1

It's a version of the Hotelling model (product differentiation).

One firm is located at the beginning of a line between 0 to 3, one at the end of the line (so one at 0, one at 3). There are $b/2$ consumers at 0, and $b/2$ at 3. Over the interval [1,2] there are $1-b$. Linear transport costs = λ per unit of distance travelled.

Utility for consumer from buying from firm 1 is $U(x_1) = v - λx - p_1$ (change x for 1-x for firm 2). The indifference condition is $U(x_1)=U(x_2)$. What are the demand functions for each firm?

My answer: $D_1(p_1,p_2) = 1/2 + (p_2-p_1)/2λ + (b/2)*p_1$

Is this correct? I don't know how to factor in the fact that there are $b/2$ consumers at either end.

martial
  • 67

1 Answers1

1

No the above is not entirely correct. Hint: depending on $v$, $\lambda$ if the price differential is to too big or too small one firm may capture the entire market. What has to be $p_1-p_2$ so that all consumers at zero buy from the firm located at one? Your demand may be discontinuous (only piecewise continuous).

Edit (too large for comment):

I think in the question change x for 1-x for firm 2 is wrong, it should be change x for 3-x for firm 2 because firm 2 is at 3.

If a consumer at zero buys from 1 her utility is: $v-p_1$. If a consumer at zero buys from 2 her utility is $v-3\lambda-p_2$. So she is indifferent if $p_1-p_2=3\lambda$.

So the demand of firm 1 is (there is one case for you to complete): $$D_1(p_1,p_2)=\begin{cases} 0 &\text{ if } \quad p_1-p_2>3\lambda\\ \dfrac{b}{2} & \text{ if } \quad\lambda <p_1-p_2<3\lambda\\ \text{ ... } & \text{ if } \quad-\lambda<p_1-p_2<\lambda\\ 1 &\text{ if } \quad p_1-p_2<-3\lambda\end{cases}$$

Sergio Parreiras
  • 3,813
  • Are you asking 'what is $p_1 - p_2$ so that the $b/2$ consumers located at 0 actually consume from firm 0 (rather than firm at 3)?' In that case, I really don't know. I've spent so long tying to work this out but I can't! – martial Jun 03 '14 at 08:30
  • I'm asking what is the lowest value of $p_1-p_2$ so all consumers buy from the firm 2 (located at 3). As $p_1-p_2$ increases, at one point no one will be buying from firm 1 (located at zero). What is the value of $p_1-p_2$ that makes a consumer located at zero indifferent between buying from firm 1 and firm 2? – Sergio Parreiras Jun 03 '14 at 12:39
  • So all $b/2$ consumers at 0 buy from firm 1 (at 0), and all $b/2$ consumers at 3 buy from firm 2 - should I assume this, or is it obvious? Then the consumers over [1,2] are indifferent if $v-λ(1+x)-p_1 = v-λ(2-(1+x))-p_2$. Is this correct? – martial Jun 05 '14 at 14:48
  • And a consumer at 0 is indifferent if $p_2-p_1=3λ$, yes? – martial Jun 05 '14 at 14:58
  • To the last: yes! See my new answer. – Sergio Parreiras Jun 05 '14 at 18:45
  • @pj241: please see updated answer – Sergio Parreiras Jun 05 '14 at 18:58
  • But what if the consumer's utility $v$ was high enough such that he would always buy? Do we just then look at the indifferent consumer from buying between each firm? – martial Jun 06 '14 at 09:54
  • even if v is high, there would always exist prices even higher so he would not buy 2) but if the difference in prices is not too high nor too low, yes you look at the consumer who is indifferent to figure out the demand
    • – Sergio Parreiras Jun 06 '14 at 12:26