Household Production: The Allocation of Time

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Household ProductionThe Allocation of Time

• EC 409
• Labor Economics
• LTC Greenwade
• Virginia Military Institute

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Household Production Model

• Assumption that utility is derived from the
  consumption of N final commodities
• U f(Zi) i 1, 2, 3, , N
• Each of these commodities is produced and
  consumed within the household by combining time
  and purchased inputs

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Household Production ModelExample of Watching a
Movie

• An individual derives utility from the act of
  watching a movie, not from purchasing a ticket to
  the movie
• The final commodity that we think of as watching
  a movie requires the use of both time and
  purchased inputs (tickets, transportation
  expenses, etc.).

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Full Cost

• The full cost of any activity involves the cost
  of both time and purchased inputs
• Similar to the movie example, the full cost of
  attending college includes the opportunity cost
  of time as well as the direct costs of tuition,
  books, supplies, etc.

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Full Cost College

• College enrollments often increase during
  recessions
• The full cost of a college degree is lower
  during a recession
• The expected opportunity cost of time (forgone
  earnings) is typically lower during a recession

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Household Production Definitions

• U f(Z1, Z2, Z3, , ZN)
• where
• U is the level of utility
• Zi is the level of production and consumption of
  activity i
• N is the number of different activities

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Definitions Production Function

• The production function
• Zi f i(ti,xi)
• produces Zi, where
• t is the amount of time devoted to producing and
  consuming commodity i
• x is the quantity of purchased inputs used to
  produce and consume commodity I
• i 1, 2, 3, , N

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Definitions Composite Commodity

• Zi f i(ti,xi)
• This is a composite commodity that is an index of
  all of the goods and services that must be
  purchased to produce Z

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DefinitionsTime Constraint

• where
• ti is the amount of time devoted to producing
  commodity i
• tw is the time spent at work
• T is the total amount of time available

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Time ConstraintTime at Work

• reduces to

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DefinitionsGoods Constraint

• where
• p is the price index for xi
• xi is the quantity of input purchased to produce
  and consume Zi
• w is the wage rate
• tw is the time spent at work

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Goods ConstraintTime at Work
Through substitution

• reduces to

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Full-Income Constraint

• Algebraic manipulation results in the full-income
  constraint

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Full-Income Constraint

• As in the neoclassical model of labor supply, the
  full-income is defined as the individuals
  maximum earning potential (wT)
• wT equals the sum of the price index for good i
  times the quantity of good i, plus the wage rate
  times the time spent in the production of good i

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Full-Income Constraint

• The first term of the summation on the RHS
  represents the explicit cost of goods and
  services to produce Zi
• The second term of the summation on the RHS
  represents the opportunity cost of time used in
  household production

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Full-Income ConstraintAlternate Statement

• Where FCi is the full cost of commodity i

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Optimal Mix of Time

• Individuals are assumed to select a mix of time
  and purchased inputs that minimizes the full cost
  of producing a given level of Zi

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Optimal Mix of Time

• This model can explain such things as
• the growth of the fast-food industry
• why convenience stores can survive while charging
  higher prices than grocery stores
• the decline in fertility
• why many people do not use coupons in grocery
  stores

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Isoquants

• Alternative combinations of time and purchased
  inputs that result in the production of a given
  number of meals of a given quality (assuming
  efficient production)

xi
Zi0
ti
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Isoquants

• In this model an isoquant curve is also an
  indifference curve since the production and
  consumption of a given level of Zi results in a
  constant level of utility

xi
Zi0
ti
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Points Across Isoquants

• At point A, an individual may prepare meals using
  basic ingredients such as flour, vegetables,
  meat, etc.
• In this case, the individual is using a large
  quantity of time, but a relatively low level of
  purchased inputs

xi
A
Zi0
ti
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Points Across Isoquants

• At B, the individual prepares meals of the same
  quality using prepackaged mixes, frozen meals,
  and other preprocessed ingredients
• This requires a larger expenditure on purchased
  inputs but less time than the method used at A

xi
B
A
Zi0
ti
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Points Across Isoquants

• The individual uses less of his or her own time
  and more purchased ingredients when producing and
  consuming meals at C
• This may involve meals consumed in restaurants or
  meals delivered to the home from restaurants

xi
C
B
A
Zi0
ti
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Isoquant Curves

• Points that lie above an isoquant correspond to
  the production of a higher level of Zi

xi
C
D
B
Zi1
A
Zi0
ti
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Isocost Curves

• The level of total costs increase as the level of
  time and purchased inputs increase
• Isocost curves have a slope equal to -w/p (the
  negative of the real wage)

xi
ti
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Least Cost Combination

• The least costly combination of time and
  purchased inputs occurs at the point of tangency
  between the isoquant curve and an isocost curve
  (point E)

xi
E
x
Zi0
ti
t
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Substitution Effect Wage Rate

• An increase in the wage results in two types of
  substitution effects
• This diagram illustrates one of the substitution
  effects resulting from a higher wage

xi
xB
B
A
xA
Zi0
ti
tA
tB
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Substitution Effect Wage Rate

• 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 (the shift
  from A to B)

xi
xB
B
A
xA
Zi0
ti
tA
tB
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Substitution Effect Time

• Time is the basis for the second type of
  substitution effect
• Some activities are inherently more
  time-intensive 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

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Substitution Effect Both

• 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 in the paid labor market

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Income Effect

• An increase in the wage, though, also increases
  the quantity of final goods (Zi) consumed

C
xC
xB
B

• This income effect tends to increase the amount
  of time required for the production and
  consumption of these goods

Zi1
Zi0
tC
tB
ti
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Backward-Bending Supply of Labor Redux

• Thus, the labor supply curve is upward sloping if
  the substitution effects are larger in magnitude
  than the income effect
• The individual operates on a backward-bending
  portion of his or her labor supply curve occurs
  if the income effect is larger than the
  substitution effects