Editing Mechanism design (section)

===Myerson ironing===

[[Image:Myerson ironing.png|thumb|325px| It is possible to solve for a goods or price schedule that satisfies the first-order conditions yet is not monotonic. If so it is necessary to "iron" the schedule by choosing some value at which to flatten the function.]]

In some applications the designer may solve the first-order conditions for the price and allocation schedules yet find they are not monotonic. For example, in the quasilinear setting this often happens when the hazard ratio is itself not monotone. By the Spence–Mirrlees condition the optimal price and allocation schedules must be monotonic, so the designer must eliminate any interval over which the schedule changes direction by flattening it.

Intuitively, what is going on is the designer finds it optimal to '''bunch''' certain types together and give them the same contract. Normally the designer motivates higher types to distinguish themselves by giving them a better deal. If there are insufficiently few higher types on the margin the designer does not find it worthwhile to grant lower types a concession (called their [[information rent]]) in order to charge higher types a type-specific contract.

Consider a monopolist principal selling to agents with quasilinear utility, the example above. Suppose the allocation schedule <math>x(\theta)</math> satisfying the first-order conditions has a single interior peak at <math>\theta_1</math> and a single interior trough at <math>\theta_2>\theta_1</math>, illustrated at right.

* Following Myerson (1981) flatten it by choosing <math>x</math> satisfying <math display="block"> \int^{\phi_1(x)}_{\phi_2(x)} \left( \frac{\partial V}{\partial x}(x,\theta) - \frac{1-P(\theta)}{p(\theta)} \frac{\partial^2 V}{\partial \theta \, \partial x}(x,\theta) - \frac{\partial c}{\partial x}(x) \right) d\theta = 0</math> where <math>\phi_1(x)</math> is the inverse function of x mapping to <math>\theta \leq \theta_1</math> and <math>\phi_2(x)</math> is the inverse function of x mapping to <math>\theta \geq \theta_2</math>. That is, <math>\phi_1</math> returns a <math>\theta</math> before the interior peak and <math>\phi_2</math> returns a <math>\theta</math> after the interior trough.
* If the nonmonotonic region of <math>x(\theta)</math> borders the edge of the type space, simply set the appropriate <math>\phi(x)</math> function (or both) to the boundary type. If there are multiple regions, see a textbook for an iterative procedure; it may be that more than one troughs should be ironed together.

====Proof====

The proof uses the theory of optimal control. It considers the set of intervals <math>\left[\underline\theta, \overline\theta \right] </math> in the nonmonotonic region of <math>x(\theta)</math> over which it might flatten the schedule. It then writes a Hamiltonian to obtain necessary conditions for a <math>x(\theta)</math> within the intervals
# that does satisfy monotonicity
# for which the monotonicity constraint is not binding on the boundaries of the interval
Condition two ensures that the <math>x(\theta)</math> satisfying the optimal control problem reconnects to the schedule in the original problem at the interval boundaries (no jumps). Any <math>x(\theta)</math> satisfying the necessary conditions must be flat because it must be monotonic and yet reconnect at the boundaries.

As before maximize the principal's expected payoff, but this time subject to the monotonicity constraint
:<math>\frac{\partial x}{\partial \theta} \geq 0</math>
and use a Hamiltonian to do it, with shadow price <math>\nu(\theta)</math>
:<math>H = \left( V(x,\theta) - \underline{u}(\theta_0) - \frac{1-P(\theta)}{p(\theta)} \frac{\partial V}{\partial \theta}(x,\theta) - c(x) \right)p(\theta) + \nu(\theta) \frac{\partial x}{\partial \theta} </math>
where <math>x</math> is a state variable and <math>\partial x/\partial \theta</math> the control. As usual in optimal control the costate evolution equation must satisfy
:<math> \frac{\partial \nu}{\partial \theta} = -\frac{\partial H}{\partial x} = -\left( \frac{\partial V}{\partial x}(x,\theta) - \frac{1-P(\theta)}{p(\theta)} \frac{\partial^2 V}{\partial \theta \, \partial x}(x,\theta) - \frac{\partial c}{\partial x}(x) \right) p(\theta) </math>
Taking advantage of condition 2, note the monotonicity constraint is not binding at the boundaries of the <math>\theta</math> interval,
:<math>\nu(\underline\theta) = \nu(\overline\theta) = 0</math>
meaning the costate variable condition can be integrated and also equals 0
:<math>\int^{\overline\theta}_{\underline\theta} \left( \frac{\partial V}{\partial x}(x,\theta) - \frac{1-P(\theta)}{p(\theta)} \frac{\partial^2 V}{\partial \theta \, \partial x}(x,\theta) - \frac{\partial c}{\partial x}(x) \right) p(\theta) \, d\theta = 0 </math>
The average distortion of the principal's surplus must be 0. To flatten the schedule, find an <math>x</math> such that its inverse image maps to a <math>\theta</math> interval satisfying the condition above.