Public-good valuation and intrafamily allocation

I derive values of marginal changes in a public good for two-person households, measured alternatively by household member i's willingness to pay (WTP) for the good on behalf of the household, WTPi(H), or by the sum of individual WTP values across family members, WTP(C). Households are assumed to allocate their resources in efficient Nash bargains over private and common household goods. WTPi(H) is then found by trading off the public good against household goods, and WTP(C) by trading the public good off against private goods. I then find that WTPi(H) is higher (lower) when member i has a high (low) marginal valuation of the public good, but on average represents WTP(C) correctly. Individuals then tend to represent households correctly on average when questioned about the household's WTP for a public good, even when they are purely selfish and answer truthfully. Counting all members' WTP answers on behalf of the household then leads to double counting. Pure and paternalistic altruism (the latter attached to consumption of the public good) move each member's WTP on behalf of the household closer to the true aggregate WTP, but only the latter raises aggregate WTP.