Neoclassical Growth Theory
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1 Chapter 2 Neoclassical Growth Theory 1.1 Solow-Swan model 1.2 Ramsey-Cass-Koopmans model 1.3 Methods: Dynamic Optimization 1 ST2011 Growth and Natural Resources
2 Neoclassical Growth Theory In contrast to endogenous growth models: long-run growth driven by exogenous technological progress Basic models: no technological progress Long-run equilibrium: zero growth, i.e. constant per capita income Transitional growth only Models with exogenous technological change: long-run growth feasible Assumptions regarding households behavior: exogenous savings rate: Solow-Swan model endogenous savings rate: Ramsey-Cass-Koopmans model 2 ST2011 Growth and Natural Resources
3 Assumptions regarding production technology: neoclassical production function with capital and labor as inputs,,, y with L = labor Y = output (y = Y/L output per capita) K = capital (k = K/L capital per capita) f(k) k Properties: constant returns to scale: (, ) positive decreasing marginal product of capital ( 0; 0) Inada conditions (lim,lim 0) Note: From now on, time indices will be dropped for simplicity whenever unambiguous. 3 ST2011 Growth and Natural Resources
4 Population growth rate: = constant Depreciation rate: = constant Capital stock dynamics (equation of motion of capital stock) aggregate: per capita: where I = investment (= saving Simplifying assumption: no technological progress 4 ST2011 Growth and Natural Resources
5 1.1 Solow-Swan Model Savings and consumption (per capita): 1 1 with s = exogenous savings rate c = consumption per capita Capital stock dynamics: 5 ST2011 Growth and Natural Resources
6 Graphical Representation of the Solow-Swan Model Assumption: initial capital stock Long-run equilibrium: k*, y* y f(k) y* c0=(1 s)y0 sf(k) y0 sy0 k0 k* k 6 ST2011 Growth and Natural Resources
7 Long-run Equilibrium General definition: in a long-run equilibrium (steady state) all variables grow at constant (possibly zero) rates (also: balanced growth path, BGP) Per capita: steady state with zero growth 0, (savings compensate for capital depreciation and population growth) Aggregate level: variables growth at rate of population growth 7 ST2011 Growth and Natural Resources
8 Comparative Statics (Exogenous) changes of savings rate, deprecation rate and rate of population growth change the equilibrium value of y and k, During the transition to the new steady state: transitionary growth BUT: increase in savings rate does not induce long-run growth Example: s rises from s1 to s2 positive growth during transition y s2 > s1 (+n) k s2f(k) s1f(k) k* k** k 8 ST2011 Growth and Natural Resources
9 Golden Rule (Optimal Consumption) Optimal consumption = maximal long-run per capita consumption ( ) Golden rule of capital accumulation: maximum of c reached, when slope of production function and depreciation function equal: : savings rate for which y (+n) k f(k) cgold sgold f(k) kgold k 9 ST2011 Growth and Natural Resources
10 Technological Progress I Falling marginal product of capital compensated by technological progress output production for a given input of labor and capital rises example: labor-augmenting technological progress (Harrod neutral technological progress), resp., with A = labor productivity (+n)k sf(k, A2) sf(k, A1) sf(k, A0) Increase of A over time (e.g. A0 < A1 < A2) production fcn rotates upward k * 0 k * 1 k * 2 k 10 ST2011 Growth and Natural Resources
11 Technological Progress II, resp. with = output in efficiency units = capital in efficiency units Economic dynamics with technological progress Long-run equilibrium assumption: labor productivity grows at a constant rate, i.e. steady state: 0 zero growth per capita in efficiency units balanced growth per capita at rate of tech. progress balanced growth of aggregates at sum of rates of tech. progress and population growth rate 11 ST2011 Growth and Natural Resources
12 1.2 Ramsey-Cass-Koopmans Model Modification compared to Solow model: households choose savings rate (and thereby today s and future consumption) for which their life-time utility is maximized. savings rate endogenous Households: Maximization of intertemporal utility subject to intertemporal budget constraint Intertemporal utility of a household: (infinite time horizon) /: instantaneous utility function (concave: 0, 0) : rate of time preference (assume for intertemporal utility to be finite) Intertemporal budget constraint: a = per capita wealth r = interest rate w = wage rate 12 ST2011 Growth and Natural Resources
13 Household optimum: from first-order conditions of utility maximization: (Keynes-Ramsey rule or Euler equation) sign and level of depends on interest rate r and the rate of time preference level of additionally depends on (= intertemporal elasticity of substitution) Example: - (Interpretation of : determines curvature of utility function the higher the faster marginal utility decreases with consumption, the more households prefer consumption smoothing over time) No Ponzi game (transversality condition): lim 0 13 ST2011 Growth and Natural Resources
14 Firms Simplifying assumption: no technological progress Maximization of profits:, (price of y normalized to 1) Firm optimum: from first-order conditions of profit maximization:,, Simultaneous optimum of households and firms 14 ST2011 Growth and Natural Resources
15 Dynamic System 1. equation of motion of consumption: 2. equation of motion of capital stock: where = savings Long-run equilibrium: growth rates: 0 0 steady state, : 15 ST2011 Growth and Natural Resources
16 Graphical Representation of the Ramsey-Cass-Koopmans Model f(k) 0 c 0 f(k) c 0 k c* c 0 16 ST2011 Growth and Natural Resources k* k
17 Transitional Dynamics and Saddle Path c 0 c* 0 k* k 17 ST2011 Growth and Natural Resources
18 Comparison of Optimal Capital Stock in Solow and Ramsey Model Ramsey model: f(k) k optimal capital stock: cgold Solow model: optimal capital stock: c 0 k as assumed that cgold c* 0 Intuition: savings lower in Ramsey model due to impatience k* kgold k 18 ST2011 Growth and Natural Resources
19 Savings Rate Savings rate endogenously determined by decisions of households In the steady state, the savings rate is constant: Along the saddle path, the savings rate changes over time as the share of consumption in output changes. 19 ST2011 Growth and Natural Resources
20 Technological Progress Assumption as in Solow-Swan model: labor-augmenting technological progress growth rate of labor productivity: Long-run equilibrium (consumption, output and capital expressed in efficiency units, e.g. ) ST2011 Growth and Natural Resources
21 1.3 Methods: Dynamic Optimization (Barro/Sala-i-Martin 2004, mathematical appendix; Chiang 1992, part 3) Fundamental problem: Maximization of an objective functional over a planning period given possible constraints (functional: see next slide) goal: find optimal magnitude of a choice variable at each point in time, i.e. find the optimal time path of the choice variable Three approaches to solve these types of maximization problems: calculus of variations dynamic programming control theory ( Pontryagin s maximum principle Hamiltonian) This lecture: control theoretical approach 21 ST2011 Growth and Natural Resources
22 The Objective Functional Function: mapping of real numbers (curves) to real ) numbers (e.g. Functional: mapping of paths (curves) to real numbers (e.g. ) Source: Chiang ST2011 Growth and Natural Resources
23 The Maximization problem I Two types of variables: Control variables: variables that can be chosen by the agent at each point in time State variables: variables that are given at each point in time (e.g. capital stock) variables that are steered by the choice of the control variables Important elements: Time horizon: planning period of the agent (can be finite or infinite) Objective functional: functional to be maximized over planning period, usually given in form of an integral Equation of motion of state variables: equations that describe the development of the state variables over time Potential further (non-)linear constraints: e.g., minimal permissible capital stock at end of planning period 23 ST2011 Growth and Natural Resources
24 The Maximization problem II subject to max 0,, control variable: c,, state variable: k = current value of objective function (i.e. value as seen from time 0), e.g. utility T = = terminal time / = average discount rate Example:,,,,, 24 ST2011 Growth and Natural Resources
25 Lagrangian of optimization problem,,,, = Lagrangian muliplier (costate variable) shadow price (value of an additional unit of k at time t in units of utility at time 0) (note: equation of motion implies continuum of constraints for each moment in time 0, equivalently: continuum of Lagrange multipliers) = Lagrangian multiplier giving the value of the terminal tock of k at T in units of utility at time 0 25 ST2011 Growth and Natural Resources
26 Procedure for static optimization problems: Maximize Lagrangian with respect to c and k for all t problem: time derivative of avoid problem by integrating by parts, giving 0 0 Recall integration general rule: ST2011 Growth and Natural Resources
27 Thus the Lagrangian can be rewritten as,,,, 0 0 Hamiltonian function:,,, Economic interpretation of Hamiltonian:,, value derived from consumption and capital at a specific instant in time,, value of an additional unit of k (that was not consumed but saved) at the same instant in time Hamiltonian represents complete contribution to utility from a specific choice of c (which entails the choice of ) for a given shadow price 27 ST2011 Growth and Natural Resources
28 Necessary conditions of intertemporal maximization can be shown to be closely related to derivatives of Hamiltonian plus an additional condition dealing with situation at terminal time T (so-called transversality condition) derivation of these conditions from Lagrangian on previous page consider the following steps: (see also Appendix A) 1. assume that and are the optimal time paths for and 2. perturb optimal path by arbitrary perturbation function giving a neighboring path which implies corresponding perturbations of and 28 ST2011 Growth and Natural Resources
29 The derivative of the Lagrangian with respect to has to be equal to zero ( 0). 3. Recall the Lagrangian,,, 0 0 After inserting the functions from step 2 for, and, we can take the first derivative of with respect to : (see also Appendix B) 29 ST2011 Growth and Natural Resources
30 4. For 0, the following conditions have to hold: 0, 0, 0: 0: first-order condition with respect to control variable first-order condition with respect to state variable, resp. equation of motion for the costate variable 5. Complementary slackness condition associated with inequality constraint 0: 0 using we get 0 Intuition in analogue to static optimization: (transversality condition) A positive capital stock at T can only be optimal if its shadow value (in terms of utility) is zero. If the shadow value is positive, then the terminal capital stock has to be zero. 30 ST2011 Growth and Natural Resources
31 Sufficient conditions of intertemporal utility maximization if,, and,, are concave necessary conditions are also sufficient Infinite time horizons for most problems dealt with in this lecture, the time horizon is infinite households maximize utility over infinite time (e.g. infinite succession of generations): subject to max 0,,,, 0 0 lim 0 FOC for control and state variables: same as for finite terminal time ( 0, 0) transversality condition now reads: lim 0 31 ST2011 Growth and Natural Resources
32 Multiple state and control variables The line of reasoning derived above for one control and one state variable holds equivalently for multiple control and state variables. Example: Additional control variables: pollution flow, resource extraction Additional state variables: human capital, stock of natural resources, stock of pollution, climate, 32 ST2011 Growth and Natural Resources
33 The Cookbook Recipe Given the maximization problem max 0,, subject to,, 0 0 lim 0 Step 1: Set up the Hamiltonian,,,, Step 2: Derive the first-order conditions for the control and state variables Step 3: Set up the transversality condition lim 0 33 ST2011 Growth and Natural Resources
34 Present and Current Value Hamiltonian I (for example of Ramsey model) Present value Hamiltonian: represents the present value of utility derived from the choice of consumption at time t: : utility derived from discounted to present time = current utility effect of consumption : value of addition to household wealth from not-consuming, i.e. from saving ( shadow price in terms of present-value prices) = future utility effect of saving 34 ST2011 Growth and Natural Resources
35 Present and Current Value Hamiltonian II Current value Hamiltonian: represents the current value total utility derived from the choice of consumption at time t: : non-discounted utility derived from : value of addition to household wealth from saving ( shadow price in terms of current-value prices) such that Modified optimality conditions: 0 lim 0 35 ST2011 Growth and Natural Resources
36 Example: The Ramsey-Cass-Koopmans model (Simplifying assumption: no technological progress) Solve the households and firms optimization problems and derive the steady state values of,,,, and s. Households: s.t. max lim 0 Firms: Cobb-Douglas production technology:, where A is a constant and capital depreciates at rate. 36 ST2011 Growth and Natural Resources
37 Appendix A: Intuition for Perturbation I Lagrangian depends on time paths of, and Maximization over entire time path of variables required Solve problem with the help of pertubation: Assume that optimal paths of variables are known (, ) Changes in, e.g., can then be represented by perturbations of the optimal path: Assume an arbitrary perturbation function and rewrite by (where is a small number) which gives a neighboring path to. Due to the perturbation also the path of and the terminal change, which gives, Inserting these functions into the Lagrangian gives the Lagrangian as a function of (as the optimal paths and perturbation functions are taken as given). As the Lagrangian is maxized for and, which are associated with 0, follows that 0 is a necessary condition for the maximum. 37 ST2011 Growth and Natural Resources 0 has to hold. It
38 Appendix A: Intuition for Perturbation II Example of a perturbation of a time path: Optimal path of a variable : Pertubation function: Perturbed optimal path: vf 38 ST2011 Growth and Natural Resources
39 Appendix B: Differentiation of a Definite Integral I Neoclassical Growth Theory Differentiation of the definite integral gives:,,,,,,, 39 ST2011 Growth and Natural Resources
40 Appendix B: Differentiation of a Definite Integral II 40 ST2011 Source: Chiang (1992) Growth and Natural Resources
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