Chess Forum and the Uncommons are the only two Greenwich Village venues with regularly scheduled summer chess camps for kids. There are 450 potential campers every week, all with overeager parents willing to pay up to $200 per week for a few blissful hours of childcare (and the false hope of admission to one of NYC’s very finest private schools). Since the two venues provide an essentially identical service, parents simply prefer to send their children to whichever is cheaper. (If they charge the same price, then they will split the market equally.)
Each venue can host at most 600 kids each week. You can assume that each venue incurs zero out of pocket expenses from this service (parents are expected to provide their children’s food and drink); however, each child displaces an adult customer who is worth about $80 in profits to the venues. Suppose that each venue takes a short-run perspective and only wants to maximize each week’s profits, and that neither one would consider exiting this business in the foreseeable future.
(a)[8 points] If you use Nash equilibrium to make a prediction, what price is each venue going to charge? Explain your reasoning.
(b)[6 points] Briefly explain two possible practical means by which the venues could earn more profits than predicted by (a).
(c)[8 points] Suppose that Chess Forum’s internet service provider, Spectrum, experiences “intermittent” service outages that knocks out their Wi-Fi, making their space less appealing to adult customers; meanwhile, the Uncommons’ Verizon-powered internet service is unaffected. How does your prediction in (a) change? Explain.
From now on, focus on the baseline setting where both venues face the same tradeoff between children and adults: each child displaces an adult worth $80 in profits. However, suppose that the market size has doubled and there are now 900 potential children, while each venue’s capacity of 600 remains unchanged. As a result, neither Chess Forum nor the Uncommons can serve the whole market alone.
(d)[5 points] Is it a Nash equilibrium for each venue to charge a price of $80 per child? Justify your answer.
(e)[5 points] Is it a Nash equilibrium for each venue to charge a price of $200 per child? Justify your answer.
Another venue, Zinc Bar, also decides to take enter this market and provide chess and other board games during the daytime. But because of a quirk in Zinc Bar’s liquor license terms, it can only host 400 campers each week. Of course, post-Covid parents don’t really care where their kids are as long as they are out of the house, so they still choose to send their children to the cheapest camp possible (splitting the market equally between any venues charging the same price). Moreover, Zinc Bar’s adult customers are no more or less profitable than Chess Forum’s or the Uncommons’.
(f)[8 points] What is the Nash equilibrium of the pricing game among these three venues. Explain.