ECONOMICS AND FINANCE
OF PENSIONS
Lecture 2
LIFE CYCLE MODELS
Dr David McCarthy
Life cycle models and how to solve them
• Last lecture we looked at the issue of inter-temporal
consumption
• We had some simple economic models of 2-period
consumption
• Also, we looked at our first OLG model, which we used
to derive insights about the macro-economics of
pensions and savings
• In the QCS we tested some of our insights about
demographic affects on asset returns of our model
© Dr David McCarthy All rights reserved
Today’s lecture
• We are interested in modelling how and why people save
over their lives (pensions are an important form of savings)
• We have to learn to walk before we can run
• So we start with our simple 3-period model, and solve it in a
new way (dynamic programming)
• We extend it to an n-period model and solve that with the
same technique
• We then introduce the Life Cycle / Permanent Income
Hypothesis and show results of empirical tests of it
• We present an alternative theory of life-cycle saving and
present empirical evidence about it
© Dr David McCarthy All rights reserved
Intertemporal consumption recalled
• Recall our two-period intertemporal consumption model:
max U (c1 , c2 )
Endowment
Consumption
c2
s.t. c1 +
=e
In time 1
1+r
Consumption in time
2, discounted at rate r
U (c1 , c2 ) = u(c1 ) + r u(c2 )
u ' ( x)  0
u ' ' ( x)  0
u ' (0)  
INADA CONDITIONS ENSURE
INTERIOR OPTIMUM
© Dr David McCarthy All rights reserved
Why stop at two periods?
•Individuals live for a long time, and make consumption
decisions quite frequently, so we should try to model that
max U (c1 ,..., cw )
•Example:
{c1 ,...cw }
 c (1  r )
1i
i
e
i
U (c1 ,..., cw )    i u (ci )
i
•(This is still a very stylised model)
•What features of “real” individuals are we missing?
© Dr David McCarthy All rights reserved
Other real-world factors which may be important
•
•
•
•
Wages (“human capital”)
Retirement
Borrowing / liquidity constraints
Risk
- Wage shocks (“risky human capital”)
- Asset returns
- Death
- Unemployment
- Health shocks
© Dr David McCarthy All rights reserved
More real-world factors
• Bequest motives
- One reason individuals save is to leave assets to their heirs,
so their own consumption is not the only thing we need to
look at
• Social security systems
• Pensions
• Families (consumption is divided between more/fewer
individuals)
• Leisure vs monetary consumption (endogenous retirement)
Not consumed
evenly across life
Often see a “dip” in
consumption during
retirement
© Dr David McCarthy All rights reserved
Road map for the next 90 minutes
• We will now build up theory to end up with a model of a
stylised individual’s entire life (called life-cycle incomeconsumption models in economics)
• We will learn how to write out the models and how to go
about solving them (sometimes not a trivial exercise)
- Solutions usually rely on numerical techniques
(analytical solutions aren’t possible so we have to do
this)
• As we build more theory, I want you to think about the
complications I have listed above and how we might
include them in our models
• The lecture will end with more applied work, so don’t
© Dr David McCarthy All rights reserved
Recall our OLG
• Let’s look more closely at the theory again.
• Recall our 3-period OLG model:
max u(c1 , c2 , c3 )
• Agent:
• B.C.:
PV of consumption
in periods 1,2 & 3
c3
c2
1
c1 +
+
= 1+
2
1 + r (1 + r )
1+ r
Income in
Periods 1 & 2
• For convenience, we write
u(c1 , c2 , c3 ) = u(c1 ) + r u(c2 ) + r 2u(c3 )
© Dr David McCarthy All rights reserved
General solution to OLG model
• There are two ways to solve this problem
• Method One (static optimisation):
• Lagrange Multipliers or constraints explicitly and
maximise
c1 (1+ r )2 + c2 (1+ r ) + c3 = (1+ r )2 + (1 + r )
• In this case we have
c3 = (1+ r )2 + (1+ r ) - c1 (1+ r ) 2 + c2 (1+ r )
max u(c1 ) + r u(c2 ) + r 2u((1+ r )2 + (1+ r ) - c1 (1+ r )2 - c2 (1+ r ))
;c2 }
•{c1Now,
we have the following:
© Dr David McCarthy All rights reserved
General solution to OLG model
•We take first order conditions w.r.t. our two variables, and
/
2
2 /
the system cbecomes:
:
u
(
c
)
+
r
(1
+
r
)
u (c3 ) = 0
1
1
u / (c1 ) = r 2 (1+ r )2 u / (c3 )
c2 : r u / (c2 ) + r 2 (1+ r )u / (c3 ) = 0
u / (c2 ) = r (1+ r )u / (c3 )
c1 = c2 = c3
•And, once again, we obtain if p(1+r)=1,
, etc
© Dr David McCarthy All rights reserved
Dynamic programming solution
• Method Two: Stochastic Dynamic Programming (DP)
•Difficult and represents a large investment in time to
understand
•However, the payoff is large, because DP is the basis of:
•
- options pricing for e.g. American options
•
- dynamic investment strategies
•
e.g. phased pension derisking, lifestyling
•
- understanding economic models of the lifecycle
© Dr David McCarthy All rights reserved
Principle of Optimality
• Aside on principle of optimality of dynamic decision
making (Bellman, 1957)
• “An optimal policy has the property that whatever the
initial
• state & initial decisions are, the remaining decisions must
• constitute an optimal policy with regard to the state
resulting
• from the first decision.”
© Dr David McCarthy All rights reserved
Example: Mornington Crescent
•
South Kensington
Green Park
Warren Street
Mornington Crescent
© Dr David McCarthy All rights reserved
Proof of principle of optimality
•Proof is easy (by contradiction)
•Assume that the shortest path from South Kensington to
Mornington Crescent goes through Warren Street and Green
Park, but that the shortest path from Green Park to
Mornington Crescent does not pass through Warren Street.
•Then the alternative route, from South Kensington to Green
Park, and from then the shortest route from Green Park to
Mornington Crescent will be shorter than the route South
Kensington – Green Park – Warren Street – Mornington
Crescent. CONTRADICTION!!!
© Dr David McCarthy All rights reserved
Using the principle of optimality
• How do we use this to solve the economic model?
• If we can express the budget constraint & the objective
function recursively, then we have dynamic problem and
can use the principle of optimality
• Let’s start with the budget constraint
• It’s clear that at any point in time, the present value of
future consumption must equal the present value of
future income plus the value of current wealth
© Dr David McCarthy All rights reserved
Recursive budget constraint
wt
3
3
• If we define
, the- jindividual’s
“wealth”
at time t as
+i
- j+ i
wi = å (1 + r )
c j - å (1 + r )
yj
j= i
Current
wealth
j= i
Present value of
future consumption
Present value of future
income
• Then we can
that,
wt +see
=
(wt + yt - ct )(1+ r )
1
• In our case
y1 = y2 = 1; y3 = 0
© Dr David McCarthy All rights reserved
Recursive objective function
• Write the objective function recursively:
3
Vi ( wi ) = max
{ci ,..., c3 }
å
j= i
3
j
r u (c j ) s.t.
å
j= i
3
(1 + r )
- j+ i
c j = wi +
å
(1 + r )-
j+ i
yj
j= i
• It’s a function of wealth, because this determines the
future value of consumption that can be supported (it is
also a function of future income, but we assume that is
fixed)
• Subject to our original budget constraint, as you can see
above
© Dr David McCarthy All rights reserved
Final period solved first
• Solve the model using “backward induction”
• We start in the final period
• Individuals then will consume all their wealth, because
they are
die u
in(cthe
next period
V3 (going
w3 ) = tomax
3 ) s.t. w4 = ( w3 - c3 )(1 + r ) = 0
{c3 }
• So
w3 = c3
• We can see that
© Dr David McCarthy All rights reserved
Then move one period back…..
• Now go one period backwards:
V2 ( w2 ) = max u(c2 ) + r V3 ( w3 )
{c2 }
s.t. w3 = (w2 + y2 - c2 )(1+ r )
• Go one period backwards again:
Discounted utility
of optimal
consumption in all
future time periods
Utility of
consumption in
period 2
V1 ( w1 ) = max u(c1 ) + r V2 ( w2 )
{c1 }
s.t. w2 = (w1 + y1 - c1 )(1+ r )
• By the principle of optimality, if we solve both of these
problems we will have obtained the optimal path.
• These eqn’s are called the “Bellman equations”
© Dr David McCarthy All rights reserved
Model solution
c3 =
• Let’s solve the model using
thewBellman
Equations:
3
V3 (w3 ) = u(c3 )
V2 ( w2 ) = max u(c2 ) + r V3 (w3 )
{c2 }
= max u(c2 ) + r V3 ((w2 + y2 - c2 )(1+ r ))
{c2 }
• F.O.C:
u / (c2 ) - (1+ r )r u / ((w2 + y2 - c2 )(1+ r )) = 0
© Dr David McCarthy All rights reserved
Takes a while…..
u / (c2 ) = (1+ r )r u / ((w2 + y2 - c2 )(1+ r ))
• (1If+ r )r = 1
thenc2 = (w2 + y2 - c2 )(1+ r )
( w2 + y2 )(1 + r )
c2 =
2+ r
( w2 + y2 )(1 + r )
c3 = ( w2 + y2 )
2+ r
( w2 + y2 )(1 + r )
c3 =
= c2
2+ r
© Dr David McCarthy All rights reserved
Now move to the first period (phew!)
V2 ( w2 ) = u (
•
So,
( w2 + y2 )(1 + r )
) + r u (××)
2+ r
( w2 + y2 )(1 + r )
= (1 + r )u (
)
2+ r
V1 ( w1 )
• Now we do it for
:
V1 ( w1 ) = max u(c1 ) + r V2 ( w2 )
{c1 }
= max u(c1 ) + r V2 ((w1 + y1 - c1 )(1+ r ))
{c1 }
© Dr David McCarthy All rights reserved
Still busy…….
•
= max u (c1 ) + r (1 + r )u(( w1 + y1 - c1 )(1 + r ) +
{c1 }
y2 (1 + r )
)
2+ r
2
2
(
w
+
y
c
)(1
+
r
)
+ y2 (1 + r )
(1
+
r
)
/
/
1
1
1
u (c1 ) = r (1 + r )
u(
)
(2 + r )
2+ r
2
(
w
+
y
c
)(1
+
r
)
+ y2 (1 + r )
/
1
1
1
= 1×u (
)
2+ r
(2 + r )c1 = (w1 + y1 - c1 )(1+ r )2 + y2 (1+ r )
© Dr David McCarthy All rights reserved
Not an easy task at all…..
(2 + r + (1+ r )2 )c1 = (w1 + y1 )(1+ r ) 2 + y2 (1+ r )
•
( w1 + y1 )(1 + r ) 2 + y2 (1 + r )
c1 =
3 + 3r + 3r 2
(1 + r )2 + (1 + r )
w1 =if 0, y1 = 1, y2 = 1
=
3 + 3r + 3r 2
•You can showc1 = c2 = c3 as before. (It’s complicated
because you need to substitute in for w2 )
•This therefore demonstrates what we already knew, and it
seems incredibly messy. (In other words, there must be an
easier way!)
© Dr David McCarthy All rights reserved
There is an easier way….
• There is an alternative method of solution of these
problems
• Uses the so-called “Envelope Theorem”
• This
Vsimply
( w ) =says
maxthat
u(cif ) + r V (w )
2
2
2
{c2 }
3
3
= max u(c2 ) + r V3 ((w2 + y2 - c2 )(1+ r ))
{c2 }
w2
c2 we can take the derivative of both sides w.r.t.
• Then
treat
means
¶
V (was) fixed.
= ¶ rThis
V ((w
+ y that:
- c )(1+ r ))
¶ w2
2
¶ w2
2
=
¶
¶ w2
3
2
2
and
2
r (1+ r )V3' (w3 )
© Dr David McCarthy All rights reserved
The Envelope Theorem
g ( y)  Vn (h(x, y))
Imagine we have the following set-up: Vn1 ( x)  max(
y
Treat optimum value of y
•
V
(
x
)
w.r.t.
x.
n

1
Take the first derivative of
as though it is fixed
Vn'1 ( x) 
d
max( g ( y)  Vn (h(x, y)) 
dx y
h ( x , y )
x
Vn' (h(x, yˆ ))
This is called the envelope theorem - because the function Vn 1 ( x)
is an envelope of the set of functions g ( y)  Vn (h(x, y)) for all y.
Vn-1(x)
g(y2)+ Vn(h(x,y2))
g(y1)+ Vn(h(x,y1))
g(y3)+ Vn(h(x,y3))
© Dr David McCarthy All rights reserved
My first Euler Equation…..
• So we have:
¶
¶ w2
V2 (w2 ) = r (1+ r ) ¶¶w3 V3 (w3 )
• The FOC gives:
• So then we must have
• This means that
u '(c2 ) = r (1+ r ) ¶¶w3 V3 (w3 )
¶
¶ w2
V2 (w2 ) = u '(c2 )
¶
¶ w3
V3 (w3 ) = u '(c3 )
u '(c2 ) = r (1+ r )u '(c3 )
EULER EQUATION
• and so
© Dr David McCarthy All rights reserved
Using Euler Equations
• So then we can formulate all the c’s in terms of, say, c1
• For example, if r (1+ r ) = 1 then c1 = c2 = c3
• Then we use the budget constraint to determine what level
of consumption can be supported by lifetime resources:
3
3
å
- j+ 1
(1 + r )
j= 1
c j = w1 +
å
(1 + r )-
j+ 1
yj
j= 1
w1 = 0, y1 = 1, y2 = 1
• If, for example,
then
1
1
1
c j (1 +
+
) = 1+
2
1 + r (1 + r )
1+ r
• and
1
1+
1+ r
cj =
1
1
1+
+
1 + r (1 + r ) 2
© Dr David McCarthy All rights reserved
We can now easily solve many models
• For instance, in an n-period model with m working periods
and n-m retired periods, if r (1+ r ) = 1 we can immediately
write
am
ci =
an
• Now you can see where the lifecycle hypothesis comes from
• You can also solve models where consumption increases or
1
decreases with time
i i r- 1
(1 + r ) r a m
• Verify at home that if u(ci ) = log(ci ) then ci =
r
a
and check that this case nests the previous
n
r (1if+ r ) = 1
•
one
(because then we don’t care about
the shape of the utility function, provided it is concave)
© Dr David McCarthy All rights reserved
Comparative statics
d
• Imagine there is an unexpected
shock to wealth (e.g. you
win the lottery), size
- What will be the effect on consumption?
+ rm
) =more
1
- Imagine wealth is zero, wages =r1(1for
periods
and you have n periods left to live,
¶ ci
1
1
am + d
=
»
ci =
¶ d an
n if r is small
a
so
n
- So therefore the effect on (annual) consumption is
~1/n
© Dr David McCarthy All rights reserved
More comparative statics
d
• Imagine there is an unexpected permanent shock to
income (e.g. you get a permanent pay rise, size )
(1 + d)am
¶ ci am
m
ci =
=
»
an
¶ d an
n if r is
so
small
- So therefore the effect on consumption is ~m/n ~ 1
© Dr David McCarthy All rights reserved
Recap: DP ingredients
V
1.
2.
3.
4.
Objective function ct
Choice variables
wt
State variables
Updating equation (budget constraint)
wt + 1 = f (wt , ct )
• We then write each period’s optimisation using a
Bellman equation, derive Euler equations and solve the
model.
• It is not always possible to use Euler equations to make
things easy, and for complicated models we must almost
always resort to computers.
© Dr David McCarthy All rights reserved
Life-cycle / Permanent Income hypothesis
• The idea that “rational”, forward-looking individuals smooth
consumption over their lives
• Originally due to Friedman, Modigliani, and Brumberg
• Individuals form an expectation about how much they can
“safely” consume over their whole lives (“Permanent Income”),
and they borrow when this level of consumption exceeds their
income, and save otherwise
• Implications (as we have seen):
• Marginal propensity to consume out of temporary wealth
shocks = 1/n
• Marginal propensity to consume out of permanent income
shocks = m/n
© Dr David McCarthy All rights reserved
Age-earnings profiles
•Age-earnings profile typically have an inverse U-shape
Peak
Retirement
•If this is true, what would the LC/PIH predict would be the
lifetime consumption profile (if r (1+ r ) = 1 )?
© Dr David McCarthy All rights reserved
Life-cycle / Permanent income hypothesis
Consumption,
Wage
• Income &
Permanent
Income
Wage Income
Assets
Consumption
Age
© Dr David McCarthy All rights reserved
Empirical tests of PI/LCH
• High frequency tests typically support the PI/LCH
• e.g. Hall (1978) Journal of Political Economy, 86(6), 971-989
• Uses quarterly US consumption and income data from 19481977
• He finds the following relationship:
ct  23  1.076ct 1  0.049 yt 1  0.051yt 2  0.023 yt 3  0.024 yt 4
(11) (0.047)
Consumption at
time t
(0.043)
(0.052)
(0.051)
(0.037)
Lagged values of income
•So consumers do a reasonably good job of smoothing out shortterm fluctuations in income
© Dr David McCarthy All rights reserved
Low-frequency tests
•
e.g. Carroll (1997) Quarterly
Journal of Economics, (112)1, 1-55
constructs 5-yearly consumptionincome profiles for different
occupational groups and shows
that average consumption closely
tracks average income until
roughly age 40-45
Thereafter, some private saving
for retirement typically begins
(its always difficult in these data
to know how pensions have been
handled)
© Dr David McCarthy All rights reserved
Stylised picture of life-time savings
Finally, around age 45
private retirement
saving typically
begins
Consumption closely
tracks income up to
middle age
Initially build up a
small stock of assets
In fact, consumption tracks disposable income closely in
the long run, and a stylised picture of lifecycle
consumption and income might look like this. Why is
this?
© Dr David McCarthy All rights reserved
Which real-world factors might explain this?
•
•
•
•
Wages (“human capital”)
Retirement
Borrowing / liquidity constraints
Risk
- Wage shocks (“risky human capital”)
- Asset returns
- Death
- Unemployment
- Health shocks
Can test with existing
theory
Need an extension to
the theory
© Dr David McCarthy All rights reserved
Borrowing constraints?
max u(c1 ) + u (c2 )
{c1 ;c2 }
c2 = (w1 - c1 + 1)
u '(cˆ1 ) = u '(cˆ2 )
ˆ
ˆ
FOC: c1 = w1 - c1 + 1
w1 + 1
cˆ1 =
2
c1- g
u (c ) =
1- g
• Let’s say
c1 £ w1
•Now, there is a constraint that
if we assume that
the
w1 + against
1
• individual
cannot
borrow
future income
cˆ = min( w ,
)
1
• Hence,
1
2
© Dr David McCarthy All rights reserved
Borrowing constraints?
•
Consumption in period 1
2
Wealth constraint
binds and c1 < c2
1.5
1
Wealth constraint does
not bind and c1 = c2
0.5
0
0
0.5
1
1.5
Wealth in period 1
2
2.5
If wealth constraint binds, individuals consume as much as they
can, and then “over” consume in the next period
Individual welfare would be improved if individuals could borrow
to smooth out their consumption across the different time periods
© Dr David McCarthy All rights reserved
More theory……
We need some more theory to include the other “complications” in
our model
• We can already include as many periods as we want
• No longer have to re-formulate the entire problem but can
easily extend it using the DP approach
More importantly, Bellman’s Principle of optimality applies in
expectation implying that we can introduce uncertainty very easily.
• This advance comes at the cost of possibly considerable mess
(mathematically speaking!)
© Dr David McCarthy All rights reserved
Example: Mornington Crescent
•
South Kensington
Green Park
Warren Street
Mornington Crescent
© Dr David McCarthy All rights reserved
Bellman Equation with Risk
•Bellman Equation with risk
max u(ci ) + E i r Vi+ 1 ( wi+ 1 )
{ci }
Conditional expectation at time i
•So, we don’t even need unconditional distributions, only
conditional ones (this is very useful if your state variables
follow Markovian processes)
© Dr David McCarthy All rights reserved
Bellman Equation with Risk
•
•
•
•
•
•
How can we use this?
Add uncertain investment returns
Add uncertain wages
Add mortality
All become manageable
wt + 1 = (wt in
+ the
yt - DP
ct framework
)(1+ rt )
How?
Asset return
uncertainty
Income uncertainty
max u(ct ) + Er p t + 1Vt + 1 (wt + 1 )
{ct }
Equals 1 if individual is alive at time t+1
and 0 otherwise
© Dr David McCarthy All rights reserved
Effect of income uncertainty
max u(c1 ) + Eu(c2 )
{c1 ;c2 }
No discounting; no interest
c2 = (w1 - c1 + y2 ), y2 U (1- e,1+ e)
•Let’s say that
c1- g
u (c ) =
1- g
•FOC:
Income next period uniformly
distributed with mean 1
u '(c1 ) = Eu '(c2 ) = Eu '((w1 - c1 + y2 ))
=
ò
1+ e
1- e
=
1
2e
1
2 e (1- g )
( w1 - c1 + y2 )- g dy2
[( w1 - c1 + 1 + e)1- g - ( w1 - c1 + 1- e)1- g ]
•Have to solve this equation numerically (Solver in Excel); let’s
set w = 2 and calculate for different values of epsilon and
© Dr David McCarthy All rights reserved
Effect of income uncertainty
1.6
As uncertainty increases,
consumption decreases, and
precautionary saving increases
Consumption in period 1
1.4
•
Optimal level of consumption
when no uncertainty = 1.5
regardless of risk aversion (why?)
1.2
1
Effect becomes more pronounced as risk
aversion increases (more risk aversion
implies more precautionary savings)
At peak, consumption is only ~70% of no
uncertainty case – significant precautionary
saving
0.8
0.6
Gamma = 2
0.4
Gamma = 5
0.2
Gamma = 8
0
0
0.2
0.4
0.6
0.8
1
Epsilon
Lower consumption in period 1 means higher average consumption in
period 2: c1 + Ec2 = w1 + Ey2 = w1 + 1
© Dr David McCarthy All rights reserved
Effect of income uncertainty
V1 (w1 ) = max u(c1 ) + Eu(c2 ) s.t. c2 = ( w1 - c1 + y2 )
• Let
{c1 ;c2 }
• Now, we can add another period exactly as before:
V0 (w0 ) = max u(c0 ) + EV1 (w1 ) s.t. w1 = (w0 - c0 + y1 )
{c0 }
•Because we cannot expressc1 analytically, we cannot derive
an analytical expression for V ( w1 ) either and therefore need
numerical methods to solve the model
•However, we can still derive an Euler equation for
consumption, in this case, it will be:
•Hence, ifc1 is reduced because of income uncertainty is
period 2, this will cause c0 to fall, even if there is no income
uncertainty in period 1: the effect is important.
u '(c0 ) = Eu '(c1 )
© Dr David McCarthy All rights reserved
Asset return uncertainty
• To illustrate the effect of investment return uncertainty on
saving, let’s solve another 2-period model
No discounting
max u (c1 ) + Eu (c2 )
{c1 ;c2 }
c2 = (w1 - c1 + 1) R2 , R2 U (1- e,1+ e)
c1- g
u (c ) =
• Let’s keep
1- g
Asset return next period uniformly
distributed with mean 1
• FOC:u '(c1 ) = ER2u '(c2 ) = ER2u '((w1 - c1 + 1) R2 )
=
•
1
2e
ò
1+ e
1- e
R2 ( R2 ( w1 - c1 + 1))- g d R2
, say- g
= ( w1 - c1 + 1)
1
2 e (2- g )
[(1 + e) 2- g - (1- e) 2- g ]
= ( w1 - c1 + 1)- g f (e, g )
© Dr David McCarthy All rights reserved
Asset return uncertainty
c1
• We can solve analytically for
-g
1
c
-g
in this case
= ( w1 - c1 + 1) f (e, g )
-
c1 = ( w1 - c1 + 1) f (e, g )
-
c1 =
( w1 + 1) f (e, g )
-
1 + f (e , g )
1
g
1
g
1
g
© Dr David McCarthy All rights reserved
•Effect of investment uncertainty apparently more severe
because it affects entire savings, not just future income (so don’t
come to the conclusion that it’s a more important risk!)
•Consequence however is qualitatively identical to uncertainty in
wages: investment uncertainty results in precautionary saving
© Dr David McCarthy All rights reserved
Effect of mortality
max u(c1 ) + Ep 2u (c2 )
{c1 ;c2 }
Equals 1 if individual is alive at
time 2 and 0 otherwise
c2 = (w1 - c1 + 1), p 2
c1- g
u (c ) =
1- g
• Let’s keep
• FOC:
Bernoulli(1- e)
Pr(p 2 = 1) = 1- e
u '(c1 ) = Ep 2u '(c2 ) = Ep 2 Eu '((w1 - c1 + 1))
c1- g = (1- e)(w1 - c1 + 1)- g
-
1
g
c1 = (1- e ) ( w1 - c1 + 1)
•
•
So,
-
c1 =
(1 + w1 )(1- e )
-
1 + (1- e )
1
g
1
g
© Dr David McCarthy All rights reserved
•Higher the chance of dying in the next period, the higher
consumption is this period
•Higher risk aversion diminished this effect
•Can you explain this intuitively?
•What about uncertainty in mortality probability?
© Dr David McCarthy All rights reserved
Called “Buffer-Stock Savings” hypothesis
• Three crucial ingredients:
• High risk aversion
• High time preference
• Considerable income uncertainty
1. Households build up precaution savings over most of LC
2. Households will not dissave much in youth
3. Households defer retirement savings until later in life
© Dr David McCarthy All rights reserved
Factors affecting savings
•
•
•
•
•
•
•
Old age/retirement
Uncertainty about longevity
Uncertainty about wages
Uncertainty about health
Uncertainty about asset returns
Time preference
Changes in family size/structure
© Dr David McCarthy All rights reserved
Life-cycle / Permanent income hypothesis
Consumption,
Wage
• Income &
Permanent
Income
Wage Income
Assets
Consumption
Age
© Dr David McCarthy All rights reserved
Conclusions
•
•
•
•
•
•
Introduced life-cycle models
Showed techniques of how to solve them (DP)
Introduced LC-PIH
Went through some tests of LC-PIH
Examined motivations for saving
Examined some deviations from LC-PIH
– Butter-stock saving
© Dr David McCarthy All rights reserved