Royal Economic Society
Gary Becker’s "A Theory of the Allocation of Time"
Royal Economic Society
Arthur Lewbel
Boston College
March 2015
Lewbel (Boston College)
03/15
1 / 11
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).
Lewbel (Boston College)
03/15
2 / 11
Example Precursors:
Mincer (1962), considered a married woman’s time trade-o¤ 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.
Lewbel (Boston College)
03/15
3 / 11
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.
Lewbel (Boston College)
03/15
4 / 11
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 di¤erent prices at, say daytime vs
nighttime or weekends vs weekdays, rather than a single wage rate.
For Becker, S is de…ned 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 .
Lewbel (Boston College)
03/15
5 / 11
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
0
0
S.
∑m
i =1 pi xi + wi Ti
Key insight: There are not two di¤erent constraints for time and
money. There is only a single budget constraint!
Many implications follow from there being just one constraint.
Lewbel (Boston College)
03/15
6 / 11
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 e¤ects of
earnings-intensive goods, like child care, could help low income
elasticity of fertility.
4. Household specialization of labor (See Pollak).
Lewbel (Boston College)
03/15
7 / 11
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.
Lewbel (Boston College)
03/15
8 / 11
One strand of literature: identi…cation and estimation?
What household level data is observable? What is needed? (A single
wage per person really helps).
Pareto e¢ cient 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,...)
Lewbel (Boston College)
03/15
9 / 11
Becker’s approach to family economics: mainstream now, but
revolutionary then.
Many were openly hostile, calling his model sterile, vacuous, cold, and
immoral.
Lewbel (Boston College)
03/15
10 / 11
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.
Lewbel (Boston College)
03/15
11 / 11
Royal Economic Society
Domestic production and matching
Economic Journal Anniversary Sessions - Becker 1965
Pierre-André Chiappori
Columbia University
Manchester, March 2015
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
1/9
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.)
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
2/9
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’
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
3/9
Becker’s framework: emphasis on domestic production
Individuals exclusively consume commodities that have been internally
produced.
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
4/9
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 ’
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
4/9
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
di¤erent members of the same household, or other special
assumptions about preferences’
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
4/9
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
di¤erent 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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
4/9
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
di¤erent 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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
4/9
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.’
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
5/9
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:
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
5/9
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?
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
5/9
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:
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
5/9
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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
5/9
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 ...
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
5/9
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 e¢ cient to match with a partner whose time is cheap
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
5/9
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 e¢ cient 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)
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
5/9
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 e¢ cient 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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
5/9
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 e¢ cient 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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
5/9
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 e¢ cient 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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
5/9
A simple example
CD preferences:
Ui = Ci Q with Q = (t1 )α1 (t2 )α2
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
6/9
A simple example
CD preferences:
Ui = Ci Q with Q = (t1 )α1 (t2 )α2
Budget constraint:
p (C1 + C2 ) = w1 (1
Chiappori
(Columbia University)
t1 ) + w2 (1
Becker 65
t2 ) with wi = Hi W
Manchester, March 2015
6/9
A simple example
CD preferences:
Ui = Ci Q with Q = (t1 )α1 (t2 )α2
Budget constraint:
p (C1 + C2 ) = w1 (1
t1 ) + w2 (1
t2 ) with wi = Hi W
Transferable utility
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
6/9
A simple example
CD preferences:
Ui = Ci Q with Q = (t1 )α1 (t2 )α2
Budget constraint:
p (C1 + C2 ) = w1 (1
Transferable utility
! surplus:
S (H1 , H2 )
Chiappori
(Columbia University)
t1 ) + w2 (1
t2 ) with wi = Hi W
=
W
max (H1 (1
p t1 ,t2
=
α1α1 α2α2
W
(H1 + H2 )1 +α1 +α2
p (α1 + α2 + 1)1 +α1 +α2 (H1 )α1 (H2 )α2
Becker 65
t1 ) + H2 (1
t2 )) (t1 )α1 (t2 )α2
Manchester, March 2015
6/9
A simple example
CD preferences:
Ui = Ci Q with Q = (t1 )α1 (t2 )α2
Budget constraint:
p (C1 + C2 ) = w1 (1
Transferable utility
! surplus:
S (H1 , H2 )
=
t1 ) + w2 (1
W
max (H1 (1
p t1 ,t2
t2 ) with wi = Hi W
t1 ) + H2 (1
t2 )) (t1 )α1 (t2 )α2
α1α1 α2α2
W
(H1 + H2 )1 +α1 +α2
p (α1 + α2 + 1)1 +α1 +α2 (H1 )α1 (H2 )α2
In particular, second cross derivative:
=
(H1 + H2 )α1 +α2
1
(α1 H2
α2 H1 )2 + α1 H22 + α2 H12
H1α1 +1 H2α2 +1
Chiappori
(Columbia University)
Becker 65
< 0 ) NAM!
Manchester, March 2015
6/9
A possible solution to the puzzle
Keep Becker’s insights:
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
7/9
A possible solution to the puzzle
Keep Becker’s insights:
Domestic production, domestic times as inputs
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
7/9
A possible solution to the puzzle
Keep Becker’s insights:
Domestic production, domestic times as inputs
PAM if complementarities in traits
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
7/9
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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
7/9
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 justi…cation: children
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
7/9
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 justi…cation: children
Then two opposite forces:
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
7/9
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 justi…cation: children
Then two opposite forces:
Educated spouse’s time is more costly ...
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
7/9
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 justi…cation: children
Then two opposite forces:
Educated spouse’s time is more costly ...
... but also more productive
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
7/9
A simple example (CCM 2015)
CD preferences:
Ui = Ci Q with Q = (H1 t1 )α1 (H2 t2 )α2
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
8/9
A simple example (CCM 2015)
CD preferences:
Ui = Ci Q with Q = (H1 t1 )α1 (H2 t2 )α2
Budget constraint:
p (C1 + C2 ) = w1 (1
Chiappori
(Columbia University)
t1 ) + w2 (1
Becker 65
t2 ) with wi = Hi W
Manchester, March 2015
8/9
A simple example (CCM 2015)
CD preferences:
Ui = Ci Q with Q = (H1 t1 )α1 (H2 t2 )α2
Budget constraint:
p (C1 + C2 ) = w1 (1
t1 ) + w2 (1
t2 ) with wi = Hi W
Transferable utility ! surplus:
S ( H1 , H2 ) =
=
Chiappori
(Columbia University)
W
max (H1 (1 t1 ) + H2 (1 t2 )) (H1 t1 )α1 (H2 t2 )α2
p t1 ,t2
α1α1 α2α2
W
(H1 + H2 )1 +α1 +α2
p ( α 1 + α 2 + 1 )1 + α1 + α2
Becker 65
Manchester, March 2015
8/9
A simple example (CCM 2015)
CD preferences:
Ui = Ci Q with Q = (H1 t1 )α1 (H2 t2 )α2
Budget constraint:
p (C1 + C2 ) = w1 (1
t1 ) + w2 (1
t2 ) with wi = Hi W
Transferable utility ! surplus:
S ( H1 , H2 ) =
=
W
max (H1 (1 t1 ) + H2 (1 t2 )) (H1 t1 )α1 (H2 t2 )α2
p t1 ,t2
α1α1 α2α2
W
(H1 + H2 )1 +α1 +α2
p ( α 1 + α 2 + 1 )1 + α1 + α2
In particular
KW
∂ 2 S ( H1 , H2 )
=
(H1 + H2 )α1 +α2
∂H1 ∂H2
p
Chiappori
(Columbia University)
Becker 65
1
>0
Manchester, March 2015
8/9
Conclusions
Technical point: complementarity between traits (matching) di¤erent
from complementarities between inputs (specialization)
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
9/9
Conclusions
Technical point: complementarity between traits (matching) di¤erent
from complementarities between inputs (specialization)
Complementarities in Becker’s contributions: clearly 73 builds on 65
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
9/9
Conclusions
Technical point: complementarity between traits (matching) di¤erent
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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
9/9
Conclusions
Technical point: complementarity between traits (matching) di¤erent
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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
9/9
Conclusions
Technical point: complementarity between traits (matching) di¤erent
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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
9/9
Conclusions
Technical point: complementarity between traits (matching) di¤erent
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.
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
9/9
Conclusions
Technical point: complementarity between traits (matching) di¤erent
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?
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
9/9
Conclusions
Technical point: complementarity between traits (matching) di¤erent
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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
9/9
Conclusions
Technical point: complementarity between traits (matching) di¤erent
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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
9/9
Conclusions
Technical point: complementarity between traits (matching) di¤erent
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
Chiappori
(Columbia University)
Becker 65
Manchester, March 2015
9/9
Royal Economic Society
Allocating Household Time:
When Does Efficiency Imply
Specialization?
Robert A. Pollak
Washington University in St. Louis
and NBER
March 31, 2015
1
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
on time allocation.
(There is lots of empirical work.)
2
Roadmap



Specialization
Toward a New New Home Economics:
Elements of a Theory of the Household
Individuals’ Production Functions and
Household Production Function
3
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
4
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.
5
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 nonCatholic 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)
6
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)
7
Still More Facts:
Labor Force Particiaption
2008 Labor Force Participation Rates for
Married men
25-34
95.3%
Married women
69.5%
Age
35-44
95.2%
73.8%
US data: CPS
8
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.
9
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.
10
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
11
is a one period model.
New Issues with MultiplePerson Households
Allocation of goods, time and commodities.
Alternative models of decision making in multipleperson 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”
12
The Theoretical Time Use
Literature
Pollak and Wachter (1975)
Gronau (1977)
One person households
No human capital
13
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.
14
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?
15
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."
16
Theory: Becker (1981, 1991) - 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.)
17
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.
18
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.
19
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
20
Three Omitted Topics
1. Human Capital
2. Joint Production and Process Preferences: Pollak and
Wachter (JPE 1975).
3. Leisure: Gronau (JPE 1977)
21
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
on human capital.
22
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. 23
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 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
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.
25
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)
26
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).
27
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.
28
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
29
Examples of Alternative
Governance Structures
1. Becker’s altruist model. One spouse as “husbandfather-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.
30
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.
31
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.
32
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. 33
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).
34
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.
35
Royal Economic Society
A simple identi…cation strategy for Gary Becker’s time
allocation model
Laurens Cherchye
Bram De Rock
(Université Libre de Bruxelles)
(University of Leuven)
Frederic Vermeulen
(University of Leuven)
Royal Economic Society Conference
March 31, 2015
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
1 / 16
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.
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
2 / 16
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.
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
3 / 16
Overview
Becker’s time allocation model.
A fundamental identi…cation problem.
A simple solution.
Conclusion.
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
4 / 16
Overview
Becker’s time allocation model.
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
5 / 16
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
k
∑ pi 0 qi
= f i (qi , ti ) with i = 1, ..., k,
= y + w m 0 tm ,
i =1
k
∑ ti + tm
= T.
i =1
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
6 / 16
Becker’s time allocation model
Budget and time constraints can be rewritten as a full income
constraint:
k
k
i =1
i =1
∑ pi 0 qi + wm 0 ∑ ti = y + wm 0 T.
Constant returns to scale and nonjointness in production imply:
k
∑ bi (pi , wi )z i = y + wm 0 T.
i =1
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
7 / 16
Becker’s time allocation model
Implies an example of Gorman’s (1959) two-stage budgeting
(Heckman, 2014).
First stage:
max u (z)
z
subject to:
k
∑ bi (pi , wi )z i = y + wm 0 T.
i =1
Second stage (i = 1, ..., k):
max f i (qi , ti )
qi ,ti
subject to:
pi 0 qi + w i 0 t i = y i .
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
8 / 16
Overview
Becker’s time allocation model.
A fundamental identi…cation problem.
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
9 / 16
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
= gqi (pi , wm , y i )
ti
= gti (pi , wm , y i ).
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:
∂c i (pi , wm , z i )
∂pi
i
i
∂c (p , wm , z i )
∂wm
CDV (Royal Economic Society Conference)
= gqi (pi , wm , c i (pi , wm , z i ))
= gti (pi , wm , c i (pi , wm , z i )).
Time allocation
March 31, 2015
10 / 16
A fundamental identi…cation problem
First stage’s optimal allocation associated with the Marshallian
demand functions:
z = g(b 1 (p1 , wm ), ..., b k (pk , wm ), y + wm 0 T).
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 v (q1 , ..., qk , l)
q1 ,...,qk ,l
subject to:
k
∑ pi 0 qi + wm 0 l = y + wm 0 T.
i =1
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
11 / 16
Overview
Becker’s time allocation model.
A fundamental identi…cation problem.
A simple solution.
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
12 / 16
A simple solution
Assume that there exists a vector s = (s 1 , ..., s k )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:
z i = f i ( qi , t i ) s i .
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.
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
13 / 16
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
b i ( pi , w i ) i
∑ s i z = y + wm 0 T.
i =1
First stage’s Marshallian demand equations:
b k ( pk , w m )
b 1 (p1 , wm )
,
...,
, y + w m 0 T).
s1
sk
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
b 1 (p1 ,wm )
b k (pk ,wm )
, ...,
, while holding constant market prices and
s1
sk
wages.
z = g(
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
14 / 16
Overview
Becker’s time allocation model.
A fundamental identi…cation problem.
A simple solution.
Conclusion.
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
15 / 16
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).
CDV (Royal Economic Society Conference)
Time allocation
March 31, 2015
16 / 16
Royal Economic Society