Household Production Model I:
The allocation of time
Household production model

In the household production model,
utility is derived from the activities (Zi)
in which people are engaged.


U=U(Z1, Z2,…, ZN)
Each final commodity is produced and
consumed within the household by
combining time and purchased inputs.
Example: college attendance

College attendance

Requires time as well as purchased inputs
(tuition, books, supplies, etc.)
Full cost


The full cost of an activity includes the
opportunity cost of time as well as the
opportunity cost of purchased inputs.
Example – college enrollments often
increase during recessions due to lower
opportunity cost of time.
Assumptions



U=U(Z1, Z2,…, ZN)
Zi=fi(ti,xi)
Where:
 ti = amount of time devoted to producing
and consuming commodity i.
 xi = amount of purchased inputs devoted
to producing and consuming commodity i.
(This is a composite commodity that is an
index of all purchased inputs used in
producing final commodities.)
Time constraint
Goods constraint
Constraints
Solving the time constraint for time at work:
Substituting this into the goods constraint results
in:
Full-income constraint


After a little algebraic manipulation, the
full income constraint is given by the
formula below.
The first time is the opportunity cost of
goods, the second is the opportunity
cost of time.
Full-income constraint (cont.)

The full-income constraint may also be
expressed as:

Where FCi = full cost of Zi:
Applications

Individuals are assumed to minimize the
full cost of consuming any commodity.
This model may explain:


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the growth of the fast-food industry,
why convenience stores can survive while
charging higher prices than grocery stores,
the decline in fertility, and
why many people do not use coupons in
grocery stores.
Isoquants

This diagram
illustrates the
possible
combinations of
time and
purchased inputs
to provide a given
quantity and
quality of meals.
Indifference curves / isoquants

An isoquant is
also an
indifference
curve since Zi is
held constant.
Points on an isoquant


At point A, an
individual may
prepare meals using
basic ingredients
such as flour,
vegetables, meat,
etc.
the individual is
using a large
quantity of time, but
a relatively low level
of purchased inputs.
Points on an isoquant (cont.)

At point B, the
individual
prepares meals of
the same quality
using
prepackaged
mixes, frozen
meals, and other
preprocessed
ingredients.
Points on an isoquant (cont.)


The individual uses less
of his or her own time
and more purchased
ingredients when
producing and
consuming meals at
point C.
This may involve meals
consumed in
restaurants or meals
delivered to the home
from restaurants.
Other isoquants

Points that lie
above an
isoquant
correspond to
the production
of a higher
level of Zi.
Isocost curves


Isocost curves
have a slope
equal to -w/p (the
negative of the
real wage).
The level of total
costs increase as
the level of time
and purchased
inputs increase.
Cost minimization


The least costly
combination of
time and
purchased inputs
occurs at the point
of tangency
between the
isoquant curve and
an isocost curve.
This occurs at
point E.
Wage increase: substitution effects
First type:


As the wage rate
increases, the
relative price of
time rises and
households
substitute
purchased inputs
for time in the
production and
consumption of a
given level of each
commodity.
Substitution effects


Second type:
 Some activities are inherently more timeintensive than other activities. When the
wage rate increases, the relative price of
time-intensive activities increases. In
response, goods-intensive activities are
substituted for time-intensive activities.
Under both types of substitution effect, a
higher wage reduces the quantity of time
used in household production and increases
the amount of time spent at work.
Income effect


xC
x
B
tB tC
An increase in the
wagealso increases
the quantity of final
commodities (Zi)
consumed.
This income effect
tends to increase the
amount of time
required for the
production and
consumption of
these commodities.
Backward-bending labor supply curve


The labor supply curve is upward
sloping if the substitution effects are
larger in magnitude than the income
effect.
An individual operates on a backwardbending portion of his or her labor
supply curve if the income effect is
larger than the substitution effects.
Specialization

If a household wishes to produce output
efficiently, each individual should specialize in
those tasks in which he or she possesses a
comparative advantage.

a household member possesses a comparative
advantage in an activity if the opportunity cost of
the activity is lower for this individual than for any
other member of the household.)
Sources of comparative advantage

A comparative advantage may exist if:


an individual is more productive in an
activity than other members of the
household (in this case an “absolute
advantage” is said to occur), or
because the individual’s time is relatively
less valuable in alternative activities.
Gender division of labor



Historically, married women have tended to
specialize in household production and
married males have tended to specialize in
market production.
Comparative advantage for women in
household production in the past?
Possible reasons:



high completed fertility rates,
high infant mortality rates, and
labor market discrimination.
Evolving gender roles

As infant mortality and completed fertility
rates decline and as female wage rates rise, it
is expected that this division of labor between
spouses will be altered.


In recent years, married women have substantially
increased the amount of time spent in the paid
labor market and have spent slightly less in
household production).
Married men now spend slightly more time in
household production than in the past.
Specialization or shared activities?


Both spouses will tend to work together
in household production tasks in which
their time is complementary
Individuals will specialize (according to
comparative advantage) when one
spouse’s time is a substitute for that of
the other spouse.
Additional worker effect




The labor force participation rate generally declines
during recessions as a result of an increase in the
number of discouraged workers.
In a household, however, one spouse may increase
his or her labor supply (or enter the labor market) if
the other spouse becomes unemployed.
This “additional worker effect” partly offsets the
“discouraged worker effect” discussed earlier.
The additional worker effect is smaller in magnitude
than the discouraged worker effect.
Additional worker effect (cont.)

The additional worker effect is relatively small
because the expected wage declines during a
recession:E(w) = pw
where:
E(w) = expected wage
p = probability of employment
w = wage rate if employed
As the unemployment rate rises during a
recession, the probability of being employed,
p, declines, leading to a reduction in the
expected wage.
Female labor supply and divorce


Married women tend to increase their labor
supply when a divorce becomes more likely.
This is partly to prepare for the reduction in
the division of labor that occurs after the
divorce.

Empirical evidence suggests that the level of per
capita consumption declines by a larger amount in
the portion of the splitoff household headed by
divorced women.
Lifetime labor supply decisions


The productivity of
time in the paid
labor force varies
over the lifecycle.
Market wages vary
over time as
productivity
changes.
Lifecycle labor supply

individuals are
expected to spend
more time working
in the paid labor
market (and less
time in household
production) when
market wage rates
are relatively high.
Labor force participation and
childrearing

Historically, many
married females
chose to reduce the
quantity of labor
supplied or leave
the labor force
during their
childbearing years.
Changes in LFPR for married women

As fertility levels
have declined and
market wage rates
have increased, a
smaller proportion
of married working
mothers exit the
labor force during
the childbearing
years today than
in past decades.
Social Security & Retirement Age

an increase
in the level
of retirement
benefits
induces
individuals to
retire earlier.
Single-parent households and welfare

Many single
parents
(typically
female) remain
out of the
labor force
Child Support Enforcement Act


the budget
constraint facing
the custodial
parent shifts
vertically
upward.
reduces state
welfare
expenditures
even if there is
no effect on
labor supply
Child Support Enforcement Act

Increases labor
supply for
some welfare
recipients who
were initially
out of the
labor force.
Child Support Enforcement Act

is expected
to reduce
labor supply
if the
custodial
parent is
initially
working.