Lecture 4' Ramsey Model

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Lecture 4. Ramsey Model

• 4.2 The Optimal Growth Problem

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Ramsey (1928) growth model

• Relax the assumptions that the saving rate is
  exogenous and constant.
• Empirical observation saving rates increase with
  economic development
• Richer dynamics of savings imply changes in
  transitional dynamics and speed of convergence

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How does this model compare to the Solow growth
model?

• Saving rate is endogenous
  

The optimizing behavior of agents is explicit
Microfoundations

• Market economy

Wages Interest
Firms
Households

• Consume
• Accumulate assets

Technical know-how
Capital Labor
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Assumptions firms

• A large number of identical firms
• A CRS production function YF(K, AL)
• The same assumptions as in the Solow model
• Firms hire workers and rent capital in
  competitive factor markets
• Output markets are competitive.
• Labor augmenting technological progressLabor
  efficiency A grows at rate g.

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Assumptions Households

• Time is infinite dynasties of households
• A large number H of identical households.
• The size of each household grows at rate n.
• Each member of household supplies 1 unit of labor
  at every point in time.
• Own firms and capital rents its capital to
  firms.
• No capital depreciation
• Initial capital K(0)/H

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The households utility function
c(t) consumption per capita L(t) total
population of the economy r rate of time
preference
The instantaneous utility function
(lHopitals rule)
coef. of relative risk aversion
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The households utility function
lHopitals rule
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Some facts about preference

• Homotheticity The function u is said to be
  homothetic if MRS(cts, ct) MRS(lcts, lct) for
  all l gt 0 and c.
• if an agents lifetime income doubles,
  optimal consumption choices will double in each
  period (income expansion paths are linear).
• CRRA? measures the curvature of the utility
  function, i.e., the elasticity of the slope of
  the utility function (marginal utility of
  consumption) with respect to ct,

how the slope of the utility function is changing
as consumption varies.
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Intertemporal elasticity of substitution 1/?

• Higher ? means a more concave utility function,
  thus a higher curvature of the utility function.
• ? 0 means risk neutral and consumer does not
  care about consumption smoothing, the utility
  function is linear and we have a constant
  marginal utility of consumption.
• As ? increases, risk aversion increases and so
  does the willingness to smooth consumption.
• As ? goes to 1, the instantaneous utility
  function goes to a logarithmic utility ln(c).
• A useful first case to consider

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Intertemporal elasticity of substitution

• Utility

Slope of the indifference curve

• h measures the curvature of the indifference
  curve
• The willingness of people to trade off
  consumption in one period to consumption in
  future period

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Transforming aggregate variables

• Express in terms of consumption per unit of
  effective labor.

Substitute into U
b
B
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Firm and household behavior

• Firm behavior is not so interesting
• At each point in time firms employ the stocks of
  labor and capital, pay them their marginal
  products, and sell the resulting output. They
  earn zero profit.

w real wage per unit of effective labor
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Linear, first-order,nonautonomous
differentialequations
I. Homogenous part
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Linear, first-order,nonautonomous
differentialequations

• II. Conjecture for the solution of the general
  equation

where D is a constant to be determined from the
initial condition
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A short cut

• A trick to solve the differential equation is to
  multiply each term by e-R(t)
• Then one can check that the LHS is the derivative
  of

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Households budget constraint An identity
For the total cumulative interest paid from time
0 to t one unit of the good invested at time
zero yields expR(t) units at time t in the
future
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No-Ponzi game condition

• We impose
• The present value of the households asset
  holding cannot be negative in the limit
• Someone cannot issue debt and rolls it over
  forever.
• Asymptotically, the level of debt can only grow
  at rate less than r.

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No-Ponzi game condition
The present value of the households asset
holdings cannot be strictly positive in the limit
No waste in the end of the day, since consumers
need to reduce wealth to zero as long as MUcgt0
TVC
Human wealth
Non-human wealth
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Household Budget constraint

• Express them in term of variables per unit of
  effective labor.
• The budget constraint
• The No-Ponzi game condition TVC

Divide by AL
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Program of the household

• We omit the constant B.

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The current-valued Hamiltonian
FOC

• Choice of c
• (current vs. future c)

shadow price of capital
C
Y
Marginal utility of c marginal value of I
I

• Equation for the costate variable

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Economic interpretation

• increase consumption today, gain utility. But
  with an opportunity cost.
• Whats the opportunity cost of increasing
  consumption today?
• the effect of changing todays consumption on
  future capital stock. The price of changing
  capital stock is measured in units of utility,
  i.e., the shadow price.

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The Euler equation

• Integrate from 0 to t

Taking exp
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• Interpretationfinite-horizon terminal
  condition
• If the present value of the marginal utility of
  the c were positive it would not be optimal to
  end up with a positive capital stock. ( you can
  always raise c by reducing K so that utility can
  be increased)
• Infinite horizon

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Optimal consumption
Rate of growth consumption per worker
Difference between the real interest rate and the
rate of time preference
Intertemporalelasticity of substitution


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Euler equation of consumption Describing how
consumption must behave over time

• The higher the MPk relative to r, the more it
  pays to depress current c to enjoy higher future
  c
• Initially K is small , MPk high, c increases
  over time on the optimal path
• The larger the elasticity of substitution the
  easier it is, in terms of utility, to forgo
  current c to increase future c.
• if c is growing (falling),
  since agents prefer to smooth c, agents demand a
  positive (negative) premium on the time discount
  factor in order to forgo consumption or the
  higher is the coeff. of relative risk aversion,
  the more premium is demanded.

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Euler equation of consumption intertemporal
budget constraint Optimal consumption path

• Given c0, and a time path for r(t), the Euler
  equation tells you how c must evolve over time in
  order for the consumption path to be an optimum.
• But what is c0?

Plug into the intertemporal budget constraint
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Euler equation of consumption intertemporal
budget constraint Optimal consumption path

• the choice of c0 is then determined by the
  requirement that the budget constraint be
  exhausted. Because the MUc is positive, if you
  find at end of time that your capital holdings
  have a non-zero time-zero value, you are not
  spending enough, and c0 needs to be raised if
  you find that you have negative capital holding
  at the end of time, you are spending too much,
  and c0 needs to be lowered.

Wealthinitial stockPV of wage income
? Propensity to consume out of wealth
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Dynamic of the economy in general equilibrium ( r
is determined by firms profit-max choice of
optimal capital)

• Using the fact that
• the Ramsey-Keynes rule
• Using the fact that
• The transversality condition
• The initial condition
• Note c(0) is endogenous

Dynamic system in (c,k)
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Phase diagram

• Iso-locus for c
• Iso-locus for k
• The steady state (c, k)

Is this an equilibrium? Yes if the TVC is
satisfied.
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Is the TVC satisfied at the steady state?
To have a well-defined discounted stream of
utility over infinite horizon, we have already
assumed that bgt0 (otherwise the integral will not
converge) ? this condition is valid
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Dynamic of c
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Dynamic of k

• When c equals to the difference between the
  actual output and break-even investment ?
• Golden Rule level of c
• c is increasing in k until
  and then decreasing. (c is a concave function of
  k)

cgtcgold
When c exceeds the level that yields , k
is falling otherwise, k is increasing
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Initial value of c
c
k
k
k(0)
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Saddle path stability

• The stable arm (the path goes through the origin
  (k0, c0) and the steady state) is the unique
  optimal path
• For c0 gt c0, curve is crossed before
  line is reached and so the economy ends
  up on a path of perpetually rising c and falling
  k ? violating continuity of the necessary
  condition, Ramsey rule.
• For c0 lt c0, the locus is reached
  first, and so the economy ends up on a path of
  falling c and rising k? violating TVC (
  )

The saddle path converging to (c, k)is the
unique equilibrium.
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Saddle Path
c
?
k
kGR
k
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Modified golden rule

• the optimal capital stock in the steady state is
  k such that,
• a golden rule relationship
• Let kGR be the level of capital that maximizes
  steady-state consumption
• Using the fact that
• we have

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Modified Golden Rule
For the modified golden rule The modification is
that the capital stock is reduced below the
golden rule by an amount that depends on the rate
of time preference. Even though society or the
family could consume more in a steady state with
the golden rule capital stock, the impatience
reflected in the rate of time preference means
that it is not optimal to reduce current
consumption in order to reach the higher golden
rule consumption level.
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Convergence Properties of the dynamic system

• Linearizing in the neighborhood of the
  steady-state
• From the transversality condition, the trajectory
  of the economy is given by the saddle-path.

The steady-state is a saddle-point
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Note first-order Taylor series expansion

• Nonlinear dynamic system

• stability of the linear system

J
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Properties of the dynamic system

• Let q1and q2 be the two given eigenvectors and
  l1lt0 and l2gt0 the two eigenvalues associated
  with the jacobian matrix J. (see A.C. Chiang,
  pp612-614)
• C20 must hold for k ? k.
• if C2gt0 violates the TVC
• if C2lt0, k ? 0
• Therefore

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Speed of adjustment

• It is given by

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Example the Cobb-Douglas production function

• Intensive form production function
• Steady-state capital stock per efficient unit of
  labour
• Saving rate

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Example the Cobb-Douglas production function

• Parameter valuesa1/3 , ?4, n2, g1,
  ?1
• r 5, s 20
• Speed of convergence
• Median lag such that
• With the same parameter values in the Solow
  growth model

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Two ways to achieve optimal allocation of goods
and resources

• Social planner ? max the aggregate welfare of all
  the households
• Indivisible hands of the markets in a
  frictionless competitive economy ? decentralized
  decision making processes
• A command optimum could reproduce the equilibrium
  allocation of a market economy provided there are
  no externalities.

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Welfare The Ramsey problem

• A social planner who can dictate the division of
  output between consumption and the
  investmentand who wants to maximize the lifetime
  utilityof a representative household
• Household/producer max
• No externality

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Competitive equilibrium Pareto optimum

• Decentralized economy Central planner
• Check for the competitive equilibrium model and
  the social planner model
• 1. budget constraint
• 2. the Euler equation of consumption

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The balance growth path

• The capital stock per unit of efficient labor is
  constant and equal to k.
• The capital stock, consumption and production
  per worker grow at the rate g .
• The economy does not converge to the path that
  yields to the maximum sustainable level of c .
• The modified golden rule capital stock k