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日期：2023-04-07 10:04

Problem Set C Microtheory II, Spring 2023

1. Consider a similar Betrand duopoly problem with differentiated products and zero production costs and the

following demand functions for each firm:

1.a) Find the best-response functions for each firm.

1.b) Carefully depict the solution graphically for the three cases where a = 1, a = 2 , and a = 4 . In each case, does

a solution exist and is it unique?

1.c) What happens to the solution in terms of prices when there are small changes in the slope of firm 2’s demand

curve?

2. A group of fishermen concentrate their activity in a certain restricted area. The fishing returns in that area

depend on the total hours worked by the whole group. Thus letting hi denote the hours worked by each fisherman

ni = ,...,1 and ∑∈ ≡ Ni

hi H the total number of hour worked by the group, it is postulated that the hourly returns of

each of them is given by the increasing and strictly concave function of 𝐻𝐻, ρ : ℜ→ℜ ++ , with 0)(limH ∞→ ρ′ H = ,

𝑙𝑙𝑙𝑙𝑙𝑙𝐻𝐻→0 𝜌𝜌′

(𝐻𝐻) = ∞, and 𝜌𝜌(0) = 0.

On the other hand, each worker incurs an individual cost (or disutility) per hour worked that is captured by some

function of hi , c : ℜ→ℜ ++ , which is identical across individuals. It is also assumed that )( i hc is increasing and

strictly convex, and satisfies ∞= ′ → )(lim iThi hc , 𝑙𝑙𝑙𝑙𝑙𝑙ℎ𝑖𝑖→0 𝑐𝑐′ (ℎ𝑖𝑖) = 0, and 𝑐𝑐(0) = 0, for some given T ≤ 24 .

Overall, the payoffs of each fisherman are given by a function ui defined as follows:

)()(),...,( 1 ii n i hchHhhu −= ρ ni = ,...,1 .

2.a) Specify the optimization problem for a social planner whose objective function is to maximize the sum of

individual utilities and characterize the solution. Label this solution by “**”. Establish that a solution exists and is

unique. If you need to, you can make additional assumptions concerning the relevant functions.

2.b) Solve for the strategic form of the game in which the fisherman independently choose their hours. Characterize

the (symmetric) Nash equilibrium and show that it exists and is unique. Label the outcome by “*”.

2.c) Compare the two solutions either graphically or mathematically.

3. A citizen must choose whether to file taxes honestly or to cheat. The tax authority decides how much effort to

invest in auditing and can choose 𝑎𝑎 ∈ [0,1]; the cost to the tax authority of investing at a level of 𝑎𝑎 is 𝑐𝑐(𝑎𝑎) =

100𝑎𝑎2. If the citizen is honest, then he receives the payoff of zero and the tax authority pays the auditing costs

without any benefit the audit, yielding the authority a payoff of −100𝑎𝑎2. If the citizen cheats, then his payoff

depends on whether he is caught or not. If he is caught, then his payoff is −100 and the tax authority’s payoff is

100 − 100𝑎𝑎2. If he is not caught, then his payoff is 50 while the tax authority’s payoff is −100𝑎𝑎2. If the citizen

cheats and the tax authority choose to audit at a level 𝑎𝑎, then the citizen is caught with probability 𝑎𝑎 and not caught

with probability 1 − 𝑎𝑎.

3.a) If the tax authority believes that the citizen is cheating for sure, what is the authority’s best-response level of 𝑎𝑎?

3.b) If the tax authority believes that the citizen is honest for sure, what is the authority’s best-response level of 𝑎𝑎?

3.c) If the tax authority believes that the citizen is honest with probability 𝑝𝑝, what is the authority’s best-response

level of 𝑎𝑎 as a function of 𝑝𝑝?

3.d) Is there a pure strategy Nash equilibrium for the game? Why or why not? (For either one, support your

answer).

3.e) Is there a mixed strategy Nash equilibrium for this game? Why or why not? (Support your answer by either

deriving the mixed strategy NE or showing that one does not exist).

4. Consider an electorate distributed uniformly along the ideological spectrum on the unit interval, from the left (a =

0) to the right (a = 1). There are two candidates (1 and 2) and the candidate with the most votes wins (i.e., the

voting rule is majority rule). If there is a tie, assume that the winner is decided by a coin toss. Each voter casts his

or her vote sincerely for the candidate that is closest to his or her ideological position. The candidates know this and

only care about winning. The payoff function 𝑢𝑢𝑖𝑖(𝑎𝑎1, 𝑎𝑎2) of candidate/player i is given by the percentage of the vote

obtained by her if the strategy profile (𝑎𝑎1, 𝑎𝑎2) is adopted by the two candidates. Assume that each candidate

simultaneously chooses a position on the ideological spectrum in order to maximize her percentage of the vote

taking as given the other candidate’s position of choice.

4.a) Candidate 1’s payoff function is given by:

Find Candidate 2’s payoff function. Then verify/explain the derivation of Candidate 1’s payoffs and its terms. Use

a diagram of the unit interval to illustrate your answer.

4.b) Find the Nash Equilibrium of this game, making sure to support your answer.

4.c) Prove that the equilibrium you found is the only one (in pure strategies).

5. There is a strategic game with 3 players, where each player’s strategy set is given by Si = [0,1], i = 1, 2, 3. The

player’s payoff functions are given by:

1 (, ,) u xyz x y z =+− 2 (, ,) u x y z x yz = − 3 (, ,) u x y z xy z = −

where the players’ strategies are given by 1s x = , 2 s y = , 3 s z = .

5.a) Show that the strategy profile (1, ,0) α where 0 1 α≤ ≤ are the only Nash equilibria of the game.

5.b) Consider the same game as above except that the players’ strategy sets are given by Si = (0,1) , i = 1, 2, 3.

Does this game have a Nash equilibrium in pure strategies? Compare your answer with respect to (5.a).

5.c) Consider the two person strategic form game with 𝑆𝑆𝑖𝑖 = ℝ, i =1,2 and the payoff functions given by

2

1 1 2 1 12 (, ) 2 u s s s ss = − and 2

2 1 2 12 2 (, ) u s s ss s = − .

Verify that this game does not have a Nash equilibrium in pure strategies, including the possible outcome

1 2 ( , ) (0,0) s s = .