Editing Mechanism design (section)

===Price discrimination===

{{harvs|txt|last=Mirrlees|year=1971|author-link=James Mirrlees}} introduces a setting in which the transfer function ''t''() is easy to solve for. Due to its relevance and tractability it is a common setting in the literature. Consider a single-good, single-agent setting in which the agent has [[quasilinear utility]] with an unknown type parameter <math>\theta</math>
:<math>u(x,t,\theta) = V(x,\theta) - t</math>
and in which the principal has a prior [[Cumulative distribution function|CDF]] over the agent's type <math>P(\theta)</math>. The principal can produce goods at a convex marginal cost ''c''(''x'') and wants to maximize the expected profit from the transaction
:<math>\max_{x(\theta),t(\theta)} \mathbb{E}_\theta \left[ t(\theta) - c\left(x(\theta)\right) \right]</math>
subject to IC and IR conditions
:<math> u(x(\theta),t(\theta),\theta) \geq u(x(\theta'),t(\theta'),\theta) \ \forall \theta,\theta' </math>
:<math> u(x(\theta),t(\theta),\theta) \geq \underline{u}(\theta) \ \forall \theta </math>
The principal here is a monopolist trying to set a profit-maximizing price scheme in which it cannot identify the type of the customer. A common example is an airline setting fares for business, leisure and student travelers. Due to the IR condition it has to give every type a good enough deal to induce participation. Due to the IC condition it has to give every type a good enough deal that the type prefers its deal to that of any other.

A trick given by Mirrlees (1971) is to use the [[envelope theorem]] to eliminate the transfer function from the expectation to be maximized,
:<math>\text{let } U(\theta) = \max_{\theta'} u\left(x(\theta'),t(\theta'),\theta \right)</math>
:<math>\frac{dU}{d\theta} = \frac{\partial u}{\partial \theta} = \frac{\partial V}{\partial \theta}</math>
Integrating,
:<math>U(\theta) = \underline{u}(\theta_0) + \int^\theta_{\theta_0} \frac{\partial V}{\partial \tilde\theta} d\tilde\theta</math>
where <math>\theta_0</math> is some index type. Replacing the incentive-compatible <math>t(\theta) = V(x(\theta),\theta) - U(\theta)</math> in the maximand,
:<math>\begin{align}& \mathbb{E}_\theta \left[ V(x(\theta),\theta) - \underline{u}(\theta_0) - \int^\theta_{\theta_0} \frac{\partial V}{\partial \tilde\theta} d\tilde\theta - c\left(x(\theta)\right) \right] \\
&{} =\mathbb{E}_\theta \left[ V(x(\theta),\theta) - \underline{u}(\theta_0) - \frac{1-P(\theta)}{p(\theta)} \frac{\partial V}{\partial \theta} - c\left(x(\theta)\right) \right]\end{align}</math>
after an integration by parts. This function can be maximized pointwise.

Because <math>U(\theta)</math> is incentive-compatible already the designer can drop the IC constraint. If the utility function satisfies the Spence–Mirrlees condition then a monotonic <math>x(\theta)</math> function exists. The IR constraint can be checked at equilibrium and the fee schedule raised or lowered accordingly. Additionally, note the presence of a [[hazard rate]] in the expression. If the type distribution bears the monotone hazard ratio property, the FOC is sufficient to solve for ''t''(). If not, then it is necessary to check whether the monotonicity constraint (see [[Mechanism design#Sufficiency|sufficiency]], above) is satisfied everywhere along the allocation and fee schedules. If not, then the designer must use Myerson ironing.