Question

Prize for the Tallest Building

For an office building in a large city, the profit increases with the number of floors for the first 40 floors at a rate of $3 per floor, and then decreases as the number of floors increases at a rate of $2 per floor. In addition, the prize or bonus profit from the tallest building is $30. Two firms, A and B, will each build one office building, and A builds first. In the event of a tie (two buildings of the same height) there is no prize.

A. The Nash equilibrium is . . .

B. The competition to be the tallest changes total building profit by $[_____]

C. Design a Pareto improvement under which the two firms share the gain equally. Assume that the number of floors is an integer. Firm A gives how much to firm B to do what

Answer 1

A). As A builds first ,he will build 54. floor .as he know B will make his decision considering his decision. Because there is loss of 2$ per floor above 40. Then considering this the highedh floor he will make is 54. And he also know that B looking at his decision ,will only build 40 floor . To beat A and get prize,B need to build 55 floor but 30$ prize will off by the loss of 2$ per floor above 40.it is in best interest for B ,given A decision ,to build 40 floor.

So in nash equilibrium A wiil build 54 floor by moving first and B will make 40 floor. The A payoff will be 122 and B payoff will be 120.

B). So if there wasn't any competition to build tallest,means cooperation ,then being the first mover or as a leader would make 41 floor and B will make 40 floor.the total profit will 120 -2+30+120=268.

But due to competition A builds 54 and B builds 40 and total profit is 122+120=242

The competition to be tallest changes total profit by 26$.

C). Pareto improvement happens when at least one person benefits by reallocation or policy change or change of stretgy without worsing any other person's current situation.

In this case a pareto improvement would happen when A will offer 26 $ B to just build 40 floor not more than that .while he himself make 41 floors..

Now A profit is 120+28=148 ,and he is giving 26$ to B.

So A PROFIT is same 148-26=122

Now B profit is now 146. Will he willing to accept this offer ?

The answer is yes.

The best alternative for B is, to get Prize which will give 30$ . To get that prize he need to build 42 floor .

So his profit will be 120+30-4=146. Which is same.

Conclusion: For pareto improvement, A gives 26$ to firm to build maximum 40 floor.