RDP 2015-03: The Value of Payment Instruments: Estimating Willingness to Pay and Consumer Surplus Equation

\begin{array}{l} \Pr (y_{i, p_{l}, p_{u}} = 1 | x_{i}) & = Pr (p_{l} \leq W T P_{i} < p_{u}) \\ = Pr (p_{l} \leq α + {x^{'}}_{i} β + ε_{i} < p_{u}) \\ = Pr (p_{l} - α - {x^{'}}_{i} β \leq ε_{i} < p_{u} - α - {x^{'}}_{i} β) \\ = Pr (p_{u} - α - {x^{'}}_{i} β > ε_{i}) - \Pr (p_{l} - α - {x^{'}}_{i} β \geq ε_{i}) \\ = F (p_{u} - α - {x^{'}}_{i} β) - F (p_{l} - α - {x^{'}}_{i} β) \end{array}