Note that _λ_<sup>ˆ</sup> = 2 _._ 7 implies a substantial wedge between ex-ante and ex-post willingnessto-pay; the median value of this wedge, expressed as the ex-post to ex-ante surplus ratio (at marginal cost), _e_<sup>(</sup><sup>_λ−_1)ˆ</sup><sup>_θi_</sup> , is 3.25 with an interquartile range of 2.35-4.40.

# **Exercise 1: Gains from a sophisticated two-part tariff**

Consider the unsophisticated social planner, who errs in assuming that preferences for piped water are exogenous and unchanging ( _λ_ = 1), but who ascertains the distribution of preferences _G_ ( _θ_ ) along with the short-run price elasticity of demand _α_ .<sup>32</sup> As noted, the optimal twopart tariff ( _F, p_ ) in this scenario was originally discussed by Auerbach and Pellechio (1978). Given alternative values of average cost _K_ , we compute ex-post social welfare at the contracts offered by the unsophisticated planner, where ex-post social welfare is the long-run ( _λ_ = 2 _._ 7) average consumer surplus net of average costs. We repeat this exercise for a sophisticated social planner, one who (correctly) assumes endogenous preferences for piped water and who knows that consumers are ex-ante unaware of having such preferences.

Figure 7 illustrates the benefits of taking endogenous preferences into account in pricing a new utility service. The percentage welfare gain attributable to sophisticated pricing, the difference between the solid and dashed curves, can be extremely large depending on the value of _K_ . To pin down the relevant scenario in our setting, we obtained actual cost figures from An Thinh water utility, consisting of their initial investment in plant and equipment (including the underground pipe network for the three villages of the commune), annual operating expenses, as well as the cost of a home connection and water meter. The water itself is pumped from a nearby river and hence is effectively free. Since the utility is privately owned and run as a business (albeit subject to government rate regulation), these figures probably represent the social opportunity cost of piped water provision in rural Vietnam quite accurately. Be that as it may, at the plant’s reported capacity of 10,000 customers and a plausible real interest rate of 5 percent, the implied _K_ works out to about 33 (in the appropriate units).<sup>33</sup>

> 32For concreteness, suppose that this planner commissions a study in which alternative ( _F, p_ ) contracts are randomly offered to potential customers and the resulting data are used to estimate take-up as a function of ( _F, p_ ). From the participation constraint, _e_<sup>_θ−αp_</sup> _≥ F_ , and a parametric assumption on _G_ , it is easy to see that both _G_ and _α_ are identified. Note that this experiment is different from the one actually conducted by Lee et al. (2020) in rural Kenya inasmuch as the latter only randomizes _F_ .

> 33Specifically, initial investment was reported to us as US$9.7 million in current dollars, annual operating costs at US$30,000, household connection cost at US$84. We convert all of these figures to monthly per customer flows. In the case of plant and equipment, we do so using a real interest rate equal to the lending rate minus the rate of inflation (both for 2019) as reported for Vietnam in World Bank statistics.

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