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A firm wants to bid for the monopoly franchise to sell hot dogs at a baseball game. It estimates the inverse demand function for hot dogs as

p = 5 0.5x

where p is the price in dollars and x is sales of hot dogs in thousands. It also estimates that it can supply the hot dogs at a constant unit of $0.50 per hot dog.

1. Write down the optimization problem this firm needs to solve.

2. Write down the first order conditions for this optimization problem. Do not solve it/them.

3.Solve the equation or system of equations you wrote in questions 2.

Explanation / Answer

1) If we put x = 0 then we get price that is p = 5 which is monopoly price. So p should be less than 5 and greater than 0.5 which is the firm's cost.

0.5 < p < 5

2) p = 5 - 0.5x

MR = 5 - x

Revenue = px = 5x - 0.5x^2

3) Equating p = 0.5

0.5 = 5 - 0.5x

4.5 = 0.5x which gives x = 9 thousand

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