Multitask, Accountability, and Institutional Design Scott Ashworth & Ethan Bueno de Mesquita Harris School of Public Policy Studies University of Chicago 1 / 32
Motivation Multiple executive tasks divided differently in different settings Unified authority: U.S. President Divided authority: U.S. States This has implications for accountability, electoral selection, and voter welfare Institutional design question When to divide tasks between multiple offices and when to unify in a single office 2 / 32
Outline The Model Equilibrium Bundling Unbundling Optimal First Period Institution Second Period Optimal Institution Conclusion 3 / 32
Timeline 1. Period 1 (i) Effort taken on two tasks (ii) Outputs observed by the Voter (iii) Election occurs 2. Period 2 (i) Effort taken on two tasks (ii) Outputs observed by the Voter 3. Game ends 4 / 32
Production Function Output on task j in period t is s t j = a t j + θ j + ɛ t j a t j is the effort on task j in period t θ j is the competence of the official on task j ɛ t j is a standard Normal shock in period t Uncertainty over competence is symmetric (all candidates are ex ante identical) An official s competence has prior distribution ( ) (( ) ( )) θ1 0 1 ρ N, 0 ρ 1 θ 2 5 / 32
Voter Payoffs Voter differentially weights each dimension of output Period t payoff is: γs t 1 + (1 γ)s t 2 No discounting 6 / 32
Officials Payoffs Bundled official s payoff if in office in period t R c(a t 1) c(a t 2) Task-1 unbundled official s payoff if in office in period t ηr c(a t 1) Task-2 unbundled official s payoff if in office in period t (1 η)r c(a t 2) c(a) = k 2 a2 7 / 32
Institutional Design Questions and Trade-Offs Compare bundled to optimal unbundled How do competence correlations and voter weights affect: Optimal institution for first period incentives Optimal institution for second period selection When are there institutional design trade-offs and when is one institution dominant for both periods? 8 / 32
Outline The Model Equilibrium Bundling Unbundling Optimal First Period Institution Second Period Optimal Institution Conclusion 9 / 32
Outline The Model Equilibrium Bundling Unbundling Optimal First Period Institution Second Period Optimal Institution Conclusion 10 / 32
Beliefs and Voting under Bundling Voter s posterior beliefs about incumbent ( m1 m 2 ) = 1 ρ 2 ( 2 ρ 2 ρ (2 ρ 2 ) 2 ρ 2 ρ 2 ρ 2 ) ( s1 a b 1 s 2 a b 2 ) Reelect iff γm 1 + (1 γ)m 2 0 equivalent to λ 1 (s 1 a b 1) + λ 2 (s 2 a b 2) 0 11 / 32
Probability of Reelection LHS from previous is Normal random variable mean = λ 1 (a 1 a b 1) + λ 2 (a 2 a b 2) σ 2 b (γ 2 + (1 γ) 2 )σ 2 m + 2γ(1 γ) cov m σ 2 m: prior variance of posterior mean on one dimension cov m : prior covariance of posterior means Probability of reelection given efforts (a 1, a 2 ): ( ) 0 λ1 (a 1 a b 1 Φ 1) λ 2 (a 2 a b 2). σ b 12 / 32
Equilibrium Efforts [ max R 1 Φ (a 1,a 2 ) ( 0 λ1 (a 1 a b 1) λ 2 (a 2 a b 2) σ b )] c(a 1 ) c(a 2 ). λ 1 R φ(0) = a b 1 and λ 2R φ(0) = a b 2 kσ b kσ b 13 / 32
Outline The Model Equilibrium Bundling Unbundling Optimal First Period Institution Second Period Optimal Institution Conclusion 14 / 32
Equilibrium Normal learning implies Voter s posterior about task-j incumbent, following s j, is λ u (s j a u j ) Reelect if posterior is greater than 0 Let σu 2 be prior variance of posterior mean ( 0 λu (a j a u ) j ) Probability of Reelection: 1 Φ σ u ηrλ u kσ u φ(0) = a u 1 and (1 η)rλ u kσ u φ(0) = a u 2 15 / 32
Optimal Unbundling ηrλ u kσ u φ(0) = a u 1 and (1 η)rλ u kσ u φ(0) = a u 2 Second Period Welfare Not a function of η because doesn t affect learning First Period Welfare Efforts are linear in η Payoffs are linear in efforts Bang-bang (η = 1 if γ > 1/2 and η = 0 if γ < 1/2) 16 / 32
Outline The Model Equilibrium Bundling Unbundling Optimal First Period Institution Second Period Optimal Institution Conclusion 17 / 32
First Period Voter Welfare: Key Trade-Off γa 1 + (1 γ)a 2 = 1 2 (a 1 + a 2 ) + 1 }{{} 2 (2γ 1) (a 1 a 2 ) }{{} Total Effort Alignment If ρ > 0 or γ (0, 1) Total effort strictly higher under bundling Alignment strictly better under unbundling 18 / 32
Total Effort Comparison 1 Marginal impact of effort on prior mean of Voter expected utility from reelecting is larger under bundling Bundled: Prior mean of γ-weighted posterior mean Unbundled: Prior mean of relevant posterior mean λ 1 + λ 2 > λ u Correlation implies more total information Voter s posteriors put less weight on priors 19 / 32
Total Effort Comparison 2 Effect of marginal change in Voter belief larger if bundled Proportional to the height of the density of the prior distribution of the (relevant) posterior φ(0) σ b > φ(0) σ u Bundling averages two imperfectly correlated random variables, so variance-reducing diversification effect σ 2 b = (γ 2 + (1 γ) 2 )σ 2 m + 2γ(1 γ) cov m σ 2 u = σ 2 m 20 / 32
Alignment Comparison Presence of correlation means Voter does not shut down incentives on less important task This is a multi-task distortion that occurs entirely through Voter learning, not cost complementarity The value of unbundling for first period welfare is the Voter commits to not reward less important task This is stark for the case of quadratic costs, but is more general 21 / 32
Main Result for First Period Welfare Proposition 5.2 There is a strictly decreasing function ˆγ(ρ), with ˆγ(ρ) 1 for all ρ, such that bundling is optimal 2 for first period welfare if and only if 1 ˆγ(ρ) γ ˆγ(ρ). The optimality is strict if the inequalities are strict. 22 / 32
First Period Optimal Institution 1.0 Γ 0.8 Unbundling Optimal 0.6 Bundling Optimal 0.4 0.2 Unbundling Optimal 0.2 0.4 0.6 0.8 1.0 Ρ 23 / 32
Some Intuitions When preference weights become more extreme Alignment becomes very important Total effort advantage gets small because diversification effect declines There are also other effects... When correlation becomes larger Distraction effect gets larger Total effort advantage gets small because diversification effect declines There are also other effects... 24 / 32
Outline The Model Equilibrium Bundling Unbundling Optimal First Period Institution Second Period Optimal Institution Conclusion 25 / 32
Second Period Voter Welfare Second period Voter welfare under bundling 1 2 E[E[γθ 1 + (1 γ)θ 2 ) γm 1 + (1 γ)m 2 0]] = σ b φ(0) Second period welfare under unbundling 1 2 (E[γE[θ 1 λ u (s 1 a u 1) 0] + (1 γ)e[θ 2 λ u (s 2 a u 2) 0]]) = σ u φ(0) 26 / 32
The Trade-offs Bundling reduces Voter flexibility Imagine an Incumbent who is good on one dimension and bad on the other Bundling increases Voter information 27 / 32
Main Result of Second Period Welfare Proposition 6.2 For any γ, there is a unique ˆρ(γ) [0, 1) such that bundling is optimal for second-period welfare if and only if: ρ ˆρ(γ). The optimality is strict if the inequality is strict. Moreover, ˆρ is strictly increasing for γ < 1/2 and strictly decreasing for γ > 1/2. 28 / 32
Second Period Optimal Institution 1.0 Γ 0.8 0.6 Unbundling Optimal Bundling Optimal 0.4 0.2 0.2 0.4 0.6 0.8 1.0 Ρ 29 / 32
Outline The Model Equilibrium Bundling Unbundling Optimal First Period Institution Second Period Optimal Institution Conclusion 30 / 32
Optimal Institution 1.0 Γ 0.8 Unbundling Optimal Unbundling Optimal for 1st Period Bundling Optimal for 2nd Period 0.6 Bundling Optimal for 1 st Period Unbundling Optimal for 2 nd Period Bundling Optimal 0.4 0.2 Unbundling Optimal Unbundling Optimal for 1st Period Bundling Optimal for 2nd Period 0.2 0.4 0.6 0.8 1.0 Ρ 31 / 32
Applications and Extensions Applications Organization of local government Scope of authority for agencies Federalism (γ as share of population in region 1) Extensions Spillovers in task-specific effort Incumbent chooses institution 32 / 32