Department of Policy and Planning Sciences
Discussion Paper Series

No.1364

An Application of
Two-Constraint Consumption Model:
Two goods case with time constraint
Revised Version

byToru MURAYAMA

December 2019

UNIVERSITY OF TSUKUBA
Tsukuba, Ibaraki 305-8573
JAPANAn Application of
Two-Constraint Consumption Model:
Two goods case with time constraint
Revised Version∗
Toru Murayama
University of Tsukuba
Dec.24 2019

∗ This paper is based on Toru Murayama’s master thesis submitted to The

University of Tokyo on Jan.4 2017.

CONTENTS
0

Abstract

1

1

Introduction

1

2

The Model

4

MRS and Optimal Point . . . . . . . . . . . . . . . . .

6Comparative Statics

8

3.1

Price Change . . . . . . . . . . . . . . . . . . . . . . .

8

3.2

Income . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.3

Disposable Time . . . . . . . . . . . . . . . . . . . . .

18

4Working Time and Income

195

Saving24

5.1

Shortage of Budget . . . . . . . . . . . . . . . . . . . .

25

5.2

Liquidity Constraint . . . . . . . . . . . . . . . . . . .

26

5.3

Other Applications . . . . . . . . . . . . . . . . . . . .

28

Concluding Remarks

28

2.1
3

6

0 Abstract
This paper conducts microeconomic analyses in which not only
budget but also another constraint can aﬀect consumer’s behavior.
We construct a model in which two constraints exist. One of theconstraints is budget and the other is time. We conduct comparative statics analyses involving two goods: time-intensive good andmoney-intensive good. In the context of transportation, an exampleof the two good is local train and high-speed railway. We also study
consumption behavior when paid work is introduced, whereby the
consumer faces the trade-oﬀ between time and money. Furthermore,
we modify the model to analyze the saving behavior of the consumer.

Keywords:
consumption behavior, modal choice, ”time-Giﬀen good”,
time-intensive good, time saving

1 Introduction
Most economics models incorporate only the budget constraint,
parameterized by prices. In a large class of environments, however,other constraints, such as time or space, bind the behavior of economic agents. For example, Hummels and Schaur (2013) estimatethat time cost of trade has an eﬀect on the demand which is equivalent to the certain amount of an advalorem tariﬀ (’each day in transit
is equivalent to an advalorem tariﬀ of 0.6 to 2.1 percent’).
There are similar cases in policy problems: space for nuclear wasteor landfill, or resources such as water and electricity. They constrain
consumption behavior, producer behavior, or government’s policy.
1

In the context of modal choice, Small (2012) and Behrens and Pels
(2012) focus on time as important aspect of economic behavior.
In this paper, we construct a model that includes the time constraint. The results of the model analysis can be extrapolated toother types of the constraint by applying.
Amongst preceding works incorporating the concept of time, Becker(1965) analyzes the optimal time allocation supposing utility is decided by the time allocation for leisure and work. DeSerpa (1971)
and Carpio and Wohlgenant (2010) regard time as a good which is
required to consume with the consumption good at the same time.
Applied studies regarding time include Fujii, Kitamura and Kumada (1998), Kono and Morisugi (2001), Kato and Imai (2005) andJara-Dı́az (2007), inter alia. Fujii, Kitamura and Kumada (1998)
conduct the empirical studies to estimate the individual traﬃc demand. Kono and Morisugi (2001) theoretically examines the changeof the value of time depending on the economic environment. Katoand Imai (2005) formulate the private travel behavior to valuate thetravel-time savings of the private trip by applying the DeSerpa’s definition of the travel time savings. Jara-Dı́az (2007) summarizes andcompares the theories on time allocation and the concepts of valueof time, and suggests the sources of improvement in the modeling of
the value of time.
This paper mainly refers to Carpio and Wohlgenant (2010) and
DeSerpa (1971). In their models, utility depends on both goods andtime. And both budget and time constraints bind. The diﬀerence
between the two models is that the amount of time to consume eachgood is expressed by an equality in Carpio and Wohlgenant (2010)
while it is expressed by an inequality in DeSerpa (1971). The advantage of the equality expression is that it facilitates the calculation.
Using this advantage, Carpio and Wohlgenant (2010) derive the basic2mathematical features of the model similar to Slutky equation and
Roy identity. On the other hand, the advantage of the inequality
is that it expresses the minimum amount of time consumption for acertain amount of good consumption. It lets us analyze the case that
additional marginal time input without the increase in good improves
the utility level, although it has drawbacks: complicated calculationand the diﬃculty of interpreting the values of time.
In my model, we assume that both the budget constraint and thetime constraint are inequalities. We also assume that the relationshipbetween a good and its time consumption is equality for each good,
and the input of the utility function is expressed only by the amountof time.
The model involves two goods, good 1 being the ”time-intensivegood” and good 2 the ”money-intensive good”. The word ”timeintensive” means that the good requires more time proportionately
to money than the other good. Conversely, ”money-intensive” meansthat the consumption needs relatively more money than time.
We analyze the case that only one of the constraints binds, which isprecluded in Carpio and Wohlgenant (2010) and DeSerpa (1971), andEvans (1972) mentioned the possibility of inequality budget but thetime constraint is equality. The proportional relationship between agood and its consumption time and the form of the utility function
allow me to figure the two-good analysis in two dimensions while
DeSerpa (1971) had to deal with four dimensions.
Using the advantages of the model, we conduct the analyses of two
goods which can be contrasted as ”fast but expensive” and ”slow but
cheap”.
The results are then extrapolated to further analyses. First is thecase that the consumer can contemporaneously exchange her disposable time for income, and vice versa. Second is an intertemporal3extension, the simplest of which is a two-period model.

2 The Model
We construct the two-good model as follows:

max U (T1 , T2 )s.t.
T 0 ≥ T1 + T2
Ti = ai Xi (∀i = 1, 2)

(1)
(2)

Y ≥ P1 X1 + P2 X2(3)Note that• Ti : the amount of time input to consume the ith good()
Ti ∈ [0, T 0 ]• T 0 > 0 : exogenous disposable time
• Xi ≥ 0 : amount of consumption of ith good
• Pi > 0 : price of ith good
• Y > 0 : exogenous income• U (T1 , T2 ) : utility function, continuous, twice-diﬀerentiable
2∂U
and, ∂T> 0 and ∂∂TU2 < 0 (∀i = 1, 2)ii• ai > 0 : the parameter which shows the necessary amount oftime to consume one unit of ith good *1(1) and (3) allow the switching of the binding constraint as discussed later. From the other point of view, it is the switch of the idle
resource.
*1 The higher the technology level, either the larger or the smaller ai , dependingon the type of the good.

4

(2) represents the assumption that the time required to consume
each unit of good is constant, while DeSerpa (1971) supposes only
the minimum amount of time for the good consumption. Ti and Xi
must be in some interval. And we assume that the interval consistsof only one element for simplicity.
We also assume that good 2 is comparatively pricey whereas good
1 is time consuming. More specifically,0<

YY< T0 <
P2 /a2P1 /a1(4)

which can be shown by the figure as follows.

Fig. 2-1
The two lines in the figure are the budget constraint line and thetime constraint line. The line, sloped -1 with intercepts T 0 is the time
constraint. The other line, sloped flatter, is the budget constraint, ofwhich the intercepts are expressed by the relative prices Pi and ai .

5The intersection point of these two lines is:
( 0
)
0
(
)
T
P
/a
−
Y
Y
−
T
P
/a
2
2
1
1
T¯1 , T¯2 =,P2 /a2 − P1 /a1 P2 /a2 − P1 /a1(5)

The consumption feasibility set is the shaded area in Fig. 2-2.

Fig. 2-2

2.1 MRS and Optimal Point
We separate the optimisation into three cases. The first case iswhen the slope of the indiﬀerence curve which goes through theintersection point, (T¯1 , T¯2 ), is flatter than the slope of the budget
constraint line at the intersection point. That is M RS(T¯1 , T¯2 ) =dU (T¯1 ,T¯2 )/dT2
1 /a1
∗
∗
− dU T¯ ,T¯ /dT < P
P2 /a2 . Then the optimal point (T1 , T2 ) is the tan( 1 2) 1
gency point of the budget line and the indiﬀerence curve. And it
locates closer to the vertical axis than the intersection point. It isshown in Fig. 2-3.
6

Fig. 2-3
Second, when the slope is steeper than -1, the optimal point is thetangency point of the time constraint line and the indiﬀerence curve.
In this case, the optimal point (T1∗ , T2∗ ) locates left of the intersection
point and closer to the horizontal axis, as illustrated:

Fig. 2-4
7

[]
P1 /a1
Finally, in the case the slope lies within −1, − P2 /a2 , the optimal
point is the intersection point of the two constraints. （T ∗ = T¯1 ,1

T2∗ = T¯2 ）Fig. 2-5

3 Comparative Statics
3.1 Price Change
3.1.1 The price of the ”time-intensive good”
When the price of good 1 rises (P1 < P1′ ), the possible consumption
set is shown by the lower shaded area in Fig. 3-1.

8

Fig. 3-1
The set can be partitioned into three areas as Fig. 3-2 shows. The
partition depends on how the slope of the binding constraint line
changes according to the price change.

Fig. 3-2
9Note that the binding constraint switches from the time constraint
to the budget constraint in area II. Consequently, the slope of thebinding constraint becomes flatter by the price change.
To explain the ”switching eﬀect”, the following two provisional demand functions of good 1 are defined.

T1T = arg max U (T1 , T2 )s.t. T 0 = T1 + T2

(6)

T1Y = arg max U (T1 , T2 )s.t. Ti = ai Xi (∀i = 1, 2)
Y = P1 X1 + P2 X2(7)

The former shows the time demand for good 1 if the consumer should
face the time constraint only. Conversely, the later is the time demand
for good 1 if the consumer should face the budget constraint only.
][
The eﬀect of the price change when T ∗ ∈ T¯1 , T¯′ can be divided
1

1

into two parts:
dT1∗
∂T1T
=
dP1
∂T 0

(

Y
− T0
′
P1 /a1)∂T1Y+∂ (P2 /a2 )

(

Y
Y
−
′
P1 /a1P2 /a2)(8)

The first and the second terms respectively correspond to (a) and (b)in the following figure:

10

(a)
(b)

Fig. 3-3
Conditions under which the optimal T1 increases when P1 rises,
can be summarized as follows.
Proposition 1[]∗¯1 , T¯′ .
∈TSuppose
T
1
1
)
(
(
)
∂T1Y
∂T T
dT1∗
Y
Y
−
If ∂T10 P ′Y/a1 − T 0 + ∂(P2 /a
>
0,
then
′ /a
)
P
P
/a
dP1 > 0.
1
2
2
2
1

1

Definition 1（”time-Giﬀen good”）

]
[
dT ∗
Good 1 is a time-Giﬀen good if T1∗ ∈ T¯1 , T¯1′ and dP11 > 0.

3.1.2 Example of a ”time-Giﬀen good”: local train and express railway
The more money and time we have, the more frequently we aﬀord
to travel. Realistically, however, we face the shortage in at least oneof these two resources; one of the constraints has a binding eﬀect on
the travel demand.
Suppose that the local train is ”time-intensive transportation” andthe express railway is ”money-intensive transportation”. Then, some11

travelers may increase the frequency of using the local train, when
the fee of the local train rises. They will aﬀord less frequent journeys
by the express railway because of the eﬀective budget shrink. And
the decrease of the express railway use relaxes the time constraint,
and then, the consumer utilizes the saved time by increasing the useof the local train. This will happen because of the time constraint.
3.1.3 The price of the ”money-intensive good”
The case that the price of good 2 increases (P2 < P2′ ) is drawn in
Fig. 3-4.

Fig. 3-4[]
The eﬀect of the price change when T1∗ ∈ T¯1 , T¯1′ is:
dT1∗
∂T1T
=
dP2
∂T 0

(

Y
− T0
P1 /a1)∂T1Y+∂ (P2 /a2 )

(

Y
Y
− ′
P1 /a1P2 /a2)(9)

The first and the second terms respectively correspond to (a) and (b)in the following figure:
12

(a)

(b)

Fig. 3-5

Definition 2（”time-substitution(/time-complementary) good” (good 1)）(i) Good 1 is the time-substitution good for good 2[]
dT ∗if T ∗ ∈ T¯1 , T¯′ and 1 > 0.
1

1

dP2

(ii) Good 1 is the time-complementary good for good 2[]
dT ∗if T ∗ ∈ T¯1 , T¯′ and 1 < 0.
1

1

dP2

(iii)Good 1 is the time-neutral good for good 2[]
dT ∗if T ∗ ∈ T¯1 , T¯′ and 1 = 0.
1

1

dP2

Proposition 2 a[].
Suppose T(1∗ ∈ T¯1 , T¯1′ )
(
)
∂T1T
∂T1Y
dT ∗
Y
Y
Y
0
(i) If ∂T 0 P1 /a1 − T + ∂(P2 /a2 ) P1 /a1 − P ′ /a2 > 0, then dP12 > 0
2(good 1(is the time-substitution)( good of good )2).
∂T T(ii)If ∂T10

Y
0
P1 /a1 − T

∂T Y

Y
Y
P1 /a1 − P2′ /a2

1+ ∂(P2 /a2)(good 1 (is the time-complementary)( good of good
) 2).
∂T T

(iii)If ∂T10

Y
0
P1 /a1 − T∂T Y

1+ ∂(P2 /a
2)YY
P1 /a1 − P2′ /a2(good 1 is the time-neutral good of good 2).

13

dT ∗

< 0, then dP12 < 0
dT ∗= 0,then dP12 = 0

Similarly, the demands for good 2 can be defined as follows:

T2T = arg max U (T1 , T2 )s.t. T 0 = T1 + T2

(10)

T2Y = arg max U (T1 , T2 )s.t. Ti = ai Xi (∀i = 1, 2)
Y = P1 X1 + P2 X2

(11)

And
∂T2T
dT2∗
=
dP2
∂T 0

(

Y
− T0
P1 /a1)∂T2Y+∂ (P2 /a2 )

(

Y
Y
− ′
P1 /a1P2 /a2)(12)

Also note that the eﬀect of P1 change on T2∗ :
dT2∗
∂T2T
=
dP1
∂T 0

(

Y
− T0
′
P1 /a1)∂T2Y+∂ (P2 /a2 )

(

Y
Y
−
′
P1 /a1P2 /a2)(13)

Definition 3（”time-substitution(/time-complementary) good” (good 2)）(i) Good 2 is the time-substitution good for good 1[]
dT ∗if T ∗ ∈ T¯1 , T¯′ and 2 > 0.
1

1

dP1

(ii) Good 1 is the time-complementary good for good 2][
dT ∗if T ∗ ∈ T¯1 , T¯′ and 2 < 0.
1

1

dP1

(iii)Good 1 is the time-neutral good for good 2[]
dT ∗if T ∗ ∈ T¯1 , T¯′ and 2 = 0.
1

1

dP1

14

Proposition 3 a[]
Suppose T(1∗ ∈ T¯1 , T¯1′ )
.
(
)
∂T2Y
∂T2T
dT2∗
Y
Y
Y
0
−
(i) If ∂T 0 P ′ /a1 − T + ∂(P2 /a
>
0,
then
′ /a
)
P
P
/a
dP1 > 0
2
1
2
2
1

1

(good 2 (is the time-substitution)(good of good 1).
)∂T T(ii)If ∂T20

Y
0
P1′ /a1 − T

∂T Y

Y
Y
P1′ /a1 − P2 /a2

2+ ∂(P2 /a
2)dT ∗< 0, then dP21 < 0(good 2 is( the time-complementary)( good of good
) 1).
∂T T

(iii)If ∂T20

Y

P1′ /a1∂T Y

2− T 0 + ∂(P2 /a2)Y

P1′ /a1− P2Y/a2dT ∗= 0,then dP21 = 0(good 2 is the time-neutral good of good 1).
The following E Y (≤ Y ) is defined as the expenditure at (T1∗ , T2∗ ):E Y = X1∗ P1 + X2∗ P2
P1
P2
= T1∗
+ T2∗
a1
a2

(14)

and
dE Y
dT1∗ P1T∗dT ∗ P2=+ 1 + 2
dP1
dP1 a1a1
dP1 a2(15)

T∗

Because Pa11 , Pa22 , and a11 are positive, if good 1 is the time-Giﬀen good;
dT ∗

Y

2
*2 then(T1∗ , T2∗ ) is in area II and dE
dP1 ≤ 0,dP1 must be negative; good

2 is the time-complementary good for good 1. It is summarized as
the following proposition.
Proposition 4 a
If good 1 is the time-Giﬀen good,then good 2 is the time-complementary good for good 1.

*2 This is because of the switching from the time constraint to the budget

constraint.

15

3.1.4 Time Consumption Parameter
The result of the change in ai is inverse to Pi . We can confirm it
easily because ai and Pi are expressed by the relative price (Pi /ai )in the figures. It can be expressed as follows:
dTi∗
∂TiT
=
da1
∂T 0

(

Y
− T0
′
P1 /a1)∂TiY+∂ (P2 /a2 )

(

Y
Y
−
′
P1 /a1P2 /a2

)
(16)

and
∂TiT
dTi∗
=
da2
∂T 0

(

Y
− T0
P1 /a1)∂TiY+∂ (P2 /a2 )

(

Y
Y
−
P1 /a1P2 /a′2)(17)

Proposition 5

]
[
dT ∗
dT ∗
Suppose T1∗ ∈ T¯1 , T¯1′ , then dali = − dPli (i = 1, 2 and l = 1, 2).

3.2 Income
Fig. 3-6 shows the case that an increase in income shifts the budget
line outward (Y → Y ′ ). The constraint line shifts outward without
rotating in area I. It is the same as the case without the time constraint. In area II, the slope steepens. And there is no change in III.
The eﬀect of the change in area II is separated as follows:
dTi∗
∂T T
= − i0
dY
∂T

(

Y
− T0
P1 /a1)∂TiY−∂ (P2 /a2 )

(

Y
Y
−
P1 /a1P2 /a2)(i = 1, 2)(18)

The first and the second term respectively correspond to (a) and (b)in Fig. 3-7.

16

Fig. 3-6

(a)

(b)

Fig. 3-7

17

3.3 Disposable Time
Fig. 3-8 shows the case that an increase in the disposable time′

shifts the time constraint line (T 0 → T 0 ).

Fig. 3-8
There is no change in area I. The slope of the constraint flattens inarea II. It is contrary to the case of income change. In area III, thechange shifts the constraint outward. In area II, the eﬀect of thedisposable time change can be shown by the two eﬀects:
dTi∗
∂TiT
=
dT 0
∂T 0

(

Y
− T0
P1 /a1)∂TiY+∂ (P2 /a2 )

(

Y
Y
−
P1 /a1P2 /a2)(i = 1, 2)(19)

And the figure is:

18

(a)

(b)

Fig. 3-9

Also note that the diﬀerence between the income change and thedisposable time is in their directions.
Proposition 6

]
[
dT ∗
dT ∗
Suppose T1∗ ∈ T¯1 , T¯1′ , then dTi0 = − dYi (i = 1, 2).

4 Working Time and Income
In this section we analyze the case in which the consumer cantransfer her income and disposable time. The consumer can earnmore money by sacrificing her time to enjoy consumption goods, orincrease the disposable time by reduction of income(/working time).
And the consumer adjusts the allocation of time as far as it improves
her utility level.

19Here we introduce working time Tw and modify the time constraint
as follows:
T 0 = Tw + Tc
Tc = T1 + T2

(20)
(21)

Increasing the income by reduction of time for consumption can be expressed by the combination of an inner shiftof the time constraint line and an outer shift of the budgetline.

In this case,( the intersection )point of the constraint lines′′
moves leftward (T¯1 , T¯2 ) → (T¯1 , T¯2 ) . The intersection point is)(()/a2 −Y (Tw ) Y (Tw )−Tc P1 /a1,. Y (Tw )rewritten as T¯1 , T¯2 = TcPP22/a
P2 /a2 −P1 /a1
2 −P1 /a1means the income is a function of the working time, Tw . Then,
∂ T¯1
∂Tw

/dTw +P1 /a1∂ T2
2 −dY /dTw
= −PP22/a< 0 and ∂T= dY/a2 −P1 /a1
P2 /a2 −P1 /a1 > 0 . It isw¯

shown in Fig. 4-1.

Fig. 4-1

20

We also show the pictures before and after the change respectively,
in Fig. 4-2 and Fig. 4-3.

Fig. 4-2

Fig. 4-3

21

The change of the binding constraint line is categorized into thefollowing four cases:
1. The constraint shifts outward in I, same as in area I whenincome increases.
2. The eﬀect of the change can be shown by
(
)
(
)
∂T1T
Y
∂T1Y
Y
Y
′
− Tc −−−,∂T 0 P1 /a1∂ (P2 /a2 ) P1 /a1
P2 /a2same as in II when income increases.
3. The eﬀect of the change can be shown by
(
)
(
)
∂T1T
Y
∂T1Y
Y
Y
′
− Tc +−,∂T 0 P1 /a1∂ (P2 /a2 ) P1 /a1
P2 /a2same as in II when disposable time increases.
4. The constraint shifts inward in IV, same as in III when disposable time decreases.
The switching happens in both II and III; time to budget, andbudget to time, respectively.
The boundary between II and III is the intersection point of thebudget line before the change and the time constraint line after thechange:(′′′′T¯1 , T¯2()=

Tc′ P2 /a2 − Y
Y − Tc′ P1 /a1,
P2 /a2 − P1 /a1 P2 /a2 − P1 /a1)(22)

We have summarized the conditions that the individual changes
her working time by locating the optimal point before the change.
Note that we have precluded the corner solution cases in I and IV for
simplicity.
22

The condition that the individual increases or decreases working
time in each area is as follows:(I) if the optimal point before the working adjustment locates on
the budget line; the optimal point exists in area I.
(II) if the switching from the budget constraint to the time constraint improves the utility level in area II.
(III) if the switching from the time constraint to the budget constraint improves the utility level in area III.
(IV) if the optimal point before the working adjustment locates on
the budget line; the optimal point exists in area IV.
If the adjustment is complete, the optimal point after the adjustment is the intersection point of the constraint lines after the change:
(
)
( ∗
∗)
′
′
T¯1 , T¯2 = T¯1 , T¯2(23)
The reason is because, the individual can improve her utility levelby reducing the working time and increasing the disposable time if
the consumption point is in III or IV; the slope of the indiﬀerence
curve which expresses the utility level at the consumption point islarger than one. Contrarily, the consumer can improve her utility bysacrificing the disposable time and working longer if the consumption
P1 /a1point is in I or II; if the slope is less than P. It is summarized by
2 /a2

the following condition:′′
M RS(T1∗ , T2∗ ) = M RS(T¯1 , T¯2 ) ∈[]P1 /a1,1P2 /a2(24)

where
M RS(T1 , T2 ) =∂U (T1 , T2 )/∂T1∂U (T1 , T2 )/∂T2

23

(25)

The optimal solution including the optimal working time is as follows:′′(T1∗ , T2∗ , Tw∗ )= (T¯1 , T¯2 , Tw∗ ))Y ′ − Tc′ P1 /a1
Tc′ P2 /a2 − Y ′0′,, T − Tc=
P2 /a2 − P1 /a1 P2 /a2 − P1 /a1[]
P1 /a1′ ¯′∗∗¯
s.t. M RS(T1 , T2 ) = M RS(T1 , T2 ) ∈,1
P2 /a2(26)(

5 Saving
Now we consider saving by augmenting my model to two periods.
Suppose T10 = T20 (; the exogenous time resources are the same in
the two period) and the consumer does not work. We modify theutility function to express the two periods. The utility function ofeach period is consistently u(・), and shown by
U (T11 , T21 , T21 , T22 ) = u (T11 , T21 ) + δu (T12 , T22 )(27)

where Tij represents the time for good i in period j (i = 1, 2 and j =1, 2). δ is the discount factor between the two periods. If there is norevenue in the second period, then the budget constraint through thetwo periods is expressed by the income in each period, Yj (j = 1, 2),saving, S, and interest rate, r:

and

(
)
P1
P2
Y2 = (1 + r)S = (1 + r) Y1 −
T11 −
T21
a1
a2

(28)

(
)
P1
P2
P1
P2
(1 + r) Y1 −
T11 −
T21 ≥
T12 +
T22
a1
a2
a1
a2

(29)

Note that time cannot be saved although the constraint is expressedin terms of times, Tij . The following condition must hold.
Tj0 ≥ T1j + T2j , ∀j = 1, 2
24

(30)

Note also that income and time are asymmetric; only income canbe saved.

5.1 Shortage of Budget
Suppose that δ = 1 and r = 0 for simplicity. Then
Y2 = Y1 −

P1
P2
T11 −
T21
a1
a2

(31)

and
Y1 −

P1
P2
P1
P2
T11 −
T21 ≥
T12 +
T22
a1
a2
a1
a2

(32)

∗
∗
The optimization for the consumer is to achieve (T1j
, T2j
) such[]P1 /a1
∗
∗
)∈ P
, T2j
that M RS(T1j, 1 (∀j = 1, 2). The optimal point is
2 /a2

not necessarily the intersection point after the adjustment by saving
or borrowing.
Obviously, the more disposable time the individual has, the moremoney is required to achieve the ideal(/possible/potential) time consumption. In this case, the ratio of disposable time, (Tcj ), and required income, (Y j ), of each period is the same. The following equality shows the relationship.
Tc1
Y
= 1
Tc2
Y2

(33)

If she has enough money that makes the budget constraints in eachperiod slack, then she spends time to each good by the same ratio in
each period:
∗
∗
T11
T12
=
∗
∗
T21
T22

25

(34)

In other words, the optimal point divides the time constraint line bya constant ratio in each period.
However, if money is scarce in at least one of the periods, theoptimal point is the tangent point of the indiﬀerent curve and thebudget constraint line, which locates left of the intersection pointin each period. Then Yj = Pa11 T1j + Pa22 T2j (j = 1, 2) holds, and themarginal utility from budget increasing is positive in each period after
the consumption smoothing. The optimal solution is decided by theequalization of marginal utility with marginal costs:∂u (T11 (Y1 ), T21 (Y1 )) ∂u (T21 (Y2 ), T22 (Y2 ))
+
=0
∂Y1
∂Y2

(35)

If δ ∈ [0, 1] and r ≥ 0, then the consumption smoothing cannot
T∗

T∗

21

22

12
*3 And the condition of the optimization
always achieve T11∗ = T∗ .

is:∂u (T11 (Y1 ), T21 (Y1 ))∂u (T21 (Y2 ), T22 (Y2 )) dY2
+δ
=0
∂Y1
∂Y2
dY1

(36)

5.2 Liquidity Constraint
Under the liquidity constraint on saving and borrowing, theconsumer cannot save money enough to carry out the consumption
smoothing, and the optimal point in the second period locates on
the budget line, left of the intersection point, and some amount of
the time resource is left unused. It can be shown by the following*3 There are some exceptions. For example, if the preferences are contempora∗T∗
T12
neously homothetic in each period, then T11∗ = T ∗ always holds.
21
22

26

conditions.
]P1 /a1,1P2 /a2(37)

P1 /a1P2 /a2(38)∂U (T1j , T2j )/∂T1j, j = 1, 2
∂U (T1j , T2j )/∂T2j

(39)

∗
∗
M RS(T11
, T21
)∈

[

and
∗
∗
M RS(T12
, T22
)<

where
M RS(T1j , T2j ) =5.2.1 Example: age-based wage
If income is fixed by age-based wage system, the increase in incomeby aging can be shown by Y1 < Y2 . Suppose that the preference ofthe consumer, the prices of the goods, (a1 , a2 ) and working time forthe consumer are consistent through the two periods. Then the time
constraint and the slope of the budget constraint remain unchanged
through the two periods. And the diﬀerence between the two periods is caused only by the diﬀerence of Yj . In this case, because theconsumer cannot adjust the working time, she tries to improve her
utility level by borrowing money in the first period and pay back fromthe excess budget in the second period, if the optimal consumption
without the borrowing is the tangency point on the budget line(; theconsumer face the budget constraint) in the first period. In otherwords, the consumer has a chance to improve her utility by utilizing the liquidity. On the other hand, if the optimal point is on thetime constraint line(; the consumer face the time constraint) in thefirst period, she cannot conduct any improvement. And the liquidity
constraint has eﬀect only in the former case.

27

5.3 Other Applications• The cases in which each period is long enough for the prices or
the time consumption parameter of each good to change fromthe first to the second periods. For example, the price of only
one of the goods changes by inflation, or the parameter change,
ai1 ̸= ai2 , by technological innovation.
• The cases that Tj has an eﬀect on aj+1 , by job training or
R&D investment.
• The cases that the necessary time input for each good is afunction, Ti = Ti (Xi ), instead of Ti = ai Xi .

6 Concluding Remarks
In this paper we have constructed a model which includes time
constraint in addition to budget constraint, and conducted the comparative statics analyses. The main result in the comparative statics
is that the switching of the binding constraint can cause an increasein the demand of the ”time-intensive” good when its price rises. We
have further analyzed the case that the consumer can adjust working
time. The consumer faces the trade-oﬀ between working(/disposabletime) and income. The adjustment of the budget constraint and thetime constraint, via exchanging time for money, improves the consumer’s utility level. Finally we extend the model to two periods.
The consumer improves her total utility through the two periods byborrowing or saving. We also find that the improvement is possible
only if the consumer has time left unconsumed in that period when
the budget constraint binds and this can be relaxed by liquidity. As
further research, application to producer or government behavior is28

expected.

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