The Household Production Approach
The household production approach is basically an application of producer
theory and consumer theory to a domain beyond a simple exchange economy. It
has several major themes:
 The commodities that enters into the utility function are not
necessarily directly available in the market. Some commodities have to be
produced within the household with a combination of market goods, human
capital, and the
consumer's time.
 The production and consumption
of commodities need not be confined to a single individual; social
interactions and social institutions are also important.
 Goods have many attributes; they affect a consumer's utility in
more than one way.
There is no better way to study the household production approach than to
look at how it is actually put to work. Consider some examples in the
economics of the family, pioneered by Gary Becker.
Division of labor within the family
Suppose there is some generic commodity Z, which is produced by a
combination of market goods x and household production time
t_{h}. Efficiency in production is parameterized by
a, with higher a indicating greater efficiency.
The household production
function is
written as
Z = f( x, at_{h} )
where a denotes the efficiency unit of household production time.
Market goods have to be purchased
using earned income. Let the market wage rate be w and let the
price of the market good be 1. Then the budget constraint is
x = A + w t_{w}
where t_{w} is the time spent on market
employment and A is unearned income.
The time use for
individual must also satisfy a time constraint
t_{h} + t_{w} = T
where T is total time endowment.
We can combine these two constraints by writing
x + w t_{h} = A + w T
The term on the right hand side is the person's full income;
it denotes his total
income if he spends full time working in the market. Maximizing Z subject
to this constraint, the FOCs include:
f_{x}  λ = 0
f_{t} a  λ w = 0
These two equations can be combined to get f_{x}/f_{t} = a/w.
Now, consider a household consisting of two persons, say Mr. 1 and
Ms. 2.
If they don't cooperate, Mr. 1 chooses x and t_{h} such that f_{x}/f_{t} =
a_{1}/w_{1},
while Ms. 2 chooses x and t_{h}
such that f_{x}/f_{t} = a_{2}/w_{2}. But
they could do better. Suppose a_{1}/w_{1} > a_{2}/w_{2}. We say that Mr. 1 has a
comparative advantage in household production,
while Ms. 2 has a
comparative advantage in market work. Total time devoted to household
production (in efficiency units) is a_{1} t_{h1} + a_{2}
t_{h2}. If Mr. 1 works 1
more hour in household production
while Ms. 2 works a_{1}/a_{2} fewer hours,
total effective time devoted to household production remains unchanged. But
the change in total wage income is
 w_{1} + w_{2} (a_{1}/a_{2})
which is positive if a_{1}/w_{1} > a_{2}/w_{2}. Thus the household will be better
off if Mr. 1 specializes in household production while Ms. 2 specializes in
market work. This is nothing but an application of standard trade theory.
Where does the comparative advantage come from?
The parameters a and w are not really exogenous. Your efficiency in
household
production (a) and efficiency in market work (w) depend on your investment
in these two types of human capital. Suppose
a_{1} = a_{0} + α
K_{a1} and w_{1} =
w_{0} + β K_{w1},
and similarly for a_{2} and
w_{2}. How would these two persons choose their
investment levels?
If there is no division of labor within the household, the model is
completely symmetrical. So Mr. 1 and Ms. 2 will
choose the same level of investments in the two types of human capital, say
K_{a}* and K_{w}*.
Therefore a_{1}/w_{1} =
a_{2}/w_{2}. Each person
will spend some of
the time working in the labor market and some of the time producing the
household commodity.
Since there is no comparative advantage, total output Z will remain
unchanged if Mr. 1 specializes in housework and Ms. 2 specializes in market
work. But when Mr. 1 specializes in housework, investing in
K_{w} is no
longer useful. So instead of incuring an investment cost of
C(K_{a}*,K_{w}*),
he could just spend C(K_{a}*,0) on investment.
With this new investment
level, his a_{1}/w_{1}
will rise and comparative advantage will emerge.
Because the returns to human capital increases with the rate of
utilization,
it pays to invest in one type of human capital only and
use it full time than to invest in both types and use each parttime.
This consideration causes exante identical agents to specialize in
different activities.
Demand for children
Wealthy individuals consume more of most goods than poorer persons. Why
don't they also have more children? One explanation is that children are a
time intensive good. Since wealthy people tend to have greater cost
of time, they face a higher cost of raising children. But it seems that
even if we hold other things constant,
the pure income effect on the demand for children is small, if not negative.
Becker studies a model in which both the quantity and the quality of
children enter into the parent's utility function. In the simplest setting,
let
U = U(n,q,Z)
where n is the number of children, q is the quality of each child, and Z
represents other commodities. The price per unit of quality is p. So the
budget constraint is
npq + Z = M
where M is money income.
The FOCs for utility maximization are
U_{n}  λ pq = 0
U_{q}  λ pn = 0
U_{Z}  λ = 0
M  npq  Z = 0
Suppose we define π_{n} = pq and π_{q} = pn. Then, the utility maximization
problem is the same as
maximize U(n,p,Z)
subject to π_{n} n + π_{q} q + Z = M + npq
Define S = M + npq to be the parent's full income.
The resulting demand function for n is
n* = n*(π_{n}, π_{q}, S)
This is a standard Marshallian demand function with all the usual
properties. Note, however, that the shadow prices π_{n} and π_{q}
as well as the full income S are not parametric. More importantly, the
shadow price of n depends on q. If for some reason, the parent wants to
choose a higher q, this will raise the shadow price of n and hence reduce
n. As n falls, π_{q} = pn also falls, so the demand for q increases more
still, which in turn will lower the demand for n. Thus the nonlinear
nature of the budget constraint introduces a kind of multiplier
effect.
Now, back to the income effect. Suppose M increases exogenously. Other
things equal, this will raise the demand for n and q (inferior goods are a
taboo in the household production function approach!). But as n and q
increase, so do the shadow prices for n and q. The rise in shadow price
will tend to lower the demand for n and q, thereby moderating the original
increase in demand. The income elasticity of demand are therefore expected
to be relatively small given the nonlinear nature of the budget constraint.
Rotten kids
Parents love their kids, but the feeling is not always reciprocal. Think of
a parent whose utility function has both own consumption and child's
consumption as argument:
U_{p} = U_{p}(C_{p},C_{k})
The kid, on the other hand, cares only about his own consumption:
U_{k} = U_{k}(C_{k})
Suppose the parent's money income is M_{p} and the kid's money income is M_{k}.
If the parent is interested in the kid's welfare, he may want to transfer
income to him. Represent the amount of this income transfer by T. The
parent's budget constraint is
C_{p} + T = M_{p}
whereas the kid's budget constraint is
C_{k} = M_{k} + T
As long as T is positive, the parent effectively controls the kid's
consumption by choosing the approapriate amount of income transfer T. Use
the two budget constraints to substitute away C_{k} and T, the parent maximizes
U_{p} = U_{p}( C_{p}, M_{p} + M_{k}  C_{p} )
Two implications follow:
 Suppose you increases taxes on parents. M_{p} falls. But with higher
taxes today, the tax liability for future generations fall. So M_{k} rises by
an equal amount because of the government's budget constraint. With M_{p} +
M_{k} unchanged, the utility maximization problem remains unchanged. So as
long as the pattern of government expenditure remains constant, changes in
the time pattern of government financing will have no real effects. This is
the essence of the Ricardian equivalence theorem.
 Suppose the rotten kid can improve his parent's income by sacrificing
his own. Say this act raises M_{p} by a greater amount than it lowers M_{k}.
Will the rotten kid do it? The answer is yes. By doing so, he raises his
parent's full income. Since C_{k} is a normal good as far as the parent is
concerned, the parent will choose a higher C_{k} (T rises by more than the
reduction in M_{k}), and both the parent and the kid are better off. So even
though altruism is onesided, the rotten kid still has the incentive to
maximize family income rather than his own. This is known as the rotten
kid theorem.

Lecture Notes

Giffen goods
Competitive Markets