# About the optimal period 1 price in a 2 period scenario for switching costs (Klemperer, 1995)

In "Competition when consumers have switching costs" (1995), Klemperer puts forward an example with 2 periods, 2 firms, switching costs and transport costs in a Hoteling model. He then shows the equilibrium prices in both period, with the equilibrium period 1 price a firm will charge being pushed down by the monopoly power the firm obtains from switching costs. The 2nd period price equals the reservation price of the consumer, again due to the monopoly power resulting from switching costs. He then states that the period 1 price is below that of period 2 price, which is logical enough (the monopoly power deriving from switching costs makes it attractive to build market share in the first period, which can be milked in the second period - it's a notion that occurs and is demonstrated throughout the paper). However, he makes the statement with an claim I just can't wrap my head around, arguing that $$p^A_1 < p^A_2$$ as long as the value a consumer puts in consuming a unit of the good, substracted by the transport cost, remains below the consumer's reservation price. This consumer value isn't part of the model of the example, so I can't understand how this would work mathematically, and also at an intuitive level I fail to grasph what he's saying. I'm posting the relevant section of the paper below, containing the mathematical framework. The example I described above is example 1, but the prior example is relevant to it, so I'm included that.

What I want to know is why the highlighted statement is true. Intuitively at a minimum.

• What do you mean with "This consumer value isn't part of the model of the example"? Jun 24, 2023 at 12:05
• I think it's a typo. It should be " ... unless all consumers' demands are considerably higher in period 1 than in period 2." Jun 24, 2023 at 12:07
• @VARulle: I meant to say that r doesn't appear in the model of example 1 (or that of 0), although I now realise the assumption is that $r - 2T > c_1$. Could $(r - T) < R$ actually be about the period 1 marginal costs? (since both statements imply $R > c_1$) Jun 24, 2023 at 12:58
• Example 1 assumes that "a consumer ... in period one values consuming one unit of the product at r less his transport cost." Later it adds the inequality you cited as an assumption. Then it calculates equilibrium prices and says that it is easy to check that r - T < R implies that p_1 < p_2. It also adds an interpretation of r - T < R (correcting what I think is a typo: demand in period one is not extremely high). I'm not sure what the problem is? Jun 25, 2023 at 12:34
• It's not clear to me why $r - t < R$ necessarily implies $p_1 < p_2$. If you substitute the formula E3 for $p_1$ and R for $p_2$ and separate R, you get $\frac{1}{1+δ}(T + \frac{2c^A_1 +c^B_1}{3} + δ\frac{2c^A_2 - c^B_2}{3}) < R$. Even if $c_1$ and $c_2$ individually are smaller than R, that doesn't seem to imply (to me) that the inequality is true. And if it does why bother to involve r? I just can't follow the logic. Jun 25, 2023 at 14:59

## 1 Answer

Note that there seems to be a typo in the second half of the highlighted sentence. It should be "... unless all consumers' demands are considerably higher in period 1 than in period 2."

Also note that $$r$$ and $$R$$ are just the consumers' reservation prices in periods 1 and 2, respectively.

To see what is claimed "easy to check", let's denote the larger of the two firms' marginal costs in period $$t$$ by $$c_t$$. Then (E3) implies $$p_1 \le T + c_1 -\delta[\cdots]$$. Since the term in square brackets is positive (by the assumption $$R>c_2$$), we have $$p_1 < T + c_1$$. Now the assumption $$r - 2T > c_1$$ implies $$T + c_1 < r - T$$, and together with the previous inequality we get $$p_1 < r - T$$. This together with $$r - T < R$$ then implies $$p_1 < R$$. Since in equilibrium $$R=p_2$$, we get $$p_1.

• Did you mean to write $c_t$ instead of $c_1$ in $p_1 < T - c_1 - δ[...]$? I don't follow how you got to that inequality. It also occurs to me you don't need that step: I believe $p_1 = T - c_1 - δ[...]$ implies $p_1 < T - c_1$. Jun 27, 2023 at 8:02
• @Nasan: I just substituted $t=1$, since (E3) is about period 1. The inequality follows from (E3) since $c_1\ge\left[\frac{2c_1^A+c_1^B}{3}\right]$. But it should have been $\le$ instead of $<$, corrected that in my answer now. Jun 27, 2023 at 8:47