Optimal fee decisions in a Two-Part Tariff with resale and third-degree price discrimination

Say I have two customers - one is a low type and one is a high type (don't worry about their demand functions but assume they are linear). In the original case, we can differentiate consumers (i.e we know each type's demand function). Both pay a fee to get into a bar and then a per-unit competitive price for each drink they purchase (the standard two-part tariff case with third-degree price discrimination).

Now, assume regulation prohibits charging a higher price for 1 group, so the bar owner decides to only sell to the high-type consumer $$-$$ he doesn't set a fee for a low type consumer, so a low-type consumer can either pay the fee for the high type (which they won't do) or not enter the bar at all.

We cannot assume that there is no resale (i.e the high type consumer can purchase drinks for the low type and then for themselves as well) - therefore, we may indirectly sell to the low type. Suppose we know the profit-maximizing production allocation of drinks for the low-type (i.e $$MR = MC$$).

If we can't prevent resale, is it logical to set the price at the profit-maximizing condition for the low type? This would decrease the fee to be less than the entire consumer surplus for the high type, but the extra profit from indirectly selling to the low-type type through the high type should outweigh the loss of high-type fee (in a linear demand function)

• Is it a realistic assumption that high types buy drinks and carry them out of the bar to sell them to low types there? And the "low firm" in the last paragraph should be a low type, right? Jun 11, 2021 at 7:55
• If there is no price discrimination, why are you worried about resale? Jun 11, 2021 at 7:58
• "the bar owner decides to only sell to the high-type consumer" Is strange. You should just solve the optimization problem, minding that the aggregate demand is perhaps not linear but has a kink, as there is a price $\bar{p}$ at which the low-type consumer stops buying. Jun 11, 2021 at 8:00
• @Giskard There is still first-degree price discrimination (i.e the bar owner can set a fee equal to the whole consumer surplus of the high-type). So if we set $p=mc$ (by optimizing the fee for the high type), the low-type will demand their competitive level and this is profit that we could potentially obtain through slightly raising the price and indirectly selling to the low-type through the high type. The quote you say is strange is an assumption in the problem I'm solving. Jun 11, 2021 at 8:08
• @VARulle yes it's a realistic assumption - yes sorry it should be "low-type" not "low firm" I've edited it Jun 11, 2021 at 8:09