<p>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! </p>

<p>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. </p>

CC BY-SA 3.0
2017-10-25T01:36:31.280Z
2017-10-25T01:36:31.280Z
JDY
<p>First, it's important to be clear about what Rochet proves. In theorem $1$ of Rochet (1987), he shows that <em>cyclical monotonicity</em> 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.)</p>

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

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

<p>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.) <a href="https://math.stackexchange.com/questions/373229/show-that-the-maximum-of-a-set-of-convex-functions-is-again-convex">This</a> may also help.</p>

CC BY-SA 3.0
2017-11-25T09:46:45.950Z
2017-11-25T09:53:09.190Z
2017-11-25T09:53:09.190Z
Theoretical Economist
<p>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 <a href="https://ideas.repec.org/a/eee/mateco/v16y1987i2p191-200.html" rel="nofollow noreferrer">Rochet (1987)</a> 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. </p>

<p>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 $t: [0,1]^n \to \mathbb{R}$. </p>

<p>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? </p>

<microeconomics><reference-request><mathematical-economics><adverse-selection>
Multidimensional screening and convexity of the surplus/rent function
JDY
# Multidimensional screening and convexity of the surplus/rent function

(Asked by: username_0 on 2017-08-17)

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)](https://ideas.repec.org/a/eee/mateco/v16y1987i2p191-200.html) 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?

**Tags:** microeconomics, reference-request, mathematical-economics, adverse-selection

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## 2 answers

### Answer by username_1 on 2017-11-25. Score: 2 | Accepted answer

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](https://math.stackexchange.com/questions/373229/show-that-the-maximum-of-a-set-of-convex-functions-is-again-convex) may also help.


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### Answer by username_0 on 2017-10-25. Score: 0

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.


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