Allocating%20Time:%20Individuals

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Allocating TimeIndividuals Technologies and
Household Technology

• Robert A. Pollak
• Washington University
• May 27, 2008

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• Introduction
• Background
• Individuals' Technologies and Household
  Technology
• Becker's Specialization Claims
• Critique of Specialization Claims
• Conclusion

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New Home Economics (NHE)

• Reid (1934) Economics of Household Production
• Mincer (1963) Grunfeld volume
• Market Prices, Opportunity Costs, and Income
  Effects
• Becker (EJ, 1965)
• A Theory of the Allocation of Time
• Becker (1981, 1991)
• A Treatise on the Family

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Time Use Categories

• Two time uses in neoclassical economics
• leisure
• market work (labor)
• Three time uses in the NHE
• leisure
• market work (labor)
• household production

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Trends in Leisure

• Shore (1991) Overworked American
• Hochschild (1997) Time Bind
• Hamermesh (REStat, 2007)
• Aguiar and Hurst (QJE, 2007)
• Ramey and Francis (NBER, 2006)

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Process Preferences

• Preferences for Market Work, Nonmarket Work, and
  Leisure
• Is household work like market work?
• Or is it like leisure?
• Labor economists should reject this dichotomy a
  continuum
• Compensating and noncompensating wage
  differentials
• Danger and/or unpleasantness
• Equalizing differences
• Adam Smith (1776) Sherwin Rosen (Hndbk, 1986)
• Pecuniary and nonpecuniary returns

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Household vs. Family

• I will not emphasize this distinction
• Who lives with whom is endogenous
• Terminology is sloppy
• Household production
• Traditional nuclear family
• Stem family

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Two Crucial Distinctions between Becker (1965)
and Becker (1981)

• single-person households vs.
• multiple-person households
• one-period models vs.
• multi-period models

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Make this a 2 x 2 table 2 or more

• Periods
• Persons 1 2 3 or
  more
• 1
• 2
• 3 or
• more

Becker (1965)

Becker (1981)
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Toward a New New Home Economics Primitives in
the New NHE

• 1. Preferences
• e.g., individuals' utility functions
• 2. Technology
• Individuals technologies and
• household technology
• production functions
• 3. Household governance structure
• e.g., altruist model Nash bargaining

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Time Use in Minnesota Macro

• Benhabib, Rogerson and Wright (JPE, 1991)
• Greenwood and Hercowitz (JPE, 1991)
• McGrattan, Rogerson, and Wright (IER, 1997)
• Greenwood, Seshadri and Yorukoglu (REStat, 2005)
  Engines of Liberation
• Gronau (NBER, 2006)

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New Time Use Data

• American Time Use Survey (ATUS)
• Hamermesh, Franzis, and Stewart (JEP, 2005)
• Australian Time Use data
• UK Time Use Survey (UKTUS)
• German Socio-Economic Panel (GSOEP)
• Danish Time Use Survey (DTUS) and
• Danish Household Expenditure Survey (DHES)
  (Browning and Gørtz, 2006)
• Consumption and Activities Mail Survey (CAMS)
• Hurd and Rohwedder

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Empirical Time Use Research

• Gronau (1986, 1997, 2006)
• Juster and Stafford (JEL, 1991)
• Robinson and Godbey (1997) Time For Life
• Gershuny (2000) Changing Times
• Presser (2003) Working in a 24/7 Economy
• Folbre and Bittman (2004) Family Time
• Jacobs and Gerson (2004) The Time Divide
• Gronau and Hamermesh (Rev IW, 2006)
• Bianchi, Robinson, and Milkie (2006) Changing
  Rhythms of American Family Life

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• Introduction
• Background
• Individuals' Technologies and Household
  Technology
• Becker's Specialization Claims
• Critique of Specialization Claims
• Conclusion

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The New Home Economics and Household Production

• 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)

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Becker (E.J.,1965) Assumptions

• commodities (outputs) measurable
• commodity (shadow) prices
• fewer commodities than market goods
• production functions (no joint production)
• constant returns to scale
• single person households
• no human capital

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Pollak and Wachter (JPE, 1975)

• critique of Becker (E.J., 1965)
• are the commodities (outputs) measurable?
• process preferences
• people care how they spend their time
• cooking vs cleaning
• process preferences imply joint production
• with joint production, commodity shadow prices
  depend on preferences
• (as well as goods prices and technology)
• single person households
• no human capital

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Becker (1981, 1991) Assumptions

• multiperson household
• specialization and division of labor
• unitary model of household decision making
• households have utility functions
• spouses time inputs perfect substitutes in
  production
• human capital
• one household commodity (sometimes)

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Crucial Changes

• Single-person household to multiple-person
  household Specialization and Division of Labor
• One-period model to multi-period model
• Human capital in market production
• Human capital in household production
• But still no physical capital in household
  production

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• Introduction
• Background
• Individuals' Technologies and Household
  Technology
• Becker's Specialization Claims
• Critique of Specialization Claims
• Conclusion

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When Does Theory Require Individuals
Technologies?

• 1. Marriage market
• Compare well-being when single to well-being in
  particular marriage
• 2. Divorce
• Compare well-being in current marriage to
  well-being if divorced
• 3. Bargaining within marriage
• Divorce as threat point in some models
• Divorce as outside option in some models

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Specifying Household Technology The Usual
Suspects

• Joint production
• Economies of scale
• Spouses' time perfect substitutes
• Human capital
• Market (i.e. purchased) inputs,
• including hired labor
• Time or labor means household time or household
  labor, not hired labor
• Physical capital

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The Meaning of Technology

• Book of blueprints vs. knowledge in the mind of
  chief engineer
• Tacit knowledge -- Michael Polanyi
• Knowing how vs. knowing that -- Gilbert Ryle
• Recipes from cookbook vs. mind of the cook
• Technology as production set vs.
• technology as production function
• Process benefits imply joint production
• Technology transfer and learning by doing

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Production Functions for Single-person
Households

• For a single-person household, denote the
  household production function for the commodity z
  by
• z f(t,y)
• where t denotes the input of household time (or
  labor) into its production and y the vector of
  market inputs.
• This assumes no joint production and, hence, no
  process preferences.
• Assumption No output without individuals time
  input
• f (0,y)0

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Production Functions for Multiple-person
Households

• z gt1,t2,y, where t1 and t2 denote the
  spouses' time inputs into the production of z.
• Denote the individual production functions of
  the spouses by
• f1(t1,y1) and f2(t2,y2)
• Household production with specialization
• gt1,0,y
• g0, t2,y

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Revelation Assumption

• Specialized household production reveals
  individuals production functions
• gt1,0,y f1(t1, y)
• g0, t2,y f2(t2, y)
• That is, we can infer individuals production
  functions from the household production function.
  
• With specialization, this is what we would
  observe.

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Interpretations of Individuals Production
Functions

• 1. Before household formation
• marriage or cohabitation
• immaculate conception of
• household technology
• 2. After household dissolution divorce
• actual divorce
• household bargaining
• threat point or outside option
• 3. After death of spouse

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Beckers Perfect Substitutes Assumption

• Becker (sometimes) assumes that spouses time
  inputs are perfect substitutes
• gt1,t2 Gt1at2
• gt1,t2 y Gt1at2, y

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Implications of Revelation Assumption and
Beckers Perfect Substitutes Assumption

• The revelation assumption and Becker's perfect
  substitutes assumption imply that the
  individuals' production functions are identical
  to each other except for an "efficiency factor."
• Proof Perfect substitutes implies that the
  household production function is of the form
• g t1,t2 y Gt1at2, y
• so
• g t1,0, y Gt1,, y f1(t1, y)
• g 0,t2 y Gat2, y f2(a t2 , y)
• where a converts the time input of spouse 2 into
  units comparable to the time input of spouse 1.
  Thus, (t1 at2) is the total time input into the
  production of the focal commodity, measured in
  "efficiency units."

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Question

• When can we infer household production function
  from individuals production functions?
• To do so requires a new assumption Additivity

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Additive Household Production Functions No
Market Inputs

• Suppose output is produced by labor (time) alone
  no market inputs
• I say a household production function is
  additive if
• gt1,t2 f1(t1) f2(t2)
• Additivity means there are no positive or
  negative interactions (externalities spillovers)
  associated with side-by-side production.

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Additive Household Production Functions Market
Inputs

• I say that the household production function is
  "additive" if the household output from the input
  vector (t1,t2,y) is equal to the maximum of the
  sum of the outputs they would realize by
  producing "side by side."
• gt1, t2, y max f1(t1,y1) f2(t2,y2)
• subject to y1 y2 y.
• This assumes that market inputs are household
  private goods and are allocated to maximize total
  output.

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Additivity

• Additivity is a more plausible assumption than
  perfect substitutes, in part because it does not
  require that the spouses' production functions be
  identical except for an efficiency factor.

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When are Additivity and Perfect Substitutes
Compatible? - 1

• When time is the only input, combining additivity
  and perfect substitues implies
• gt1,t2 Gt1 at2 f1(t1) f2(t2).
• Making use of the assumption that a time input of
  0 implies 0 output, we obtain
• Gt1 f1(t1)
• Gat2 f2(t2).
• so
• Gt1 at2 Gt1 Gat2.
• This is Cauchy's functional equation.

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Additivity and Perfect Substitutes - 2

• Differentiating with respect to t1 we obtain
• G't1 at2 Gt1
• which implies that the functions g is linear.
• Assuming that a time input of 0 implies an output
  of 0 yields
• Gt ct
• Gt1 at2 c(t1 at2).
• f1 (t1) ct1
• f2 ( t2 ) c a t2

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Additivity and Perfect Substitutes - 3

• When time is the only input, additivity and
  perfect substitutes are compatible if and only if
  the production functions are linear.
• With nonlabor inputs, additivity and perfect
  substitutes are compatible if the production
  functions are homogeneous of degree one and
  differ only by an efficiency factor.

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Why Linear Technology is Unappealing

• Highly restrictive
• But it is implied by
• output produced by time alone,
• household's technology is additive, and
• time inputs are perfect substitutes.
• The first two assumptions are consistent with
  fatigue or boredom causing productivity to
  decline as time inputs increase. Combining them
  with the perfect substitutes assumption rules
  this out.

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Additivity and Perfect Substitutes

• Additivity and perfect substitutes imply
• G2t,2y 2Gt,y.
• PROOF Gt1 at2,y Gt1 ,y1 Gat2,.y2,
• where y is allocated to maximize output.
• Let t2 t1 / a so
• Gt1 at2,y Gt1 t1,y Gt1,y1
  Gt1,.y2,
• maximizing output implies y1 y2 y/2.
  Hence
• Gt1 t1,y G2t1,y 2Gt1,y/2
• Define t t1 and y y/2. Then
• G2t,2y 2Gt,y.

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Theorem

• If Gt,y is concave and
• G2t,2y 2Gt,y, then
• G(?t,?y) ?G(t,y) for all ? gt 0.
• That is, if the household production function is
  concave, the spouses time inputs are perfect
  substitutes, additivity and essentiallity hold,
  then the household production function is
  homogeneous of degree 1.
• Massimo Marinacci and Luigi Montrucchio proved
  this

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Decreasing Returns to Scale

• If the individuals' production functions exhibit
  decreasing returns to scale (e.g., because
  individuals become less productive as they become
  tired or bored), then Becker's specialization
  conclusion need not hold.
• If the household production function is
  additive, then efficiency is compatible with
  side-by-side production and does not imply
  specialization.

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Example

• Suppose spouses' production functions are of the
  form
• f1(t1) A1 (t1)s1 and f2 (t2) A2
  (t2)s2
• where the parameters ?1 and ?2 indicate the
  returns to scale
• properties of the individuals technologies.
  Imposing additivity and the assumption that the
  spouses' time inputs are perfect substitutes
  implies
• ?1 ?2 1.
• That is, imposing Becker's perfect substitutes
  assumption rules out decreasing returns to scale
  and implies that both spouses' production
  functions collapse to the constant returns to
  scale case, and
• gt1, t2 A1 t 1A2 t2

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• Introduction
• Background
• Individuals' Technologies and Household
  Technology
• Becker's Specialization Claims
• Critique of Specialization Claims
• Conclusion

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Becker on Specialization and the Division of
Labor - 1

• "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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Becker on Specialization and the Division of
Labor - 2

• "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."

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Where do Becker's 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,
  1938

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Beckers Theorems

• Becker's statements of the theorems do not
  include all of the assumptions. Becker
  explicitly states some additional assumptions in
  the nearby text, but other assumptions are left
  unstated. In Pollak (REHO, 2003) I argue that
  this style -- conclusions presented as "theorems"
  without explicit statements of their hypotheses
  -- is vintage Becker.
• Bergstrom (JPE 1989), A Fresh Look at the Rotten
  Kid Theorem-and Other Household Mysteries

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General Points about Beckers Theorems

• The theorems are not restricted to married couple
  households.
• Efficiency in household production is explicitly
  assumed.
• Human capital -- market human capital and
  household human capital -- appear to play a
  central role.

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Three Specific Points - 1

• 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 (LE, 2005)
• 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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Three Specific Points - 2

• 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 is additive, then
  efficiency may call for side-by-side production
  rather than specialization.

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Three Specific Points III

• Sometimes Becker assumes that the time inputs of
  husbands and wives are "perfect substitutes."
  (But not in the statement of these two theorems.)

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• Introduction
• Background
• Individuals' Technologies and Household
  Technology
• Becker's Specialization Claims
• Critique of Specialization Claims
• Conclusion

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The Meaning of Specialization

• Specialization by sector
• (home vs. market)
• Specialization by activity
• (e.g., cooking vs. cleaning)
• What is an activity?
• cooking?
• grilling?
• grilling fish?

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Role of Unstated Special Assumptions

• 1. how many commodities?
• activities?
• Lundberg (LE, 2005)
• 2. suppose you become tired or bored
• implications for technology
• decreasing returns to scale
• 3. suppose you become tired or bored
• implications for preferences
• process preferences
• implications for specialization
• depend on household governance structure

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Beckers Explanation of Specialization

• Becker emphasizes role of family specific,
  relationship specific, sector specific, or
  activity specific human capital in household
  production
• Household human capital accumulates during
  marriage.
• Household human capital investments before
  marriage these cannot be family specific
• gender as coordinating mechanism
• Family specific vs. general (household) human
  capital
• Evidence for household human capital

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Alternative Explanations of Specialization

• Technology (perfect substitutes)
• Human Capital
• Market human capital
• Household human capital
• Sector specific or activity specific
• Family specific
• Process Gender Norms (Doing Gender)
• Preferences
• Economics of Scope
• Economics of Scale
• Transaction Costs coordination and monitoring

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• Introduction
• Background
• Individuals' Technologies and Household
  Technology
• Becker's Specialization Claims
• Critique of Specialization Claims
• Conclusion

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Conclusion

• What do we mean by specialization?
• sector vs. activity
• Becker's specialization claims depend on special
  and implausible assumptions about technology
  and tastes fatigue and boredom
• Why specialization?
• Is it perfect substitutes and human capital?
• If it is human capital, is it household rather
  than market human capital?

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Relationship between Individuals' and Household
Technologies

• Household formation
• marriage or cohabitation
• immaculate conception of household
  technology
• Household dissolution
• actual divorce
• household bargaining
• threat point or outside option
• Death of spouse
• Disability of spouse

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Additivity and Perfect Substitutes

• Additivity
• gt1 , t2 f1(t1 ) f2 (t2 )
• Perfect substitutes
• gt1 , t2 Gt1 a t2

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Implications of Additivity and Perfect Substitutes

• gt1 , t2 f1(t1 ) f2(t2 )
• and
• gt1 , t2 Gt1 a t2
• implies
• gt1 , t2 ct1 c a t2

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Alternative Explanations of Specialization

• Technology (perfect substitutes)
• Human Capital
• household human capital
• market human capital
• Gender Norms (Doing Gender)
• Economics of Scope
• Transaction Costs
• coordination
• monitoring

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