<p>Consider the incentive and participation constraints. That is, look at which buyer types would choose which option.</p>

<p>A buyer does not buy anything if 
<span class="math-container">$$v_a &lt; p_a \quad v_b &lt; p_b \quad v_a +v_b &lt; p_{ab}.$$</span>
A buyer chooses only good <span class="math-container">$i \in \{a,b\}$</span> with <span class="math-container">$j\neq i$</span> if
<span class="math-container">$$v_i - p_i \geq 0 \quad v_i - p_i \geq v_i + v_j - p_{ab}.$$</span>
Otherwise she chooses the bundle.
With the assumption of independent uniformly distributed values, it then requires some caution to derive the probabilities of each event.</p>

<p>Finally, you want to maximize your profit given by
<span class="math-container">\begin{align}
(1-p_a)(p_{ab}-p_a) p_a + (1-p_b)(p_{ab}-p_b) p_b + \\
\left( \frac{(p_a + p_b - p_{ab})^2}{2} + (1 - p_a) (1 + p_a - p_{ab}) + (1 - p_b) (1 + p_b - p_{ab}) \\
- (1 - p_a) (1 - p_b) \right) p_{ab},
\end{align}</span>
which is super annoying, but Mathematica suggests that your solution is correct.</p>

CC BY-SA 4.0
2019-03-20T14:04:45.247Z
2019-03-20T17:07:56.050Z
2019-03-20T17:07:56.050Z
Bayesian
<p>Consider an example of a risk-neutral seller that has two distinct indivisible goods for sale. The seller wants to maximize the expected revenue. The buyer's utility is <span class="math-container">$$I_av_a+I_bv_b-t,$$</span> where <span class="math-container">$I_a,I_b$</span> are indicators of whether the buyer gets good <span class="math-container">$a$</span> or <span class="math-container">$b$</span> and <span class="math-container">$t$</span> is the monetary transfer. Both <span class="math-container">$v_a,v_b$</span> are the types of the buyer (willingness to buy the good).
<em>Let <span class="math-container">$v_a$</span> and <span class="math-container">$v_b$</span> be i.i.d. draws from a uniform distribution on <span class="math-container">$[0,1]$</span>.</em></p>

<p>Assume a seller quotes three prices: <span class="math-container">$p_a,p_b,p_{ab}$</span>, where <span class="math-container">$p_{ab}\leq p_a+p_b$</span>. What is the optimal choice of <span class="math-container">$p_a,p_b,p_{ab}$</span>?</p>

<p>The textbook (T. Borgers, An Intro to the theory of Mechanism Design) I am studying simply says that this is a simple calculus problem and gives the solution as <span class="math-container">$$p_a=p_b=2/3\\
p_{ab}=\frac{4-\sqrt{2}}{3}.$$</span></p>

<p>I am not sure how to do that. I'd appreciate any help or hint. Thank you.</p>

CC BY-SA 4.0
2019-03-20T13:31:27.193Z
I suggest not to close this question, because it is rather a conceptual one than a simple homework question. Moreover, it is more advanced than the usual undergrad textbook question. We should encourage everyone who is interested in economics on that level.
Let me add a missing assumption in your question (in italics).
Bayesian
<game-theory><mechanism-design>
Screening theory: Bundling problem
johnny09
# Screening theory: Bundling problem

(Asked by: username_0 on 2019-03-20)

Consider an example of a risk-neutral seller that has two distinct indivisible goods for sale. The seller wants to maximize the expected revenue. The buyer's utility is $$I\_av\_a+I\_bv\_b-t,$$ where $I\_a,I\_b$ are indicators of whether the buyer gets good $a$ or $b$ and $t$ is the monetary transfer. Both $v\_a,v\_b$ are the types of the buyer (willingness to buy the good).
*Let $v\_a$ and $v\_b$ be i.i.d. draws from a uniform distribution on $[0,1]$.*

Assume a seller quotes three prices: $p\_a,p\_b,p\_{ab}$, where $p\_{ab}\leq p\_a+p\_b$. What is the optimal choice of $p\_a,p\_b,p\_{ab}$?

The textbook (T. Borgers, An Intro to the theory of Mechanism Design) I am studying simply says that this is a simple calculus problem and gives the solution as $$p\_a=p\_b=2/3\\
p\_{ab}=\frac{4-\sqrt{2}}{3}.$$

I am not sure how to do that. I'd appreciate any help or hint. Thank you.

**Tags:** game-theory, mechanism-design

> **Comments:**
> **username_1**: I suggest not to close this question, because it is rather a conceptual one than a simple homework question. Moreover, it is more advanced than the usual undergrad textbook question. We should encourage everyone who is interested in economics on that level. Let me add a missing assumption in your question (in italics).  

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## 1 answers

### Answer by username_1 on 2019-03-20. Score: 6 | Accepted answer

Consider the incentive and participation constraints. That is, look at which buyer types would choose which option.

A buyer does not buy anything if
$$v\_a < p\_a \quad v\_b < p\_b \quad v\_a +v\_b < p\_{ab}.$$
A buyer chooses only good $i \in \{a,b\}$ with $j\neq i$ if
$$v\_i - p\_i \geq 0 \quad v\_i - p\_i \geq v\_i + v\_j - p\_{ab}.$$
Otherwise she chooses the bundle.
With the assumption of independent uniformly distributed values, it then requires some caution to derive the probabilities of each event.

Finally, you want to maximize your profit given by
\begin{align}
(1-p\_a)(p\_{ab}-p\_a) p\_a + (1-p\_b)(p\_{ab}-p\_b) p\_b + \\
\left( \frac{(p\_a + p\_b - p\_{ab})^2}{2} + (1 - p\_a) (1 + p\_a - p\_{ab}) + (1 - p\_b) (1 + p\_b - p\_{ab}) \\
- (1 - p\_a) (1 - p\_b) \right) p\_{ab},
\end{align}
which is super annoying, but Mathematica suggests that your solution is correct.


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