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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?

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  • $\begingroup$ 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. $\endgroup$ – Giskard Nov 19 '15 at 10:16
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    $\begingroup$ This seems to be closely related to the literature on social choice theory, it one replaces family with society and family member with voter. $\endgroup$ – Ubiquitous Nov 19 '15 at 14:06
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    $\begingroup$ A Classic reference for this type of problem is the book "Economics of the Family" by Browning and Chiaporri. $\endgroup$ – ChinG Nov 19 '15 at 19:58
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I think any (and none) of your answers is correct. The market demand simply depends on how the family makes its purchasing decisions.

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 $u_i$.

Compare this with the case where the utility functions are cardinally comparable and the household is utilitarian. In this case, the family utility function is the sum of all utility functions.

More generally speaking, if purchase decisions are made via some game, then you have to solve the game first in order to obtain the demand functions. But whether a utility representation exists for the family utility and what it would look like crucially depends on the game structure.

One special case which has been discussed in the literature is the case where the household income is somehow distributed among family members and each family member spends separately. There then exists a class of utility functions for which the family expenditure will be independent of the distribution of income among the family members. For details, see https://en.wikipedia.org/wiki/Gorman_polar_form

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