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Hi, everyone! I'd appreciate some help with this problem, please.

Suppose that a seller puts a one-unit good on the market, and a buyer comes around with an unknown valuation $θ$. This buyer has a quasi-linear utility function $θq-p,$ and costs are quadratic with $c(q) = \dfrac{1}{2}q^2.$

Now suppose that $θ$ is uniformly distributed on $[0,1].$

Find the optimal direct mechanism $(p(·),q(·)).$

At the moment, I understand that we would need to maximize the seller's expected revenue, though I am not sure to navigate this problem. All help is appreciated.

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  • $\begingroup$ It seems that no trade may be a feasible outcome. $\endgroup$
    – Nazarene
    Dec 9, 2023 at 5:16

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Let $(q(\theta), p(\theta))$ be the mechanism. Let us show that the optimal mechanism will define a cutoff $\theta^\ast$ such that $q(\theta) = 1$ and $p(\theta) = \theta^\ast$ if and only if $\theta \ge \theta^\ast$.

The IC constraint will require that for all $\theta$ and $\theta' \in [0,1]$:

$$ \theta q(\theta) - p(\theta) \ge \theta q(\theta') - p(\theta'). \tag{IC} $$

The PC (outside option normalized to $0$) tells us that: $$ \theta q(\theta) - p(\theta) \ge 0. \tag{PC} $$

Now assume that the mechanism specifies to sell to type $\theta$, so $q(\theta) = 1$

This requires that: $$ \theta - p(\theta) \ge 0 \to p(\theta) \le \theta. $$

Now, take any type $\theta' > \theta$. Then the (IC) constraint specifies that: $$ \theta' q(\theta') - p(\theta') \ge \theta' - p(\theta)\ge \theta' - \theta > 0. $$ If $p(\theta') \ge 0$ this requires that $q(\theta') = 1$.

Then using the (IC) constraint again, we have: $$ \theta'- p(\theta') \ge \theta' - p(\theta), $$ which implies that $p(\theta') \le p(\theta)$.

If $q(\theta) = 0$, then the (PC) constraint tells us that $p(\theta) = 0$.

Let $\theta^\ast$ be the lowest value of $\theta$ for which $q(\theta) = 1$. Summarizing, we have the following:

  • If $\theta \ge \theta^\ast$, then $q(\theta) = 1$ and $p(\theta) \le p(\theta^\ast)$.
  • If $\theta < \theta^\ast$ then $q(\theta) = 0$ and $p(\theta) = 0$.
  • For all $\theta$ with $q(\theta) = 1$, $p(\theta) \le \theta$

Note that, as the firm tries to maximize profits, it will decide to put $p(\theta) = p(\theta^\ast) = \theta^\ast$ for all $\theta \ge \theta^\ast$.

This implies the following rule.

  • For all $\theta \ge \theta^\ast$, set $q(\theta) = 1$ and $p(\theta) = \theta^\ast$.
  • For all $\theta < \theta^\ast$, set $q(\theta) = 0$ and $p(\theta) = 0$.

The profits will then be equal to: $$ \theta^\ast (1 - \theta^\ast) - \frac{1}{2}(1- \theta^\ast)^2. $$ Taking first order conditions gives: $$ 1-2\theta^\ast + (1- \theta^\ast) = 2 - 3\theta^\ast = 0 $$ The second order condition is negative so the objective function is concave. This gives a value of, $\theta^\ast = 2/3$.

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  • $\begingroup$ Thank you! I really appreciate your help. $\endgroup$
    – Nazarene
    Feb 1 at 15:08

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