ + n + g A ) θ (δ + ρ + θg A) k k [ α ( x) k α (δ + n + g A )] }, ] ( θ x ) α k α (32) = ( x) k α (δ + n + g A ), (33) The locus of the first equation is then x = ψ = ( ) k α + ψ θ α, [ ] θ (δ + ρ + θg A) α (δ + n + g A ) Depending on the sign of ψ it is either increasing or decreasing with k If ψ = 0 then x = ( θ ) It 33

34 Setting k = 0 the locus of the second equation is x = (δ + n + g A ) k α, which is decreasing in k The Jacobian of the system is then ( α k α ( J = θ x) α ( α) k α ( x) ( α) k α 2 k α 2 ) The determinant of J is [ det J = α k α ( x) ( α) k α 2] + ( θ ) x α ( α) k α 2 kα = θ α ( α) k 2α 3 < 0 This means that x and the saving rate ( x) are saddle-path stable since the eigenvalues have alternating signs Therefore, depending on ψ, x and x either increase or decrease to their steady-state levels In the special case when ψ = 0 x = θ The constant savings rate assumed in the Solow-Swan model is a special case of the Ramsey model, which is true under these specific parameter values Note that when θ = (logarithmic utility) the inter-temporal substitution and income effects from changes in interest rate exactly cancel each other When θ < consumers tolerate large deviations from smooth consumption profile and substitution effect is dominant In particular the propensity to consume declines with interest rate Whereas when θ > consumers do not tolerate large deviations from smooth consumption profile and income effect is dominant In such a case propensity to consume increases with interest rate Evaluation of the model Predictions of the Solow-Swan model remain under this more general framework With exogenous technological change, the model performs well in accounting for facts -5 However, it cannot explain the facts and 2 Its explanatory power is not improved for fact 6 34

35 Extensions Government in the Ramsey model Assume that there is a government in the economy described by the Ramsey model The government consumes amount G of final goods For now, suppose that these purchases have neither utility nor productive effects (We will allow later government purchases to positively affect firms output) The government also makes amount V lump-sum transfers to households The government runs a balanced budget through taxes that it collects The taxes are proportional and time-invariant levies on wage inome, τ w, asset income, τ a, consumption, τ c, and firms earnings ( π), τ f Therefore, the government s budget constraint is G + V = τ w wl + τ a rw + τ c C + τ f π These taxes modify the representative household s budget constraint It is now ϖ = [( τ a ) r n] ϖ + ( τ w ) w ( + τ c ) c + v Assuming CIES utility function, this implies that the household s written in terms of current value Hamiltonian is max H = c θ + q {[( τ a ) r n] ϖ + ( τ w ) w ( + τ c ) c + v}, c θ st ϖ(0) > 0 - given, Therefore, the optimal rules are [c] : c θ = ( + τ c ) q, [ϖ] : q = q [( τ a ) r ρ], [T V C] : lim ϖ(τ)q(τ) exp [ (ρ n) t] = 0 τ + From the first two conditions it follows that now familiar Euler equation is ċ c = θ [( τ a) r ρ] Notice that τ c does not affect consumption path, though τ a does so This happens because τ c is time-invariant and reduces equally both present and future consumption Meanwhile, although τ a is also time invariant it affects capital accumulation and, therefore, income and consumption in the future Suppose that the firms taxable earnings are equal to output less wage payments and depreciation π = F (K, L) wl δk 35

36 Therefore, firms profits after tax are notice that R = r + δ This implies that firms optimal rules are π = ( τ f ) [F (K, L) wl δk] rk [K] : [ ] F (K, L) r = ( τ f ) δ, K [L] : F (K, L) w =, L which imply that firms make zero profits These conditions in per capita terms are r = ( τ f ) [ f (k) δ ], w = f (k) kf (k) From the budget constraints of the government and household it follows that ϖ = (r n) ϖ + w c + τ f [f (k) w δk] ĝ In equilibrium assets and capital holdings are the same since there is no debt Therefore, using the conditions above and Euler equation we have k = f (k) (n + δ) k c ĝ, ċ c = { ( τ a ) ( τ f ) [ f (k) δ ] ρ } θ Notice that τ w does not show up in these modified equilibrium rules This happens because the household supplies its labor inelastically Assume now that ĝ = τ a = τ f = 0 Clearly, in such a case this system of differential equations is equivalent to (27), (28), and (29) with g A = 0 and phase diagram of the system is the one offered above Consider an unanticipated increase in ĝ keeping τ a = τ f = 0 This increase is financed with a combination of τ c and τ w In such a case k shifts downward, which implies a new and lower level of steady-state capital and consumption After this unanticipated increase in ĝ the model economy following TVC makes a discrete jump to the stable-arm and converges gradually to the new steady-state The phase diagram of this transition is offered below (dashed lines are for the system where ĝ = 0!) 36

37 In turn, consider an unanticipated increase in any of the tax rates τ a or τ f Such an increase shifts ċ c to the right Therefore, it implies lower steady-state level of capital and consumption Similarly, after this unanticipated increase in a tax rate the model economy following TVC makes a discrete jump to the stable-arm and converges gradually to the new steady-state diagram of this transition is offered below (solid lines are for the system where τ a = τ f = 0!) The phase Productive public expenditures in the Ramsey model Consider the set up offered above However, suppose instead that government purchases are dedicated for creating productive public services for each firm The production function of the firms takes a form of Y = F ( K, L; G ), Y G where G is total public expenditure, Y is public services per output of a firm, F ( K, L; G Y homogenous of degree one in K and L, and proportional to ( ) G γ Y where γ (0, ), ie, ( ) G γ Y = F (K, L), Y 37 ) is

38 Public services are productive means that holds since γ > 0 Y (G/Y ) > 0, 2 Y K (G/Y ) > 0, and 2 Y L (G/Y ) > 0, which Suppose, further that () firms take G Y as a parameter, (2) assets are capital K, (3) depreciation is paid by the household, (4) public expenditure G is financed through a proportional and timeinvariant tax on households gross income τ = τ a = τ w, τ (0, ), and (5) transfers are V = 0 G = τ (wl + RK), This implies that the budget constraint of the households is given by K = ( τ) (RK + wl) δk C Therefore, the representative household s budget constraint is k = ( τ) (Rk + w) (n + δ) k c Assuming CIES utility function, this implies that the household s written in terms of current value Hamiltonian is max H = c θ + q [( τ) (Rk + w) (n + δ) k c], c θ st k(0) > 0 - given, Therefore, the optimal rules are [c] : c θ = q, [ϖ] : q = q [( τ) R δ ρ], [T V C] : lim ϖ(τ)q(τ) exp [ (ρ n) t] = 0 τ + From the first two conditions it follows that now familiar Euler equation is ċ c = [( τ) R δ ρ] θ In turn, firms optimal rules are [K] : R = F ( K, L; G ) Y, K [L] : w = F ( K, L; G ) Y, L 38

39 which implies that G = τy Given that G = τy these conditions in per capita terms are This implies that R = f (k) τ γ, w = f (k) τ γ kf (k) τ γ k = ( τ) τ γ f (k) (n + δ) k c, ċ c = [ ( τ) τ γ f (k) δ ρ ] θ Suppose that the government sets tax rate τ in order to maximize ċ c Given that γ (0, ) function φ (τ) = ( τ) τ γ is strictly concave in τ, ie, there is a laffer curve Therefore, the maximum of ċ c is attained at τ = γ γ + It interesting alsowhat is the optimal tax rate for the Social Planner In order to derive the optimal tax rate solve the Social Planner s problem: max H = c θ + q [( τ) τ γ f (k) (n + δ) k c], c,τ θ st k(0) > 0 - given, The resulting optimal rules for consumption and savings are [c] : c θ = q, [k] : q = q (ρ n) [ ( τ) τ γ f (k) (n + δ) ], which imply that consumption and capital follow ċ c = [ ( τ) τ γ f (k) δ ρ ], θ k = ( τ) τ γ f (k) (n + δ) k c This system of equations is apparently the same as the system of equations that governs the behaviour of the economy in decentralized equilibrium Maximizing H with respect to τ, in turn, yields Therefore, with τ = γ γ+ τ = γ γ + the decentralized equilibrium yields the first-best outcome 39

40 The AK model - Spillovers à la Romer (986) We started from the standard Solow-Swan growth model This model, as well as the Ramsey model, has neoclassical production function for the final good With neoclassical production function these models cannot endogenously generate long run growth since the returns on capital decline with accumulation of capital The model presented below assumes a neoclassical production function - at the "individual level" In addition, it assumes that the labor augmenting technology is a function of average percapita capital stock While doing so, it has in mind some "learning by doing" effects/spillovers as in Arrow (962), ie, the workers learn/become more productive while working with desks, computers, etc Main assumptions The level of technology/ effi ciency that augments the labor input in the production is a function of the average capital-labor ratio in the economy The motivation for this is that investment of a firm brings productivity gains from its use of the labor The firm builds up the knowledge (technical expertise) of how to effi ciently use the capital by accumulating it, ie, there is "learning-by-doing" (eg, production lines) This learning-by-doing effect has an aggregate impact, when any individual firm s technical effi ciency is public knowledge, so that all firms can benefit from the technological advance for the use of capital in the production This gives the link between A i (ie, some ith firm s effi ciency) and the capital-labor ratio in the economy The level of technology is assumed to be A λk, where k is the average of per-capita capital stock and λ > 0 measures the effi ciency of use of the capital There are continuum of (ex ante) identical firms of mass one The production ( ) function in a firm takes the form: Y i = F (K i, AL i ) = L i F (k i, λk) L i λkf ki λk Thus, there are decreasing returns to capital at the firm-level since the firm is so small that it does not take into account its impact on A However, there are constant returns to capital in symmetric equilibrium since k i = k One-sector model of growth: K = Y C δk From the consumption-side, the representative household (HH) chooses its consumption and next period assets to maximize its lifetime utility U = + 0 u (c) exp [ (ρ n) t] dt, subject to its budget constraint: ϖ = (r n)ϖ + w c, where u (c) = c θ θ Further assumptions Population grows at exogenous rate n 40