Two countries, 1 and 2, simultaneously set their respective rates of capital taxation, t1 and t2, where 0 t1, t2 1. The total amount of capital...

Q.1.Two countries, 1 and 2, simultaneously set their respective rates of 

capital taxation, t1 and t2, where 0 ≤ t1, t2 ≤ 1. The total amount of capital 

invested in country 1 is k1 = 1−t1 + 0.5t2, and in country 2 is k2 = 1−t2 + 0.5t1. 

Each country wants to maximise its total revenue from capital taxation, where revenue to country i is Ri = tiki, where i = 1, 2.(i) Find the unique Nash equilibrium of this game, by first deriving each country’s reaction (or best response) function. 

(ii) Derive the tax rates that maximise the sum, R1 + R2, of the revenues. (iii) Show that both countries strictly prefer the outcome obtained in part (ii) than the Nash equilibrium tax rates, and then discuss the implications of these results.

Q.2.

Two farmers (1 and 2) have to decide the number of goats (x1 and x2, repectively) that they will purchase. The cost of each goat is 4. The value to a farmer of each goat depends on the total number of goats that will be purchased (which will then be grazed on the village green). The value per goat is 100 − G, where G denotes the total number of goats purchased (x1 + x2). Each farmer is a profit maximiser.(i) Assuming that the two farmers make their respective purchases simulta- neously, derive the number of goats purchased by each farmer in the Nash equilibrium.(ii) Now suppose that farmer 2 dies. In this circumstance, how many goats 

would be purchased by farmer 1?(iii) Using the result obtained in part (i), or otherwise, show that the Nash equilibrium outcome obtained in part (i) is Pareto-inefficient. Discuss the intuition behind this observation.(iv) Now suppose that there are N farmers (where N > 2) in the village. Derive the Nash equilibrium quantities of goats purchase by each farmer as- suming that the purchases are made simultaneously. And then show that the total number of goats purchased in equilibrium increases as N increases. Discuss these results.

Q3. 

Consider a population of voters uni- formly distributed along the ideological spectrum from left (x = 0) to right (x = 1). Each of the two candidates for a single office simultaneously chooses a campaign platform (i.e., a point on the line between x = 0 and x = 1). The voters observe the candidates’ choices, and then each voter votes for the candidate whose platform is closest to the voter’s position on the spectrum. (For example, if they choose x1 = 0.3 and x2 = 0.6, then all those to the left of x = 0.45 vote for candidate 1, all those to the right of x = 0.45 vote for candidate 2, and therefore candidate 2 wins the election with 55 percent of the vote). Assume that the candidates care only about being elected - they do not really care about their platforms at all!Find the unique NE, assuming that if the candidates choose the same plat- form, then each wins the election with probability one-half.(Hint: Use the method of elimination).