Answer

Solution

1. 1

   The monopolist's optimum lies at one of the buyers' valuations (any higher price

2. 2

   loses an additional buyer without gain, any lower price wastes surplus). Profit at each:

3. 3

   \begin{itemize}

4. 4
   • Price $v_A$: only $A$ buys, profit $= v_A$.
5. 5
   • Price $v_B$: $A$ and $B$ buy, profit $= 2 v_B$.
6. 6
   • Price $v_C$: all three buy, profit $= 3 v_C$.
7. 7

   \end{itemize}

8. 8

   Compare:

9. 9
   $$2 v_B \;\;\text{vs}\;\; v_A: \quad v_A < \tfrac{7 v_B}{6} < 2 v_B, \quad\text{so}\quad 2 v_B > v_A.$$
10. 10
    $$2 v_B \;\;\text{vs}\;\; 3 v_C: \quad v_B > \tfrac{8 v_C}{5}, \quad\text{so}\quad 2 v_B > \tfrac{16}{5} v_C > 3 v_C.$$
11. 11

    Thus charging $v_B$ yields the highest profit.

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