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Motivating Loyal Bureaucrats in Sequential Agency

  • Alexander Rodivilov EMAIL logo and Dongsoo Shin
Published/Copyright: June 19, 2024
The B.E. Journal of Theoretical Economics
From the journal The B.E. Journal of Theoretical Economics Volume 24 Issue 2

Abstract

We study a principal–agent model in which a legislature and a bureaucrat sequentially play the principal’s role. In the first stage, the legislature offers a menu of transfer payments to the bureaucrat for implementing a public project. In the second stage, the bureaucrat offers a menu of the project attribute levels to the legislature. Then, the legislature decides whether to go forward with the project and aggregates information on the public’s valuation of the project. The key trade-off in this paper is information versus the bureaucrat’s loyalty. We show that when the bureaucrat is loyal enough to the public, taking advantage of his loyalty requires inducing him to make his decision independent of the public’s valuation.

Keywords: principal – agent; false moral hazard; sequential agency
JEL Classification: D82; D86; H11; H40
Corresponding author: Alexander Rodivilov, School of Business, Stevens Institute of Technology, Hoboken, NJ 07030, USA, E-mail: 

Acknowledgements

We thank the participants at the Econometric Society Asian Meeting 2022 and the EARIE conference 2021 for comments and discussions. The financial support from Leavey Research Grant is acknowledged. We also thank the editor, Ronald Peeters, and two anonymous referees.

Appendix A: Proof of Proposition 1

Combining (IC H ) and (IC L ) in P b gives:

t H − t L v L ≥ x H − x L ≥ t H − t L v H ,

implying that if t H = t L , then 0 ≥ x H − x L ≥ 0. It follows that x H * = x L * if t H = t L .

Appendix B: Proof of Proposition 2

We first describe the project attribute levels x H * ( t H , t L | λ ) and x L * ( t H , t L | λ ) chosen by the bureaucrat in case t H = t L . Then we describe the project attribute levels x H * ( t H , t L | λ ) and x L * ( t H , t L | λ ) chosen by the bureaucrat in case t H > t L .

  1. Equal Transfers (t H = t L ): Consider first equal transfers (t H = t L = t) chosen by the legislature. We proved in Proposition 1 that if t H = t L , then the bureaucrat will choose equal project attribute levels, i.e. x H = x L = x. This implies that the (IC H ) and (IC L ) constraints are automatically satisfied. In addition, the (PC H ) constraint is implied by (PC L ): v H x − t > v L x − t ≥ 0. The bureaucrat’s optimization problem then simplifies to:

    max x t − β x 2 2 − ( 1 − β ) x 2 2 + λ [ β ( v H x − t ) + ( 1 − β ) ( v L x − t ) ] ,

    subject to (PC L ): v L x − t ≥ 0. Labeling μ as the Lagrange multiplier associated with (PC L ) constraint, the optimization problem has the following Lagrangian:

    L = t − β x 2 2 − ( 1 − β ) x 2 2 + λ [ β ( v H x − t ) + ( 1 − β ) ( v L x − t ) ] + μ [ v L x − t ] = ( 1 − λ ) t − x 2 2 + λ E [ v i ] x + μ [ v L x − t ] ,

    which gives the following optimality condition:

    (A1) ∂ L ∂ x = λ E [ v i ] − x + μ v L = 0 .

    If μ > 0, then the (PC L ) constraint is binding and, as a result, x* = t/v L . This is the case if μ = x * − λ E [ v i ] / v L > 0 or, equivalently, if λ < t/(v L E[v i ]). If μ = 0, then from (A1) it must be that x* = λE[v i ]. This is the case if (PC L ) is slack, i.e. if v L λE[v i ] − t ≥ 0, which can be rewritten as λ ≥ t/(v L E[v i ]).

  2. Different Transfers (t H ≠ t L ): Consider next different transfers (t H ≠ t L ) chosen by the legislature. First, (PC H ) is implied by the (IC H ) and (PC L ) since: v H > v L ⇒ v H x L − t L > v L x L − t L ≥ 0. Also, (IC H ) implies v H x H − t H ≥ 0. Thus, (PC H ) is automatically satisfied and can be ignored in the analysis. We will show that, depending on the transfer payments chosen by the legislature, six cases are possible: any two of the three constraints, (PC L ), (IC H ) and (IC L ), may be binding simultaneously, and each of them might be slack. Labeling μ, μ H and μ L as the Lagrange multiplier associated with (PC L ), (IC H ) and (IC L ) constraints respectively, the optimization problem has the following Lagrangian:

    L = β t H − β x H 2 2 + β λ ( v H x H − t H ) + ( 1 − β ) t L − ( 1 − β ) x L 2 2 + ( 1 − β ) λ ( v L x L − t L ) + μ [ v L x L − t L ] + μ H x H − x L − t H v H + t L v H + μ L t H v L − t L v L − x H + x L .

    The optimality conditions are:

    (A2) ∂ L ∂ x H = β ( λ v H − x H ) + μ H − μ L = 0 ,

    (A3) ∂ L ∂ x L = ( 1 − β ) ( λ v L − x L ) + μ v L − μ H + μ L = 0 .

    We examine each of six cases below.

Case 1: μ L = μ = 0 and μ H > 0.

Claim 1 t H − t L > λv H (v H − v L ), v L λ E [ v i ] ≥ ( 1 − β v L v H ) t L + β v L v H t H if and only if μ L = μ = 0 and μ H > 0.

Proof.

Suppose μ L = μ = 0 and μ H > 0. Then conditions (A2) and (A3) can be rewritten respectively as:

(A4) β ( λ v H − x H ) + μ H = 0 ,

(A5) ( 1 − β ) ( λ v L − x L ) − μ H = 0 .

Combining (A4) and (A5) we have λE[v i ] − βx H − (1 − β)x L = 0. Also, binding (IC H ) implies that x H = x L + (t H − t L )/v H . Combining the two equation together we have:

(A6) x H * ( t H , t L | λ ) = λ E [ v i ] + ( 1 − β ) t H − t L v H  and  x L * ( t H , t L | λ ) = λ E [ v i ] − β t H − t L v H .

Next, we establish parameters necessary for μ L = μ = 0 < μ H . From (A4) we have μ H = β(x H − λv H ), and form (A5) we have μ H = (1 − β)(λv L − x L ). Thus, μ H > 0 is equivalent to x L * ( t H , t L | λ ) < λ v L and x H * ( t H , t L | λ ) > λ v H , which can be rewritten as t H − t L > λv H (v H − v L ). Condition μ = 0 is equivalent to v L x L * ( t H , t L | λ ) − t L ≥ 0 , which can be rewritten as v L λ E [ v i ] ≥ ( 1 − β v L v H ) t L + β v L v H t H .□

Case 2: μ L = μ = μ H = 0.

Claim 2 λv L (v H − v L ) ≤ t H − t L ≤ λv H (v H − v L ) and t L ≤ λ v L 2 if and only if μ L = μ = μ H = 0.

Proof.

Suppose μ L = μ = μ H = 0. Then conditions (A2) and (A3) can be rewritten as:

x H * ( t H , t L | λ ) = λ v H  and  x L * ( t H , t L | λ ) = λ v L .

Next, we establish parameters necessary for μ L = μ = μ H = 0. Condition μ = 0 is equivalent to t L ≤ λ v L 2 . Condition μ L = 0 is equivalent to λv L (v H − v L ) ≤ t H − t L . Condition μ H = 0 is equivalent to t H − t L ≤ λv H (v H − v L ).□

Case 3: μ H = μ = 0 and μ L > 0.

Claim 3 t H − t L < λv L (v H − v L ) and v L λE[v i ] ≥ βt H + (1 − β)t L if and only if μ H = μ = 0 and μ L > 0.

Proof.

Suppose μ H = μ = 0 and μ L > 0. Then conditions (A2) and (A3) are rewritten respectively as:

(A7) β ( λ v H − x H ) − μ L = 0 ,

(A8) ( 1 − β ) ( λ v L − x L ) + μ L = 0 .

Combining (A7) and (A8) we have λE[v i ] − βx H − (1 − β)x L = 0. Next, the binding (IC L ) implies x H = x L + (t H − t L )/v L . Combining the two equations we have:

(A9) x H * ( t H , t L | λ ) = λ E [ v i ] + ( 1 − β ) t H − t L v L  and  x L * ( t H , t L | λ ) = λ E [ v i ] − β t H − t L v L .

Next, we establish parameters necessary for μ H = μ = 0 < μ L . From (A7) we have μ L = β(x H − λv H ), and form (A8) we have μ L = (1 − β)(λv L − x L ). Thus, μ L > 0 is equivalent to x L * ( t H , t L | λ ) < λ v L and x H * ( t H , t L | λ ) > λ v H , which can be rewritten as t H − t L < λv L (v H − v L ). Condition μ = 0 is equivalent to v L x L * ( t H , t L | λ ) − t L ≥ 0 , which can be rewritten as v L λE[v i ] ≥ βt H + (1 − β)t L .□

Case 4: μ L = 0, μ > 0 and μ H > 0.

Claim 4 t H v H + t L 1 v L − 1 v H > λ v H and β t H v H + t L 1 v L − β v H > λ E [ v i ] if and only if μ L = 0, μ > 0 and μ H > 0.

Proof.

Suppose μ L = 0, μ > 0 and μ H > 0. Then conditions (A2) and (A3) can be rewritten as:

(A10) β ( λ v H − x H ) + μ H = 0 ,

(A11) ( 1 − β ) ( λ v L − x L ) + μ v L − μ H = 0 .

Binding (IC H ) implies x H = x L + (t H − t L )/v H , and binding (PC L ) implies x L = t L /v L . Combining the two equations gives:

x H * ( t H , t L | λ ) = t H v H + t L 1 v L − 1 v H  and  x L * ( t H , t L | λ ) = t L v L .

Next, we establish parameters necessary for μ L = 0, μ > 0 and μ H > 0. From (A10) we have μ H = β(x H − λv H ), and from (A11) we have μv L = μ H + (1 − β)(x L − λv L ). Therefore, μ H > 0 is equivalent to t H v H + t L 1 v L − 1 v H > λ v H . Condition μ > 0 is equivalent to β t H v H + t L 1 v L − β v H > λ E [ v i ] .□

Case 5: μ L = μ H = 0 and μ > 0.

Claim 5 t L > λ v L 2 , t H v H + t L 1 v L − 1 v H ≤ λ v H , λ v H ≤ t H v L if and only if μ L = μ H = 0 and μ > 0.

Proof.

Suppose μ L = μ H = 0 < μ. Then conditions (A2) and (A3) can be rewritten as:

(A12) β ( λ v H − x H ) = 0 ,

(A13) ( 1 − β ) ( λ v L − x L ) + μ v L = 0 .

Binding (PC L ) implies x L = t L /v L . Combining with the (A12) gives:

x H * ( t H , t L | λ ) = λ v H  and  x L * ( t H , t L | λ ) = t L v L .

Next, we establish parameters necessary for μ L = μ H = 0 and μ > 0. From (A13) we have μv L = (1 − β)(x L − λv L ). Thus, μ > 0 is equivalent to t L > λ v L 2 . Conditions μ H = μ L = 0 is equivalent to t H v H + t L 1 v L − 1 v H ≤ λ v H and λ v H ≤ t H v L .□

Case 6: μ H = 0, μ L > 0 and μ > 0.

Claim 6 t H < λv L v H and λv L E[v i ] < βt H + (1 − β)t L if and only if μ H = 0, μ L > 0 and μ > 0.

Proof.

Suppose μ H = 0, μ L > 0 and μ > 0. Then conditions (A2) and (A3) can be rewritten as:

(A14) β ( λ v H − x H ) − μ L = 0

(A15) ( 1 − β ) ( λ v L − x L ) + μ v L + μ L = 0 .

The binding (PC L ) implies x L = t L v L , and the binding (IC L ) implies x H = x L + ( t H − t L ) v L . Combining the two equations together we have:

x H * ( t H , t L | λ ) = t H v L  and  x L * ( t H , t L | λ ) = t L v L .

Next, we establish parameters necessary for μ H = 0, μ L > 0 and μ > 0. From (A14) we have μ L = β(λv H − x H ). Thus, μ L > 0 is equivalent to t H < λv L v H . From (A15) we have μv L = −(1 − β)(λv L − x L ) − μ L . Thus, μ > 0 is equivalent to λv L E[v i ] < βt H + (1 − β)t L .□

Note that only in Cases 1, 3, 4 and 6 either (IC H ) or (IC L ) are binding. Also, (PC L ) is non-binding only in Cases 1 and 3. That is, only relevant cases in which inducing the legislature’s truthful report is an issue and the public gets a rent when i = L are Cases 1 and 3. From (A6) and (A9) in the proofs of Claims 1 and 3, ∂ x i * ( t H , t L | λ ) ∂ t i = 0 if and only if t H = t L .

Appendix C: Proof of Proposition 3

Following the proof of Proposition 2, we will examine the two scenarios (i) when the legislature chooses identical transfers, t L = t H , and (ii) different transfers, t H ≠ t L , separately. For each of the two cases, we first determine the optimal transfer policy, t H * and t L * , and the corresponding values of the public’s expected payoff. Then, we establish conditions when one transfer policy delivers a higher value of the objective function than the other. Finally, we prove that if λ ≥ λ ̄ , then t H * = t L * = 0 and x H * = x L * = λ E v i .

  1. Equal Transfers (t H = t L ): In proving Lemma 2 (Appendix B), we established that if t L = t H = t, then the bureaucrat chooses the project’s attribute level x, where

x = t v L     if  λ < t v L E [ v i ] , λ E [ v i ]     if  λ ≥ t v L E [ v i ] .

The public’s expected payoff is then:

  β t H − x H * ( t H , t L | λ ) 2 2 + λ [ v H x H * ( t H , t L | λ ) − t H ] + ( 1 − β ) t L − x L * ( t H , t L | λ ) 2 2 + λ [ v L x L * ( t H , t L | λ ) − t L ] = β v H v L − 1 t     if  λ < t v L E [ v i ] , λ ( E [ v i ] ) 2 − t     if  λ ≥ t v L E [ v i ] .

Thus, the bureaucrat’s expected payoff becomes:

  β t − x 2 2 + λ ( v H x − t ) + ( 1 − β ) t − x 2 2 + λ ( v L x − t ) = 1 − λ + λ E [ v i ] v L t − t 2 2 v L 2     if  λ < t v L E [ v i ] , ( 1 − λ ) t + ( λ E [ v i ] ) 2 2     if  λ ≥ t v L E [ v i ] .

The legislature’s optimization problem then becomes:

max t t β v H v L − 1 1 t ∈ D 1 + λ E v i 2 − t 1 t ∈ D 2 ,  where 1 t ∈ T = 1     if  t ∈ T , 0     if  t ∉ T , D 1 ≡ g t : λ < t v L E [ v i ] , 1 − λ + λ E [ v i ] v L t − t 2 2 v L 2 ≥ 0 , t ≥ 0 , D 2 ≡ g t : λ ≥ t v L E [ v i ] , ( 1 − λ ) t + ( λ E [ v i ] ) 2 2 ≥ 0 , t ≥ 0 .

Since D1 ∩ D2 = ∅, we first determine the highest value of the objective function if t ∈ D 1. Then we find the highest value of the objective function if t ∈ D 2. Finally, we compare the two values. We examine each case of t ∈ D 1 and t ∈ D 2 in turn.

Suppose t ∈ D 1. Note tat D 1 is determined by the following inequalities: t > λv L E[v i ], t 1 − λ + λ E [ v i ] / v L − t / 2 v L 2 ≥ 0 and t ≥ 0. Hence:

D 1 = ∅   if  λ v L E [ v i ] ≥ 2 v L 2 1 − λ + λ E [ v i ] v L , λ v L E [ v i ] < t ≤ 2 v L 2 1 − λ + λ E [ v i ] v L   if  λ v L E [ v i ] < 2 v L 2 1 − λ + λ E [ v i ] v L .

Since the objective function is increasing in t for t ∈ D 1, the optimal transfer is given by t 1 * = 2 v L 2 ( 1 − λ + λ E [ v i ] / v L ) , which is relevant if and only if

(A16) λ v L E [ v i ] < 2 v L 2 ( 1 − λ + λ E [ v i ] / v L ) ,

or, equivalently, if either

(A17) Δ v < v L β  and  0 ≤ λ < 2 v L 2 v L − E [ v i ] ,

(A18) or  Δ v ≥ v L β ,

where (A17) and (A17) are derived from solving the inequality in (A16) with respect to λ. The value of the objective function in this case is:

(A19) 2 β v L 2 1 − λ + λ E [ v i ] v L v H v L − 1 .

That is, if either (A17) or (A18) holds, then there is a unique optimal transfer chosen by the legislature, t 1 * = 2 v L 2 ( 1 − λ + λ E [ v i ] / v L ) , which yields the expected payoff in (A19).

Suppose now t ∈ D 2. Then clearly the objective function is decreasing in t, so the optimal transfer is t 2 * = 0 if

(A20) Δ v < v L β  and  λ ≥ 2 v L 2 v L − E [ v i ] .

The value of the objective function is:

(A21) λ ( E [ v i ] ) 2 .

We now compare the public’s expected payoffs in (A19) and (A21). Note that:

(A22) λ E v i 2 < 2 β v L 2 1 − λ + λ E [ v i ] v L v H v L − 1 ,

if and only if either

(A23) ( E [ v i ] ) 2 − 2 ( β Δ v ) 2 ≤ 0 ,  or

(A24) ( E [ v i ] ) 2 − 2 ( β Δ v ) 2 > 0  and  λ < 2 β v L Δ v ( E [ v i ] ) 2 − 2 ( β Δ v ) 2 ≡ λ ̄ ,

where (A23) and (A24) are derived from solving the inequality in (A22) with respect to λ. Therefore, if either (A23) and (A24) holds, then the legislature’s optimal transfer payment to the bureaucrat is: t 1 * = 2 v L 2 ( 1 − λ + λ E [ v i ] / v L ) .

From (A20) and (A24), note that:

( E [ v i ] ) 2 − 2 ( β Δ v ) 2 > 0 ⟺ Δ v < v L ( 2 − 1 ) β , where  v L ( 2 − 1 ) β > v L β  for any  β .

Thus, t* = 0 if v L / β < Δ v < v L / ( 2 − 1 ) β and λ ≥ λ ̄ , and t * = 2 v L 2 ( 1 − λ + λ E [ v i ] / v L ) > 0 if Δ v ≥ v L / ( 2 − 1 ) β .

Finally, it is also optimal to set t * = 2 v L 2 ( 1 − λ + λ E [ v i ] / v L ) > 0 if Δv < v L /β and λ < min  { λ ̄ , 2 v L / 2 v L − E [ v i ] } . Since λ ̄ < 2 v L / 2 v L − E [ v i ] for all parameters,[22] the last condition simplifies to Δv < v L /β and λ < λ ̄ . Thus, if Δ v < v L / ( 2 − 1 ) β and λ ≥ λ ̄ , then the optimal transfer is t* = 0, and the corresponding expected payoff to the public is:

π A = λ ( E [ v i ] ) 2 .

In case either Δ v < v L / ( 2 − 1 ) β and λ < λ ̄ or Δ v ≥ v L / ( 2 − 1 ) β , then t * = 2 v L 2 ( 1 − λ + λ E [ v i ] / v L ) , and the corresponding expected payoff to the public is:

π B = 2 β Δ v ( v L − λ v L + λ E [ v i ] ) .

Different Transfers (t H ≠ t L ): Since the legislature’s optimization problem depends on values of x L * ( t H , t L | λ ) and x H * ( t H , t L | λ ) , we analyze with each of the six cases for t H ≠ t L in Claim 1 ∼ Claim 6 from Appendix B.

Case 1: μ L = μ = 0 and μ H > 0.

The legislature chooses t H and t L to maximize:

λ ( E [ v i ] ) 2 − β β + ( 1 − β ) v L v H t H − ( 1 − β ) 1 + β 1 − v L v H t L ,  s.t.

t H − t L > λ v H ( v H − v L ) ,

v L λ E [ v i ] ≥ 1 − β v L v H t L + β v L v H t H ,

β t H − λ E [ v i ] + ( 1 − β ) t H − t L v H 2 2 + λ v H λ E [ v i ] + ( 1 − β ) t H − t L v H − t H + ( 1 − β ) t L − λ E [ v i ] − β t H − t L v H 2 2 + λ v L λ E [ v i ] − β t H − t L v H − t L ≥ 0 ,

where the last constraint is (PC). Note that the value of the objective function is at most λ ( E [ v i ] ) 2 − β ( β + ( 1 − β ) v L / v H ) ε , which is achieved if t L = 0 and t H = ɛ > 0. This is strictly less than π B = λ ( E [ v i ] ) 2 , which is achieved if t L = t H = 0. Thus, we can disregard Case 1 from the consideration.

Case 2: μ L = μ = μ H = 0.

The legislature chooses t H and t L to maximize:

λ E v i 2 − β t H − ( 1 − β ) t L ,  s.t.

t L + λ v L ( v H − v L ) ≤ t H ,

t H ≤ t L λ v H ( v H − v L ) ,

t L ≤ λ v L 2 ,

( 1 − λ ) ( β t H + ( 1 − β ) t L ) + λ E v i 2 2 ≥ 0 ,

where the last constraint is (PC). Note that we can ignore the bureaucrat’s participation constraint (PC) since it is automatically satisfied. Next, the highest value of the objective function is achieved at:

t L * = 0 and t H * = λ v L ( v H − v L ) ,

and the corresponding expected payoff to the public in this case becomes:

π D = λ β v H 2 − β v L v H + v L 2 .

Case 3: μ H = μ = 0 and μ L > 0.

The legislature chooses t H and t L to maximize:

λ ( E [ v i ] ) 2 − β t H − ( 1 − β ) t L + β ( 1 − β ) v H v L − 1 ( t H − t L ) ,  s.t.

t H − t L < λ v L ( v H − v L ) ,

v L λ E [ v i ] ≥ β t H + ( 1 − β ) t L ,

  ( 1 − λ ) β t H + ( 1 − β ) t L + λ β ( 1 − β ) v H v L − 1 ( t H − t L ) − β ( 1 − β ) 2 v L 2 ( t H − t L ) 2 + ( λ E [ v i ] ) 2 2 ≥ 0 ,

where the last constraint is (PC). Note that the objective function is decreasing in t L and, therefore, t L * = 0 . Next, the objective function is decreasing (increasing) in t H if 1 − ( 1 − β ) v H / v L − 1 ≥ 0 ( < 0 ) or, equivalently, if Δv ≤ v L /(1 − β) (Δv > v L /(1 − β)).

If Δv ≤ v L /(1 − β), therefore, the value of the objective function is strictly less than λ ( E [ v i ] ) 2 ( < π B = λ ( E [ v i ] ) 2 ), which is achieved if t L = t H = 0. Thus, we can disregard this case from the consideration. If Δv > v L /(1 − β), then the optimal pricing policy is to set t H as close to v L λE[v i ]/β as possible, which is Case 2 we considered before. Thus, we can disregard Case 3 from the consideration.

Case 4: μ L = 0, μ > 0 and μ H > 0.

The legislature chooses t H and t L to maximize:

β t L v H v L − 1 ,  s.t.

t H v H + t L 1 v L − 1 v H > λ v H ,

t H v H + t L 1 β v L − 1 v H > λ v H + ( 1 − β ) μ v L ,

β t H + 1 + β λ v H v L − 1 − 1 t L − β 2 t H − t L v H + t L v L 2 − ( 1 − β ) 2 t L v L 2 ≥ 0 .

Consider first a (relaxed) optimization problem of choosing t H > t L to maximize function β t L v H / v L − 1 subject to (PC) constraint only. Labeling γ as the Lagrange multiplier associated with the (PC) constraint, the optimization problem has the following Lagrangian:

L = β t L v H v L − 1 + . γ β t H + 1 + β λ v H v L − 1 − 1 t L − β 2 t H − t L v H + t L v L 2 − 1 − β 2 t L v L 2

The optimality conditions are:

∂ L ∂ t H = γ β 1 − 1 v H t H − t L v H + t L v L = 0 ,

∂ L ∂ t L = β v H v L − 1 + γ 1 + β λ v H v L − 1 − 1 − β t H − t L v H + t L v L Δ v v H v L − 1 − β v L 2 t L = 0 .

The expression for ∂ L / ∂ t L = 0 implies that γ > 0 and, thus we have:

(A25) t H = v H 2 − t L v H v L − 1 .

Binding (PC) implies:

(A26) β t H + 1 + β λ v H v L − 1 − 1 t L − β 2 t H − t L v H + t L v L 2 − 1 − β 2 t L v L 2 = 0 .

Combining (A25) and (A26) together, we have a quadratic equation in t L :

− 1 − β 2 v L 2 t L 2 + 1 − β − β ( 1 − λ ) v H v L − 1 t L + β v H 2 2 = 0 ,

which gives:

(A27) t L * = v L − β [ v H − λ Δ v ] + ( v L − β [ v H − λ Δ v ] ) 2 + β ( 1 − β ) v H 2 ( 1 − β ) / v L .

Since t H * = v H 2 − v H / v L − 1 t L * , this case exists if and only if v H 2 − v H / v L − 1 t L * > t L * , or

λ < Δ v − v L 2 β Δ v ≡ λ ̂ .

Finally, the two constraints we ignored in the relaxed problem are automatically satisfied with t L * and t H * obtained above. The expected payoff to the public in this case is:

(A28) π C = β v H v L − 1 t L * ,

which exists if and only if λ < λ ̂ .

Case 5: μ L = μ H = 0 and μ > 0.

The legislature chooses t H and t L to maximize:

β λ v H 2 − t H ,  s.t.

t L > λ v L 2 ,

t H v H + t L 1 v L − 1 v H ≤ λ v H ,

t H ≥ λ v H v L ,

  β t H − ( λ v H ) 2 2 + λ ( v H ( λ v H ) − t H ) + ( 1 − β ) t L − ( t L / v L ) 2 2 + λ v L t L v L − t L ≥ 0 ,

where the last constraint is (PC). Note that since t H ≥ λv H v L , the value of the objective function is at most β λ v H 2 − β λ v H v L = β λ v H 2 − v H v L . This is strictly less than π D = λ β v H 2 − β v L v H + v L 2 from Case 2. Therefore, we can disregard Case 5 from the consideration.

Case 6: μ H = 0, μ L > 0 and μ > 0.

The legislature chooses t H and t L to maximize:

β t H v H v L − 1 , s.t.

t H < λ v L v H ,

λ v L E [ v i ] < β t H + ( 1 − β ) t L ,

β t H − ( t H / v L ) 2 2 + λ v H t H v L − t H + ( 1 − β ) t L − ( t L / v L ) 2 2 + λ v L t L v L − t L ≥ 0 ,

where the last constraint is (PC). Note that since t H < λv L v H , the value of the objective function is at most β λ v H v L v H / v L − 1 = β λ v H 2 − v H v L . This is strictly less than π D = λ β v H 2 − β v L v H + v L 2 from Case 2. Therefore, we can disregard Case 6 from the consideration.

We have now established conditions when the legislature chooses different transfers (t H ≠ t L ) over identical transfer (t H = t L ) in order to maximize the public’s expected payoff. The equilibrium value of the public’s expected payoff is π = max{π A , π B , π C , π D }, where:

π A = λ ( E [ v i ] ) 2 ,

π B = 2 β Δ v ( v L − λ v L + λ E [ v i ] ) ,

π C = β Δ v v L − β [ v H − λ Δ v ] + ( v L − β [ v H − λ Δ v ] ) 2 + β ( 1 − β ) v H 2 ( 1 − β ) ,

π D = λ β v H 2 − β v L v H + v L 2 .

The remaining task is to compare the values of π A , π B , π C and π D and determine conditions when each of them might be higher than the others.

We first compare π A and π B when t L = t H . We established earlier that if Δ v < v L / ( 2 − 1 ) β and λ ≥ λ ̄ , then π A > π B . In case either Δ v < v L / ( 2 − 1 ) β and λ < λ ̄ , or Δ v ≥ v L / ( 2 − 1 ) β , then π A ≤ π B . For λ ̄ to be positive, however, Δ v < v L / ( 2 − 1 ) β must hold. Thus, if Δ v ≥ v L / ( 2 − 1 ) β , then λ ̄ ≤ 0 , which in turn implies that π A ≥ π B for any λ (at Δ v = v L / ( 2 − 1 ) β , π A = π B ).

We next compare π C and π D when t L ≠ t H . First we define:

λ ̃ ≡ 2 β v L Δ v β ( 1 − 2 β ) v H 2 + β ( 4 β − 1 ) v L v H + v L 2 ( 1 − 2 β 2 )

Next, π D > π C is explicitly expressed as:

(A29)   λ β v H 2 − β v L v H + v L 2 > β Δ v v L − β [ v H − λ Δ v ] + ( v L − β [ v H − λ Δ v ] ) 2 + β ( 1 − β ) v H 2 ( 1 − β ) ,

which is simplified as:

λ > λ ̃ .

Recall from (A28) that the public’s expected payoff is π C and it exists if and only if λ < λ ̂ . Therefore:

λ < min { λ ̂ , λ ̃ } ⇒ π C > π D .

Finally, we compare max{π A , π B } and max{π C , π D } to find the highest value of the public’s expected payoff. That π D > π A is explicitly written as:

λ β v H 2 − β v L v H + v L 2 > λ ( E [ v i ] ) 2 ,

which is simplified to:

(A30) v H > 2 − β 1 − β v L .

Thus, π D > π A if and only if (A30) holds. Next, π D > π B is explicitly expressed as:

λ β v H 2 − β v L v H + v L 2 > 2 β Δ v ( v L − λ v L + λ E [ v i ] ) ,

which can be rewritten as:

Φ > 2 β v L Δ v ,

where Φ ≡ β ( 1 − 2 β ) v H 2 + β ( 4 β − 1 ) v L v H + v L 2 ( 1 − 2 β 2 ) . Thus, π B > π D if and only if either:

Φ < 0 , or  Φ > 0  and  λ < λ ̃ .

Solving Φ < 0 in v H , the conditions can be rewritten as follows: π B > π D if and only if either:

β > 1 2  and  Δ v > v ̃ L ,

β < 1 2  and  λ < λ ̃ ,

or  β > 1 2 ,  Δ v < v ̃ L  and  λ < λ ̃ ,

where v ̃ L ≡ β ( 4 β − 1 ) + β ( 9 β − 4 ) − 2 β ( 2 β − 1 ) v L / 2 β ( 2 β − 1 ) . We now consider π C and π B . That π C > π B is written explicitly as:

β Δ v v L − β [ v H − λ Δ v ] + ( v L − β [ v H − λ Δ v ] ) 2 + β ( 1 − β ) v H 2 ( 1 − β ) > 2 β Δ v ( v L − λ v L + λ E [ v i ] ) ,

which is simplified as:

2 β λ Δ v + ( Δ v − 2 v L ) 2 > 0 ,

and this inequality holds for all parameters. Since the relevant set of parameters (when π C > π B ) for which we consider π C is when λ < λ ̂ , we have π C > π B if and only if λ < λ ̂ . Lastly, we consider π C and π D . Note that π C > π D if and only if:

  β Δ v v L − β [ v H − λ Δ v ] + ( v L − β [ v H − λ Δ v ] ) 2 + β ( 1 − β ) v H 2 ( 1 − β ) > λ β v H 2 − β v L v H + v L 2 ,

β < 1 2 and λ < λ ̄ .

To sum up, we established that:

  1. π A > max{π B , π C , π D } if and only if λ > λ ̄ .

  2. π B > max{π A , π C , π D } if and only if either β ≥ 1 2 and λ ̄ ≤ λ < λ ̂ , or β < 1 2 and λ < min { λ ̄ , λ ̃ } .

  3. π C > max{π A , π B , π D } if and only if either β ≥ 1 2 and λ < λ ̂ , or β < 1 2 and λ < min { λ ̂ , λ ̃ } .

  4. π D > max{π A , π B , π C } if and only if β < 1 2 and max { λ ̂ , λ ̃ } ≤ λ < λ ̄ .

Therefore, if λ ≥ λ ̄ , then t H * = t L * = 0 and x H * = x L * = λ E v i . This concludes the proof of Proposition 3.

Appendix D: Proof of Proposition 4

After substituting for x H * ( t L , t H ) = t L 1 v L − 1 v H + t H v H and x L * ( t L , t H ) = t L v L in the objective function and the constraint in P l , the legislature’s problem is rewritten as:

max t H , t L β t L v H v L − 1 ,  subject to

t H − t L ≥ 0 and (PC): β t H + ( 1 − β ) t L − β 2 Δ v t L v H v L + t H v H 2 − 1 − β 2 t L v L 2 ≥ 0 .

The Lagrangian of the problem then is:

L = β t L v H v L − 1 + μ 1 [ t H − t L ] + μ 2 β t H + ( 1 − β ) t L − β 2 Δ v t L v H v L + t H v H 2 − 1 − β 2 t L v L 2 ,

and the relevant Kuhn-Tucker conditions for the optimization problem with respect to t L and t H , respectively, are:

(A31) β v H v L − 1 − μ 1 + μ 2 ( 1 − β ) − β Δ v t L v H v L + t H v H Δ v v H v L − 1 − β v L 2 t L = 0 ,

and

(A32) μ 1 + μ 2 β − β t L 1 v L − 1 v H + t H v H 1 v H = 0 .

Since β t L ( v H v L − 1 ) > 0 , having μ 1 = μ 2 = 0 contradicts (A31). Having μ 1 > 0 and μ 2 = 0 is impossible as well since this would contradict (A32). Therefore, a solution must have either μ 1, μ 2 > 0 (both are positive) or μ 2 > μ 1 = 0.

Case 1: Suppose μ 1 > 0 and μ 2 > 0. First, μ 1 > 0 implies t H = t L = t. Second, μ 2 > 0 implies that (PC) is binding and gives the following expression:

β t + ( 1 − β ) t − β 2 Δ v t L v H v L + t v H 2 − ( 1 − β ) 2 t v L 2 = t 1 − t 2 v L 2 = 0 ,

which implies either t = 0 or t = 2 v L 2 . Adding (A31) and (A32) we obtain:

β v H v L − 1 − μ 1 + μ 2 ( 1 − β ) − β Δ v t v H v L + t v H Δ v v H v L − 1 − β v L 2 t + μ 1 + μ 2 β − β Δ v t v H v L + t v H 1 v H = β v H v L − 1 + μ 2 1 − t v L 2 = 0 .

If t = 0, then we have a contradiction since β v H v L − 1 + μ 2 = 0 is impossible – the contradiction emerges since v H v L − 1 > 0 and μ 2 > 0 by assumption. If t = 2 v L 2 , then β v H v L − 1 + μ 2 1 − t v L 2 = 0 implies μ 2 = β v H v L − 1 > 0 . From (A32) we obtain:

μ 1 = μ 2 β t v L v H − 1 = β 2 v H v L − 1 2 v L v H − 1 ,

which is positive if only if 2 v L v H − 1 > 0 . Thus, if Δv < v L , there exists a solution with t H = t L = t = 2 v L 2 > 0 .

Case 2: Suppose now μ 1 = 0 and μ 2 > 0. First, (A32) condition simplifies to:

(A33) μ 2 β − β Δ v t L v H v L + t H v H 1 v H = 0 ,

implying that t H = v H 2 − v H v L − 1 t L . Second, μ 2 > 0 implies that (PC) constraint is binding, and together with t H = v H 2 − v H v L − 1 t L , the binding constraint gives:

  β t H + ( 1 − β ) t L − β 2 Δ v t L v H v L + t H v H 2 − 1 − β 2 t L v L 2 = − 1 − β 2 v L 2 t L 2 + 1 − β v H v L t L + β 2 v H 2 = 0 ,

a quadratic equation with respect to t L . The discriminant of this quadratic equation is D = 1 − β v H v L 2 + β ( 1 − β ) ( v H / v L ) 2 > 0 . Therefore, the relevant root of the quadratic equation is:

(A34) t L = 1 − β v H v L + 1 − β v H v L 2 + β ( 1 − β ) v H 2 v L 2 ( 1 − β ) / v L 2 > 0 .

Adding (A31) and (A33) we obtain:

  β v H v L − 1 + μ 2 ( 1 − β ) − β Δ v t L v H v L + t H v H Δ v v H v L − 1 − β v L 2 t L + μ 2 β − β Δ v t L v H v L + t H v H 1 v H = β v H v L − 1 + μ 2 1 − β v H v L − 1 − β v L 2 t L = 0 ,

which implies that μ 2 = β v H v L − 1 / 1 − β v H v L − 1 − β v L 2 t L . Since 1 − v H v L < 0 , we have μ 2 > 0 if and only if 1 − β v H v L − 1 − β v L 2 t L < 0 , which holds true for any parameters given the t L expressed in (A34).■

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Received: 2023-11-26
Accepted: 2024-06-03
Published Online: 2024-06-19

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Global Dynamics and Optimal Policy in the Ak Model with Anticipated Future Consumption
  4. Offsetting Distortion Effects of Head Starts on Incentives in Tullock Contests
  5. Collusive Price Leadership Among Firms with Different Discount Factors
  6. Motivating Loyal Bureaucrats in Sequential Agency
  7. Disclosure of Product Information After Price Competition
  8. Uncertain Commitment Power in a Durable Good Monopoly
  9. Optimal Trade Policy in a Ricardian Model with Labor-Market Search-and-Matching Frictions
  10. Consumer-Benefiting Transport Costs: The Role of Product Innovation in a Vertical Structure
  11. Information Disclosure by Informed Intermediary in Double Auction
  12. Notes
  13. Strategic Partial Inattention in Oligopoly
  14. The Role of Informative Advertising in Aligning Preferences Over Product Design
  15. To Bequeath, or Not to Bequeath? On Labour Income Risk and Top Wealth Concentration
https://doi.org/10.1515/bejte-2023-0122
Keywords for this article
principal – agent; false moral hazard; sequential agency

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Global Dynamics and Optimal Policy in the Ak Model with Anticipated Future Consumption
  4. Offsetting Distortion Effects of Head Starts on Incentives in Tullock Contests
  5. Collusive Price Leadership Among Firms with Different Discount Factors
  6. Motivating Loyal Bureaucrats in Sequential Agency
  7. Disclosure of Product Information After Price Competition
  8. Uncertain Commitment Power in a Durable Good Monopoly
  9. Optimal Trade Policy in a Ricardian Model with Labor-Market Search-and-Matching Frictions
  10. Consumer-Benefiting Transport Costs: The Role of Product Innovation in a Vertical Structure
  11. Information Disclosure by Informed Intermediary in Double Auction
  12. Notes
  13. Strategic Partial Inattention in Oligopoly
  14. The Role of Informative Advertising in Aligning Preferences Over Product Design
  15. To Bequeath, or Not to Bequeath? On Labour Income Risk and Top Wealth Concentration
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