  In continuous-choice settings, consumers decide not only on whether to
purchase a product, but also on how much to purchase. Thus, firms optimize a
full price schedule rather than a single price point. This paper provides a
methodology to empirically estimate the optimal schedule under
multi-dimensional consumer heterogeneity. We apply our method to novel data
from an educational-services firm that contains purchase-size information not
only for deals that materialized, but also for potential deals that eventually
failed. We show that this data, combined with identifying assumptions, helps
infer how price sensitivity varies with "customer size". Using our estimated
model, we show that the optimal second-degree price discrimination (i.e.,
optimal nonlinear tariff) improves the firm's profit upon linear pricing by at
least 5.5%. That said, this second-degree price discrimination scheme only
recovers 5.1% of the gap between the profitability of linear pricing and that
of infeasible first degree price discrimination. We also conduct several
further counterfactual analyses (i) empirically quantifying the magnitude by
which incentive-compatibility constraints impact the optimal pricing and
profits, (ii) comparing the role of demand- v.s. cost-side factors in shaping
the optimal price schedule, and (iii) studying the implications of fixed fees
for the optimal contract and profitability.
