Infinite-Horizon and Overlapping Generations Models

1
Chapter 2

• Infinite-Horizon and Overlapping Generations
  Models

2
Part A

• The Ramsay-Cass-Koopmans (Representative Agent)
  Model

3
2.1 Assumptions
4
Firms

• Large number of profit maximizing firms
• Perfectly competitive on the output side
• Perfectly competitive on the input side
• Owned by households
• Identical, with constant-returns-to-scale
• Cobb-Douglass production is the easiest
• Use exogenous technology, A

5
Households

• Large, fixed number of H identical households
• Dynasties
• Households live forever
• Population, L, and household size both grow at
  rate n
• Everyone supplies 1 unit of labor
• No labor-leisure choice
• Households can own capital, which they rent to
  firms
• No depreciation (for simplicity it can be
  harder to make plausible assumptions that allow a
  lot of cancellation in the math if we keep it in
  the model)

6
Households

• Utility
• U?e-?tuc(t)L(t)H-1dt
• Period/instantaneous utility, uc(t) depends on
  current consumption only
• Discount at rate ?
• Typically, there needs to be a condition (like ?
  gt n) so that utility doesnt go to infinity as
  population size increases
• Multiplied by household size, L/H
• So it wont be utility improving to beggar
  future generations to increase savings today
• Infinite time horizon

7
2.2 The Behavior of Households and Firms
8
Firms

• Extensive form production function YF(K,AL)
• Intensive form production function yf(k)
• Capital is paid its marginal product or rental
  rate
• r(t)fk(t)
• Labor is paid its marginal product. For a
  competitive firm, this is what is left over after
  paying capital
• Wf(k)-kf(k), (this identity is quicker than
  calculus)
• Actual labor is paid wAW

9
Households Budget Constraint

• The household takes r and w as given
• The household is not liquidity constrained
• In general, C W, and ?C ?W
• The size of the household, L/H doesnt affect
  anything since both sides get multiplied by it
• The household discounts at the continuously
  compounded instantaneous rate of interest
  R(t)?rdt
• The households own the initial capital stock,
  K(0)/H

10
Households Budget Constraint

• The model doesnt make sense if capital can go to
  zero as time goes to infinity
• This is why we dont use capital in a Fisher
  model with finite time, why would you have
  capital at all in the second period?
• We require that the discounted value of the
  capital stock be non-negative in the limit
• lim(s ?8)e-R(s)k(s)?0

11
Households Maximization Problem

• Its useful to assume a CRRA utility function
• Consumption for an individual is C(t)
• Instantaneous utility is u(t)C(t)(1-?)/(1-?)
• Using LHopitals rule (that the limit of a
  fraction is the same as the limit of a second
  fraction comprised of the derivatives of the
  numerator and denominator of the first fraction)
  we can show that u(t)lnc(t), when ? 1
• This is useful because it is simple, and because
  estimates of ? are near 1.

12
Households Maximization Problem

• We need to extract the effects of the intial
  value of technology
• Consumption per effective worker is
  c(t)C(t)/A(t)
• So,
• u(t)A(t)(1-?)c(t)(1-?)/(1-?)
• u(t)A(0)egt(1-?)c(t)(1-?)/(1-?)
• u(t)A(0)(1-?)egt(1-?)c(t)(1-?)/(1-?)
• Note that A(0)(1-?) is a constant

13
Households Maximization Problem

• U?e-?tuc(t)L(t)H-1dt
• Now we need to get the starting value for L out
  of the equation using L(t)L(0)ent
• So
• U?e-?tA(0)(1-?)egt(1-?)c(t)(1-?)/(1-?)L(0)en
  tH-1dt
• Extract constants and gather exponents to get
• UA(0)(1-?)L(0)H-1?e-?tegt(1-?)c(t)(1-?)/(1-?
  )entdt
• UA(0)(1-?)L(0)H-1?e-?g(1-?)ntc(t)(1-?)/(
  1-?)dt

14
Households Maximization Problem

• It looks complex, but the utility function has a
  general form of
• UB?e-ßtc(t)(1-?)/(1-?)dt
• Where B A(0)(1-?)L(0)H-1 is a constant
  capturing a weighted version of starting
  technology per household
• Where ß--?g(1-?)n is a composite discount
  rate that needs to be positive (just like any
  discount rate in continuous time)
• This is asserted without proof on pg. 49, but
  justified on pg. 53

15
Households Maximization Problem

• In the budget constraint
• Total income depends on the wage per worker,
  times technology, times population, divided by
  the number of households
• Total consumption clearly depends on population
  and households. It also depends on technology,
  although this might not be so obvious.
• These can be factored out once we account for
  starting capital

16
Households Maximization Problem

• Starting capital per effective worker is capital
  per household times population, divided by
  households.
• Divide through by starting technology to get
  capital per worker
• So, starting capital per household can be
  expressed as
• k(0)A(0)L(0)/H

17
Households Maximization Problem

• Discounted infinite lifetime consumption is
• ?e-R(t)c(t)A(t)L(t)/Hdt
• ?e-R(t)c(t)entA(0)egtL(0)/Hdt
• Discounted infinite lifetime income is
• ?e-R(t)w(t)A(t)L(t)/Hdt
• ?e-R(t)w(t)entA(0)egtL(0)/Hdt
• Factoring out A(0)L(0)/H, we get constraint
• ?e-R(t)c(t)e(ng)tdt k(0)
  ?e-R(t)w(t)e(ng)tdt

18
Household Behavior

• The maximization problem is now fairly simple
• The FOCs are then the budget constraint and
• Be-ßtc(t)-??e-R(t)e(ng)t, for each dt
• Take logs to get
• logB ßt ?lnc(t) log? R(t) nt gt
• Take the derivative with respect to time to get
• ß ?(dc/dt)/c(t) r(t) n g
• Substitute out ß to get the Keynes-Ramsay rule
• (dc/dt)/c(t) r(t)-?-?g/?

19
Household Behavior

• The above is consumption per effective worker. To
  get consumption per capita we need
• (dC/dt)/C(t)(dA/dt)/A(t)(dc/dt)/c(t)
• (dC/dt)/C(t) g r(t)-?-?g/?
• (dC/dt)/C(t) r(t)-?/?
• Either one of these relationships can be thought
  of as the Euler equation for the problem

20
Household Behavior

• The Euler equation says that consumption will
  grow if
• The rate of return on capital exceeds the
  discount rate
• The former is what you get for giving up
  consumption
• The latter is how willing you are to give up
  consumption
• If you are not willing to give up some
  consumption now, you wont be able to grow.
• But, it is only sensible to give up some of your
  consumption if you can earn a reasonable rate of
  return on it.

21
2.3 The Dynamics of the Economy
22
The Dynamics of c

• We are interested in describing when consumption
  is increasing, and when it is decreasing
• In order for dc/dt/c(t) to be equal to zero,
  the numerator of the RHS must be zero
• r(t)-?-?g0
• r(t)fk(t), so there is a single value of k
  for which this is true
• Call it k

23
The Dynamics of c

• For k gt k,
• f(k) , f(k) or r(t) lt r(t)
• This implies that for high value of k that c will
  fall
• By the same argument, for low values of k, c will
  be increasing

24
The Dynamics of k

• What will dk/dt look like?
• New investments in capital are f(k)-c.
• The amount of capital that needs to be purchased
  to supply new effective labor is (ng)k
• So, dk/dt f(k)-c-(ng)k
• This is analogous to the distance between the
  sf(k) and (ngd)k lines in the Solow model
• Its sort of parabolic

25
The Dynamics of k

• Below this curve, k is large relative to c, so k
  will be increasing
• Above this curve, c is large relative to k, so k
  will be decreasing

26
The Phase Diagram

• The intersection of the line and the curve is the
  steady state
• This type of intersection is called a saddlepoint
• A saddlepoint is an equilbrium that can only be
  reached along a single path
• There is a single path from the bottom-left to
  the top-right that leads to this steady state
• This is called the saddlepath

27
The Phase Diagram

• Is the steady state at the golden rule level of
  capital (the one where c is maximized)?
• The steady state is where r ??g
• The golden rule is where r ng
• We already know that the composite discount rate
  must have -?g(1-?)n lt 0
• This can only hold where ng lt ??g
• This is where the rate of return is higher at the
  steady state, or the level of capital is lower

28
The Initial Value of c

• There are two rules for thinking about phase
  diagrams
• Paths that start at points in the same vertical
  or horizontal line cant cross
• Each locus of points where a variable doesnt
  grow can only be crossed going horizontally or
  vertically

29
The Saddle Path

• Because of the rules for movement in a phase
  diagram, there is only one point immediately
  before the steady state that will lead into the
  steady state
• By induction there is only one point immediately
  preceding that point that will connect ultimately
  to the steady state

30
Welfare

• By construction, the equilibrium of a
  Ramsay-Cass-Koopmans model is Pareto-optimal
• It satisfies the First Welfare Theorem
• Its markets are competitive and complete
• There are no externalities
• There are a finite number of agents (thus the
  fixed number of households)

31
2.5 The Balanced Growth Path
32
Properties of the Balanced Growth Path

• It turns out that once the steady state is
  reached, the properties of the model are the same
  as the Solow growth model where the savings rate
  is fixed and sub-optimal
• Namely, that growth in per capita income depends
  on growth in technology

33
The Balanced Growth Path and the Golden-Rule
Level of Capital

• Since the actual golden rule level of capital
  where consumption is maximized cannot be
  achieved, and
• Since the steady state is optimal
• The saddlepath is sometimes called a
  modified-golden-rule path

34
2.6 The Effect of a Fall in the discount rate
35
Qualitative Effects

• This section focuses on the discount rate as an
  example of comparative dynamics.
• The discount rate only effects the dc/dt 0
  locus
• A fall must be balanced by a fall in f(k), which
  can only be produced by a higher k

36
Qualitative Effects

• The new steady state is to the right of the
  initial one
• It will have a new saddlepath as well
• The old saddlepath is now explosive
• Capital cannot jump instantaneously, but
  consumption can
• So, consumption drops until the new saddlepath is
  reached

37
Rate of Adjustment and the Slope of the Saddlepath

• See the spreadsheet

38
The Speed of Adjustment

• See the spreadsheet

39
2.7 The Effects of government purchases
40
Adding Government to the Model

• Start out by eliminating just about everything
  that is important about government
• Government does nothing for utility, or budget
  constraints
• Those are relaxed in later chapters
• Government does is place a drag on the economy
  (it isnt perfectly efficient)
• We model this by subtracting G(t) from dk/dt
• This shifts its graph downward

41
The Effects of Permanent Changes In Government
Purchases

• A permanent increase/decrease in G changes the
  position of the steady-state and the saddlepath
• The old saddlepath is now explosive
• Capital cannot change instantaneously
• Consumption can change instantaneously , and it
  changes by the negative of the change in G to
  move the economy to the new saddlepath

42
The Effects of Temporary Changes In Government
Purchases

• A temporary increase/decrease in G cannot take
  use to the new saddlepath and then back to the
  old one
• Since the change is temporary, the new
  saddlepath is never there
• Instead consumption jumps instantaneously to a
  point connected to an explosive path that
  intersects the saddlepath at the time when the
  temporary purchase cease

43
Empirical Application Wars and Real Interest
Rates

• Wars induce temporary changes in government
  spending
• The temporary transit on an explosive path
  corresponds to higher real interest rates
• Barro 87 offers some evidence in support of a
  correlation between temporary military spending
  and real rates

44
Part B The Diamond Model
45
2.8 Assumptions
46
Assumptions
Period In Model Period In Model Period In Model Period In Model
Generation Born at 1 2 3 4
1 Young Old
2 Young Old
3 Young Old

• Note that the richness of behavior caused by
  having an infinite number of finitely-lived
  individuals instead of one infinitely-lived
  individual makes OLG models contain RA models
  as a special case.

47
Assumptions

• The model is discrete
• Agents live for 2 periods
• New agents enter the model each period
• It is the sense in which the number of optimizers
  is infinite that gives the Diamond overlapping
  generations model its qualitative differences
  from the Ramsay-Cass-Koopmans representative
  agent model

48
Assumptions

• Lt young agents are born in each period
• The rate of population growth is still n
• Agents supply 1 unit of labor when young, and 0
  when old
• Because agents have finite lives, no limit needs
  to be placed on the discount rate of the model
• Depreciation rate is 100

49
Household Behavior

• An agents income in period 1 is their per capita
  wage times aggregate technology, wtAt
• Savings earn the exogenous rate rtf(kt)
• This equals the marginal product of capital
• The per capita wage is wt
• Consumption of an old agent is
• C2t1(1rt1)(wtAt-C1t)
• The lifetime budget constraint is then
• C1t1(1rt1)C2t1(1rt1)wtAt

50
Household Behavior

• With CRRA utility, the Lagrangian is
• L(C1t)(1-?)/(1-?) 1/(1?)(C2t1)(1-?)/(1-?)
  ?(1rt1)wtAt-C1t1(1rt1) - C2t1
• Maximization of the Lagrangian yields the Euler
  equation
• C2t1(1rt1)(-1/?)C1t(1?)(-1/?)
• Since C1t-? - ?(1rt) 0
• And 1/(1?)C2t1-? ? 0

51
Household Behavior

• Rearrange the Euler equation and budget
  constraint to get a solution for C1t
• C2t1(1rt1)(-1/?)C1t(1?) (-1/?)
• C1t(1rt1)C2t1(1rt1)wtAt
• C2t1(1rt1)wtAt-C1t(1rt1)
• wtAt-C1t(1rt1)(?-1)/?C1t(1?)(-1/?)
• (1?)(1/?)wtAt-C1t(1rt1)(?-1)/?C1t
• C1t(1?)(1/?)/(1?)(1/?)(1rt1)(?-1)/?wtA
  t

52
Household Behavior

• So, first period consumption is a not a simple
  function
• Let
• C1t1-s(rt1)wtAt, where
• 1-s(rt1) (1?)(1/?)/(1?)(1/?)(1rt1)(?-
  1)/?
• The saving rate depends on ?
• Increasing in r for ? lt 1
• Decreasing in r for ? gt 1

53
2.10 the dynamics of the economy
54
The Equation of Motion of k

• For aggregate capital
• Kt1 s(rt1)LtAtwt
• rt1 is paid at t1 based on decisions made at t
• Divide by next periods effective labor
• Kt1/Lt1At1 s(rt1)LtAtwt/Lt1At1
• But Lt/Lt11/(1n) and At/At1 1/(1g)
• kt1 s(rt1)wt/(1n)(1g)
• Or, in terms of capital only
• kt1 sf(kt1f(kt)-ktf(kt)/(1n)(1g)

55
The Evolution of k

• Because the equation for k includes multiple
  functions of k, it can be highly non-linear
• This leads to a lot of interesting theoretical
  behavior if there are multiple steady-states
• For example, if the gross investment curves to
  create multiple steady-states, they each have to
  be evaluated as sinks, sources or saddlepoints.
• In this case, it is possible to have more than
  one stable steady-state, and to then rank them as
  better or worse so poverty traps are possible.

56
Logarithmic Utility and Cobb-Douglas Production

• In this case, the dynamics of k are simple, and
  very much like the Solow model

57
The Speed of Convergence

• An important difference with this model is that
  convergence to the steady state is much faster
• See the spreadsheet

58
The General Case

• The phase diagram relating kt1 to kt need not be
  monotonic
• This means there can be multiple steady states
• Some sources
• Some sinks
• Self-fulfilling prophecies are possible
• See the spreadsheet

59
2.11 The Possibility of Dynamic Inefficiency

• Even though they are competitive, OLG models need
  not be Pareto efficient
• The capital stock can exceed the golden rule
  level of capital this occurs over time so it is
  known as dynamic inefficiency.
• Recall that the Solow and Ramsay-Cass-Koopmans
  models end up with capital below the golden rule
  level so it is typically dynamically efficient
• This means that everyone can be made better off
  by reducing capital and consuming instead

60
2.11 The Possibility of Dynamic Inefficiency

• In particular, since there are an infinite number
  of generations, each larger than the last, it is
  possible to
• Transfer from the current young to the current
  old to make them better off
• Transfer from the future young to the future old
  (who are the current young) to make them better
  off too
• Repeat infinitely

61
2.11 The Possibility of Dynamic Inefficiency

• Dynamic inefficiency is really about
  overinvestment in capital
• The economy would be more efficient if capital
  was cut because consumption would go up instead
  of down
• Alternatively, the rate of return on capital is
  too low

62
Empirical Application Are Modern Economies
Dynamically Inefficient?

• Naïve Approach
• Measure the real rate of return and the real rate
  of growth (or the nominal rates)
• If the real rate of return is lower, then the
  economy is dynamically inefficient
• The problem is what asset to use
• Safe ones suggest dynamic inefficiency
• Risky ones suggest dynamic efficiency
• Either way, the naïve approach confirms dynamic
  inefficiency

63
Empirical Application Are Modern Economies
Dynamically Inefficient?

• Better Approach
• You cant be overinvesting if some of your income
  from capital is not being reinvested in more
  capital
• The difference must be going to consumption
• Evidence suggests that capital income exceeds
  investment, and therefore economies are not
  dynamically inefficient
• This means that we do not have to use OLG models
  to capture this feature of reality. But, we still
  might use them for other reasons.

64
2.12 Government In the Diamond Model

• The result is similar to the Ramsay-Cass-Koopmans
  model
• The deadweight loss of government shifts the
  capital locus (however graphed) towards lower
  capital growth