Question

Consider a small town, with two restaurants: a tapas bar and a bistro. Both restaurants have a set menu and the cost of serving each customer is $8 for both restaurants. Tapas bar chooses its price for the menu denoted by Pt, and bistro chooses its price for the menu denoted by Pb. Both restaurants are trying to maximize their profits. Tapas bar faces the demand curve Qt = 44 − 2Pt + Pb and bistro faces the demand curve Qb = 44 − 2Pb + Pt.

a) Write down the profit function of the tapas bar.

b) Write down the profit function of the bistro.

c) Find the best reply function of the tapas bar.

d) Find the best reply function of the bistro.

e) Find the Nash equilibrium using the best reply functions you derived in part c and d.

f) Draw the best reply functions you found in part c and d in the same graph. Show the Nash 2 equilibrium on your graph.

g) Does the Nash equilibrium you found generate the highest possible total profits (that is the sum of profits of the tapas bar and the profits of the bistro) for the two firms? If so, argue why this is the case. If not, argue why this is not the case and find the point that generates the highest possible total profits for the two firms.

Answer 1

a. Profit Function For Tapas Bar = Total Revenue of Tapas Bar – Total Cost

Given the Demand Function for Tapas Bar: Qt = 44 − 2Pt + Pb, we can find its total revenue:

TR(T) = Qt * Pt

TR(T) = (44 − 2Pt + Pb)Pt

TR(T) = 44Pt – 2Pt2 + PbPt

Total Cost for Tapas bar = $8

Thus, Profit of Tapas Bar : Pr(T) = TR(T) – TC

Pr(T) = 44Pt – 2Pt2 + PbPt – 8

b. Profit Function For Bistro = Total Revenue of Bistro – Total Cost

Given the Demand Function for Bisrto: Qt = 44 − 2Pb + Pt, we can find its total revenue:

TR(B) = Qb * Pb

TR(B) = (44 − 2Pb + Pt)Pb

TR(B) = 44Pb – 2Pb2 + PbPt

Total Cost for Tapas bar = $8

Thus, Profit of Bistro = Pr(B) = TR(B) – TC

Pr(B) = 44Pb – 2Pb2 + PbPt – 8

c. For the best reply function of tapas bar, we differentiate Pr(T) with respect to Pt and equate it to zero.

Pr(T) = 44Pt – 2Pt2 + PbPt – 8

dPr(T)/dPt = 44 – 4Pt + Pb

dPr(T)/dPt = 0

44 – 4Pt + Pb = 0

44 + Pb = 4Pt

Pt = 11 + 0.25Pb

d. For the best reply function of the bistro, we differentiate Pr(B) with respect to Pb and equate it to zero.

Pr(B) = 44Pb – 2Pb2 + PbPt – 8   

dPr(B)/dPb = 44 – 4Pb + Pt

dPr(B)/dPb = 0

44 – 4Pb + Pt = 0

44 + Pt = 4Pb

Pb = 11 + 0.25Pt

e. To find the Nash equilibrium, we use the value of Pb and put that into the best response function of Tapas Bar:

Pt = 11 + 0.25Pb

Pt = 11 + 0.25(11 + 0.25Pt)

Pt = 11 + 2.75 + 0.0625Pt

Pt - 0.0625Pt = 13.75

0.9375Pt = 13.75

Pt = 13.75 / 0.9375

Pt = 14.6

Using Pt and putting in Bistros best reply function:

Pb = 11 + 0.25Pt

Pb = 11 + 0.25 (14.6)

Pb = 11 + 3.6

Pb = 14.6