# Demand function of a family

Different family members have different utility functions, but All family members consume the same bundle.

For example, consider a family $F$ that has to select a bundle of funiture ($x$) and electronic equipment ($y$). Each member $i\in F$ has a different utility function $u_i(x,y)$. The family has a budget $I$. How can the demand of the family be calculated?

I thought of several options:

1. Calculate an aggregate utility function, e.g:

$$u_F(x,y) = \min_{i\in F} u_i(x,y)$$

Then, calculate the demand in the usual way: select a bundle $(x_F,y_F)$ that maximizes the aggregate utility $u_F$ in the budget-set.

A problem with this method is that it requires to normalize the members' utility functions to the same scale.

1. Calculate the optimal bundle of each family member separately: each member selects a bundle $(x_i,y_i)$ that maximizes his utility function $u_i$ given the family's income $I$. Then, calculate the family bundle as an average of the members' bundles:

$$(x_F,y_F) = \frac{1}{|F|}\sum_{i\in F} (x_i,y_i)$$ If the budget-set is convex, then this bundle is also in the budget set.

1. Divide the family income $I$ among the family members, such that each member $i\in F$ receives an income $I/|F|$. Then, let each member select a bundle $(x_i',y_i')$ that maximizes his utility function $u_i$ in given his fraction of the income. Then, calculate the family bundle as a sum of the members' bundles:

$$(x_F,y_F) = \sum_{i\in F} (x_i',y_i')$$

Each definition probably has different implications on results such as competitive equilibrium, welfare theorem, etc.

What is a good reference on this problem?

• I think you need to look up optimality conditions for "public goods". The exact solution would still depend on the decision making structure of the family (as @HRSE pointed out in his answer) but you could derive all the Pareto-optimal consumption bundles. – Giskard Nov 19 '15 at 10:16
• This seems to be closely related to the literature on social choice theory, it one replaces family with society and family member with voter. – Ubiquitous Nov 19 '15 at 14:06
• A Classic reference for this type of problem is the book "Economics of the Family" by Browning and Chiaporri. – ChinG Nov 19 '15 at 19:58

For example, suppose the matriarch $i$ decides on every purchase of the family and acts selfishly. Then obviously the utility function of the family is simply $u_i$.