Demand derivation in vertical differentiation with a bad characteristic

Question

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

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.