<p>Take a consumer with utility
<span class="math-container">$$
U = x - x^2 - \theta x d,
$$</span>
where <span class="math-container">$x$</span> is the amount of information given to the firm and <span class="math-container">$d$</span> is the disclosure set by the firm.</p>
<p>The optimal level of <span class="math-container">$x$</span> is given by the first order condition:
<span class="math-container">$$
1 - 2 x - \theta d = 0 \to x = \frac{1 - \theta d}{2}
$$</span>
A consumer will buy from the firm if her utility is greater than zero. Inserting the optimal value of <span class="math-container">$x$</span> into the utility function gives:
<span class="math-container">$$
x(1 - \theta d - x) = \frac{(1 - \theta d)^2}{4},
$$</span>
which is always greater or equal to zero. So everyone will buy from the firm. If <span class="math-container">$\theta \sim U[0,1]$</span> then the total amount disclosed is:
<span class="math-container">$$
\int_0^1 \frac{1 - \theta d}{2} d\theta = \frac{1}{2}[\theta]^1_0 - \frac{d}{2}\left[\frac{\theta^2}{2}\right]^1_0 = \frac{1}{2} - \frac{d}{4}
$$</span>
From this, we see that profits of the firm are given by:
<span class="math-container">$$
d\int_0^1 x_\theta d \theta = \left(\frac{1}{2} - \frac{d}{4}\right) d.
$$</span>
Maximizing this with respect to <span class="math-container">$d$</span> gives,
<span class="math-container">$$
\frac{1}{2} - \frac{d}{2} = 0 \to d = 1.
$$</span>
If there is a price <span class="math-container">$P&gt; 0$</span> things change. In this case, the consumer will buy from the firm if:
<span class="math-container">$$
\frac{(1 - \theta d)^2}{4} \ge P,\\
\to (1 - \theta d) \ge 2 \sqrt{P},\\
\to \theta \le \frac{1 - 2 \sqrt{P}}{d}
$$</span>
If the right hand side is between zero and one, then the amount of consumers that will buy from the firm is given by:
<span class="math-container">$$
\frac{1 - 2 \sqrt{P}}{d}
$$</span>
The total amount disclosed by these consumers is given by:
<span class="math-container">$$
\begin{align*}
&amp;\int_0^{\frac{1 - 2 \sqrt{P}}{d}} \frac{1 - \theta d}{2} d\theta = \frac{1}{2}\frac{1 - 2 \sqrt{P}}{d} - \frac{d}{4}\left(\frac{1 - 2 \sqrt{P}}{d}\right)^2,\\
&amp;= \frac{1}{2}\left(\frac{1 - 2\sqrt{P}}{d}\right)\left(\frac{1 + 2 \sqrt{P}}{2}\right),\\
&amp;= \frac{1}{4d}(1 - 4P)
\end{align*}
$$</span>
Then total profits of the firm are given by the total mass of consumers buying from the firm times the price plus the benefit from disclosure:
<span class="math-container">$$
\frac{1}{4d}(1 - 4P)d + \frac{1 - 2 \sqrt{P}}{d}P,\\
= \frac{1}{4}(1 - 4P) + \frac{P}{d} - 2\frac{P^{3/2}}{d}
$$</span>
The first order condition with respect to <span class="math-container">$P$</span> is given by:
<span class="math-container">$$
\begin{align*}
&amp;-1 + \frac{1}{d} - 3 \frac{\sqrt{P}}{d} = 0,\\
\to &amp;\sqrt{P} = \frac{1}{3}(1 - d),\\
\to &amp;P = \frac{1}{9}(1 - d)^2
\end{align*}
$$</span>
Then profits are equal to:
<span class="math-container">$$
\frac{1}{4}\left(1 - \frac{4}{9}(1 - d)^2\right) + \frac{1}{9}\frac{(1-d)^2}{d} - 2 \frac{1}{27d}(1 - d)^3
$$</span>
This should be maxmized with respect to <span class="math-container">$d$</span>, taking into account that we required <span class="math-container">$\frac{1 - 2\sqrt{P}}{d}$</span> to be between zero and one.</p>
<p>The cases where either <span class="math-container">$\frac{1 - 2\sqrt{P}}{d}$</span> is bigger than 1 or smaller than zero can then also be considered. The firm then takes the case where profits are maximal.</p>

CC BY-SA 4.0
2021-06-28T08:11:47.657Z
Thank you, I had a mistake in the indifferent consumer. It would have been enough to show me that ;)
KArrow'sBest
CC BY-SA 4.0
2021-06-28T09:03:01.300Z
I only have one further question, that is what gave me a lot of confusion. The indifferent consumer should also solve the equation for optimal information $x^* = 0 $ don't you think? I'd interpret that as the consumer $\theta$ that would give 0 information so he would be indifferent in buying or not buying.
KArrow'sBest
CC BY-SA 4.0
2021-06-28T10:08:06.063Z
@KArrow'sBest I think the indifferent consumer is the one that is indifferent between buying from the firm (and offering her optimal amount of info) and not buying from the firm. So the fact that this consumer chooses $x$ in the optimally is already taken into account.
tdm
CC BY-SA 4.0
2021-06-28T10:26:14.847Z
I think there is a mistake in the part: $\frac{1}{4}\left(1 - \frac{4}{9}(1 - d)^2\right) + \frac{1}{9}\frac{(1-d)^2}{d} - 2 \frac{1}{27}(1 - d)^3$ that should be $\frac{1}{4}\left(1 - \frac{4}{9}(1 - d)^2\right) + \frac{1}{9}\frac{(1-d)^2}{d} - 2 \frac{1}{27 d}(1 - d)^3$
KArrow'sBest
CC BY-SA 4.0
2021-06-28T13:31:06.817Z
@KArrow'sBest You are correct. Stupid mistake. This makes the problem considerably more complicated.
tdm
CC BY-SA 4.0
2021-06-28T15:55:34.557Z
Thank you anyway. Not having access to the department and face-to-face exchange with colleagues you helped already a lot. I think given the complexity (I am not using std uniform) I will solve the model numerically to see what changes with respect to the published article. ;)
KArrow'sBest
CC BY-SA 4.0
2021-06-25T11:18:46.970Z
2021-06-28T13:30:04.650Z
2021-06-28T13:30:04.650Z
tdm
<p>Today's question is about a variant of Tirole vertical differentiation framework. I am stuck thinking about the demand and profit function derivation where consumers can pick the level of <span class="math-container">$x$</span> at their last stage.
Suppose the following utility:
<span class="math-container">$$ U = x - x^2 -\theta xd_i - P_i $$</span></p>
<p>To simplify here let's assume there are no prices (<span class="math-container">$P_i = 0$</span>) and the firm revenues are a function (say multiplicative) of <span class="math-container">$\pi(x_k, d_i)=\sum_kx_kd_i$</span> where <span class="math-container">$k$</span> indicates the consumers and <span class="math-container">$i$</span> the firm.</p>
<p>In this case we'd have <span class="math-container">$$ 1-2x_k^*-\theta_k d_i = 0 \iff x_k^* = \frac{1-\theta_k d_i}{2}$$</span></p>
<p>Then I find the indifferent consumer as <span class="math-container">$$ x(1 - x -\theta' d_i) =0$$</span></p>
<p>Inserting the optimal level of <span class="math-container">$x$</span> this becomes:
<span class="math-container">$$ \theta' =1/d_k$$</span></p>
<p>(this is also the solution at <span class="math-container">$x_k^* = 0$</span>)</p>
<p>If I go on and suppose the standard uniform as a distribution for <span class="math-container">$\theta \in [0,1]$</span> the demand would be given the <span class="math-container">$\frac{\partial U}{\partial \theta}&lt;0$</span>:
<span class="math-container">$$D= 1/d_k$$</span></p>
<p>Is there a problem with this approach that is recursive? What could be a strategy to solve this kind of problem, that is non-standard in the vertical differentiation framework?
What stupid things am I saying?</p>
<p>Related question on the original model <a href="https://economics.stackexchange.com/questions/45601/profit-function-in-vertical-differentiation-model-competing-with-privacy">here</a></p>

CC BY-SA 4.0
2021-06-23T08:57:32.513Z
what is $d_i$? If $i$ varies over the firms, why does the utility function only depend on one $d_i$ (i.e; only one firm)? What is $x$? Can you give some more info on what, and how, consumers make a choice. Maybe a link to the relevant article?
tdm
CC BY-SA 4.0
2021-06-23T10:05:11.907Z
Thanks for the comment. Only monopolist. The issue is conceptual, but I would provide more input: I am playing with the functions in "Competing with privacy" of Casadeus-Masanell management science. The only difference is that I want consumers x* (in their paper y*) dependent on the consumer type, so basically that optimal x* is a function of $\theta$.
KArrow'sBest
CC BY-SA 4.0
2021-06-23T10:25:54.493Z
https://dash.harvard.edu/bitstream/handle/1/13135313/casadesus-masanell,hervas-drane_competing-with-privacy05.pdf?sequence=1
KArrow'sBest
<mathematical-economics><product-differentiation>
Demand derivation in vertical differentiation with a bad characteristic
KArrow'sBest
# Demand derivation in vertical differentiation with a bad characteristic

(Asked by: username_0 on 2021-06-22)

Today's question is about a variant of Tirole vertical differentiation framework. I am stuck thinking about the demand and profit function derivation where consumers can pick the level of $x$ at their last stage.
Suppose the following utility:
$$ U = x - x^2 -\theta xd\_i - P\_i $$

To simplify here let's assume there are no prices ($P\_i = 0$) and the firm revenues are a function (say multiplicative) of $\pi(x\_k, d\_i)=\sum\_kx\_kd\_i$ where $k$ indicates the consumers and $i$ the firm.

In this case we'd have $$ 1-2x\_k^\*-\theta\_k d\_i = 0 \iff x\_k^\* = \frac{1-\theta\_k d\_i}{2}$$

Then I find the indifferent consumer as $$ x(1 - x -\theta' d\_i) =0$$

Inserting the optimal level of $x$ this becomes:
$$ \theta' =1/d\_k$$

(this is also the solution at $x\_k^\* = 0$)

If I go on and suppose the standard uniform as a distribution for $\theta \in [0,1]$ the demand would be given the $\frac{\partial U}{\partial \theta}<0$:
$$D= 1/d\_k$$

Is there a problem with this approach that is recursive? What could be a strategy to solve this kind of problem, that is non-standard in the vertical differentiation framework?
What stupid things am I saying?

Related question on the original model [here](https://economics.stackexchange.com/questions/45601/profit-function-in-vertical-differentiation-model-competing-with-privacy)

**Tags:** mathematical-economics, product-differentiation

> **Comments:**
> **username_1**: what is $d_i$? If $i$ varies over the firms, why does the utility function only depend on one $d_i$ (i.e; only one firm)? What is $x$? Can you give some more info on what, and how, consumers make a choice. Maybe a link to the relevant article?  
> **username_0**: Thanks for the comment. Only monopolist. The issue is conceptual, but I would provide more input: I am playing with the functions in "Competing with privacy" of Casadeus-Masanell management science. The only difference is that I want consumers x* (in their paper y*) dependent on the consumer type, so basically that optimal x* is a function of $\theta$.  
> **username_0**: https://dash.harvard.edu/bitstream/handle/1/13135313/casadesus-masanell,hervas-drane_competing-with-privacy05.pdf?sequence=1  

---

## 1 answers

### Answer by username_1 on 2021-06-25. Score: 2 | Accepted answer

Take a consumer with utility
$$
U = x - x^2 - \theta x d,
$$
where $x$ is the amount of information given to the firm and $d$ is the disclosure set by the firm.

The optimal level of $x$ is given by the first order condition:
$$
1 - 2 x - \theta d = 0 \to x = \frac{1 - \theta d}{2}
$$
A consumer will buy from the firm if her utility is greater than zero. Inserting the optimal value of $x$ into the utility function gives:
$$
x(1 - \theta d - x) = \frac{(1 - \theta d)^2}{4},
$$
which is always greater or equal to zero. So everyone will buy from the firm. If $\theta \sim U[0,1]$ then the total amount disclosed is:
$$
\int\_0^1 \frac{1 - \theta d}{2} d\theta = \frac{1}{2}[\theta]^1\_0 - \frac{d}{2}\left[\frac{\theta^2}{2}\right]^1\_0 = \frac{1}{2} - \frac{d}{4}
$$
From this, we see that profits of the firm are given by:
$$
d\int\_0^1 x\_\theta d \theta = \left(\frac{1}{2} - \frac{d}{4}\right) d.
$$
Maximizing this with respect to $d$ gives,
$$
\frac{1}{2} - \frac{d}{2} = 0 \to d = 1.
$$
If there is a price $P> 0$ things change. In this case, the consumer will buy from the firm if:
$$
\frac{(1 - \theta d)^2}{4} \ge P,\\
\to (1 - \theta d) \ge 2 \sqrt{P},\\
\to \theta \le \frac{1 - 2 \sqrt{P}}{d}
$$
If the right hand side is between zero and one, then the amount of consumers that will buy from the firm is given by:
$$
\frac{1 - 2 \sqrt{P}}{d}
$$
The total amount disclosed by these consumers is given by:
$$
\begin{align\*}
&\int\_0^{\frac{1 - 2 \sqrt{P}}{d}} \frac{1 - \theta d}{2} d\theta = \frac{1}{2}\frac{1 - 2 \sqrt{P}}{d} - \frac{d}{4}\left(\frac{1 - 2 \sqrt{P}}{d}\right)^2,\\
&= \frac{1}{2}\left(\frac{1 - 2\sqrt{P}}{d}\right)\left(\frac{1 + 2 \sqrt{P}}{2}\right),\\
&= \frac{1}{4d}(1 - 4P)
\end{align\*}
$$
Then total profits of the firm are given by the total mass of consumers buying from the firm times the price plus the benefit from disclosure:
$$
\frac{1}{4d}(1 - 4P)d + \frac{1 - 2 \sqrt{P}}{d}P,\\
= \frac{1}{4}(1 - 4P) + \frac{P}{d} - 2\frac{P^{3/2}}{d}
$$
The first order condition with respect to $P$ is given by:
$$
\begin{align\*}
&-1 + \frac{1}{d} - 3 \frac{\sqrt{P}}{d} = 0,\\
\to &\sqrt{P} = \frac{1}{3}(1 - d),\\
\to &P = \frac{1}{9}(1 - d)^2
\end{align\*}
$$
Then profits are equal to:
$$
\frac{1}{4}\left(1 - \frac{4}{9}(1 - d)^2\right) + \frac{1}{9}\frac{(1-d)^2}{d} - 2 \frac{1}{27d}(1 - d)^3
$$
This should be maxmized with respect to $d$, taking into account that we required $\frac{1 - 2\sqrt{P}}{d}$ to be between zero and one.

The cases where either $\frac{1 - 2\sqrt{P}}{d}$ is bigger than 1 or smaller than zero can then also be considered. The firm then takes the case where profits are maximal.

> **Comments:**
> **username_0**: Thank you, I had a mistake in the indifferent consumer. It would have been enough to show me that ;)  
> **username_0**: I only have one further question, that is what gave me a lot of confusion. The indifferent consumer should also solve the equation for optimal information $x^* = 0 $ don't you think? I'd interpret that as the consumer $\theta$ that would give 0 information so he would be indifferent in buying or not buying.  
> **username_1**: @username_0 I think the indifferent consumer is the one that is indifferent between buying from the firm (and offering her optimal amount of info) and not buying from the firm. So the fact that this consumer chooses $x$ in the optimally is already taken into account.  
> **username_0**: I think there is a mistake in the part: $\frac{1}{4}\left(1 - \frac{4}{9}(1 - d)^2\right) + \frac{1}{9}\frac{(1-d)^2}{d} - 2 \frac{1}{27}(1 - d)^3$ that should be $\frac{1}{4}\left(1 - \frac{4}{9}(1 - d)^2\right) + \frac{1}{9}\frac{(1-d)^2}{d} - 2 \frac{1}{27 d}(1 - d)^3$  
> **username_1**: @username_0 You are correct. Stupid mistake. This makes the problem considerably more complicated.  
> **username_0**: Thank you anyway. Not having access to the department and face-to-face exchange with colleagues you helped already a lot. I think given the complexity (I am not using std uniform) I will solve the model numerically to see what changes with respect to the published article. ;)  

---

