III.
ON THE THEORY OF ACCUMULATION AND INTERTEMPORAL
CONSUMPTION CHOICE BY HOUSEHOLDS IN AN ENVIRONMENT OF
CERTAINTY
Begin the study of Finance with the analysis of an economy where all future outcomes are
known with certainty, but households receive income (their endowments) and consume at
different points in time. In particular, it is shown how the consumption-saving decision is made
and why the introduction of a capital market and financial securities can improve consumer
welfare.
As was discussed in the Introduction, the major decisions of the financial manager are to
choose which (physical) investments to make and to choose the appropriate means for financing
them. It is assumed that the "correct" policies chosen will be those that maximize some criterion
function (or performance index) specified by the firm. We prepare for the study of corporate
finance by deducing here and in Section IV a rational criterion function for the firm and the
management rules which optimize this criterion function in the simplified world of perfect
markets and certainty. Despite the simplicity of the model relative to the "real" world, the results
derived from this model form a basis for the rationalization of the more complex decision rules
developed later.
Hence, while the manifest functions of the analysis are to show how
intertemporal allocations are made and to show what role capital markets play in these
allocations, an important latent function of the analysis is to provide a foundation for corporate
financial theory.
We begin the analysis by solving the two-period problem and then extend it in a natural
fashion to the general case of many periods.
Consumer Behavior: The Two-Period Case
The four assumptions of Section II (A.II.1) - (A.II.4), are maintained throughout the
analysis. It is further assumed that each consumer has a well-behaved utility function expressing
his preferences between current consumption,
C0 ,
and next period's consumption,
C1.
Because the emphasis is on the intertemporal allocation of consumption, it is assumed that there
34
Finance Theory
is a single consumption good in each period. The consumer's utility function is denoted by
U[C0,C1]. Because both period's consumptions are considered goods (in contrast to "bads"), it is
assumed that U1[C0,C1] ≡ ∂U[C0,C1]/ ∂C0 > 0 and U2[C0,C1] ≡ ∂U[C0,C1]/∂C1 > 0. By
assuming the strict inequality, we rule out the possibility of satiation. I.e., consumers will always
strictly prefer more to less of either C0 or C1. We also assume sufficient regularity and
concavity of U to ensure existence of unique interior maximums.
An indifference curve is the set of all combinations of current and next-period
consumption,
(C0,C1),
such that the consumer is indifferent among these alternative
combinations i.e., they are curves of equal utility or iso-utility curves. Formally, it is the
functional relationship between C0 and C1 such that U[C0,C1] = U , where U
is a constant.
Figure 1 illustrates the general shape of the indifference curves, and as they are drawn,
1
2
3
U >U >U .
Analytically, by the Implicit Function Theorem or heuristically, by using
differentials, we have that d U = 0 = U 1 d C 0 + U 2 d C 1 , or that
(III.1)
⎛ dC 1 ⎞
⎜⎜
⎟⎟ = - U 1 [ C 0 ,C 1 ]/ U 2 [ C 0 ,C 1 ] < 0,
⎝ dC 0 ⎠U =U
_
where (dC1/dC0) is the slope of the indifference curve defined by U[C0,C1] = U at the point
(C0,C1). As shown in Figure 1, this slope is always strictly negative.
Case 1. The Simplest Capital Market: Pure Exchange
For this case, we assume that there are no means of physical production. I.e., there is no
way of using the current period's goods to produce additional goods next period. However,
suppose there does exist a market for trading current period's goods in return for a claim on
goods next period. So, an individual can go to the market and exchange current period goods for
"pieces of paper" which, in turn, can be exchanged next period for goods. Alternatively, he can
35
Robert C. Merton
receive current period goods by issuing "pieces of paper" which he must redeem for goods next
period. In effect, in the former case, he is lending and in the latter, he is borrowing.
If, by convention, the price per unit of current period goods is set equal to one (i.e., a unit
of current period goods is numeraire), then the (current) price per unit of next period goods, P,
is the rate of exchange for claims on next period goods in terms of current period goods. So, P
units of current period goods can buy a claim on one unit of next period goods.
In an
intertemporal context, this price is also written as P ≡ 1/(1+ r) where r is the rate of interest.
Hence, one unit of current goods can be exchanged for (1 + r) units of goods delivered next
period.
Figure III.1
Indifference Curves
36
A consumer's endowment of exogenous income is denoted by (y0,y1) where y0 is the
number of units of current goods he owns and y1 is the number of units of goods that he will
receive next period. The consumer's current wealth, W0, is equal to the value of his endowment
i.e., W0 = y0 + Py1. The consumer's feasible consumption set is the set of all combinations
(C0,C1)
which he can afford to buy.
Thus, if
(C0,C1)
are in the consumer's feasible
consumption set, then the cost of that consumption program, C0 + PC1, can be no larger than his
wealth W0. Moreover, as long as a consumer prefers more consumption to less, he would never
choose a program which costs less than his wealth. Hence, if it is assumed that the consumer
will choose the most preferred feasible consumption program, then he will act so as to maximize
U[C0,C1] subject to his budget constraint that W0 = C0 + PC1.
Substituting for C0 in U from the budget constraint, we can write the consumer choice
problem as
Max U[ W
(III.2)
0
- PC 1 , C 1 ]
C1
which leads to the first-order condition for an interior maximum
dU
(III.3)
dC 1
*
= 0 = - U 1 [ W 0 - PC*1 ,C*1 ]P + U 2 [ W 0 - PC*1 ,C*1 ] ,
*
*
*
where (C0 ,C1 ) is the optimal consumption program. Noting that C0 = W0 - PC1 , we can
rewrite (III.3) as
( dC 1 / dC 0 )U = U * = - 1/P = - (1 + r) ,
(III.4)
*
*
*
where U ≡ U [C0 ,C1 ] is the maximum feasible value of utility. Hence, the optimum occurs at
the point where an indifference curve is tangent to the budget constraint as shown in Figure 2.
Note that in arriving at the optimality condition (III.3), we have used assumption (A.II.4)
that the consumer acts as a pure competitor or price-taker. So, in solving for his most preferred
consumption program, the consumer treats the price (or interest rate) as a given number which
does not change in response to the different consumption choices that he might make.
37
Robert C. Merton
Figure III.2
In the absence of an exchange market and without physical storage of goods through time,
the optimal consumption program for the consumer will simply be to consume current income.
I.e.,
Co = yo and C1 = y1 .
Hence, if the solution to (III.3) yields
*
C0 ≠ y0
(and
*
therefore, C1 ≠ y1 ), then the consumer will be better off as a result of the creation of an
exchange market. Moreover, he can be no worse off because he always has the option not to use
the market and choose C0 = y0 and C1 = y1 which is called the autarky point.
Even if physical storage of goods is feasible, then in the absence of an exchange market,
the feasible consumption choices are constrained to have C0 ≤ y0. That is, physical storage
38
Finance Theory
allows one to "move" goods "forward" in time for consumption, but it does not allow one to
"move" goods "backward" in time.
So, for example, suppose that one had an income stream of (y0 =) ten bushels of wheat
this period and (y1 =) fifty bushels of wheat next period. In the absence of an exchange market,
there is no way that he can consume more than ten bushels of wheat this period even if costless
storage of wheat were available. However, in the presence of an exchange market, in addition to
the ten bushels he has, he could consume up to 50/(1+r) bushels of wheat in the initial period
where r is the market interest rate. Even if his endowment had been y0 = 50 and y1 = 10, then
he would still be better off to save wheat for next period through the exchange market rather than
by storage provided that the interest rate is positive.
Problem III.1:
Choosing an Optimal Consumption Allocation:
Suppose that one has a
preference function given by U[C0,C1] = log(C0) + log(C1)/(1+δ) and an endowment of y0 = y1
= y.
*
If
r
is the market rate of interest, then what is the optimal allocation
*
*
*
*
*
*
*
(C0 , C1 )? From (III.3), we have that U1[C0 ,C1 ]/U2[C0 ,C1 ] = (1+δ)C1 /C0 = 1+r, or that
C1 = (1+r)C0 /(1+δ ). W0 = y + Py = (2+r)y/(1+r). From
*
*
C 0 = W0 - PC1 = ⎡
⎣( 2+r ) y − C1 ⎤⎦ (1 + r ) . Substituting
*
*
*
into
the
the
*
C1 from the optimality condition, we have that
(III.5a)
*
C 0 = { (1 + δ )(2 + r)/[(1 + r)(2 + δ )] } y
and
(III.5b)
*
C 1 = (2 + r)y/(2 + δ ) .
39
budget
budget
constraint,
constraint
for
Robert C. Merton
Time Preference
A consumer is said to have a positive time preference if for every (a,b) such that b > a,
U[b,a] > U[a,b]. He has no time preference if U[b,a] = U[a,b], and a negative time preference
if U[b,a] < U[a,b].
In the example of preferences used in Problem III.1,
δ
can be interpreted as the
consumer's rate of time preference. If δ > 0, then he has positive time preference. If δ = 0,
then he has no time preference, and if δ < 0, then he has negative time preference. Note that in
*
that example, if the interest rate exceeds his rate of time preference (r > δ), then C0 < y , and
he will save some of his current period's income to consume next period. If r < δ, then
*
C0 > y and he borrows against next period's income to consume more than his current income.
*
If r = δ, then C0 = y, and he does not trade, but consumes exactly his income in each period.
Suppose that the consumer in this example were the only person in the economy (i.e., a
"Robinson Crusoe" economy). Because he can only trade with himself, the autarky solution is
the only feasible solution. However, we can compute the "equilibrium" rate of interest consistent
with autarky and that rate clearly must be r = δ. Hence, by this example, we have illustrated one
of the possible explanations for a positive rate of interest: namely, consumers' impatience to
consume or a positive time preference.
Case 2. A No-Exchange Market Economy: Pure Production
As in the first case, we assume that the consumer has an endowment of exogenous income
(y0,y1), but in addition, he has the opportunity to use some of his current income to produce
next-period goods. One may wish to think of the "good" as seed which can either be eaten
(consumed) or planted (invested). However, because there is no exchange market, physical
production is the only means he has to increase his next period's consumption beyond next
period's income. Moreover, because there is no exchange market, the only way that he can
40
Finance Theory
produce is by forgoing some current consumption i.e., if X0 denotes the amount he invests in
production, then
X 0 = y 0 - C 0 > 0.
(III.6)
The technology available to him is described by a production function f, such that X0
units of current goods invested will produce X1 = f(X0) units of the good next period. It is
assumed that f(0) = 0 and df/dX0 ≡ f′ (X0) > 0. It is further assumed that the production
technology exhibits non-increasing returns to scale (i.e., d f / dX 0 ≤ 0 ) . Figure 3 illustrates
2
2
the production function for decreasing returns to scale, and for 0 ≤ X0 ≤ y0, describes his
Production Possibility Frontier. The maximum output that he can produce is
max
X1
= f(y0)
which corresponds to X0 = y0 and C0 = 0. Hence, f(y0) ≥ X1 ≥ 0. His next period's
consumption can be written as
(III.7)
C 1 = y1 + X 1
= y1 + f( y 0 - C 0 )
41
Robert C. Merton
Figure III.3
Production Function
which for y0 - C0 ≥ 0, describes his feasible consumption set of Consumption Possibility
Frontier. Because there is no exchange market and therefore, no prices, the consumer does not
have a budget constraint of the type in Case 1.
However, his consumption choices are
constrained by (III.7) which is called a technological budget constraint. Hence, if, as in Case 1,
it is assumed that the consumer will choose the most-preferred feasible consumption program,
then he will act so as to maximize U[C0,C1] subject to his technological budget constraint.
Substituting for C1 in U from (III.7), we can write the consumer choice problem as
Max U[ C
0
, y 1 + f( y 0 - C 0 )]
C0
which leads to the first-order condition for an interior maximum
42
Finance Theory
0 = U 1 [ C*0 , y 1 + f( y 0 - C*0 )] - f ′( y 0 - C*0 )U 2 [ C*0 , y 1 + f( y 0 - C*0 )].
(III.8)
Assuming that the optimum is interior, we can rewrite (III.8) as
*
*
*
*
*
U 1 [ C 0 ,C 1 ]/ U 2 [ C 0 ,C 1 ] = f ′( X 0 )
(III.9)
where
*
*
C1 = y1 + f(X 0 )
*
*
and the optimal amount to plant, X 0
is given by
*
X 0 = y 0 - C0 . As was done in Case 1, we have from (III.1) that (III.9) can be rewritten in
*
*
terms of the slope of an indifference curve through (C0 ,C1 ) as
( dC 1 / dC 0 )U =U * = - f ′( X *0 )
(III.10)
*
*
*
where U = U[C0 ,C1 ]. Figure 4 plots the Consumption Possibility Frontier along with a
*
*
*
graphical solution of the optimal consumption-production program (C0 ,C1 ,X0 ). Because
there is no exchange market, he cannot "borrow" against next period's income, y1, to consume
more in the current period. (i.e., C1 ≥ y1 and C0 ≤ y0). Hence, as shown in Figure 4, the
Consumption Possibility Frontier has a vertical portion for C1 ≤ y1.
43
Robert C. Merton
Figure III.4
Although there is no market rate of interest in Case 2, we can define an "implied" or
*
"technological" rate of interest, r , by 1 + r ≡ f (′ X 0 ). By comparing (III.10) with (III.4), we
see that r serves as a surrogate for the market rate r, and hence illustrates a second reason for
a positive rate of interest: namely, the productivity of (physical) investment.
Case 3. Production Within an Exchange Market Economy
We maintain the same assumptions about the consumer's endowment of exogenous income
and a production technology as in Case 2. However, we now allow for an exchange market as in
Case 1 where the current market price of next period's goods is P = 1/(1+r). In this environment
his current wealth, W0, can be written as
44
Finance Theory
W 0 = y0 + Py1 + Pf( X 0 ) - X 0
(III.11)
where the first two terms on the right-hand side represent the current value of his endowment of
exogenous income and the last two terms represent the net current value of operating his
production technology with an input intensity of X0. That is, if he buys inputs today with a
current value of X0, then he will receive an output next period of f(X0) which has a current
value of Pf(X0). The difference between the two is the net increment to his current wealth from
operating the technology at that intensity. Note that unlike in Case 1, the consumer's current
wealth is affected by one of his decisions: namely, the amount of physical production he
undertakes, X0.
As in Cases 1 and 2, the consumer chooses an investment-consumption program,
(X0,C0,C1), so as to maximize U[C0,C1] subject to the budget constraint that W0 = C0 + PC1.
Because there now exists an exchange market, (III.6) in Case 2 is no longer a constraint i.e., the
consumer can borrow against future income to either consume or invest in physical production in
the current period. Substituting for C0 from the budget constraint, we can write the consumer
choice problem as
Max U[ y
0
+ Py1 + Pf( X 0 ) - X 0 - PC 1 , C 1 ]
{ X 0 ,C 1 }
which leads to the set of first-order conditions for an interior maximum
∂U/ ∂ X 1 = 0 = U 1[ C *0 ,C*1 ](Pf '( X *0 ) - 1)
(III.12a)
and
∂U/∂ C 1 = 0 = - PU 1 [ C*0 , C*1 ] + U 2 [ C*0 , C*1 ] ,
(III.12b)
where
*
*
*
(X0 ,C0 ,C1 ) denotes the quantities chosen for the optimal investment-consumption program, and
45
Robert C. Merton
*
*
*
*
C0 = y 0 + Py1 + Pf(X 0 ) - X 0 - PC1 . Because the consumer is assumed never to be
*
*
satiated, U1[C0 ,C1 ] > 0, and we can rewrite (III.12a) as
f (′ X *0 ) = 1/P
= 1+ r .
(III.13a)
By inspection of (III.13a), we see that, unlike in (III.9) of the Robinson Crusoe Case 2, the
optimal amount to invest in physical production,
*
X 0 , does not depend either upon the
consumer's preferences, U, or his endowment, (y0,y1). Hence, two consumers with quite
different preferences between current and future consumption and with quite different
endowments, but who face the same market rate of interest and have the same production
technologies, will choose the same level of physical investment in their technologies,
*
X 0 . Such a result about physical production is called an efficiency condition because it is
independent of either preferences or endowments, and hence independent of who owns the
production technology.
One interpretation of the optimality condition (III.13a) can be derived as follows: as
previously noted, the current wealth of the consumer is affected by the choice of production
**
X0
intensity. I.e., W0 can be written as W0(X0). If
denotes that amount of physical
investment which maximizes the current wealth of the consumer, then from (III.11),
**
X0 is the solution to the problem:
Max [ y
0
+ P y 1 + Pf( X 0 ) - X 0 ]
{ X0}
which leads to the first-order condition for an interior maximum
(III.14)
**
dW0 /dX 0 = 0 = Pf '( X 0 ) - 1
46
Finance Theory
**
*
which is identical to (III.13a). i.e., X 0 = X 0 . Hence, optimality condition (III.13a) can be
interpreted as saying "Choose physical investment so as to maximize one's current wealth." This
is called the Value Maximization Rule and it has significant implications for the theory of
Finance. However, discussion of these implications is postponed until Section IV.
Consider now the second optimality condition (III.12b). From (III.1), it can be rewritten in
*
*
terms of the slope of an indifference curve through the point (C0 , C1 ) as
(III.13b)
( dC 1 / dC 0 )U =U * = - 1/P = - (1 + r) .
Comparing (III.13b) with (III.4), we find that it is identical to the optimality condition in the Pure
*
*
*
Exchange Case 1 if we use as current wealth, W0 ≡ y 0 + Py1 + Pf(X 0 ) - X 0 . Hence, one
can describe the solution of the optimal investment-consumption program for the consumer as
taking place in two steps. First, choose physical investment so as to maximize current wealth.
Second, as in the case of pure exchange, use the exchange market to borrow or lend (against this
maximized wealth) so as to achieve the most-preferred, feasible consumption allocation. Figure
5 provides a graphical solution of the problem, and is, in essence, a composite of Figures 3 and 4.
As inspection of Figure 5 clearly demonstrates, the consumer is better off in the presence of an
exchange market than he was in the Robinson Crusoe framework of Case 2.
Hence, the existence of an exchange or capital market will not only affect the patterns of
consumption chosen but also will alter the allocation of physical investment among the various
technologies, and in so doing affect the total output for the economy.
47
Robert C. Merton
Figure III.5
Note: By trading, he reaches a higher indifference curve.
The Multi-Period Consumption and Allocation Decision: The T-Period Case
We now extend the previous analysis to a consumer who lives for T-periods with a utility
function for lifetime consumption described by
U[C0,C1,...,CT-1,CT]
where
Ct
is his
consumption in period t, t = 0,...,T. Let yt denote the exogenous income he will receive in
period t, t = 0,...,T. There exists an exchange market which is open each period and allows for
trading the current period's consumption good and claims on consumption goods in the future.
th
Specifically, at each point in time, there are (T+1) different claims traded in units where the τ
48
Finance Theory
such claim gives its owner the right to one unit of the consumption good payable τ periods from
the date at which it is issued, τ = 0,...,T. In effect, these claims are pure discount loans as
defined in Section II. Let Pt(τ) denote the price at date t of a discount loan which pays one unit
of the consumption good τ periods from date t (i.e., at date t + τ). If, by convention, the
current period's (or "spot") price of the consumption good is taken as numeraire', then Pt(0) = 1,
for all t.
In the absence of any production capabilities, the consumer's current wealth, W0, at date t
= 0 can be written as
W 0 = ∑ P0 ( τ
T
(III.15)
τ =0
) yτ .
As in the two-period analysis, the consumer's feasible consumption set is the set of all
consumption programs that he can afford to buy. Hence, for a consumption program to be
T
∑ P ( τ )C ≤ W
feasible, it must satisfy
0
τ
0
which defines the feasible consumption set.
τ =0
Provided that satiation is ruled out, the T-period consumer allocation problem is formulated as
maximize U[C0,C1,...,CT] subject to the budget constraint that W 0 = ∑ P0 ( τ ) Cτ
T
.
Noting
τ =0
that P0(0) = 1, we can substitute for C0 from the budget constraint, and rewrite the problem as
U[ W 0 - ∑ P0 ( τ )C τ ,C 1 ,C 2 ,...,C T ]
T
Max
{ C 1 ,C 2 ,...,C T }
τ =1
which leads to T first-order conditions
(III.16)
0 = -U 1 [ C*0 ,C*1 ,...,C*T ] P0 ( τ ) + U τ +1 [ C*0 ,C*1 ,...,C*T ],τ = 1,2,...,T,
th
where Uτ ≡ ∂U[C0,C1,...,CT]/∂Cτ-1 denotes the partial derivative of U with respect to its τ
argument
and
*
*
*
(C 0,C 1,...,C T)
is
the
49
optimal
consumption
program
with
Robert C. Merton
T
*
0
C =W 0 -
∑ P ( τ )C .
*
τ
0
In words, (III.16) says that at the optimum, the ratio of the marginal
τ =1
utility of consumption should just equal the ratio of the marginal cost of consumption in period τ
to the marginal utility of current consumption in period τ, P0(τ), to the marginal cost of current
consumption, P0(0) = 1. From (III.16), we have that
*
*
*
*
*
*
U t+1[ C 0 , C 1 ,...,C T ]/ U s+1[ C 0 ,C 1 ,..., C T ]
(III.17)
= P 0(t)/ P 0(s), s, t = 0,1,...,T .
As with Case 3 of the two-period analysis, we now expand the analysis of the T-period
case to allow for production. Generalizing the production function description of the technology
from the two-period case, let ft(X0t,X1t,...,Xt-1,t) denote the production function for output in
period t, (t=1,2,...,T) where Xjt is the amount of input required to be invested in period j,
(j=0,1,2,...t–1), in order to produce output ft in period t. In an analogous fashion to (III.11) in
the two-period case, we can write the current wealth of the consumer as
T
T
τ =0
τ =1
W 0 = ∑ P0 ( τ ) yτ + ∑ P0 ( τ ) f τ ( X 0τ , X 1τ ,..., X τ -1,τ )
(III.18)
- ∑ P0 ( τ
T -1
)Xτ
τ =0
T
where X τ ≡
∑
X τ j is the total amount of inputs required in period τ to allow production
j=τ +1
plan {ft}, τ = 0,...,T–1. Define the net increment to the consumer's current wealth of production
plan {ft}, V0, by
(III.19)
V 0 ≡ ∑ P0 ( τ ) f τ ( X 0τ , X 1τ ,..., X τ -1,τ ) - ∑ P0 ( τ
T
T -1
τ =1
τ =0
50
) Xτ .
Finance Theory
The combined investment-consumption choice problem is formulated as choose the production
and consumption program so as to maximize U[C0,C1,...,CT] subject to the budget constraint
T
T
τ =0
τ =0
that W 0 = ∑ P0 ( τ ) yτ + V 0 = ∑ P0 ( τ ) Cτ . Substituting for C0 from the budget constraint, the
problem can be rewritten as choose (C1,...,CT) and (Xjt, j = 0,...,t-1 and t=1,...,T-1) so as to
⎡T
Max U ⎢∑ P0 ( τ
⎣τ =0
T
) yτ + V - ∑ P ( τ ) Cτ , C
τ
0
0
=1
1
⎤
,..., C T ⎥
⎦
which leads to T(T+1)/2 first-order conditions for the production choices
∂U/ ∂ X jt = 0 = U 1[ C *0 , C *1 ,..., C *T ]
(III.20a)
∂V0 / ∂ X jt , j = 0,1,...,t - 1 and t = 0,...,T - 1
and T first-order conditions for the consumption choices
(III.20b)
∂U/∂ C τ = 0 = - U 1 [ C*0 , C*1 ,..., C*T ] P0 ( τ )
+ U τ +1 [ C*0 , C*1 ,..., C*T ] , τ = 1,...,T .
Noting that U1 > 0, the first-order conditions (III.20a) can be rewritten as ∂V0/∂Xjt = 0, j =
0,1,...,t-1 and t = 0,1,...,T-1, and in that form, are simply the generalization of condition
(III.13a) in the two-period case. Indeed, the interpretation given to (III.13a) in the two-period
case of choosing a physical production program so as to maximize the consumer's current wealth
carries over exactly to the T-period case. From (III.18) and (III.19), the set of {Xjt} which
maximizes W0 are the ones that maximize V0. But, the set of first-order conditions that
maximize V0 are simply ∂V0/∂Xjt = 0. Hence, (III.20a) simply says choose physical production
so as to maximize current wealth, and therefore the Value Maximization Rule applies in the
general T-period case.
Inspection of (III.20b) shows that it is identical to the first-order conditions for the pureexchange case (III.16) where the level of current wealth used is the maximized value,
51
Robert C. Merton
*
W 0.
Hence, as was shown in the two-period case, the solution of the T-period optimal
investment-consumption program for the consumer can be described as taking place in two steps:
namely, first, choose physical investments so as to maximize current wealth. Second, use the
exchange market to borrow or lend so as to achieve the most preferred feasible consumption
allocation.
On the Connection Between the T-Period and Two-Period Analyses
While the T-period consumer choice is a more realistic description of the world than the
two-period formulation, the analysis is more complex and is burdened by a barrage of notation.
Moreover, it does not readily lend itself to the relatively intuitive graphical display of the
solution. We have already shown that the fundamental behavioral characteristics (such as the
Value Maximization Rule) deduced in the two-period case carry over to the general T-period
case. We now show that, in essence, the general T-period problem can always be structured so as
to "look like" a two-period problem. Not only does this connection between the two problems
make the analysis of the T-period problem more tractable, but it also provides the appropriate
framework for studying the intertemporal consumption-investment choice problem in an
uncertain environment.
In the previous analysis, we solved the entire lifetime consumption choice problem by
having the consumer choose at date t = 0, (C0,C1,...,CT) so as to maximize U[C0,C1,...,CT]
subject to his budget constraint W 0 = ∑ P0 ( τ ) Cτ
T
.
Suppose we move ahead one period to
τ =0
date t = 1. Suppose further that the consumer consumed C 0 units at date t = 0. The consumer
choice problem at date t = 1 can be formulated as choose (C1,C2,...,CT) so as to maximize
T
U[C0 ,C1 ,...,CT ] subject to his budget constraint W 1 = ∑ P1 ( τ - 1) Cτ where W1 is his
τ =1
52
Finance Theory
C0 is not a choice variable at t = 1 because whatever was
wealth at date t = 1. Note:
consumed at time t = 0 is now past history.
We can solve the optimal choice problem at t=1 in the same way that the problem was
solved at t = 0, and in analogous fashion to (III.16), we arrive at the (T-1) first-order conditions
that
0 = - U 2 [ C 0 ,W 1 - ∑ P 1 ( τ
T
τ =2
(III.21)
- 1) Cτ ,C ,...,C ] P ( τ - 1)
*
*
2
*
T
1
+ U τ [ C ,C ,...,C ] , τ = 2,...,T,
+1
0
1
*
T
T
*
*
C 1 = W 1 - ∑ P1 ( τ - 1) Cτ .
where
From (III.21), it is clear that the optimal solution
τ =2
*
*
*
(C1 , C 2 ,...,CT ) will depend upon the amount of wealth
{P1 (1),...,P1 (T-1)}, C0 ,
W1,
the
prices
and the form of the utility function U.
Define the function J by
(III.22)
J[ C 0 ,W 1; P 1(1),..., P 1(T - 1)] ≡ U[ C 0 , C *1 ,...,C *T ] .
*
*
*
J is the "level" of utility associated with a consumption program of ( C 0 ,C 1 ,C 2 ,...,C T ) , and is
the maximal level of utility (corresponding to the most preferred feasible program) conditional
on consuming C0 units at date t = 0 and having wealth W1 at time t = 1. Because the prices
{P1(τ)}
are not affected by the choices made by the consumer, they can be treated as
parameters. Hence, a shortened form for J is simply to write it as " J[ C 0 ,W 1 ]."
Return now to the original problem of selecting an optimal consumption program at time
t = 0. Of course, at t = 0, the consumer is free to choose any (feasible) level for C0. A
necessary condition for a consumption program to be optimal is that whatever level of
consumption is chosen for C0, the choices made for C1,C2,...,CT must be the best one can do
53
Robert C. Merton
conditional
*
*
on
having
chosen
level
C0 .
That,
of
course,
is
exactly
what
*
C1 ,C 2 ,...,CT represent in the t = 1 problem just solved where they represent the best the
consumer can do conditional on having chosen to consume C0 at time t = 0. Further, we have
that wealth at time t = 1, W1, can be expressed in terms of wealth at time t = 0 as
(III.23)
W 1 = [ W 0 - C 0 ]/ P0 (1) .
I.e., whatever part of current wealth that is not currently consumed will grow in one period by the
one-period interest rate.
Hence, having solved the conditional (on
t = 0
consumption)
optimization problem as of t = 1, we can reformulate the consumer choice problem at t = 0 as:
Choose current consumption, C0, so as to maximize J[C0,W1] subject to the budget constraint
W0 = C0 + P0(1)W1. Expressed in this way, except for some notational differences, this problem
is in essence the same as the two-period choice problem solved in Case 1 of this section where
the utility function "J" replaces "U[Co,C1]" and "W1" replaces next period consumption "C1"
i.e., the T-period consumption problem can be reformulated as a two-period problem.
Although in the formulation the utility function, J, has utility depending upon (next
period's) wealth, the consumer still only gets direct utility from consumption. In effect, wealth
W1 acts as a "surrogate" for future consumption so that the utility "tradeoff" between C0 and
W1 is really a tradeoff between current and future consumption. J is sometimes called the
indirect or derived utility function, and provided that the direct utility function, U[C0,C1,...,CT],
is a well behaved, (quasi) concave function, J will be a well behaved, (quasi) concave function
in (C0,W1).
To solve the problem, we substitute for W1 using the budget constraint to get:
Max J [ C
0
,( W 0 - C 0 )/ P0 (1)]
C0
which leads to the first-order condition
(III.24)
0 = J 1 [ C*0 ,W *1 ] - J 2 [ C*0 ,W *1 ]/ P0 (1)
54
Finance Theory
where subscripts denote partial derivatives of J with respect to the appropriate arguments and
*
*
W1 ≡ (W0 - C0 )/P0 (1). As in (III.4), we can rewrite (III.24) in terms of the slope of an
*
*
indifference curve through the point (C 0,W 1) as
( dW *1 /d C 0 )J = J * = - 1/ P0 (1),
(III.25)
*
*
*
where J ≡ J[C 0,W 1]. Figure 6 provides the graphical solution which is
Figure III.6
analogous to the one displayed in Figure 2 for the two-period problem. Although the derivation
presented here is more descriptive than rigorous, the analysis can be made rigorous by using the
mathematical technique of dynamic programming.
While the explicit development of this
technique is more appropriately the subject of an advanced treatment of Finance, the interested
55
Robert C. Merton
reader can find its development in the context of this problem in Fama [American Economic
Review, March 1970].
In summary, we have solved the general intertemporal consumption-investment problem in
a certainty environment. In so doing, we have shown that the creation of financial securities and
an exchange market will make the consumer better off.
In particular, we showed that an
exchange market was the only means by which an individual consumer can convert future
income or output into current consumption. While this manifest function of the exchange market
more than justifies its existence (indeed, if such markets did not exist, we would have to invent
them), it has an important latent function as the means for permitting an efficient organization of
the economy's production. This important latent function is the topic of the next section.
56