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Yuppietown has two food stores, La Boulangerie, which sells bread, and La Fromagerie, which sells cheese. It costs $1 to make a loaf of bread and $2 to make a pound of cheese. If La Boulangerie’s price is P1 dollars per loaf of bread and La Fromagerie’s price is P2 dollars per pound of cheese, their respective weekly sales, Q1 thousand loaves of bread and Q2 thousand pounds of cheese, are given by the following equations: Q1 = 14 − P1 − 0.5P2 and Q2 = 19 − 0.5P1 − P2. (a) For each store, write its profit as a function of P1 and P2. Then find their best-response

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Answer:

Step-by-step explanation:

Given that:

Q1 = 14 − P1 − 0.5P2

Q2 = 19 − 0.5P1 − P2

Recall that:

Profit = Total revenue - Total cost

π₁ = P₁Q₁ MCQ₁

π₁ = P₁(14 - P₁ - 0.5P₂) - Q₁

π₁ = 14P₁- P₁² - 0.5 P₁P₂ - 14 + P₁ + 0.5P₂

π₁ = 15P₁ - P₁² - 0.5P₁P₂ + 0.5P₂ - 14

π₂ = P₂ (19 - 0.5P₁ - P₂) - 2(19 - 0.5P₁ - P₂)

π₂ = 19P₁ - 0.5P₁P₂ - P₂² - 38 + P₁ + 2P₂

π₂ = 21P₂ - P₂² - 0.5 P₁ P₂ + P₁ - 38

The profit functions are π₁  & π₂

The best-response function from firm 1  is;

π₁ = 15P₁ - P₁² - 0.5P₁P₂ + 0.5P₂ - 14

[tex]\mathbf{\dfrac{\partial \pi_1}{\partial P_1}= 15 - 2P_1 -0.5P_2 =0}[/tex]

[tex]\mathbf{P_1= \dfrac{15-0.5P_2}{2}}[/tex]

The best-response function from firm 2  is;

π₂ = 21P₂ - P₂² - 0.5 P₁ P₂ + P₁ - 38

[tex]\mathbf{\dfrac{\partial \pi_2}{\partial P_2}= 21 - 2P_2 -0.5P_1 =0}[/tex]

[tex]\mathbf{P_2= \dfrac{21-0.5P_1}{2}}[/tex]

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Here we must find the profit for each one of the two stores.

  • For La Boulangerie the profit is: (14 − P1 − 0.5*P2)*(P1 - $1)
  • For La Fromagerie the profit is: (19 − 0.5P1 − P2)*(P2 - $2).

Remember that profit is defined as the difference between the revenue and the cost.

For La Boulangerie we know that they sell Q1 thousand loaves. Each loaf costs $1. Each loaf is sold by P1

Then the profit is just:

P = Q1*(P1 - $1) = (14 − P1 − 0.5*P2)*(P1 - $1)

Which is already written only in terms of P1 and P2.

Similarly, for the other store we will get the profit:

P' = Q2*(P2 - $2) = (19 − 0.5P1 − P2)*(P2 - $2)

Now we must get their best response, which is given by deriving it with respect to the price and making it equal to zero:

For the first store we have:

dP/dP1 = $1 + 14 - 0.52*P2 - 2*P1 = 0

Now we solve it for P1:

P1 = ($1 + 14 - 0.52*P2)/2

Similarly, for the other store we have:

dP'/dP2 = 19 - 0.5*P1 - 2*P2 + $2 = 0

Solving it for P2 it gives:

P2 = (19 - 0.5*P1 + $2)/2

If you want to learn more about profit, you can read:

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