# Marginal and Nash

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Three airlines (A, B and C) are competing for passengers on a lucrative long-haul air route. At present, the carriers are charging identical fares ($225 for a one-way ticket), the result of a truce in recent price wars. The airlines currently compete for market share via the number of scheduled daily departures they offer. Each airline must make its decision for the desired number of daily departures for the coming month-without knowing its rivals' plans ahead of time. Each airline is aware of the following facts:

? The size of the total daily passenger market is stable irrespective of the number of departures offered. At current prices, an estimated 2,000 passengers fly the route each day.

? Each airline's share of these total passengers equals its share of the total number of flights offered by the three airlines.

? The airlines fly identical planes and have identical operating costs. Each plane holds a maximum of 200 passengers. Regardless of the plane's loading, each one-way trip on the route costs the airline $20,000.

As the manager of one of these airlines, how many daily departures should you schedule for the upcoming month? After seeing the first month's results (your rivals' choices and the resulting profits), what decisions would you make for the second and subsequent months?

Towards answering these questions:

a. Denote the airlines' number of departures by a, b and c, respectively. Show that airline A's profit can be expressed as

PROFITa = 450[a/(a+b+c)] - 20a .

b. Using this formula, fill in the profits in Airline A's Payoff Table given in the attached spreadsheet. In each column, highlight Airline A's best response. Offer some conclusions.

Suppose the industry finds itself in this situation month after month.

c. Equilibrium behaviour. Argue that in a Nash equilibrium each airline will offer 1/3 of the flights. Now look at the Payoff Table to determine the unique Nash equilibrium for this industry. For this equilibrium, determine each airline's profit, industry profits and the total number of flights.

d. Collusive behaviour. First determine the number of daily departures that maximize the industry's profit. Now as the manager of Airline A, what would you propose? Why would this proposal not be an equilibrium?

e. Why do hubs exist?

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The solution applies the marginal and nash to answer the given questions with all details provided.

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a. Denote the airlines' number of departures by a, b and c, respectively. Show that airline A's profit can be expressed as

PROFITa = 450[a/(a+b+c)] - 20a .

In the whole market, the total daily revenue is $225*2,000 = $450,000

Airline A's share of the revenue equals its share of the total number of flights offered , which is a/(a+b+c), then its revenue is $450000[a/(a+b+c)]

while its cost is $20,000a.

Then A's profit is =450000[a/(a+b+c)] - 20,000a

Or = 450[a/(a+b+c)] - 20a (in thousands)

b. Using this formula, fill in the profits in Airline A's Payoff Table given in the attached spreadsheet. In each column, highlight Airline A's best response. Offer some conclusions.

Please refer to the attached EXCEL for calculation.

I doubt if we should highlight A's best response in each ROW, rather than column. Because obviously A's ...

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