neoclassical growth theory

Neoclassical Growth Theory

1 ST2011 Growth and Natural Resources

Chapter 2

Neoclassical Growth Theory 1.1 Solow-Swan model

1.2 Ramsey-Cass-Koopmans model

1.3 Methods: Dynamic Optimization



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



Assumptions regarding production technology: neoclassical production function with capital

and labor as inputs

, , → ,

with L = labor Y = output (y=Y/L output per capita) K = capital (k=K/L capital per capita)

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.

f(k)

y

k



Population growth rate: = constant

Depreciation rate: = constant

Capital stock dynamics (equation of motion of capital stock)

aggregate: where I = investment (= saving

per capita:

Simplifying assumption: no technological progress



1.1 Solow-Swan Model

Savings and consumption (per capita):

1 1 with s = exogenous savings rate c = consumption per capita

Capital stock dynamics:



Graphical Representation of the Solow-Swan Model

Assumption: initial capital stock

Long-run equilibrium: k*,y*

y

k

f(k)

sf(k)c0=(1‐s)y0

sy0

y0

k0

y*

k*



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



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

s2f(k)

s1f(k)

(+n) k

k* k**

s2 >s1

k

y



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 func-

tion and depreciation function equal: → : savings rate for which ∗

cgold

y

k

f(k)

(+n)k

sgoldf(k)

kgold



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

k*1 k*2

sf(k,A1)

sf(k,A2)

k*0

(+n)k

sf(k,A0)

k

Increase of A over time (e.g. A0<A1<A2) → production fcn rotates upward



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



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 maxim-ized.

→ 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



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



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 →



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 ∗, ∗: ∗

∗ ∗ ∗



Graphical Representation of the Ramsey-Cass-Koopmans Model

0 ⟺ ∗ ∗ ∗

0 ⟺ ∗

k

f(k)

k

c

k*

c* 0 c

c

0

f(k)



Transitional Dynamics and Saddle Path

k

c

k*

c*0

0



kgold

cgold

cgold

k*

c*

k

c 0k

f(k) k

0

Comparison of Optimal Capital Stock in Solow and Ramsey Model Ramsey model:

• optimal capital stock: ∗ Solow model:

• optimal capital stock:

⟹ ∗ as assumed that

⟹ ∗

Intuition: savings lower in Ramsey model due to impatience



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.



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. )

0 ⟺

0⟺



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 possi-

ble 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

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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



The Maximization problem II

max 0 , ,

subject to

, ,

0 0

∙ 0 0 = current value of objective function (i.e. value as seen from time 0), e.g. utility

= / = average discount rate

T = terminal time

Example: , , , , ,

control variable: c

state variable: k



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



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: ∙ | ∙

∙ 0 ∙ 0 ∙



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



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

∗ ∙ ∗ ∙



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)

∙ ∙ ∙

⟺ ∙ ∙ ∙ ∙ ∙ ∙ ∙



4. For 0, the following conditions have to hold:

→ 0: first-order condition with respect to control variable

→ 0: 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:

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.

0, 0, ∙ ∙

(transversality condition)



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):

max 0 , ,

subject to , ,

0 0

lim→

∙ 0 → FOC for control and state variables: same as for finite terminal time ( 0, 0)

→ transversality condition now reads: lim → 0



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,…



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

0 ⟺ ∙ 0

0 ⟺ ∙ Step 3: Set up the transversality condition lim → 0



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



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



Example: The Ramsey-Cass-Koopmans model (Simplifying assumption: no technological progress) Households:

max 0 1

s.t.

0 0

lim→

0

Firms: Cobb-Douglas production technology:

, where A is a constant and capital depreciates at rate .

Solve the households’ and firms’ optimization problems and derive the steady state values of , , , , and s.



Appendix A: Intuition for Perturbation I Lagrangian depends on time paths of , and → Maximization over entire time path of variables re-

quired 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, 0 has to hold. It

follows that 0 is a necessary condition for the maximum.

38

A E

AppendixExample of a

Optima

Pertuba

Perturb

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bed optimal p

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Appendix B: Differentiation of a Definite Integral I Differentiation of the definite integral

,

gives:

,,

,,

,

,

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AAppendixx B: Differentiati

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neoclassical growth theory

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