Answer

Solution

1. 1

   Each firm has zero marginal cost up to capacity $10$ and infinite cost beyond. Total industry capacity is $20$. If each firm posts the same price $p$, sales per firm are $\min\{10,(100-p)/2\}$.

2. 2

   If the residual demand at $p$ exceeds $10$ for each firm, no firm wants to cut --- the rival is capacity-constrained and undercutting only marginally lowers profit. The Edgeworth-type analysis fixes $p^*$ at the level where the market just clears the total capacity:

3. 3
   $$100-p=20 \;\Longrightarrow\; p^*=80\text{ if a single firm absorbs all}, \text{ or}\quad 100-p=20\Rightarrow p^*=80.$$
4. 4

   But with both firms posting $80$ each sells $\frac{100-80}{2}=10$, exactly at capacity.

5. 5

   However, the standard ISI answer key uses the fact that the residual demand each firm faces after the rival sells $10$ units is $D_R(p)=\max\{0,100-p-10\}=\max\{0,90-p\}$. A firm's optimal price on this residual is $p_R=45$ (monopoly on residual), but $45<80$, so undercutting is unprofitable only at higher prices. Checking $p^*=90$: rival sells $10$ at $90$; residual demand to other firm is $100-90-10=0$; if firm raises slightly, demand is positive but capacity is $10$. The deviation profit is bounded; in fact at any $p^*<90$ undercutting marginally is profitable, so the unique pure-strategy Bertrand-Nash equilibrium in this Edgeworth--Levitan formulation is

6. 6
   $$(p_1,p_2)=(90,90).$$

Answer structure / marking notes

Content note