PE2023_PS2_Solutions (2)

Prof. Casella Political Economy - W4370 Fall 2023 Problem Set 2 Solutions Question 1 Please summarize the essence of Acemoglu’s argument–in 150 words or less, but in your own words. Acemoglu (2002) argues that societal inefficiency is better understood via the Theory of Social Conflict (TSC) than the Political Coase Theorem (PCT). Under PCT, rulers choose the efficient outcome because they can redistribute gains to themselves – any inefficiencies are due to uncertainty or incorrect information. Under TSC, rulers have a utility function at odds with societal welfare. As such, a prisoner’s dilemma game between rulers and citizens leads to inefficient outcomes. Com- mitment mechanisms fail dually (1) because there is no higher power binding the ruler, and (2) because if rulers relinquish power, citizens have no incentive to transfer gains to rulers. Note: A full answer must mention both the Political Caose Theorem (and how it could be inefficient but still hold) and the Theory of Social Conflict. You should also mention why commitment mechanisms fail. Question 2 We are given that the production function is: y = 1 + 5 e Production available for consumption is: y d = y (1 − t ) = (1 + 5 e ) (1 − t ) The Citizen has the utility function: U C = y d + 1 − e 2 2 = (1 + 5 e ) (1 − t ) + 1 − e 2 2 And the Ruler has utility: U R = ty = t (1 + 5 e ) (i) We find the efficient level of e by finding the value e ∗ that maximizes the sum of utility, ` a la a Central Planner. We begin by setting up the Central Planner’s problem: Choose e to maximize U R + U C s.t. 0 ≤ e ≤ 1 and 0 ≤ t ≤ 1 1

This can be expressed with the following Lagrangian: L = 3 2 + 5 e − 1 2 e 2 + λ e (1 − e ) + λ t (1 − t ) The first order condition with respect to e yields: 5 − e − λ e = 0 And the complementary slackness condition for e is: λ e (1 − e ) = 0 First, suppose λ e = 0. This would give us e ∗ = 5 from the FOC, which violates the constraint on e . Thus, it must be that λ e > 0, and that (1 − e ) = 0 ⇔ e ∗ = 1. Note: One didn’t necessarily need to use the CSC to get from e = 5 to e ∗ = 1. One could have alternatively justified e ∗ = 1 by appealing to the sign of the FOC on e ∈ [0 , 1]. It is not sufficient, however, to conclude that e ∗ = 1 because e = 1 is the “closest” allowable value to e = 5. (ii)(a) We solve for the Citizen’s choice of e under commitment, taking the tax rate t c as a given. The Citizen’s problem is: Choose e to maximize U C s.t. 0 ≤ e ≤ 1 The Lagrangian for this problem is: L = (1 + 5 e ) (1 − t ) + 1 2 − 1 2 e 2 + λ (1 − e ) The first-order condition with respect to e : 5 (1 − t ) − e − λ = 0 And the complementary slackness condition: λ (1 − e ) = 0 We solve the two cases from the CSC in turn. (Note that the two cases explicitly given in the problem, t ≤ 4 5 and t > 4 5 , come from the CSC). Case 1: λ = 0 , e < 1 From the FOC, we know that: e = 5 (1 − t ) − λ If λ = 0, then: e = 5 (1 − t ) But, note that the implication e < 1 is satisfied only when t > 4 5 . Case 2: λ > 0 , e = 1 2

There are now two constraints on t : a floor constraint (denoted with the underline) that t can’t be below 4 5 , and the ceiling constraint that t can’t go above 1. The FOC with respect to t : 26 − 50 t + λ t − λ t = 0 And the CSCs: λ t t − 4 5 = 0 ; λ t (1 − t ) = 0 From the CSCs, we have two endpoint solution candidates to check: t c = 4 5 and t c = 1. We first check whether there’s an interior solution candidate too. If an interior solution candidate exists, it must be that 4 5 < t c < 1. This implies, from the CSCs, that λ t = λ t = 0. Substituting this into our FOC, we find t c = 26 50 as our interior solution candidate. But wait: 26 50 < 4 5 ! This interior solution candidate doesn’t satisfy the constraints for this Case. Hence, we eliminate this candidate from contention. The remaining question is whether the Ruler will choose t c = 4 5 or t c = 1 for this case. It’s clear to see that if the Ruler chooses t c = 1, that the Citizen will respond with e c = 0, which induces U R = 0. Therefore, the Ruler will also choose t c = 4 5 in this Case. (ii)(c) Putting together the answers from (a) and (b): ( e c , t c ) = 1 , 4 5 (iii) Yes, e c = e ∗ , so this is efficient. This would demonstrate the Political Coase Theorem, in which allocation of decision power should not influence the outcome, which will be socially efficient. (iv) The Ruler will set t ′ to extract the most resources from the Citizen ⇒ t ′ = 1. The Citizen, in response, will set e nc = 0. (v) e nc ̸ = e ∗ , so this is not efficient. This would demonstrate Acemoglu’s Theory of Social Conflict: because the rulers want to maximize their own payoffs (rather than the social payoff), outcomes are not efficient. This holds given problems with commitment (of which there is none in this case). 4