3rd Degree Price Discrimination I came across a True/False question in my economics problem set:


  Is the following statement true or false?
  "A monopolist can practice
  third-degree price discrimination in the two markets it serves.
  Quantity demanded at any positive price is always higher in market 1
  than in market 2. Therefore, when the monopolist profit maximizes, it
  will certainly charge a higher price in market 1 than in market 2."


The answer is false. However, I do not understand why it is false. 

I considered two cases, case 1 where the monopolist charges both markets a uniform price. If the monopolist practices price discrimination, profits will increase for the monopolist, and price charged in market 1 will be higher than market 2. 

Case 2 is where only market 1 (since it is the "strong" market) is served under uniform pricing. By price discriminating, monopolist will also serve market 2 as it generates profit, and price in market 1 is higher than in market 2. 

Why then is the statement false? Am I missing out another case? 

Any help is greatly appreciated! Thank you.
Here is a counterexample (i.e. an example showing that the given statement is false). Assumptions:


The cost of producing each unit of the good is \$1.
All consumers are exactly identical, with each consumer willing to pay up to \$10 for one unit of the good. (Also, each consumer is willing to pay at most \$0 for a second unit of the good.)
Market 1 has 200 consumers, while Market 2 has 100. (And so, it is true that "quantity demanded at any positive price is always higher in market 1 than in market 2."†)


Since all the consumers are exactly identical, the profit-maximizing price in both markets is simply the maximum each consumer is willing to pay, namely \$10 (and the producer will produce 300 units of the good). Thus, it is false that the monopolist "will certainly charge a higher price in market 1 than in market 2".



†This statement is actually wrong. At a sufficiently-high price, the quantity demanded of a good in any two markets will be equal to zero. To correct this statement, one would either change "always higher" to "no lower" or append to the start of this statement "if quantity demanded is non-zero, then ..."