Property rights enforcement with unverifiable incomes

Abstract

I study the extent of secure property rights a planner can implement. Agents can produce output, appropriate others’ output, or work in property rights enforcement. The planner pays enforcement personnel using taxes collected from producers who can hide income from taxation at a cost. The planner implements perfectly secure property rights by incentivizing production through redistributive taxation and absorbing potential appropriators as enforcement personnel. Both taxation and employment in enforcement institutionalize redistribution that would otherwise take place through appropriation. Higher costs of hiding income permit more redistributive taxation and less enforcement, leading to more production and higher welfare.

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Notes

  1. 1.

    See Internal Revenue Service (2016). An estimate of the contribution of underreporting to the gap after enforced payments is not available. Tax compliance between 2008 and 2010 was comparable to that in 2006 (see pp. 6–8). See, e.g., Andreoni et al. (1998) on tax underpayment and evasion; see, e.g., Quintin (2008) on the link between limited contract enforcement, taxation, and informality.

  2. 2.

    Tullock (1967), Rose-Ackerman (1975), and Becker (1968) started large literatures on some such activities. The availability of these activities affects the allocation of talent and resources. See, e.g., Baumol (1990), Murphy et al. (1991, 1993), Acemoglu (1995), and Acemoglu and Verdier (1998).

  3. 3.

    On inequality, redistribution, and crime, see, e.g., Benoît and Osborne (1995), Imrohoroğlu et al. (2000).

  4. 4.

    Following Lacker and Weinberg (1989), this assumption is without loss of generality (see Appendix 1).

  5. 5.

    At the solution of the planner’s problem, agents of type l do not want to display \(w_h\), even if it is costless.

  6. 6.

    There is a \(\underline{\underline{\phi }}<\underline{\phi }\) so that if \(\phi \in [\underline{\underline{\phi }},\underline{\phi }]\), no feasible regime improves on anarchy, but some avoid appropriation.

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Authors

Corresponding author

Correspondence to Jan U. Auerbach.

Additional information

I thank Costas Azariadis for many discussions as well as three anonymous referees and participants at the Midwest Macro Meeting at Washington University in St. Louis, the International Meeting of the Association for Public Economic Theory at the University of Luxembourg, the Meeting of the European Public Choice Society at the University of Freiburg, the WINIR Symposium at the University of Bristol, the Midwest Economic Theory Conference at the University of Rochester, and the InsTED Workshop at Indiana University Bloomington. Errors are mine.

Appendices

Appendices

A Tax schedules

I briefly replicate the argument given in Lacker and Weinberg (1989) for this environment. A mechanism consists of a message space M and a tax schedule t that maps the message and the income displayed into \({\mathbb {R}}\). A producer w chooses a message \(m(w)\in M\) and an income display z(w) to maximize \(w-t(m(w),z(w))-\psi (w-z(w))\). Suppose two agents with different productivities \(w_1\) and \(w_2\ne w_1\) were to send different messages \(m(w_1)=m_1\) and \(m(w_2)=m_2\ne m_1\) but display the same income \(z(w_1)=z(w_2)=\hat{z}\). By optimality of message and display, for agent \(w_1\), \(t(m_1,\hat{z})\le t(m_2,\hat{z})\), while for agent \(w_2\), \(t(m_1,\hat{z})\ge t(m_2,\hat{z})\), so that \(t(m_1,\hat{z})=t(m_2,\hat{z})\). The same income display implies the same tax payment, irrespective of the message, which justifies focusing on tax schedules that only depend on displayed income.

B The planner’s objective function

Using the payoff expressions (2)–(4), the balanced budget (5), and the definitions of the probabilities (1), which the planner understands, the planner’s objective function is given by

$$\begin{aligned}&\chi _l^p\mu _l\varphi _l(\sigma )+\chi _h^p\mu _h\varphi _h(\sigma )+\left( \chi _l^e\mu _l+\chi _h^e\mu _h\right) w_e+(\chi _l^a\mu _l+\chi _h^a\mu _h)\nu (\sigma )\\&\quad =(1-\theta p)[\chi _l^p\mu _l(w_l-t_l)+\chi _h^p\mu _h(w_h-t(\zeta ))]-(1-\theta p)\chi _h^p\mu _h\psi (w_h-\zeta )\\&\qquad +[\chi _l^p\mu _l t_l+\chi _h^p\mu _h t(\zeta )]+\theta p \left[ \chi _l^p\mu _l (w_l-t_l)+\chi _h^p\mu _h(w_h-t(\zeta ))\right] \\&\qquad -\theta p\chi _h^p\mu _h\psi (w_h-\zeta )\\&\quad =\chi _l^p\mu _l(w_l-t_l)+\chi _h^p\mu _h(w_h-t(\zeta ))-\chi _h^p\mu _h\psi (w_h-\zeta )+\chi _l^p\mu _l t_l+\chi _h^p\mu _h t(\zeta )\\&\quad =\chi _l^p\mu _lw_l+\chi _h^p\mu _hw_h-\chi _h^p\mu _h\psi (w_h-\zeta ). \end{aligned}$$

C Proofs

Proposition 1

Proof

In anarchy, \(\sigma =\left( \chi _l^p,0,\chi _l^a,\chi _h^p,0,\chi _h^a,0,0,0\right) \), \(\varphi _i(\sigma )=(1-p)w_i\), \(i=l,h\), and \(\nu (\sigma )=(\chi _l^p\mu _lw_l+\chi _h^p\mu _hw_h)\), where \(p=\chi _l^a\mu _l+\chi _h^a\mu _h\) and \(\chi _i^a=1-\chi _i^p\), \(i=l,h\). First, in equilibrium, \(\chi _l^p=0\). Suppose for a contradiction that \(\chi _l^p>0\). Then, it must hold that \(\varphi _h(\sigma )>\varphi _l(\sigma )\ge \nu (\sigma )\) so that all agents of type h produce, \(\chi _h^p=1\). It follows that \(\nu (\sigma )=(\chi _l^p\mu _lw_l+\mu _hw_h)>(\chi _l^p\mu _l+\mu _h)w_l=(1-p)w_l=\varphi _l(\sigma )\), a contradiction. Thus, \(\chi _l^p=0\) and \(\chi _l^a=1\). For any \(\chi _h^p\in [0,1]\), \(\varphi _h(\sigma )=(1-p)w_h=(1-\mu _l-\chi _h^a\mu _h)w_h=(\mu _h-\chi _h^a\mu _h)w_h=(1-\chi _h^a)\mu _hw_h=\chi _h^p\mu _hw_h=\nu (\sigma )\ge \chi _h^p\mu _hw_l=(1-p)w_l\). That is, any profile of occupational choices that has any share of agents of type h producing and all other agents appropriating maximizes all agents’ expected payoffs, given all others’ occupational choices, and thus is an equilibrium. \(\square \)

Proposition 2

Proof

The objective function in Problem (FBP) is less than or equal to \(\bar{w}=\mu _lw_l+\mu _hw_h\) and equals \(\bar{w}\) if and only if \(\chi _l^p=\chi _h^p=1\) so that \(\chi _l^e=\chi _h^e=0\), implying \(\chi _l^p\mu _l t_l+\chi _h^p\mu _h t_h=0\). \(\square \)

Proposition 3

Proof

The objective function in Problem (\(\hbox {FBP}^\prime \)) is less than or equal to \(\bar{w}=\mu _lw_l+\mu _hw_h\) and equals \(\bar{w}\) if and only if \(\chi _l^p=\chi _h^p=1\) so that \(\chi _l^e=\chi _h^e=0\), implying \(\chi _l^p\mu _l t_l+\chi _h^p\mu _h t_h=0\). Consider the regime given by \(\chi _l^p=\chi _h^p=1\), \(\chi _l^e=\chi _h^e=\chi _l^a=\chi _h^a=0\), \(t_l=w_l-\bar{w}\), \(t_h=w_h-\bar{w}\), and \(w_e=0\). The tax receipts equal \(\mu _l (w_l-\bar{w})+\mu _h (w_h-\bar{w})=\mu _l w_l+\mu _h w_h-(\mu _l+\mu _h)\bar{w}=\bar{w}-\bar{w}=0\). As \(p=0\), \((1-\theta p)(w_i-t_i)=\bar{w}\); as \(\theta =1\), \(\theta \left[ \chi _l^p\mu _l (w_l-t_l)+\chi _h^p\mu _h(w_h-t_h)\right] =\bar{w}\). Thus, this regime satisfies all constraints of Problem (\(\hbox {FBP}^\prime \)) and implements the first-best. It is the only regime that does so: if \(w_l-t_l\ne w_h-t_h\), then constraint (13) is violated for one of the two types of agents as \(\theta \left[ \chi _l^p\mu _l (w_l-t_l)+\chi _h^p\mu _h(w_h-t_h)\right] =\mu _l (w_l-t_l)+\mu _h(w_h-t_h)>\min \{w_l-t_l,w_h-t_h\}\); if \(w_l-t_l=w_h-t_h\), then the balanced budget, together with zero tax receipts after transfers, implies that \(w_l-t_l=w_h-t_h=\bar{w}\). \(\square \)

Lemma 1

Proof

First, production by all agents of typehcan be attained. Consider the regime \(\sigma \) that has all agents of type h produce and is given by \(\chi _h^p=\chi _l^a=1\), \(\chi _h^e=\chi _h^a=\chi _l^p=\chi _l^e=0\), \(t_l=t_h=0\), \(w_e=0\). Then, \(\theta =1\) and \(p=\mu _l\) so that \(\varphi _h(\sigma )=(1-\mu _l)w_h=\mu _hw_h\), \(\varphi _l(\sigma )=(1-\mu _l)w_l=\mu _hw_l\), and \(\nu (\sigma )=\mu _hw_h\). Therefore, \(\varphi _h(\sigma )\ge \max \{w_e,\nu (\sigma )\}\) and \(\nu (\sigma )\ge \max \{\varphi _l(\sigma ),w_e\}\). Thus, \(\sigma \) satisfies all constraints of Problem (PP) and is thus attainable.

Second, if any agents of typelproduce, then all agents of typehproduce. Consider a candidate regime \(\sigma \) and suppose for a contradiction that \(\chi _l^p>0\) and \(\chi _h^p<1\). Then, \(\theta p<1\) so that \(\varphi _h(\sigma )=(1-\theta p)\max \{w_h-t_h, w_h-t_l-\phi \}\ge (1-\theta p)(w_h-t_l-\phi )>(1-\theta p)(w_h-t_l-(w_h-w_l))=(1-\theta p)(w_l-t_l)=\varphi _l(\sigma )\), because \(w_h-w_l>\phi \). But, as \(\chi _l^p>0\), constraint (8) requires that \(\varphi _l(\sigma )\ge \max \{w_e,\nu (\sigma )\}\). That is, \(\varphi _h(\sigma )>\max \{w_e,\nu (\sigma )\}\), which violates at least one of the constraints (9) and (10), as \(\chi _h^p<1\) implies via constraint (11) that either \(\chi _h^e>0\), or \(\chi _h^a>0\), or both. Thus, \(\sigma \) is not in the constraint set of Problem (PP), a contradiction that completes the proof. \(\square \)

Lemma 2

Proof

Consider a candidate regime \(\sigma \) and suppose for a contradiction that \(\chi _l^p>0\) and \(\chi _l^e=0\). As \(\chi _h^p=1\) by Lemma 1, \(\chi _h^e=\chi _l^e=0\) and thus \(\theta =1\). Therefore, \(p=1-q\) so that \(\varphi _l(\sigma )=(1-p)(w_l-t_l)=q(w_l-t_l)\). At the same time, \(\nu (\sigma )=\chi _l^p\mu _l(w_l-t_l)+\mu _h\max \{w_h-t_h, w_h-t_l-\phi \}\ge \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_l-\phi )>\chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_l-(w_h-w_l))=\chi _l^p\mu _l(w_l-t_l)+\mu _h(w_l-t_l)=q(w_l-t_l)\), because \(w_h-w_l>\phi \). That is, \(\nu (\sigma )>\varphi _l(\sigma )\), which violates constraint (8), because \(\chi _l^p>0\). Thus, \(\sigma \) is not in the constraint set of Problem (PP), a contradiction that completes the proof. \(\square \)

Proposition 4

Proof

Consider any regime \(\sigma \) with \(\chi _h^p=1\), \(\chi _l^p>0\), and, by Lemma 2, \(\chi _l^e>0\) that satisfies all constraints of Problem (PP) and induces producers of type h to hide income from taxation. That is, \(w_h-t_l-\phi >w_h-t_h\). The value of the objective function is \(\chi _l^p\mu _lw_l+\mu _hw_h-\mu _h\phi \). I show that there is an alternative regime \(\hat{\sigma }\) in the constraint set which has the same occupation assignments but a different tax schedule that prevents income from being hidden and so increases welfare, and thus dominates \(\sigma \). There are two cases: \(\chi _l^a=0\) and \(\chi _l^a>0\).

First, suppose \(\chi _l^a=0\). Then, \(p=0\), \((1-\theta )=\chi _l^e\mu _l=1-q\), and from constraints (8) and (9) for agents of type l, \(\varphi _l(\sigma )=w_e\ge \nu (\sigma )\), or

$$\begin{aligned} w_l-t_l=w_e&=\left( \chi _l^e\mu _l\right) ^{-1}\left( \chi _l^p\mu _lt_l+\mu _ht_l\right) \\&\ge \left( 1-\chi _l^e\mu _l\right) \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_l-\phi ) \right] . \end{aligned}$$

Let \(\epsilon =\frac{\mu _h}{\mu _l+\mu _h}\phi \) and let \(\hat{\sigma }\) be given by \(\hat{\chi }_i^j=\chi _i^j\) for all ij, \(\hat{t}_h=t_l+\phi -\epsilon \), \(\hat{t}_l=t_l-\epsilon \), and \(\hat{w}_e=w_e+\epsilon \). Then, producers of type h do not hide income as \(w_h-\hat{t}_h=w_h- t_l-\phi +\epsilon =w_h-\hat{t}_l-\phi \), which is also greater than \(w_l-\hat{t}_l\), because \(w_h-w_l>\phi \), so that \(\varphi _h(\hat{\sigma })>\varphi _l(\hat{\sigma })\). Also, \(\varphi _l(\hat{\sigma })=w_l-\hat{t}_l=w_l-t_l+\epsilon =w_e+\epsilon =\hat{w}_e\) and the tax receipts increase to

$$\begin{aligned} \chi _l^p\mu _l\hat{t}_l+\mu _h\hat{t}_h&=\chi _l^p\mu _l(t_l-\epsilon )+\mu _h(t_l+\phi -\epsilon )\\&=\chi _l^p\mu _lt_l+\mu _ht_l+\mu _h\phi -q\epsilon \\&=\chi _l^e\mu _lw_e+(1-q)\epsilon \end{aligned}$$

or \(\chi _l^e\mu _lw_e+\chi _l^e\mu _l\epsilon \), because \(\mu _h\phi =(\mu _l+\mu _h)\epsilon =\epsilon \) from \(\epsilon =\frac{\mu _h}{\mu _l+\mu _h}\phi \) and \(1-q=1-\theta \), which is strictly greater than zero and exactly enough to pay all \(\chi _l^e\mu _l\) enforcers \(\hat{w}_e\), while

$$\begin{aligned} \nu (\hat{\sigma })&=\left( 1-\chi _l^e\mu _l\right) \left[ \chi _l^p\mu _l(w_l-t_l+\epsilon )+\mu _h(w_h-t_l-\phi +\epsilon )\right] \\&=\left( 1-\chi _l^e\mu _l\right) \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_l-\phi )\right] +\left( 1-\chi _l^e\mu _l\right) q\epsilon \\&\quad <\nu (\sigma )+\epsilon . \end{aligned}$$

That is, \(\varphi _l(\hat{\sigma })=\hat{w}_e>\nu (\hat{\sigma })\) so that \(\hat{\sigma }\) satisfies all constraints in Problem (PP) but, saving the cost of hiding income, yields a higher objective function value, \(\chi _l^p\mu _lw_l+\mu _hw_h\), than \(\sigma \), and thus dominates it.

Second, suppose \(\chi _l^a>0\). Then, \(p>0\) and, as \(\sigma \) obeys all constraints, \(\varphi _l(\sigma )=w_e=\nu (\sigma )\) or

$$\begin{aligned} (1-\theta p)(w_l-t_l)=w_e&=\left( \chi _l^e\mu _l\right) ^{-1}\left( \chi _l^p\mu _lt_l+\mu _ht_l\right) \\&=\theta \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_l-\phi ) \right] . \end{aligned}$$

Let \(\epsilon _l=\frac{\mu _h}{\mu _l+\mu _h}\phi (1+(1-\theta )\theta )^{-1}(\theta ^{-1}-p)^{-1}>0\) and \(\epsilon _h=\frac{\mu _l+\mu _h}{\mu _h}(\theta ^{-1}-p-\chi _l^p\mu _l)\epsilon _l\) and let \(\hat{\sigma }\) be given by \(\hat{\chi }_i^j=\chi _i^j\) for all ij, \(\hat{t}_h=t_l+\phi -\epsilon _h\), \(\hat{t}_l=t_l-\epsilon _l\), and \(\hat{w}_e=w_e+(1-\theta p)\epsilon _l\). Then, \(\epsilon _h>\epsilon _l\) because \(\mu _h^{-1}(\theta ^{-1}-p-\chi _l^p\mu _l)>1\) as \(\theta ^{-1}>1>\chi _l^a\mu _l+\chi _l^p\mu _l+\mu _h\), so that producers of type h do not hide income as \(w_h-\hat{t}_h=w_h- t_l-\phi +\epsilon _h>w_h- t_l-\phi +\epsilon _l=w_h-\hat{t}_l-\phi \), which is also greater than \(w_l-\hat{t}_l\), because \(w_h-w_l>\phi \), so that \(\varphi _h(\hat{\sigma })>\varphi _l(\hat{\sigma })\). Also, \(\varphi _l(\hat{\sigma })=(1-\theta p)\left( w_l-\hat{t}_l\right) =(1-\theta p)(w_l-t_l)+(1-\theta p)\epsilon _l=w_e+(1-\theta p)\epsilon _l=\hat{w}_e\) and the tax receipts increase to, using the definition of \(\epsilon _l\) to replace \(\mu _h\phi =(\mu _l+\mu _h)(1+(1-\theta )\theta )(\theta ^{-1}-p)\epsilon _l\), the expression for \(\epsilon _h\), the budget equation, and the fact that \((1-\theta )=\chi _l^e\mu _l\),

$$\begin{aligned} \chi _l^p\mu _l\hat{t}_l+\mu _h\hat{t}_h&=\chi _l^p\mu _l(t_l-\epsilon _l)+\mu _h(t_l+\phi -\epsilon _h)\\&=\chi _l^p\mu _lt_l+\mu _ht_l+\mu _h\phi -\mu _h\epsilon _h-\chi _l^p\mu _l\epsilon _l\\&=(\chi _l^p\mu _lt_l+\mu _ht_l)+(1+(1-\theta )\theta )(\theta ^{-1}-p)\epsilon _l\\&\quad -(\theta ^{-1}-p-\chi _l^p\mu _l)\epsilon _l-\chi _l^p\mu _l\epsilon _l\\&=\chi _l^e\mu _lw_e+(1-\theta )\epsilon _l-(1-\theta )\theta p\epsilon _l+\chi _l^p\mu _l\epsilon _l-\chi _l^p\mu _l\epsilon _l\\&=\chi _l^e\mu _lw_e+\chi _l^e\mu _l(1-\theta p)\epsilon _l\\&=\chi _l^e\mu _l(w_e+(1-\theta p)\epsilon _l)\\&=\chi _l^e\mu _l\hat{w}_e, \end{aligned}$$

which is strictly greater than zero and exactly enough to pay all \(\chi _l^e\mu _l\) enforcers \(\hat{w}_e\), while

$$\begin{aligned} \nu (\hat{\sigma })&=\theta \left[ \chi _l^p\mu _l(w_l-t_l+\epsilon _l)+\mu _h(w_h-t_l-\phi +\epsilon _h)\right] \\&=\theta \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_l-\phi )\right] +\theta (\chi _l^p\mu _l\epsilon _l+\mu _h\epsilon _h)\\&=\nu (\sigma )+\theta (\chi _l^p\mu _l\epsilon _l+(\theta ^{-1}-p-\chi _l^p\mu _l)\epsilon _l)\\&=\nu (\sigma )+(1-\theta p)\epsilon _l. \end{aligned}$$

That is, \(\varphi _l(\hat{\sigma })=\hat{w}_e=\nu (\hat{\sigma })\) so that \(\hat{\sigma }\) satisfies all constraints in Problem (PP) but, saving the cost of hiding income, yields a higher objective function value, \(\chi _l^p\mu _lw_l+\mu _hw_h\), than \(\sigma \), and thus dominates it, which completes the proof. \(\square \)

Lemma 3

Proof

Consider any regime \(\sigma \) with \(\chi _h^p=1\), \(\chi _l^p>0\), and, by Lemma 2, \(\chi _l^e>0\) that satisfies all constraints of Problem (PP). Following Proposition 4, \(\sigma \) also induces \(\zeta =w_h\), \(\psi (w_h-\zeta )=0\), and \(t(\zeta )=t_h\). First, as \(\chi _l^p>0\) and \(\chi _l^e>0\), combining constraints (8) and (9) gives

$$\begin{aligned} \varphi _l(\sigma )\ge \max \left\{ w_e, \nu (\sigma )\right\} \ge w_e\ge \max \left\{ \varphi _l(\sigma ), \nu (\sigma )\right\} \ge \varphi _l(\sigma ), \end{aligned}$$

implying that \(\varphi _l(\sigma )=w_e\). Second, if \(\chi _l^a>0\), then the inequality in constraint (10) has to be satisfied and combining it with constraint (9) yields

$$\begin{aligned} \nu (\sigma )\ge \max \left\{ \varphi _l(\sigma ), w_e\right\} = w_e\ge \max \left\{ \varphi _l(\sigma ), \nu (\sigma )\right\} \ge \nu (\sigma ), \end{aligned}$$

implying that \(\varphi _l(\sigma )=w_e=\nu (\sigma )\). Finally, suppose that \(\chi _l^a=0\). Then, the inequality in constraint (10) may or may not be satisfied. Suppose that \(w_e>\nu (\sigma )\). I show that there is another regime \(\hat{\sigma }\) with \(\hat{w}_e=\nu (\hat{\sigma })\) that is associated with higher welfare and thus dominates \(\sigma \). Given the regime \(\sigma \), \(\chi _l^e=1-\chi _l^p>0\), \(p=0\), \(\theta =1-\chi _l^e\mu _l\), and \(\varphi _l(\sigma )=w_e>\nu (\sigma )\) or

$$\begin{aligned} w_l-t_l=w_e&=\left( \chi _l^e\mu _l\right) ^{-1}\left( \chi _l^p\mu _lt_l+\mu _ht_h\right) \\&\quad>\left( 1-\chi _l^e\mu _l\right) \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] >0. \end{aligned}$$

The tax receipts implied by \(\sigma \) are bounded away from zero by \(\nu (\sigma )\) as, by inequality (20), \(w_h-t_h>w_l-t_l\ge 0\). There exists an \(\epsilon >0\) such that, using \(q=(\chi _l^p\mu _l+\mu _h)\),

$$\begin{aligned} w_l-t_l+\epsilon&>\left( \chi _l^e\mu _l\right) ^{-1}\left( \chi _l^p\mu _lt_l+\mu _ht_h-q\epsilon \right) \\&>\left( 1-\chi _l^e\mu _l\right) \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)+q\epsilon \right] \end{aligned}$$

(the tax receipts are still bounded away from zero). There exists a \(\delta >0\), \(\delta <\chi _l^e\) such that

$$\begin{aligned} w_l-t_l+\epsilon&>\left( \chi _l^e\mu _l-\delta \mu _l\right) ^{-1}\left( \chi _l^p\mu _lt_l+\mu _ht_h-q\epsilon +\delta \mu _l(t_l-\epsilon )\right) \\&>(1-\chi _l^e\mu _l+\delta \mu _l)\\&\quad \times \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)+q\epsilon +\delta \mu _l(w_l-t_l+\epsilon )\right] \end{aligned}$$

(the tax receipts are still bounded away from zero). Then, there exists a \(\kappa >0\) such that

$$\begin{aligned}&w_l-t_l+\epsilon >\left( \chi _l^e\mu _l-\delta \mu _l\right) ^{-1}\left( \chi _l^p\mu _lt_l+\mu _ht_h-q\epsilon +\delta \mu _l(t_l-\epsilon )-\mu _h\kappa \right) \\&\quad =(1-\chi _l^e\mu _l+\delta \mu _l)\left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)+q\epsilon \right. \\&\qquad \left. +\,\delta \mu _l(w_l-t_l+\epsilon )+\mu _h\kappa \right] \end{aligned}$$

(the receipts are still positive). Then, there are \(\gamma _l>0\) and \(\gamma _h=-\frac{\left( \chi _l^p+\delta \right) \mu _l}{\mu _h}\gamma _l<0\) such that

$$\begin{aligned} w_l-t_l+\epsilon -\gamma _l&=\left( \chi _l^e\mu _l-\delta \mu _l\right) ^{-1}\left( \chi _l^p\mu _lt_l+\mu _ht_h-q\epsilon +\delta \mu _l(t_l-\epsilon )-\mu _h\kappa \right. \\&\quad \left. +\left( \chi _l^p+\delta \right) \mu _l\gamma _l+\mu _h\gamma _h\right) \\&=(1-\chi _l^e\mu _l+\delta \mu _l)[\chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)+q\epsilon \\&\quad +\delta \mu _l(w_l-t_l+\epsilon )+\mu _h\kappa \\&\quad -\left( \chi _l^p+\delta \right) \mu _l\gamma _l-\mu _h\gamma _h], \end{aligned}$$

because \(\left( \chi _l^p+\delta \right) \mu _l\gamma _l+\mu _h\gamma _h=0\) (the tax receipts have not changed) or, more compactly,

$$\begin{aligned} w_l-(t_l-\epsilon +\gamma _l)&=((\chi _l^e-\delta )\mu _l)^{-1}\Big (\left( \chi _l^p+\delta \right) \mu _l(t_l-\epsilon +\gamma _l)\\&\quad +\mu _h(t_h-\epsilon -\kappa +\gamma _h)\Big )\\&=(1-(\chi _l^e-\delta )\mu _l)\left[ \left( \chi _l^p+\delta \right) \mu _l(w_l-(t_l-\epsilon +\gamma _l))\right. \\&\quad \left. +\,\mu _h(w_h-(t_h-\epsilon -\kappa +\gamma _h))\right] , \end{aligned}$$

or, even more compactly,

$$\begin{aligned} w_l-\hat{t}_l&=\left( \hat{\chi }_l^e\mu _l\right) ^{-1}\left( \hat{\chi }_l^p\mu _l\hat{t}_l+\mu _h\hat{t}_h\right) \\&=\left( 1-\hat{\chi }_l^e\mu _l\right) \left[ \hat{\chi }_l^p\mu _l\left( w_l-\hat{t}_l\right) +\mu _h\left( w_h-\hat{t}_h\right) \right] , \end{aligned}$$

where \(\hat{\chi }_l^p=\chi _l^p+\delta >0\), \(\hat{\chi }_l^e=\chi _l^e-\delta >0\), \(\hat{\chi }_l^a=\chi _l^a=0\), \(\hat{t}_l=t_l-\epsilon +\gamma _l\), \(\hat{t}_h=t_h-\epsilon -\kappa +\gamma _h\). Letting \(\hat{w}_e=\left( \hat{\chi }_l^e\mu _l\right) ^{-1}\left( \hat{\chi }_l^p\mu _l\hat{t}_l+\mu _h\hat{t}_h\right) \) and \(\hat{\sigma }=(\hat{\chi }_l^p,\hat{\chi }_l^e,\hat{\chi }_l^a,1,0,0,\hat{t}_l,\hat{t}_h,\hat{w}_e)\) gives \(\varphi _l(\hat{\sigma })=\hat{w}_e=\nu (\hat{\sigma })\). As \(w_h-t_h\ge w_h-t_l-\phi \) by inequality (18), \(w_h-\hat{t}_h\ge w_h-\hat{t}_l-\phi >w_l-\hat{t}_l\) is satisfied, implying both no hidden income and \(\varphi _h(\hat{\sigma })>\varphi _l(\hat{\sigma })\), and the tax receipts are strictly positive. Therefore, \(\hat{\sigma }\) satisfies all constraints of Problem (PP) but yields a higher objective function value than \(\sigma \), because \(\hat{\chi }_l^p=\chi _l^p+\delta >\chi _l^p\), which completes the proof. \(\square \)

Lemma 4

Proof

Consider any solution \(\sigma \) to Problem (\(\hbox {PP}^{\prime }\)). From constraint (22), it has to hold that \(t_l+\phi \ge t_h\). The following argument is independent of whether \(p=0\) or \(p>0\). Suppose for a contradiction that \(t_l+\phi >t_h\), implying that \((1-\theta p)(w_h-t_h)>(1-\theta p)(w_h-t_l-\phi )\). Then, there exist an \(\epsilon _h>0\) and (as \(\chi _l^p>0\)) an \(\epsilon _l=-\frac{\mu _h}{\chi _l^p\mu _l}\epsilon _h<0\) such that \(t_l+\phi +\epsilon _l=t_h+\epsilon _h\), or

$$\begin{aligned} t_l+\phi =t_h+\left( 1+\frac{\mu _h}{\chi _l^p\mu _l}\right) \epsilon _h. \end{aligned}$$

Let \(\tilde{t}_i=t_i+\epsilon _i\). As \(\tilde{t}_l+\phi =t_l+\phi +\epsilon _l=t_h+\epsilon _h=\tilde{t}_h\), as \(\phi <(w_h-w_l)\),

$$\begin{aligned} (1-\theta p)(w_h-\tilde{t}_h)=(1-\theta p)(w_h-\tilde{t}_l-\phi )>(1-\theta p)(w_l-\tilde{t}_l) \end{aligned}$$
(27)

and, starting from \(\varphi _l(\sigma )=w_e=\nu (\sigma )\), as \(\tilde{t}_l<t_l\), using \(\epsilon _l=-\frac{\mu _h}{\chi _l^p\mu _l}\epsilon _h\) and \(\theta =1-\chi _l^e\mu _l\),

$$\begin{aligned} (1-\theta p)(w_l-\tilde{t}_l)>w_e=&\left( \chi _l^e\mu _l\right) ^{-1}\left( \chi _l^p\mu _l\tilde{t}_l+\mu _h\tilde{t}_h\right) \\ =&\left( \chi _l^e\mu _l\right) ^{-1}\left( \chi _l^p\mu _lt_l+\mu _ht_h+\chi _l^p\mu _l\epsilon _l+\mu _h\epsilon _h\right) \\ =&\left( \chi _l^e\mu _l\right) ^{-1}\left( \chi _l^p\mu _lt_l+\mu _ht_h\right) \\ =&\nu (\sigma )\\ =&\left( 1-\chi _l^e\mu _l\right) \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right) ]\\ =&\left( 1-\chi _l^e\mu _l\right) \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right. \\&\left. -\chi _l^p\mu _l\epsilon _l-\mu _h\epsilon _h\right] \\ =&\left( 1-\chi _l^e\mu _l\right) \left[ \chi _l^p\mu _l(w_l-\tilde{t}_l)+\mu _h(w_h-\tilde{t}_h)\right] , \end{aligned}$$

because \(\chi _l^p\mu _l\epsilon _l+\mu _h\epsilon _h=0\). The tax receipts implied by \(\sigma \) are bounded away from zero by \(\nu (\sigma )\) as, by inequality (20), \(w_h-t_h>w_l-t_l\ge 0\). In fact, due to \(\varphi _l(\sigma )=w_e=\nu (\sigma )>0\), \(w_l-t_l>0\). The last set of inequalities and equalities can be summarized as

$$\begin{aligned} (1-\theta p)(w_l-\tilde{t}_l)&>\left( \chi _l^e\mu _l\right) ^{-1}\left( \chi _l^p\mu _l\tilde{t}_l+\mu _h\tilde{t}_h\right) \\&=\left( 1-\chi _l^e\mu _l\right) \left[ \chi _l^p\mu _l(w_l-\tilde{t}_l)+\mu _h(w_h-\tilde{t}_h)\right] >0. \end{aligned}$$

Then, there exists a \(\gamma >0\), \(\gamma <w_l-\tilde{t}_l\) (\(<w_h-\tilde{t}_h\) by (27)), such that, letting \(\bar{t}_i=\tilde{t}_i+\gamma \),

$$\begin{aligned} (1-\theta p)\left( w_l-\bar{t}_l\right)&>\left( \chi _l^e\mu _l\right) ^{-1}\left( \chi _l^p\mu _l\bar{t}_l+\mu _h\bar{t}_h\right) \\&>\left( 1-\chi _l^e\mu _l\right) \left[ \chi _l^p\mu _l\left( w_l-\bar{t}_l\right) +\mu _h(w_h-\bar{t}_h)\right] >0. \end{aligned}$$

Then, irrespective of whether \(p=0\) or \(p>0\), there exists a \(\bar{\delta }_1>0\), \(\bar{\delta }_1<\chi _l^e\), such that, using \(\theta =1-\chi _l^e\mu _l\), for all \(\delta _1\in [0,\bar{\delta }_1]\),

$$\begin{aligned} (1-(1-(\chi _l^e-\delta _1)\mu _l)p)\left( w_l-\bar{t}_l\right) >((\chi _l^e-\delta _1)\mu _l)^{-1}\left( (\chi _l^p+\delta _1)\mu _l\bar{t}_l+\mu _h\bar{t}_h\right) , \end{aligned}$$

(note: \(\bar{t}_l\) could be negative) as well as a \(\bar{\delta }_2>0\), \(\bar{\delta }_2<\chi _l^e\), such that, for all \(\delta _2\in [0,\bar{\delta }_2]\),

$$\begin{aligned}&((\chi _l^e-\delta _2)\mu _l)^{-1}\left( \left( \chi _l^p+\delta _2\right) \mu _l\bar{t}_l+\mu _h\bar{t}_h\right) \\&\quad>(1-(\chi _l^e-\delta _2)\mu _l)\left[ \left( \chi _l^p+\delta _2\right) \mu _l\left( w_l-\bar{t}_l\right) +\mu _h(w_h-\bar{t}_h)\right] >0. \end{aligned}$$

Let \(\delta =\min \{\bar{\delta }_1,\bar{\delta }_2\}\). Then,

$$\begin{aligned} (1-(1-(\chi _l^e-\delta )\mu _l)p)\left( w_l-\bar{t}_l\right)&>((\chi _l^e-\delta )\mu _l)^{-1}\left( \left( \chi _l^p+\delta \right) \mu _l\bar{t}_l+\mu _h\bar{t}_h\right) \\&>(1-(\chi _l^e-\delta )\mu _l)\\&\quad \times \left[ \left( \chi _l^p+\delta \right) \mu _l\left( w_l-\bar{t}_l\right) +\mu _h(w_h-\bar{t}_h)\right] \\&>0. \end{aligned}$$

Let \(\hat{\chi }_l^p=\chi _l^p+\delta >0\), \(\hat{\chi }_l^e=\chi _l^e-\delta >0\), \(\hat{\chi }_l^a=\chi _l^a\ge 0\), \(\hat{p}=\hat{\chi }_l^a\mu _l=\chi _l^a\mu _l=p\), and \(\hat{\theta }=\left( 1-\hat{\chi }_l^e\mu _l\right) \). Then, there exists an \(\eta >0\) such that, letting \(\breve{t}_i=\bar{t}_i-\eta \),

$$\begin{aligned} \left( 1-\hat{\theta }\hat{p}\right) \left( w_l-\breve{t}_l\right)&>\left( \hat{\chi }_l^e\mu _l\right) ^{-1}\left( \hat{\chi }_l^p\mu _l\breve{t}_l+\mu _h\breve{t}_h\right) \\&=\hat{\theta }\left[ \hat{\chi }_l^p\mu _l\left( w_l-\breve{t}_l\right) +\mu _h\left( w_h-\breve{t}_h\right) \right] >0. \end{aligned}$$

Then, there are a \(\kappa _l>0\) and a \(\kappa _h=-\frac{\hat{\chi }_l^p\mu _l}{\mu _h}\kappa _l<0\) such that, letting \(\hat{t}_i=\breve{t}_i+\kappa _i\),

$$\begin{aligned} \left( 1-\hat{\theta }\hat{p}\right) \left( w_l-\hat{t}_l\right)&=\left( \hat{\chi }_l^e\mu _l\right) ^{-1}\left( \hat{\chi }_l^p\mu _l\hat{t}_l+\mu _h\hat{t}_h\right) \\&=\left( \hat{\chi }_l^e\mu _l\right) ^{-1}\left( \hat{\chi }_l^p\mu _l\breve{t}_l+\mu _h\breve{t}_h+\hat{\chi }_l^p\mu _l\kappa _l+\mu _h\kappa _h\right) \\&=\left( \hat{\chi }_l^e\mu _l\right) ^{-1}\left( \hat{\chi }_l^p\mu _l\breve{t}_l+\mu _h\breve{t}_h\right) \\&=\hat{\theta }\left[ \hat{\chi }_l^p\mu _l\left( w_l-\breve{t}_l\right) +\mu _h\left( w_h-\breve{t}_h\right) \right] \\&=\hat{\theta }\left[ \hat{\chi }_l^p\mu _l\left( w_l-\breve{t}_l\right) +\mu _h\left( w_h-\breve{t}_h\right) -\hat{\chi }_l^p\mu _l\kappa _l-\mu _h\kappa _h\right] \\&=\hat{\theta }\left[ \hat{\chi }_l^p\mu _l\left( w_l-\hat{t}_l\right) +\mu _h\left( w_h-\hat{t}_h\right) \right] >0, \end{aligned}$$

because \(\hat{\chi }_l^p\mu _l\kappa _l+\mu _h\kappa _h=0\), or, summarizing these equations,

$$\begin{aligned} \left( 1-\hat{\theta }\hat{p}\right) \left( w_l-\hat{t}_l\right)&=\left( \hat{\chi }_l^e\mu _l\right) ^{-1}\left( \hat{\chi }_l^p\mu _l\hat{t}_l+\mu _h\hat{t}_h\right) \\&=\hat{\theta }\left[ \hat{\chi }_l^p\mu _l\left( w_l-\hat{t}_l\right) +\mu _h\left( w_h-\hat{t}_h\right) \right] >0. \end{aligned}$$

Letting \(\hat{w}_e=\left( \hat{\chi }_l^e\mu _l\right) ^{-1}\left( \hat{\chi }_l^p\mu _l\hat{t}_l+\mu _h\hat{t}_h\right) \) and \(\hat{\sigma }=(\hat{\chi }_l^p,\hat{\chi }_l^e,\hat{\chi }_l^a,1,0,0,\hat{t}_l,\hat{t}_h,\hat{w}_e)\), the last equation implies \(\varphi _l(\hat{\sigma })=\hat{w}_e=\nu (\hat{\sigma })\). The tax receipts are still bounded away from zero. Since \(t_l+\epsilon _l+\phi =t_h+\epsilon _h\), as established in the very beginning, and \(\kappa _l>0\) while \(\kappa _h<0\), it holds that \(t_l+\epsilon _l+\gamma -\eta +\kappa _l+\phi >t_h+\epsilon _h+\gamma -\eta +\kappa _h\) so that \(\hat{t}_l+\phi >\hat{t}_h\). Hence, constraint (22) is satisfied and \(w_h-\hat{t}_h>w_l-\hat{t}_l\). Therefore, \(\hat{\sigma }\) satisfies all constraints of Problem (\(\hbox {PP}^{\prime }\)) but yields a higher objective function value than \(\sigma \), because \(\hat{\chi }_l^p=\chi _l^p+\delta >\chi _l^p\), a contradiction. \(\square \)

Proposition 5

Proof

Consider any solution \(\sigma \) to Problem (\(\hbox {PP}^{\prime }\)) and suppose for a contradiction that some agents of type l engage in appropriation. I show in two steps that there is another regime in the constraint set that attains a higher objective function value and does not have any appropriators. First, starting from \(\sigma \), I show that the planner can reallocate all appropriators to enforcement and find a tax increase for all producers as well as a wage in enforcement such that no agent wants to deviate to an occupation different from the amended occupation prescriptions. Second, starting from this amended regime, I show that the planner can reallocate some of these enforcers to production and find a tax schedule as well as a wage in enforcement such that all constraints of Problem (\(\hbox {PP}^{\prime }\)) are satisfied, while more agents produce, which increases the objective function value.

As \(\chi _l^p>0\), \(\chi _l^e>0\) by Lemma 2. Following Lemma 4, \(t_h=t_l+\phi \). Then, writing out the payoffs \(\varphi _l(\sigma )\) and \(\nu (\sigma )\) in constraints (23) and (24), using constraint (26) to replace p with \(\chi _l^a\mu _l\) and \(\chi _l^e\mu _l\) with \((1-\theta )\) and constraint (25) to replace \(\chi _l^a\) with \(1-\chi _l^p-\chi _l^e\), so that \(p=\chi _l^a\mu _l=(1-\chi _l^p-\chi _l^e)\mu _l=\mu _l-\chi _l^p\mu _l-(1-\theta )=\theta +\mu _l-\chi _l^p\mu _l-\mu _l-\mu _h=\theta -(\chi _l^p\mu _l+\mu _h)=\theta -q\), where the last equality follows from constraint (26), the solution \(\sigma \) to Problem (\(\hbox {PP}^{\prime }\)) satisfies

$$\begin{aligned}&\displaystyle t_h=t_l+\phi , \end{aligned}$$
(28)
$$\begin{aligned}&\displaystyle (1-\theta (\theta -q))(w_l-t_l)=w_e,\end{aligned}$$
(29)
$$\begin{aligned}&\displaystyle (1-\theta )w_e=\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) ,\end{aligned}$$
(30)
$$\begin{aligned}&\displaystyle w_e=\theta \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] ,\end{aligned}$$
(31)
$$\begin{aligned}&\displaystyle \theta -q\ge 0. \end{aligned}$$
(32)

Inequality (32) ensures that \(\chi _l^a\ge 0\). The assumption that some agents of type l engage in appropriation implies that \(\chi _l^a>0\) and thus \(\theta >q\). Combining Equations (29)–(31), we have

$$\begin{aligned} \begin{aligned} (1-\theta (\theta -q))(w_l-t_l)&=(1-\theta )^{-1}\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) \\&=\theta \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] >0, \end{aligned} \end{aligned}$$
(33)

where \(w_e=(1-\theta )^{-1}\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) \) from (30), as \(\chi _l^e>0\) implies that \((1-\theta )>0\). All payoffs are strictly positive as \(w_h-t_h>w_l-t_l\ge 0\) by Equation (20), implying that \(\theta \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] >0\), and thus \(w_l-t_l>0\).

Step 1.First, reallocating all \((\theta -q)\) appropriators to enforcement, keeping all other agents in their occupation, implies that \(\tilde{\theta }=q<\theta \), \(1-\tilde{\theta }=1-q>1-\theta \), so that the payoffs satisfy

$$\begin{aligned} w_l-t_l>\max \left\{ (1-q)^{-1}\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) ,q\left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] \right\} >0, \end{aligned}$$
(34)

because, compared to (33), \(1>(1-\theta (\theta -q))\), \((1-q)^{-1}<(1-\theta )^{-1}\), and \(q<\theta \). There are two cases: either, case (a), \(\theta (1-\theta )\ge q(1-q)\) or, case (b), \(\theta (1-\theta )<q(1-q)\).

Case (a). Suppose that \(\theta (1-\theta )\ge q(1-q)\). In this case, inequality (34) can be written as

$$\begin{aligned} w_l-t_l>(1-q)^{-1}\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) \ge q\left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] , \end{aligned}$$
(35)

because from (33),

$$\begin{aligned} \left( \chi _l^p\mu _l t_l+\mu _h t_h\right) =(1-\theta )\theta \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] , \end{aligned}$$

so that

$$\begin{aligned} (1-q)^{-1}\left( \chi _l^p\mu _l t_l+\mu _h t_h\right)&=(1-q)^{-1}(1-\theta )\theta \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] \\&\ge q\left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] , \end{aligned}$$

as \(\theta (1-\theta )\ge q(1-q)\). Therefore, from (35), there exists an \(\epsilon >0\) such that

$$\begin{aligned} \begin{aligned} w_l-t_l-\epsilon&=(1-q)^{-1}\left( \chi _l^p\mu _l (t_l+\epsilon )+\mu _h (t_h+\epsilon )\right) \\&>q\left[ \chi _l^p\mu _l(w_l-t_l-\epsilon )+\mu _h(w_h-t_h-\epsilon )\right] . \end{aligned} \end{aligned}$$
(36)

All payoffs are still strictly positive as \(\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) >0\) from (33), so that, for \(\epsilon >0\), \((\chi _l^p\mu _l (t_l+\epsilon )+\mu _h (t_h+\epsilon ))>0\), implying that \(w_h-t_h-\epsilon>w_l-t_l-\epsilon >0\). That is, taxing all producers that additional \(\epsilon >0\) allows to reallocate all those who were assigned to appropriation in the original regime \(\sigma \) to enforcement, and to pay all, now \((1-q)\), enforcement personnel the wages required, i.e., \((1-q)^{-1}(\chi _l^p\mu _l (t_l+\epsilon )+\mu _h (t_h+\epsilon ))\), such that no agent wants to deviate to appropriation and all agents of type l are indifferent between production and enforcement.

Case (b). Suppose that \(\theta (1-\theta )<q(1-q)\). In this case, Equation (34) can be written as

$$\begin{aligned} w_l-t_l>q\left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] >(1-q)^{-1}\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) , \end{aligned}$$
(37)

by a similar argument as above. There exists an \(\epsilon _1>0\) such that, using \(q=\chi _l^p\mu _l+\mu _h\),

$$\begin{aligned} w_l-t_l-\epsilon _1&=(1-q)^{-1}\left( \chi _l^p\mu _l (t_l+\epsilon _1)+\mu _h (t_h+\epsilon _1)\right) \\&=(1-q)^{-1}\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) +(1-q)^{-1}q\epsilon _1. \end{aligned}$$

Multiplying by \((1-q)\) and using \(\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) =(1-\theta )(1-\theta (\theta -q))(w_l-t_l)\) from (33),

$$\begin{aligned} \epsilon _1=\left[ (1-q)-(1-\theta )(1-\theta (\theta -q))\right] (w_l-t_l). \end{aligned}$$

Similarly, there exists an \(\epsilon _2>0\) such that, again using \(q=\chi _l^p\mu _l+\mu _h\),

$$\begin{aligned}&(1-q)^{-1}\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) +(1-q)^{-1}q\epsilon _2\\&\quad =(1-q)^{-1}\left( \chi _l^p\mu _l (t_l+\epsilon _2)+\mu _h (t_h+\epsilon _2)\right) \\&\quad =q\left[ \chi _l^p\mu _l(w_l-t_l-\epsilon _2)+\mu _h(w_h-t_h-\epsilon _2)\right] \\&\quad =q\left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] -q^2\epsilon _2. \end{aligned}$$

Multiplying both sides by \((1-q)\) and collecting terms gives

$$\begin{aligned} (q+(1-q)q^2)\epsilon _2=(1-q)q\left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] -\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) , \end{aligned}$$

and using \(\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) =(1-\theta )(1-\theta (\theta -q))(w_l-t_l)\) and \([\chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)]=\theta ^{-1}(1-\theta (\theta -q))(w_l-t_l)\) from (33) gives

$$\begin{aligned} \epsilon _2=\frac{1}{q(1+(1-q)q)}\left[ (1-q)q\theta ^{-1}-(1-\theta )\right] (1-\theta (\theta -q))(w_l-t_l). \end{aligned}$$

We have that \(\epsilon _1>\epsilon _2\), because

$$\begin{aligned}&\left[ (1-q)-(1-\theta )(1-\theta (\theta -q))\right] \\&\quad >\frac{1}{q(1+(1-q)q)}\left[ (1-q)q\theta ^{-1}-(1-\theta )\right] (1-\theta (\theta -q)). \end{aligned}$$

To verify this claim, multiply both sides by \(\theta q(1+(1-q)q)\) to get

$$\begin{aligned}&(1+(1-q)q)\left[ \theta q(1-q)-q \theta (1-\theta )(1-\theta (\theta -q))\right] \\&\quad >\left[ (1-q)q-\theta (1-\theta )\right] (1-\theta (\theta -q)) \end{aligned}$$

and note that, as \(q\in (0,1)\) and \((1-\theta (\theta -q))<1\),

$$\begin{aligned}&(1+(1-q)q)\left[ \theta q(1-q)-q \theta (1-\theta )(1-\theta (\theta -q))\right] \\&\quad>\left[ \theta q(1-q)-q \theta (1-\theta )(1-\theta (\theta -q))\right] \\&\quad>\left[ \theta q(1-q)-q \theta (1-\theta )\right] \\&\quad>\left[ (1-q)q-\theta (1-\theta )\right] \\&\quad >\left[ (1-q)q-\theta (1-\theta )\right] (1-\theta (\theta -q)), \end{aligned}$$

where the third inequality holds because, as \(q\in (0,1)\) and \(1-\theta >0\),

$$\begin{aligned} \theta q(1-q)-q \theta (1-\theta )>(1-q)q-\theta (1-\theta )\\ \Leftrightarrow \quad \theta (1-\theta )-q \theta (1-\theta )>(1-q)q-\theta q(1-q)\\ \Leftrightarrow \quad \theta (1-\theta )(1-q)>(1-q)q(1-\theta )\\ \Leftrightarrow \quad \theta >q, \end{aligned}$$

which holds by assumption. Therefore, as \(\epsilon _1>\epsilon _2\),

$$\begin{aligned} w_l-t_l-\epsilon _1&=(1-q)^{-1}\left( \chi _l^p\mu _l (t_l+\epsilon _1)+\mu _h (t_h+\epsilon _1)\right) \\&\quad>(1-q)^{-1}\left( \chi _l^p\mu _l (t_l+\epsilon _2)+\mu _h (t_h+\epsilon _2)\right) \\&=q\left[ \chi _l^p\mu _l(w_l-t_l-\epsilon _2)+\mu _h(w_h-t_h-\epsilon _2)\right] \\&\quad >q\left[ \chi _l^p\mu _l(w_l-t_l-\epsilon _1)+\mu _h(w_h-t_h-\epsilon _1)\right] , \end{aligned}$$

so that there is an \(\epsilon =\epsilon _1>0\) such that

$$\begin{aligned} \begin{aligned} w_l-t_l-\epsilon&=(1-q)^{-1}\left( \chi _l^p\mu _l (t_l+\epsilon )+\mu _h (t_h+\epsilon )\right) \\&\quad >q\left[ \chi _l^p\mu _l(w_l-t_l-\epsilon )+\mu _h(w_h-t_h-\epsilon )\right] . \end{aligned} \end{aligned}$$
(38)

Again, all payoffs are still strictly positive as \(\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) >0\) from (33), so that, for \(\epsilon >0\), \((\chi _l^p\mu _l (t_l+\epsilon )+\mu _h (t_h+\epsilon ))>0\), implying that \(w_h-t_h-\epsilon>w_l-t_l-\epsilon >0\). That is, again, taxing all producers that additional \(\epsilon >0\) allows to reallocate all those who were assigned to appropriation in the original regime \(\sigma \) to enforcement, and to pay all, now \((1-q)\), enforcement personnel the wages required, i.e., \((1-q)^{-1}\left( \chi _l^p\mu _l (t_l+\epsilon )+\mu _h (t_h+\epsilon )\right) \), such that no agent wants to deviate to appropriation and all agents of type l are indifferent between production and enforcement. As summarized by Equations (36) and (38), irrespective of whether \(\theta (1-\theta )\ge q(1-q)\) or \(\theta (1-\theta )<q(1-q)\), there exists an \(\epsilon >0\) such that increasing the taxes for all producers by that amount allows to support the wage bill for all enforcers, once all appropriators are reallocated to enforcement, and no agent wants to deviate from the planner’s amended occupational prescriptions.

Step 2.Second, starting from the planner’s amended occupational prescriptions and taxes that satisfy (38), which is the same as (36), there is a tax decrease \(\gamma >0\) such that

$$\begin{aligned} w_l-t_l-\epsilon +\gamma&>(1-q)^{-1}\left( \chi _l^p\mu _l (t_l+\epsilon -\gamma )+\mu _h (t_h+\epsilon -\gamma )\right) \\&>q\left[ \chi _l^p\mu _l(w_l-t_l-\epsilon +\gamma )+\mu _h(w_h-t_h-\epsilon +\gamma )\right] >0. \end{aligned}$$

Then, using \(q=\chi _l^p\mu _l+\mu _h\), there exists a \(\bar{\delta }_1>0\), \(\bar{\delta }_1<\chi _l^e\), such that, for all \(\delta _1\in [0,\bar{\delta }_1]\),

$$\begin{aligned} w_l-t_l-\epsilon +\gamma&>(1-((\chi _l^p+\delta _1)\mu _l+\mu _h))^{-1}\\&\quad \times \left( (\chi _l^p+\delta _1)\mu _l (t_l+\epsilon -\gamma )+\mu _h (t_h+\epsilon -\gamma )\right) \end{aligned}$$

(note: \(t_l+\epsilon -\gamma \) could be negative) as well as a \(\bar{\delta }_2>0\), \(\bar{\delta }_2<\chi _l^e\), such that, for all \(\delta _2\in [0,\bar{\delta }_2]\),

$$\begin{aligned}&\left( 1-\left( \left( \chi _l^p+\delta _2\right) \mu _l+\mu _h\right) \right) ^{-1} \left( \left( \chi _l^p+\delta _2\right) \mu _l (t_l+\epsilon -\gamma )+\mu _h (t_h+\epsilon -\gamma )\right) \\&\quad >(\left( \chi _l^p+\delta _2\right) \mu _l+\mu _h)\left[ \left( \chi _l^p+\delta _2\right) \mu _l(w_l-t_l-\epsilon +\gamma )+\mu _h(w_h-t_h-\epsilon +\gamma )\right] . \end{aligned}$$

Then, letting \(\delta =\min \{\bar{\delta }_1,\bar{\delta }_2\}>0\), so that \(0<\delta <\chi _l^e\), reallocating enforcers to production,

$$\begin{aligned} w_l-t_l-\epsilon +\gamma&>\left( 1-\left( \left( \chi _l^p+\delta \right) \mu _l+ \mu _h\right) \right) ^{-1}\\&\quad \times \left( \left( \chi _l^p+\delta \right) \mu _l (t_l+\epsilon -\gamma )+\mu _h (t_h+\epsilon -\gamma )\right) \\&>\left( \left( \chi _l^p+\delta \right) \mu _l+\mu _h\right) \\&\quad \times \left[ \left( \chi _l^p+\delta \right) \mu _l(w_l-t_l-\epsilon +\gamma )+\mu _h(w_h-t_h-\epsilon +\gamma )\right] . \end{aligned}$$

Now, there is a tax decrease \(\eta >0\) such that

$$\begin{aligned}&w_l-t_l-\epsilon +\gamma +\eta \\&\quad> \left( 1-\left( \left( \chi _l^p+\delta \right) \mu _l+\mu _h\right) \right) ^{-1}\left( \left( \chi _l^p+\delta \right) \mu _l (t_l+\epsilon -\gamma -\eta )\right. \\&\qquad +\mu _h (t_h+\epsilon -\gamma -\eta )\big )\\&\quad =\left( \left( \chi _l^p+\delta \right) \mu _l+\mu _h\right) \left[ \left( \chi _l^p+\delta \right) \mu _l(w_l-t_l-\epsilon +\gamma +\eta )\right. \\&\qquad \left. +\mu _h(w_h-t_h-\epsilon +\gamma +\eta )\right] \\&\quad >0. \end{aligned}$$

Then, there are a tax changes \(\kappa _l>0\) and \(\kappa _h=-\frac{\left( \chi _l^p+\delta \right) \mu _l}{\mu _h}\kappa _l<0\) such that

$$\begin{aligned}&w_l-t_l-\epsilon +\gamma +\eta -\kappa _l\\&\quad =\left( 1-\left( \left( \chi _l^p+\delta \right) \mu _l+\mu _h\right) \right) ^{-1}\\&\qquad \times \left( \left( \chi _l^p+\delta \right) \mu _l (t_l+\epsilon -\gamma -\eta +\kappa _l)+\mu _h (t_h+\epsilon -\gamma -\eta +\kappa _h)\right) \\&\quad =\left( 1-\left( \left( \chi _l^p+\delta \right) \mu _l+\mu _h\right) \right) ^{-1} \left( \left( \chi _l^p+\delta \right) \mu _l (t_l+\epsilon -\gamma -\eta )\right. \\&\qquad +\mu _h (t_h+\epsilon -\gamma -\eta )+\left. \,\left( \chi _l^p+\delta \right) \mu _l\kappa _l+\mu _h\kappa _h\right) \\&\quad =\left( 1-\left( \left( \chi _l^p+\delta \right) \mu _l+\mu _h\right) \right) ^{-1} \left( \left( \chi _l^p+\delta \right) \mu _l \left( t_l+\epsilon -\gamma -\eta \right) \right. \\&\qquad +\mu _h \left( t_h+\epsilon -\gamma -\eta \right) +\left. \,\left( \chi _l^p+\delta \right) \mu _l\kappa _l-\left( \chi _l^p+\delta \right) \mu _l\kappa _l\right) \\&\quad =\left( \left( \chi _l^p+\delta \right) \mu _l+\mu _h\right) \left[ \left( \chi _l^p+\delta \right) \mu _l(w_l-t_l-\epsilon +\gamma +\eta )\right. \\&\qquad \left. +\mu _h(w_h-t_h-\epsilon +\gamma +\eta )\right] \\&\quad =\left( \left( \chi _l^p+\delta \right) \mu _l+\mu _h\right) [\left( \chi _l^p+\delta \right) \mu _l(w_l-t_l-\epsilon +\gamma +\eta )\\&\qquad +\mu _h(w_h-t_h-\epsilon +\gamma +\eta )\\&\qquad -\left( \chi _l^p+\delta \right) \mu _l\kappa _l-\mu _h\kappa _h]\\&\quad =\left( \left( \chi _l^p+\delta \right) \mu _l+\mu _h\right) \\&\qquad \times \left[ \left( \chi _l^p+\delta \right) \mu _l \left( w_l-t_l-\epsilon +\gamma +\eta -\kappa _l\right) +\mu _h \left( w_h-t_h-\epsilon +\gamma +\eta -\kappa _h\right) \right] . \end{aligned}$$

Therefore, letting \(\hat{\chi }_l^p=\chi _l^p+\delta \), \(\hat{\chi }_l^e=1-\hat{\chi }_l^p\), \(\hat{\chi }_l^a=0\), \(\hat{t}_l=t_l+\epsilon -\gamma -\eta +\kappa _l\), \(\hat{t}_h=t_h+\epsilon -\gamma -\eta +\kappa _h\), and \(\hat{w}_e=\left( \hat{\chi }_l^e\mu _l\right) ^{-1}\left( \hat{\chi }_l^p\mu _l \hat{t}_l+\mu _h \hat{t}_h\right) =(1-\left( \hat{\chi }_l^p\mu _l+\mu _h\right) )^{-1}(\hat{\chi }_l^p\mu _l \hat{t}_l+\mu _h \hat{t}_h)>0\), the regime \(\hat{\sigma }=(\hat{\chi }_l^p,\hat{\chi }_l^e,\hat{\chi }_l^a,1,0,0,\hat{t}_l,\hat{t}_h,\hat{w}_e)\) implies a balanced budget and satisfies

$$\begin{aligned} w_l-\hat{t}_l=\hat{w}_e&=\left( \hat{\chi }_l^e\mu _l\right) ^{-1}\left( \hat{\chi }_l^p\mu _l \hat{t}_l+\mu _h \hat{t}_h\right) \\&=\left( \hat{\chi }_l^p\mu _l+\mu _h\right) \left[ \hat{\chi }_l^p\mu _l\left( w_l-\hat{t}_l\right) +\mu _h\left( w_h-\hat{t}_h\right) \right] , \end{aligned}$$

i.e., \(\varphi _l(\hat{\sigma })=\hat{w}_e=\nu (\hat{\sigma })\), as well as \(\hat{t}_l+\phi \ge \hat{t}_h\), because \(t_l+\phi =t_h\) and \(\kappa _l>0>\kappa _h\) so that

$$\begin{aligned} t_l+\epsilon -\gamma -\eta +\kappa _l+\phi \ge t_h+\epsilon -\gamma -\eta +\kappa _h, \end{aligned}$$

and, finally, \(\hat{\chi }_l^p=\chi _l^p+\delta >0\) and \(\hat{\chi }_l^e=1-\hat{\chi }_l^p=1-\chi _l^p-\delta =\chi _l^a+\chi _l^e-\delta>\chi _l^a>0\) as \(\chi _l^p>0\), \(\chi _l^e>0\), \(\chi _l^a>0\) and \(0<\delta <\chi _l^e\). Thus, \(\hat{\sigma }\) satisfies all constraints of Problem (\(\hbox {PP}^{\prime }\)) but yields a higher objective function value than \(\sigma \), because \(\hat{\chi }_l^p=\chi _l^p+\delta >\chi _l^p\), a contradiction. \(\square \)

Proposition 6

Proof

Following Lemma 1, any solution satisfies \(\chi _h^{p*}=1\), \(\chi _h^{e*}=\chi _h^{a*}=0\). Following Lemmas 23 and Proposition 4, and the details of their proofs, if a solution \(\sigma \) satisfies \(\chi _l^{p}>0\), then it has to solve Problem (\(\hbox {PP}^{\prime }\)): the objective function is aggregate output, which is maximized by maximizing \(\chi _l^p\), and for all regimes \(\sigma \) with \(\chi _l^p>0\) in the constraint set of Problem (PP), either \(\sigma \) or another regime \(\hat{\sigma }\) with \(\hat{\chi }_l^p>0\) that is associated with a higher objective function value than \(\sigma \) is in the constraint set of Problem (\(\hbox {PP}^{\prime }\)). Problem (\(\hbox {PP}^{\prime }\)) has a solution if the constraint set is nonempty, because the objective function is continuous and the constraint set is a closed subset of \(\varSigma ^\prime \), which is compact. There are two cases.

Case 1. Suppose there is no regime with \(\chi _l^p>0\) in the constraint set of Problem (\(\hbox {PP}^{\prime }\)), which includes the case in which the constraint set is empty. By Lemmas 13 and Proposition 4, and the details of their proofs, for all regimes \(\sigma \) with \(\chi _l^p>0\) in the constraint set of Problem (PP), either \(\sigma \) or another regime \(\hat{\sigma }\) with \(\hat{\chi }_l^p>0\) that is associated with a higher objective function value than \(\sigma \) is in the constraint set of Problem (\(\hbox {PP}^{\prime }\)). Thus, if there is no regime with \(\chi _l^p>0\) in the latter, then there are no regimes with \(\chi _l^{p}>0\) in the constraint set of Problem (PP). In this case, as there is no feasible regime with \(\chi _l^{p}>0\), the solution to Problem (PP) is any regime that has \(\chi _l^{p*}=0\) and all agents of type h produce, \(\chi _h^{p*}=1\), and attains an objective function value of \(\mu _hw_h\). The planner cannot do better than the best anarchy regime.

Case 2. Suppose there is a regime \(\sigma \) with \(\chi _l^p>0\) in the constraint set of Problem (\(\hbox {PP}^{\prime }\)). Then, the constraint set is nonempty, Problem (\(\hbox {PP}^{\prime }\)) has a solution \(\sigma ^*\), and any such solution \(\sigma ^*\) has \(\chi _l^{p*}>0\), because \(\chi _l^p>0\) is attainable. In addition, following Lemma 4 and Proposition 5, any such solution \(\sigma ^*\) satisfies \(t_h^*=t_l^*+\phi \) and \(\chi _l^{a*}=0\). Thus, \(\chi _l^{p*}>0\), \(\chi _l^{e*}=1-\chi _l^{p*}>0\), and \(\chi _l^{a*}=0\), while \(\chi _h^{p*}=1\) and \(\chi _h^{e*}=\chi _h^{a*}=0\), so that there are no appropriators and, as \(\theta =q\) and \((1-\theta )=(1-q)\), all \((1-q)\) agents that do not produce are employed in enforcement. Combining these insights with constraints (21)–(26), the following holds: If there is a regime\(\sigma \)with\(\chi _l^p>0\)in the constraint set of Problem (\(\hbox {PP}^{\prime }\)), then a solution\(\sigma ^*\)to Problem (\(\hbox {PP}^{\prime }\)) exists and satisfies both\(\chi _l^{p*}>0\)and the system of Equations (39)–(43), where

$$\begin{aligned}&\displaystyle t_h=t_l+\phi , \end{aligned}$$
(39)
$$\begin{aligned}&\displaystyle w_l-t_l=w_e,\end{aligned}$$
(40)
$$\begin{aligned}&\displaystyle (1-\theta )w_e=\left( \chi _l^p\mu _l t_l+\mu _h t_h\right) ,\end{aligned}$$
(41)
$$\begin{aligned}&\displaystyle w_e=\theta \left[ \chi _l^p\mu _l(w_l-t_l)+\mu _h(w_h-t_h)\right] ,\end{aligned}$$
(42)
$$\begin{aligned}&\displaystyle \theta =\chi _l^p\mu _l+\mu _h. \end{aligned}$$
(43)

Equations (39)–(43) are five equations in five unknowns \(\chi _l^p\), \(t_l\), \(t_h\), \(w_e\), and \(\theta \). For a solution to this system to be consistent with \(\chi _l^{p*}>0\), it has to satisfy \(\theta =\chi _l^p\mu _l+\mu _h>\mu _h\). Using (39), (40), and (43) to replace \(t_h\), \(w_e\), and \((\chi _l^p\mu _l+\mu _h)\) in (41) and (42) gives

$$\begin{aligned} (1-\theta )(w_l-t_l)&=\theta t_l+\mu _h\phi , \end{aligned}$$
(44)
$$\begin{aligned} w_l-t_l&=\theta [\chi _l^p\mu _lw_l+\mu _hw_h]-\theta ^2t_l-\theta \mu _h\phi . \end{aligned}$$
(45)

Rewriting (44) yields

$$\begin{aligned} t_l=(1-\theta )w_l-\mu _h\phi , \end{aligned}$$
(46)

which using (40) gives

$$\begin{aligned} w_e=\theta w_l+\mu _h\phi , \end{aligned}$$
(47)

both as functions of parameters and \(\theta \). Plugging \(t_l\) into (45) gives

$$\begin{aligned}&w_l-(1-\theta )w_l+\mu _h\phi =\theta [\chi _l^p\mu _lw_l+\mu _hw_h]-\theta ^2((1-\theta )w_l-\mu _h\phi )-\theta \mu _h\phi \\&\quad \Leftrightarrow \theta w_l+\mu _h\phi =\theta [\chi _l^p\mu _lw_l+\mu _hw_h]-\theta ^2w_l+\theta ^2\theta w_l+\theta ^2\mu _h\phi -\theta \mu _h\phi \\&\quad \Leftrightarrow \left( 1-\theta ^2\right) (\theta w_l+\mu _h\phi )=\theta [\chi _l^p\mu _lw_l+\mu _hw_h-\theta w_l-\mu _h\phi ]\\&\quad \Leftrightarrow \left( 1-\theta ^2\right) (\theta w_l+\mu _h\phi )=\theta [\chi _l^p\mu _lw_l+\mu _hw_h-(\chi _l^p\mu _l+\mu _h) w_l-\mu _h\phi ]\\&\quad \Leftrightarrow \left( 1-\theta ^2\right) (\theta w_l+\mu _h\phi )=\theta \mu _h(w_h-w_l-\phi ), \end{aligned}$$

i.e., one equation in one unknown, \(\theta \). Thus, suppressing parameters, let \(h:(0,1)\rightarrow {\mathbb {R}}\) be

$$\begin{aligned} h(\theta )=\theta \mu _h(w_h-w_l-\phi )-\left( 1-\theta ^{2}\right) (\theta w_l+\mu _h\phi ). \end{aligned}$$
(48)

For all \(\phi \in (0,(w_h-w_l))\), h is a strictly convex function of \(\theta \) with

$$\begin{aligned} \lim _{\theta \rightarrow 0}h(\theta )=-\mu _h\phi <0\quad \text{ and }\quad \lim _{\theta \rightarrow 1}h(\theta )=\mu _h(w_h-w_l-\phi )>0, \end{aligned}$$

and thus has a unique root \(\theta ^*\in (0,1)\) such that \(h(\theta ^*)=0\). An increase in \(\phi \) decreases \(h(\theta )\) for all \(\theta \), and thus increases \(\theta ^*\). Via (43), (46), (39), and (47), the unique \(\theta ^*\) implies unique

$$\begin{aligned}&\displaystyle \chi _l^{p*}=\mu _l^{-1}\left( \theta ^*-\mu _h\right) , \end{aligned}$$
(49)
$$\begin{aligned}&\displaystyle t_l^*=\left( 1-\theta ^*\right) w_l-\mu _h\phi ,\end{aligned}$$
(50)
$$\begin{aligned}&\displaystyle t_h^*=t_l^*+\phi ,\end{aligned}$$
(51)
$$\begin{aligned}&\displaystyle w_e^*=\theta ^*w_l+\mu _h\phi , \end{aligned}$$
(52)

completing the unique solution to system (39)–(43). As \(\theta ^*\in (0,1)\) and \(\phi \in (0,(w_h-w_l))\), this unique solution satisfies \(\chi _l^{p*}<1\), \(t_l^*\in [-w_h,w_l]\), \(t_h^*\in [-w_h,w_h]\), and \(w_e^*\in [0,w_h]\). It is consistent with \(\chi _l^{p*}>0\) if and only if \(\theta ^*>\mu _h\). If \(\theta ^*\le \mu _h\), then the unique solution to (39)–(43) does not satisfy \(\chi _l^{p*}>0\). Then, by the statement in italics above, there is no regime with \(\chi _l^p>0\) in the constraint set of Problem (\(\hbox {PP}^{\prime }\)), and case 1 applies. The condition \(\theta ^*>\mu _h\) is satisfied and case 2 applies (is violated and case 1 applies) if, plugging \(\mu _h\) into h,

$$\begin{aligned} h(\mu _h)&=\mu _h^2(w_h-w_l-\phi )-(1-\mu _h^{2})(\mu _h w_l+\mu _h\phi )< (\ge )\;0\\&\Leftrightarrow \mu _h(w_h-w_l-\phi )<(\ge )\;(1-\mu _h^{2})(w_l+\phi )\\&\Leftrightarrow (1+\mu _h(1-\mu _h))\phi>(\le )\;\mu _hw_h-(1+\mu _h(1-\mu _h))w_l\\&\Leftrightarrow (1+\mu _h\mu _l)\phi>(\le )\;\mu _hw_h-(1+\mu _h\mu _l)w_l\\&\Leftrightarrow \phi >(\le )\;\underline{\phi }\equiv \frac{\mu _hw_h-(1+\mu _l\mu _h)w_l}{(1+\mu _l\mu _h)}, \end{aligned}$$

which is the stated condition. If \(\phi >\underline{\phi }\), combining the unique solution to system (39)–(43) with \(\chi _l^{e*}=1-\chi _l^{p*}>0\), \(\chi _l^{a*}=0\), \(\chi _h^{p*}=1\), and \(\chi _h^{e*}=\chi _h^{a*}=0\) gives the unique solution \(\sigma ^*\).

Finally, suppose \(\phi >\underline{\phi }\). Then, from (50), \(t_l^*=(1-\theta ^*)w_l-\mu _h\phi \). From the definition of h in (48), with \(\phi \rightarrow (w_h-w_l)\), \(h(\theta )\rightarrow -(1-\theta ^{2})(\theta w_l+\mu _h(w_h-w_l))<0\) for all \(\theta <1\), so that \(\theta ^*\rightarrow 1\) with \(\phi \rightarrow (w_h-w_l)\): for all \(\epsilon >0\), \(h(1-\epsilon )<0\) for some \(\phi \in (0,(w_h-w_l))\), \(\phi \) close enough to \((w_h-w_l)\), so that \(\theta ^*>1-\epsilon \). Thus, \(t_l^*\rightarrow -\mu _h(w_h-w_l)<0\) with \(\phi \rightarrow (w_h-w_l)\). It follows that for high enough \(\phi \), \(t_l^*<0\), which completes the proof. \(\square \)

Proposition 7

Proof

The function h as defined in Equation (48) is a strictly convex function of \(\theta \), negative valued for \(\theta \) approaching zero and switching sign with \(\theta \) approaching one, therefore has a positive derivative at its unique root \(\theta ^*\), and a higher \(\phi \) decreases it for all \(\theta \). Hence, \(\theta ^*\) increases with \(\phi \). As \(\phi >\underline{\phi }\), (49)–(52) apply. From (49), \(\chi _l^{p*}\) increases, so that \(\chi _l^{e*}=1-\chi _l^{p*}\) decreases; from (52), \(w_e^*\) increases, as it increases in both \(\theta ^*\) and \(\phi \); from (50), \(t_l^*\) decreases, as it decreases in both \(\theta ^*\) and \(\phi \); and from (51), the difference \(t_h^*-t_l^*=\phi \) increases with \(\phi \). \(\square \)

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Auerbach, J.U. Property rights enforcement with unverifiable incomes. Econ Theory 68, 701–735 (2019). https://doi.org/10.1007/s00199-018-1141-9

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Keywords

  • Costly falsification
  • Institutions
  • Property rights
  • Enforcement

JEL Classification

  • O17
  • P14