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Allocating Time: Does Economic Theory Imply Specialization in Efficient Married Couple Households?

Robert A. PollakWashington University in St.

LouisMay 17, 2011

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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.

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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.

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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)

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Still More Facts:Labor Force Particiaption

2008 Labor Force Participation rates for

Married men Married women 25-34 95.3% 69.5%

Age 35-44 95.2% 73.8%

US data: CPS

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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.

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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 FamilyBecker (1981, 1991) differs from Becker

(1965) and from Michael and Becker (1973)

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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

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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

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Specialization: Becker and his CriticsTime 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)

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Facts and TheoryIn 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?

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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."

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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.)

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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.

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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.

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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?

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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.

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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

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A Very Brief Digression:Three Omitted Topics1. 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.

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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.

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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).

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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.

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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.

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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.

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Notation: Production Functions in Multiple-Person Householdsz = 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],

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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

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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).

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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.

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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.

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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.

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The Additivity Assumption with Market InputsAA: 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.

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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…)

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The Market Sector - 1Intuition

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.

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The Market Sector - 2Intuition

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.

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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.

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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.

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Toward a New New Home EconomicsPrimitives in the New NHE - 1

Four components: 1. Preferences, 2. Constraints/ technology, 3. Governance structure (e.g., Nash

bargaining) 4. Transaction costs

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Toward a New New Home Economics: Primitives in the New NHE - 2

1. PreferencesIndividuals' utility functions

2. Constraints/ opportunities Budget constraint; time constraints Technology

Individuals’ technologies and household technology

Production functions3. Governance structure

e.g., altruist model; Nash bargaining4. Transaction costs (e.g., coordination costs).

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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.

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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 marriageDivorce as threat point in some modelsDivorce as outside option in most models

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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.

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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 bargaining3. Other cooperative and noncooperative bargaining models (e.g., repeated games, two stage games)4. Chiappori’s “collective model” as a reduced form.

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Transaction CostsCoordination 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.

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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.

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Production Functions in Single-Person HouseholdsIn 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.)

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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.

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Interpretations of Individuals’ Production FunctionsSingle-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.

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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]

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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.

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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

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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.

.

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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.

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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.

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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.

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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.

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How Rabbits Got In - 5Both 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.

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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?

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Different Comparative Advantages and SpecializationThe 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.

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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.

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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)

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Individuals' and Household TechnologyWe need individuals’ technologies as well as household

technology.Household formation:

marriage or cohabitationimmaculate conception of household technology

Household dissolution: actual divorceHousehold 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

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Explanations of Specialization Not Based on TechnologyPreferences/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 marketNash bargaining within marriage

Transaction Costs Inability to make binding commitments Asymmetric information and monitoring Coordination and monitoring