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:

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 th. Efficiency in production is parameterized by a, with higher a indicating greater efficiency. The household production function is written as

Z = f( x, ath )
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 tw
where tw is the time spent on market employment and A is unearned income. The time use for individual must also satisfy a time constraint
th + tw = T
where T is total time endowment.

We can combine these two constraints by writing

x + w th = 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:
fx - λ = 0
ft a - λ w = 0
These two equations can be combined to get fx/ft = 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 th such that fx/ft = a1/w1, while Ms. 2 chooses x and th such that fx/ft = a2/w2. But they could do better. Suppose a1/w1 > a2/w2. 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 a1 th1 + a2 th2. If Mr. 1 works 1 more hour in household production while Ms. 2 works a1/a2 fewer hours, total effective time devoted to household production remains unchanged. But the change in total wage income is

- w1 + w2 (a1/a2)
which is positive if a1/w1 > a2/w2. 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 a1 = a0 + α Ka1 and w1 = w0 + β Kw1, and similarly for a2 and w2. 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 Ka* and Kw*. Therefore a1/w1 = a2/w2. 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 Kw is no longer useful. So instead of incuring an investment cost of C(Ka*,Kw*), he could just spend C(Ka*,0) on investment. With this new investment level, his a1/w1 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 part-time. This consideration causes ex-ante 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

Un - λ pq = 0
Uq - λ pn = 0
UZ - λ = 0
M - npq - Z = 0
Suppose we define πn = pq and πq = pn. Then, the utility maximization problem is the same as
maximize U(n,p,Z)
subject to πn n + πq q + Z = M + npq
Define S = M + npq to be the parent's full income. The resulting demand function for n is
n* = n*(πn, πq, S)
This is a standard Marshallian demand function with all the usual properties. Note, however, that the shadow prices πn and πq 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, πq = pn also falls, so the demand for q increases more still, which in turn will lower the demand for n. Thus the non-linear 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 non-linear 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:

Up = Up(Cp,Ck)
The kid, on the other hand, cares only about his own consumption:
Uk = Uk(Ck)
Suppose the parent's money income is Mp and the kid's money income is Mk. 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
Cp + T = Mp
whereas the kid's budget constraint is
Ck = Mk + 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 Ck and T, the parent maximizes
Up = Up( Cp, Mp + Mk - Cp )

Two implications follow:

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