Allocating Time:
Does Economic Theory Imply
Specialization in Efficient
Married Couple Households?
Robert A. Pollak
Washington University in St. Louis
May 17, 2011
1
Motivation: What Does Economic
Theory Teach About Specialization?
Does the economic theory of time use imply that
efficiency requires specialization in multiple-person
households (e.g., married couples).
This is what Becker claimed in the Treatise on the
Family (1981).
Why focus on Becker?
Because there isn’t much subsequent theoretical work
on time allocation.
2
The Meaning of Specialization
With two sectors (commodities, activities)
1. Strong (complete) specialization: each spouse
allocates time to only one sector
2. Weak (partial) specialization = specialization: one
spouse allocates time to one sector, the other spouse
allocates time to one sector or to both sectors
3. Nonspecialization: both spouses allocate time to
both sectors
How do we define specialization when there are more
than two activities? More about this later.
3
Some Facts:
Household Production
“Traditional gender roles do persist in the allocation of
time within households. Total hours of housework in
married couple households fell more than 20 percent
between 1965 and 1995 (Bianchi, Milkie, Sayer, and
Robinson, 2000) but, though husbands’ hours of
housework increased substantially, wives still
performed most of the housework at the end of this
period. In the 2005 American Time Use Survey,
married women reported an average of 16 hours per
week of ‘household activities’ compared to less than
11 hours for men.” Lundberg and Pollak (JEP, 2007)
4
Still More Facts:
Labor Force Particiaption
2008 Labor Force Participation rates for
Married men
25-34
95.3%
Married women
69.5%
Age
35-44
95.2%
73.8%
US data: CPS
5
Widespread Inefficiency?
If the economic theory of time use implies that
efficiency requires specialization in multiple person
(e.g., married couple) households, then the
prevalence of married-couple households in which
both husbands and wives allocate time to both the
market sector and household sector is evidence of
widespread inefficiency.
Becker makes a further claim that the pattern of
specialization will be gendered, with wives
specializing in the household and husbands in the
market. I ignore this further claim.
6
The Household Production Model
and the New Home Economics
Becker (EJ, 1965)
"A Theory of the Allocation of Time“
Becker wrote: households are "assumed to
combine time and market goods to produce
more basic commodities that directly enter
their utility functions.“
Michael and Becker (Swedish J of Ec, 1973)
Becker (1981, 1991) A Treatise on the Family
Becker (1981, 1991) differs from Becker (1965)
and from Michael and Becker (1973)
7
The Household Production Model
Becker’s household production model remains the lens
through which virtually all economists and many
other social scientists view time allocation.
We can differ about details, and details are important,
but as a conceptual framework, Becker’s household
production model is the only game in town.
Time allocation
between market and nonmarket work
among nonmarket work activities
between work and leisure
among leisure activities
8
Differences between Becker (1965)
and Becker’s Treatise (1981, 1991)
Multiple-person households in the Treatise
(vs single-person households in 1965)
Human capital: both market and household human
capital in the Treatise (vs no human capital in the 1
period model in 1965)
Three new issues:
1. Allocation in multiple person households
(bargaining?)
2. “Specialization” and division of labor in multiple
person households
3. Human capital in household production
9
Specialization:
Becker and his Critics
Time use theory is essentially Becker.
Becker’s claim in the Treatise: Efficiency requires
specialization, regardless of preferences or bargaining
power.
Further claim: wives in household, husbands in market.
Theoretical critiques of Becker on efficiency and
specialization:
Folbre (“A Theory of the Misallocation of Time,” in
Folbre and Bittman, 2004)
Lundberg (“Gender in Household Decision Making”
2008)
10
Facts and Theory
In the light of the facts about labor force participation
and about housework cited above, the theoretical
claim that efficiency requires specialization is (or
should be) an embarrassment, unless we accept that
many married couple households are inefficient.
Does the economic theory of time allocation really imply
specialization conclusions that are not consistent with
the data?
What does Becker actually say about specialization?
11
Theory: Becker (1981) - 1
Treatise on the Family: ”Theorem 2.1. If all
members of an efficient household have
different comparative advantages, no more
than one member would allocate time to both
the market and household sectors. Everyone
with a greater comparative advantage in the
market than this member's would specialize
completely in the market, and everyone with
a greater comparative advantage in the
household would specialize completely there."
12
Theory: Becker (1981) - 2
Treatise on the Family: "Theorem 2.3. At most
one member of an efficient household would
invest in both market and household capital
and would allocate time to both sectors.”
(This is complete statement of Theorem 2.3.)
13
Theory: Becker (1981) - 3
Treatise on the Family: "Theorem 2.4. If
commodity production functions have
constant or increasing returns to scale, all
members of efficient households would
specialize completely in the market or
household sectors and would invest only in
market or household capital.”
This sounds like a result from trade theory, but
I will argue that the trade theory analogy is
seriously flawed.
14
Perfect Substitutes - 1
Treatise on the Family: Chapter 2 (p. 32), "Since all
persons are assumed to be intrinsically identical, they
supply basically the same kind of time to the
household and market sectors. Therefore, the
effective time of different members would be perfect
substitutes, even if they accumulate different
amounts of household capital..." (italics in original;
underline added for emphasis).
Perfect substitutes is a distraction - a red herring.
Notice that it is NOT mentioned in any of the
specialization theorems.
15
Perfect Substitutes - 2
Specialization is plausible if we assume perfect
substitutes.
In fact, it is more than plausible: efficiency and perfect
substitutes assumption imply specialization. No
additional assumptions are necessary. None of
Becker’s 5 specialization theorems assumes perfect
substitutes; if they, additional assumptions would be
unnecessary.
How plausible is the perfect substitutes assumption?
16
Perfect Substitutes - 3
The perfect substitutes assumption is a highly
restrictive, ad hoc assumption to which economics
has no commitment.
“Therefore” is a problem: “intrinsically identical” does
not imply “perfect substitutes.” For example, if
spouses have identical Cobb-Douglas or CES
production functions, their time inputs are NOT
perfect substitutes unless output is linear in the time
inputs.
17
Where do the Specialization
Results Come from?
"Pure economics has a remarkable way of
producing rabbits out of a hat -apparently a priori propositions which
apparently refer to reality. It is
fascinating to try to discover how the
rabbits got in; for those of us who do
not believe in magic must be convinced
that they got in somehow." J. R. Hicks,
Value and Capital, 1939
18
A Very Brief Digression:
Three Omitted Topics
1. Joint Production and Process Preferences: Pollak and
Wachter (JPE 1975). I assume that individuals care
about the nominal outputs of home production (a
clean house; a home cooked meal) but not about
how they spend their time (cleaning; cooking).
2. Leisure: Gronau (JPE 1977) extends Becker (1965) to
include leisure. I ignore leisure.
3. Human Capital: Becker (1981) emphasizes human
capital in the Treatise, although it does not play an
explicit role in the hypotheses of the specialization
theorems. More about human capital later.
19
Meanings of Specialization
With two sectors (commodities, activities)
1. Strong (complete) specialization: each spouse
allocates time to only one sector
2. Weak (partial) specialization: one spouse allocates
time to one sector, the other spouse allocates time to
one sector or to both sectors
3. Nonspecialization: both spouses allocate time to
both sectors
How do we define specialization when there are more
than two activities? More about this later.
20
Does Efficiency in Production
Imply Specialization?
Becker’s claim: Efficiency in production requires
specialization, regardless of preferences or bargaining
power.
Becker’s assumes two “sectors,” a market sector and a
household sector.
I begin by showing that the specialization conclusion
does not hold when there are two household sectors
(e.g., hunting and gathering) and no market sector.
I then use this result to show that the specialization
conclusion does not hold with a market sector and a
household sector. (Then I introduce human capital).
21
Comment: Different
Comparative Advantages - 1
The hypothesis: “If all members of a household have
different comparative advantages…”
This hypothesis is easily misinterpreted as an
assumption about the household technology. It is
not.
Except in very special cases, different comparative
advantages is an hypothesis about (efficient) time
allocation within the household.
Efficiency production with both spouses allocating time
to both activities requires equal comparative
advantages.
22
Comment : Different
Comparative Advantages - 2
So the theorem says: “If we don’t have an interior
solution (ie., both spouses allocating time to both
sectors), then we have a corner solution” (at least
one spouse does not allocate time to both sectors).
This is not a technical criticism. Theorems are supposed
to be tautologies.
But if you thought that “equal comparative advantages”
was extremely unlikely -- perhaps a set of measure 0
– then you misunderstood the hypothesis of the
theorem.
23
Comment: Constant or
Increasing Returns to Scale
The hypothesis: “If commodity production functions
have constant or increasing returns to scale…”
In economics, the usual returns to scale assumption is
that production functions are concave, which implies
constant or decreasing returns to scale.
Even if we assume increasing returns to scale, the
specialization conclusion does not follow unless we
also impose an “additivity assumption.”
The “additivity assumption” is plausible in the context of
international trade, but not in the context of
household production.
24
Notation: Production Functions
in Multiple-Person Households
z = g[th,tw,y], where th and tw denote the husband’s and
wife’s time inputs into the production of z.
Define the “unilateral household production functions”
g[th,0,y] and g[0, tw,y].
WEA: Weak Essentiality Assumption: no output without
some strictly positive time input: g[0,0,y] = 0.
SEA: Strong Essentially Assumption: if th tw = 0, then
g[th,tw,y] = 0.
Introducing notation for the activity or sector:
zi = gi[thi,twi,y],
25
Examples
All examples assume output is produced by time alone.
Strong Essentiality:
Cobb-Douglas g[th,tw] = A (t1)σ1(t2)σ2
Fixed-coefficient g[th,tw] = [min {thi, twi}]σi
(And CES cases between)
Weak essentiality:
Additive case: g[th,tw] = Ah (th)σh + Aw (tw)σw
26
No Joint Production
NJP: No Joint Production: Each activity produces one
and only one output.
“Joint production” means that an activity produces two
or more “commodities” (outputs). Following Becker,
assume no joint production. No joint production
implies a one-to-one correspondence between
activities and commodities. Each activity has its own
production function, so we don’t have to use
production sets to characterize technology.
“No joint production” rules out process preferences and
economies of scope (important with more activities).
27
Perfect Substitutes
Assumption (PSA)
PSA: Perfect Substitutes Assumption
g[th,tw,y]= G[th+αtw,y]
where α converts the time input of spouse w into
units comparable to the time input of spouse h. Thus
[th + αtw]
is the total time input into the production of the focal
commodity, measured in "efficiency units."
The perfect substitutes assumption is a boundary case:
linear isoquants in the (th,tw) subspace.
Different efficiency factors for different activities.
The PSA leads directly to specialization, without any
additional assumptions.
28
Increasing Returns to Scale Does
Not Imply Specialization
Example: If two activities have Cobb-Douglas or fixed
coefficient technologies with increasing returns
g[th,tw] =[min {thi, twi}]σi
i > 1
g[th,tw] = A (t1)σ1(t2)σ2
then positive output of both commodities is possible
only with nonspecialization: both spouses allocate
time to both activities.
The nonspecialization conclusion holds for the
intermediate CES technologies.
The specialization requires not only increasing returns
to scale but also an additivity assumption.
29
The Additivity Assumption
(AA) with No Market Inputs
Suppose output is produced by time alone; no market
inputs
AA: The household production function satisfies the
additivity assumption if the household output from
the input vector (thi,twi) is the sum of the outputs
they would realize with unilateral production):
gi[thi,twi] = gi[thi,0] + gi[0, twi]
The additivity assumption means there are no positive
or negative “spillovers” (within household
externalities) associated with bilateral production in
the household. The AA is a strong restriction.
30
The Additivity Assumption
with Market Inputs
AA: The household production function satisfies the
additivity assumption if the household output from
the input vector (thi,twi,y) is the maximum of the sum
of the outputs they would realize with unilateral
production):
gi[thi, twi, yi] = max {gi(thi,0,yhi) + gi(0,twi,ywi)}
subject to yhi + ywi  yi.
This assumes that market inputs are allocated to
maximize total output. It applies regardless of
whether inputs are household private goods or
household public goods. Inputs that cannot be
reallocated between spouses cause no problem
because they can be absorbed into the unilateral
production functions.
31
The Additivity Assumption and
Increasing Returns to Scale
A version of the additivity assumption is standard in
international economics: if both countries produce
the same good, each uses its own technology and
world output is the sum of their separate outputs.
With the additivity assumption, increasing RTS lead
to specialization.
The additivity assumption is plausible for international
trade but not for households.
Spouses may be able to do better than using their
unilateral technologies by dividing activities into
subactivities and specializing. (Or they may get in
each other’s way and do worse: “Too many cooks…) 32
The Market Sector - 1
Intuition
Suppose there is a market sector and a household
sector, and that the spouses have equal wage rates
and identical production functions with decreasing
returns to scale satisfying the property that as time
inputs go to 0, marginal products go to infinity. Then
efficiency requires that both spouses allocate time to
both the market sector and the household sector.
Furthermore, the spouses allocate the same amount
of time of the market sector (th1= tw1) and the same
amount of time to the household sector (th2 = tw2).
That is, the only efficient allocations are symmetric.
33
The Market Sector - 2
Intuition
Now suppose that we perturb the spouses wage rates
so they are slightly different: does efficiency now
require specialization?
Because the production functions satisfy the property
that as time inputs go to 0, marginal products go to
infinity, both spouses allocate time to the hh sector.
IF there is specialization, it must mean that the spouse
with the lower wage rate now allocates 0 time to the
market and works only in the hh sector. Despite
decreasing RTS, the low wage spouse increases the
time allocated to hh production enough to maintain
production of the household commodity.
34
Can Human Capital Save the
Specialization Theorems?
Clearly not unless we rule out strong essentiality (rule
out Cobb-Douglas, fixed-coefficient, CES between).
In the discussion of specialization in the Treatise,
Becker assumes that
(1) there are two kinds of human capital: market
human capital and household human capital.
(2) household human capital is “labor augmenting” and
(3) sector specific human capital is essential for
production -- a sector specific human capital
essentiality assumption.
35
Economies of Scope
"One common example of economies of scope is child
care and house-based chores: many chores can be
completed while at the same time attending to a
child” Fafchamps and Quisumbling (Handbook of
Development Economics, 2008, p. 3198).
A plausible story about economies of scope requires
more than one commodity in the household sector.
Technical point: to represent technology exhibiting
economies of scope, we need production sets rather
than production functions.
36
Toward a New New Home Economics
Primitives in the New NHE - 1
Four components:
1. Preferences,
2. Constraints/ technology,
3. Governance structure (e.g., Nash
bargaining)
4. Transaction costs
37
Toward a New New Home
Economics: Primitives in the New
NHE - 2
1. Preferences
Individuals' utility functions
2. Constraints/ opportunities
Budget constraint; time constraints
Technology
Individuals’ technologies and
household technology
Production functions
3. Governance structure
e.g., altruist model; Nash bargaining
4. Transaction costs (e.g., coordination costs).
38
Preferences
Preferences (utility functions) for both spouses.
Preferences for market goods and home produced
commodities (home cooked meal; clean house).
Following Becker, assume there are no “process
preferences.”
With process preferences, people care not only about
home cooked meals and a clean house, but also care
whether they spend time cooking or cleaning.
Any specialization/ unilateral production conclusion
depends on assuming away process preferences.
39
When Do We Need Individuals’
Technologies?
1. Single-person households are intrinsically
interesting
2. Marriage market: Compare well-being when
single to well-being in particular potential
marriage
3. Divorce: Compare well-being in current
marriage to well-being if divorced
4. Bargaining within marriage
Divorce as threat point in some models
Divorce as outside option in most models
40
Individuals’ Technologies
Single-person household are interesting for their own
sake and because their technologies shed light on
time allocation in multiple-person household.
Recognizing that individuals’ technologies play a role in
multiple-person households is new.
My original title was: Allocating Time:
Individuals’ Technologies, Household Technology,
Specialization, and the Household Division of Labor.
Empirical literature has overemphasized time allocation
in multiple-person households and under emphasized
time allocation in single-person households.
41
Governance Structure
Examples of alternative governance structures:
1. Becker’s altruist model (One spouse as
husband-father-dictator-patriarch who makes
the decisions.)
2. Nash bargaining
3. Other cooperative and noncooperative
bargaining models (e.g., repeated games,
two stage games)
4. Chiappori’s “collective model” as a reduced
form.
42
Transaction Costs
Coordination
Asymmetric information and monitoring.
Becker devotes a section of Chapter 2 to “Shirking,
Household Size, and the Division of Labor” in which
he discusses “[s]hirking, pilfering, or other
malfeasance”
Why does the spouse who cares most about how a
particular task is done wind up doing it herself?
Without asymmetric information and monitoring
costs, which spouse does a task is independent of
which spouse wants it done.
43
When Do Assumptions about
Technology Imply Conclusions about
Specialization/ Unilateral Production?
It takes very strong assumptions about
technology to imply conclusions about
“specialization” or “unilateral production” that
hold for all possible assumptions about
preferences, governance structures,
transaction costs and market wage rates.
44
Production Functions in
Single-Person Households
In a single-person household, denote the household
production function for the commodity z by
z = f(t,y)
where t denotes the input of the individuals’ time into
its production and y the vector of market inputs.
EA: Essentiality Assumption: No output without strictly
positive input of individual’s time: f (0,y)=0
Denote the individual production functions of the
spouses by f1(t1,y1) and f2(t2,y2).
(Superscripts and subscripts refer to the spouses, not
to the commodity, z.)
45
Correspondence Assumption (CA)
CA: Unilateral household production functions
corresponds to individuals’ production functions
g[t1,0,y] = f1(t1, y) and g[0, t2,y] =f2(t2, y).
That is,
(1) If we know the individuals’ production function,
then we can infer the unilateral household production
functions and
(2) If we know the unilateral household production
functions, then we can infer individuals’ production
functions.
The CA assumption bites when there is lots of unilateral
production.
46
Interpretations of Individuals’
Production Functions
Single-person households (one adult household).
Unilateral production in multiple-person households
(e.g., married couples).
The standard analysis assumes nothing about the
relationship between individuals’ technologies and
household technology: immaculate conception of
household technology.
CA is an assumption about the household production
function at the time of household formation, at least
about portion corresponding to unilateral production.
CA is also an assumption about the genesis of the
individual production functions following divorce.
Recall: we need individual production functions to talk
about marriage, divorce, and household bargaining. 47
Implications of Correspondence
Assumption and the Perfect
Substitutes Assumption
The CA and the PSA together imply that the
individuals' production functions are identical
to each other except for the "efficiency
factor,” a
f1[at, y] = f2[t, y].
An alternative and more transparent gender
neutrality assumption is that all individuals
have identical or proportional individual
production functions:
a f1[t, y] = f2[t, y]
48
Toward the
Additivity Assumption
Is bilateral production plausible? To assess this, I want
to look at some examples.
"The theory of permutations, like everything else, is
best understood by staring hard at some non-trivial
examples." Halmos, Finite-Dimensional Vector
Spaces, (1958, p 42)
I want a procedure for generating nontrivial examples.
The additivity assumption does this. It is also the
additional assumption needed for the validity of the
claim that constant or increasing returns to scale
implies specialization.
49
The Additivity Assumption and
the Correspondence Assumption
The AA and the CA imply
g[t1, t2, y] = max {f1(t1,y1) + f2(t2,y2)}
subject to y1 + y2  y.
The AA is consistent with (but does not require)
bilateral production - could have unilateral production
Why would an efficient household choose bilateral
rather than unilateral production?
Perhaps because they become tired or bored and, as a
result, become less productive: medical interns and
residents; airline pilots; truck drivers
50
Decreasing Returns to Scale
The example shows that with additivity, when
the unilateral production functions exhibit
decreasing returns to scale (e.g., because
individuals become less productive when they
become tired or bored), then efficiency may
require bilateral production and the
“specialization” conclusion need not hold.
.
51
How Rabbits Got In - 1
Different comparative advantages…
For a wide class of assumptions about technology,
production efficiency implies equal comparative
advantages.
Different comparative advantages rules out interior
solutions (i.e., those in which both spouses allocating
time to the production of both goods). Hence,
different comparative advantages implies
specialization: a corner solution.
Unless marginal products are constant, unequal
comparative advantages is not simply an assumption
about technology.
52
How Rabbits Got In - 2
Constant or Increasing Returns to Scale:
Concavity is the usual assumption in economics.
The assumption of constant or increasing returns to
scale rules out the possibility that an individual who
devotes more time to an activity becomes less
productive (e.g., as a result of fatigue or boredom). If
both spouses experience reduced productivity due to
fatigue or boredom and the household technology
satisfy the AA, then efficiency may require bilateral
production rather than unilateral production.
53
How Rabbits Got In - 3
Constant or increasing returns to scale…
Without the additivity assumption, constant or
increasing RTS do not imply specialization.
For example, consider the household production
function
z = [min {t1, t2}]σ
 > 1.
To get the specialization conclusion requires imposing
the additivity assumption.
The additivity assumption is why increasing returns
implies specialization in international trade.
54
How Rabbits Got In - 4
Perfect substitutes…
If the spouses time imputs are perfect substitues, then
efficiency in production implies specialization without
any additional assumptions.
55
How Rabbits Got In - 5
Both theorems assume that there are only two
"sectors"-- home and market.
This assumption is crucial for Becker's conclusion about
the efficiency of wives specializing in the home and
husbands in the market. Lundberg (2008)
If there are m household commodities then, for
households in which both husbands and wives
participate in the market, Becker's reasoning implies
that husbands specialize in the production of m* of
these home-produced commodities and the wives in the
production of the remaining m-m* commodities.
56
Meaning of Activity/Commodity:
Specialization and Unilateral
Production.
Specialization by sector
(home vs. market)
Specialization by activity
(e.g., cooking vs. cleaning)
What is an activity? How many activities?
cooking?
grilling?
grilling fish?
grilling salmon?
57
Different Comparative
Advantages and Specialization
The claim that different comparative
advantages implies specialization or unilateral
production is correct.
But in many plausible cases, production
efficiency requires equal comparative
advantages.
Different comparative advantages implies no
interior solution and, hence, a corner
solution.
Different comparative advantages is not simply
an assumption about household technology.
58
Perfect Substitutes and
Specialization
The assumption that spouses’ time inputs are
perfect substitutes in production implies
specialization or unilateral production, but the
perfect substitutes assumption is difficult to
motivate.
In conjunction with the AA, the PSA implies
CRS, ruling out cases in which fatigue or
boredom reduce productivity.
59
The Meaning of Specialization
What do we mean by specialization?
Unilateral vs bilateral production
sector vs. activity
How many commodities?
How many activities?
Lundberg (2008)
60
Individuals' and Household
Technology
We need individuals’ technologies as well as household
technology.
Household formation:
marriage or cohabitation
immaculate conception of household technology
Household dissolution: actual divorce
Household bargaining: threat point or outside option
Study individual production functions, not just
household production functions.
Relationship between individuals’ technologies and
household technology: the correspondence
assumption
61
Explanations of Specialization
Not Based on Technology
Preferences/Norms:
Process preferences
Gender Norms
“Doing Gender”; “Gender Display”; “Identity”
Akerlof and Kranton Identity Economics (2010)
Governance structure and bargaining power
altruist model
binding agreements in the marriage market
Nash bargaining within marriage
Transaction Costs
Inability to make binding commitments
Asymmetric information and monitoring
Coordination and monitoring
62