Lecture 4: Ramsey Model

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Lecture 4 Ramsey Model

• 4.1 A math review of
• Optimal Control

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Frank Plumpton Ramsey, 1903-1930

• "A Contribution to the Theory of Taxation", 1927,
  EJ
• "A Mathematical Theory of Saving", 1928, EJ.

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Pre-Ramsey Economic Thoughts about Savings

• Böhm-Bawerk (1889) people are "myopic" in the
  sense that they tend to underestimate their
  future needs and desires and therefore "discount"
  their future utilities.
• Pigou (Economics of Welfare, 1920) if agents
  tend to underestimate their future utility, they
  will probably not make proper provision for their
  future wants and thus personally save less than
  they would have wished had they made the
  calculation correctly. In other words, savings,
  as a whole, are less than what is "optimal".
• Yet in order to confirm that the rate of savings
  thrown up by a market system with myopic agents
  was indeed suboptimal, one must first determine
  what the optimal savings rate might be.

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Solow vs. Ramsey

• Solow model agents in the economy (or the
  dictator) follow a simplistic linear rule for
  consumption and investment.
• Ramsey model agents (or the dictator) choose
  consumption and investment optimally so as to
  maximize their individual utility (or social
  welfare).
• establishing the main characteristics of modern
  dynamic macroeconomics.

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Ramsey (1928) optimal growth model

• How much a nation should save?
• A framework for studying the optimal
  intertemporal allocation of resources.
• Microfoundations
• ? The optimizing behavior of agents is
  explicit
• ?The saving rate is endogenous

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Optimization over Time

• Two modern ways of solving dynamic problem
• ltClassical Approachgt
• calculus of variations (Euler equations)
• Ramsey was among the first to introduce into
  economics in his 1928 paper.
• (1) Optimal Control (Lagrangian methods)
• - a simple two-period example
• - infinite horizon
• (2) Dynamic Programming
• - discrete problem Bellman equations

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A simple two-period consumption model

• A simple consumer's optimization problem
• Max u(c1)u(c2)
• s.t. p1c1 p2c2 x
• Set up a Lagrangian and solve for all optimal
  choices simultaneously.
• But what if we have a 100-period problem to
  solve?
• So instead of using brute force to find the
  solutions of all the cs in one step, we
  reformulate the problem.

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Important terminology

• Definition
• x1 endowment which is available in period 1,
• x2 endowment left in period 2.
• Budget constraint x2 x1-p1c1, with x20.
• x2 defines the state that the decision maker
  faces at the start of period 2.
• x2-x1-p1c1 the state equation, equation of
  motion or the transition equation describing the
  change in the x from period 1 to period 2,

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Important terminology

• State variable xt (the state faced by the
  decision-maker in period t)
• It is parametric (i.e., it is taken as given) to
  the decision-maker's problem in t and xt1 is
  determined by the choices made in t. When we get
  to t1, then xt1 is parametric (i.e., taken as
  given).
• The state variables in a problem are the
  variables upon which a decision maker bases his
  or her choices in each period.
• What you do now will influence the value of the
  state variable in the future.

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State variable

• Stock variable or state variable
• A variable for which there is a law of motion
• the relationship between the actions taken and
  the evolution of the states
• A variable whose level at any point in time is
  given (like a predetermined variable)
• e.g. capital stock (k)
• ? you can choose only the rate of change

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Important terminology

• ct is the vector of tth period choice variables.
• Choice variables determine the (expected) payoff
  in the current period and the (expected) state
  next period.
• These variables are also referred to as control
  variables. Their levels can be chosen at any
  point in time, e.g. consumption, labor supply,
  etc.

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Optimization
Optimization problem with four constraints, two
of them are inequality constraints
How can we solve it?
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The optimal control way of solving the problem

• OC makes use of
• Pontryagin's
• maximum principle.
• The first step incorporates
• the state equation into
• the intra-temporal
• constraints so that
• we now have

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The optimal control way of solving the problem

• Envelope theorem
• its optimal value the marginal value of an
  additional unit of the state variable, xt.

Co-state variable Lagrange multipliers
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The FOCs
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Hamiltonian

• To simplify the problem and get some intuition,
  we define a Hamiltonian function
• Hamiltonian aims to transform a dynamic problem
  into a static one
• Notes
• The tth Hamiltonian includes only ct and lt
• Unlike a Lagrangian, only the RHS of state eq.

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Lagrangian ? Hamiltonian
H maximized at every point in time w.r.t. the
control variable
costate variable changes over time at a rate
equal to the marginal value of the state variable
to the Hamiltonian.
state equation must always be satisfied
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Fourth condition Transversality condition

• Transversality condition how we transverse over
  to the world beyond t1,2
• It is a terminal condition for infinite horizon
• In this two-period case the condition that x3 0
  serves that purpose.
• We'll discuss the transversality condition in
  more detail later on

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Conclusion

• These four conditions are the starting points for
  solving most optimal control problems.
• If we want an explicit solution, then we would
  solve this system of equations.
• Although we'll usually solve OC problems in
  continuous time, the parallels should be obvious
  when we get there.

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Quick review

• OC problems
• objective function
• the state equation(s),
• zt the (set of ) choice variable(s),
• xt the (set of ) state variable(s),
• x0 the initial condition for the state
• Intratemporal constraint g(x,z,t)0

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Quick review

• As we saw in the two-period discrete-time model,
  OC problems can be solved more easily using
  Hamiltonian. Generally, the Hamiltonian takes the
  form
• H U(x,z,t)ltf(x,z,t)
• The maximum principle due to Pontryagin states
  that the following conditions, if satisfied,
  guarantee a solution to the problem

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Quick review
Two sets of differential equations
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Transversality condition

• What happens as we transverse to time outside the
  planning horizon.
• Think about the Kuhn Tucker conditions!
• l(T)0 the condition for a problem in which
  there is no binding constraint on the terminal
  value of the state variable(s). (xT0 ? lTxT0)
• If you could end up with any x you want, then at
  the optimum you shouldn't want to accumulate (or
  get rid of) any x at the end.

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Optimal Control in continuous time

• Reference
• Barro and Sala-i-Martin, Economic Growth, pp.
  498-510
• What is the meaning of l?
• Hamiltonian dynamic generalization of the
  Lagrange method, so the dynamic Lagrange
  multipliers l have a similar economic
  interpretation as shadow prices
• For a simple proof, see Kamien and Schwarz (1981,
  pp. 125)

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Economic interpretation of the optimal control
theory

• Intuitively, why do we need to impose the TVC?
• Consumers are impatient since rgt0, consumers
  tend to put off asset accumulation until a later
  date.
• Giving this propensity of the households, it is
  possible that households end up with lots of
  asset at a terminal date, which appears useless.
• The TVC says that the value of households
  assets must approach zero as time approaches
  infinity. That is if you decide to hold assets at
  the terminal period, it is worth zero today.

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Example optimal lifetime consumption

• The CP problem
• in every period, a household receives labor
  income y and capital income rw(t), where w(t) is
  the cumulated wealth
• she has to choose optimally consumption c(t) and
  savings s(t) at every instant along a finite
  horizon
• preferences are of the log family
• no discounting
• zero initial wealth and final wealth

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The optimal lifetime consumption

• The program of the agent can be written as
• How do we deal with the problem?

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Hamiltonian function

• Idea behind
• If agent decides to modify his control variable
  c, it has two consequences
• It modifies the current utility u(c(t))
• It affect the state variable in the future
  periods,
• But how to value this effect on ?
• Use the shadow price(Lagrange multiplier) m
• When m is correctly chosen, the effect of current
  choices on the future is summarized by

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The first-order conditions

• (1).
• (2).
• (3).

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Solving this first order differential
equation, Optimal path of consumption
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Life-cycle pattern

• Consumption is increasing over time
• In first period of life, household will save to
  guarantee his future consumption
• As the end of the horizon becomes closer, starts
  consuming more than y, hence will be decumulating
  wealth
• Typical life-cycle pattern first saving, then
  dissaving

c
y
time
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Homework consider the following continuous time
infinite horizon model

• Set up the Hamiltonian H(c,k,t,l)
• Write down the TVC and then derive the first
  order conditions try to interpret these
  condition in economic intuitions
• Derive the Ramsey rule

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Time Preference

• Exponential discounting leads to a fast decay at
  the beginning of the period

exp(-rt)
t
0
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A summary of the procedure of deriving the FOCs

• Step 1 construct a Hamiltonian function
• Step 2 take the derivatives of the Hamiltonian
  w.r.t the control variables and set it to zero.
• Step 3 take the derivatives of the Hamiltonian
  w.r.t the state variables and set it to the
  negative of the multiplier with time derivative
• Step 4 solve a time path for c from the equation
  obtained from steps 2 and 3.

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What is the interpretation of the Lagrange
multiplier m?
If you decide to save 1 unit of income at date
t, how much was it worth at date 0? If you hold 1
unit saving at date t ? you sacrifice 1 unit of
consumption at date t whose utility value is
Hence mt is the price at date 0 of a unit of
saving to be hold at date t 27
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The Ramsey rule
At optimal, consumption grows at a constant rate
r. 27
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Keynes-Ramsey rule

• Ramsey actually worked on an idea put forward by
  Keynes more savings today imply more consumption
  tomorrow. So we must contrast the cost to
  postpone our consumption today with the benefit
  of enjoying it tomorrow.
• This is the Keynes-Ramsey rule. It lies in the
  heart of the golden and modified golden rule of
  capital accumulation. 27