Quick Notes on OG Models
Professor: Alan G. Isaac
November 13, 2013
Contents
1 Two-Period Consumption Decision 3
1.1 OG Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Firms 9
2.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Social Security 12
3.1 Partial Equilibrium Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 General Equilibrium Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 The Golden Rule 15
5 More Cohorts 17
5.0.1 Retired Individuals . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.0.2 Working Individuals . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.1 Path Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1
CONTENTS CONTENTS
5.3 HM’s Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.3.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Be sure to read the entire Blanchard and Fischer treatment of social security in the
OG model. These notes are purely supplementary.
We turn to the overlapping generations (OG) model at this point because of two
primary virtues.
� it allows exploration of the implications of life-cycle savings in an intergenerational
world. We will focus in particular on the savings and investment repercussions of
social security systems.
� it shows how competitive equilibrium may not be Pareto optimal.
Copyright © 2013 by Alan G. Isaac 2
1 TWO-PERIOD CONSUMPTION DECISION
1 Two-Period Consumption Decision
We initially simplify in three important ways: two-periods lives, no uncertainty, and
no labor/leisure trade-off. Inelastically supplied labor means that in the individual’s
optimization problem we can treat labor income as exogenous.
An individual born at time t lives two periods, t and t + 1. Given initial wealth a0,
her basic optimization problem is to
maxc0,c1,a2
U(c0, c1, a2) (1)
subject to the constraints
a1 = R0,1(a0 + y0 − c0) a2 = R1,2(a1 + y1 − c1) (2)
Here yt is income and ct is consumption. These take place at the beginning of the period,
and Rt,t+1 is the gross interest rate applying over the period. (Later we will just call this
Rt+1, but for now we want to emphasize the timing.)
Arithmetically, we can combine these two constraints by eliminating a1:
c0 + c1/R0,1 + a2/R0,1R1,2 = a0 + y0 + y1/R0,1 (3)
That is, in present value terms, the total value of “expenditures” must equal the total
value of available resources (w0 = a0 + y0 + y1/R0,1, i.e., initial wealth plus labor income).
If capital markets allow unrestricted borrowing and lending at the prevailing interest rate,
so that there is no effective constraint on the value of a1, then this single constraint is
adequate for the consumer optimization problem. This means that the timing of con-
sumption does not depend on the timing of income: only the present value of the income
stream is relevant to the consumption decision.
Copyright © 2013 by Alan G. Isaac 3
1 TWO-PERIOD CONSUMPTION DECISION
We should consider two additional constraints: ct ≥ 0 and a2 ≥ 0. The first is an
essentially a physical constraint: negative consumption is not possible. We generally
(directly or indirectly) assume that this constraint is not binding. We may need the
second constraint to rule out consumer plans to die in debt. For now we will simplify the
utility function to U(c0, c1) and set a0 = a2 = 0. Our problem becomes
maxc0,c1
U(c0, c1) s.t. c0 + c1/R0,1 = y0 + y1/R0,1 (4)
Note that when we can aggregate the single period budget constraints, it becomes
clear that there is nothing inherently dynamic in the optimization problem. Think of the
price of first period consumption as 1 and the price of second period consumption as 1/R.
Then it is essentially our familiar two-good consumer choice problem.
maxcU(c) s.t. p · c = w (5)
where
c = (c0, c1) p = (1, 1/R0,1) w = a0 + y0 + y1/R0,1 (6)
As an algebraic convenience, let us transform this constrained optimization problem
into an unconstrained optimization by defining s = y0 − c0. Our problem becomes
maxsU(y0 − s, R0,1s+ y1) (7)
The first-order necessary condition for an optimum is then
−U0 +R0,1U1 = 0 (8)
This is just the standard Fisher equality between the marginal rate of substitution and
Copyright © 2013 by Alan G. Isaac 4
1 TWO-PERIOD CONSUMPTION DECISION
the gross return to investment.
U0/U1 = R0,1 (9)
It has become more common to write this condition as
U0 = R0,1U1 (10)
and call it an Euler condition. The Euler condition relates the marginal utility of con-
sumption in adjacent periods.
Now we will add structure to our model. Our consumer works when young and lives
off her savings when old. This further simplifies the buget contraint: there is no labor
income when old.
maxU(c0, c1) s.t. c0 + c1/R0,1 = y0 (11)
Figure 1 presents the solution of this optimization problem for two different levels
of the interest rate. Clearly an increase in the interest rate generate has two conflicting
effects on first period consumption. First, it is a change in the relative price: consumption
when young now involves a larger sacrifice of consumption when old, so there is a tendency
to postpone consumption. On the other hand, a higher interest rate raises real income,
which tends to increase consumption in both periods. On net, we do not know whether
consumption rises or falls in response to an interest rate increase. Since saving is just
s(w,R) = w − c0(w,R), we also do not know whether saving increases or decreases.
The effects of an exogenous increase in wage income are easier to pin down. An
increase in y0 simply shifts the budget constraint outward, without changing the slope.
As long as both goods are normal, some of this increase in income will be allocated toward
consumption when old. As a result, saving is increasing in w.
Copyright © 2013 by Alan G. Isaac 5
1.1 OG Comparative Statics 1 TWO-PERIOD CONSUMPTION DECISION
c1
c0
eeeeeeeeeeee U
JJJJJJJJJJJJJJJ
U ′
c0 y0
Ry0
R′y0
Figure 1: Two Period Consumption Decision
1.1 OG Comparative Statics
Next we will examine the comparative statics algebraically. It will simplify things a bit if
we adopt the following popular time-separable utility function:
U(c0, c1) = u(c0) + βu(c1)
=1∑t=0
βtu(ct)(12)
Note that u(·) is the same each period. We will assume that in each period marginal
utility is positive and decreasing: u′ > 0 and u′′ < 0. Note that we follow Irving Fisher in
assuming people are impatient: consumption in the present period yields more utility than
the same consumption in a future period (from the point of view of present planning).
Here 0 < β < 1 is the discount factor.1 The assumption that β < 1 is the assumption that
the rate of time preference is positive: consumers would prefer to shift a perfectly smooth
1The discount factor β is related to the rate of time preference θ according to β = 1/(1 + θ). Theassumption that β < 1 is the assumption that the rate of time preference is positive: consumers wouldprefer to shift a perfectly smooth consumption stream toward the present (if there were no costs in totalconsumption).
Copyright © 2013 by Alan G. Isaac 6
1.1 OG Comparative Statics 1 TWO-PERIOD CONSUMPTION DECISION
consumption stream toward the present (if there were no costs in total consumption).
Exercise 1
Consumption of a normal good is increasing in income. Prove that additivity along with
concavity of u(·) implies that each period’s consumption is normal.
With time separable, discounted utility, the consumer’s maximization problem be-
comes
maxsu(y0 − s) + βu(R0,1s) (13)
with first-order condition
u′0/βu′1 = R0,1 (14)
or equivalently
u′0/u′1 = R0,1β (15)
Exercise 2
Show that a second-order sufficient condition is satified.
So whether the consumption stream is actually tilted toward youth depends on the
relative size of R and β. Equivalently, if we write R = 1 + r and β = 1/(1 + θ), it depends
on the size of the interest rater (r) relative to the rate of time preference (θ). If the
interest rate is high enough, it can pay to tilt your consumption profile toward old age:
total physical consumption increases enough to offset impatience. If θ = r (i.e., Rβ = 1)
then consumption is constant across the two periods.
Exercise 3
Note that separability and concavity of u(·) imply each period’s consumption is normal.
(You should prove this.) Do we use separability anywhere? Or just normality?
More explicitly, the first order condition is
−u′(y0 − s) +R0,1βu′(R0,1s) = 0 (16)
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1.1 OG Comparative Statics 1 TWO-PERIOD CONSUMPTION DECISION
Satisfaction of the first-order condition implicitly defines savings as a function of y0 and
R0,1.
s = s(y0, R0,1) (17)
where
sR ≶ 0, sy0 > 0 (Given concavity) (18)
Exercise 4
Show that if u(·) is increasing and strictly concave then we know 1 > sw > 0 but sr is
unsigned. Can you provide a condition to sign sr? Why is it ambiguous? [Hint: consider
the elasticity of substitution.]
Comment on timing: the young receive the current wage wt for the labor supplied
this period, but they receive next period’s marginal product of capital rt+1 for period t
saving. This is natural in this model, for capital takes a period to put in place. Thus we
can write saving as
s = s[wt, Rt+1] (19)
Comparative Statics:
If u(·) is strictly increasing and strictly concave, so that u′ > 0 and u′′ < 0, then we
know 1 > sw > 0 but sr ≶ 0.
−u′′0dw + u′′0ds+R2βu′′1ds+ βu′1dR + sRβu′′1dR = 0 (20)
→ ∂s/∂w = 1/(1 +R2βu′′1/u′′0) (21)
→ ∂s/∂R = −(βu′1 + sRβu′′1)/(u′′0 +R2βu′′1) (22)
Copyright © 2013 by Alan G. Isaac 8
2 FIRMS
2 Firms
All costs of adjustment are ignored. Individual firms choose their optimal level of N and
K each period. (Note, however, that in the aggregate N and K are predetermined.)
maxK,N
F (K,N)− wN − rK (23)
yields the FOCs
FK(Kd, Nd) = r (24)
FN(Kd, Nd) = w (25)
which can be solved for
Nd = N(w, r) (26)
Kd = K(w, r) (27)
Assume CRTS production: F (λK, λN) ≡ λF (K,N). In this case, total product exactly
exhausted when factors are paid their marginal products.
Define k ≡ K/N and f(k) ≡ F (k, 1) = (1/N)F (K,N). Then
F (K,N) ≡ Nf(K/N)
FK = N1
Nf ′ = f ′
FN = f −Nf ′K/N2 = f − kf ′
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2 FIRMS
So we can also write our FOCs as
f ′(kd) = r
f − kdf ′ = w
yielding f ′(k) = r and f(k) − kf ′(k) = w as a complete description of factor market
equilibrium.
Thus given K and N , and therefore k, we can solve
r = r(k) r′ < 0
w = w(k) w′ > 0
At this point you should look at the FIGURE in Blanchard and Fischer: r = slope =
tan θ
Timing and accumulation: This model assumes that this period’s saving by the young
determines next period’s supply of capital.
Kt+1 = Ntst
kt+1 ≡ Kt+1/Nt+1
= (Nt/Nt+1)st
= st/(1 + n)
Copyright © 2013 by Alan G. Isaac 10
2.1 Comments 2 FIRMS
2.1 Comments
1) Only the young save, the old dissave. So aggregate saving is
St ≡ Yt − Ct
= F (Kt, Nt)−Ntcyt −Nt−1cot
= F (Kt, Nt)−Nt[wt − st]−Nt−1[RtKt/Nt−1]
= F (Kt, Nt)− wtNt − rtKt︸ ︷︷ ︸0
+Ntst −Kt since Rt = 1 + rt
= Ntst −Kt
IF factor markets clear.
Now St ≡ It is an identity ; Kt+1 −Kt = It is an assumption (about units of measure-
ment and depreciation). So
Kt+1 −Kt = Ntst −Kt (28)
2) Given the previous period’s saving decision, you might be tempted to think of both
capital and labor as inelastically supplied in period t. But that saving decision was made
with (perfect foresight) knowledge of rt.
2.2 Dynamics
Recall
kt+1 = st/(1 + n) and st = s(wt, rt+1) (29)
so that
kt+1 = s(wt, rt+1)/(1 + n) (30)
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3 SOCIAL SECURITY
where
wt = f(kt)− ktf ′(kt)⇔ wt = w(kt) with w′ > 0
rt+1 = f ′(kt+1)⇔ rt+1 = r(kt+1) with r′ < 0
Thus the dynamic behavior of the economy can be summarized in the following non-
linear difference equation in k.
kt+1 = s[w(kt), r(kt+1)]/(1 + n) (31)
We can derive the slope in (kt+1, kt)-space as follows.
(1 + n)dkt+1 = sww′dkt + srr
′dkt+1
dkt+1 = [sww′t/(1 + n− srr′t+1)]dkt
Don’t Forget: w′ = dw/dkt r′ = dr/dkt+1
Note that dkt+1/dkt > 0 unambiguously IF we make the classical assumption that
sr > 0.
The model doesn’t guarantee existence, uniqueness, or stability of a steady state.
For the moment, let’s assume there is a unique, non-zero k steady state, k∞, and that
1 > dkt+1/dkt > 0 at k∞.
At this point you should look at the relevant GRAPH in Blanchard and Fischer:
3 Social Security
Here we will consider a pay as you go system, which is roughly the US system. Current
collections from young workers δt are fully disbursed to the current old. So the young pay
δt into social security when working and receive (1 + n)δt+1 when retired.
Copyright © 2013 by Alan G. Isaac 12
3.1 Partial Equilibrium Effects 3 SOCIAL SECURITY
We continue to think of st as per capita saving by the young at time t out of their
disposable income, which is now wt − δt. We now need to keep track of the consumption
of both the young and the old each period, so we modify our notation accordingly.
The budget constraints becomes
cyt = wt − δt − st cot+1 = Rt+1st + (1 + n)δt+1 (32)
Therefore the problem is
maxsu[wt − δt − st] + βu[Rt+1st + (1 + n)δt+1] (33)
FOC:
−u′0 + βRt+1u′1 = 0⇔ st = s(wt, δt, Rt+1, δt+1) (34)
Let’s assume δt constant at δ so that st = s(wt, Rt+1, δ). Given w and r, we can totally
differentiate the first order condition with respect to s and δ.
u′′yds+ u′′ydδ + βR2u′′ods+ βR(1 + n)u′′odδ = 0 (35)
Using this, we find
∂s
∂δ= −
u′′y + β(1 + n)Ru′′ou′′y + βu′′oR
2(36)
3.1 Partial Equilibrium Effects
1. ∂s/∂δ < 0! Social security decreases individual private saving. [HW: intuition for
this. Hint: note that when δt 6= δt+1, changing either one has the same qualitative
effect on saving, so graph the effect of each of these changes separately.]
2. |∂s/∂δ|><1 as n><r
Copyright © 2013 by Alan G. Isaac 13
3.2 General Equilibrium Effects 3 SOCIAL SECURITY
3.2 General Equilibrium Effects
We know
↑ δ ⇔↓ st ⇔↓ kt+1 ⇔↑ rt+1 (37)
but the crucial feedback from rt+1 to st is theoretically ambiguous.
Given kt, what is net (general equilibrium) effect of increase social security payments
(↑ δ) on st (and therefore on kt+1)?
st = s[w(kt), R(kt+1), δ] (38)
Recall
(1 + n)kt+1 = st (39)
So
(1 + n)dkt+1 = sddδ + sRR′dkt+1 (40)
Rearranging, we get
dkt+1/dδ = sd/[1 + n− sRR′] (41)
Since sd < 0, the sign of dkt+1/dδ depends on the denominator 1 +n− sRR′. This can be
interpreted in terms of the slope of the saving and “investment” functions.
At this point you should GRAPH s/(1 + n) vs R and kt+1 vs f ′, noting that the first
has slope sR/(1 + n) and the second has slope 1/R′.
We will assume the denominator is positive (1 + n− sRR′ > 0), so that dkt+1/dδ < 0.
Justification: unique, stable, and non-oscillatory equilibrium requires 0 < dkt+1/dkt < 1.
Recall that
dkt+1
dkt=
sww′t
1 + n− sRR′t+1
(42)
[or, since w′ = −kR′, dkt+1/dkt = −swktR′t/(1+n−sRR′t+1)] so that (1+n−sRR′t+1) > 0
Copyright © 2013 by Alan G. Isaac 14
4 THE GOLDEN RULE
is necessary for dkt+1/dkt > 0.
At this point you should GRAPH the shift of the equilibrium accumulation curve
under these assumptions.
Capital accumulation slows, steady state k decreases. Is this bad? Depends on dy-
namic efficiency. See the discussions p. 113 of B&F.
4 The Golden Rule
Recall
Ct = Ntcyt +Nt−1cot (43)
so
c ≡ Ct/Nt = cyt + cot/(1 + n) (44)
or
St = F (Kt, Nt)− Ct ⇔ ct = f(kt)− [syt − kt] (45)
Recall also that
(1 + n)kt+1 = st ⇔ ct = f(kt)− [(1 + n)kt+1 − kt] (46)
So c∞ = f(k∞)− nk∞
Note
dc∞dk∞
= f ′(k∞)− n><0 as f ′(k∞)><n (47)
Golden Rule: f ′(k∞) = n Maximizes Steady State Consumption Per Capita.
HW: Suppose f ′(k)− n > 0. What happens if we steal dk from capital stock at time
t? Show p.103 B & F ct+i > c∞ ∀i > 0.
So we can give dk to period t consumers: all better off
Copyright © 2013 by Alan G. Isaac 15
4 THE GOLDEN RULE
Exercise 5
Summarize Blanchard and Fischer’s discussion in section 3.2 of a fully funded social
security system.
Exercise 6
Let F (λK, λN) ≡ λF (K,N). Show that this first degree homogeneity (constant returns
to scale) implies that
FKK + FNN = F (K,N) (48)
so total product exactly exhausted when factors are paid their marginal products. [Hint:
Differentiate w.r.t. λ and evaluate at λ = 1.]
Suggested reading: Blanchard & Fischer ch.3; Silberberg chapter 12.; Allais 1947,
Economie et Interet; Samuelson 1958 JPE 66(6); Diamond 1965 AER 55(5).
Copyright © 2013 by Alan G. Isaac 16
5 MORE COHORTS
5 More Cohorts
In this section, we consider a simplified version of Auerbach and Kotlikoff (1987), following
Heer and Maussner (2009).
Households live Tw + T r economic years (working and retirement). The total popula-
tion is normalized to one. So each generation has measure 1/(Tw + T r.
Consider a 60 generation OG model, where Tw = 40 and T r = 20. so each generation
has measure 1/60.
Copyright © 2013 by Alan G. Isaac 17
5 MORE COHORTS
5.0.1 Retired Individuals
The value function of an individual depends on the aggregate variables K and N , as wellas on their individual holding k.
Consider a retired individual of generation s. They want to maximize the value ofconsumption over the rest of their lifetime.
V s(ks, Kt, Nt) = maxcu(cs, 1) + βV s+1(ks+1, Kt+1, Nt+1) (49)
For a retired indvidual, the transition equation for k is
ks+1 = (1 + r)ks + b− cs (50)
Plugging into the recursive value equation gives us our Bellman equation:
V s(ks, Kt, Nt) = maxcs
u(cs, 1) + βV s+1((1 + r)ks + b− cs, Kt+1, Nt+1) (51)
First-order condition:0 = ucs − βV s+1
ks+1 (52)
Envelope condition:V sks = (1 + r)βV s+1
ks+1 (53)
Joint implication:(1 + r)ucs = V s
ks (54)
Looking at that one period ahead:
(1 + r)ucs+1 = V s+1ks+1 (55)
Putting these together,ucs = (1 + r)βucs+1 (56)
Note that this is a second-order difference equation in k, as we see by substittuting for cfrom the transition equation.
ucs [(1 + r)ks + b− ks+1, 1] = (1 + r)βucs+1 [(1 + r)ks+1 + b− ks+2, 1] (57)
This is our Euler equation.
Copyright © 2013 by Alan G. Isaac 18
5 MORE COHORTS
Suppose an agent is in his first year of retirement, with wealth kTw+1. Then we are
trying to plan 19 values of ks (s = 42, ..., 60) with 19 Euler equations. (Remember,k61 = 0.)
Solution strategy for the retirement path, given a target ks. (Here our target will bethe k for the first year of retirement that is implied by being borh with no beques. (I.e.,our underlying target is k0 = 0.
� set k61 = 0 (dead; no bequests)
� guess k60
� use our Euler equation to work backwards to implied ks
� if implied ks not equal to ks, adjust guess and try again
Given one-period-ahead values for k and c, we can compute deviation from the Eulercondition as follows:
def r f o l d ( k0 , k1 , c1 ) :””” Return f l o a t , d e v i a t i o n from Eu l e r c o n d i t i o n . ( See HM p . 4 6 0 . )
Use t h i s t o f i n d k0 so t h a t Eu l e r c o n d i t i o n s a t s i f i e d .
”””
global pen , beta , rc0 = (1+r ) *k0 + pen − k1return u c ( c0 , 1 ) / beta − u c ( c1 , 1 ) *(1+ r )
Use this to choose k0 so that Euler condition satsified. That is, use the Euler equationerror as a root function in an optimization.
( k0 , ) , , jcode , = f s o l v e ( r f o l d , k1 , args=(k1 , c1 ) , f u l l o u t pu t=True )
Copyright © 2013 by Alan G. Isaac 19
5 MORE COHORTS
5.0.2 Working Individuals
We handle working individuals essentially the same way, but they must also choose theirlabor supplies.
Assume there are no bequests, so they are born with 0 wealth.Consider a working individual of generation s. They want to maximize the value of
consumption and leisure over the rest of their lifetime.
V s(ks, `s, Nt, Kt) = maxcu(cs, `s) + βV s+1(ks+1, `s+1, Kt+1, Nt+1) (58)
For a working indvidual, the transition equation for k is
ks+1 = (1 + r)ks + w(1− τ)(1− `)− cs (59)
Plugging into the recursive value equation gives us our Bellman equation:
V s(ks, `s, Nt, Kt) = maxcs,`s
u(cs, `) +βV s+1((1 + r)ks +w(1− τ)(1− `)− cs, `s+1, Kt+1, Nt+1)
(60)We now have two first-order conditions:
0 = ucs − βV s+1ks+1
0 = u`s − βw(1− τ)V s+1ks+1
(61)
The envelope condition is unchanged:
V sks = (1 + r)βV s+1
ks+1 (62)
Our earlier work showed us
(1 + r)ucs+1 = V s+1ks+1 (63)
so our first order conditions imply
ucs = β(1 + r)ucs+1
u`s = w(1− τ)ucs(64)
We can again plug in our transition equation to get a difference equation system in k thatwe will call our Euler equations. Given one-period-ahead values for k, n, and c, we canagain compute deviation from the Euler conditions:
def r f young ( kn0 , k1 , n1 , c1 ) :global beta , rk0 , n0 = kn0c0 = (1+r ) *k0 + (1−tau ) *w*n0 − k1r f 1 = u c ( c0 ,1−n0 ) /beta − u c ( c1 ,1−n1 ) *(1+ r )r f 2 = (1−tau ) *w*u c ( c0 ,1−n0 ) − u e l l ( c0 ,1−n0 )return ( r f1 , r f 2 )
Copyright © 2013 by Alan G. Isaac 20
5 MORE COHORTS
Once again, use this to choose k0 so that Euler condition satsified. That is, use theEuler equation error as a root function in an optimization.
( k0 , ) , , jcode , = f s o l v e ( r f young , ( k1 , n1 ) ,\args=(k1 , c1 ) , f u l l o u t pu t=True )
Copyright © 2013 by Alan G. Isaac 21
5.1 Path Computation 5 MORE COHORTS
5.1 Path Computation
def pathupdate ( kbar , nbar ) :global nq1 , pen , w, rw = y n ( kbar , nbar ) #c om p e t i t i v e wage
r = y k ( kbar , nbar ) − de l t a #c om p e t i t i v e i n t e r e s t r a t e
#compute p e n s i o n ( g i v e n r e p l a c emen t r a t i o & income t a x r a t e )
pen = rep * (1−tau ) * w * nbar * 3/2 .0
k0 1 , k0 = 0 , 0 #j u s t a d e c l a r a t i o n ; t h e s e v a l u e s aren ' t u s ed
#born w/o wea l t h , so s e e k ( bac kward i n d u c t i o n ) p a t h where k [ 0 ]=0
q1 = 0 ; #maximum i t e r a t i o n s i s nq1 ( i . e . , 30 ) ; f o r c e burn i n
while ( q1 < nq1 ) and ( ( q1<=5) or ( abs ( k0 )>=to l k ) ) :q1 = q1 + 1i f q1 > nq1 :
print "WARNING: kpath(kbar,nbar) convergence failed"
breaki f verbose : print "iteration {} over q1:" . format ( q1 )
#upda t e f i n a l a s s e t l e v e l ( u s e HM ' s i n i t i a l v a l s )
i f q1==1: k60 = 0.15e l i f q1==2: k60 1 , k60 = k60 , 0 . 2else : k60 1 , k60 = k60 , updat e k l a s t ( k60 1 , k0 1 , k60 , k0 )
# NOW, g i v e n k60 , work b a c kwa r d s t o k0 ( h o p i n g f o r 0) .
### FIRST , compute o p t im a l r e t i r e d d e c i s i o n s ( wo r k i n g
. . . b a c kwa r d s ) .
kpath = kpath o ld ( r f o l d , k60 , 0 , tr −1) #( no b e q u e s t )
### SECOND, compute t h e o p t im a l d e c i s i o n s w h i l e wo r k i n g .
### (Remember , r e t i rmemen t y e a r s d e t e rm i n e d a bo v e . )
knpaths = kpath yng ( r f young , kpath [−1] , kpath [−2] , 40)kpath . extend ( knpaths [ 0 ] )
k0 1 , k0 = k0 , kpath [−1]i f True : print ”””
q : { q } q1 : { q1 } k0 : { k0 } k l a s t : { k l a s t }””” . format (q=q , q1=q1 , k0=kpath [−1] , k l a s t=kpath [ 1 ] )return kpath [ : : − 1 ] , knpaths [ 1 ] [ : : − 1 ]
Copyright © 2013 by Alan G. Isaac 22
5.2 Equilibrium 5 MORE COHORTS
5.2 Equilibrium
Aggregate production: each period is
Y = KαN1−α (65)
Equilibrium requires
� Consistency of individual behavior and aggregate outcomes:
Nt =Tw∑s=1
nstTw + T r
Kt =Tw+T r∑s=1
kstTw + T r
Note that these sums are weighted by the appropriate population masses.
� Competitive factor markets clear each period:
w = (1− α)(K/N)α r = α(N/K)1−α − d
� Satisfaction of the consumers’ Euler conditions.
� Goods market clearingF (N,K) = C + I
where
C =T 2+T r∑s=1
cs
TW + T rI = K+1 − (1− d)K
� The government budget constraint is satisfied
τwN = bT r
T r + Tw
Note on the government budget constraint: The budget is balanced each period. Taxesfinance retirement pensions of b per retiree. Note total pension expenditure is multipliedby the fraction of the population receiving benefits, because the population has beennormalized to 1.
Copyright © 2013 by Alan G. Isaac 23
5.3 HM’s Simulation 5 MORE COHORTS
Macro solution strategy:
� guess K and N ,
� iteration step: solve optimization for the implied Knew and Nnew
� while the change is too big, set (K,N)← (Knew,Nnew) and repeat the iteration step
kpath , npath = pathupdate ( kbar , nbar )#s t o r e o l d K & N, t h en up d a t e K & N ( b u t no t t o o f a s t )
kold , nold = kbar , nbarkbar = phi * kold + (1−phi ) *np .mean( kpath )nbar = phi *nold + (1−phi ) *np .mean( npath ) *2/3
5.3 HM’s Simulation
In HM, tility is dervied from consumption (c) and leisure (`) according to
u(c, `) =((c+ ψ)`γ)1−η − 1
1− η(66)
5.3.1 Calibration
HM’s parameter values
� η = 2
� β = 0.99
� α = 0.3
� d = 0.1
� Tw = 40
� T r = 20
� γ = 2.0
� rep = 0.3
� τ = rep/(2 + rep)
� ψ = 0.001
Comment on the replacement ratio:
rep =b
(1− τ)wn(67)
is set to 0.3. The pension b is an implied value.
Copyright © 2013 by Alan G. Isaac 24