Macroeconomics
Class 5: Optimal consumption and saving decisions.
Fabio Canova
ICREA-UPF, BSGE, CREI, CREMed, and CEPR
January 2012
Outline
Household consumption-saving decisions
Finitely lived, certain economies.
In nitely lived, uncertain economies.
1
What is wrong with the Solow model?
- So far, the savings rate s (the marginal propensity to save) was assumed
to be a constant fraction of income.
- Since agents can consume or save their income yt = ct + st, this assumption implies that ct = (1 s)yt = cyt; where c = (1 s) is the marginal
propensity to consume.
- Many economists think that such a speci cation is problematic.
i) Are household really looking only at current income when they decide
consumption? (many borrow to buy durable goods, cars, houses, etc.).
ii) Why are people saving? Presumably to have future consumption. Given
the pro le of earnings, does it make sense to save a constant fraction of
your income all your life?
iii) What do people do in recessions? Try to stabilize their consumption
running down their savings. Empirically, there is evidence that households
tend to smooth their consumption (that is, personal income is typically
more volatile than personal consumption).
2
Permanent income theory
- Illustrate the theory with a two period model and partial equilibrium
(endowment) economy.
- We will extend it later to include production decisions and to have agents
living more than 2 periods (in nitely lived agents).
- Basic idea: consumption-saving decisions determined by the optimizing
behavior of individuals.
- This is a major building block of macro models. A large literature uses
this framework (growth, education decisions, asset holding, monetary and
scal policy decisions).
The problem
- There is a large number of identical households living two periods.
- They know when they are born they will receive endowment w0 and w1
in the two periods of life.
- They start with zero wealth in the rst period.
- Can borrow/save between the two periods at the gross real rate r = 1+ R
(principal plus interest), which is known in the rst period.
max U (co) + U (c1)
c0 ;c1
(1)
c0 + S0 = w0
c1 = w1 + rS0
(2)
(3)
subject to
where S0 are savings (if positive) or borrowing (if negative),
1 is the
discount factor - future utility may be worth less than current one, and r
is given.
- Solving for S0 from the period 0 budget constraint and putting it in the
period 1 constraint we have
c1 = w1 + r(w0
or dividing both sides by r we have
c0 )
(4)
w
c
c0 + 1 = w0 + 1
r
r
(5)
- (5) is the intertemporal budget constraint. It says that the ow of total
consumption in the two periods must be equal to the ow of total income
in the two periods (where the future is discounted by r).
- Agents can borrow and lend in the rst period of life but, in the end,
they must consume what is their lifetime (permanent) income.
- 1r is the price of future consumption relative to current consumption.
- Any combination (C(0), C(1)) within the triangle is feasible, given the
interest rate and the two endowments.
- Agents problem: Find the highest indi erence curve that is tangent to
the intertemporal budget constraint.
Example 1
- No future discounting ( = 1), r = 1
max U (co) + U (c1)
(6)
c0 + c1 = w0 + w1
(7)
c0 ;c1
subject to
How do you solve this problem?
1) Form the Lagrangian L = U (co) + U (c1) + (c0 + c1
maximize with respect to (c0; c1; ).
w0
w1) and
2) For this problem there is an easier way to nd the solution. Solve for
c1 = w0 + w1 c0 from the budget constraint, plug it into the utility
function, and maximize with respect to c0, i.e.
max U (co) + U (w0 + w1
c0
c0)
(8)
Taking the derivatives with respect to c0 we have
@U0
@c0
@U1
=0
@c0
(9)
An optimal decision implies that the marginal utility of consumption is
equalized in the two periods.
If U (c0) = log c0 and U (c1) = log c1, the optimality condition implies
1 = 1 : consumption is also equalized across periods, so that c = c =
0
1
c0
c1
0:5(w0 + w1).
Consumption in each period is half of the permanent income (consumption smoothing).
How does this compare with the standard consumption function?
- Standard consumption function : c0 = cw0; c1 = cw1.
- Optimal consumption function : c0 = c1 = 0:5(w0 + w1).
i) What is the e ect of an increase in w0 (temporary increase in income)?
- Standard consumption function:
c0 = c
- Optimal consumption function :
c0 =
w0;
c1 = 0.
c1 = 0:5 w0.
Any (temporary change) in income is smoothed across periods. If you
unexpectedly receive an inheritance you do not spend all of it in the same
period! You increase consumption from that point on.
ii) What is the e ect on an increase in w0 and in w1 (permanent increase
in income)?
- Standard consumption function:
c0 = c
- Optimal consumption function :
c0 =
w0;
c1 = c
c1 = 0:5( w0 +
w1.
w1).
The increase in (optimal) consumption is larger when the change in income
is permanent than when it is transitory. Why?
When the increase is permanent need to worry less about the future, i.e.
consume more and save less than when the change is temporary.
Example 2
- Future discounting ( < 1), r 6= 1
max U (co) + U (c1)
c0 ;c1
(10)
c1
w1
c0 +
= w0 +
r
r
(11)
@U (c0)
@U (c1)
= r
@c0
@c0
(12)
subject to
Optimality condition:
- intertemporal rate of time preferences ( 1 = subjective one period
interest rate). It measures impatience in consumption.
- r is the gross real interest rate received between two adjacent periods.
r is the e ective time discounting. If
r = 1 consumption is
constant over time (as in the previous example): discounting the future
and the return from saving just cancel each other out.
- If
r > 1; consumption path is upward sloping (MU of consumption
today is higher than the MU of consumption tomorrow).
- If
r < 1 consumption path is downward sloping (MU of consumption
today is lower than the MU of consumption tomorrow).
Depending on the relative magnitude of ( ; r), consumption path will
have di erent properties.
c1 1=
1 1=
- If U (c) =
(constant relative risk aversion (CRRA) utility) the
optimality condition is
c1 = ( r) c0
-Typically
1, so ( r)
1 if
r
What happens if discount rate falls (
1
lower) ?
What happens if the interest rate increases?
(13)
3
Consumption-saving problems when T > 2
How do you compute permanent income if T > 2?
- Suppose the world starts at t=0 and agents have wealth equal Q(0). In
each period of their life, they receive an endowment w(t).
- Let the gross interest rate prevailing from period t to period t + 1 be
r(t).
Permanent income (the present value of all future income at the beginning of time):
w(0)
w(1)
w(2)
P I = Q(0) +
+
+
+ :::
r(0)
r(0)r(1) r(0)r(1)r(2)
(14)
Example 3.1 Suppose Q(0) = 0; r(0) = r(1) = r(2) = r and T = 3.
w(0)
w(1)
w(2)
Then permanent income as of the beginning of time is r + r2 + r3
(Notice that in previous examples permanent income was computed as of
w(1)
w(2)
time 0. If we had done this here we would have w(0) + r + r2 .)
Present value of consumption:
c(0)
c(1)
c(2)
PV C =
+
+
+ :::
r(0) r(0)r(1) r(0)r(1)r(2)
(15)
Intertemporal budget constraint:
PI
PV C
(16)
You can not consume over your lifetime (in present value terms) more
than what you earn (i.e. you permanent income).
Evolution of wealth over time:
Q(1) = r(0)Q(0) + (w(0) c(0))
Q(2) = r(1)Q(1) + (w(1) c(1))
...
...
Q(t) = r(t 1)Q(t 1) + (w(t 1) c(t 1))
...
...
T
T
i
Y
X1 Y
Q(T + 1) =
r(j)Q(0) +
(
r(T j))(w(T i)
j=0
c(T
i)) + (w(T )
c(T ))
i=1 j=0
Q(T + 1) can not be negative. Why?
- If this was allowed, agents could set Q(T + 1) = 1 (borrow in nite
amount during their lifetime and never repay it) and enjoy in nite utility.
- Thus Q(T + 1)
0. As we will see, Q(T + 1) > 0 is not optimal.
Agents could increase consumption (and utility) by accumulating less in
every period without a ecting their Permanent Income.
- Hence, only Q(T + 1) = 0 is possible: we will see how we can derive this
more formally next.
Consumer problem:
max
T
X
c(t);Q(t+1) t=0
t U (c(t))
= U (c(0)) + U (c(1)) + 2U (c(2)) + : : : (17)
subject to
Q(t + 1) = r(t)Q(t) + (w(t)
Q(T + 1)
c(t))
0
(18)
(19)
Equivalently, the maximization is subject to the constraint
w(0)
w(1)
w(2)
P I = Q(0) +
+
+
+ :::
r(0)
r(0)r(1) r(0)r(1)r(2)
c(0)
c(1)
c(2)
PV C =
+
+
+ :::
r(0) r(0)r(1) r(0)r(1)r(2)
Assume that U (c) is strictly increasing and strictly concave.
(20)
Geometric discounting of utility.
Problem of nding f(c(t); Q(t + 1)gT
t=0 can be split into a set of smaller
problems since decisions about C (t); Q(t + 1) a ects only utility at t and
t + 1. That is, we replicate the two period solution for adjacent periods.
- Form the Lagrangian:
L=
T
X
t [U (c(t)) +
(t)(Q(t + 1)
r(t)Q(t) + (w(t)
c(t))] + Q(T + 1)
t=0
- The rst order (necessary) conditions (valid for every 0 < t
c(t) : U 0(t) = (t)
Q(t + 1) : (t) =
(t + 1)r(t)
(t) : Q(t + 1) = r(t)Q(t) + (w(t)
c(t))
T)
(21)
(22)
(23)
and (this is valid at t = T )
Q(T + 1) :
(T ) =
(24)
Since (19) is an inequality constraint, we also have the complementary
slackness condition (from Kuhn-Tucker theorem)
Q(T + 1) = 0
(25)
(25) means that either = 0; Q(T + 1) > 0 or that > 0; Q(T + 1) = 0.
The rst can only happen if the utility function has a satiation point (you
do not care if you bring assets in the grave....).
- Since we have assumed that u(:) is increasing in consumption, the only
possible solution is
Q(T + 1) = 0
(26)
- Using (24), we can rewrite the complementary slackness condition (25)
in a di erent way:
(T )Q(T + 1) = 0
(27)
Note that equation (22) implies
(t + 1) =
=
(t 1)
(t)
= 2
= :::
r(t)
r(t)r(t 1)
(0)
t Qt
j)
j=0 r (t
(28)
Assuming (0) = 1 (27) can also equivalently be written as
Q(T + 1)
T QT r (t
j=0
j)
=0
(29)
The complementary slackness conditions says that the present discounted value of the wealth at T + 1 must be zero.
Combining (21) and(22) we have
U 0(t) = r(t)U 0(t + 1)
(30)
This equation is called Euler equation: it describes how consumption is
optimally related across time.
- If
r(t) = 1 marginal utility of consumption is constant over time.
- If r(t) < (>)1 marginal utility of consumption is increasing (decreasing)
over time.
Conclusion: Optimal solution for the problem for nite T > 2 is obtained
by a sequence of two period (overlapping) problems.
3.1
Consistency of optimal plans
- Suppose agents maximize their utility at time t=1 forever by choosing
(c(0)c(1); c(2); c(3); : : : ; c(T ); : : :)
- Suppose at time equal t=2, they are given the option of remaximize utility
from time t=2 onward by choosing (c(0); c(1); c (2); c (3); : : : ; c (T ); : : :).
- The optimal plan is consistent at t=2 if (c(0); c(1); c(2); c(3); : : : ; c(T ); : : :)
= (c(0); c(1); c (2); c (3); : : : ; c (T ); : : :).
- The optimal plan is consistent at any t if (c(0); c(1); c(2); c(3); : : : ; c(t); : : :)
= (c(0); c(1); c(2); c(3); : : : ; c (t); : : :).
- Basically a plan is consistent if agents have no incentive to reoptimize as
time goes by.
- Why is it this relevant? There are going to be situations when agents may
want to reoptimize as they move forward, for example reneging promises
made before (e.g. defaulting).
- This is the insight that lead Kydland and Prescott to get the Nobel prize
a few years ago.
3.2
What happens if agents are in nitely lived?
- Why in nitely lived agents? Dynasties. Care about o -springs.
- Useful approximation for long horizon problems.
The mathematical formulation of the problem is the same. But now we
do not have a condition like Q(T + 1) = 0 since time is in nite.
- What are conditions we impose on the wealth if agents live forever?
The condition (29) is now
lim
T !1
Q(T + 1)
T QT r (T
j=0
j)
=0
(31)
From t = 0 point of view, the present value of wealth in the in nite future
must be positive.
- (31) is called no-Ponzi condition (if you borrow during your lifetime, you
must repay it in the in nite future).
- The complementary slackness condition (25), now called Transversality
condition is
lim
T !1
(T )Q(T + 1) = 0
(32)
- If the above condition does not hold, consumption path is not optimal.
If limT !1 Q(T + 1) = a > 0, it possible to increase consumption at any
t by a without a ecting PI.
3.3
What happens if there is uncertainty about the future?
- So far agent maximize their utility knowing all future path of endowments
(or in a production economy, knowing the path of future technological
improvements).
- Unrealistic. What happens if agents do not know the future?
- Rational expectation assumption: Agents do not know future endowments (future technological progress) but know the distribution from where
they are drawn.
Example 3.2 Suppose agents know that future endowments are described by the following law of motion
w(t) = w(t
1) + e(t) e(t)
(0;
2
)
(33)
Then E(t)w(t + 1) = w(t) (since E(t) e(t+1)=0) and var(t)w(t + 1) = (w(t + 1)
E(t)w(t + 1))2 = 2 .
Consumer problem:
max
1
X
c(t);Q(t+1) t=0
subject to
E(t) tU (c(t))
Q(t + 1) = r(t)Q(t) + (w(t)
c(t))
(34)
(35)
Optimality condition (Euler equation):
U 0(t) = r(t)E(t)U 0(t + 1)
(36)
No-Ponzi condition
Q(T + 1)
=0
lim E(0) QT
T !1
r
(
T
j
)
j=0
(37)
and the complementary slackness condition is
lim E(0) (T )Q(T + 1) = 0
T !1
(38)
Same conditions as before, but they now hold in expectations.
The No-Ponzi condition and the complementary slackness condition have
expectations dated at time zero since they refer to the present value of
wealth.
Under uncertainty we use the certainty equivalence principle: solve the
perfect foresight problem (the problem where you know the value of all
future variables). Insert expectations in front of all future variables after
you have derived the optimality conditions for the perfect foresight problem.
Example 3.3 Suppose agents receive during their in nite lifetime a random
endowment w(t) and that future endowments are unknown, but they are
known to be drawn from a distribution with mean w and variance 2w .
P
t log(c(t)).
Suppose their preferences are characterized by 1
t=0
The optimal consumption saving path is now characterized by the equation
1
r(t + 1)
= Et
8t
c(t)
c(t + 1)
(39)
If r(t + 1) = r = 1 , the above equation implies E (c(t + 1)) = c(t) or
c(t + 1) = c(t) + v (t), where v (t) is a mean zero expectation error.
- Consumption is a random walk: expected change in consumption is zero.
- This does not mean that consumption will not change (ex-post).
Conclusion
What characterizes the optimal consumption/saving path for an in nitely
lived agent which is faced with uncertainty?
1) Euler equation
2) The budget constraint
3) The transversality condition
Alternatively, the optimal consumption/saving path is characterized by 1)
and the present value budget constraint.
4
Asset pricing
- The optimal consumption decision (Euler equation) can be used to characterize equilibrium asset prices. How?
- Euler equation
U 0(t) = r(t)E (t)U 0(t + 1)
(40)
If we assume that consumption path is given, (40) can be used to price
asset which promise to pay r(t) one period from now. Thus:
1
=
r(t)
E (t)U 0(t + 1)
U 0(t)
(41)
The ratio of expected marginal utilities of consumption in two adjacent
periods can be used to price a bond which promise r(t) next period (or
1 of a bond which pays one unit of
equivalently to determine the price r(t)
consumption next period).
- If e.g. c(t) = y (t) (no saving), the above is an equation determining
the equilibrium interest rate as a function of the output in two subsequent
periods.
- The same logic can be applied to price n period bonds, i.e. to price
discount bonds which will give one unit of consumption after n periods,
or contingent assets, i.e. assets that pay one unit of consumption only in
certain state of nature (for example, insurance policies).
Example 4.1 Consider a two periods bond which pays r2(t) units of consumption at maturity. Then the Euler equation corresponding to this bond
would be
E (t)U 0(t + 2)
1
2
=
(42)
0
r2(t)
U (t)
If we buy this bond one period after it has been issued and hold it to
maturity its price will be
1
=
r(t + 1)
E (t + 1)U 0(t + 2)
U 0(t + 1)
(43)
where r(t + 1) is the return you can get at maturity. Combining (41) and
(42) we get the no-arbitrage condition
1
1
=
r2(t)
r(t)
1
r(t + 1)
(44)
Conclusion: Form the point of view of (optimal) agents) holding a two
period bonds to maturity is equivalent to holding one period bond from t
to t+1 and then investing the proceeds to buy another one period bond.
Homework
1) (Forced savings) Consider a two period problem where agents start with
wealth equal Q(1), face an exogenous gross interest rate equal to r(1) and
receive endowments equal to w(1) and w(2). Suppose preferences are described by U (c(1); c(2)) = log(c(1))+ log(c(2)) and that the government
taxes agents by taking away a portion of their endowment at time t = 1
and give them back at time t = 2. That is disposable income in the two
periods are W (1) T and W (2) + T .
i) Derive the optimal consumption (saving) decision using calculus.
ii) Graphically show how optimal consumption (saving) is chosen.
iii) How would your answer in i) and ii) would change is rather than taxing
lump sump the government taxes proportionally to income? That, is the
endowment in the two periods are now (1 t)W (1) and W (2) + tW (1).
2) (Bequest) Suppose agents start with wealth equal Q(1), face an exogenous gross interest rate equal to r between period 1 and 2, and receive
endowments equal to w(1) and w(2) in their two periods of life. Assume
that when they die, they want to bequest Q(3) to their o spring and that
this bequest gives them utility. Suppose preferences of agents are described
by U (c(1); c(2); Q(3)) = U (c(1)) + U (c(2)) + U (Q(3)).
i) Derive the rst order condition of the problem and interpret them.
ii) Assume U (c(1); c(2); Q(3)) = log(c(1)) + log(c(2)) + log(Q(3)). Find
the equilibrium c(1); c(2); Q(3), given r(1); Q(1); w(1); w(2). What would
happen to the bequest Q(3) if the initial wealth Q(1) is higher?
iii) Compare two economies, one where is high (say close to one), and
one where is low (say close to zero). Which economy will consume more?
Which will bequest more?
3) (Insurance policy) Consider a farmer who has a random stream of endowments and wants to buy an insurance policy from other farmers, in case
his endowment falls below some level w in certain periods. The policy Z (t)
will pay r(t) if w(t) < w and 0 otherwise. Assume that all farmers are
identical. What would Z (t) be in equilibrium? What would the price of
the insurance policy be in equilibrium?
4) In example of the production economy, derive the capital accumulation
equation and study, either analytically or numerically, what happens to the
equation when and increase.