Royal Economic Society Gary Beckers "A Theory of the Allocation - - PowerPoint PPT Presentation

Royal Economic Society

Gary Becker’s "A Theory of the Allocation of Time"

Royal Economic Society

Arthur Lewbel

Boston College

March 2015

Setting the table: Before "Treatise on the Family" (1981, 1991), before "Theory of Social Interactions (1974)," Gary Becker (1965) "A Theory of the Allocation of Time." The Economic Journal, 75(299) 493-517. Goal was to provide, "a basic theoretical analysis of choice that includes the cost of time on the same footing as the cost of market goods" Economists before him accounted for foregone earnings from time for human capital investment but, "economists have not been equally sophisticated about other non-working uses of time" (p. 493-94).

Example Precursors: Mincer (1962), considered a married woman’s time trade-off between housework and paid work. Gorman (1956, not cited), proposed and analyzed a household production function (but not with time). Becker’s models are now THE foundational modeling framework for household level analyses of consumption and time use. So natural it’s hard to believe it had to be invented.

Becker’s framework: Utility function U (Z1,...,Zm). Each commodity Zi produced by a household production function

Zi = fi (xi, Ti).

Each xi is a bundle of goods purchased at the vector of prices pi. Each Ti is a bundle of time use quantities, at vector of prices wi. Time use and purchased goods create commodities (home production), commodities produce utility, maximized under an overall budget constraint.

Time for Becker Becker noted full income S is easily calculated and interpreted when there is only a single wage rate that doesn’t depend on T. This is the standard modeling assumption today, but Becker said this case was "special and unlikely," and did not impose it. Becker thought of time as having different prices at, say daytime vs nighttime or weekends vs weekdays, rather than a single wage rate. For Becker, S is defined by maximizing an "earnings" function

W (Z1,...,Zm) subject to the single budget constraint and to the

production functions for each commodity. Marginal costs, which determine behavior, need not equal the average time costs wi.

With multiple consumption goods, and multiple types of time, still get two stage decomposition:

• 1. Calculate S.
• 2. Maximize household utility U (f1 (x1, T1) ,...,fm (xm, Tm)) under

∑m

i=1 p ixi + w i Ti ≤ S.

Key insight: There are not two different constraints for time and

• money. There is only a single budget constraint!

Many implications follow from there being just one constraint.

Becker doesn’t identify or estimate the model, but draws implications (casual empiricism). Examples:

• 1. Must consider shadow cost of time as a cost of commuting to

work.

• 2. As wages rise, people waste more food to save on shopping and

food prep time.

• 3. Variation in time use price (e.g. wage rates) across households

induces variation in the shadow price of goods. Implication: Engel curves underestimate the true income effects of earnings-intensive goods, like child care, could help low income elasticity of fertility.

• 4. Household specialization of labor (See Pollak).

Following Becker (1965) Dynamic, forward looking optimization. Combine with Becker’s (1974), "A Theory of Social Interactions." A collective household model instead of his unitary model (though still just one household budget constraint!). Power now becomes relevant. See Heckman (2015) for many more.

One strand of literature: identification and estimation? What household level data is observable? What is needed? (A single wage per person really helps). Pareto efficient household and distribution factors (Chiappori with many coathors including Browning, Ekeland,...) Revealed Preference bounds on household resource shares (Vermeulen, Cherchye, De Rock,...) Restrictions to identify household resource shares, including children as people instead of just public goods (Lewbel, Pendakur,...)

Becker’s approach to family economics: mainstream now, but revolutionary then. Many were openly hostile, calling his model sterile, vacuous, cold, and immoral.

Couple’s time combines with purchased goods to jointly create household utility. Likewise, in the last 50 years, our time has combined with Becker’s models to jointly create enormous social utility.

Royal Economic Society

Domestic production and matching

Economic Journal Anniversary Sessions - Becker 1965 Pierre-André Chiappori

Columbia University

Manchester, March 2015

Becker on Matching

Seminal, 1973 JPE paper: “Yet, one type of behavior has been almost completely ignored by economists, although scarce resources are used and it has been followed in some form by practically all adults in every recorded society. I refer to marriage.” Becker concludes: “Therefore, the neglect of marriage by economists is either a major oversight or persuasive evidence of the limited scope of economic analysis.” (Ibid.)

Becker on Matching (cont.)

Main insights: Marital choices as rational decisions → the economic approach is relevant Household as a small economy, with domestic production → reference to the 65, EJ paper Men and women compete between them for a spouse; the outcome of these interactions is an equilibrium. Consequences: Marital sorting — who marries whom — has an important, economic component Depends on ‘complementarity’ or ‘substituability’ of male and female traits within the household production function The intra-household allocation of resources determined by the equilibrium prevailing on the ‘marriage market’

Becker’s framework: emphasis on domestic production

Individuals exclusively consume commodities that have been internally produced.

Becker’s framework: emphasis on domestic production

Individuals exclusively consume commodities that have been internally produced. ‘All commodities can be combined into a single aggregate Z’

Becker’s framework: emphasis on domestic production

Individuals exclusively consume commodities that have been internally produced. ‘All commodities can be combined into a single aggregate Z’ ‘Our concentration on the output and distribution of Z does not presuppose transferable utilities, the same preference function for different members of the same household, or other special assumptions about preferences’

Becker’s framework: emphasis on domestic production

Individuals exclusively consume commodities that have been internally produced. ‘All commodities can be combined into a single aggregate Z’ ‘Our concentration on the output and distribution of Z does not presuppose transferable utilities, the same preference function for different members of the same household, or other special assumptions about preferences’ Individual traits are ‘complement’ if the ‘marginal productivity’ of one spouse’s trait increase with the partner’s

Becker’s framework: emphasis on domestic production

Individuals exclusively consume commodities that have been internally produced. ‘All commodities can be combined into a single aggregate Z’ ‘Our concentration on the output and distribution of Z does not presuppose transferable utilities, the same preference function for different members of the same household, or other special assumptions about preferences’ Individual traits are ‘complement’ if the ‘marginal productivity’ of one spouse’s trait increase with the partner’s Modern versions consider a more general framework

Is matching assortative on Human Capital?

Becker’s claim: Negative Assortative Matching (NAM) ‘... the correlation between mates for wage rates or for traits of men and women that are close substitutes in household production will tend to be negative.’

Is matching assortative on Human Capital?

Becker’s claim: Negative Assortative Matching (NAM) ‘... the correlation between mates for wage rates or for traits of men and women that are close substitutes in household production will tend to be negative.’ Argument:

Is matching assortative on Human Capital?

Becker’s claim: Negative Assortative Matching (NAM) ‘... the correlation between mates for wage rates or for traits of men and women that are close substitutes in household production will tend to be negative.’ Argument:

Positive or Negative Assortative Matching (PAM or NAM): are traits complement or substitute?

Is matching assortative on Human Capital?

Becker’s claim: Negative Assortative Matching (NAM) ‘... the correlation between mates for wage rates or for traits of men and women that are close substitutes in household production will tend to be negative.’ Argument:

Positive or Negative Assortative Matching (PAM or NAM): are traits complement or substitute? EJ 65:

Is matching assortative on Human Capital?

Becker’s claim: Negative Assortative Matching (NAM) ‘... the correlation between mates for wage rates or for traits of men and women that are close substitutes in household production will tend to be negative.’ Argument:

Positive or Negative Assortative Matching (PAM or NAM): are traits complement or substitute? EJ 65:

crucial inputs are husband’s and wife’s time spent in domestic production

Is matching assortative on Human Capital?

Becker’s claim: Negative Assortative Matching (NAM) ‘... the correlation between mates for wage rates or for traits of men and women that are close substitutes in household production will tend to be negative.’ Argument:

Positive or Negative Assortative Matching (PAM or NAM): are traits complement or substitute? EJ 65:

crucial inputs are husband’s and wife’s time spent in domestic production High wage means that time input is costly ...

Is matching assortative on Human Capital?

Becker’s claim: Negative Assortative Matching (NAM) ‘... the correlation between mates for wage rates or for traits of men and women that are close substitutes in household production will tend to be negative.’ Argument:

Positive or Negative Assortative Matching (PAM or NAM): are traits complement or substitute? EJ 65:

crucial inputs are husband’s and wife’s time spent in domestic production High wage means that time input is costly ...

→ therefore efficient to match with a partner whose time is cheap

Is matching assortative on Human Capital?

Becker’s claim: Negative Assortative Matching (NAM) ‘... the correlation between mates for wage rates or for traits of men and women that are close substitutes in household production will tend to be negative.’ Argument:

Positive or Negative Assortative Matching (PAM or NAM): are traits complement or substitute? EJ 65:

crucial inputs are husband’s and wife’s time spent in domestic production High wage means that time input is costly ...

→ therefore efficient to match with a partner whose time is cheap Note that this is not the specialization logic (substituability between time inputs in the production function, see Pollak 2013)

Is matching assortative on Human Capital?

Becker’s claim: Negative Assortative Matching (NAM) ‘... the correlation between mates for wage rates or for traits of men and women that are close substitutes in household production will tend to be negative.’ Argument:

Positive or Negative Assortative Matching (PAM or NAM): are traits complement or substitute? EJ 65:

crucial inputs are husband’s and wife’s time spent in domestic production High wage means that time input is costly ...

→ therefore efficient to match with a partner whose time is cheap Note that this is not the specialization logic (substituability between time inputs in the production function, see Pollak 2013)

Problem: counterfactual

Is matching assortative on Human Capital?

Becker’s claim: Negative Assortative Matching (NAM) ‘... the correlation between mates for wage rates or for traits of men and women that are close substitutes in household production will tend to be negative.’ Argument:

Positive or Negative Assortative Matching (PAM or NAM): are traits complement or substitute? EJ 65:

crucial inputs are husband’s and wife’s time spent in domestic production High wage means that time input is costly ...

→ therefore efficient to match with a partner whose time is cheap Note that this is not the specialization logic (substituability between time inputs in the production function, see Pollak 2013)

Problem: counterfactual

PAM even 50 years ago

Is matching assortative on Human Capital?

Becker’s claim: Negative Assortative Matching (NAM) ‘... the correlation between mates for wage rates or for traits of men and women that are close substitutes in household production will tend to be negative.’ Argument:

Positive or Negative Assortative Matching (PAM or NAM): are traits complement or substitute? EJ 65:

crucial inputs are husband’s and wife’s time spent in domestic production High wage means that time input is costly ...

→ therefore efficient to match with a partner whose time is cheap Note that this is not the specialization logic (substituability between time inputs in the production function, see Pollak 2013)

Problem: counterfactual

PAM even 50 years ago In particular, educated women do not marry uneducated husbands

A simple example

CD preferences: Ui = CiQ with Q = (t1)α1 (t2)α2

A simple example

CD preferences: Ui = CiQ with Q = (t1)α1 (t2)α2 Budget constraint: p (C1 + C2) = w1 (1 − t1) + w2 (1 − t2) with wi = HiW

A simple example

CD preferences: Ui = CiQ with Q = (t1)α1 (t2)α2 Budget constraint: p (C1 + C2) = w1 (1 − t1) + w2 (1 − t2) with wi = HiW Transferable utility

A simple example

CD preferences: Ui = CiQ with Q = (t1)α1 (t2)α2 Budget constraint: p (C1 + C2) = w1 (1 − t1) + w2 (1 − t2) with wi = HiW Transferable utility

→ surplus: S (H1, H2) = W p max

t1,t2 (H1 (1 − t1) + H2 (1 − t2)) (t1)α1 (t2)α2

= W p αα1

1 αα2 2

(α1 + α2 + 1)1+α1+α2 (H1 + H2)1+α1+α2 (H1)α1 (H2)α2

A simple example

CD preferences: Ui = CiQ with Q = (t1)α1 (t2)α2 Budget constraint: p (C1 + C2) = w1 (1 − t1) + w2 (1 − t2) with wi = HiW Transferable utility

→ surplus: S (H1, H2) = W p max

t1,t2 (H1 (1 − t1) + H2 (1 − t2)) (t1)α1 (t2)α2

= W p αα1

1 αα2 2

(α1 + α2 + 1)1+α1+α2 (H1 + H2)1+α1+α2 (H1)α1 (H2)α2 In particular, second cross derivative: − (H1 + H2)α1+α2−1 (α1H2 − α2H1)2 + α1H2

2 + α2H2 1

• Hα1+1

1

Hα2+1

2

< 0 ⇒ NAM!

A possible solution to the puzzle

Keep Becker’s insights:

A possible solution to the puzzle

Keep Becker’s insights:

Domestic production, domestic times as inputs

A possible solution to the puzzle

Keep Becker’s insights:

Domestic production, domestic times as inputs PAM if complementarities in traits

A possible solution to the puzzle

Keep Becker’s insights:

Domestic production, domestic times as inputs PAM if complementarities in traits Individuals characterized by their HC

A possible solution to the puzzle

Keep Becker’s insights:

Domestic production, domestic times as inputs PAM if complementarities in traits Individuals characterized by their HC

Additional ingredient: HC as an input in domestic production process → obvious justification: children

A possible solution to the puzzle

Keep Becker’s insights:

Domestic production, domestic times as inputs PAM if complementarities in traits Individuals characterized by their HC

Additional ingredient: HC as an input in domestic production process → obvious justification: children Then two opposite forces:

A possible solution to the puzzle

Keep Becker’s insights:

Domestic production, domestic times as inputs PAM if complementarities in traits Individuals characterized by their HC

Additional ingredient: HC as an input in domestic production process → obvious justification: children Then two opposite forces:

Educated spouse’s time is more costly ...

A possible solution to the puzzle

Keep Becker’s insights:

Domestic production, domestic times as inputs PAM if complementarities in traits Individuals characterized by their HC

Additional ingredient: HC as an input in domestic production process → obvious justification: children Then two opposite forces:

Educated spouse’s time is more costly ... ... but also more productive

A simple example (CCM 2015)

CD preferences: Ui = CiQ with Q = (H1t1)α1 (H2t2)α2

A simple example (CCM 2015)

CD preferences: Ui = CiQ with Q = (H1t1)α1 (H2t2)α2 Budget constraint: p (C1 + C2) = w1 (1 − t1) + w2 (1 − t2) with wi = HiW

A simple example (CCM 2015)

CD preferences: Ui = CiQ with Q = (H1t1)α1 (H2t2)α2 Budget constraint: p (C1 + C2) = w1 (1 − t1) + w2 (1 − t2) with wi = HiW Transferable utility → surplus: S (H1, H2) = W p max

t1,t2 (H1 (1 − t1) + H2 (1 − t2)) (H1t1)α1 (H2t2)α2

= W p αα1

1 αα2 2

(α1 + α2 + 1)1+α1+α2 (H1 + H2)1+α1+α2

A simple example (CCM 2015)

CD preferences: Ui = CiQ with Q = (H1t1)α1 (H2t2)α2 Budget constraint: p (C1 + C2) = w1 (1 − t1) + w2 (1 − t2) with wi = HiW Transferable utility → surplus: S (H1, H2) = W p max

t1,t2 (H1 (1 − t1) + H2 (1 − t2)) (H1t1)α1 (H2t2)α2

= W p αα1

1 αα2 2

(α1 + α2 + 1)1+α1+α2 (H1 + H2)1+α1+α2 In particular ∂2S (H1, H2) ∂H1∂H2 = KW p (H1 + H2)α1+α2−1 > 0

Conclusions

Technical point: complementarity between traits (matching) different from complementarities between inputs (specialization)

Conclusions

Technical point: complementarity between traits (matching) different from complementarities between inputs (specialization) Complementarities in Becker’s contributions: clearly 73 builds on 65

Conclusions

Technical point: complementarity between traits (matching) different from complementarities between inputs (specialization) Complementarities in Becker’s contributions: clearly 73 builds on 65 Domestic production function crucially important, especially regarding HC formation for children

Conclusions

Technical point: complementarity between traits (matching) different from complementarities between inputs (specialization) Complementarities in Becker’s contributions: clearly 73 builds on 65 Domestic production function crucially important, especially regarding HC formation for children

Growth

Conclusions

Technical point: complementarity between traits (matching) different from complementarities between inputs (specialization) Complementarities in Becker’s contributions: clearly 73 builds on 65 Domestic production function crucially important, especially regarding HC formation for children

Growth Inequality

Conclusions

Technical point: complementarity between traits (matching) different from complementarities between inputs (specialization) Complementarities in Becker’s contributions: clearly 73 builds on 65 Domestic production function crucially important, especially regarding HC formation for children

Growth Inequality ... etc.

Conclusions

Technical point: complementarity between traits (matching) different from complementarities between inputs (specialization) Complementarities in Becker’s contributions: clearly 73 builds on 65 Domestic production function crucially important, especially regarding HC formation for children

Growth Inequality ... etc.

Empirical implications?

Conclusions

Technical point: complementarity between traits (matching) different from complementarities between inputs (specialization) Complementarities in Becker’s contributions: clearly 73 builds on 65 Domestic production function crucially important, especially regarding HC formation for children

Growth Inequality ... etc.

Empirical implications?

the form of domestic production functions has a potentially crucial impact on matching

Conclusions

Technical point: complementarity between traits (matching) different from complementarities between inputs (specialization) Complementarities in Becker’s contributions: clearly 73 builds on 65 Domestic production function crucially important, especially regarding HC formation for children

Growth Inequality ... etc.

Empirical implications?

the form of domestic production functions has a potentially crucial impact on matching → conversely, observed matching patterns may tell us something about domestic production function

Conclusions

Technical point: complementarity between traits (matching) different from complementarities between inputs (specialization) Complementarities in Becker’s contributions: clearly 73 builds on 65 Domestic production function crucially important, especially regarding HC formation for children

Growth Inequality ... etc.

Empirical implications?

the form of domestic production functions has a potentially crucial impact on matching → conversely, observed matching patterns may tell us something about domestic production function

Still much to learn from considering several Beckerian insights jointly

Royal Economic Society

1

Allocating Household Time: When Does Efficiency Imply Specialization?

Robert A. Pollak Washington University in St. Louis and NBER March 31, 2015

2

What Does Economic Theory Teach About Time Allocation?

Does the economic theory of time use imply that efficiency requires specialization in multiple-person households (e.g., married couples; cohabiting couples)? This seems to be what Becker claims in the Treatise on the Family (1981; 1991). Why focus on Becker? Because there isn’t much subsequent theoretical work

• n time allocation.

(There is lots of empirical work.)

3

Roadmap

 Specialization  Toward a New New Home Economics:

Elements of a Theory of the Household

 Individuals’ Production Functions and

Household Production Function

4

Meaning of Specialization - 1

With two sectors (home and market)

• 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

5

Meaning of Specialization - 2

The sector specialization claim is not that husbands spend more time in market work than wives, and wives spend more time in household work than husbands. This is wrong for two reasons

• 1. It introduces a gendering that is not part of the

definition of specialization.

• 2. Sector specialization has a specific technical meaning
• - we never observe both spouses working in both

sectors.

6

Meaning of Sector Specialization - 3

The theoretical claim is that efficiency requires that at least one spouse allocates zero time to one sector or the other. The time allocations of married men and married women have become more similar over the last 50 years . Both spouses typically spend time in both market work and household work. In rich non- Catholic countries, total time allocated to work by married men and married women is about the same. Burda, Hamermesh, and Weil, “Total Work and Gender: Facts and Possible Explanations” (2012)

7

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)

8

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

9

Widespread Inefficiency?

If the economic theory of time use implied that efficiency required specialization in 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 would be evidence of widespread inefficiency. An additional claim: Becker makes a further claim that the efficient pattern of specialization is gendered, with wives specializing in the household and husbands in the market. I ignore this further claim about gendering.

10 10 10 10

The Household Production Model and the New Home Economics

Becker (Economic Journal, 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.“ Becker (1981, 1991) A Treatise on the Family Becker’s household production model remains the lens through which virtually all economists and many other social scientists view time allocation.

11 11 11

The Household Production Model

There is more than one version of the household production model. Becker (1981, 1991) differs from the earlier versions, Becker (1965) and Michael and Becker (1973) Multiple-person households in the Treatise

• vs. single-person households in Becker (1965)

Human capital: both market and household human capital in the Treatise

• vs. no human capital in Becker (1965) which

is a one period model.

12

New Issues with Multiple- Person Households

Allocation of goods, time and commodities. Alternative models of decision making in multiple- person households: Binding commitments in the marriage market Becker’s altruist model Bargaining within marriage Chiappori’s collective model as reduced form The allocation of goods, time and commodities may or may not correspond to “specialization”

13

The Theoretical Time Use Literature

Pollak and Wachter (1975) Gronau (1977) One person households No human capital

14

Specialization

Becker dominates time use theory in economics. Becker’s claim: efficiency implies sector specialization, regardless of preferences or bargaining power Becker’s further claim about gendering: husbands in the market; wives at home.

15 15 15

Facts and Theory

In the light of the facts about labor force participation and about housework cited above, the theoretical claim that efficiency implies sector specialization is (or should be) an embarrassment to economists, unless we accept that many married couple households are inefficient. I return to this later. What does Becker actually say about sector specialization?

16 16 16

Theory: Becker (1981, 1991) - 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."

17 17 17

Theory: Becker (1981, 1991) - 2

Treatise on the Family: "Theorem 2.3. At most

• ne 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.)

18 18 18

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 are NOT mentioned in any of the specialization theorems. But perfect substitutes are mentioned in the text and are crucial.

19

Perfect Substitutes - 2

With perfect substitutes, efficiency implies

• specialization. No additional assumptions are

necessary (well, only one -- the absence of “process preferences”) None of Becker’s specialization theorems explicitly assume perfect substitutes; if they did, additional assumptions would be unnecessary. The perfect substitutes assumption is a highly restrictive, ad hoc assumption to which economics has no commitment.

20 20 20 20

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

21 21 21

Three Omitted Topics

• 1. Human Capital
• 2. Joint Production and Process Preferences: Pollak and

Wachter (JPE 1975).

• 3. Leisure: Gronau (JPE 1977)

22 22 22

Does Efficiency in Production Imply Specialization?

Becker’s claim: Efficiency in production requires sector specialization, regardless of preferences or bargaining power. Becker’s assumes two “sectors,” a market sector and a household sector. Human capital appears to play a critical role in Becker’s specialization theorems I show that with perfect substitutes, Becker’s default assumption, the specialization results do not depend

• n human capital.

23 23 23

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 household technology. It is not. Except in very special cases, comparative advantage depends on the allocation of time within the household. Different comparative advantages is an hypothesis about (efficient) time allocation within the household. Efficient production with both spouses allocating time to both activities requires equal comparative

• advantages. Think about the first order conditions.

24 24

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

• ne spouse does not allocate time to both sectors).

This is not a technical criticism of the theorem. Theorems are supposed to be tautologies. But if you thought that “equal comparative advantages” was an hypothesis about household technology and extremely unlikely -- perhaps a set of measure 0 – then you misunderstood the hypothesis.

24

25

Many Commodities

Suppose there are many household commodities. Lundberg (2008) points out that if there are m household commodities then, for households in which both husbands and wives participate in the market, perfect substitutes 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. This may lead to activity specialization, but not sector specialization unless m* = 0 or m* = m. Economies of scope provide incentives for the same spouse to engage in all household activities.

26 26 26

Toward a New New Home Economics Primitives in the New NHE - 1

Four components:

• 1. Preferences
• 2. Constraints/ (including technology)
• 3. Governance structure (e.g., Becker’s altruist

model; cooperative Nash bargaining)

• 4. Information structure (leading to transaction

costs)

27 27 27 27

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) determines “distribution factors”

• 4. Information structure (transaction costs; coordination

costs).

28 28 28

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 whether they spend time cooking or cleaning. Any specialization/ unilateral production conclusion depends on assuming that process preferences are absent or too weak to upset sector specialization.

29 29 29 29

Why We Need Individuals’ Technologies as well as Household Technology

• 1. Single-person (one adult) households are

intrinsically interesting

• 2. Marriage market: Compare well-being when

single with well-being in particular potential marriage

• 3. Divorce: Compare well-being in current

marriage with well-being if divorced

• 4. Allocation within marriage (e.g., bargaining)

Divorce as outside option in most models Divorce as threat point in some models

30

Examples of Alternative Governance Structures

• 1. Becker’s altruist model. One spouse as “husband-

father-dictator-patriarch” who makes all decisions (Pollak, 1988).

• 2. Nash bargaining (Manser and Brown; McElroy and

Horney; Lundberg and Pollak)

• 3. Other cooperative and noncooperative bargaining

models (e.g., repeated games, two stage games)

• 4. Chiappori’s “collective model” as a reduced form

corresponding to any bargaining model with a unique, Pareto-efficient solution.

31 31 31

Information Structure and Transaction Costs

Asymmetric information and monitoring. Coordination and household management: Mrs Beeton 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 person who cares most about how a particular task is done so often decide to do it herself? Without asymmetric information and monitoring costs, which spouse does a task is independent of which spouse wants it done.

32 32 32

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 and market wage rates.

33 33 33

How Rabbits Got In - 1

Unless marginal products are constant, comparative advantages depend on time allocation. “Different comparative advantages” is assumption about efficient time allocation, not just about technology. For a wide class of assumptions about technology and wage rates, production efficiency requires nonspecialization and implies equal comparative advantages. Different comparative advantages rule out interior solutions (i.e., those in which both spouses allocate time to both sectors). Different comparative advantages implies specialization: a corner solution.

34 34

How Rabbits Got In - 2

Perfect substitutes If the spouses time inputs are perfect substitutes, then efficiency in production implies specialization without any additional assumptions (e.g., about additivity, returns to scale, or human capital).

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How Rabbits Got In - 3

The theorems assume that there are only two "sectors"-- home and market. But 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 (Lundberg, 2008). This is a kind of specialization, but it is not sector specialization.

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A simple identi…cation strategy for Gary Becker’s time allocation model

Laurens Cherchye

(University of Leuven)

Bram De Rock

(Université Libre de Bruxelles)

Frederic Vermeulen

(University of Leuven) Royal Economic Society Conference

March 31, 2015

Introduction

50 years ago, Gary Becker published “A theory of the allocation of time” in the Economic Journal. Laid the foundations of household production theory, together with Gorman (1956) and Lancaster (1966). Households combine market goods and time to produce nonmarket goods, which provide utility. Enormous in‡uence on the literature: Muth (1966), Gronau (1970), Grossman (1972), Michael (1973), Willis (1973), Pollak and Wachter (1975), Rosenzweig and Schultz (1983), Cunha and Heckman (2007), etc.

Introduction

Empirical implementation hampered by the lack of values (‘prices’) for di¤erent time uses. Usual approach: prices of female and male time uses are equal to their respective market wages. This implies a fundamental identi…cation problem. We present a simple solution to this identi…cation problem. Based on variables (‘production shifters’) that are related to the total factor productivities associated with the production of nonmarket goods.

Overview

Becker’s time allocation model. A fundamental identi…cation problem. A simple solution. Conclusion.

Overview

Becker’s time allocation model.

Becker’s time allocation model

Households derive utility from nonmarket goods, like a clean home, child rearing or eating. Nonmarket goods produced by means of market goods and time. Constant returns to scale and nonjointness in production (Pollak and Wachter, 1975). The household’s maximization problem: max

z,q1,...,qk,t1,...,tk,tm u(z)

subject to: zi = f i(qi, ti) with i = 1, ..., k,

k

∑

i=1

pi0qi = y + wm0tm,

k

∑

i=1

ti + tm = T.

Becker’s time allocation model

Budget and time constraints can be rewritten as a full income constraint:

k

∑

i=1

pi0qi + wm0

k

∑

i=1

ti = y + wm0T. Constant returns to scale and nonjointness in production imply:

k

∑

i=1

bi(pi, wi)zi = y + wm0T.

Becker’s time allocation model

Implies an example of Gorman’s (1959) two-stage budgeting (Heckman, 2014). First stage: max

z

u(z) subject to:

k

∑

i=1

bi(pi, wi)zi = y + wm0T. Second stage (i = 1, ..., k): max

qi,ti f i(qi, ti)

subject to: pi0qi + wi0ti = yi.

Overview

Becker’s time allocation model. A fundamental identi…cation problem.

A fundamental identi…cation problem

Prices of di¤erent time uses usually not observable. Assumption that values of time uses equal the market wages. Second stage’s optimal input choices to produce nonmarket good i are observable Marshallian functions: qi = gi

q(pi, wm, yi)

ti = gi

t(pi, wm, yi).

Integrability results in standard demand analysis imply that f i can be recovered up to a monotone increasing transformation (that satis…es homotheticity) if and only if Slutsky conditions are satis…ed: ∂ci(pi, wm, zi) ∂pi = gi

q(pi, wm, ci(pi, wm, zi))

∂ci(pi, wm, zi) ∂wm = gi

t(pi, wm, ci(pi, wm, zi)).

A fundamental identi…cation problem

First stage’s optimal allocation associated with the Marshallian demand functions: z = g(b1(p1, wm), ..., bk(pk, wm), y + wm0T). No independent variation of price indices given changes in prices and market wages. Preferences cannot be disentangled from technologies: continuum of utility and production functions gives rise to observationally equivalent behavior (see Chiappori and Lewbel, 2014, and Chiappori and Mazzocco, 2014). Standard labor supply model belongs to that continuum (Heckman, 2014): max

q1,...,qk,l v(q1, ..., qk, l)

subject to:

k

∑

i=1

pi0qi + wm0l = y + wm0T.

Overview

Becker’s time allocation model. A fundamental identi…cation problem. A simple solution.

A simple solution

Assume that there exists a vector s = (s1, ..., sk)0 of ‘production shifters’ that a¤ect overall productivity but not the optimal relative input choices. The variables s are basically total factor productivities. This implies production functions (i = 1, ..., k) of the form: zi = f i(qi, ti)si. Examples: minus the average age of children in the production function of child rearing (Cunha and Heckman, 2007; Cunha, Heckman and Schennach, 2010) or education (Michael, 1973). Relation with Stigler and Becker (1977): di¤erences in observed behavior not explained by ad-hoc taste di¤erences but through di¤erences in the household production functions that impact the income and prices faced by households.

A simple solution

Second stage same as before: the production functions can be identi…ed through variation in pi and wm (the production shifters do not play any role in the optimal input allocation to produce the nonmarket goods). First stage’s full income constraint now equals:

k

∑

i=1

bi(pi, wi) si zi = y + wm0T. First stage’s Marshallian demand equations: z = g(b1(p1, wm) s1 , ..., bk(pk, wm) sk , y + wm0T). Utility function u can be identi…ed up to a monotone increasing transformation if and only if Slutsky conditions are satis…ed through variation in the production shifters s, and thus the prices

b1(p1,wm) s1

, ..., bk (pk,wm)

sk

, while holding constant market prices and wages.

Overview

Becker’s time allocation model. A fundamental identi…cation problem. A simple solution. Conclusion.

Conclusion

Becker’s time allocation model with uniform prices for di¤erent time uses can be identi…ed by means of production shifters. We assumed a unitary model. A related identi…cation strategy (with more general technologies) can be used in collective models that account for intra-household allocation issues (see Cherchye, De Rock and Vermeulen, 2012).

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