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

  • $\begingroup$ 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? $\endgroup$
    – tdm
    Commented Jun 23, 2021 at 8:57
  • $\begingroup$ 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 $\theta$. $\endgroup$ Commented Jun 23, 2021 at 10:05
  • $\begingroup$ dash.harvard.edu/bitstream/handle/1/13135313/… $\endgroup$ Commented Jun 23, 2021 at 10:25

1 Answer 1


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.

  • $\begingroup$ Thank you, I had a mistake in the indifferent consumer. It would have been enough to show me that ;) $\endgroup$ Commented Jun 28, 2021 at 8:11
  • $\begingroup$ 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. $\endgroup$ Commented Jun 28, 2021 at 9:03
  • $\begingroup$ @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. $\endgroup$
    – tdm
    Commented Jun 28, 2021 at 10:08
  • $\begingroup$ 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 $\frac{1}{4}\left(1 - \frac{4}{9}(1 - d)^2\right) + \frac{1}{9}\frac{(1-d)^2}{d} - 2 \frac{1}{27 d}(1 - d)^3$ $\endgroup$ Commented Jun 28, 2021 at 10:26
  • $\begingroup$ @KArrow'sBest You are correct. Stupid mistake. This makes the problem considerably more complicated. $\endgroup$
    – tdm
    Commented Jun 28, 2021 at 13:31

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