1
$\begingroup$

I'm trying to find the Nash equilibria in the following problem:

Two firms produce an identical product with no fixed costs. The marginal cost of each firm $i=1,2$ is $c_i\in (0,1)$. Customers are distributed uniformly on $[1,2]$. The location of firm $i$ is $i$. The firms simultaneously choose prices. We denote by $p_i\in [0,\bar{p}]$ the price chosen by firm $i$, where $\bar{p}>0$ is some large number. Customers have linear transportation costs: if customer $x\in [1,2]$ buys from firm 1 then he incurs a cost of $x-1$, and if he buys from firm 2 then he incurs a cost of $2-x$. The value of a unit of the product is $V>0$ for each customer, and each customer buys one unit of the product, from the firm that maximizes his utility. This is a game in normal form with two players (the firms).

How do I find the best response functions for both firms? Right now I have solved that: The net value of buying from firm 1 is $V_1=V-(x-1)-p_1$, and the net value of buying from firm 2 is $V_2=V-(2-x)-p_2$. For consumers located at $x=1, V_1=V-p_1, V_2=V-(1+p_2)$. If $p_1>1+p_2$, then even for consumers living at $x=1$, it's too expensive to buy from firm 1. Thus, all consumers buy from firm 2. For consumers located at $x=2, V_1=V-(1+p_1), V_2=V-p_2$. If $p_2>1+p_1$, then even for consumers living at $x=2$, it's too expensive to buy from firm 2. Thus, all consumers buy from firm 1.

Then if $p_2-1<p_1<1+p_2\implies p_2-p_1\in [-1,1]$, then when $V_1=V_2$, we have $x^*=\frac{3+p_2-p_1}{2}$. Then for all consumers located on the right of $x^*$, they buy from firm 2; for all consuemrs located on the left of $x^*$, they buy from firm 1.

How do I proceed from this?

edit: ok... so from what I've gathered, the payoff function should be \begin{align} u_i(p_i,p_{-i})=\begin{cases} 0\;\text{if }p_i\geq p_{-i}+1\\ (p_i-c_i)\cdot 1 \;\text{if }p_i\leq p_{-i}-1\\ (p_i-c_i)(\frac{1+p_{-i}-p_i}{2})\;\text{if }p_i\in (p_{-i}-1,p_{-i}+1)\end{cases} \end{align} Then the best response function should be \begin{align} BR_i(p_{-i})=\begin{cases} \frac{1+p_{-i}+c_i}{2}\;\text{(you can compute this by FOC of the 3rd case from $u$)} \;\text{if $p_{-i}$ is in a range}\\ p_{-i}-1\;\text{if $p_{-i}$ is in a range}\end{cases} \end{align} but I'm having trouble finding the range of $p_{-i}$.

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.