Water Price Q2 Q1 Water Quantity NSC In Figure 3.1 b, the current social cost of water is the area of the OPZCQmax rectangle, or P2 times Qmax. Let's ignore the shape of the demand {or marginal benefit) curve for now. Itwater would be priced according to each resident's consumption instead, at a price of P1 per unit, it is expected that the average consumption would decrease to Q1, and there will be a gain in social surplus equivalent to the area BCDE. Indeed, at a new average consumption point like E, where the quantity consumed dropped to Q1 and consumers are paying a marginal private cost (MPC)of P1, the previously 'wasted' surplus marked as 35 gain is recovered. However, to calculate this gain, we need to make some assumptions about the (unknown) true shape of D, For instance, we can assume demand tor residential water is linear (i.e., it has a constant slope), or loglinear (i.e., it has a constant price-elasticity). Let's look briefly at these two cases. IfD is assumed to be a linear function relating the price and the quantity of water, it will be ofthe form: 9 = a l [3 X p, where is the quantityeaxis intercept of the inverse demand curve, while is its slope. In Figure 3.l , a: : Qmm, while ,6 is the unknown slope of demand for residential water services. How can we estimate this linear demand ifthe slope is unknown? One common way in practice is to look at the literature for studies of other communities and 'borrow' their slope estimates. Ideally, you would want the socioeconomic characteristics of various communities to match as closely as possible, in order to achieve what is called external validity. Suppose we nd a study of several comparable communities that found an average slope of demand to be Bye. Then, we can go ahead and plot the equation 41 = QM + 5mm X pas representing the demand curve. Note that in order to plot it in the format of Figure 3.l , we need to represent the equation of the inverse demandcurve (price as a function of qua ntity), or: 5:\".qu Q...) '53:? 1 w ave where time is a negative constant. This inverse demand curve has its horizontal intercept in the 'flat fee' consumption point of me, which corresponds top : l]. The rest of the demand curve is drawn using the slope estimate from the literature ne, which for instance implies a quantity demanded of Q1 when the price is P1. Notice that at a consumption-based price equal to P2, which is the flat fee charged by the city under the previous pricing scheme, quantity demanded is actually only Q2. The elimination of'wasteful consumption' of [me Q1) through pricing water at P1 then translates into a social surplus gain of SSgain' p