ECOS3035: Economics of Political Institutions
Homework II
1. Consider the two-period agency model of elections from class. There are three players: two politicians (an incumbent and a challenger), and a voter. A politician can be either of high quality H with probability p or of low quality L with probability 1−p, where p = 0.2. Quality is privately observed by the politician. A politician in office in period t, t ∈ {1, 2}, exerts effort et ∈ [0, 1] which she privately observes and which costs her .
In each period t, a politician in office can deliver two outcomes: good or bad. The type H politician delivers the good outcome for sure. The type L politician, by contrast, delivers the good outcome in period t with probability et
, and the bad outcome with the complementary probability.
A politician gets a payoff of B ∈ (0, 1) for each period that she is in office, and otherwise she gets a payoff of 0. The voter gets a payoff of 1 for each period where the outcome is good, otherwise he gets a payoff of 0.
The timing of the game is as follows. In period 1, the incumbent chooses e1. The outcome for period 1 is realized. The voter observes the outcome and chooses whether to reelect the incumbent or to replace the incumbent with the challenger. In period 2, the politician in office chooses e2. The outcome for period 2 is realized. The game ends.
(a) What is the equilibrium level of effort for the high type politician and the low type politician in the second period?
(b) Consider the stage of the first period where the voter observes the outcome (good or bad). For each of these outcomes compute the posterior probabilities that the voter assigns to the type of incumbent (H or L). How does this depend on the conjectured effort of the incumbent? Why?
(c) Specify the voter’s optimal reelection strategy in period 1 following the realization of the outcome.
(d) Compute the incumbent’s equilibrium level of effort in period 1 for each type.
(e) Given this equilibrium level of effort in period 1, compute the probability of a good outcome at the start of the game.
2. Consider Case I in Banerjee, Hanna and Mullainathan (2012). Suppose H = h = yH > L = l = yL > 0, and NH < 1. Suppose there is no testing.
(a) Find the efficient (social welfare maximizing) probabilities πH and πL for this case.
(b) Consider the mechanism:
Verify that the slot probabilities satisfy the slot constraint (total number of slots equals 1). Verify that pL satisfies the affordability constraint for L. Also find the maximum value of ϵ (call this ϵmax) that satisfies both the affordability and incentive constraints for type H. (All of these should convince you that the mechanism is feasible for the bureaucrat to use – it satisfies incentives, participation, the slot constraint, and the affordability constraint).
(c) Suppose the government imposes a rule that the bureaucrat has to use the mechanism
Find conditions under which there is no corruption – that is, the rule is never broken by the bureaucrat.
(d) Suppose the government imposes a rule that the bureaucrat has to use the mechanism
where ϵmax is the same as in the part above. Find conditions under which there is corruption – that is, the rule is broken if the bureaucrat has a very low cost of breaking the rule. Which type of agent does the bureaucrat get a bribe from? What is the value of the bribe?
(e) Consider the same mechanism above but replace yL with y < yL. Would there be more corruption or less corruption by the bureaucrat – in terms of breaking the rule and the bribe charged? Be very brief.