# Multidimensional screening and convexity of the surplus/rent function

I'm starting to read the literature of multidimensional screening models for monopolists selling $n$ goods to a continuum of buyers with $m=n$ dimensional types, and Rochet (1987) proves that a mechanism is implementable if and only if the surplus function is convex. Let $s(\theta)$ be the surplus function defined as $\max\{ u(x,\theta) -t(\theta)\}$ where $x \in \mathbb{R}^n$, and $t$ is the tariff function the monopolist announces by the Taxation Principle.

For example, the monopolist solves the problem of maximizing the following functional (assuming 0 cost of producing the good) $$\max J(t) = \int_D t(\theta) \mathrm{d} \theta = \int_D u(x,\theta)-s(\theta) \mathrm{d} \theta$$ where $\theta \in [0,1]^n$ and $t: [0,1]^n \to \mathbb{R}$.

My question is: what is the intuition behind the fact that $s(\theta)$ must be convex? What happens if the function is piecewise linear (and hence trivially convex) but globally concave, i.e. there are kinks along null-measure sets? Can you point me to any resources?

## 2 Answers

First, it's important to be clear about what Rochet proves. In theorem $1$ of Rochet (1987), he shows that cyclical monotonicity is equivalent to the implementability of a mechanism for general environments (i.e. arbitrary allocation/type spaces). (See the statement of theorem $1$, or a standard text such as Borgers (2015), for the definition of cyclical monotonicity.)

The convexity result (proposition $2$) applies to the case when $u(x,\theta)$ is linear in $\theta$. In this case, we have that

$$V(\theta) = \max_{\theta^\prime} \, u\left(x\left(\theta^\prime\right),\theta\right) - t\left(\theta^\prime \right) .$$

The convexity of $V$ follows from the fact that it is the maximum of a family of linear functions. I'm not sure there's much more intuition to be given here, but a simple drawing may help. (That is, draw two straight lines that intersect. The maximum over the two straight lines will be convex, while the minimum will be concave.) This may also help.

This is an old question, and my apologies for surfacing this again. For anyone curious about the answer, I think the result is quite intuitive and unless I am missing something completely, the answer seems clear enough that I apologize for wasting your time!

In essence, incentive compatibility requires that truth-telling yields a higher payoff. Now we require this to be the case along all dimensions of the buyer's type. This is exactly akin to convexity, and we can see this through the Taylor Series expansion. Any movement along any other dimension is an underestimator.