Negotiations and love songs:
The dynamics of bargained household decisions
Stephen Lich-Tyler†
Department of Economics
University of Michigan at Ann Arbor
Ann Arbor, Michigan 48109-1220
[email protected]
Abstract
This study contains theoretical and empirical examinations of the
allocations resulting from a multi-period household bargaining problem.
A household has three possible “intertemporal procedure” it can use for
cooperative decisions: solving the lifetime problem period-by-period (a
repeated static game); deciding all allocations simultaneously at the
beginning (a multi-stage game with full commitment); or solving it using
backwards induction (a multi-stage game without commitment). Though
these procedures are equivalent for individual decisions, household
decisions are almost never invariant to the procedure used. Assuming that
the household uses an arbitrary set of rules resulting in a continuous and
efficient outcome (a generalized “bargaining solution”), I derive a unique
modified Euler equation for each procedure. Using data on married
couples from the Panel Study of Income Dynamics from 1976-1986, I
examine the impact of future and past reservation utilities on current
consumption. Results suggest that different households use different
intertemporal procedures. Using a latent class model, I determine which
variables affect the probability that a household uses a procedure.
Children, divorce costs, and the age of the marriage are factors that
substantially affect these probabilities. (JEL classifications: D91, D19, J12)
With disagreements about the meaning of a marriage contract, conversations are hard and wild.… Two
people are playing the game: negotiations and love songs are often mistaken for one and the same.
Paul Simon, “Train in the Distance”
†
Research fellow and visiting assistant professor, University of Michigan–Ann Arbor. For useful
comments, I thank Dan Hamermesh, Dan Slesnick, Aloysius Siow, Max Stinchcombe, William
Thomson, and seminar participants at several institutions. I presented versions of this study at the
2002 Society of Labor Economists Meeting and the 2003 American Economic Association Meeting.
Household bargaining & time
1.
2
INTRODUCTION
Household bargaining models by Manser and Brown (1980), McElroy and
Horney (1981), Lundberg and Pollak (1993), as well as the more general collective
approach of Chiappori (1988a), have greatly developed our understanding of
pluralistic household behavior. A number of empirical studies show that we can
clearly reject unitary models of household behavior (Horney and McElroy [1988],
Lundberg et al. [1997], Browning and Chiappori [1998], Thomas [1990, 1999], for
instance).1 However, the literature has focused the household allocation problem in
a static setting—it has not addressed the issue of how the bargaining process works
over a number of periods. Households (for the most part) endure for more than a
single period, though, and the multi-period aspect of the game can potentially affect
the solution substantially. Furthermore, in order to make valid economic inferences
about household behavior from consumption dynamics or savings behavior, it is
necessary to view bargaining in a multi-period setting.
In order to do this, we need to state explicitly how we believe households
solve multi-period bargaining problems—an issue which has received only brief and
informal mention in the literature. Browning and Chiappori (1998) suggest that the
household decision-making process is a “repeated game.” This is the implicit
assumption behind a number of empirical studies that examine the change in
consumption between two periods as a function of the change in those periods’ threat
points (such as Lundberg, Pollak, and Wales). Carlin (1991) makes this assumption
explicit when examining human capital investment decisions from a household
perspective.
However, this “repeated game” approach is not the only way one could model
the multi-period bargaining procedure, and the model does not even specify how the
household makes decisions about how to allocate resources between periods.2 Recent
papers have explored the savings behavior of the pluralistic household. Browning
(2000) models the savings decision as a specific non-cooperative game. In examining
the “retirement-consumption puzzle,” Lundberg, Startz, and Stillman (2001) note
that the allocation of goods over the life-cycle of the household can depend on
whether the household is able to make binding intertemporal agreements.
1
The most familiar models can all be found in Haddad, Hoddinott, and Alderman
(1997). This book also contains a number of empirical studies, focusing on the
intrahousehold distributive effects of economic development.
2
Because of legal institutions, the savings decisions essentially must be a joint
decision—since most of the household’s assets are considered community property,
agents are held responsible for their partners’ debts, and so forth.
Household bargaining & time
3
Examining intertemporal allocation issues is fundamentally important to our
understanding of household behavior. How are savings and investment decisions
made? Does it make sense to say that a household has a “rate of time preference?”
Can one make valid inferences about the bargaining process by examining one period
in isolation of others? Is the household decision-making rule homogeneous over
time, or does it evolve over the “lifecycle” of the household? The household
intertemporal allocation problem is more complex than the individual problem. In
single-person decision making, intertemporally separable preferences imply that a
multi-period utility maximization problem is essentially solved through a two-step
procedure. In the first stage, the individual makes an intertemporal budgeting
decision; this decision depends on his or her rate of time preference and the interest
rate. Afterwards, he or she chooses in each period the bundle of goods that
maximize the period’s instantaneous utility function. The decision within the period
is made in isolation from decisions about other periods, and one can analyze it in the
same manner as one would analyze a static model. Modeling the household
intertemporal allocation decision is more complex. Two problems arise immediately
in this two-step: the household itself is not endowed with a rate of time preference,
and those of the agents do not necessarily agree. Furthermore, it would seem that
the agents’ preferences are not the only factors affecting its savings
decision—“strategic” considerations may arise.3 For example, one household
member might favor high household savings if he knows he has relatively better
outside options later in life—though he might actually be relatively impatient.4
At a first glance, it seems intuitive that an observed change in a bargained
household decision (from one year to the next) reflects a change in the agents’
outside options). This would indeed be the outcome if the household uses a twostep procedure as outlined above: first allocating resources intertemporally, then
bargaining over the choice set within each period separately. But this sort of
procedure has other implications that might not conform to our perceptions about
household behavior. First, it is fundamentally inefficient, since agents will have
different marginal rates of substitution between goods in different periods. Second,
3
In divorce settlements, women typically receive a disproportionate share of the
household’s assets. This could easily affect individuals’ savings incentives. In this
paper, I do not consider that the state of assets may affect agents’ threat points.
4
Similar circumstances—the fact that women tend to outlive their husbands—are
behind the saving and consumption behavior examined by Browning (2000), by
Lundberg and Ward-Batts (2000), and by Lundberg et al. (2001).
Household bargaining & time
4
agents will their partners’ transitory “rough years,” increasing their own utility in
response to a temporary reduction in their partners’ outside options.
Typically, we view the household as an institution which exists to increase
surplus though economies of scale, specialization, and gains from trade. It also
serves a role in providing social insurance during unusual circumstances. Partners
help smooth over each other’s rough spots, like child-rearing or unemployment.
While this smoothing behavior is not incompatible with a household bargaining
model, it is incompatible with a model in which the household’s decisions are made
in each period independently of other periods.
The purpose of this paper is to illustrate that household decisions depend on
how the intertemporal aspect is treated. This affects how we should interpret
observed behavior. In my analysis of a multi-period decision problem, the concept
of “household bargaining” is treated as generally as possible. I assume that through
some unspecified decision-making process, the household picks a particular outcome
from each set of feasible utilities. The set of rules generating this decision, called a
“bargaining solution,” is like the notion of an abstract choice structure belonging to
the household.5 Models in the existing literature—including Manser/Brown,
McElroy/Horney, and Chiappori—are easily understood in this framework.
To motivate my argument that the difference between the bargaining
procedures is non-trivial, the first section of this paper provides an example of a
simple two-good, two-period problem.6 This gives some insight into how and why
the solution to a multi-period allocation problem need not reflect the solution to the
allocation problem for each year individually.
The following section develops three procedures for solving a multi-period
household bargaining problem. The essential distinction between these procedures
is what the household views as the problem over which they are bargaining. The first
is called the myopic procedure: agents solve the allocation problem in each period
independently, not regarding past or future bargaining problems. The second is the
contractual procedure: the household makes allocation decisions for all periods
simultaneously, before the first period, viewing the entire lifetime as a single
5
However, bargaining solutions often violate properties desired in choice structures.
Lundberg/Pollak (2001) also illustrate that the outcome of a multi-period household
bargaining problem may depend on how the household approaches the problem.
They look at the relocation decisions of dual-earner couples. When offered a job
that is better for one spouse but worse for the other, whether or not the couple
moves might depend on whether or not the winner can commit to compensating the
loser after settling into the new location. These scenarios are comparable to the
“contractual” and “prescient” approaches suggested in this essay.
6
Household bargaining & time
5
bargaining problem. After solving this lifetime bargaining problem, agents have no
ability to renegotiate or to change the allocation later. Finally, I present the prescient
procedure in which agents solve a lifetime bargaining problem, but they are conscious
that the chosen allocation should not give anyone incentive to reopen negotiations
later. The household has a different consumption dynamics under each procedure;
each of these is characterized generally in terms of the chosen bargaining solution
(which could be Nash, or Kalai-Smorodinsky, or egalitarian, for example). In the
following section, I discuss some of the important distinctions between the models.
My analysis closes with this result: for any bargaining solution which considers
disagreement points, bargained household decisions cannot be invariant to the
intertemporal procedure. Therefore, one can generally always identify the procedure
by examining the household’s consumption decisions over time, regardless of the
bargaining solution. In the empirical portion of this paper, I use two methods to
test which intertemporal procedure describes household behavior.
Since each intertemporal procedure implies a unique specification for
consumption dynamics, I first test which of these specifications can be rejected. For
a sample of married couples from the Panel Studies of Income Dynamics (PSID),
three different cases show that I can reject all three intertemporal bargaining
procedures (as well as the unitary model) at almost any confidence level. However,
when I divide my sample into couples with and without children, I find that the
intertemporal procedure may differ among households: the prescient procedure
works remarkably well for couples with children, while the myopic procedure works
well for those without children.
Since it appears that the correct specification may differ among couples, I
then model this as a “latent class” problem. Using this approach, I can identify the
parameters of all three specifications at the same time, and also identify the
probability that a particular observation falls into each “class.” In addition to the
number of children in the family, variables affecting divorce probabilities have a
great deal of power in explaining the intertemporal procedure used by the household.
2.
A MOTIVATING EXAMPLE
This section presents a two-period bargaining problem, which can be solved
in multiple manners. In one case, agents solve the bargaining problem for each
period separately; in the other, the procedure is to solve the allocation for both
periods simultaneously . The allocations satisfying the Nash bargaining solution under
Household bargaining & time
6
these procedures do not agree. This example shows that if the household is solving a
multi-period problem, we cannot necessarily make any inferences from examining
the periods in isolation. The example also illustrates how, in a multi-period model,
agents make Pareto gains by smoothing over fluctuations in their partners’ outside
options over the lifecycle.
Hrothgar and Wealhtheow are a young Anglo-Saxon couple contemplating
matrimony—and as is often the case, negotiations accompany their love songs.
Money is a major concern of the first two years. Wealhtheow’s wages as a
handmaiden give her an income of 10 sceat7 in the first year, but her decision to raise
a child during the second year means that she receives nothing. Hrothgar’s earnings
profile is reversed: he receives no income while training to be a warrior in the first
year, but anticipates obtaining plunder worth 10 sceat in the second. Finally—if they
can manage to live together in peace, the household8 gains a large surplus valued at
90 sceat in each of two years from economies of scale in housing.
As far as this young couple is concerned, honey and wine are the only two
goods worth purchasing. Hrothgar alone enjoys honey (which is fermented to
produce mead). He has no regard for wine, which is Wealhtheow’s tipple.
Preferences of these agents can be represented as:
(2.1)
(2.2)
Uw ( xw1 , xw 2 ) := uw1 ( xw1 ) + uw 2 ( xw1 ) = 0.1 xw1 + 0.1 xw 2
U h ( xh1 , xh 2 ) := uh1 ( xh1 ) + uh 2 ( xh 2 ) = 0.1 xh1 + 0.1 xh 2
where xlt ∈[0, 1] is the expenditure at time t ∈ {1, 2 } devoted to consumption of good
l ∈ { h, w} . The function Ui the lifetime utility function of agent i, and uit is the r
instantaneous utility function.
The absence of a reliable banking system in Anglo-Saxon England prohibits
the couple from saving. The individual earnings of the agents mean that they can
guarantee themselves utility of (ew1 , eh1 ) = (1, 0) at t=1 and (ew 2 , eh 2 ) = (0, 1) at t=2 should
they fail to come to an agreement. If they fail to make an agreement in both periods,
they have the lifetime reservation utility ( Ew , Eh ) := (ew1 , eh1 ) + (ew 2 , eh 2 ) = (1, 1) .
There are several ways the couple could solve the two-period bargaining
problem. One possibility to solve each period separately, as it occurs. In this case,
the Nash solution dictates that the household pick the feasible allocation that solves:
7
8
Weighing 22 grains apiece, there were approximately 240 sceat in a pound of silver.
A mead-hall name Heorot.
7
Household bargaining & time
(2.3)
max xwt ,xht
∏ ( u( x ) − e )
it
it
i∈{ w ,h }
in each of the years. A little algebra reveals to the couple that they should allocate
their expenditures (6.5, 3.5) in the first year. This is reversed in the second period:
(3.5,6.5 ) . These can be described as allocations satisfying yearly solutions.
An alternative procedure has Hrothgar and Wealhtheow agreeing (before t=1)
to an allocation for each period. In bargaining, they will be considering their lifetime
utility and their lifetime reservations. The solution to this problem dictates a
feasible allocation solving the problem:
(2.4)
max xw 1 ,xh 1 ,xw 2 ,xh 2
∏ (U (x , x ) − E )
i1
i2
i
i∈{ w ,h }
Since the agents have symmetric intertemporal utility functions and lifetime
reservation utilities, it should be no surprise that the lifetime solution produces
symmetric utilities (and hence, expenditures as well). The household contracts to
purchase (5, 5) in the first year and (5, 5) in the following year.
Clearly the choice of bargaining procedure affects the household decision.
Leading into the next section of this paper, I would like to highlight some of the
differences and to explain why one might find a particular model compelling (or not).
For the yearly solutions, allocations preserve the intuition that the change in allocation
is the result of a change in threat points. The allocation dictated as a solution to the lifetime
problem does not respond to a transitory change in threat points. If an outsider observes only
the reservations and final allocations and tries to infer the decision-making
mechanism, he might mistakenly guess that there is no strategic interaction—after
all, Wealhtheow receives exactly the same share of pie when she has a good outside
option as when she has a poor outside option.9
The yearly solutions are not Pareto optimal. Quite clearly, Hrothgar substitutes
the second pie for the first at a much higher rate than does Wealhtheow. Because
gains from trade remain, it seems implausible as the final outcome of the bargaining
process. In contrast, the lifetime solution results in an efficient allocation.
9
If discounting is added into the problem so that Wealhtheow values the second
year relatively more than Hrothgar, the lifetime solution gives her a larger allocation
when her threat point is lower.
Household bargaining & time
8
On the other hand, the lifetime solution is not “time consistent.” Suppose the players
have agreed to this solution. After the first purchases of honey and wine have been
consumed and digested, Hrothgar might realize that he has a strong bargaining
position if he were to re-open negotiations. The first year’s actions are irrevocable at
this point, after all. Without a enforcement mechanism, this might be an
unreasonable outcome from the bargaining process.10
According to the yearly solutions, Hrothgar and Wealhtheow take advantage of their
partners’ “bad years”; they assist in smoothing their partners’ consumption under the lifetime
solution. That is to say, when Hrothgar’s income goes up, the household’s
consumption on his favored good changes by more than his income in the one-period
solutions—he receives a disproportionate increase in welfare. In the two-period
solution, this does not happen. Instead, Wealhtheow supports Hrothgar during his
years of military training, and he supports her during her years of child-rearing.
For individual utility maximization, it is irrelevant whether we choose to solve
the intertemporal problem period-by-period, all periods simultaneously, or (as a third
alternative) backwards by dynamic programming. However, this example shows that
this is certainly not true of household decisions. Modeling the multi-period
bargaining problem differently can generate radically different results.
One should therefore be cautious when deciding how to model the lifetime
bargaining procedure. Three specifications of a multi-period household bargaining
process are examined in the following section.
3.
THE MULTI-PERIOD BARGAINING PROBLEM
In household bargaining problems, a number of agents must agree to a
common consumption bundle. In my model, the bargaining problem is extended to
a number of time periods: the household must choose a consumption bundle in each
period, so agents must also agree to decisions about the intertemporal allocation of
resources. Time makes the game richer, and time allows several ways of modeling
the problem—with different consequences for household behavior. In this section I
examine three possible bargaining procedures; the following section provides a
summary of the distinctions among the procedures.
10
The lifetime solution need not even be individually rationally at any point—which
causes one to wonder why an agent would voluntarily stay in the household when his
guaranteed outside option is preferred to the bargained decision.
9
Household bargaining & time
The household is comprised of two agents,11 denoted by i ∈ { w, h} . Agents
have perfect foresight. The entire allocation problem spans a length of time called a
lifetime and represented by the unit interval. This is divided into periods of length
∆t , called years. Years are indexed by t ∈ T : = {k∆t : k ∈ N, k ≤ 1 ∆t } .
The agents share a vector of consumption goods xt ∈ R L in that year. These
goods may be enjoyed by one or both agents—though one agent’s “enjoyment” may
come only indirectly by “caring” about the other’s private consumption. Each agent
has intertemporally separable preferences over the lifetime allocation x = ( xt )t ∈T .
Naturally, agents may have different preferences over these commodities—some may
be “husband-specific” or “wife-specific” goods, some may be pure public goods, and
others may be disliked by one person. Preferences over these allocations can be
represented by the lifetime utility function:
(3.1)
Ui ( x ) =
∑δ
it
ui ( xt ) ∆t
t ∈T
The strictly positive but decreasing δ it is the individual’s discount function (with δ it set
equal to one), and ui : R l → R is his instantaneous utility function.12 This should be
regarded as a “total utility function,” incorporating altruism or any other
interdependence of preferences. The instantaneous utility functions are twice
continuously differentiable and strictly monotone and concave. The household
jointly owns assets At at time t. Subject to some given initial assets and a terminal
nonnegativity constraint, it faces the budget constraint:
(3.2)
∆At := At + ∆t − At = ( rt At − pt ⋅ xt )∆t ≡ ( rt At − mt )∆t
where At represents the household’s assets at time t, rt the rate of return on assets,
and pt the price vector. Total household expenditures in year t are denoted by mt .
11
Extension to an arbitrary number of agents is straightforward, though it provides
no new results for my paper—and probably unnecessary, since most households
consist of only one or two active decision makers. However, additional concerns
about properties of the bargaining solution arise when examining more agents (see
Lensberg [1987] and Thomson [1994]); it is interesting to note that the Nash solution
is one of the few solutions satisfying consistency among larger groups.
12
Strict concavity is a convenience used to ensure that the bargaining solution and
demands are single-valued. Note that this assumption is technically incompatible
with “assignable” or person-specific goods, which are discussed only for illustrative
purposes only. One can resolve this conflict by assuming single-valuedness directly,
or by viewing assignable goods as having microscopic impact on the partner’s utility.
10
Household bargaining & time
Each individual is endowed with a stream of unspecified “outside options” or “threat
points” yielding utility of (eit )t∈T , called the year t outside utility.
Given a particular Walrasian budget set denoted by B( pt , mt ) and a particular
disagreement utility dit for each of the agents,13 a one-period household allocation problem
is described by the set of feasible utilities:
(3.3)
{
H( pt , mt , dt ) : = (v1t , v2t ) ∈ R 2+ : ( ∃xt ∈ B( pt , mt )) vit ≤ ui ( xt ) − dit
}
This assumes free disposability of utility above the disagreement point. The domain
of all allocation problems, with potential “gains from marriage” for both partners,14 is
defined as:
(3.4)
H : = { H( pt , mt , dt ) : pt ∈ R l+ + , mt > 0, ( ∃v ∈ H( pt , mt , dt ) v >> 0}
Let S02 denote the collection of all compact, convex, and comprehensive sets in the
nonnegative quadrant. A two person bargaining solution can be described as some
function F: S02 → R +2 , such that F (S) ∈ S for all S ∈ S02 . This can be seen as some
abstract set of rules determining the choice of the household over the feasible
outcomes. The household allocation problems are a subset of S02 . This means that
any bargaining solution is defined over H , associating a unique utility vector with
every household budget set and disagreement point. It is assumed that the
bargaining solution satisfy these three axioms:
Disagreement monotonicity: if di′ ≥ di and d′j = d j , then Fi ( S − d′ ) ≥ Fi ( S − d ) .
Continuity: if Sn → S , then F (Sn ) → F (S) .
Efficiency: ∀S ∈ S02 : F (S) ∈ PO(S) : = {vt ∈ S: ( ∃/ ut ∈ S) ut >> vt } .
The first two assumptions are, for the most part, explicit and formal
characterizations of preconceptions about “bargaining” and decision-making.
Disagreement monotonicity plays no technical role, but serves as a formal statement of
the conventional wisdom on household decision-marking: everything else equal, an
agent receives a weakly preferred allocation when his threat point increases.
Continuity assumes that the household’s response to small changes in the feasible set
is reasonably small. In part this is a mathematical nicety—but it is also requiring that
13
A distinction between the outside utility and the disagreement utility is relevant
for some of the bargaining procedures.
14
The following notation is used for vector inequalities: for x, y ∈ R n , x >> y means
xm > ym in all dimensions m = 1, 2,K, n ; x ≥ y indicates xm ≥ ym for m = 1, 2,K, n .
11
Household bargaining & time
the household’s behavior exhibit somewhat systematic behavior (similar to
requesting individuals’ preferences to be continuous). The strongest of these
assumptions is efficiency, that negotiations do not stop until all available gains have
been exhausted. Since this is a common assumption in economics and in cooperative
game theory, I feel it needs no further justification.
Given some bargaining solution F, a bargained household decision can be
described as the allocation xt* ( pt , mt , dt ; F ) ∈ B( pt , mt ) such that F ( Ht ) ≡ ui ( xt* ) − dit .
The awkwardness of general “bargaining solutions” is that they map from utility sets
into utility vectors; in examining household decisions, we are more interested in a
mapping from budget sets to allocations. Outside of a class of bargaining solutions
satisfying “independence of irrelevant alternatives,” few bargained household
decisions can be represented as the solution to any maximization problem.15
To see why this is a property associated with maximizing social welfare
functions, consider the problem of maximizing some W over a feasible budget set B .
If some x* is the solution to this problem, then for any B′ ⊆ B such that x* ∈ B′ , x*
is once more the argument which maximizes W. In single-person choice theory, this
property is called the weak axiom of revealed preference.16
Axiomatic bargaining theory usually expresses how utility payoffs respond to
changes in feasible utility sets; convex analysis tells us how support functions respond
to changes in values. Combining these two ideas, I find it convenient to express and
analyze the bargained household decision as the solution of an expenditure
minimization problem. This means that the following arguments, and modified Euler
equations, are appropriate for all bargaining solutions.
The household bargained decision is a bundle which minimizes household
expenditures, subject to constraints that each agent’s utility not fall below the utility
levels dictated by the bargaining solution:
(3.5)
{
Naturally, there exists a vector in the budget set satisfying both of these constraints.
Strict concavity of the utility functions, along with efficiency, implies uniqueness.
From continuity of the bargaining solution, the demands are continuous for all
pt ∈ R l+ + and mt > 0 . Assuming that the composite function F o H is continuously
15
Thomson
16
}
xt* : = arg min xt pt ⋅ xt + µwt ( Fw ( Ht ) − uh ( xt ) + dwt ) + µ ht ( Fh ( Ht ) − uh ( xt ) + dht )
(1994) calls this “contraction [of the feasible set] irrelevance.”
Samuelson (1947) shows how preference maximization, along with the weak axiom,
is equivalent to maximization of some function.
12
Household bargaining & time
differentiable with respect to the arguments of H, then demand functions are
differentiable as well.
Provided that the bundle is an interior optimum, then household allocation
satisfies the condition:
(3.6)
(
)
(
λt pt′ = µƒwt Duw ( xt* ) + µƒht Duh ( xt* )
)
Both sides of (3.6) have been multiplied by a positive variable λt , which may be
interpreted as the “household marginal family welfare of wealth,” and the µƒit have
been redefined accordingly17.
At this point, equation (3.6) illustrates that this framework generalizes a
number of common approaches to modeling household behavior. It closely
resembles the collective approach of Chiappori, making the additional connection to
a bargaining solution and incorporating disagreement points. Duality immediately
identifies the relationship between (3.6) and common bargaining solutions:
µƒit ≡ Fj ( pt , mt , dt ) for the Nash solution, while µƒit ≡ µƒjt for the (symmetric) utilitarian
rule. Traditional unitary models can be interpreted as the case where µƒit ≡ 0 for one
of the agents.
A lifetime household allocation problem, denoted by ( H( pt , mt , dt ) , δ t )t ∈T , is a
collection of one-period allocation problems as well as agents’ discounting of utility
in each period. Just as H( pt , mt , dt ) was the set of feasible utility gains in year t, there
is a lifetime feasible set:18
(3.7)
H T (( pt , mt , dt ,δ t )t ∈T ) : =
∑ δ H( p , m , d )
t
t ∈T
{
t
t
t
}
: = (Vwt , Vht ) ∈ R 2+ : ( ∃(vt ∈ H( pt , mt , dt ))t ∈T )Vit = ∑t ∈T (δ it vit )
In other words, this is the sum of all the yearly bargaining problems, rescaled
according to the discount function. Since H T ∈ S02 , bargaining solution are clearly
defined for this class of problems as well as for the yearly problems. While H T
represents all the utility vectors which could be achieved through bargaining, not all
will be possible under all multi-period procedures; in other words, what is feasible
through multi-period bargaining procedures is a subset of H T . At this point, I
discuss three approaches to the lifetime bargaining problem.
17
Nonnegativity of multipliers comes from efficiency. With a normalization, the
right-hand side of (3.6) is a convex combination of the utility functions’ gradients.
18
Given a set S ⊆ R 2 and a vector α ∈ R 2+ + , then αS represents the rescaling of S by
α ; that is, αS : = {(t1 , t2 ) ∈ R 2 :(∃s ∈ S) tn = α n sn } .
Household bargaining & time
13
3.1 The myopic procedure: a repeated static game
The myopic model is essentially the “repeated model” of household decisionmaking. In this approach, the household bargaining problem is solved on a year-byyear basis. Through some unspecified procedure, agents initially decided how to
budget their lifetime wealth between periods. Considering only the yearly feasible
set and the yearly outside utilities (in other words, agents take dt = et ), agents
negotiate over the household allocation for the year.
The household has budgeted some amount mt to spend in the year.19 The
bargaining solution awards the utility payoffs F ( H( pt , mt , et )) to agents in each year.
Between two years, the change in the agents’ utility—yearly “gains from marriage,” in
the words of McElroy/Horney—can be described:
(3.8)
∆Ft ≅ Dp ( F o Ht )∆pt + Dm ( F o Ht )∆mt + Dd ( F o Ht )∆et
This will be one of the factors influencing the dynamics of the household
allocation. Another is the contribution of each agent to the “family marginal utility”
expressed in equation (3.6). It is convenient to define this relative preference
contribution, π itN , in the following manner:20
(3.9)
π itN : =
µit Dui ( xt* )
[ µit Dui ( xt* ) + µ jt Du j ( xt* )]
This index π lit is negative for any commodity which is a “bad” for agent i, and
positive when seen as a good. In this paper, a commodity l is said to be assignable to
agent i if π ljt = 0 and π lit = 1 ; that is, if ∂u j ( xt ) ∂xlt ≡ 0 .21
Totally differentiating equation (3.6) generates an intertemporal path for the
bargained household allocation, the so-called “Euler equation”:
19
The budgeting process need not be specified. A possibility is that the household
decided on a two-step process: at t = 0 , they would decide how much to spend in
each year, and they would pick the actual allocation when the year arrives. Another
is that (in extreme myopia) they live from paycheck to paycheck. Of course, many
other possibilities remain.
20
The following notation is used throughout the paper: “ ≅ ” means “differing by a
term of o(∆t) ”; for a vector z ∈ R l , [z] denotes the l × l diagonal matrix with z on
its
diagonal; “division” of some y by a matrix or vector z means z −1 y or [z]−1 y .
21
Note that this definition of assignability, a strong one, also requires that agents are
“egoistic”—at least regarding consumption of this particular good.
14
Household bargaining & time
(3.10)
∆xt*  N
≅  [π wt ] σ w ( xt* )
*
xt
(
)
−1
(
)
−1
+ [π htN ] σ h ( xt* ) 
−1
 ∆pt  N DF µwt

D µ 
+ π htN F ht  ∆Ft 
 p −  π wt µ
µ ht 
 t

wt
An agent’s l × l substitution matrix (when evaluated at the household allocation) is
denoted by σ i ( xt ) = [ Dui ( xt )] ([ xt ] D 2ui ( xt )) , which is assumed to be nonsingular.
As the problem has been formulated, the effects of changes in threat points
and prices on the bargaining solution enter into the allocation dynamics by changing
the weights µit . Convexity of the household cost function ensures that ∂µit ∂Fit ≥ 0 ,
but says nothing about the sign of ∂µit ∂Fjt . Changes in the multipliers ∆µit are
dictated by the change in the bargaining solution; this is decomposed in equation
(3.8) into wealth, price, and outside options effects.
As a special case of (3.9), suppose that a particular good l is assignable to
individual i and additively separable in his utility function. In this case, the
bargained household decision dictates the following path:
(3.11)
∆xlt
≅ σ li ( xt* )
xlt

 ∆plt DF µit
−
∆Ft 

µit

 plt
Though only this one agent cares about consumption of the good, its consumption
does not depend only on his preferences—namely, yearly changes in the bargaining
solution are also an important factor. Generally speaking, this means that the
intertemporal consumption decision does not reflect the agent’s rate of time
preference, so the allocation is far from optimal: (∀t ∈ T) vt ∈PO( H( pt , mt , dt )) ⇒
/
T
∑t∈T (δ t vt ) ∈ PO( H (( pt , mt , dt ,δ t )t∈T ) . This inherent inefficiency in the year-by-year,
myopic approach might prompt agents to attempt to solve the lifetime bargaining
problem directly.
3.2 The contractual procedure: a multi-stage game with full commitment
In the contractual model, the bargaining procedure changes so that agents
tackle the multi-period problem by the lifetime bargaining problem directly. At
t = 0 , feasible (lifetime) gains from marriage are the set H T (( pt , mt , dt ,δ t )t ∈T ) , which is
the rescaled sum of all years’ feasible sets. The relevant “disagreement point” is that
agents lose all years’ gains from marriage if the negotiation fails—that is, that they
receive their outside option in each of the following periods, or ∑t ∈T δ it eit . The
bargaining solution grants F ( H T (( pt , mt , dt ,δ t )t ∈T ) as a lifetime utility to each of the
agents. As a modification of (3.6), the lifetime expenditure minimization problem is:
15
Household bargaining & time
( xt )t ∈T : = arg min
(3.12)
∑ (1 + r )
−1
(
+ µ ( F (H ) − ∑ δ
ptT xt + µw Fw ( H T ) − ∑ δ wt uw ( xt ) + ∑ δ wt ewt
t ∈T
t ∈T
t ∈T
T
h
h
t ∈T
ht
uh ( xt ) + ∑ δ ht eht
t ∈T
)
)
This specifies an allocation in each year. After agreeing to this at t = 0 , agents
cannot make later attempts to change the solution—perhaps legal or social
institutional marriage contracts binding, or perhaps players can threaten to punish
deviations with divorce or other punishment.
Similar to (3.9), a relative preference contribution is created to show how
much of the household’s preferences come from each agent at a particular time:
(3.13)
π itC : =
µiδ it Dui ( xt* )
[ µiδ it Dui ( xt* ) + µ jδ jt Du j ( xt* )]
The definition of the substitution matrix is unchanged. From the optimality
condition of (3.12), one can infer that the household allocation follows the path:
(3.14)
∆xt  C
≅  [π wt ] σ w ( xt* )
xt
(
)
−1
(
)
−1
+ [π htC ] σ h ( xt* ) 
−1
 ∆pt
 ∆δ wt C ∆δ ht C  
 p − rt −  δ π wt + δ π ht  
 wt
 t

ht
In the contractual model, the constraints µi are constant over the lifetime. This
means that—as in the example of the previous section—the dynamics of the
household allocation do not reflect yearly changes in outside options, or any other
“strategic effects” of price or wealth changes.
The household’s intertemporal substitution matrix ends up being essentially a
weighted average of the two agents’ matrices, and the household’s rate of time
preference is likewise a weighted average of the agents’ discount rates. In this sense,
the contractual model of decision-making looks the most like “sharing” or
“cooperating” over the lifecycle of the household.
The dynamics of private goods are particularly interesting in this model. For
a good l , assignable to i and separable in his utility function, then:
(3.15)
 ∆p
∆xlt
∆δ it 
≅ σ l*i ( xt )  lt − rt −
δ it 
xlt
 plt
The bargained household decision dictates that the Euler equation for this good be
exactly the same as this agent would choose when acting singly—a result which is
invariant to any particular bargaining solution, provided that agents are operating on
Household bargaining & time
16
the lifetime efficiency frontier. Intuitively, this is sensible: if this good enters the
utility function of only this agent and affects nothing else, then the most efficient
action for the household is to let the agent choose the consumption path which
maximizes his preferences.
The contractual model of multi-period household bargaining results in the
most efficient outcomes. However, one might object that it is generally not timeconsistent. Agents may well have the incentive to renegotiate the household
decision in some year after the beginning—especially if one agent’s remaining utility
from the household allocation falls below what he could get from his outside option
in all remaining periods: nothing rules out the possibility that ∑ s≥t δ is eis > ∑ s≥t δ isui ( x*s )
at some time t. While agents do not exploit intertemporal efficiency gains in the
myopic model, they forsake time consistency by taking full advantage of the gains in
the contractual model.
3.3 The prescient procedure: a multi-stage game without commitment
This third model of household decision making lies between the earlier two.
In the myopic model, agents are extremely myopic—the yearly bargaining solution is
unaffected by the past or the future. In particular, by solving each problem
independently, agents must believe that a failure to successfully negotiate a
household decision in any one period has no effect on the following years’
negotiations. It is doubtful that family members realistically expect that they can
resort to their “outside option” in any year—whether divorce, separation, or noncooperative cohabitation—without impacting future negotiations. One can
interpret the “prescient model” as a version of the myopic model in which agents
realize that if ever they take their outside option in one year, then they will remain
there forever after—forfeiting all future gains from bargaining.
Another way to understand the prescient procedure is that it attempts to find
a solution to the lifetime bargaining problem that satisfies the following timeconsistency condition: if ( xt* )t∈T is the allocation chosen when the lifetime problem is
H( pt , mt , dt ))t ∈T , then for any τ ∈ T the allocation ( xt* )t ∈T\{ s <τ } would also be chosen for
the “truncated” lifetime problem H( pt , mt , dt ))t ∈T\{ s<τ } . This constraint means that the
household effectively considers only the future when picking a year’s allocation—in
contrast to the contractual model, in which the yearly allocation depends on factors
throughout the lifetime. In early years, these two procedures should produce similar
allocations, since most of the lifetime is in the future.
17
Household bargaining & time
However, the allocations of the prescient procedure will closely resemble
those of the myopic model as t approaches one. In the final year, agents will
certainly receive F ( H( p1 , m1 , e1 )) as the bargaining solution. Working backwards,
agents know that if their surplus (or “gains from marriage”) remaining at t + ∆t is
indisputably F ( Ht + ∆t ) , then in the previous year the relevant feasible utilities set for
the truncated lifetime problem is ( H( pt , mt , et ) + (δ t + ∆t δ t ) F ( Ht + ∆t ) + R 2− ) ∩ R 2+ ; that is,
the comprehensive hull of the year t bargaining problem, shifted by the discounted
value of the surplus.
The solution to this truncated problem is denoted by F ( Ht ) . The household
allocation chosen, for this year and beyond, must satisfy:
(3.16)
Fi ( Ht ) =
∑
∑
δ is
δ is
ui ( x*s ) −
e
δ
δ it
s∈T\{ s <t } it
s∈T\{ s <t } it
Since the same will be true in the following year (advancing dates forward by ∆t ), the
following must also be true:
(3.17)
Fi ( Ht ) −
δit +∆t
δit
(
)
Fi ( Ht +∆t ) = ∑ δδisit ui ( x*s ) − ∑ δδisit eis −
s ≥t
s ≥t
δit +∆t
δit
(∑
δ is
δ
s >t it + ∆t
ui ( x*s ) − ∑ δitδ+is∆t eis
s >t
)
Simplifying this expression allows me to characterize the household allocation for
year t as the solution to this expenditure minimization problem:
(3.18)
{
This is identical to problem (3.5), with agents taking the disagreement utility to be
dit = (δ it + ∆t δ t ) Fi ( Ht +∆t ) + eit . The constraints in (3.19) are the same as those of the
contractual model when t = 0 , and are those of the myopic model when t = 1 . If the
values of the µit multipliers also agree with the respective models at these times,
then the relative preference contribution of the agents:
(3.19)
}
xt* : = arg min xt pt ⋅ xt + µwt ( Fw ( Ht ) − uh ( xt ) + dwt ) + µ ht ( Fh ( Ht ) − uh ( xt ) + dht )
π itP : =
µit Dui ( xt* )
[ µit Dui ( xt* ) + µ jt Du j ( xt* )]
will coincide with Π0i from the contractual model in the first year, and π it from the
myopic model in the last. Using this, one can describe the dynamics of the bargained
household decision as:
18
Household bargaining & time
(3.20)
∆xt*  P
≅  [π wt ] σ w ( xt* )
*
xt
(
)
−1
(
)
−1
+ [π htP ] σ h ( xt* ) 
−1
 ∆pt  P DF µwt

D µ 
+ π htP F ht  ∆Ft 
 p −  π wt µ
µ ht 
 t

wt
This resembles the dynamics of the myopic model, though changes in the yearly
gains from marriage will be (in contrast to [3.8]):
∆Ft ≅ Dp ( F o Ht )∆pt + Dm ( F o Ht )∆mt
(3.21)
(
)
(
)


− Dd ( F o Ht ) u( xt* ) − et + ( ∆δ t δ t ) ∑ δtδ+s∆t u( x*s ) − es  
 { s >t


In other words, changes in the disagreement point are not the yearly changes in the
outside option, but rather the changing value of future gains from marriage. Part of
the change comes from discounting, and another part comes from the diminishing
future. Discounting is the larger factor earlier. This effect suggests that agents’
behavior when young is much more cooperative, more sharing—because of the large
marital surplus lost if negotiations fail. Towards the end of the lifetime, discounting
becomes relatively small—the diminishing future becomes much more important. In
these years, the household decision is affected more by yearly changes.
Inasmuch as the prescient Euler equation looks like the myopic one, the
dynamics of assignable goods are also similar (except that ∆Ft is different, as
distinguished in [3.21]). As a consequence, it should be apparent that the household
decision is not necessarily efficient under the prescient procedure.
4.
THE LIFE-CYCLE OF THE HOUSEHOLD
Under the various procedures, bargained household decisions respond
differently to factors like discount rates, threat points, and preferences. As shown
above, they have different welfare implications. At this point, it is appropriate to
make some brief remarks distinctions between the models.
Welfare of the household members. Because the contractual procedure
results in a lifetime efficient household allocation—in contrast to the other two
procedures—it may be surprising that this procedure generally does not imply that
both agents would benefit from signing a contract. In fact, if F is a bargaining
solution for a two-person allocation problem such that the super-additivity condition
(4.1)
(
)
Fi H T ( pt , mt , dt ,δ t )t ∈T ) ≥ ∑t ∈T δ it Fi ( H( pt , mt , dt ))
Household bargaining & time
19
holds for all discount rates δ i : T → [0, 1] , then F is necessarily the Perles-Maschler
(1981) bargaining solution. Restricting the discount rate of the agents to be identical,
the egalitarian and utilitarian solutions also satisfy this condition (Thomson 1994).
However, a similar statement cannot be made for a household consisting of
more than two members. In short, it seems unlikely that all agents would
unanimously prefer one bargaining procedure to another.
Threat points. Under the contractual and prescient procedures, the
bargained household decision does not respond to yearly changes in outside options.
This is not surprising for the contractual model, since the allocation problem is
solved from a lifetime perspective. As such, one might describe behavior as very
“considerate”: agents are not exploiting each other’s rough years, choosing instead to
help each other smooth.
When agents are prescient, they anticipate future years’ outside options.
Because the current decision is a function of all future threat points, the yearly
change in the household allocation can be described more appropriately as “minus
eit ,” rather than “change in eit .” Under the myopic procedure, the change in xt* is
directly a function of the change in outside options, as illustrated in equation (3.9).
This “inconsiderate” behavior is essentially the source of the model’s inefficiencies.
Future marital surplus. Under the prescient procedure, the real “threat” is
the loss of the gains from marriage in future years. Because the size of these gains
varies with the size of the future, we might expect to see some “lifecycle of the
married household.” In the early years, household agents are more autonomous in
their private decisions (as represented by “assignable” goods). Household behavior is
considerate and cooperative, resembling the more efficient contractual model: there
is a great deal to be gained from successfully working out the problems facing the
household in these years. This is less true toward the end of the lifetime, since there
is less to lose from disagreement.22 Decisions respond more to transitory changes in
the household bargaining problem.
Discount rates. Individual discount rates play no role in determining
household consumption under the myopic procedure, except perhaps through the
budgeting decision. This implies that the household will appear to discount all goods
22
I am reminded of my grandparents, who lived out the cliché of “bickering like an
old married couple.”
Household bargaining & time
20
at equal rates. In contract, the contractual household has no single rate of time
preference across all goods—rather, the discounting of each good will be determined
by the relative preferences of the agents. This means that goods favored by the
impatient partner are consumed relatively earlier in the “life-cycle” of the household.
This is also true, but to a lesser extent, under the prescient procedure. In this
model, all goods are implicitly discounted at the same rate through the unspecified
budgeting process, but another factor is added to this. Because the less patient
partner value his future marital surplus less, he receives disproportionately higher
utility in earlier years to assuage his impatience. Naturally, this means that the goods
favored by this agent are consumed more in these earlier years. This “strategic
effect” of discount rates becomes smaller as the household ages.
Social norms, institutions, and marriage.
The procedure for
intertemporal bargained household decisions may not be universal. Just as whether
divorce or non-cooperative cohabitation is the relevant outside option, this
procedure may vary across time and cultures—and even vary across households
within the same culture.
In a more traditional society, religion or community could impose effective
marital contracts—rules about how to allocate resources within the household over
the lifetime. Efficiency gains could be one motivation for social enforcement,
though once more we note that efficiency can come at one agent’s expense.
But when household are dissolvable and contracts not binding, then another
model might be more appropriate. If divorce is permissible but it brings a
permanent end to the “gains from marriage” of a cooperative household, then we
might expect that the prescient model accurately reflects behavior. On the other
hand, if divorce and immediate remarriage are quick and costless, then the myopic
model may accurately describe this repeated game.
Another possibility is that the threat point of marriage is non-cooperative
cohabitation, as in the “separate spheres model.” If this is the case, then agents
might be able to reconcile their disagreements after some time. Since moving to the
threat point is not a permanent change, the myopic model might once again be the
appropriate way to model behavior.
In actuality, there may be a great deal of variation in how families view
problems and threat points. There may be a number of factors which influence how
couples make joint decisions and what happens when they fail to agree.
Household bargaining & time
21
Equivalence of the procedures. Though they appear to give different
conditions, it is not impossible that the three procedures discussed in this paper
actually generate the same household demands. If this is the case, one can
choose—without loss of generality—any procedure to model multi-period household
bargaining problems. For instance, if Dd F ≡ 0 , then (3.8) is exactly the same
expression as (3.19). This is, in fact, a condition necessary for equivalence: that the
bargaining solution not depend on disagreement utilities at all. While this already
fails to conform with our notions of “bargained” household decisions, the exact
condition for general equivalence is even stronger. These procedures yield the same
allocations if and only if the household is dictatorial, or agents discount identically
and the household maximizes a generalized utilitarian social welfare function.23
This result should serve as a warning. Except in one special case, our
interpretation of household decisions is subject to how we think the household
solves the multi-period problem. However, it also produces a strong positive result:
regardless of the bargaining solution, we can identify the intertemporal procedure
econometrically.
5.
TWO ECONOMETRIC TESTS
In this section of the paper, I develop the econometric model which will be
used to distinguish among these three intertemporal decision-making procedures.
To derive the empirical specifications, I need to make several simplifying
assumptions about the preferences and the bargaining solution.24 Next, I take two
approaches to identify the intertemporal procedure used by households.
The first is a simple specification test. Each of the procedures suggests a
different specification, and I can test formally whether this specification can be
rejected. Afterwards, I take a “latent class” approach: I assume that each of the
procedures may describe the behavior of some households, though I cannot
distinguish which households fall into which “class” (or “regime”). The three models
23
Myerson (1981) derives this result for social choice under uncertainty. In the
bargaining literature, Ponsati and Watson (1997) use a comparable equivalence (for
multi-issue problems), along with efficiency and symmetry, as an axiomatic
characterization of the symmetric utilitarian rule.
24
Eventually these identify the Nash bargaining solution. However, I present each
assumption as it is needed and in the context in which it is used. I feel that this
bargaining solution should be understood as the implication of empirically palatable
assumptions, and not as an a priori assumption.
22
Household bargaining & time
are estimated at the same time, along with a “switching equation” which assigns to
each observation a probability of being in each class.
5.1 Specifying the model
Changing from the notation of the previous section, I now set the “lifetime”
of the marriage to 2 + 2 y years, with the length of each year normalized to one. I will
be examining the changes in variables from year t to t + 1 , while considering the
previous y years of history and the subsequent y years.
There are two assumptions about preferences. First, I assume that agents
discount their lifetime utility discount geometrically:
s =2 y + 2
(5.1)
Ui ( x) =
∑β
s
i
ui ( xs ) ,
s =1
where βi is agent i’s discount rate.25 Additionally, I assume that there are two goods
(denoted by l and k) that are separable in both agents’ utility functions.
In this paper, I have not addressed how the household makes decisions about
budgeting money between periods.26 These income effects could cloud empirical
results by introducing unintended correlation.27 However, if I am willing to assume
that the bargaining solution can be represented as the maximum to some (sufficiently
smooth) household social welfare function, I can essentially sweep these income
effects under the rug. This means that I can devise a test which does not rely on any
assumptions about household budgeting. This household social welfare function is
W, defined as:
(5.2)
25
( F ( H ), F ( H )) ≡ arg max
w
t
h
t
( Vwt ,Vht )∈Ht
W (Vwt , Vht )
The subscripts i and j refer to the two members of an arbitrary household; to
avoid confusion, I do not index the households.
26
As mentioned in footnote 18.
27
In the standard model of individual utility maximizing behavior, marginal utility of
wealth is constant over the life-cycle: ∆ ln λt equals the interest rate. This is
generally not a consequence of pluralistic household behavior, and the analog to
marginal utility of wealth is likely to be determined by strategic factors. The
problem is complicated by uncertainty and liquidity constraints, which can make
∆ ln λt differ from the interest rate even for individuals . These constraints and
volatility are likely correlated with work history and future wages.
23
Household bargaining & time
where Vit* : = ui ( xt* ) − eit represents the “gains from marriage” for person i at time t.
Assuming the existence of this function is equivalent to the bargaining axiom usually
called “independence of irrelevant alternatives” or “contraction independence.” The
econometric appeal of this assumption is clear, since we are limited by data which
reports only chosen allocations and not potentially relevant (but unchosen) ones.28
Working with the household welfare maximization problem rather than its
dual, I express the optimality condition as:
(5.3)
(
∂W Vwt* , Vht*
∂Vw
) ( Du
w
)
(x ) +
*
t
(
∂W Vwt* , Vht*
∂Vh
) ( Du ( x )) = λ p
h
*
t
t
t
This expression conveniently gives a functional form for the µƒit multipliers in
(3.6)—they are the derivatives of the social welfare functions—and it captures wealth
effects in λt . Because the budgeting process remains unspecified, λt is regarded as an
unknown and bothersome parameter.
After differentiating (5.3) with respect to time, the household Euler equation
can be represented in the following form:
(5.4)

( ∆ ln x ) σ π ( x
*
lt
Q
lwt
lw
*
lt
)
+
π lQht 
= ∆ ln plt + ∆ ln λt − Φ Qlt + η lt ,
* 
σ lh ( xlt ) 
where Q is the intertemporal procedure used: N for the myopic approach, C for the
contractual approach, and P for the prescient approach. The relative preference
contributions of each agent to the purchase of good l are captured by π litQ , as defined
in section three for each of the procedures. The term Φ Qlt captures anything that is
unique to procedure Q (I will refer to these as the “strategic effects”). The error
term η lt represents the usual unobservable idiosyncrasies and approximation error.
For convenience, I will the term
(5.5)
 π lQwt
π lQht 
σ ( x ) := 
+
*
* 
 σ lw ( xlt ) σ lh ( xlt ) 
Q
l
−1
*
lt
to stand for the household’s intertemporal elasticity of substitution when using
procedure Q .
28
This assumption is equivalent to several other types of “invariance,” two of which
are quite applicable to household behavior. See Ponsati and Watson (1997) for
“simultaneous implementation agenda invariance,” or Lensberg (1987) for consistency
among an arbitrary number of agents.
24
Household bargaining & time
By taking the difference between the consumption paths of two goods, the
unknown term ∆ ln λt is eliminated. This equation looks like:
(5.6)
∆ ln x*lt ∆ ln x*kt
−
= ( ∆ ln plt − ∆ ln pkt ) − Φ Qlt − Φ Qkt + (η lt − η kt )
σ lQ ( x*lt ) σ kQ ( x*kt )
(
)
If the household’s intertemporal elasticities of substitution for the two goods are
approximately equal, then the left-hand side reduces to a simple double difference.29
Regardless, one can quite simply estimate the equation:
(5.7)
 x* 
p 
∆ ln * lt σ 2  = σ 1 ∆ ln lt  − σ 1 Φ Qlt − Φ Qlt + ε lkt
 pkt 
 ( xkt ) 
(
)
and identify how “strategic effects” affect the bargained household decision. Any
wealth effects have been completely differenced from this specification, which is
critically important.
5.2 A specification test
While the three procedures all take the same general form, none is nested
within the others. However, equation (5.7) can be nested within another model—and
that means that the three procedures can all be nested within the same model.
Let me return to the specific Euler equations generated by the three
intertemporal procedures. For the myopic, contractual, or prescient household,
equation (5.4) takes the form:30
(5.8)
(5.9)
(5.10)
29
 π lNwt
∆ ln x*lt
π lNht  ′
2
=
+
−
∆
ln
p
∆
ln
,
λ
lt
t
 ∂W ∂V ∂W ∂V  DV W
σ lN ( x*lt )


h
w
(
) ( ∆V ) + η
*
t
lt


∆ ln x*lt
πC
πC
= ∆ ln plt + ∆ ln λt −  t lwt
, t lht
 ′ ( DV W )(1 − β ) + η lt
*
C
σ l ( xlt )
 βw ∂W ∂Vw β h ∂W ∂Vh 
s =t + y + 1


 π lPwt
∆ ln x*lt
π lPht  ′ 2
s −t
*
*
=
+
−
1
−
β
β
V
−
V
∆
ln
p
∆
ln
,
D
W
λ
(
)
 + η lt

V
l
s
t
t
t



σ lP ( x*lt )
 ∂W ∂Vw ∂W ∂Vh 


s =t + 1
(
)
∑
A double difference between goods and across time is the approach also used by
Lundberg/Pollak/Wales to identify whether a policy change impacts bargained
household decisions.
30
As in the previous sections, the “multiplication” of vectors is element-by-element,
representing a rescaling.
25
Household bargaining & time
Differencing the consumption path for good k from that for good l gives the
following specific forms for equation (5.6):
(5.11)
(5.12)
(5.13)
p 
∆ ln x*lt ∆ ln x*kt
− N * = ∆ ln lt  − ϖ wtlk ,ϖ htlk ′ DV2 W ∆Vt* + (η lt − η kt )
*
N
σ l ( xlt ) σ k ( xkt )
 pkt 
*
*
p 
∆ ln xlt ∆ ln xkt
− C * = ∆ ln lt  − ϖ wtlk ,ϖ htlk ′ ( DV W )(1 − β ) + (η lt − η kt )
*
C
σ l ( xlt ) σ k ( xkt )
 pkt 
(
) (
(
)
p 
∆ ln x*lt ∆ ln x*kt
− P * = ∆ ln lt  − ϖ wtlk ,ϖ htlk ′ DV2 W
*
P
σ l ( xlt ) σ k ( xkt )
 pkt 
(
)(
)(
)
)
s =t + y + 1


*
*
s −t
1
−
β
β
V
−
V
)
 (
s
t  + (η lt − η kt )



s =t + 1
∑
The term capturing “strategic effects” (that is, Φzlt − Φzkt ) consists of two parts. The
first of these is the parameter ϖ itlk , which reflects agent i‘s preference to substitute
consuming more of good l instead of k. In all three equations, this is a term which
can be defined as:
(5.14)
ϖ itlk ∝
Dl ui ( x*lt ) Dk ui ( x*kt )
−
pl
pk
For a utility maximizing individual, ϖ itlk is identically zero—the marginal utility per
dollar spent is the same for all goods. For a pluralistic household, this will generally
not be zero unless agents have the same preferences over the two goods. However,
the greatest ability to see the influence of “strategic effects” on household decisions
comes when the difference between agents’ marginal rates of substitution is largest.
The remaining part of Φ Qlt − Φ Qkt depends on the gains from marriage and
derivatives of the social welfare function. As a proxy for each person’s gains from
marriage in each year, I use a function of the wage rate.31 Regardless of what the
threat point is, it seems quite reasonable to assume that it is concave and increasing
in the wage rate, wit . Since the bargaining solution is assumed to be monotonic in
the threat point, it is clear that Vit* is increasing in wit . If the rate of increase
becomes smaller as wit gets larger (which must eventually be true), then Vit* is also
concave in wit . Both of these properties are captured by assuming that Vit* = α i ln wit .
In order to express the derivatives of the social welfare function, I assume
that household decisions are symmetric, and that they are invariant to scaling of the
31
Furthermore, conventional wisdom says that the balance of power within the
household is linked to relative earnings. To quote a recent newspaper article on
division of chores within the household: “The biggest bargaining chip is the size of
the pay packet, and the partner with the biggest wedge calls the shots” (T h e
Guardian, 9 July 2001, p.19).
26
Household bargaining & time
utility functions. This implies that the first derivative of W (Vw , Vh ) must be in direct
proportion to the vector (Vh , Vw ) , and that the second derivative must be directly
proportional to 12×2 − I2×2 . For a household using this Nash bargaining solution,
equations (5.11) – (5.13) are:
∑(

 − ∑ϖ

(5.14)
 x* 
p 
∆ ln * lt σ 2  = σ 1 ∆ ln lt  −
ϖ jtlkα iσ 1 ln wi ,t +1 − ln wit
 pkt  i∈{ w,h }
 ( xkt ) 
(5.15)
 x* 
p
∆ ln * lt σ 2  = σ 1 ∆ ln lt
 pkt
 ( xkt ) 
(5.16)
s =t + y + 1


 x 
p 
∆ ln * σ 2  = σ 1 ∆ ln lt  −
ϖ jtlkα iσ 1  (1 − βi ) ∑ βis−t ln wis − ln wit  + ε lkt
s =t + 1
 pkt  i∈{ w,h }
 ( xkt ) 


*
lt
(
lk
jt
α iσ 1 (1 − β j )
s =t + y + 1
i∈{ w ,h }
∑
s =t − y
)) + ε
lkt
βis ln wis + ε lkt
∑
In reduced form, the parameters of these equations are described by equations (5.17)
– (5.19), below. Additionally, I present a fourth possible specification. If the unitary
model accurately describes household behavior, then the Euler equation should
contain no “strategic terms.” As noted before, strategic terms also disappear when
household members have the same marginal rates of substitution (when ϖ itlk = 0 ), so
this relationship does not necessarily imply the unitary model: equation (5.20) could
mean that there is no need to bargain because preferences agree.
)
∑(

 − ∑ ∑ (γ β ln w ) + ε

(5.17)
 x* 
p 
∆ ln * lt σ 2  = σ 1 ∆ ln lt  −
γ i ln wi ,t +1 − γ i ln wit + ε lkt
 pkt  i∈{ w,h }
 ( xkt ) 
(5.18)
 x* 
p
∆ ln * lt σ 2  = σ 1 ∆ ln lt
 pkt
 ( xkt ) 
s =t + y + 1
i∈{ w ,h }
s =t − y
i
s
i
is
lkt
(5.19)
s =t + y + 1


 x*lt 
 plt 
∆ ln * σ 2  = σ 1 ∆ ln
γ i ,t +1 ∑ βis−t −1 ln wis − γ it ln wit  + ε lkt
−

s =t + 1
 pkt  i∈{ w,h } 
 ( xkt ) 

(5.20)
 x* 
p 
∆ ln * lt σ 2  = σ 1 ∆ ln lt  + ε lkt
 pkt 
 ( xkt ) 
∑
Since these specifications are not nested, I cannot compare them directly.
However, each of these equations is a special case of the following equation:
(5.21)
s =t + y + 1
 x* 
 p  s =t + y +1
∆ ln * lt σ 2  = σ 1 ∆ ln lt  + ∑ γ ws ln wws + ∑ γ hs ln whs + ε lkt
s =t − y
 pkt  s =t −y
 ( xkt ) 
By comparing this equation to the specification given by each bargaining procedures,
I am asking the question, “Out of all possible (linear) relationships between the
current variables and the two wage profiles, can we reject that the coefficients fit the
27
Household bargaining & time
pattern generated by procedure Q ?” This is more challenging, and more informative,
test than using non-nested hypothesis testing methods to compare (5.17) – (5.20).
These methods may select a winner, when in fact none of the three specifications
works well. Instead, I will test each of the specifications. Whenever I find that
three models can easily be rejected but that the fourth cannot be, I will take this as
support for the non-rejected specification.
Calculating test statistics is straightforward. Assuming that the error terms
are independent and have a common distribution ε lkt ~ N (0,φ 2 ) , then the likelihood
ratio is described by the logarithm of the ratios of sum of squared residuals. It is
asymptotically distributed chi-squared with degrees of freedom equal to the number
of restrictions imposed by the specification:
(5.22)
HN :
n ln( RSSN RSSU ) → χ 2 (4 y + 2)
(5.23)
HC :
n ln( RSSC RSSU ) → χ 2 (4 y)
(5.24)
H P:
n ln( RSSP RSSU ) → χ 2 (4 y − 2)
(5.25)
H∅ :
n ln( RSS∅ RSSU ) → χ 2 (4 y + 4 )
d
d
d
d
The last of these corresponds with equation (5.20).
These specification tests do not exactly tie household behavior to a particular
bargaining procedure. However, these specifications are quite restrictive, so the
tests do tell us whether the behavior conforms with a fairly unusual pattern.
These tests can be more compelling if we notice different patterns of
behavior for different groups of people—provided we have reason to believe that
these groups should be using different procedures for solving multi-period bargaining
problems. The latent class model will pursue this idea.
5.3 A Latent Class Approach
Throughout this paper, I have given reasons why we might find each
intertemporal bargaining procedure to be a sensible way of modeling behavior, at
least for some people. In fact, one model might not fit everyone: cultural or religious
factors, family composition, or legal institutions might determine which approach
the household takes.
Formally, I treat this as a latent class problem: there is heterogeneity among
households with respect to the model of behavior, although I cannot observe which
28
Household bargaining & time
model applies to which observations.32 However, I believe that the likeliness of a
household using a particular intertemporal procedure depends on some observable
characteristics. Suppose that a household could use one of three procedures to solve
a multi-period bargaining problem. Of the three possibilities, there is some chance
that the household uses procedure “ Q .” This means that the correct model is:
(5.26)
(
)
(
)
∆ ln x*lt ( x*kt )σ 2 = σ 1 ∆ ln( plt pkt ) − σ 1 Φ Qlt − Φ Qkt + ε lkt with probability P Q
I maintain the assumption that the error terms are i.i.d., but with different variance
for the three classes: ε lkt ~ N (0,φ Q2 ) . As a consequence, there is a probability P Q that
a particular residual has the following density:
(5.27)

σ
∆ ln xl*t ( xkt* ) − σ 1 ∆ ln( plt pkt ) + σ (Φ lQt − Φ ktQ )

f (ε lkt ) =
exp

−φ Q2
2 πφ Q2

1
( (
2
)
)  w.p. P
2


Q
Summing over all the possible classes, the likelihood of this observation is:
(5.28)


σ
∆ ln xl*t ( xkt* ) − σ ∆ ln( plt pkt ) + σ (Φ lQt − Φ ktQ )
 PQ

f (ε lkt ) =
exp


−φ Q2
2 πφ Q2
Q∈ { N ,C, P } 


∑
( (
2
)

) 
2


Then I assume that the chance that this observation belongs to class Q depends on a
set of observable characteristics, denoted by zt . I use a multinomial logit to specify
this probability:
(5.29)
PQ =
exp(θQ ⋅ zt )
∑ Q exp(θQ ⋅ zt )
Inserting this into (5.28), the log-likelihood function for an observation is:
(5.30)


Q
Q
*
* σ
 exp(θ Q⋅ zt )
 ∆ ln xlt ( xkt ) − σ∆ ln( plt pkt ) + σ (Φ lt − Φ kt )
ln
+ ln ∑
exp

φ ∑ Q exp(θ Q⋅ zt )
−φ Q2
2π
Q  Q


1
( (
2
)

) 
2


This equation can be estimated using standard maximum likelihood techniques.
Because the logit allows one normalization, I set the numerator of (5.29) equal to one
for the myopic model.
32
Greene (2001) has a discussion of latent class models and their applications in
economics. They are most familiar as “regime-switching” models. In my model,
however, I am not observing individuals switching from one class to another.
Household bargaining & time
6.
29
DATA ON HOUSEHOLD EXPENDITURES
These tests require a data set following households for a sufficiently long
period, reporting expenditures on a variety of goods and earnings potentials of
household members. Unfortunately, most longitudinal data sets are quite limited in
their consumption data.
The Panel Study of Income Dynamics (PSID) has detailed information on the
composition of households and labor market characteristics of each member. This
provides a good wage history (and future) for most individuals, and relatively good
estimates (based on education, location, and past work history) for years when they
do not participate in the labor market.
The PSID also reports household expenditures on food and utilities for
eleven consecutive years, from 1976 to 1986. Positioned in the center of the panel,
these years allow me to see individuals’ wages quite a few years into the future and
past. I examine the relationship between yearly changes in consumption of food and
utilities (from t to t + 1 , with both of these years being between 1976 and 1986) and
wages in these years as well as the previous and subsequent five years.33
Each household in my sample consists of a couple who stay married for these
twelve consecutive years, and who have no other adults living in the home in any of
these years. These requirements ensure that the household has the same set of
active decision-makers in all the years, and that they are successful in reaching some
household decision. Both the husband and the wife must be between the ages of 22
and 65 in year t, and they must reside in the continental U.S. Approximately 1000
households meet these criteria in any year; a household appears in my sample for
each year in which it meets the requirements. The 10,506 observations in my sample
come from 1,701 unique households.
Characteristics of the household, other than work-related variables, are
reported in Table 6.1. Means and standard deviations are calculated using PSID
family sampling weights. Table 6.1 classifies the sample by race of the head, region of
residence, and educational attainment of the parents of the household head and wife.
33
In the first years (from 1969 until 1972), consumption information in the PSID
included food, utilities, alcohol, tobacco, childcare, and housework. Though the last
four of these goods would be prime candidates for studying household bargaining,
many families reported not purchasing these goods at all. Looking at this period also
means that I cannot assess the impact of past variables on current consumption.
Household bargaining & time
30
To capture possible differences in attitudes toward household decisionmaking, I include a set of variables for the religious preference of the head, region
where the head grew up, and two variables capturing the socio-economic background
of the household members. The first of these is a set of class indicators based on
whether the head reports his childhood status as “poor,” “average” or variable, or
“pretty well-off.” A second set of variables indicate the educational attainment of
the parents of the head and the wife.
This table also contains several variables constructed to capture the location
of the couple in the life-course of the marriage and their likelihood of divorce. Two
of these indicate whether the couple was married in the three years prior to the
period examined, and whether they divorced during the three years afterwards.34
Since the likeliness of divorce could easily affect how the household makes
intertemporal decisions, I create two variables that have a known correlation with
divorce rates. Sander (1985) finds that couples living on farms are less likely to
divorce (possibly due to greater gains from intrahousehold specialization of labor).
The variable included in Table 6.1 is based on whether the household head identifies
himself as a farmer (this question is asked in the PSID). Second, Peters (1986) notes
that divorce rates are higher in states with laws permitting “no-fault” divorce.
Women’s labor supply behavior is also different. To determine whether this could
be explained by the intertemporal bargaining procedure adopted by the household, I
include a variable indicating whether the state of residence is one that Peters lists as
granting no-fault (or relatively easy) divorce.35
Next, Table 6.2 reports the household expenditure on food, both at home and
away from home, and utilities. Average prices for food and utilities come from the
CPI-U, and are based on region and (whenever available) city size.36 The correlation
between changes in the prices of the two goods is approximately 0.56. These price
indices are used to calculate the change in quantity consumed from the change in
expenditures.
Work-related characteristics of the individuals are listed in Table 6.3:
experience, tenure, and education. The variables in this table include the reported
actual number of years that each person was working, as well as the number of those
34
That is: married during the years t − 8 through t − 6 , or divorced in years t + 7
through t + 9.
35
Based on the laws in place in 1979, which is roughly in the middle of my sample.
36
Because the CPI-U does not calculate utilities prices indices prior to 1977, the
change in the price of “all goods” is used for utilities in 1976 and 1977.
Household bargaining & time
31
years which are full-time work experience. Tenure is given both as number of years
with the same employer, and as number of years in the same job.37 Table 6.3 also lists
a number of indicators for highest educational attainment of each person.
Twelve years of average hourly earnings, calculated from reported annual
labor earnings and reported hours worked, are listed in Table 6.4. However, not all
household members work in every year: during any given year, approximately 90% of
men in the sample are working, compared with around 60% of the women. The first
column of numbers in Table 6.4 reports the number of observations reporting
nonzero labor earnings. The means and standard deviation of these observed
wages—which are conditional on the person selecting into the labor force—are
reported in the following column.
For years when the wage rate is unobserved, I impute an expected value. In
order to correct for selectivity in participation, I estimate the wage using a standard
Heckman procedure. The wage equation is modeled as a quadratic in age, both
measures of work experience, and both measures of tenure (when available); I control
for education, race, and region. In the selection probit, I include the individual’s
own age and the partner’s age, years of (full- or part-time) experience, number of
children, race, region, and religious preference. Thus, the probability of
participating is identified by the partner’s age, as well as by religious preference.
Using the estimated coefficients from these equations, I impute the wage rate
for all individuals whose wages are missing. The final column reports the means and
standard deviations of these variables—that is, the wage rate when observed, and the
estimated value otherwise. These means are lower than the observed means,
suggesting positive selection into the labor force. Including imputed wages naturally
tends to decrease the variance in wages, as is also seen in Table 6.5.
As a reference group, another sample consisting of single-headed households
is created from the PSID. The criteria for this group are the same as those for the
married couples, except that the household head must remain unmarried and living
without any other adults for the twelve consecutive years.
This group is small, consisting of 1,782 observations. Its characteristics are
summarized in Table 6.5. It is strikingly different from the main sample: the sample
of single-headed households is predominantly female-headed and is
disproportionately black. It is considerably older, and only around 21% have children
37
The PSID does not ask either person about employer tenure in 1977, 1978, or 1979.
In 1978, it does not report job tenure for the wife, either.
Household bargaining & time
32
living at home. Expenditures are also included in Table 6.5, while work-related
variables are listed in Table 6.6. Both observed and imputed wages are reported in
Table 6.7; these imputed wages are estimated in the same manner as for
households.38 Though this group is not the focus of my study, they provide an
opportunity to double-check the results from the married couples. Since they are
individuals who make their own decisions, we would expect that the unitary model
describes their behavior. If instead we observe behavior that resembles one of the
other specifications, we might infer that the pattern is generated by a phenomenon
other than intra-household bargaining.
7.
RESULTS
Controlling for regional price changes, age, number of children, region, year,
race, and religious preference, I estimate the unrestricted equation (5.21) for the
difference between consumption of food and utilities for three cases: using total food
expenditures, food away from home, and food at home. This is compared to the
specifications given by the three intertemporal bargaining procedures and the unitary
model. The p-values at which these specifications can be rejected are reported in
Table 7.1.
For all married couple households, none of the models describes behavior
well. Although the prescient model works slightly better than the others, we can
easily reject all three bargaining procedures, as well as the unitary model. However,
when the sample is divided into households with children and households without
children , a different pattern emerges. For married couples with children (Table 7.2),
the prescient model fits remarkably well—much better than the other specifications.
For instance, consider total food expenditures versus utilities. Except for the
prescient model, all of the specifications have p-values that can be rejected at all
conventional confidence levels (even below 0.5%). On the other hand, the prescient
model is barely rejected at the 10% level. This twenty-fold increase in P-value lends
substantial support to the prescient model for this group.
For couples without children (Table 7.3), the myopic approach appears to be
the winner. In two of the three cases, we observe that this specification fits
remarkably better than the others. In the third case, it is still the hardest model to
reject, although there is not any substantial difference from the others—but we
cannot reject that the “unitary” model works as well as any of the other
38
Naturally, the selection criteria do not include the spouse’s age for these people.
Household bargaining & time
33
specifications. This could indicate that childless couples have almost identical
preferences over eating out and utilities—and when preferences agree, there is no
need to bargain. It could mean, instead, that one partner makes all decisions. It
could also mean that the data are too noisy to see any systematic behavior.
Table 7.4 shows the results for the unmarried, single-adult households. In
two of the three cases, unitary behavior cannot be rejected at the 5% level—neither
the unrestricted model nor the three bargaining procedures provides a significantly
better fit. However, Table 7.4 shows that for “food away from home” for singles,
there is a correlation with wages that does not fit the pattern predicted by unitary
behavior or by any of the bargaining procedures.
These specification tests reveal that there may well be differences in how
households solve multi-period bargaining problems. Furthermore, for the most part
the unitary specification seems reasonable for single-adult households—and
certainly, we do not see this group behaving in the same manner as married couples.
The latent-class model has three components: an equation for each of the
three bargaining procedures, and two selection (or “switching”) equations. The
variables common to all three procedures (including the change in relative prices and
controls for year, region, race, and religion) are restricted to be the same under each
bargaining procedure. In order to simplify the problem, I use a discount rate of
βw = β h = 0.9 to calculate the appropriate right-hand side variables for each of the
three procedures: change in log wages if myopic, change in the present discounted
value of future log wages if prescient, or discounted value of lifetime log wages if
contractual. The latent class approach allows observations to sort themselves into
the set of explanatory variables that best describe their behavior. This sorting is
assumed to be determined by a number of selection variables.
The same set of variables is used in the two selection equations, except for
“Head is a farmer” which was dropped from the second equation.39 The selection
probabilities are identified by the number of children, recent marriage, future
divorce, and characteristics relating to the background of the head and the wife.
In Table 7.5, I report estimation results. From the selection equations, it is
immediately apparent that the contractual procedure appears very unlikely for most
39
Due to other characteristics (children and region, in particular), there is a very
high probability that farm couples exhibit prescient behavior, which made it difficult
to identify whether farmers were more likely to be naïve or contractual.
Household bargaining & time
34
individuals.40 Because of this, the parameter estimates from that procedure turn out
to be rather imprecise. However, estimates from the other two procedures are quite
reasonable. For both the myopic and prescient procedures, coefficients on variables
for the wife are smaller and noisier than for the head. This can be attributed to
greater measurement error in wives’ wages, since the values of wages are imputed for
a much larger share of women.41
The estimates in Table 7.5 suggest a (compensated) price elasticity of –0.47
for food, and –0.12 for utilities. These elasticitities are estimated indirectly: when
including both the change in relative prices and the change in utilities as explanatory
variables, the coefficient on ∆ ln( pfood / putil ) is interpreted as the price elasticity of
food, and the coefficient on ∆ ln( xutil ) is the ratio of price elasticities.
Estimates suggest that for a myopic household, a 10% increase in the head’s
wage rate from t to t+1 results in a modest increase in the consumption of food,
relative to utilities, by 0.14%. For a prescient household, a 10% increase in the
present discounted value of the head’s wages results in a 0.20% increase.42 In either
case, an increase in the wife’s wages generates a slightly smaller effect in the opposite
direction.43 Since food and utilities are nearly public goods, it is not surprising that
the effects are so small—but we can infer that men have a slight preference for food
relative to utilities. This is consistent with my expectations for two reasons. Since
the man typically spends less time at home, he is simply not as inclined to keeping
the temperature as comfortable. Furthermore, utilities expenditures are related to
the size of the house, and thus it is correlated with the “amount of housing”
consumed—if the woman spend more time at home, she gains more utility from
consuming housing.
From the selection equations, we can calculate the probabilities that a
particular household uses each procedure. Examples provided in Tables 7.6 and 7.7
show the sizes of these effects more clearly—for a rather typical household in Table
40
However, the contractual model is not superfluous: I can reject the hypothesis
that the naïve and prescient procedures are the only two “classes.”
41
When considering only couples who report wages in every year, the magnitude of
effects for men and women appears to be almost the same.
42
For all purposes, these effects are the same size, a comforting result—either
model says that a 10% increase in the head’s “disagreement utility” produces about a
0.15-0.20% increase in food relative to utilities.
43
In part, this is due to more measurement error in wives’ wages. The difference is
smaller when examining only couples reporting wages, but it does not disappear
entirely. It may well be that the wage rate is just a worse measure of disagreement
utility for women ( α w < α h , in the notation of §5.1), which would produce this effect.
Household bargaining & time
35
7.6, and for a young household with two children in Table 7.7, where probabilities
can change quite a bit in response to changes in the explanatory variables.
The likeliness of using the prescient procedure is determined largely by the
number of children, whether the couple divorces afterwards, whether no-fault
divorce is allowed, and whether the head is a farmer. Children have a large, positive
effect on selection into the prescient procedure. The two variables used to capture
the cost of divorce, farming and no-fault divorce, show that costlier divorce
substantially increases the probability of using the prescient procedure as well.
Surprisingly, couples who divorce years in the future are also more likely to be
using the prescient procedure. Does this suggest that these households already
anticipate a particular terminal date, and that they solve the allocation problem
backwards from that point? Might these couples divorce because they are unusually
forward-looking, and they recognize when marital gains are zero? At any rate, these
are not couples with an “easy come, easy go” attitude toward marriage: to be included
in the sample, they must have been married for at least twelve years; most of them
have been married for at least fifteen.44
There is a statistically significant, but modest, effect of the region where the
household head grew up. Households originally from the northeast use the myopic
approach more often, while westerners are more likely to exhibit prescient behavior.
Parents’ education and religious preference seem to have little effect on the
procedure used by the household.
For most groups, the contractual model does not work very well at all—but
there are exceptions. Table 7.7 is intended to show the most likely candidates,
recently married couples and southerners. Once there is a sufficiently high
probability of falling into this class, the effect of an additional child or of no-fault
divorce laws is to increase further the probability of exhibiting contractual behavior.45
The representative household decides whether to use the myopic or prescient
procedure in solving its multi-period bargaining problems. Which procedure is
chosen depends largely on the number of children, the cost of divorce, and the
length of time married or until divorce. Religious preference and socio-economic
44
Children are present in an unusually large share of these families, 87.8%. (Overall,
65.0% of the households in my sample have children). Aside from that, this group is
not remarkably different from others in my sample.
45
For the profile shown in Table 7.7, religious preference seems to have a rather
large effect on the selection probabilities. This is a bit misleading, since there is
actually very little variation in the religious preference: over 75% of recently married
couples from the south belong to a Protestant group. Among these groups, there is
little difference in their probabilities.
Household bargaining & time
36
background are largely irrelevant. These factors may have some effect for the small
group of individuals exhibiting behavior more similar to the contractual model.
8.
CONCLUSIONS
In this paper, I have argued that we need to think carefully about how we
model intertemporal household bargained decisions. Depending on the
intertemporal procedure used by the household, we may not be able to make valid
inferences about behavior from examining one period in isolation from the others.
Since the implied behavior is generally different, each of the procedures generates a
unique Euler equation. I use these competing specifications to determine which
procedure, if any, accurately describes household behavior.
The first empirical tests suggest three things. First, it seems that there is no
one intertemporal bargaining procedure that describes household behavior.
However, it is hard to reject that couples with children use the prescient approach,
while childless couples take the myopic approach. This is sensible, since the myopic
model might describe a world in which divorce and remarriage are relatively quick
and costless. Finally, while married couples’ behavior is consistent with these two
intertemporal bargaining procedures, single-adult households do not fit into any of
these patterns. In two of three cases, we cannot reject the hypothesis of unitary
behavior. This lends strong support for the belief that the observed effect is indeed
due to intrahousehold bargaining.
Estimates from the latent-class model give us some insight into how different
couples view their marriages. These results confirm that children are important in
determining the intertemporal procedure used by the household. Despite its
efficiency advantage, the contractual procedure is used by few households. As would
be expected when contracts are not enforceable, most couples appear to follow the
myopic or prescient procedures. Any variable that increases the difficulty of
dissolving the marriage—children, stricter divorce laws, living on a farm—drives the
household toward the prescient procedure. Religion and socio-economic
background appear to have little influence. How a household approaches a multiperiod bargaining problem depends on economic, and not cultural, factors.46
The changes in food and utilities consumption due to changes in the threat
points are quite small. Since food and utilities are generally regarded as public goods,
46
Apparently, this is also true for a bird called a Plymouth dover (The Economist, 31
March 2001, page 74).
Household bargaining & time
37
this is not surprising—I conclude that preferences over these goods are closely
aligned. In a follow-up to this paper, I wish to examine goods that are more
“assignable.” In fact, for households using the myopic procedure, a short panel is
sufficient to identify the “assignability” of household goods.47
Except for newlyweds and couples with many children, it appears that few
households take a contractual approach to household bargained decisions. For some,
marriage appears to be a repeated game: each year’s decision is made on its own, with
no regard for the future, as if agents know that they can easily change partners. For
others—especially households with children—multi-period household bargaining
problems are solved by backwards induction, knowing that renegotiations can and
will occur. To conclude, again I quote the same Paul Simon song: “What is the point of
this story? What information pertains? The thought that life could be better is indelibly woven
into our hearts and our brains.”
47
Rethinking “risk-aversion” and household responses to uncertainty constitutes a
second extension of this paper. When faced with uncertainty about a future
variable, does the household decide a complete plan for each state in advance, or
does bargaining occur after the uncertainty is resolved? In analogue to the
“contractual” and “naïve” intertemporal procedures, these different ways of
bargaining under uncertainty generally do not produce the same outcome. Questions
about households and uncertainty abound: do husbands and wives provide labor
market insurance for one another? how does the household handle fertility risk? what
are the effects of uncertainty about future income, threat points? what factors affect
the risk of divorce?
Household bargaining & time
9.
38
REFERENCES
Martin Browning. (2000). “The savings behaviour of a two-person household.”
Scandinavian Journal of Economics 102.
Martin Browning and Pierre-Andre Chiappori. (1998) “Efficient intra-household
allocations: A general characterization and empirical tests.” Econometrica 66.
P.S. Carlin. (1991). “Intra-family bargaining and time allocation.” Research in
population economics, volume 7. Greenwich CT: JAI.
Pierre-Andre Chiappori. (1988a). “Rational household labor supply.” Econometrica 56.
Pierre-Andre Chiappori. (1992). “Collective labor supply and welfare.” J. of Political
Econ. 100.
Pierre-Andre Chiappori. (1997). “’Collective’ models of household behavior: the
sharing rule approach.” Intrahousehold resource allocation in developing countries.
L. Haddad, J. Hoddinott, and H. Alderman, eds. Baltimore: Johns Hopkins
UP.
William Dickens and Kevin Lang. (1985). “A test of dual labor market theory.”
Amer. Econ. Rev. 75.
William Dickens and Kevin Lang. (1988). “Labor market segmentation and the
union wage premium.” Rev. of Econ. Stat. 70.
Larry Elliott. “Who else will do a woman’s work?” The Guardian. 9 July 2001, p. 19.
William Greene. (2001). “Fixed and random effects in nonlinear models.” New York
University working paper.
Lawrence Haddad, John Hoddinott, and Harold Alderman, eds. (1997) Intrahousehold
resource allocation in developing countries: models, methods, and policy. Baltimore:
Johns Hopkins UP.
Mary Jean Horney and Marjorie McElroy. (1988). “The household allocation
problem: Empirical results from a bargaining model.” Research in population
economics, volume 6. Greenwich CT: JAI.
Terje Lensberg. (1987). “Stability and collective rationality.” Econometrica 55.
Shelly Lundberg and Robert Pollak. (1993). “Separate spheres bargaining and the
marriage market.” J. of Polit. Econ 101.
Shelly Lundberg and Robert Pollak. (2001). “Efficiency in marriage.” Working
paper, University of Washington.
Shelly Lundberg, Robert Pollak, and Terence Wales. (1997). “Do husbands and
wives pool their resources? Evidence from the UK child benefit.” J. of
Human Resources 32.
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39
Shelly Lundberg, Richard Startz, and Steven Stillman. (2001). “The retirementconsumption puzzle: a marital bargaining approach.” Working paper,
University of Washington.
Shelly Lundberg and Jennifer Ward-Batts. (2000). “Saving for retirement:
household bargaining and household net worth.” Working paper, University
of Washington.
Marjorie McElroy. (1990). “The empirical content of Nash-bargained behavior.” J. of
Human Resources 25.
Marjorie McElroy and Mary Jean Horney. (1981). “Nash-bargained household
decisions: towards a generalization of the theory of demand.” Intl. Econ. Rev.
22.
Marilyn Manser and Murray Brown. (1980). “Marriage and household decision
making: a bargaining analysis.” Intl. Econ. Rev. 21.
Roger Myerson. (1981). “Utilitarianism, egalitarianism, and the timing effect in
social choice problems.” Econometrica 49.
M. A. Perles and Michael Maschler (1981). “The super-additive solution for the Nash
bargaining game.” International J. of game theory 10.
Elizabeth Peters. (1986) “Marriage and divorce: informational constraints and
private contracting.” Amer. Econ. Rev. 76.
Clara Ponsati and Joel Watson. (1997). “Multiple-issue bargaining and axiomatic
solutions.” International J. of game theory 26.
William Sander. (1985) “Women, work, and divorce.” Amer. Econ. Rev. 75.
Paul Samuelson. (1947) Foundations of economic analysis. Cambridge MA: Harvard UP.
Duncan Thomas. (1990). “Intra-household resource allocation: an inferential
approach.” J. of Human Resources 25.
Duncan Thomas, Dante Contreras, and Elizabeth Frankenberg. (1999). “Distribution
of power within the household and child health.” Indonesian Family Life
Survey (IFLS) Working Paper.
William Thomson. (1994). “Cooperative models of bargaining.” Handbook of game
theory, vol. 2. R. J. Aumann and S. Hart, eds. New York: Elsevier.
40
Household bargaining & time
Table 6.1:
Household characteristics: married couples (N=10,506)
Means and standard deviations, using family sampling weights
Variable Description
Age of head
Age of wife
Number of children at home, aged 0-17
Race of head: Black
Race of head: Latino
Region (current): Northeastern United States
Region (current): North-central United States
Region (current): Southern United States
Region (current): Western United States
Religious preference: Catholicism
Religious preference: Judaism
Religious preference: Conservative Protestantism48
Religious preference: Moderate Protestantism49
Religious preference: Liberal Protestantism50
Whether children at home, aged 0-17
Whether married recently beforehand
Whether divorced shortly thereafter
Whether head is a farmer
Whether state granted no-fault divorce in 1978
Region of head’s childhood: Northeastern U.S.
Region of head’s childhood: North-central U.S.
Region of head’s childhood: Southern U.S.
Region of head’s childhood: Western U.S.
Head was upper class as a child
Head was middle class as a child
Head was lower class as a child
Education of head’s father: less than high school
Education of head’s father: high school or technical
Education of head’s father: post-secondary
Education of wife’s father: less than high school
Education of wife’s father: high school or technical
Education of wife’s father: post-secondary
48
Mean
(St. dev.)
44.90
(10.51)
42.67
(10.38)
1.34
(1.25)
3.6%
1.9%
24.4%
31.5%
28.4%
15.7%
25.4%
5.48%
19.2%
19.7%
13.6%
64.9%
7.67%
2.32%
3.32%
49.2%
26.4%
33.2%
26.3%
14.2%
17.4%
46.6%
36.0%
64.7%
19.1%
12.5%
59.1%
25.2%
13.8%
Includes Baptists and miscellaneous denominations.
Includes Lutherans and Methodists.
50
Includes Episcopalians, Presbyterians, and nondenominational Protestants.
49
41
Household bargaining & time
Table 6.2:
Household expenditures (N=10,506)
Means and standard deviations, using family sampling weights
Food at home, t+1
(expenditures in 1990$)
Food at home, t
(expenditures in 1990$)
Food away from home, t+1
(expenditures in 1990$)
Food away from home, t
(expenditures in 1990$)
Utilities, t+1
(expenditures in 1990$)
Utilities, t
(expenditures in 1990$)
Percent change in food prices
Percent change in utilities prices
5709.34
(2980.20)
5814.86
(3022.44)
1325.28
(1358.39)
1301.40
(1354.57)
2111.93
(1292.34)
2144.48
(1557.20)
0.0571
(0.0316)
0.0728
(0.0560)
Household bargaining & time
Table 6.3:
Work-related characteristics (N=10,506)
Means and standard deviations, using family sampling weights
Variable Description
Mean
(St. dev.)
Work experience, head
21.74
(Years)
(10.96)
Full-time work experience, head
20.51
(Years)
(11.34)
Tenure in job, head
8.13
(Years)
(8.65)
Tenure with employer, head51
9.25
(Years)
(10.22)
Work experience, wife
9.55
(Years)
(8.43)
Full-time work experience, wife
7.35
(Years)
(7.71)
Tenure in job, wife52
3.05
(Years)
(5.07)
Tenure with employer, wife4
3.53
(Years)
(5.90)
Education of head: some high school
20.2%
Education of head: high school or equivalency
18.3%
Education of head: some college or technical training
27.9%
Education of head: four-year university degree
9.35%
Education of head: professional degree
4.10%
Education of wife: some high school
12.9%
Education of wife: high school or equivalency
20.8%
Education of wife: some college or technical training
19.8%
Education of wife: four-year university degree
5.55%
Education of wife: professional degree
1.60%
51
52
42
Employer tenure is not available for either member in 1977, 1978, or 1979.
Job tenure is not available for the wife in 1978.
43
Household bargaining & time
Table 6.4:
Current, future, and past wages of household head and wife
Variable Description
Head
Average hourly earnings, t+6
(1990$)
Average hourly earnings, t+5
(1990$)
Average hourly earnings, t+4
(1990$)
Average hourly earnings, t+3
(1990$)
Average hourly earnings, t+2
(1990$)
Average hourly earnings, t+1
(1990$)
Average hourly earnings, t
(1990$)
Average hourly earnings, t-1
(1990$)
Average hourly earnings, t-2
(1990$)
Average hourly earnings, t-3
(1990$)
Average hourly earnings, t-4
(1990$)
Average hourly earnings, t-5
(1990$)
Wife
Average hourly earning, t+6
(1990$)
Average hourly earnings, t+5
(1990$)
Average hourly earnings, t+4
(1990$)
Average hourly earnings, t+3
(1990$)
Average hourly earnings, t+2
(1990$)
Average hourly earnings, t+1
(1990$)
Average hourly earnings, t
(1990$)
Average hourly earnings, t-1
(1990$)
Average hourly earnings, t-2
(1990$)
Average hourly earnings, t-3
(1990$)
Average hourly earnings, t-4
(1990$)
Average hourly earnings, t-5
(1990$)
53
Number
observed
Observed mean
(St. dev.)
Imputed mean53
(St. dev.)
8,852
18.03
(12.07)
18.05
(11.75)
18.12
(11.68)
18.19
(11.54)
18.28
(11.45)
18.32
(11.36)
18.30
(11.11)
18.19
(10.79)
18.14
(10.68)
18.10
(10.49)
18.08
(10.33)
18.06
(10.58)
17.58
(11.22)
17.67
(11.07)
17.78
(11.12)
17.91
(11.07)
17.98
(11.05)
18.01
(11.02)
18.09
(10.79)
18.08
(10.53)
18.03
(10.43)
17.98
(10.25)
17.97
(10.11)
17.93
(10.34)
10.84
(7.96)
10.79
(7.95)
10.76
(8.08)
10.68
(7.96)
10.70
(8.01)
10.71
(8.00)
10.86
(8.36)
10.93
(8.22)
11.16
(8.74)
11.30
(8.73)
11.46
(8.43)
11.64
(8.45)
10.14
(6.50)
9.77
(6.58)
9.80
(6.62)
9.67
(6.51)
9.68
(6.49)
9.40
(6.53)
9.76
(6.65)
9.57
(6.64)
9.85
(6.89)
9.73
(6.89)
9.90
(6.64)
10.31
(6.52)
9,142
9,320
9,480
9,613
9,745
9,818
9,870
9,919
9,961
9,945
9,899
6,611
6,830
6,793
6,706
6,598
6,442
6,257
6,100
5,922
5,754
5,606
5,454
This column reports the actual wage when observed, and an imputed value for
individuals who report no labor earnings.
44
Household bargaining & time
Table 6.5:
Characteristics of household heads: Singles (N=1,782)
Means and standard deviations, using family sampling weights
Variable Description
Female-headed household
Age of head
Number of children at home, aged 0-17
Whether children at home, aged 0-17
Race of head: Black
Race of head: Latino
Religious preference: Catholicism
Religious preference: Judaism
Religious preference: Conservative Protestantism54
Religious preference: Moderate Protestantism55
Religious preference: Liberal Protestantism56
Region: Northeastern United States
Region: North-central United States
Region: Southern United States
Region: Western United States
Food at home, t+1
(expenditures in 1990$)
Food at home, t
(expenditures in 1990$)
Food away from home, t+1
(expenditures in 1990$)
Food away from home, t
(expenditures in 1990$)
Utilities, t+1
(expenditures in 1990$)
Utilities, t
(expenditures in 1990$)
54
Mean
(St. dev.)
84.4%
48.3
(11.8)
0.36
(0.85
20.6%
17.4%
1.12%
19.2%
2.67%
22.9%
19.6%
16.6%
24.8%
25.3%
29.4%
20.4%
2767.41
(2114.36)
2840.73
(2147.04)
822.19
(1098.17)
832.97
(1141.17)
1389.80
(1265.26)
1408.02
(1473.67)
Includes Baptists and miscellaneous denominations.
Includes Lutherans and Methodists.
56
Includes Episcopalians, Presbyterians, and nondenominational Protestants.
55
Household bargaining & time
Table 6.6:
Work-related characteristics (Singles)
Means and standard deviations, using family sampling weights
Variable Description
Mean
(St. dev.)
Work experience, head
18.37
(Years)
(11.89)
Full-time work experience, head
15.62
(Years)
(11.87)
Tenure in job, head
5.58
(Years)
(7.09)
Tenure with employer, head57
6.50
(Years)
(7.92)
Education of head: some high school
16.6%
Education of head: high school or equivalency
17.0%
Education of head: some college or technical training 32.8%
Education of head: four-year university degree
9.34%
Education of head: professional degree
2.29%
57
45
Employer tenure is not available for either member in 1977, 1978, or 1979.
Household bargaining & time
Table 6.7:
46
Current, future, and past wages of household head (Singles)
Means and standard deviations, using family sampling weights
Number Observed Imputed58
observed (St. dev.) (St. dev.)
Average hourly earnings of head, t+6 1,129
12.71
11.50
(1990$)
(10.16)
(8.58)
Average hourly earnings of head, t+5 1,182
12.47
11.57
(1990$)
(9.46)
(8.17)
Average hourly earnings of head, t+4 1,214
12.19
11.00
(1990$)
(8.79)
(7.80)
Average hourly earnings of head, t+3 1,246
12.25
11.09
(1990$)
(8.88)
(7.96)
Average hourly earnings of head, t+2 1,275
12.22
1120
(1990$)
(8.51)
(7.71)
Average hourly earnings of head, t+1 1,295
12.16
11.06
(1990$)
(8.03)
(7.45)
Average hourly earnings of head, t
1,316
12.24
11.10
(1990$)
(8.18)
(7.70)
Average hourly earnings of head, t-1 1,313
12.26
11.62
(1990$)
(8.04)
(7.37)
Average hourly earnings of head, t-2 1,309
12.10
11.13
(1990$)
(7.54)
(7.08)
Average hourly earnings of head, t-3 1,298
12.17
11.23
(1990$)
(7.51)
(7.00)
Average hourly earnings of head, t-4 1,275
12.18
11.16
(1990$)
(7.50)
(7.02)
Average hourly earnings of head, t-5 1,243
12.40
11.24
(1990$)
(7.87)
(7.32)
Variable Description
58
This column reports the actual wage when observed, and an imputed value for
individuals who report no labor earnings.
47
Household bargaining & time
Table 7.1:
Food versus utilities: All married couples
Sum of squared residuals (P-values in parentheses)
Total food
N=10,506
Food, out
N=9,314
Food, in
N=10,468
Unrestricted model
1506.883
5317.247
1649.368
Myopic bargaining
(22 restrictions)
1512.910
(0.0064)
5339.847
(0.0123)
1655.654
(0.0113)
Contractual bargaining
(20 restrictions)
1512.944
(0.0026)
5339.855
(0.0057)
1654.764
(0.0249)
Prescient bargaining
(18 restrictions)
1511.738
(0.0134)
5335.480
(0.0227)
1654.374
(0.0237)
Unitary behavior
(24 restrictions)
1513.717
(0.0029)
5344.285
(0.0031)
1655.863
(0.0161)
Total sum of squares
1562.606
5404.234
1702.377
Table 7.2:
Food versus utilities: Married, with children
Sum of squared residuals (P-values in parentheses)
Total food
N=7,526
Food, out
N=6,712
Food, in
N=7,493
Unrestricted model
1026.795
3593.905
1058.655
Myopic bargaining
(22 restrictions)
1032.733
(0.0042)
3611.783
(0.0344)
1064.674
(0.0055)
Contractual bargaining
(20 restrictions)
1032.847
(0.0014)
3613.564
(0.0071)
1064.523
(0.0032)
Prescient bargaining
(18 restrictions)
1030.422
(0.0881)
3602.238
(0.4734)
1061.974
(0.1738)
Unitary behavior
(24 restrictions)
1033.220
(0.0034)
3614.853
(0.0159)
1064.966
(0.0066)
Total sum of squares
1067.172
3654.640
1097.700
48
Household bargaining & time
Table 7.3:
Food versus utilities: Married, without children
Sum of squared residuals (P-values in parentheses)
Total food
N=2,980
Food, out
N=2,602
Food, in
N=2,975
Unrestricted model
471.053
1703.234
580.017
Myopic bargaining
(22 restrictions)
475.289
(0.2236)
1718.308
(0.4058)
585.312
(0.2099)
Contractual bargaining
(20 restrictions)
476.721
(0.0169)
1717.900
(0.3239)
587.106
(0.0148)
Prescient bargaining
(18 restrictions)
476.559
(0.0106)
1718.061
(0.2083)
586.748
(0.0116)
Unitary behavior
(24 restrictions)
477.455
(0.0202)
1720.150 588.271
(0.3677) (0.0128)
Total sum of squares
492.978
1749.214
Table 7.4:
601.869
Food versus utilities: Singles
Sum of squared residuals (P-values in parentheses)
Total food Food, out Food, in
N=1,782
N=1,115
N=1,738
Unrestricted model
558.124
859.977
510.382
Myopic bargaining
(11 restrictions)
563.569
(0.0993)
881.828
(0.0033)
515.337
(0.1142)
Contractual bargaining 563.913
(10 restrictions)
(0.0488)
884.299
(0.0006)
515.523
(0.0656)
Prescient bargaining
(9 restrictions)
563.610
(0.0424)
880.835
(0.0016)
516.191
(0.0201)
Unitary behavior
(12 restrictions)
564.088
(0.0899)
884.817
(0.0015)
516.241
(0.0702)
Total sum of squares
576.288
902.485
528.350
49
Household bargaining & time
Table 7.5:
Estimates of the three models, all married couples
Dependent variable: ∆ ln xfood
Variable
Selection into prescient
Constant
Number of children
Whether married recently
Whether divorced afterward
No-fault divorce allowed
AGEh − AGEw
Head comes from Northeast
Head comes from Midwest
Head comes from South
Head from upper-class
Head from lower-class
Head is a farmer
Head’s father: HS education
Head’s father: more than HS
Wife’s father: HS education
Wife’s father: more than HS
Catholic
Jewish
Protestant, liberal
Protestant, moderate
Protestant, conservative
Coefficient Stand. Err. P-value
0.5150
-0.1035
0.9560
-0.5015
0.0260
-0.6933
-0.3994
-0.3576
-0.3179
-0.1420
0.6952
-0.1513
0.1125
-0.0468
0.1257
0.0501
0.0877
-0.0278
0.0643
-0.2545
0.1932
0.1124
0.2373
0.2803
0.0997
0.0136
0.1594
0.1389
0.1586
0.1442
0.1038
0.2414
0.1309
0.1531
0.1144
0.1417
0.1541
0.2264
0.1674
0.1542
0.1685
0.000
0.663
0.001
0.000
0.056
0.000
0.004
0.024
0.027
0.171
0.004
0.248
0.462
0.682
0.375
0.745
0.699
0.868
0.677
0.131
-4.047
0.5709
0.9509
-0.2715
0.2987
-0.0448
0.8376
0.6051
1.243
-0.2330
-0.0821
-0.5910
-0.6615
-0.3039
-0.8461
-0.8573
-0.3954
-0.5884
-0.7790
-0.6095
0.5743
0.0613
0.3029
0.7994
0.2123
0.0302
0.5446
0.5154
0.5070
0.2890
0.2182
0.2902
0.3821
0.2211
0.4006
0.3224
0.4916
0.3837
0.3385
0.2477
0.000
0.000
0.002
0.734
0.159
0.139
0.124
0.240
0.014
0.420
0.707
0.042
0.083
0.169
0.035
0.008
0.421
0.125
0.021
0.014
0.0629
0.745
Selection into contractual
Constant
Number of children
Whether married recently
Whether divorced afterward
No-fault divorce allowed
AGEh − AGEw
Head comes from Northeast
Head comes from Midwest
Head comes from South
Head from upper-class
Head from lower-class
Head’s father: HS education
Head’s father: more than HS
Wife’s father: HS education
Wife’s father: more than HS
Catholic
Jewish
Protestant, liberal
Protestant, moderate
Protestant, conservative
50
Household bargaining & time
Table 7.5, continued:
Dependent variable: ∆ ln xfood
Common parameters
(
∆ ln pfood / putil
∆ ln xutil
)
-0.4675
0.2676
0.2579
0.0458
Included
Included Unimportant
Included Unimportant
Included Unimportant
Year dummies
Region dummies
Race dummies
Religion dummies
Myopic parameters
Constant
∆ ln Wh
∆ ln Ww
Standard error of observation
Prescient parameters
Constant
∑
∑
s =t + 6
s =t + 1
s =t + 6
s =t + 1
(0.9
(0.9
s −t −1
s −t −1
)
(0.9
ln W ) − ∑
(0.9
ln Wh − ∑
w
s =t + 6
s =t
s =t + 6
s =t
s −t
ln Wh
s −t
ln Ww
Standard error of observation
Contractual parameters
Constant
∑
∑
s =t + 6
s =t − 5
s =t + 6
s =t − 5
(0.9
(0.9
s
s
ln Wh
ln Ww
)
)
Standard error of observation
Goodness of fit
Number of parameters estimated
Log likelihood
)
)
0.081
0.000
-0.0310
0.0139
-0.00788
0.1991
0.0112
0.00724
0.0254
0.00347
0.006
0.055
0.754
0.000
-0.0661
0.0195
-0.012
0.4549
0.0439
0.00541
0.00721
0.0119
0.132
0.000
0.089
0.000
0.7798
-0.0155
0.0249
1.263
0.6408
0.0303
0.0375
0.0457
0.224
0.610
0.507
0.000
74
-2821.858
0.000
51
Household bargaining & time
Table 7.6:
Changes in the probabilities of belonging to classes due to
changes in explanatory variables (typical household)
Myopic
Contractual Prescient
Procedure Procedure
Procedure
Reference individual:
AGEh = 44.9 , AGEw = 42.4 , 1 child
Head from Midwest, Catholic
0.4232
0.0091
0.5677
+1 child
-0.1186
+0.0025
+0.1161
-1 child
+0.1282
-0.0024
-0.1258
Married recently
+0.0183
+0.0155
-0.0337
Divorced afterwards
-0.2013
-0.0055
+0.2067
No-fault divorce allowed
+0.1199
+0.0066
-0.1265
+10 to AGEh
-0.0600
-0.0041
+0.0641
-10 to AGEh
+0.0604
+0.0072
-0.0675
Head from Northeast
+0.0701
+0.0043
-0.0744
Head from South
-0.0133
+0.0076
+0.0057
Head from West
-0.0912
-0.0052
+0.0964
Head was upper-class
+0.0785
-0.0006
-0.0780
Head was lower-class
+0.0347
-0.0000
-0.0347
Head is a farmer
-0.1537
-0.0033
+0.1570
Head’s father: high school degree
+0.0387
-0.0036
-0.0351
Head’s father: more than high school
-0.0252
-0.0047
+0.0298
Wife’s father: high school degree
+0.0123
-0.0022
-0.0102
Wife’s father: more than high school
-0.0280
-0.0055
+0.0334
No religious preference
+0.0066
+0.0127
-0.0793
Jewish
-0.0112
+0.0050
0.0062
Protestant, liberal
+0.0175
+0.0033
-0.0208
Protestant, moderate
-0.0037
+0.0007
+0.0031
Protestant, conservative
+0.0726
+0.0046
-0.0772
52
Household bargaining & time
Table 7.7:
Changes in the probabilities of belonging to classes due to
changes in explanatory variables (young couple)
Myopic
Contractual Prescient
Procedure Procedure
Procedure
Reference individual:
AGEh = 30.0 , AGEw = 30.0 , 2 children
Head from South, recently married
0.2966
0.1456
0.5579
+1 child
-0.1120
+0.0889
+0.0231
-1 child
+0.1357
-0.0636
-0.0721
Divorced afterwards
-0.1370
-0.0859
+0.2229
No-fault divorce allowed
+0.0604
+0.0907
-0.1511
+10 to AGEh
-0.0301
-0.0620
+0.0922
-10 to AGEh
+0.0141
+0.0931
-0.1073
Head from Northeast
+0.0775
-0.0231
-0.0544
Head from Midwest
+0.0299
-0.0609
+0.0311
Head from West
-0.0355
-0.1087
+0.1442
Head was upper-class
+0.0661
-0.0046
-0.0615
Head was lower-class
+0.0277
+0.0010
-0.0287
Head is a farmer
-0.1065
-0.0523
+0.1587
Head’s father: high school degree
+0.0496
-0.0515
+0.0019
Head’s father: more than high school
+0.0012
-0.0702
+0.0690
Wife’s father: high school degree
+0.0202
-0.0309
+0.0107
Wife’s father: more than high school
+0.0025
-0.0826
+0.0801
Catholic
+0.0173
-0.0802
+0.0629
Jewish
-0.0011
-0.0479
+0.0490
Protestant, liberal
+0.0258
-0.0577
+0.0319
Protestant, moderate
+0.0129
-0.0759
+0.0630
Protestant, conservative
+0.0704
-0.0477
-0.0227