Tax policy design in a hierarchical model with occupational decisions

Abstract

This research examines the impact of occupational choices and tax evasion on the tax administration policy in a hierarchical tax model. The economy has two sectors, wage-earners and self-employment, with evasion only possible in the latter. Incorporating occupational decisions produces a smaller marginal tax rate and a larger budget for the IRS. However, the resources are still insufficient to audit all self-employed, resulting in distortions in occupational choices favoring self-employment. These distortions prevent production efficiency from achieving the optimum level, indicating that the Diamond-Mirrlees theorem is not applicable in this context. Finally, applying differential taxation represents a Pareto improvement, but it results in higher taxes for self-employment.

1 Introduction

Increasing tax revenue has been an essential topic for governments. For instance, after the economic crisis of 2008, developed countries focused on combating tax evasion and understanding its mechanisms to increase tax revenues (Slemrod, 2019). This is mainly because fighting against tax evasion increases tax revenue through effectiveness instead of changing tax policy. Along this line, Keen and Slemrod (2017) study the characteristics of a tax administration policy beyond its design, pointing out the relevance of the relationship between enforcement cost and tax revenue. Therefore, a central scope in tax policy is finding ways to make compliance policy more efficient.

However, different occupations do not provide the same tax evasion opportunities. Kleven et al. (2011) and Slemrod (2007) show that evasion in third-party income reports is low, and the problem arises mainly in occupations with income self-reporting. Self-reporting could produce bunching (see Kleven 2016, for a review), which could be connected with agents’ ability (Bastani & Waldenström, 2021), and strategic declaration to avoid forceful audits (Almunia & Lopez-Rodriguez, 2018). Hence, occupations with income self-reporting are prone to evasion since agents can strategically declare income to pay fewer taxes. For this reason, in the design of a tax administration policy, it is necessary to consider different evasion opportunities across occupations and agents’ occupational choices. Still, the literature has not placed enough emphasis on inquiring about both issues in the same context, producing a lack of information about distortions because of coexisting evasion and different occupations.

The current study tries to close this gap by incorporating occupational decisions - self-employment and wage-earner - and tax evasion into a tax policy design model. Considering both elements in a tax policy design model captures the idea that different sectors have different evasion opportunities. The main interest is to study distortions in audits, the marginal tax rate, and the budget for the IRS, as well as its consequences for occupational decisions. The setting is a hierarchical model similar as in Sánchez and Sobel (1993), in which a government interacts with a tax agency (the IRS henceforth), and this agency with taxpayers. Since hierarchical models help perform normative analyses (Melumad & Mookherjee, 1989), this paper focuses on the normative implications of including evasion and occupational choices in tax policy design.

This paper solves a three-stage game in which the government, the IRS, and a continuum of risk-neutral taxpayers interact. Taxpayers choose to be self-employed or wage-earners, and evasion is only possible in self-employment. In the first stage, a social welfare-maximizing government commits to a marginal tax rate, a public good provision, and a budget for the IRS. In the second stage, the IRS commits to an audit function. The IRS is entrusted to maximize the expected tax collection. Finally, in the third stage, taxpayers maximize their utility by choosing an occupation. If they are self-employed, they also decide their income declaration.

The IRS finds a direct incentive-compatible mechanism composed of an audit function and effective taxes, which are the agents’ tax liability, to enforce a true income declaration. Since productivity is private information, the revelation principle (Myerson, 1979, 1981) is used to find a mechanism that incentivizes agents to reveal their income. The audit function depends on the budget of the IRS but always takes the maximum possible value. All self-employed are audited with a large enough budget, but only low and middle-income self-employed are audited in an underbudgeted situation. If an agent is audited, its effective tax is the same as the government committed, otherwise it pays the same tax as the last audited taxpayer. In the underbudgeted case, those results distort occupational decisions for (potential) high-income self-employed in favor of self-employment. The same results are obtained if the audit cost is monotonically non-decreasing in the self-employment income and with a fine rate increasing in the self-employment income but with a slope bounded from above.

The public good provision depends on the government’s redistributive motives. Redistribute motives are captured by the social weight that shows the relevance of the agent’s consumption change in public funds terms. When the average social weight is different than one, the public good provision is biased from the first-best.

The marginal tax rate is lower than in the case without occupational decisions and can be characterized by two effects: welfare and revenue. This means taxes are upward biased if the government does not consider occupational decisions. The difference emerges because higher taxes raise tax revenue and downward welfare, increasing the incentive to evade. Hence, welfare and revenue effects due to occupational decisions produce lower taxes. Differential taxation (one marginal tax rate for each occupation) is a Pareto improvement, but it implies a higher marginal tax rate in self-employment. This is explained by a rise in self-employment taxes inducing taxpayers to move to the wage-earner sector, reducing evasion incentives.

The IRS budget is larger than in the case without occupational decisions, and three effects characterize its level: behavioral, mechanical, and welfare. This result comes from the marginal gains for an extra dollar/euro to the IRS. When audit increases, more taxpayers decide their occupation efficiently, paying higher taxes and increasing the effective tax for those not audited. However, this effect is decreasing in the extra budget, making marginal costs higher than benefits at some point. Therefore, it is not optimal to audit the entire self-employment sector.

Distortions through audits mean that the Diamond-Mirrlees theorem (Diamond & Mirrlees, 1971a1971b ) does not apply in this setting. This is because attaining production efficiency in a second-best tax scheme is impossible. At the equilibrium, the IRS is underbudgeted, and (potential) high-income self-employed have incentives to be self-employed. Differential taxation does not solve this result, producing more distortions under this scheme. Hence, because of the institutional design, high-skilled wage-earner workers move to their less productive occupation, which in this setting is self-employment. The critical element for this result is the non-optimality of giving enough resources to the IRS.

This study brings together two strands of literature. First, it expands the theoretical tax administration literature by demonstrating and explaining distortions due to coexisting evasion and occupational decisions. In this area, all papers build on the classical contributions of Allingham and Sandmo (1972) and Srinivasan (1973), and the extension of Yitzhaki (1974). Most papers focus on the evasion effect on self-employment, modeling only one sector and demonstrating the consequences of auditing and taxation.¹ Examples of models that assume auditing depends on declared income, similar to this paper, are Reinganum and Wilde (1985, 1986); Scotchmer (1987); Border and Sobel (1987). In this line, Cremer et al. (1990) and Sánchez and Sobel (1993) model hierarchical situations focusing on one occupation. Particularly, Sánchez and Sobel (1993) characterized the solution to the IRS’s problem and the tax policy, being the closest benchmark for our results. General conclusions from those studies are that audits have to be strongest in sectors more prone to finding evaders, and the tax system becomes regressive because of evasion. Regarding the tax administration models, Chander and Wilde (1998) provides a general characterization, and Keen and Slemrod (2017) study its components, showing the trade-offs beyond it and the elements that must be analyzed in its design. This paper expands this literature by incorporating occupational choices and showing that this produces lower taxes and a larger budget for the IRS, and differential taxation is a Pareto improvement.

The second strand of literature relies on the evasion effect on the labor market, particularly on the extensive margin decision, in a setting with tax administration policy. This line began with Sandmo (1981), later followed by theoretical developments by Boadway et al. (1991); Cowell (1985) and Pestieau and Possen (1991), Parker (1999) includes a simulation for the UK, and Watson (1985) and Kesselman (1989) use a general equilibrium framework. Recently, Casamatta (2021) incorporated occupational decisions in an optimal income taxation model similar to Chander and Wilde (1998), but without an IRS. Therefore, this paper contributes by studying distortions in occupational decisions in a hierarchical model. In this sense, the underbudgeted IRS result is critical because it raises the incentives to become self-employed. Moreover, and in a similar vein as Gomes et al. (2017) and Best et al. (2015), it is demonstrated that those distortions affect production efficiency in the second-best optimal taxation setting.

The rest of the paper is organized as follows. Section 2 presents the model and provides the first-best as a benchmark. Section 3 solves the IRS’s problem, showing the characterization of the audit policy and the effective tax. Section 4 solves the government’s problem, obtaining the marginal tax rate, the IRS budget, and the provision of public goods. Section 5 explains the consequences of the equilibrium and connects it with some empirical results. Section 6 describes the optimality of differential taxation and extensions over the audit cost and fine rate. Finally, Sect. 7 concludes.

2 Model

The economy has three agents: the government, the IRS, and a continuum of risk-neutral taxpayers of mass one. Two sectors exist: self-employment and wage-earner, indexed by $i \in \lbrace s, d \rbrace$, where taxpayers decide where to work. While in sector d, employers report taxpayers’ income; in sector s, taxpayers self-report their income. It is assumed that in sector d, cooperation between firms and workers to underreport wages and divide the evaded amount is impossible. However, tax evasion is possible in sector s. The IRS knows perfectly which sector each taxpayer works in, and, because of evasion, needs to incur a cost to monitor income declarations in sector s. It is assumed that the government has no resources (technology, knowledge, and expertise, among other things) to perform the audit process efficiently. Therefore, a specialized agency to enforce tax compliance is necessary.

An interesting way to analyze this setting is through differential taxation: one tax scheme for each occupation. This approach could encompass each occupation’s inherent differences, for instance, evasion. Along this line, Gomes et al. (2017) find the optimal differential taxation in an occupational decisions model without evasion. Appendix A uses Gomez and coauthors’ paper as a benchmark to demonstrate that introducing evasion in this setting is worth it. Indeed, evasion necessitates a tax compliance policy, which alters optimal taxes and increases government costs.

2.1 Agents

Nature randomly chooses a productivity $n = (n^{s}, n^{d})$ for each taxpayer. Each term represents productivity in the self-employment and wage-earner sectors, respectively. Formally, n comes from a joint distribution function F with support $[\underline{n}^{i}, \overline{n}^{i}] \times [\underline{n}^{j}, \overline{n}^{j}]$, where $i, j \in \lbrace s, d \rbrace$ and $i \ne j$. It is assumed that F is twice continuously differentiable (positive derivative), and the distribution of each productivity is positive correlated. In particular, we assume that $\overline{n}^{s} > \overline{n}^{d}$.² Also, let us define $F_{i}$ as the marginal distribution respect to sector i (and $f_{i}$ its density) and $F_{i\mid j}$ the conditional distributions (and $f_{i\mid j}$ its density), with $i, j \in \lbrace s, d \rbrace$ and $i \ne j$. These functions are common knowledge in the model.

Taxpayers consume a homogeneous good whose production is linear in labor, and its price is normalized to one. Also, it is assumed that this market is perfectly competitive. Hence, the wage in each sector is equal to one, and the agent’s income is the same as the agent’s productivity. Given this, we hereafter denominate $n^{i}$ the taxpayer’s income in sector $i \in \lbrace s, d \rbrace$.

Taxpayers decide their occupation by comparing both sectors and choosing which gives them the highest utility. Each taxpayer offers an inelastic labor supply equal to one in each sector. Since taxpayers are risk-neutral, their utility can be written as a quasi-linear function in consumption, $C:[\underline{n}^{i}, \overline{n}^{i}] \rightarrow \mathbb {R}_{+}$, with $i \in \lbrace s, d \rbrace$. Agents pay taxes over their reported income. Let us assume a linear tax scheme composed of a marginal tax rate $\tau \le 1$.³ Let $R \in \mathbb {R}_{+}$ be a certain level of public goods, and $\phi : R \rightarrow \mathbb {R}_{+}$ the function that captures the benefit for its provision, that is increasing and concave ($\phi ^{\prime } > 0$ and $\phi ^{\prime \prime } < 0$). From these, the agents’ utility is $U(n^{i}, R) = C(n^{i}) + \phi (R)$.

In the wage-earner sector, consumption equals the after-tax income, $n^{d} (1 - \tau )$. Consequently, the utility in this sector is $U^{d}(n^{d}, R) = n^{d} (1 - \tau ) + \phi (R)$. Note that since wage-earners do not make any decision, $U^{d}(n^{d}, R)$ is their indirect utility.

In the self-employment sector, taxpayers make an income declaration $w: [\underline{n}^{s}, \overline{n}^{s}] \rightarrow \mathbb {R}_{+}$. The IRS does not reward over-reporting, implying that any declaration above real productivity is a dominated strategy, hence $w \le n^{s}$. Self-employed face an audit with probability $\alpha \in [0,1]$, and if the IRS audits them, it immediately discovers whether they have evaded. Audited self-employed pay the penalty $\rho \in \mathbb {R}$, composed by a fine rate $\pi > 1$ over the evaded taxes. The penalty takes the following form

$$\begin{aligned} \rho (n^{s})= \left\{ \begin{array}{ll} \pi \tau \left( n^{s} - w \right) &{}\,\, {\hbox {if }} \,\, w < n^{s} \\ 0 \,\, &{} {\hbox {if}} \,\, w \ge n^{s} \end{array} \right. \end{aligned}$$

The penalty function establishes that the penalty is zero if a taxpayer declares their true income. Otherwise, the penalty is the evaded taxes $\tau \left( n^{s} - w \right)$ expanded by the fine rate $\pi$.

Let us define $\bar{U}^{s}(n^{s}, R)$ as the utility with the optimal income declaration in sector s. Formally, this utility is given by the following

$$\begin{aligned} \bar{U}^{s}(n^{s}, R) = \max \limits _{w\le n^{s}} \left[ n^{s} - \tau w - \alpha (w) \max \left\{ \pi \tau \left( n^{s} - w \right) , 0 \right\} + \phi (R) \right] \end{aligned}$$

where $w(n^{s})$ is the solution to this problem. In this case, $C(n^{s}) = n^{s} - \tau w - \alpha (w) \max \left\{ \pi \tau \left( n^{s} - w \right) , 0 \right\}$.

2.2 IRS

The IRS chooses the audit function to maximize the expected tax collection. Define the audit function as $\alpha : [\underline{n}^{s}, \overline{n}^{s}] \rightarrow [0,1]$. The IRS has a budget B to finance the cost of auditing, where the cost of auditing a taxpayer is c, linear, and constant. If the IRS does not use its entire budget, the excess must be returned to the government along with the tax collection. The problem that the IRS solves is as follow

$$\begin{aligned}&\max _{\alpha (n^{s})} \; \int \limits _{\mathcal {N}^{d}} \tau n^{d} \mathrm{{d}}F(n^{d},n^{s}) \, + \, \int \limits _{\mathcal {N}^{s}} \left( \tau w + \rho (n^{s}) \right) \mathrm{{d}}F(n^{d},n^{s}) \\ s.t.&\\&B \ge \int \limits _{\mathcal {N}^{s}} c \cdot \alpha (w) \mathrm{{d}}F(n^{d},n^{s}) \\ \end{aligned}$$

where $\mathcal {N}_{i}$ is the set of taxpayers that decide to work in the sector $i = \lbrace s,d \rbrace$.

2.3 Government

The government chooses the marginal tax rate $\tau \in [0,1]$, a public good provision $R \in \mathbb {R}_{+}$, and the budget for the IRS $B \in \mathbb {R}_{+}$, to maximize a social welfare function (SWF). Let us define G(U) as the SWF, and assume that $G^{\prime } > 0$ and $G^{\prime \prime } \le 0$. In this case, G depends on the agent’s utility instead of their sum. This assumption captures the government’s concern for each possible income realization.

The government has a budget constraint where the cost of providing one unit of a public good is one. Given this fact, the problem of the government is as follows

$$\begin{aligned}&\max _{\tau , R, B} \; \int \limits _{\mathcal {N}^{d}} G \left( U^{d}(n^{d}, R)\right) \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\mathcal {N}^{s}} G \left( \bar{U}^{s}(n^{s}, R) \right) \mathrm{{d}}F(n^{d},n^{s}) \\ s.t.&\\ (BC)&\; \int \limits _{\mathcal {N}^{d}} \tau n^{d} \mathrm{{d}}F(n^{d},n^{s}) \, + \, \int \limits _{\mathcal {N}^{s}} \left( \tau w + \rho (n^{s}) \right) \mathrm{{d}}F(n^{d},n^{s}) \\&+ \left( B - \int \limits _{\mathcal {N}^{s}} c \cdot \alpha (w) \mathrm{{d}}F(n^{d},n^{s})\right) \ge B + R \\ (LL)&\; \; U^{d}(n^{d}, R) \ge 0 , \; \bar{U}^{s}(n^{s}, R) \ge 0 \end{aligned}$$

where $\mathcal {N}_{i}$ is the set of taxpayers that decide to work in the sector $i = \lbrace s,d \rbrace$. The first restriction is the budget constraint (BC), which states that the expected tax collection must be at least equal to the expenses in the public goods provision and the budget for the IRS. The final restriction is the limited liability (LL), which states that any agent has a non-negative utility.

2.4 Timing

In the first stage, the government chooses a marginal tax rate, $\tau$, a budget for the IRS, B, and a level of public goods provision, R, to maximize social welfare. The IRS selects an audit function in the second stage, $\alpha$, in order to maximize the expected tax collection. In the third stage, taxpayers decide in which sector to work, s or d, and if self-employed, also choose their income declaration w. Finally, taxpayers pay their taxes. Wage-earner and non-audited self-employed only pay taxes on their declared income and audited self-employed also pay penalties. Figure 1 shows this game.

Fig. 1

Timing of the Game

2.5 Full-information solution

The first-best solution is obtained to provide a benchmark with some results. The government decides the sector in which each agent works, its consumption level, and a public good provision. Let us define a consumption function, $C: [\underline{n}^{d}, \overline{n}^{d}]\times [\underline{n}^{s}, \overline{n}^{s}] \rightarrow \mathbb {R}_{+}$, a public good provision, $R \in \mathbb {R}_{+}$, and an occupational choice function, $Z: [\underline{n}^{d}, \overline{n}^{d}]\times [\underline{n}^{s}, \overline{n}^{s}] \rightarrow \lbrace s, d \rbrace$. The occupational choice function determines the sector in which each agent will optimally work.

Proposition 1

The first-best solution is given by $\lbrace Z_{s}, Z_{d},$ $R, C_{i}(n^{i}, n^{j}) \rbrace$, with $i\ne j$ and $i,j \in \lbrace s,d\rbrace$, and takes the following form

$$\begin{aligned} Z_{d} & = \lbrace d \: | \: \forall (n^{d}, n^{s}) \in Z_{d} \; \Rightarrow \; n^{s} < n^{d} \rbrace \\Z_{s} & = \lbrace s \: | \: \forall (n^{d}, n^{s}) \in Z_{s} \; \Rightarrow \; n^{s} \geq n^{d} \rbrace \\ 1 & = \phi^{\prime}(R) \\R & = \int \limits_{Z_{d}}\left( n^{d} - C_{d}(n^{d}, n^{s}) \right) dF(n^{d},n^{s}) + \int \limits_{Z_{s}} \left( n^{s} - C_{s}(n^{d}, n^{s}) \right) dF(n^{d},n^{s}) \\ & \dfrac{d G \left( U(C_{d}(n^{d}, n^{s}), R) \right)}{d U(C_{d}(n^{d}, n^{s}), R)} = \dfrac{d G \left( U(C_{s}(n^{d}, n^{s}), R) \right)}{d U(C_{s}(n^{d}, n^{s}), R)} \end{aligned}$$

Proof

See Appendix D$\square$

Proposition 1 shows the first-best solution in this setting. The first and second equations show that each agent works in its most productive occupation. The next equation reflects the optimal provision of public goods, indicating that the marginal cost of providing the public goods equals the sum among agents of the marginal rate of substitution (MRS) between the public and consumption goods. This result is known as the Bowen-Lindahl-Samuelson (henceforth BLS) rule (Samuelson, 1954, 1955). The last two equations relate to the optimal consumption choice, meaning that two agents in different occupations with the same income should have the same consumption. This establishes an equality condition related to horizontal equality between occupations in this setting.

2.6 Mechanism design approach

Since self-employed agents can hide information, the IRS needs to use a method that incentivizes them to declare their true income. Considering the revelation principle (Myerson, 1979, 1981), finding a direct incentive-compatible (IC) mechanism is without loss of generality. The mechanism $\mathcal {M}: \lbrace \alpha , \mathcal {T} \rbrace$ is composed by $\alpha$, the probability to audit an agent, and $\mathcal {T}$ the effective tax paid, namely $\mathcal {T}(n^{s}) = \tau w(n^{s})$.⁴ Notice that if the audit function takes the value of $\frac{1}{\pi }$, self-employed always reveal their real income (Scotchmer, 1987).⁵ This result allows us to redefine the support for the audit function as $\alpha \in [0, \frac{1}{\pi }]$. The direct IC mechanism is as follows

$$\begin{aligned} \mathcal {M} = \left\{ \begin{array}{rl} \alpha : &{} [\underline{n}^{s}, \overline{n}^{s}] \rightarrow [0,\frac{1}{\pi }] \\ \mathcal {T}: &{} [\underline{n}^{s}, \overline{n}^{s}] \rightarrow [0, \overline{n}^{s}] \end{array} \right. \end{aligned}$$

A direct IC mechanism in this setting results in self-employed declaring their real income, $w(n^{s}) = n^{s}$. This means that the IRS perfectly knows the real income of all self-employed at the equilibrium, and this is given by the incentive structure behind the mechanism $\mathcal {M}$.

Unless the mechanism follows the implementability requirements, some self-employed hide part of their income from the IRS. Let us define $V^{s}(n^{s}, R; \mathcal {M})$ as the indirect utility in sector s given a mechanism $\mathcal {M}$. Any implementable direct IC mechanism has the following characteristics

Lemma 2.1

The IC mechanism $\mathcal {M}$ is implementable if and only if

• 1. $\alpha (n^{s})$ is non-increasing in $n^{s}$

• 2. $\mathcal {T}(n^{s}) = \dfrac{n^{s} ( 1 - \tau \alpha (n^{s})\pi ) + \phi (R) }{1 - \pi \alpha (n^{s})} - \dfrac{\int \limits _{j = \underline{n}}^{n^{s}} \left( 1 - \alpha (j) \pi \tau \right) dj}{1 - \pi \alpha (n^{s})} - \dfrac{V^{s}(\underline{n}, R; \mathcal {M})}{1 - \pi \alpha (n^{s})}$

where $\underline{n}$ is the first income level where the mechanism applies.

Proof

See Appendix B$\square$

The above Lemma describes the mechanism’s requirements to incentivize agents to reveal their true income. The first requirement is given by the incentive to underreport income, saying that audits cannot be increasing in self-employment wages.⁶ In the opposite case, the IRS will audit high income reports more intensively, increasing the probability of non-auditing at low income levels and promoting evasion there. Therefore, the IRS must audit those income declaration levels where it is prone to find evaders.

The second requirement establishes the tax scheme incentivizing agents to reveal their actual income. This requirement shows the effect of evasion on the tax scheme and the impact of auditing on it. To promote truth-telling, the IRS should reward some declarations, depending on the audit level, allowing smaller tax payments than what taxpayers owe.

By using requirements in Lemma 2.1, it is possible to define the occupational choice rule and the threshold function in the agents’ occupational decisions. These elements are helpful in understanding distortions by IRS policy, as well as in simplifying the IRS’s problem.

Definition 1

(Occupational Choice Rule in an IC Mechanism). For any direct IC mechanism $\mathcal {M}$, $\mathcal {O}^{i}(\mathcal {M})$ is the set that results from agents’ occupational decisions and is defined by

$$\begin{aligned} \mathcal {O}^{s}(\mathcal {M}) =&\left\{ (n^{d}, n^{s}) \in [\underline{n}^{d}, \overline{n}^{d}] \times [\underline{n}^{s}, \overline{n}^{s}] \; \mid \; V^{s}(n^{s}, R; \mathcal {M}) \ge U^{d}(n^{d}, R) \right\} \\ \mathcal {O}^{d}(\mathcal {M}) =&\left\{ (n^{d}, n^{s}) \in [\underline{n}^{d}, \overline{n}^{d}] \times [\underline{n}^{s}, \overline{n}^{s}] \; \mid \; V^{s}(n^{s}, R; \mathcal {M}) < U^{d}(n^{d}, R) \right\} \end{aligned}$$

This definition implicitly states that the mechanism $\mathcal {M}$ defines agents’ occupational decisions. Agents compare their potential utility in each sector to decide where to work. The utility level given by $\mathcal {M}$ determines the attractiveness of sector s compared to d, showing that the mechanism also determines occupational decisions.

Lemma 2.2

For any direct IC mechanism $\mathcal {M}$, there exists a threshold function $\kappa : [\underline{n}^{s}, \overline{n}^{s}] \rightarrow \mathbb {R}_{+}$ that establishes the minimum income to work as a wage-earner. Moreover, in equilibrium, the threshold function is characterized by

$$\begin{aligned} \kappa ^{\prime }(n^{s}) = \dfrac{V^{s}_{n^{s}}(n^{s}, R; \mathcal {M})}{U^{d}_{\kappa (n^{s})}(\kappa (n^{s}), R)} \end{aligned}$$

where $X_{i}$ denotes the derivative of variable X with respect to i, and

$$\begin{aligned} \kappa (n^{s}) = \overline{n}^{d} \quad \text {for} \quad \hat{w} \le n^{s} \le \overline{n}^{s} \end{aligned}$$

where $\hat{w}$ is the first income level that produces $\kappa (n^{s}) = \overline{n}^{d}$.

Proof

See Appendix C$\square$

Consequently, the IRS incentivizes self-employed agents to reveal their real wages by choosing an audit function that maximizes the expected collection and holds conditions in Lemma 2.1. Formally, the problem of the IRS is as follows

$$\begin{aligned}&\max _{\alpha (n^{s})} \; \int \limits _{\mathcal {O}^{d}(\mathcal {M})} \tau n^{d} \mathrm{{d}}F(n^{d},n^{s}) \, + \, \int \limits _{\mathcal {O}^{s}(\mathcal {M})} \mathcal {T}(n^{s}) \mathrm{{d}}F(n^{d},n^{s}) \\ s.t.&\\&B \ge \int \limits _{\mathcal {O}^{s}(\mathcal {M})} c \cdot \alpha (n^{s}) \mathrm{{d}}F(n^{d},n^{s}) \\&\alpha (n^{s}) \; \text {non-increasing in } n^{s} \\&\mathcal {T}(n^{s}) = \dfrac{n^{s} ( 1 - \tau \alpha (n^{s})\pi ) + \phi (R) }{1 - \pi \alpha (n^{s})} - \dfrac{\int \limits _{j = \underline{n}}^{n^{s}} \left( 1 - \alpha (j) \pi \tau \right) dj}{1 - \pi \alpha (n^{s})} - \dfrac{V^{s}(\underline{n}, R; \mathcal {M})}{1 - \pi \alpha (n^{s})} \\&U^{d}(\kappa (n^{s}), R) = V^{s}(n^{s}, R; \mathcal {M}) \end{aligned}$$

The first restriction is the budget constraint, which asserts that the audit cost cannot be larger than the budget for the IRS. The second and third conditions are the implementability requirements for the mechanism $\mathcal {M}$. Finally, the fourth condition is the threshold function’s definition, which defines the occupational choice rule. Also, since the direct IC mechanism is used, the IRS function does not have penalties and uses the effective tax instead.

3 IRS problem

The IRS chooses an audit function that maximizes the expected tax collection and holds with the IC conditions. As Lemma 2.1 states, the implementability conditions for any direct IC mechanism $\mathcal {M}$ involve an audit function and an effective tax scheme. When choosing an audit function, the tax scheme is determined, and it is enough to design the monitoring policy to commit a specific tax scheme. Also, the monitoring process is required only on sector s because wage-earners report their incomes through their employers. This means that auditing affects only self-employment, and indirectly the wage-earner sector through agents’ occupational decisions.

The control theory is used to solve this problem.⁷ This procedure is similar to one used in the optimal taxation literature (For instance Mirrlees 1971). This is motivated by the necessity for the self-employed indirect utility to follow a specific function, $V^{s} _{n^{s}} (n^{s}, R; \mathcal {M}) = 1 - \pi \alpha (n^{s}) \tau$, a requirement for mechanism $\mathcal {M}$. To incentivize self-employed real income declaration, the IRS needs to ensure a certain level of utility for each possible earnings, or that the increase in self-employed utility follows a specific pattern. Defining $p(n^{s})$ as the adjoint function associated with the self-employed utility and $\mu$ as the Lagrange multiplier associated with the budget constraint, the Lagrangian of this problem is

$$\begin{aligned}&\mathcal {L}(V^{s}(n^{s}, R; \mathcal {M}); V^{s}_{n^{s}}(n^{s}, R; \mathcal {M})) = \int \limits _{\underline{n}^{s}}^{\overline{n}^{s}} \int \limits _{\kappa (n^{s})}^{\overline{n}^{d}} \tau n^{d} \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\underline{n}^{s}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} \left[ n^{s} - V^{s}(n^{s}, R; \mathcal {M}) \right] \mathrm{{d}}F(n^{d},n^{s}) \\&+ p(n^{s}) V^{s}_{n^{s}}(n^{s}, R; \mathcal {M}) - \mu \left( \int \limits _{\underline{n}^{s}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} c \cdot \left[ \dfrac{1-V^{s}_{n^{s}}(n^{s}, R; \mathcal {M})}{\pi \tau } \right] \mathrm{{d}}F(n^{d},n^{s}) - B \right) \end{aligned}$$

The linear effect of auditing gives the solution to this problem. Theoretically, this means that the audit should take the maximum or minimum value, depending on the budget constraint. The IRS audits with the maximum intensity until it spends all its budget. Consequently, all the budget B is used to audit and the IRS does not return money to the government. Also, this solution implies that some taxpayers are not audited if the budget is small enough. From this point comes the need to further explain this solution.

The audit function means that the IRS must monitor those earning levels that are prone to evasion more intensively. In this model, the self-employed have an incentive to underreport income. Without auditing, all agents will report the minimum possible earnings, $\underline{n}^{s}$, and full misreporting exists. Taxpayers will report more earnings if the IRS audits them. The minimum level to discourage evasion is $\frac{1}{\pi }$, meaning that any level between zero and this limit prevents full misreporting but does not incentive a real earning declaration. Let us assume that the IRS audits less than $\frac{1}{\pi }$ in $\underline{n}^{s}$ and does not use all its budget. The tax authority could increase tax collection by spending more resources on auditing and discouraging evasion at $\underline{n}^{s}$. Suppose we repeat this exercise along the earning distribution. In that case, the IRS starts to discourage evasion at earning levels where it is more prone to find evaders until it uses all the budget B. Therefore, the IRS needs to audit more intensively at the earnings levels where agents are more incentivized to misreport. Examples of the same explanation modeling the same motivation to misreport are (Paramonova) Kuchumova, Y. (2017) and Sánchez and Sobel (1993), and studies using different incentives include Bigio and Zilberman (2011) and Zilberman (2016).

The budget constraint defines the last audited earning level when the IRS is underbudgeted (B is insufficient to audit all self-employed). This level is defined as $w^{*}$. Hence, the IRS spends all its budget auditing until $w^{*}$. Since this is the last income level audited, it separates the audit function between its extreme values. Below $w^{*}$, the IRS audits with maximum intensity, $\alpha (n^{s}) = \frac{1}{\pi }$. Above $w^{*}$, the IRS does not audit, or $\alpha (n^{s})=0$. Scotchmer (1987); Border and Sobel (1987) and Sánchez and Sobel (1993) find the same solution, showing that the pattern is related to the IRS’s objective and the agents’ incentive to evade. The following equation characterizes $w^{*}$.

$$\begin{aligned} \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} f(n^{d},n^{s}) dn^{d} dn^{s} = \dfrac{B \pi }{c} \end{aligned}$$

(1)

Proposition 2 formalizes the abovementioned elements and incorporates the effective tax formula. It is defined as a solution for the IRS, the mechanism $\mathcal {M}$ that maximizes tax collection and fulfills the implementability requirements in Lemma 2.1.

Proposition 2

The direct IC mechanism $\mathcal {M}$ which solves this problem is

$$\begin{aligned} \alpha \left( {n^{s} } \right) = \left\{ {\begin{array}{*{20}c} {\frac{1}{\pi }} & {if} & {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n} ^{s} \le n^{s} \le w^{ * } } \\ 0 & {if} & {w^{ * } < n^{s} \le \bar{n}^{s} } \\ \end{array} } \right. \end{aligned}$$

(2)

$$\begin{aligned}{\mathcal{T}}\left( {n^{s} } \right) = \left\{ {\begin{array}{*{20}c} {\tau n^{s} } & {if} & {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n} ^{s} \le n^{s} \le w^{*} } \\ {\tau w^{*} } & {if} & {w^{*} < n^{s} \le \bar{n}^{s} } \\ \end{array} } \right. \end{aligned}$$

(3)

where the threshold $w^{*}$ is defined by equation (1)

Proof

See Appendix E$\square$

The audit function determines the effective tax scheme. The role of this tax scheme is to ensure that the agents’ utility is the same as a truth-telling agent, and in this sense is a tool to incentivize this behavior. Also, this scheme helps us see the effect of evasion, capturing the tax gap by comparing it with the tax owed. The effective tax equals the tax owed below the threshold. This comes from the audit level, which eliminates the incentive to evade taxes. On the contrary, agents with earnings higher than $w^{*}$ pay fewer taxes than they owe. This comes from the zero audits in this zone, implying that to eliminate the incentive to misreport, agents pay the same as the last audited agent. Paying this effective tax level equals declaring an income equal to $w^{*}$. Therefore, allowing them to pay this tax level causes the evasion gains to fall to zero, incentivizing a true income declaration and becoming a more regressive tax scheme. Cremer et al. (1990); Erard and Feinstein (1994) and Reinganum and Wilde (1986) obtain similar theoretical results about the regressivity of the system because of the existence of tax evasion.

Fig. 2

The IRS Solution

Figure 2 shows the results from Proposition 2 assuming a budget that does not allow auditing all self-employed. Panel (a) shows the audit function, Panel (b) the effective taxes, and Panel (c) the threshold function $\kappa (n^{s})$ where $\hat{w}$ is the first level that produces $\kappa (n^{s}) = \overline{n}^{d}$. Given the audit function, agents with earnings above $w^{*}$ pay the same taxes. This creates an increasing tax gap ($\tau (n^{s} - w^{*})$) because tax payment is the same for non-audited agents. The threshold $w^{*}$ helps us to demonstrate why the audit function maximizes the expected tax collection. Let us move $w^{*}$ to the left in Panel (a), causing more agents to not be audited. This makes $\tau w^{*}$ decrease in the y-axis in Panel (b). Both effects end up in a smaller collection since a set of agents pays fewer taxes than in the initial situation. Therefore, auditing with the maximum intensity at each possible income level maximizes tax collection.

The occupational decisions are distorted because of the audit function and effective taxes. Panel (c) shows that the incentive to be self-employed rises for agents with income above $w^{*}$. This comes from the tax gap. Since that agent pays fewer taxes, smaller earnings in the self-employment sector are required to equalize utilities between sectors. In particular, agents compare $n^{d}(1-\tau )$ with $n^{s} - \tau w^{*}$, generating the minimum wage-earner income to equalize utilities between sectors rises. This effect results in a distortion in occupational choices in incomes higher than $w^{*}$, producing that some agents do not work in their most productive occupation.

The audit result drives the effect of the effective tax scheme and the distortions on occupational decisions. However, the result relies on an underbudgeted situation. This situation critically depends on audit costs and the budget for the IRS. Corollary 1 shows that if there is no audit cost, it is optimal for the IRS to audit the entire self-employment sector, and distortions vanish. This highlights the relevance of giving enough resources to the IRS to diminish the distortions in the labor market or improve the monitoring process’s effectiveness (similar to reducing the audit cost in the model).

Corollary 1

If the IRS has no audit cost (i.e., $c = 0$), the direct IC mechanism $\mathcal {M}$ will be $\alpha (n^{s}) = \frac{1}{\pi }$ and $\mathcal {T}(n^{s}) = \tau n^{s}$, $\forall n^{s} \in \left[ \underline{n}^{s}, \overline{n}^{s} \right]$.

Proof

See Appendix F$\square$

Summarizing, the best action for the IRS is to audit until $w^{*}$. This level depends on three exogenous variables for the IRS: the audit cost, the fine rate, and its budget. Suppose those variables’ conjugation implies that auditing the entire self-employment sector is not optimal. In that case, distortions in occupational decisions will appear and come into play only for agents with earnings greater than $w^{*}$. The result of these distortions is that the higher the marginal tax rate, the more agents decide to become self-employed (see Eq. (5)). This comes from the fact that the IRS needs to incentivize taxpayers to declare their real wages, and to do this, the effective tax for non-audited agents is constant. This increases the utility of working as self-employed, provoking distortions in occupational decisions.

4 Government problem

The social welfare-maximize government chooses a public good provision, the marginal tax rate, and a budget for the IRS. The procedure to solve the government problem uses the budget constraint and incorporates the rest of the restrictions into the solutions. Let $\delta$ be the Lagrange multiplier for the government budget constraint. The Lagrangian for this problem is

$$\begin{aligned} \mathcal {L} =&\int \limits _{\mathcal {O}^{d}(\mathcal {M})} G \left( U^{d}(n^{d}, R)\right) \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\mathcal {O}^{s}(\mathcal {M})} G \left( V^{s}(n^{s}, R; \mathcal {M})\right) \mathrm{{d}}F(n^{d},n^{s}) \nonumber \\&- \delta \left\{ B + R - \tau \int \limits _{\mathcal {O}^{d}(\mathcal {M})} n^{d} \mathrm{{d}}F(n^{d},n^{s}) - \tau \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} n^{s} \mathrm{{d}}F(n^{d},n^{s}) - \tau \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*} \mathrm{{d}}F(n^{d},n^{s})\right. \nonumber \\&\left. - \tau \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*} \mathrm{{d}}F(n^{d},n^{s}) \right\} \end{aligned}$$

(4)

Before showing the government’s policy, let us describe the threshold function to simplify the explanation of each policy instrument. The threshold function takes the following form

$$\begin{aligned} \kappa (n^{s}) = \left\{ \begin{array}{cl} n^{s} &{}\,\, {\hbox {if}} \,\, n^{s}< w^{*} \\ \dfrac{n^{s} - \tau w^{*}}{1-\tau } &{} \,\,{\hbox {if}}\,\, w^{*} \le n^{s} \le \hat{w} \\ \overline{n}^{d} &{} \,\,{\hbox {if}} \,\, \hat{w} < n^{s} \end{array}\right. \end{aligned}$$

(5)

Both instruments, taxes and the IRS budget, modify the shape of the threshold function and the level where it takes the maximum, $\hat{w}$. Any change in these instruments produces two effects: distortions in the incentives to be in some occupation, and changes in the mass of agents that always prefer being self-employed. Figure 6 Panel (b), in Appendix C, shows that all agents with productivity higher than $\hat{w}$ always prefer being self-employed because their utility in sector s is always greater than in sector d. Finally, $w^{*}$ could be equal to $\hat{w}$, but this case depends on the budget for the IRS.

4.1 Optimal public goods provision

The public good provision is characterized by the Bowen-Lindhal-Samuelson (BLS) rule. To give a better interpretation, let us define $g(n^{s})$ as the social value of consumption for an agent with income $n^{i}$ expressed in terms of public funds. Formally, $g(n^{d}) = \frac{G^{\prime }(U^{d})}{\delta }$, and $g(n^{s}) = \frac{G^{\prime }(V^{s})}{\delta }$. This expression indicates the importance of an increase in the agent’s consumption concerning public funds. The following proposition shows the BLS rule in this setting.

Proposition 3

The Bowen-Lindahl-Samuelson rule is

$$\begin{aligned} \phi ^{\prime } (R) \left( \int \limits _{\mathcal {O}^{d}(\mathcal {M})} g(n^{d}) \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\mathcal {O}^{s}(\mathcal {M})} g(n^{s}) \mathrm{{d}}F(n^{d},n^{s})\right) = 1 \end{aligned}$$

(6)

Proof

See Appendix G.1$\square$

The public good provision could bias from the first-best in this setting. Proposition 3 shows that the average social weight $g(n^{i})$ is critical to see if the public good provision is the same as the first-best. To simplify, define the average social weight as $\mathbb {E}(g)$, Proposition 3 shows that $\phi ^{\prime }(R) = \frac{1}{\mathbb {E}(g)}$. We recover the first-best solution if the average social weight reaches one. If the average social weight is greater than one, there is an under-provision, and if it is smaller than one there is an over-provision. This result critically depends on the tax structure. If the tax includes a demogrant or a lump-sum transfer, the BLS rule is the same as in the first-best.

This result states that public good provision relies on the government’s redistributive intentions. Let us provide some examples. If the government has a Utilitarian motive, $g(n^{i}) = 1$, the BLS is the same as in the first-best. This means that the government is interested in the size of the economy, and to maximize it as much as possible, the best policy is to attain efficient public good provision. In another case, if the government follows a Rawlsian intention, $g(n^{i}) = 1$ only for the person with the minor utility, there will be an over-provision of the public good. Considering that the mass of people at this income level is smaller than one, the average social weight is lower than one, producing over-provision. Since the government wants to increase the utility of the most impoverished agent, it is optimal to provide more public good than in the first-best. The public good is under-provided if the average weight is larger than one. In this case, the average social weight means that the government has a large worry about agents’ consumption changes. A way to diminish those fluctuations is to decrease public goods spending, producing less pressure on tax collection. Therefore, the public good provision responds to the government’s redistribution sense and the value it places on consumption fluctuation.

4.2 Optimal marginal tax rate

The marginal tax rate (MTR) captures the forces associated with efficiency and equity, and the marginal gains/losses from movements in the extensive labor margin. The design of the MTR involves equity and efficiency elements, named welfare and revenue, respectively, but also the inclusion of occupational decisions. Occupational decisions produce a trade-off for the tax design. On the one hand, increased MTR motivates agents not to be wage-earners, eroding revenue, but increasing revenue through higher taxes. On the other hand, reduces the MTR recovers efficiency (for a given audit function) but drives tax collection to zero. These forces are displayed in Proposition 4.

Proposition 4

The optimal marginal tax rate is characterized by

$$\begin{aligned}&\dfrac{\tau }{1-\tau } \int \limits _{w^{*}}^{\hat{w}} (n^{s}-w^{*}) \kappa _{\tau }^{\prime }(n^{s}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} = \int \limits _{\mathcal {O}^{d}(\mathcal {M})} (1-g(n^{d})) n^{d} \mathrm{{d}}F(n^{d},n^{s}) \nonumber \\&+ \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} (1-g(n^{s})) n^{s}\mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} (1-g(n^{s})) w^{*} \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} (1-g(n^{s})) w^{*} \mathrm{{d}}F(n^{d},n^{s}) \end{aligned}$$

(7)

Proof

See Appendix G.2. $\square$

The marginal tax rate can be decomposed into two effects: welfare and revenue. The welfare effect shows the impact of change in the agent’s utility through the effect of taxes on consumption. When taxes rise, all agents face a larger tax liability than before and consume less. However, the reduction in their consumption depends on each consumer’s wage. This effect must be measured in public funds, and $g(n^{i})n^{i}$ captures it. The revenue effect has two components. First, a rise in taxes increases the marginal tax liability, which is equal to the taxpayer’s wage. Hence, $(1-g(n^{i}))n^{i}$ captures the net monetary effect on agents’ welfare. Second, an increase in taxes causes some agents with income higher than $w^{*}$ to change their occupation and pay a different tax level. When taxes rise, a taxpayer with a former wage $\kappa (n^{s})$ moves from the wage-earner to the self-employment sector and pays $\tau w^{*}$. The term which captures this effect is $\tau (\kappa (n^{s}) - w^{*}) \kappa _{\tau }(n^{s}) f^{d}(\kappa (n^{s}))$. This effect comes from the introduction of occupational decisions. In equilibrium, the sum of all effects must be zero.

Fig. 3

Effect of Raising the Marginal Tax Rate

Figure 3 shows the forces behind the design of the MTR. Panel (a) shows the consequences of increasing the MTR on the threshold function and occupational decisions. The red line indicates the initial situation, whereas the dashed green line illustrates a rise in the marginal tax. All agents with a productivity below the red (green) line (dashed line) are self-employed. An increase in taxes results in agents with productivity between $w^{*}$ and $\hat{w}^{1}$ changing their occupational choices, which is depicted by the blue area. However, $w^{*}$ does not change because neither does the IRS budget.⁸ These results yield an increase in the mass of self-employed agents. Additionally, an increase in the MTR reduces the productivity level $\hat{w}$, from $\hat{w}^{1}$ to $\hat{w}^{2}$. The effects are twofold. Firstly, some agents change their occupations, modifying the proportion of agents in each occupation. Secondly, agents who change their occupation face new taxes, impacting the tax revenue and their utility.

Figure 3b shows the consequences of an increase in marginal tax on the taxes paid by agents and tax revenue. The red line represents the initial situation, and the green line refers to a tax rise. The solid line is for taxes paid in self-employment, and the dashed line is for the wage-earner sector. When taxes rise, all agents pay higher taxes than before, represented by the difference between the red and green lines, increasing tax revenue. However, some agents change their occupations and face fewer taxes. The agents who change their occupation are those who do not face audits: they bear taxes equal to $\tau w^{*}$ instead of $\tau n^{d}$. This change produces a revenue loss for the government, depicted by the blue zone. Moreover, the more significant the tax increase, the greater the revenue loss. Therefore, tax revenue rises and welfare loss, adding to the occupational choices effect on revenue (decrease) and welfare (increase), are the forces behind the optimal tax rate.

Including occupational choices results in a smaller MTR than in a situation with only self-employment. To see this, notice that we can rewrite Eq. (7) as $\tau = \frac{A}{A+B}$, where A is the sum after the equal sign, and B is integral in the left-hand side. B is strictly positive since incomes are larger than threshold $w^{*}$ and the threshold function increases with taxes (see Eq. (5)) in this zone. Therefore, the term B produces the tax rate decrease, showing that occupational decisions result in a smaller MTR. Particularly, $\tau <1$. A situation where only one sector exists produces only similar terms as A and an MTR equal to one. Border and Sobel (1987) and Sánchez and Sobel (1993) obtain this result when only self-employment is considered. Therefore, occupational decisions create a downward force on the MTR, yielding $\tau <1$.

The result of the MTR shows that not considering occupational decisions induces an upward bias in the tax rate. This is due to the forces behind the welfare and revenue effects, which go in opposite directions. Raising taxes mechanically increases tax collection but also reduces agents’ welfare. This yields a movement of agents to self-employment to pay fewer taxes and increase their utility, eroding tax revenue. This finding is evidence of the forces behind the Laffer curve. The marginal tax will increase if total revenue and welfare increase due to the effect of occupational decision changes. Otherwise, the marginal tax will decrease.

4.3 Optimal IRS budget

The budget for the IRS delimits the size of the tax authority. This is the channel through which this policy instrument affects tax compliance and agents’ decisions. Changes in the IRS budget induce changes in occupational choices and tax revenue through changes in the mass of audited agents. These are the forces behind the optimal budget for the IRS.

Proposition 5

The optimal budget for the IRS is characterized by

$$\begin{aligned} 1 =&\, \tau \int \limits _{w^{*}}^{\hat{w}} (w^{*} - \kappa (n^{s}))\dfrac{\mathrm{{d}} w^{*}}{\mathrm{{d}} B} \kappa ^{\prime }_{B} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \nonumber \\&+ \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} \left( 1-g(n^{s}) \right) \tau \dfrac{\mathrm{{d}} w^{*}}{\mathrm{{d}} B} \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} \left( 1-g(n^{s}) \right) \tau \dfrac{\mathrm{{d}} w^{*}}{\mathrm{{d}} B} \mathrm{{d}}F(n^{d},n^{s}) \end{aligned}$$

(8)

Proof

See Appendix G.3. $\square$

The budget comprises three effects: behavioral, mechanical, and welfare. The behavioral effect relates to the capacity to audit more (fewer) taxpayers, producing changes in agents’ occupational decisions. The term that captures the behavioral effect is $\tau (w^{*} - \kappa (n^{s}))$, and comes from the inclusion of occupational decisions. The mechanical effect reflects the impact on non-audited agents’ taxes ($\tau \frac{d w^{*}}{d B}$) and the cost of changing the budget, which is equal to one. Since the threshold level has risen, taxpayers who do not face an audit pay more taxes than before, which is purely mechanical. This produces a fall in their consumption, captured by the budget effect on taxes, and also creates a welfare loss, which must be measured in terms of public funds ($\tau \frac{\mathrm{{d}} w^{*}}{\mathrm{{d}} B} g(n^{s})$). In equilibrium, the sum of the three effects must be zero, which provides the characterization in Proposition 5.

When the government gives a larger budget to the IRS, the audit threshold $w^{*}$ rises. This means more agents are audited, increasing the zone where agents decide their work sector efficiently. Revenue increases because those agents pay higher taxes as wage-earners than as self-employed, and the tax paid in the zero-audit zone also rises since $w^{*}$ is larger than before. Nevertheless, this also produces a welfare cost because those agents consume less. Hence, behind the optimal budget lie revenues raised from the behavioral and mechanical effects and costs from welfare and mechanical effects.

Since the government has a new channel to increase revenue, resources for the IRS are larger than in a context with only one sector. Let us use Eq. (8) to see this. This equation’s left-hand side (LHS) is the marginal cost for an increase of B, and the right-hand side (RHS) is the marginal benefit. The first term in the RHS is positive but decreasing.⁹ In a model with only one sector, the first term in the RHS does not exist. Without this term, the RHS is smaller than the LHS, or the marginal benefits are smaller than the cost. Since the net benefits decrease after the optimal budget, this result shows that the budget in this setting is smaller than with occupational decisions.

Fig. 4

Optimal IRS Budget Comparison

Figure 4 shows the last result. The blue line is the net benefit curve without occupational decisions and the green line is when occupational choices are considered. The optimal budget, $B^{2}$, is larger than when occupational decisions are not considered, $B^{1}$. This is because the behavioral effect on occupational decisions is positive, and the marginal cost of increasing the budget does not change compared to the case without occupational choices. In other words, because of more auditing process, agents are incentivized to be wage-earners at the margin, and the government can increase its revenue through this. However, this increase in the budget cannot attain the level of auditing all self-employed since the marginal cost lower-bound is one, and the marginal benefit is zero at this point.

In summary, the optimal budget shows relevant results. First, like in Sánchez and Sobel (1993), the IRS always has a smaller budget than would allow it to audit the entire self-employment sector. This is because the lower bound of the marginal cost for increasing the budget is always one, and the marginal gains are zero at that budget level. Second, as established in Slemrod and Yitzhaki (1987), the government must consider the effect on the agents’ tax burden when deciding the IRS budget, producing an impact on welfare. Finally, the budget is larger than the case without occupational decision because of the marginal gains due to the behavioral effect.

5 Equilibrium

The equilibrium in the model shows the effect of including occupational decisions and evasion in the design of tax policies. The marginal tax rate is lower than the tax without occupational choices and is smaller than one. This is due to the intent to diminish tax evasion gains. The budget for the IRS is larger than the case without occupational decisions, but not enough to audit the entire self-employment sector. This results in high-income self-employed not being audited, which increases the attractiveness of this occupation. Therefore, at the optimum, occupational decisions are distorted by the tax policy, and the tax system becomes regressive due to the incentives for truth-telling in the self-employment sector.

The equilibrium is sensitive to the government’s redistributive intentions, captured by $g(n^{i})$ with $i \in \left\{ s,d \right\}$. For a redistribution given by $g(n^{i}) = 1$, which is a Utilitarian SWF, the public good provision reaches the first-best, but the MTR and the budget for the IRS are zero. Conversely, when $g(n^{i}) = 1$ for the minimum utility and zero otherwise, a Ralwasian SWF, there is an over-provision of the public good, and the MTR and budget for the IRS reach the maximum possible value. Hence, when the redistribution intention increase, the public good, the MTR, and the budget for the IRS rise. The public good is the government’s redistribution instrument, thus redistribution implies increasing its provision. Regarding taxes and the size of the IRS, since $g(n^{i})$ captures the relevance for the government of agents’ consumption changes, the more significant the average, the more minor the MTR and the budget for the IRS in order not to distort consumption.¹⁰ This also shows that a progressive redistributive intention that gives more relevance to lower incomes produces larger taxes and budget for the IRS. To counteract the negative impact of tax evasion by high-income individuals, the IRS could receive a larger budget, which supports a progressive redistributive agenda.

At the optimum, the tax policy distorts agents’ occupational decisions in favor of self-employment. The connection between taxes and self-employment decisions has been studied extensively, showing a positive relationship.¹¹ However, the effect of evasion is always behind this fact. The forces behind the model’s equilibrium distortions help explain this evidence. The further the tax administration policy is from the optimum, the more extensive the distortions in occupational decisions, and this comes from the effect of taxes on evasion. Higher taxes and weak tax compliance capacity promote self-employment by increasing evasion’s marginal gains and facilitating this behavior. The model demonstrates that the lack of resources to audit all self-employed critically induces this mechanism.

However, it is possible to improve these distortions when evasion and occupational decisions are considered in the tax policy design. First, the budget for the IRS should be larger to reduce the attractiveness of self-employment through stronger auditing. This result is in line with those of Jung et al. (2022) but still rests on the equality between marginal cost and benefits, producing the underbudget result. This policy produces an increase in revenue, reducing distortions in evasion and occupational decisions and improving the regressivity of the tax system. Second, the IRS must use audits to enforce tax compliance and as a threat to avoid income misreporting. Audits might be stronger at those income levels where it is likely to find evaders. This result is similar to other theoretical studies ((Paramonova) Kuchumova, Y., 2017; Sánchez & Sobel, 1993), and empirically, Almunia and Lopez-Rodriguez (2018) show that firms make strategic declarations to avoid more forceful audits. Moreover, if low-productivity agents have the motivation to report greater productivity or income, Bigio and Zilberman (2011) and Zilberman (2016) show that audits increase with the agents’ productivity. This reinforces the idea that auditing zones where there is more probability of finding evaders.

The effect on occupational decisions may explain the empirical relation between tax rates and income tax evasion. If the marginal tax falls, the threshold function will take smaller values, reducing the incentive to evade taxes and resulting in less incentive to be self-employed. Empirically, Berger et al. (2016) and Kleven et al. (2011) show a positive relationship between tax rates and evasion. This relationship indicates that the government must reduce incentives for evasion by cutting taxes or increasing the IRS budget. Along this line, this paper demonstrates that occupational decisions produce lower optimal taxes and a larger budget for the IRS. This mechanism is similar to previous findings in the general equilibrium literature (Watson, 1985; Kesselman, 1989), reinforcing the importance of taxes in discouraging tax evasion. The critical issue behind this result is distortions by auditing on occupational decisions resulting from the lack of resources for the IRS. Therefore, it is necessary to align the tax administration policy to disincentivize evasion and improve the labor market’s efficiency.

Distortions of occupational decisions affect the equality condition over consumption in Proposition 1, revealing the evasion effect on inequality. Because of the incentives to declare their actual income, high-income self-employed are gifted with lower effective taxes, increasing their consumption compared to similar wage-earners. This affects horizontal equality and could distort the income distribution that the authority sees. Through an allowed biased declaration, the authority does not see the actual distribution. Instead, some excess mass appears at some points (bunching). Evasion distorts the optimal allocation of consumption and goes against horizontal equality. Since consumption and income have different distributions because of bunching in the latter, some policy designs would be biased. Also, through an increase in consumption, some agents increase their welfare. This could imply a Pareto improvement if evasion does not raise taxes, dropping other agents’ consumption.¹²

Agents have more incentives to be self-employed at the equilibrium, affecting production efficiency. If the system provides incentives to move resources from one sector to another, there would be a lack of productivity where the resources were first. In this model, firms would not find enough high-skilled workers in the wage-earner sector. This means the Diamond-Mirrlees theorem (Diamond & Mirrlees, 1971a, b) does not apply in this setting, in the sense that efficiency in productivity is not attained in the second-best. Although this takes place in a different environment, it is motivated by the underbudget result, since, as Corollary 1 pointed out, with enough resources, the IRS will audit all self-employed, and occupational decisions are not distorted. Gomes et al. (2017) find a similar conclusion in a model without tax evasion and IRS, and Best et al. (2015) find empirical evidence of the gains from inefficient productivity policies for improvement in tax compliance and tax revenue. In this case, the difference arises because providing enough resources to the IRS to conduct audits in the self-employment sector is not optimal.

This result, as established in Best et al. (2015), is based on the differences in tax capacity. Environments with lower tax capacities could benefit from inefficient productivity policies if these increased tax compliance and tax collection. Tax evasion is critical in explaining the distance from production efficiency. Taxpayers choose occupations or sectors, seeking to increase their utility, and higher evasion opportunities alter this choice. Also, since distortions persist at the equilibrium, a joint strategy between institutions (IRS and Governments) and policies (tax schemes, audits, penalties, among others) is necessary to recover productivity efficiency.

6 Extensions

Although the equilibrium in this model has already been shown, some questions are still open. Firstly, this section looks into imposing two different linear tax rates, one for each occupation. This extension could help to recover efficiency in allocating workers. Finally, the problem of achieving the same audit result with nonlinear audit costs and imposing an increasing fine rate is resolved.

6.1 Differential taxation

Let us define differential taxation as one MTR for each sector. Define $t_{s}$ and $t_{d}$ as the tax rates in the self-employment and the wage-earner sector, respectively. Also, assume that no rate can be higher than one. Using differential taxation, the threshold function changes, having the following form

$$\begin{aligned} \kappa (n^{s}) = \left\{ \begin{array}{cl} \dfrac{n^{s}(1 - t_{s})}{1- t_{d}} &{}\,\, {\hbox {if}} \,\, n^{s}< w^{*} \\ \dfrac{n^{s} - t_{s} w^{*}}{1- t_{d}} &{}\,\, {\hbox { if}}\,\, w^{*} \le n^{s} < \hat{w} \\ \overline{n}^{d} &{} \,\,{\hbox {if}} \,\, \hat{w} \le n^{s} \end{array} \right. \end{aligned}$$

Although this specification changes the threshold function, it does not alter the form of the audit function. This result comes from the conditions for solving the IRS problem. Differential taxation only changes the applicable tax rate to determine the threshold. This is to say, audit the entire self-employment sector or until $w^{*}$.¹³

The government solves a Lagrangian as in Sect. 4, choosing both tax rates. Since the focus is only on differential taxation, neither the optimal budget for the IRS nor public good provision is obtained.

Proposition 6

In a hierarchical model with occupational choices, differential taxation implies a higher marginal tax rate in the sector where evasion is possible. Moreover, differential taxation is a Pareto improvement from the situation with the same marginal tax rate for each occupation.

Proof

See Appendix H. $\square$

Figure 5 shows the basic idea behind Proposition 6. The red line represents the threshold function, and the blue dotted line represents the same wage in both sectors. Zone A represents revenue losses from agents who become wage-earners but pay higher taxes as self-employed. In contrast, zone B shows the increase in tax collection from agents who pay higher taxes in the wage-earner sector than in the self-employment sector. Differential taxation is possible only if zone A is smaller than zone B. Thus, differential taxation occurs if the increase in tax collection from agents above $w^{*}$ is greater than the losses from agents below it. When the difference in the marginal taxes rises, the threshold function falls, increasing A and decreasing B. Thus, it is not optimal for the government to separate both marginal taxes significantly. Consequently, differential taxation aims to solve tax collection losses due to non-audited taxpayers on the top rather than improving the allocation of agents. This ends up distorting occupational decisions even more than in the former setting.

Fig. 5

Differential Taxation

A higher tax in the self-employment sector means using tax incentives to eliminate the motivation to evade. A rise in the self-employment tax rate increases the incentive to be a wage-earner, decreasing evasion. The similarity with past policy recommendations on cutting taxes as a tax compliance policy relies upon this fact. However, in a context with different occupations and evasion possibilities, the incentive to misreport means that taxes must rise in the occupation with the greatest opportunities to evade.

Proposition 6 shows that dividing the tax system regarding the different occupations is a Pareto improvement. The reason comes from the evasion effect and the use of taxes to discourage it. Since evasion affects the marginal gains from evading, disparities in evasion opportunities should be treated individually depending on the context. This results in tax policy aligning with audits, discouraging evasion, and moving resources to occupations where evasion does not exist.

Differential taxation produces a worse allocation of workers than the former setting, generating a puzzle regarding production efficiency. As has been pointed out, distortions in occupational decisions mean that the Diamond-Mirrlees theorem does not apply in this setting. This extension does not resolve this. Instead, differential taxation makes the difference deeper. This result is produced by the linear tax system and the usefulness of moving taxpayers to the wage-earner sector. The linear tax assumption makes the government distort all agents to resolve problems in the zero-audit zone. Also, incentivizing workers to be wage-earners helps to solve evasion and can increase agents’ welfare due to lower taxes, which is a good policy for the government. Regarding this point, Gomes et al. (2017) show the optimality of moving taxpayers to a specific work sector in a nonlinear tax setting. Therefore, the welfare gains from this kind of policy against evasion could exist even in nonlinear tax policy models.

Corollary 2

In a hierarchical model with occupational choices and linear tax schemes, it is impossible to recover the efficiency of worker allocation.

Corollary 2 reflects that it is impossible to reconcile incentives for taxpayers to tell the truth with the incentive to choose their occupations efficiently. This result is essential for designing tax and audit policies. The possible solution to the audit scheme’s inefficiencies through differential taxation allows the government to improve economic efficiency and increase revenue. Hence, the government needs to find other policies to improve efficiency and increase revenue.

Finally, this result also provides evidence that a limited tax compliance context would benefit from distortionary policies regarding productivity, similar to Best et al. (2015). This arises because separating the tax system produces an increase in revenue and welfare but increases the inefficiency in productivity. In this case, the necessity to use taxes as a dissuasive tool for evasion causes the optimality of differential taxation. This increases productivity misallocation but implies a Pareto improvement.

6.2 Audit results

6.2.1 Audit costs

This section extends the analysis by allowing the audit cost to depend on the agents’ earnings. It is assumed that the IRS knows the cost when the agent reveals its income. The audit cost is defined as $c(n^{s})$, without any restriction a priori. From Appendix E, the condition to determine the solution’s form is the first necessary condition to solve the Lagrangian. In this condition, only the audit cost changes because neither the control variable nor the state variable changes.

A similar condition to a single-crossing is found. Let us derive the new first necessary condition regarding the self-employment wage. The following equation establishes the requirement for the same solution over the audit function.

$$\begin{aligned} -\dfrac{c^{\prime}(n^{s})}{c(n^{s})} < \dfrac{\pi \tau}{\mu c(n^{s})} + \dfrac{f_{d}(\kappa(n^{s}))k^{\prime}(n^{s})}{F_{d}(\kappa(n^{s}))} \end{aligned}$$

where $n^{s}$ is the level that produces $\frac{\mathrm{{d}} \mathcal {L}}{\mathrm{{d}} V^{s}_{n^{s}}}=0$. The condition is held with a positive marginal cost or a strictly increasing cost because the right side is positive by definition. Hence, the threshold solution is maintained if the audit cost is monotonically non-decreasing in the self-employment income.

An increasing audit cost means that the IRS is auditing fewer taxpayers. Assuming the condition derived, auditing self-employed with larger incomes is more expensive, making the IRS use all its budget to audit fewer taxpayers (see Eq. (5)). This means that fewer high-income self-employed are audited, and the problem in the allocation of workers increases. However, it is natural to think that taxpayers with more resources find more sophisticated ways to shelter their income, and the auditing process becomes more complex or costly. This point shows that the problems discussed in Sect. 5 are more serious when we consider increasing auditing costs.

6.2.2 Fine rate

Now we analyze the case where the fine rate is not linear. This case regards the intention to deter tax evasion through high penalties. Two options are explored: the fine rate increase with the amount evaded, and the fine rate rise with the real income. These specifications allow the IRS to audit more taxpayers at a lower cost because a higher fine rate replaces the audit’s deterrence effect.

Given the direct IC mechanism, agents honestly report their income and any instrument that depends on an evaded level vanishes. Thus, a fine rate that depends on the amount of evaded taxes collapses to a linear fine rate in equilibrium. This situation is the same as in the model and results in an equal audit function as presented earlier.

As for a fine rate that depends on the real earnings, the result is different. In this case, a similar analysis of the audit cost is performed. Now, the fine rate depends on self-employment income. Deriving the new first necessary condition with respect to the self-employment income allows us to obtain the single-crossing condition.

$$\begin{aligned} \dfrac{\pi^{\prime}(n^{s})}{\pi(n^{s})} < \dfrac{\pi(n^{s}) \tau}{\mu c} + \dfrac{f_{d}(\kappa(n^{s}))\kappa^{\prime}(n^{s})}{F_{d}(\kappa(n^{s}))} \end{aligned}$$

As stated before, $n^{s}$ is the wage level that produces $\frac{d \mathcal {L}}{d V^{s}_{n^{s}}} = 0$. This expression shows that the marginal fine rate must be bounded from above to obtain the threshold form in the audit function. Therefore, imposing an increasing fine rate to place higher penalties on higher incomes, audit more agents, and improve the IRS’s effectiveness is possible.

7 Conclusion

This paper investigates distortions in the tax administration policy by jointly introducing occupational decisions and tax evasion. To do this, we modeled a hierarchical situation where the government entrusts tax compliance policy to the IRS and decides a marginal tax rate, a budget for the IRS, and a public good provision. The IRS collects taxes and conducts audits, interacting with taxpayers who choose between being self-employed or wage-earners. Income tax evasion is possible only in self-employment because such agents self-report their income. This means that tax compliance policy is conducted entirely in the self-employment sector.

The coexistence of occupational decisions and evasion distorts the tax administration policy. First, not considering occupational decisions produces an upward bias on taxes. Second, the budget for the IRS is larger than if occupational choices are not considered, but not enough to audit all self-employed. Both elements prove that a non-optimal policy favors evasion and distorts agents toward self-employment. This comes from audit distortions. The impossibility of auditing all self-employment results in the high-income self-employed paying the same taxes, increasing their utility, and distorting occupational decisions in this zone. Differential taxation is a Pareto improvement in this setting but implies higher taxes in the self-employment sector. This result distorts the allocation of workers even more than the former scenario. Finally, at the optimum, the optimal allocation of workers is slanted in favor of self-employment, and production efficiency is not attained in the second-best tax optimum. This result shows that the Diamond-Mirrlees theorem does not apply in this setting.

The previous results highlight the relevance of considering occupational decisions in tax administration design. If tax administration policy does not consider agents’ occupational choices, distortion in favor of evasion and self-employment appears. This paper demonstrates how the policy should be, providing clear policy recommendations. First, audits should be more vigorous at income levels where evaders are more prone to declare. Besides, supposing all occupations face the same tax scheme, the marginal tax rate should be lower, and the IRS size should be larger than the setting that does not consider occupational decisions. However, if differential taxation is allowed, the tax system should differ by occupation or group depending on the opportunity to evade taxes. Moreover, taxes must be higher in occupations where evasion incentives are greater.

Some problems remain at equilibrium, like tax system regressivity and productivity inefficiencies. It is essential to inquire about new instruments, mechanisms, and strategies to increase enforcement policies’ effectiveness at a lower economic cost. Examples can be focused on persuasion policies that accompany auditing. Since taxes and audits affect expected consumption and occupational choices, a non-pecuniary policy affects only the latter. Indeed, shaming policies (See Perez-Truglia and Troiano (2018) for instance) can be an excellent example of persuasion policies toward tax compliance.

Appendices

Appendix A Occupational choice model vs. Evasion model

This section compares a differential taxation model with a hierarchical model with tax evasion. First the differential tax model is presented, and some assumptions are made to nest with the hierarchical one. Finally, the hierarchical model is presented and the conditions under which both models could give the same result are described.

The economy has two sectors, self-employed and wage-earner, s and d, respectively. Each agent has an ability tuple $n = (n_{s}, n_{d})$, which represents its productivity in each sector. Each productivity comes from a distribution function, twice continuously differentiable, and with support $[\underline{n}_{i}, \overline{n}_{i}]$. Agents choose the labor supply $h_{i}$, implying an effective labor $h_{i} n_{i}$ in sector $i \in \lbrace s, d \rbrace$. Labor income in sector i is $y_{i} = w_{i} h_{i} n_{i}$, where $w_{i}$ is the wage. In the first model, the government imposes two different tax schemes, where $T_{i}$ is the income tax in sector i. Given these facts, the agent’s utility in sector i is $u_{i}(n_{i}) = w_{i} h_{i} n_{i} - \psi (h_{i}) - T_{i}(w_{i} h_{i} n_{i})$

where $\psi$ is the convex cost of labor. The following assumption is made to nest this frame with a hierarchical model with tax evasion. Let us assume that the labor supply in the sector d is inelastic and equals one, that is $h_{d} = \overline{h}_{d} = 1$. By using this, the first-order conditions with respect to the labor supply in sector s and the derivatives with respect to productivity in each sector are

$$\begin{aligned} \begin{array}{cccl} \langle h_{s} \rangle &{} w_{s} n_{s} \left[ 1-T^{\prime }_{s}(w_{s} h_{s} n_{s}) \right] &{} = &{} \psi ^{\prime }(h_{s}) \\ \langle n_{s} \rangle &{} w_{s} h_{s} \left[ 1 - T^{\prime }_{s}(w_{s} h_{s} n_{s})\right] &{} = &{} u^{\prime }_{s}(n_{s}) \\ \langle n_{d} \rangle &{} w_{d} \left[ 1 - T^{\prime }_{d}(w_{d} n_{d})\right] &{} = &{} u^{\prime }_{d}(n_{d}) \\ \end{array} \end{aligned}$$

Also, let us define the threshold function $c: [\underline{n}_{s}, \overline{n}_{s}] \rightarrow [\underline{n}_{d}, \overline{n}_{d}]$. This reflects the agent’s minimum ability in sector d to choose to be a wage-earner, given its ability in sector s. In that sense, it is a threshold level that separates the extensive margin decision. By definition, this function holds with $u_{s}(n_{s}) = u_{d}(c(n_{s}))$, implying the following condition

$$\begin{aligned} c^{\prime }(n_{s}) = \dfrac{u^{\prime }_{s}(n_{s})}{u^{\prime }_{d}(c(n_{s}))} = \dfrac{w_{s} h_{s} \left[ 1 - T^{\prime }_{s}(w_{s} h_{s} n_{s})\right] }{w_{d} \left[ 1 - T^{\prime }_{d}(w_{d} c(n_{s}))\right] } \end{aligned}$$

(A1)

This function is equal to the one obtained in Gomes et al. (2017). Therefore, this results from a differential taxation model without evasion.

For the model with tax evasion, let us assume that in the self-employment and wage-earner sectors, the labor supply is inelastic and equal to one, and in sector s agents declare their income. In this case, the government decides the audit function and only one income tax scheme for all agents, T. With these assumptions, both problems are similar because the government chooses two instruments in each case. In the case of evasion, the government chooses the audit probability and the tax scheme, and in the model without evasion, it decides on two tax schemes, one for each occupation.

Now, define $x_{s}$ as the income report in sector s, and $\rho (x_{s})$ as the audit function that depends on the declared income in the sector s. If an evader is discovered, it pays a linear fine on the evaded amount with a penalty rate $\pi > 1$. The agent’s utility in each sector is

$$\begin{aligned} \begin{array}{ccl} u_{d}(n_{d}) &{} = &{} w_{d} n_{d} - T(w_{d} n_{d}) - \psi (\overline{h}_{d}) \\ u_{s}(n_{s}) &{} = &{} w_{s} n_{s} - T(x_{s}) - \rho (x_{s}) \pi \left[ T(w_{s} n_{s}) - T(x_{s}) \right] - \psi (\overline{h}_{s}) \end{array} \end{aligned}$$

In this model, only the derivatives with respect to each sector’s ability are obtained. These derivatives are the following

$$\begin{aligned} \begin{array}{cccl} \langle n_{s} \rangle &{} w_{s} \left[ 1 - \rho (x_{s}) \pi T^{\prime }(w_{s} n_{s})\right] &{} = &{} u^{\prime }_{s}(n_{s}) \\ \langle n_{d} \rangle &{} w_{d} \left[ 1 - T^{\prime }(w_{d} n_{d})\right] &{} = &{} u^{\prime }_{d}(n_{d}) \\ \end{array} \end{aligned}$$

The threshold function in this case is characterized by the following

$$\begin{aligned} c^{\prime }(n_{s}) = \dfrac{u^{\prime }_{s}(n_{s})}{u^{\prime }_{d}(c(n_{s}))} = \dfrac{w_{s} \left[ 1 - \rho (x_{s}) \pi T^{\prime }(w_{s} n_{s})\right] }{w_{d} \left[ 1 - T^{\prime }(w_{d} c(n_{s}))\right] } \end{aligned}$$

(A2)

Let us compare Eq. (A1) with Eq. (A2) to find the conditions under which the characterization of both threshold functions is identical. To simplify the analysis, assume the same wage in each model. Now, suppose that the tax scheme in the model without evasion is optimal for a particular distribution of ability. Since the wage-earner sector is the same in both models, let us define $T = T_{d}$, and impose that the threshold function c is the same in both models. Otherwise, both models do not fit well since extensive margin decisions will differ. With these assumptions, the denominators in Eqs. (A1) and (A2) are the same.

Now, to obtain the same solution, we require the same numerators in Eqs. (A1) and (A2). This is obtained with the following condition over the audit function

$$\begin{aligned} \rho (x_{s}) = \dfrac{1 - h_{s}\left[ 1- T^{\prime }_{s}(w_{s} h_{s} n_{s})\right] }{\pi T^{\prime }(w_{s} n_{s})} \end{aligned}$$

where $h_{s}$ is the labor supply in the differential taxation model. However, if the audit strategy takes other forms, both solutions are not the same. For instance, suppose the following

$$\begin{aligned} \rho (x_{s}) = \left\{ \begin{array}{cc} \frac{1}{\pi } &{}\,\, {\hbox {if}} \,\, n_{s} < n^{*} \\ 0 &{} \,\,{\hbox {if}} \,\, n^{*} \le n_{s} \end{array} \right. \end{aligned}$$

where $n^{*}$ is the threshold up to which the IRS audits. This strategy, optimal under a hierarchical model (Sánchez & Sobel, 1993), shows that it is not always possible to obtain the same optimal solution in both models. To obtain the same solution, the requirement is that $\frac{1 - h_{s}\left[ 1- T^{\prime }_{s}(w_{s} h_{s} n_{s})\right] }{T^{\prime }(w_{s} n_{s})} = 1$ for $n^{s} < n^{*}$ and 0 otherwise. Therefore, assuming that both models can be nested is not trivial. In addition, both problems are different since the government faces auditing costs, which produce more expenditure, and needs to consider incentive-compatible conditions to incentivize agents to declare their real type.

Appendix B Proof of Lemma 2.1

To obtain a direct incentive-compatible mechanism, it is necessary to analyze two restrictions on self-employment, participation (P) and incentive compatibility (IC).

$$\begin{aligned} (P)&\; U^{s}(n^{s}, R) = n^{s} - \mathcal {T}(n^{s}) - \alpha (n^{s})\pi \left[ \tau n^{s} - \mathcal {T}(n^{s})\right] + \phi (R) \ge 0 \\ (IC)&\; U^{s}(n^{s}, R; n^{s}) \ge U^{s}(n^{s}, R; w) \\ (IC)&\; n^{s} - \mathcal {T}(n^{s}) - \alpha (n^{s})\pi \left[ \tau n^{s} - \mathcal {T}(n^{s})\right] + \phi (R) \ge n^{s} - \mathcal {T}(w) - \alpha (w)\pi \left[ \tau n^{s} - \mathcal {T}(w)\right] + \phi (R) \; \; \forall w \le n^{s} \end{aligned}$$

The participation constraint is the limited liability condition, which refers to the non-negative utility for each taxpayer. Following the mechanisms design literature, the incentive compatibility restriction can be replaced by the following

$$\begin{aligned} V(n^{s}, R; \mathcal {M}) =&\max \limits _{w} \left\{ n^{s} - \mathcal {T}(w) - \alpha (w) \pi \left( \tau n^{s} - \mathcal {T}(w) \right) \right\} \end{aligned}$$

where $V(\cdot )$ is the indirect utility. Now, let us demonstrate the first statement following Guesnerie and Laffont (1984). First, we obtain the first-order condition (FOC) for the optimal income declaration in $V(n^{s}, R; \mathcal {M})$.

$$\begin{aligned} \dfrac{V(n^{s}, R; \mathcal {M})}{\partial w} = - \mathcal {T}^{\prime }(w) - \alpha ^{\prime }(w) \pi (\tau n^{s} - \mathcal {T}(w)) + \alpha (w) \pi \mathcal {T}^{\prime }(w) = 0 \end{aligned}$$

Second, we obtain the second-order condition (SOC). This must be zero or negative because of the quasi-concavity of the indirect utility.

$$\begin{aligned} \dfrac{\partial ^{2} V(n^{s}, R; \mathcal {M})}{\partial w^{2}} = - \mathcal {T}^{\prime \prime }(w) - \alpha ^{\prime \prime }(w) \pi (\tau n^{s} - \mathcal {T}(w)) + 2 \alpha ^{\prime }(w) \pi \mathcal {T}^{\prime }(w) + \alpha (w) \pi \mathcal {T}^{\prime \prime }(w) \le 0 \end{aligned}$$

Third, the FOC is derived with respect to $n^{s}$ and assumed truth-telling, $w = n^{s}$.

$$\begin{aligned} \dfrac{\partial ^{2} V(n^{s}, R; \mathcal {M})}{\partial w \partial n^{s}} = - \alpha ^{\prime }(n^{s}) \pi \tau \underbrace{- \mathcal {T}^{\prime \prime }(n^{s}) - \alpha ^{\prime \prime }(n^{s}) \pi (\tau n^{s} - \mathcal {T}(n^{s})) + 2 \alpha ^{\prime }(n^{s}) \pi \mathcal {T}^{\prime }(n^{s}) + \alpha (n^{s}) \pi \mathcal {T}^{\prime \prime }(n^{s})}_{SOC \le 0} = 0 \end{aligned}$$

All terms captured by the underbrace are the SOC, so being zero or negative. Hence, the first term must be positive or zero to hold with the equality to zero of the whole equation. The marginal tax is positive or zero by assumption. Therefore the audit must be non-increasing in the self-employed wage. This is the first condition of the Proposition.

For the effective tax, let us derive $V(n^{s}, R; \mathcal {M})$ with respect to $n^{s}$

$$\begin{aligned} \dfrac{\partial V(n^{s}, R; \mathcal {M})}{\partial n^{s}} = 1 - \alpha (w)\pi \tau - \dfrac{\partial w}{\partial n^{s}}\underbrace{\left[ \dfrac{\partial \mathcal {T}(w)}{\partial w}\left( 1 - \alpha (w)\pi \right) + \dfrac{\partial \alpha (w)}{\partial w}\pi \left( \tau n^{s} - \mathcal {T}(n^{s}) \right) \right] }_{\dfrac{\partial V(n^{s}, R; \mathcal {M})}{\partial w} = 0} \end{aligned}$$

Using the envelope theorem and the optimal condition on the income declaration, we can simplify the above expression. As we seek agents’ truth-telling, let us impose this condition, i.e., $w = n^{s}$.

$$\begin{aligned} \dfrac{\partial V(n^{s}, R; \mathcal {M})}{\partial n^{s}} = 1 - \alpha (n^{s})\pi \tau \end{aligned}$$

Now, by using the fundamental theorem of calculus, let us rewrite the indirect utility for the self-employed. We equalize this with the definition of the indirect utility assuming truth-telling.

$$\begin{aligned} \int \limits _{j = \underline{n}}^{n^{s}} \left( 1 - \alpha (j) \pi \tau \right) dj + V(\underline{n}, R; \mathcal {M}) = V^{s}(n^{s}, R; \mathcal {M}) =&\, n^{s} - \mathcal {T}(n^{s})- \alpha (n^{s}) \pi \left( \tau n^{s} - \mathcal {T}(n^{s}) \right) + \phi (R) \end{aligned}$$

where $V(\underline{n}, R; \mathcal {M})$ is the indirect utility of an agent with income $\underline{n}$, or the first agent for whom the mechanism is applied.

By doing algebra the effective tax formula is obtained.

$$\begin{aligned} \mathcal {T}(n^{s}) =&\dfrac{n^{s} ( 1 - \tau \alpha (n^{s})\pi ) + \phi (R) }{1 - \pi \alpha (n^{s})} - \dfrac{\int \limits _{j = \underline{n}}^{n^{s}} \left( 1 - \alpha (j) \pi \tau \right) dj}{1 - \pi \alpha (n^{s})} - \dfrac{V^{s}(\underline{n}, R; \mathcal {M})}{1 - \pi \alpha (n^{s})} \end{aligned}$$

Appendix C Proof of Lemma 2.2

Suppose agents A and B exist. Agent A has higher income in the wage-earner sector than agent B but the same in self-employment: that is $n^{d}_{A} > n^{d}_{B}$ and $n^{s}_{A} = n^{s}_{B}$. Since the utility in the wage-earner sector is increasing in income ($\frac{\mathrm{{d}} U^{d}}{\mathrm{{d}} n^{d}} = 1-\tau \ge 0$), agent A has higher utility in the wage-earner sector than agent B but the same in the self-employment sector. Now, assume agent B is wage-earner. Since agent A has greater utility in the wage-earner sector, it also works in the wage-earner sector. Therefore, every agent with higher productivity in the wage-earner sector and the same productivity in the self-employed sector as agent B is a wage-earner.

Now, let us assume that exist an agent C who has less income in the wage-earner sector and the same in the self-employment sector as agent B. That is, $n^{d}_{B} > n^{d}_{C}$ and $n^{s}_{B} = n^{s}_{C}$. Agent C’s income is such that $U^{d}(n^{d}_{C}, R) = U^{s}(n^{s}_{C}, R)$. By Definition 1, agent C is self-employed. Hence, any agent with lower income in the wage-earner sector and the same in self-employment as agent C is self-employed. This is because utility in the wage-earner sector increases with income, resulting in those agents having strictly higher utility in the self-employment sector than in the wage-earner.

Therefore, for each productivity in the self-employment sector there exists an income level in the wage-earner sector which equalizes the sectors’ utility. This wage determines a switch point in the occupational decision. Formally, define this level as $\kappa (n^{s})$, with $V^{s}(n^{s}, R;\mathcal {M}) = U^{d}(\kappa (n^{s}), R )$, and every agent with $n^{d} \le \kappa (n^{s})$ works in the self-employment sector. This demonstrates the existence of the threshold function.

The second part of Lemma 2.2 comes from deriving $V^{s}(n^{s}, R;\mathcal {M}) = U^{d}(\kappa (n^{s}), R )$ with respect to $n^{s}$.

$$\begin{aligned} V^{s}_{n^{s}}(n^{s}, R; \mathcal {M}) =&U^{d}_{\kappa }(\kappa (n^{s}), R) \kappa ^{\prime }(n^{s}) \\ \kappa ^{\prime }(n^{s}) =&\dfrac{V^{s}_{n^{s}}(n^{s}, R; \mathcal {M})}{U^{d}_{\kappa }(\kappa (n^{s}), R )} \ge 0 \end{aligned}$$

where the expression $X_{i}$ refers to the derivative of X with respect to i.

Finally, given the assumption that $\overline{n}^{s} > \overline{n}^{d}$, and that $\kappa ^{\prime }(n^{s}) \ge 0$, $\kappa (n^{s})$ reaches the maximum income in the wage-earner sector at a self-employment income smaller than $\overline{n}^{s}$. Figure 6, Panel (b) shows this situation. After an earning level defined by $\hat{w}$, the threshold function takes only one value, the maximum possible income in the wage-earner sector.

Fig. 6

Productivity Distribution Cases. Note: For both figures, it is assumed an audit function given by Proposition 2

Panel (a) shows the case where the earning differences are at the minimum earning level. In this case, the minimum income in one sector is compared with a set of income levels in another. However, since utility in sector d is increasing in $n^{d}$, if $\underline{n}^{d} < \underline{n}^{s}$, all agents would prefer to be self-employed because they attain higher utility levels. For this reason, the focus is on the case with $\overline{n}^{s} > \overline{n}^{d}$ which are shown in the Panel (b).

Appendix D Proof of Proposition 1

The problem that the government solves is as follows

$$\begin{aligned}& \max_{Z_{d}, Z_{s}, C_{d}(n^{d}, n^{s}), C_{s}(n^{d}, n^{s}), R} \; \int \limits_{Z_{d}} \left[ G \left( U(C_{d}(n^{d}, n^{s}), R)\right) \right] dF^{d}(n^{d},n^{s}) + \int \limits_{Z_{s}} \left[ G\left( U(C_{s}(n^{d}, n^{s}), R)\right) \right] dF^{d}(n^{d},n^{s}) \\s.t. & \\ & \int \limits_{Z_{d}} n^{d} dF^{d}(n^{d},n^{s}) + \int \limits_{Z_{s}} n^{s} dF^{d}(n^{d},n^{s}) \geq \int \limits_{Z_{d}} C_{d}(n^{d}, n^{s}) dF^{d}(n^{d},n^{s}) + \int \limits_{Z_{s}} C_{s}(n^{d}, n^{s}) dF^{d}(n^{d},n^{s}) + R \end{aligned}$$

where $Z_{i}$ is the set of agents in the work side i and $C_{i}$ is the consumption function for agents in sector i, with $i \in \lbrace s, d \rbrace$. The government has a resource constraint, which states that production in both sectors must be equal to consumption and spending on public goods.

To prove the first two results, suppose that a mass of agents greater than zero is assigned to work in their less productive sector. This mass of agents has $n^{s} > n^{d}$ but work in sector d. From the resource constraint, redirecting all these agents to sector s increases economic resources, which can be assigned to consumption or the provision of public goods, increasing social welfare. Hence, a reallocation of agents that provides higher social welfare exists, and the initial solution cannot be optimal. Therefore, agents work in the sector where they have higher productivity.

The fifth expression comes from the maximization with respect to consumption in each sector. The first-order condition (FOC) over consumption is

$$\begin{aligned} \dfrac{\mathrm{{d}} G (U(C_{i}(n^{d}, n^{s}), R))}{\mathrm{{d}} U(C_{i}(n^{d}, n^{s}), R)} f(n^{d},n^{s}) - \gamma f(n^{d},n^{s}) = 0, \qquad \text {for} \, i \in {s,d} \end{aligned}$$

where $\gamma$ is the Lagrange multiplier for the resource constraint. The fifth expression is obtained by assuming an interior solution and using $\gamma$ to equalize conditions in both sectors.

The third expression comes from the maximization with respect to public goods. The FOC is

$$\begin{aligned} \int \limits _{Z_{d}} \dfrac{\mathrm{{d}} G}{\mathrm{{d}} U} \times \dfrac{\mathrm{{d}} \phi (R)}{\mathrm{{d}} R} \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{Z_{s}} \dfrac{\mathrm{{d}} G}{\mathrm{{d}} U} \times \dfrac{\mathrm{{d}} \phi (R)}{\mathrm{{d}} R} \mathrm{{d}}F(n^{d},n^{s}) - \gamma = 0 \end{aligned}$$

Both sides can be divided by $\gamma$ and replaced by the FOC from consumption in each sector.

$$\begin{aligned} \int \limits _{Z_{d}} \dfrac{\dfrac{\mathrm{{d}} G}{\mathrm{{d}} U} \times \dfrac{\mathrm{{d}} \phi (R)}{\mathrm{{d}} R}}{\dfrac{\mathrm{{d}} G}{\mathrm{{d}} U} \times \dfrac{\mathrm{{d}} U}{\mathrm{{d}} C_{d}}} \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{Z_{s}} \dfrac{\dfrac{\mathrm{{d}} G}{\mathrm{{d}} U} \times \dfrac{\mathrm{{d}} \phi (R)}{\mathrm{{d}} R}}{\dfrac{\mathrm{{d}} G}{\mathrm{{d}} U} \times \dfrac{\mathrm{{d}} U}{\mathrm{{d}} C_{s}}} \mathrm{{d}}F(n^{d},n^{s}) = 1 \end{aligned}$$

Notice that $\frac{\mathrm{{d}} U}{\mathrm{{d}} C_{i}} = 1$ because agents are risk-neutral, and $\frac{\mathrm{{d}} U}{\mathrm{{d}} R} = \phi ^{\prime }(R)$. Those facts give us the following, which is the third expression.

$$\begin{aligned} \phi ^{\prime }(R) = \int \limits _{Z_{d}} \phi ^{\prime }(R) \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{Z_{s}} \phi ^{\prime }(R) \mathrm{{d}}F(n^{d},n^{s}) = 1 \end{aligned}$$

Finally, the fourth expression shows the resources needed to finance the optimal provision of public goods R and comes from the resource constraint.

Appendix E Derivation of the IRS policy

Following the literature on optimal taxation (see for instance Mirrlees (1971) and Gomes et al. (2017)), the control theory is used to solve this problem. Let us use $V^{s} (n^{s}, R; \mathcal {M})$ as the state variable, and $V^{s}_{n^{s}}(n^{s}, R; \mathcal {M})$ as the control variable. Both variables are chosen since the IC requirements in Lemma 2.1 state a condition over changes in the indirect utility of the self-employed when income rises. To perform this problem, it is necessary to make some changes. Firstly the audit function is replaced using the IC condition, $\frac{d V^{s} (n^{s}, R; \mathcal {M})}{d n^{s}} = V^{s} _{n^{s}} (n^{s}, R; \mathcal {M}) = 1 - \pi \alpha (n^{s}) \tau$. Secondly, replace the effective tax using the equality $\mathcal {T}(n^{s}) = n^{s} - V^{s}(n^{s}, R; \mathcal {M})$. This specification requires replacing the condition over audits for a condition over the marginal indirect utility, which must be non-decreasing and whose support is $[1-\tau , 1]$. Third, since $\kappa ^{\prime }(n^{s}) > 0$, it is possible to work with double integrals. By using those changes, the Lagrangian of this problem is

$$\begin{aligned}&\mathcal {L}(V^{s}(n^{s}, R; \mathcal {M}); V^{s}_{n^{s}}(n^{s}, R; \mathcal {M})) = \int \limits _{\underline{n}^{s}}^{\overline{n}^{s}} \int \limits _{\kappa (n^{s})}^{\overline{n}^{d}} \tau n^{d} \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\underline{n}^{s}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} \left[ n^{s} - V^{s}(n^{s}, R; \mathcal {M}) \right] \mathrm{{d}}F(n^{d},n^{s}) \\&+ p(n^{s}) V^{s}_{n^{s}}(n^{s}, R; \mathcal {M}) - \mu \left( \int \limits _{\underline{n}^{s}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} c \cdot \left[ \dfrac{1-V^{s}_{n^{s}}(n^{s}, R; \mathcal {M})}{\pi \tau } \right] \mathrm{{d}}F(n^{d},n^{s}) - B \right) \end{aligned}$$

where $p(w^{s})$ is the adjoint function associated with the state variable, and $\mu$ is the Lagrange multiplier associated with the budget constraint. Notice that the problem now depends on the state and control variable instead of the audits and effective tax scheme.

Since the constraints have inequalities and the control variable is involved in them, this is a mixed constraint problem. For this reason, Chapter 4 in Seierstad and Sydsaeter (1986) is followed to obtain the necessary conditions for an optimum. Also, since the Hamiltonian (first two elements in the above equation) and the Lagrangian constraints are concave and quasi-concave on the state and control variable, respectively, it is ensured that the problem is maximized at the optimum.

Following Theorem 3 in Chapter 4 in Seierstad and Sydsaeter (1986), the necessary conditions to solve this problem are the following

• 1. $\dfrac{d \mathcal{L}(V^{s}(n^{s}, R; \mathcal{M}); V^{s}_{n^{s}}(n^{s}, R; \mathcal{M}))}{d V^{s}_{n^{s}}(n^{s}, R; \mathcal{M})} = p(n^{s}) + \mu \left(\int \limits_{\underline{n}^{s}}^{\overline{n}^{s}} \int \limits_{\underline{n}^{d}}^{\kappa(n^{s})} \left[ \dfrac{c}{\pi \tau} \right] dF(n^{d},n^{s}) \right) = p(n^{s}) + \mu \left(\int \limits_{\underline{n}^{d}}^{\kappa(n^{s})} \left[ \dfrac{c}{\pi \tau} \right] f_{d}(n^{d}) dn^{d} \right) \gtreqqless 0$

• 2. $\dfrac{d p(n^{s})}{d n^{s}} = - \dfrac{d \mathcal{L}(V^{s}(n^{s}, R; \mathcal{M}); V^{s}_{n^{s}}(n^{s}, R; \mathcal{M}))}{d V^{s}(n^{s}, R; \mathcal{M})} = \int \limits_{\underline{n}^{s}}^{\overline{n}^{s}} \int \limits_{\underline{n}^{d}}^{\kappa(n^{s})} f(n^{d},n^{s}) dn^{d} dn^{s} = \int \limits_{\underline{n}^{d}}^{\kappa(n^{s})} f_{d}(n^{d}) dn^{d}$

• 3. $V^{s}(\overline{n}^{s})$ is free, implying that $p(\overline{n}^{s}) = 0$.

• 4. $\mu \ge 0$, $\mu \left( \int \limits _{\underline{n}^{s}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} c \cdot \left[ \dfrac{1-V^{s}_{n^{s}}(n^{s}, R; \mathcal {M})}{\pi \tau } \right] \mathrm{{d}}F(n^{d},n^{s}) - B \right) = 0$

Condition 1 does not depend on the control variable. This means that the solution implies the control variable takes its maximum or minimum value. Condition 2 relates to the slope of the adjoint function. This condition is positive and monotone increasing, showing that the adjoint function has a monotone-increasing slope. Condition 3 states that the value of the adjoint function in the maximum income level is zero. The last two results show that the adjoint function has a negative value and increases up to zero at $\overline{n}^{s}$.

Using condition 1 allows us to find the sign of $\mu$. The term in parentheses is strictly positive, and the adjoint function is negative or zero. Using both facts to find the sign of $\mu$ yields

$$\begin{aligned} \mu = \dfrac{- p(n^{s})}{\int \limits _{\underline{n}^{s}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} \left[ \dfrac{c}{\pi \tau } \right] \mathrm{{d}}F(n^{d},n^{s})} > 0 \end{aligned}$$

From condition 4, $\mu \ge 0$, and the above inequality states that $\mu = 0$ only at $p(\overline{n}^{s})$. Now, let us use the second part of condition 4 to find a solution. Let us define the income that equalizes the budget for the IRS with the auditing cost as $w^{*}$. From Lemma 2.1, audits must be non-increasing in self-employed income, and from condition 1, the control variable takes only its extreme values. Those facts result in for $n^{s} \le \hat{w}$, the control function takes the minimum value, or the audit function reaches its maximum value. For the rest of the earnings, $n^{s} > \hat{w}$, the budget is equal to the audit cost, and the control variable takes its maximum variable, or the audit function is equal to zero. Therefore, for $n^{s} \le \hat{w}$, the solution implies $V^{s}_{n^{s}}(n^{s}, R; \mathcal {M}) = 1-\tau$ or $\alpha (n^{s}) = \frac{1}{\pi }$, otherwise $V^{s}_{n^{s}}(n^{s}, R; \mathcal {M}) = 1$ or $\alpha (n^{s}) = 0$. The following equation characterizes $w^{*}$.

$$\begin{aligned} \int \limits _{\underline{n}^{s}}^{w^{\ast}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} f(n^{d},n^{s}) dn^{d} dn^{s} = \dfrac{B \pi }{c} \end{aligned}$$

Appendix F Proof of Corollary 1

If the audit cost is zero ($c=0$), the cost of auditing is smaller than the budget for the IRS. Hence, it is always optimal to choose the maximum possible limit to audit, or $w^{*} = \overline{n}^{s}$. Another way to see this is the following. If $c = 0$, the ratio $\frac{B \pi }{c} = \infty$ for $B>0$. Hence, the term $\int \limits _{\underline{n}^{s}}^{w^{\ast}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} f(n^{d},n^{s}) dn^{d} dn^{s}$ could be as big as possible. For the case of $B=0$, the audit cost is equal to the budget for every possible level of $w^{*}$, and $w^{*} = \overline{n}^{s}$ maximize the expected revenue, being the solution. Therefore, for the case of a zero audit cost, the solution is to audit all self-employment sector.

Appendix G Derivation of the government policy

1.1 G.1 Optimal BLS rule

The first-order condition of Eq. (4) with respect to R is

$$\begin{aligned}&\int \limits _{\mathcal {O}^{d}(\mathcal {M})} \dfrac{G^{\prime } \left( U^{d}(n^{d}, R) \right) }{\delta } \phi ^{\prime }(R) \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\mathcal {O}^{s}(\mathcal {M})} \dfrac{G^{\prime } \left( V^{s}(n^{s}, R; \mathcal {M})\right) }{\delta } \phi ^{\prime }(R) \mathrm{{d}}F(n^{d},n^{s}) = 1 \end{aligned}$$

Using the definition of $g(n^{d}) = \frac{G^{\prime } \left( U^{d}(n^{d}, R) \right) }{\delta }$ and $g(n^{s}) = \frac{G^{\prime } \left( V^{s}(n^{s}, R; \mathcal {M})\right) }{\delta }$, and factorizing by $\phi ^{\prime }(R)$ yields

$$\begin{aligned} \phi ^{\prime } (R) \left( \int \limits _{\mathcal {O}^{d}(\mathcal {M})} g(n^{d}) \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\mathcal {O}^{s}(\mathcal {M})} g(n^{s}) \mathrm{{d}}F(n^{d},n^{s})\right) = 1 \end{aligned}$$

1.2 G.2 Optimal marginal tax rate

The first-order condition of Eq. (4) with respect to $\tau$ is

$$\begin{aligned}&\int \limits _{\underline{n}^{s}}^{w^{*}} (-\kappa ^{\prime }_{\tau }) G \left( U^{d}(\kappa (n^{s}), R) \right) f_{s\mid d}(n^{s} \mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} + \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits _{\kappa (n^{s})}^{\overline{n}^{d}} G^{\prime }_{U^{d}} \dfrac{\mathrm{{d}} C_{d}}{\mathrm{{d}} \tau } \mathrm{{d}}F(n^{d}, n^{s}) \\&+ \hat{w}^{\prime }_{\tau } \int \limits _{\kappa (\hat{w})}^{\overline{w}^{d}} G \left( U^{d}(n^{d}, R) \right) f_{s\mid d}(\hat{w} \mid n^{d}) f_{d}(n^{d})dn^{d} + \int \limits _{w^{*}}^{\hat{w}} (-\kappa ^{\prime }_{\tau }) G \left( U^{d}(\kappa (n^{s}), R) \right) f_{s\mid d}(n^{s} \mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} \\&+ \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\kappa (n^{s})}^{\overline{n}^{d}} G^{\prime }_{U^{d}} \dfrac{\mathrm{{d}} C_{d}}{\mathrm{{d}} \tau } \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\underline{n}^{s}}^{w^{*}} \kappa ^{\prime }_{\tau } G \left( V^{s}(n^{s}, R ;\mathcal {M})\right) f_{s\mid d}(n^{s} \mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} \\&+ \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} G^{\prime }_{V^{s}} \dfrac{\mathrm{{d}} C_{s}}{d \tau } \mathrm{{d}}F(n^{d},n^{s}) + \hat{w}^{\prime }_{\tau } \int \limits ^{\kappa (\hat{w})}_{\underline{n}^{d}} G \left( V^{s}(\hat{w}, R; \mathcal {M})\right) f_{s\mid d}(\hat{w} \mid n^{d}) f_{d}(n^{d})dn^{d} \\&+ \int \limits _{w^{*}}^{\hat{w}} \kappa ^{\prime }_{\tau } G \left( V^{s}(n^{s}, R; \mathcal {M})\right) f_{s\mid d}(n^{s} \mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} + \int \limits _{w^{*}}^{\hat{w}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} G^{\prime }_{V^{s}} \dfrac{\mathrm{{d}} C_{s}}{\mathrm{{d}} \tau } \mathrm{{d}}F(n^{d},n^{s}) \\&+ (- \hat{w}^{\prime }_{\tau }) \int \limits ^{\kappa (\hat{w})}_{\underline{n}^{d}} G \left( V^{s}(\hat{w},R; \mathcal {M})\right) f_{s\mid d}(\hat{w} \mid n^{d}) f_{d}(n^{d})dn^{d} + \int \limits _{\hat{w}}^{\overline{n}^{s}} \kappa ^{\prime }_{\tau } G \left( V^{s}(n^{s},R; \mathcal {M})\right) f_{s\mid d}(n^{s} \mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} \\&+ \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} G^{\prime }_{V^{s}} \dfrac{\mathrm{{d}} C_{s}}{\mathrm{{d}} \tau } \mathrm{{d}}F(n^{d},n^{s}) - \delta \left\{ - \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits _{\kappa (n^{s})}^{\overline{n}^{d}} n^{d} \mathrm{{d}}F(n^{d},n^{s}) - \tau \int \limits _{\underline{n}^{s}}^{w^{*}} (- \kappa ^{\prime }_{\tau }) \kappa (n^{s}) f_{s\mid d}(n^{s} \mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} \right. \\&- \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\kappa (n^{s})}^{\overline{n}^{d}} n^{d} \mathrm{{d}}F(n^{d},n^{s}) - \tau \hat{w}^{\prime }_{\tau } \int \limits _{\kappa (\hat{w})}^{\overline{n}^{d}} n^{d} f_{s\mid d}(\hat{w} \mid n^{d}) f_{d}(n^{d})dn^{d} - \tau \int \limits _{w^{*}}^{\hat{w}} (-\kappa ^{\prime }_{\tau }) \kappa (n^{s}) f_{s\mid d}(n^{s} \mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} \\&- \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} n^{s} \mathrm{{d}}F(n^{d},n^{s}) - \tau \int \limits _{\underline{n}^{s}}^{w^{*}} n^{s} \kappa ^{\prime }_{\tau } f_{s\mid d}(n^{s} \mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} - \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*} \mathrm{{d}}F(n^{d},n^{s}) \\&- \tau \hat{w}^{\prime }_{\tau } \int \limits _{\underline{n}^{d}}^{\kappa (\hat{w})} w^{*} f_{s\mid d}(\hat{w} \mid n^{d}) f_{d}(n^{d})dn^{d} - \tau \int \limits _{w^{*}}^{\hat{w}} w^{*} \kappa ^{\prime }_{\tau } f_{s\mid d}(n^{s} \mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} - \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*} \mathrm{{d}}F(n^{d},n^{s}) \\&\left. - \tau (-\hat{w}^{\prime }_{\tau }) \int \limits _{\underline{n}^{d}}^{\kappa (\hat{w})} w^{*} f_{s\mid d}(\hat{w} \mid n^{d}) f_{d}(n^{d})dn^{d} - \tau \int \limits _{\hat{w}}^{\overline{n}^{s}} w^{*} \kappa ^{\prime }_{\tau } f_{s\mid d}(n^{s} \mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} \right\} = 0 \end{aligned}$$

where $x^{\prime }_{y} = \frac{\partial x}{\partial y}$ except for densities f. Some result are used to simplify the above expression. First, using the definition of social valuation of the agent’s consumption, $g(n^{s}) = \frac{G_{V^{s}}}{\delta }$ and $g(n^{d}) = \frac{G_{U^{d}}}{\delta }$. Second, from the threshold function definition, $U^{d}\left( \kappa (n^{s}), R \right) = V^{s}(n^{s}, R; \mathcal {M})$, implying that some terms are canceling among them. Third, by assumption $\kappa (\hat{w}) = \overline{n}^{d}$, causing some integrals collapse to zero. Fourth, as a result of the audit function and the assumption over earning distributions, $\kappa ^{\prime }_{\tau } = 0$ for $n^{s} \le w^{*}$ and $n^{s} \ge \hat{w}$, resulting in some terms being equal to zero. Finally, the derivatives of consumption with respect to marginal tax in each sector are

$$\begin{aligned} \dfrac{\mathrm{{d}} C_{d}}{\mathrm{{d}} \tau } = - n^{d} , \qquad \dfrac{\mathrm{{d}} C_{s}}{\mathrm{{d}} \tau } = \left\{ \begin{array}{cc} - n^{s} &{} \,\,{\hbox {if}} \,\, n^{s} < w^{*} \\ - w^{*} &{} \,\,{\hbox {if}} \,\, w^{*} \le n^{s} \end{array} \right. \end{aligned}$$

Using these facts and doing algebra yields

$$\begin{aligned}&\int \limits _{\mathcal {O}^{d}(\mathcal {M})} g(n^{d}) (-n^{d}) \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} g(n^{s}) (-n^{s}) \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{w^{*}}^{\hat{w}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} g(n^{s}) (- w^{*}) \mathrm{{d}}F(n^{d},n^{s}) \\&+ \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} g(n^{s}) (- w^{*}) \mathrm{{d}}F(n^{d},n^{s}) = - \int \limits _{\mathcal {O}^{d}(\mathcal {M})} n^{d} \mathrm{{d}}F(n^{d},n^{s}) + \tau \int \limits _{w^{*}}^{\hat{w}} \kappa ^{\prime }_{\tau } \kappa (n^{s}) f_{s\mid d}(n^{s} \mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} \\&- \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} n^{s} \mathrm{{d}}F^{s}(n^{d},n^{s}) - \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*} \mathrm{{d}}F(n^{d},n^{s}) - \tau \int \limits _{w^{*}}^{\hat{w}} w^{*} \kappa ^{\prime }_{\tau } f_{s\mid d}(n^{s} \mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} - \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*} \mathrm{{d}}F(n^{d},n^{s}) \end{aligned}$$

By making the replacement $\kappa (n^{s}) - w^{*} = \dfrac{n^{s} - w^{*}}{1-\tau }$ for $n^{s} \in [w^{*}, \hat{w}]$, and doing algebra, the optimal marginal tax formula is

$$\begin{aligned}&\dfrac{\tau }{1-\tau } \int \limits _{w^{*}}^{\hat{w}} (n^{s}-w^{*}) \kappa _{\tau }^{\prime }(n^{s}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} = \int \limits _{\mathcal {O}^{d}(\mathcal {M})} (1-g(n^{d})) n^{d} \mathrm{{d}}F(n^{d},n^{s}) \\&+ \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} (1-g(n^{s})) n^{s}\mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} (1-g(n^{s})) w^{*} \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} (1-g(n^{s})) w^{*} dF(n^{d},n^{s}) \end{aligned}$$

1.3 G.3 Optimal budget for the IRS

The first-order condition of Eq. (4) with respect to B is

$$\begin{aligned}&w^{*\prime }_{B} \int \limits _{\kappa (w^{*})}^{\overline{n}^{d}} G \left( U^{d}(n^{d}, R) \right) f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d})dn^{d}+ \int \limits _{\underline{n}^{s}}^{w^{*}} (\kappa ^{\prime }_{B}) G \left( U^{d}(\kappa (n^{s}), R) \right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} \\&+ (- w^{*\prime }_{B}) \int \limits _{\kappa (w^{*})}^{\overline{n}^{d}} G \left( U^{d}(n^{d}, R) \right) f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d})dn^{d} + \hat{w}^{\prime }_{B} \int \limits _{\kappa (\hat{w})}^{\overline{n}^{d}} G \left( U^{d}(n^{d}, R) \right) f_{s\mid d}(\hat{w}\mid n^{d}) f_{d}(n^{d})dn^{d} \\&+ \int \limits _{w^{*}}^{\hat{w}} (-\kappa ^{\prime }_{B}) G \left( U^{d}(\kappa (n^{s}), R) \right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} + w^{*\prime }_{B} \int \limits ^{\kappa (w^{*})}_{\underline{n}^{d}} G \left( V^{s}(w^{*}, R; \mathcal {M})\right) f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d})dn^{d} \\&+ \int \limits _{\underline{n}^{s}}^{w^{*}} \kappa ^{\prime }_{B} G \left( V^{s}(n^{s}, R; \mathcal {M})\right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} + (-w^{*\prime }_{B}) \int \limits ^{\kappa (w^{*})}_{\underline{n}^{d}} G \left( V^{s}(w^{*}, R; \mathcal {M})\right) f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&+ \hat{w}^{\prime }_{B} \int \limits ^{\kappa (\hat{w})}_{\underline{n}^{d}} G \left( V^{s}(\hat{w}, R; \mathcal {M})\right) f_{s\mid d}(\hat{w}\mid n^{d}) f_{d}(n^{d}) dn^{d} + \int \limits _{w^{*}}^{\hat{w}} \kappa ^{\prime }_{B} G \left( V^{s}(n^{s}, R; \mathcal {M})\right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} \\&+ \int \limits _{w^{*}}^{\hat{w}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} G^{\prime }_{V^{s}} \dfrac{\mathrm{{d}} C_{s}}{\mathrm{{d}} B} \mathrm{{d}}F(n^{d},n^{s}) + (- \hat{w}^{\prime }_{B}) \int \limits ^{\kappa (\hat{w})}_{\underline{n}^{d}} G \left( V^{s}(\hat{w}, R; \mathcal {M})\right) f_{s\mid d}(\hat{w}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&+ \int \limits _{\hat{w}}^{\overline{n}^{s}} \kappa ^{\prime }_{B} G \left( V^{s}(n^{s}, R; \mathcal {M})\right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} + \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} G^{\prime }_{V^{s}} \dfrac{\mathrm{{d}} C_{s}}{\mathrm{{d}} B} \mathrm{{d}}F(n^{d},n^{s}) \\&- \delta \left\{ 1 - \tau w^{*\prime }_{B} \int \limits _{\kappa (w^{*})}^{\overline{n}^{d}} n^{d} f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} - \tau \int \limits _{\underline{n}^{s}}^{w^{*}} (-\kappa ^{\prime }_{B}) \kappa (n^{s}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} \right. \\&- \tau (-w^{*\prime }_{B}) \int \limits _{\kappa (w^{*})}^{\overline{n}^{d}} f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d})dn^{d} - \tau \hat{w}^{\prime }_{B} \int \limits _{\kappa (\hat{w})}^{\overline{n}^{d}} n^{d} f_{s\mid d}(\hat{w}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&- \tau \int \limits _{w^{*}}^{\hat{w}} (- \kappa ^{\prime }_{B}) \kappa (n^{s}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} - \tau w^{*\prime }_{B} \int \limits _{\underline{n}^{d}}^{\kappa (w^{*})} w^{*} f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&- \tau \int \limits _{\underline{n}^{s}}^{w^{*}} n^{s} \kappa ^{\prime }_{B} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} - \tau (- w^{*\prime }_{B}) \int \limits _{\underline{n}^{d}}^{\kappa (w^{*})} w^{*} f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&- \tau \hat{w}^{\prime }_{B} \int \limits _{\underline{n}^{d}}^{\kappa (\hat{w})} w^{*} f_{s\mid d}(\hat{w} \mid n^{d}) f_{d}(n^{d}) dn^{d} - \tau \int \limits _{w^{*}}^{\hat{w}} \kappa ^{\prime }_{B} w^{*} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} - \tau \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} \dfrac{\mathrm{{d}} w^{*}}{\mathrm{{d}} B} \mathrm{{d}}F(n^{d},n^{s}) \\&\left. - \tau (- \hat{w}^{\prime }_{B}) \int \limits _{\underline{n}^{d}}^{\kappa (\hat{w})} w^{*} f_{s\mid d}(\hat{w} \mid n^{d}) f_{d}(n^{d}) dn^{d} - \tau \int \limits _{\hat{w}}^{\overline{n}^{s}} w^{*} \kappa ^{\prime }_{B} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s}))dn^{s} - \tau \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} \dfrac{\mathrm{{d}} w^{*}}{\mathrm{{d}} B} \mathrm{{d}}F(n^{d},n^{s}) \right\} = 0 \end{aligned}$$

where $x^{\prime }_{y} = \frac{\partial x}{\partial y}$ except for densities f. Some results and assumptions are used to simplify the above equation. First, by using the social valuation of the agent’s consumption, $g(n^{s}) = \frac{G_{V^{s}}}{\delta }$ and $g(n^{d}) = \frac{G_{U^{d}}}{\delta }$. Second, by definition $U^{d}(\kappa (n^{s}), R ) = V^{s}(n^{s}, R; \mathcal {M})$, making some terms cancel among them. Third, $\kappa ^{\prime }_{B} = 0$ for $n^{s} \le w^{*}$ and $n^{s} > \hat{w}$, resulting in some terms being equal to zero. Fourth, by assumption $\kappa (\hat{w}) = \overline{n}^{d}$, collapsing some terms to zero. Finally, the derivative of consumption in the self-employment sector with respect to the budget is

$$\begin{aligned} \dfrac{\mathrm{{d}} C_{s}}{\mathrm{{d}} B} = \left\{ \begin{array}{cc} 0 &{}\,\,{\hbox { if}} \,\, n^{s} < w^{*} \\ - \tau \dfrac{\mathrm{{d}} w^{*}}{\mathrm{{d}} B} &{} \,\,{\hbox {if}} \,\, w^{*} \le n^{s} \end{array} \right. \end{aligned}$$

Using these facts and doing algebra yields

$$\begin{aligned} 1 =&\, \tau \int \limits _{w^{*}}^{\hat{w}} (w^{*} - \kappa (n^{s}))\dfrac{\mathrm{{d}} w^{*}}{\mathrm{{d}} B} \kappa ^{\prime }_{B} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \\&+ \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} \left( 1-g(n^{s}) \right) \tau \dfrac{\mathrm{{d}} w^{*}}{\mathrm{{d}} B} \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} \left( 1-g(n^{s}) \right) \tau \dfrac{\mathrm{{d}} w^{*}}{\mathrm{{d}} B} \mathrm{{d}}F(n^{d},n^{s}) \end{aligned}$$

Appendix H Differential taxation

The Lagrangian of the government problem is

$$\begin{aligned} \mathcal {L} =&\int \limits _{\mathcal {O}^{d}(\mathcal {M})} G \left( U^{d}(n^{d}, R) \right) \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\mathcal {O}^{s}(\mathcal {M})} G \left( V^{s}(n^{s}, R; \mathcal {M})\right) \mathrm{{d}}F(n^{d},n^{s}) \\&- \delta \left\{ B + R - t_{d} \int \limits _{\mathcal {O}^{d}(\mathcal {M})} n^{d} \mathrm{{d}}F(n^{d},n^{s}) - t_{s} \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} n^{s} \mathrm{{d}}F(n^{d},n^{s}) - t_{s} \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*} \mathrm{{d}}F(n^{d},n^{s}) \right. \\&\left. - t_{s} \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*} \mathrm{{d}}F(n^{d},n^{s}) \right\} \end{aligned}$$

For simplicity, the notation over the Lagrange multiplier for the budget constraint is maintained. The first-order condition regarding $t_{d}$ is

$$\begin{aligned}&w^{*\prime }_{t_{d}} \int \limits _{\kappa (w^{*})}^{\overline{n}^{d}} G \left( U^{d}(n^{d}, R) \right) f_{s\mid d}(w^{*} \mid n^{d}) f_{d}(n^{d}) dn^{d} + \int \limits _{\underline{n}^{s}}^{w^{*}} (-\kappa ^{\prime }_{t_{d}}) G \left( U^{d}(\kappa (n^{s}), R) \right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \\&+ \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits _{\kappa (n^{s})}^{\overline{n}^{d}} G^{\prime }_{U^{d}} \dfrac{\mathrm{{d}} C_{d}}{\mathrm{{d}} t_{d}} \mathrm{{d}}F(n^{d},n^{s}) + (- w^{*\prime }_{t_{d}}) \int \limits _{\kappa (w^{*})}^{\overline{n}^{d}} G \left( U^{d}(n^{d}, R) \right) f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&+\hat{w}^{\prime }_{t_{d}} \int \limits _{\kappa (\hat{w})}^{\overline{n}^{d}} G \left( U^{d}(n^{d}, R) \right) f_{s\mid d}(\hat{w}\mid n^{d}) f_{d}(n^{d}) dn^{d} + \int \limits _{w^{*}}^{\hat{w}} (-\kappa ^{\prime }_{t_{d}}) G \left( U^{d}(\kappa (n^{s}), R) \right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \\&+ \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\kappa (n^{s})}^{\overline{n}^{d}} G^{\prime }_{U^{d}} \dfrac{\mathrm{{d}} C_{d}}{\mathrm{{d}} t_{d}} \mathrm{{d}}F(n^{d},n^{s}) + w^{*\prime }_{t_{d}} \int \limits _{\underline{n}^{d}}^{\kappa (w^{*})} G\left( V^{s}(w^{*}, R; \mathcal {M})\right) f_{s\mid d}(w^{*} \mid n^{d}) f_{d}(n^{d}) dn^{d} \\&+ \int \limits _{\underline{n}^{s}}^{w^{*}} \kappa ^{\prime }_{t_{d}} G \left( V^{s}(n^{s}, R; \mathcal {M})\right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} + \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} G^{\prime }_{V^{s}} \dfrac{\mathrm{{d}} C_{s}}{\mathrm{{d}} t_{d}} \mathrm{{d}}F(n^{d},n^{s}) \\&+ (- w^{*\prime }_{t_{d}}) \int \limits _{\underline{n}^{d}}^{\kappa (w^{*})} G\left( V^{s}(w^{*}, R; \mathcal {M})\right) f_{s\mid d}(w^{*} \mid n^{d}) f_{d}(n^{d}) dn^{d} + \hat{w}^{\prime }_{t_{d}} \int \limits ^{\kappa (\hat{w})}_{\underline{n}^{d}} G \left( V^{s}(\hat{w}, R; \mathcal {M})\right) f_{s\mid d}(\hat{w}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&+ \int \limits _{w^{*}}^{\hat{w}} \kappa ^{\prime }_{t_{d}} G \left( V^{s}(n^{s}, R; \mathcal {M})\right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} + \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} G^{\prime }_{V^{s}} \dfrac{\mathrm{{d}} C_{s}}{\mathrm{{d}} t_{d}} dF(n^{d},n^{s}) + \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} G^{\prime }_{V^{s}} \dfrac{\mathrm{{d}} C_{s}}{\mathrm{{d}} t_{d}} \mathrm{{d}}F(n^{d},n^{s}) \\&+ (-\hat{w}^{\prime }_{t_{d}}) \int \limits ^{\kappa (\hat{w})}_{\underline{n}^{d}} G \left( V^{s}(\hat{w}, R; \mathcal {M})\right) f_{s\mid d}(\hat{w} \mid n^{d}) f_{d}(n^{d}) dn^{d} + \int \limits _{\hat{w}}^{\overline{n}^{s}} \kappa ^{\prime }_{t_{d}} G \left( V^{s}(n^{s}, R; \mathcal {M})\right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \\&- \delta \left\{ - \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\kappa (n^{s})}^{\overline{n}^{d}} n^{d} \mathrm{{d}}F(n^{d},n^{s}) - t_{d} w^{*\prime }_{t_{d}} \int \limits _{\kappa (w^{*})}^{\overline{n}^{d}} n^{d} f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} - t_{d} \int \limits _{\underline{n}^{s}}^{w^{*}} (-\kappa ^{\prime }_{t_{d}}) \kappa (n^{s}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \right. \\&- \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\kappa (n^{s})}^{\overline{n}^{d}} n^{d} \mathrm{{d}}F(n^{d},n^{s}) - t_{d} (- w^{*\prime }_{t_{d}}) \int \limits _{\kappa (w^{*})}^{\overline{n}^{d}} n^{d} f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} - t_{d} (-\hat{w}^{\prime }_{t_{d}}) \int \limits _{\kappa (\hat{w})}^{\overline{n}^{d}} n^{d} f_{s\mid d}(\hat{w}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&- t_{d} \int \limits _{w^{*}}^{\hat{w}} (-\kappa ^{\prime }_{t_{d}}) \kappa (n^{s}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} - t_{s} (w^{*\prime }_{t_{d}}) \int \limits _{\underline{n}^{d}}^{\kappa (w^{*})} w^{*} f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&- t_{s} \int \limits _{\underline{n}^{s}}^{w^{*}} \kappa ^{\prime }_{t_{d}} n^{s} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} - t_{s} \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*\prime }_{t_{s}} dF(n^{d},n^{s}) - t_{s} (- w^{*\prime }_{t_{d}}) \int \limits _{\underline{n}^{s}}^{\kappa (w^{*})} w^{*} f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&- t_{s} \hat{w}^{\prime }_{t_{d}} \int \limits _{\underline{n}^{d}}^{\kappa (\hat{w})} w^{*} f_{s\mid d}(\hat{w}\mid n^{d}) f_{d}(n^{d}) dn^{d} - t_{s} \int \limits _{w^{*}}^{\hat{w}} \kappa ^{\prime }_{t_{d}} w^{*} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} - t_{s} \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{s}}^{\kappa (n^{s})} w^{*\prime }_{t_{d}} \mathrm{{d}}F(n^{d},n^{s}) \\&\left. - t_{s} (-\hat{w}^{\prime }_{t_{d}}) \int \limits _{\underline{n}^{s}}^{\kappa (\hat{w})} w^{*} f_{s\mid d}(\hat{w}\mid n^{d}) f_{d}(n^{d}) dn^{d} - t_{s} \int \limits _{\hat{w}}^{\overline{n}^{s}} \kappa ^{\prime }_{t_{d}} w^{*} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \right\} = 0 \end{aligned}$$

where $x^{\prime }_{y} = \frac{\partial x}{\partial y}$ except for densities f. Some assumptions and results are used to simplify the above equation. First, the social valuation of the agent’s consumption is $g(n^{s}) = \frac{G_{V^{s}}}{\delta }$ and $g(n^{d}) = \frac{G_{U^{d}}}{\delta }$. Second, by definition $U^{d}(\kappa (n^{s}), R) = V^{s}(n^{s}, R; \mathcal {M})$, making some terms cancel among them. Third, by definition $\kappa (\hat{w}) = \overline{n}^{d}$, collapsing some terms to zero. Fourth, from the threshold function, $\kappa ^{\prime }_{t_{d}} = \frac{\kappa (n^{s})}{(1-t_{d})}$ for $n^{s} \le w^{*}$, $\kappa ^{\prime }_{t_{d}} = -\dfrac{t_{s} w^{*\prime }_{t_{d}}}{1 - t_{d}} + \frac{\kappa (n^{s})}{(1-t_{d})}$ for $w^{*} < n^{s} \le \hat{w}$, and $\kappa ^{\prime }_{t_{d}} = 0$ for $n^{s} > \hat{w}$. Fifth, $\frac{\mathrm{{d}} C_{d}}{\mathrm{{d}} t_{d}} = - n^{d}$. Sixth, $\frac{\mathrm{{d}} C_{s}}{\mathrm{{d}} t_{d}} = 0$ for $n^{s} \le w^{*}$, and $\frac{\mathrm{{d}} C_{s}}{\mathrm{{d}} t_{d}} = -\dfrac{\mathrm{{d}} w^{*}}{\mathrm{{d}} t_{d}} t_{s}$ for $w^{*}< n^{s} \le \hat{w}$. Using these facts and doing algebra yields

$$\begin{aligned} t_{d} =&\dfrac{\overbrace{\left( \int \limits _{\mathcal {O}^{d}(\mathcal {M})} (1-g(n^{d})) n^{d} \mathrm{{d}}F(n^{d},n^{s}) \right) }^{A} (1 - t_{d}) }{\underbrace{\int \limits _{\underline{n}^{s}}^{w^{*}} \kappa (n^{s})^{2} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} + \int \limits _{w^{*}}^{\hat{w}} \left( \kappa (n^{s}) - t_{s} w^{*\prime }_{t_{d}} \right) \kappa (n^{s}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s}}_{B}} \\&+ \dfrac{\overbrace{w^{*\prime }_{t_{d}}\left( \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} (1-g(n^{s})) \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} (1-g(n^{s})) \mathrm{{d}}F(n^{d},n^{s}) \right) }^{D} (1 - t_{d}) t_{s} }{\underbrace{\int \limits _{\underline{n}^{s}}^{w^{*}} \kappa (n^{s})^{2} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} + \int \limits _{w^{*}}^{\hat{w}} \left( \kappa (n^{s}) - t_{s} w^{*\prime }_{t_{d}} \right) \kappa (n^{s}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s}}_{B}} \\&+ \dfrac{t_{s} \overbrace{ \left( \int \limits _{\underline{n}^{s}}^{w^{*}} n^{s} \kappa (n^{s}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} + \int \limits _{w^{*}}^{\hat{w}} w^{*} \left( \kappa (n^{s}) - t_{s} w^{*\prime }_{t_{d}} \right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \right) }^{C}}{\underbrace{\int \limits _{\underline{n}^{s}}^{w^{*}} \kappa (n^{s})^{2} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} + \int \limits _{w^{*}}^{\hat{w}} \left( \kappa (n^{s}) - t_{s} w^{*\prime }_{t_{d}} \right) \kappa (n^{s}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s}}_{B}} \end{aligned}$$

Some parts are denoted with letters to simplify the exposure. The optimal tax rate is

$$\begin{aligned} t_{d} = \dfrac{A + t_{s} D}{A + B + t_{s} D} + t_{s} \dfrac{C}{A + B + t_{s} D} \end{aligned}$$

In order to fulfill the limited liability constraint, the marginal tax rate in the wage-earner sector cannot be higher than one. Thus, it is necessary that $t_{s} \le B/C$.

Now, the tax rate in the self-employed sector will be obtained. The first-order condition with respect to $t_{s}$ is

$$\begin{aligned}&w^{*\prime }_{t_{s}} \int \limits _{\kappa (w^{*})}^{\overline{n}^{d}} G\left( U^{d}(n^{d}, R) \right) f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} + \int \limits _{\underline{n}^{s}}^{w^{*}} (-\kappa ^{\prime }_{t_{s}}) G \left( U^{d}(\kappa (n^{s}), R) \right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \\&+ (-w^{*\prime }_{t_{s}}) \int \limits _{\kappa (w^{*})}^{\overline{n}^{d}} G\left( U^{d}(n^{d}, R) \right) f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} + \hat{w}^{\prime }_{t_{s}} \int \limits _{\kappa (\hat{w})}^{\overline{n}^{d}} G \left( U^{d}(n^{d}, R) \right) f_{s\mid d}(\hat{w}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&+ \int \limits _{w^{*}}^{\hat{w}} (-\kappa ^{\prime }_{t_{s}}) G \left( U^{d}(\kappa (n^{s}), R) \right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} + w^{*\prime }_{t_{s}} \int \limits _{\underline{n}^{d}}^{\kappa (w^{*})} G \left( V^{s}(w^{*}, R; \mathcal {M}) \right) f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&+ \int \limits _{\underline{n}^{s}}^{w^{*}} \kappa ^{\prime }_{t_{s}} G \left( V^{s}(n^{s}, R; \mathcal {M})\right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} + \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} G^{\prime }_{V^{s}} \dfrac{\mathrm{{d}} C_{s}}{\mathrm{{d}} t_{s}} \mathrm{{d}}F(n^{d},n^{s}) \\&+(- w^{*\prime }_{t_{s}}) \int \limits _{\underline{n}^{d}}^{\kappa (w^{*})} G \left( V^{s}(w^{*}, R; \mathcal {M}) \right) f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} + \hat{w}^{\prime }_{t_{s}} \int \limits ^{\kappa (\hat{w})}_{\underline{n}^{d}} G \left( V^{s}(\hat{w}, R; \mathcal {M})\right) f_{s\mid d}(\hat{w}\mid n^{s}) f_{d}(n^{s}) dn^{s} \\&+ \int \limits _{w^{*}}^{\hat{w}} \kappa ^{\prime }_{t_{s}} G \left( V^{s}(n^{s}, R; \mathcal {M})\right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} + \int \limits _{w^{*}}^{\hat{w}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} G^{\prime }_{V^{s}} \dfrac{\mathrm{{d}} C_{s}}{\mathrm{{d}} t_{s}} \mathrm{{d}}F{d}(n^{d},n^{s}) + \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} G^{\prime }_{V^{s}} \dfrac{\mathrm{{d}} C_{s}}{\mathrm{{d}} t_{s}} \mathrm{{d}}F(n^{d},n^{s}) \\&+ (-\hat{w}^{\prime }_{t_{s}}) \int \limits ^{\kappa (\hat{w})}_{\underline{n}^{d}} G \left( V^{s}(\hat{w}, R; \mathcal {M})\right) f_{s\mid d}(\hat{w}\mid n^{d}) f_{d}(n^{d}) dn^{d} + \int \limits _{\hat{w}}^{\overline{n}^{s}} \kappa ^{\prime }_{t_{s}} G \left( V^{s}(n^{s}, R; \mathcal {M})\right) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \\&- \delta \left\{ - t_{d} w^{*\prime }_{t_{s}} \int \limits _{\kappa (w^{*})}^{\overline{n}^{d}} n^{d} f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} - t_{d} \int \limits _{\underline{n}^{s}}^{w^{*}} (-\kappa ^{\prime }_{t_{s}}) \kappa (n^{s}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \right. \\&- t_{d} (-w^{*\prime }_{t_{s}}) \int \limits _{\kappa (w^{*})}^{\overline{n}^{d}} n^{d} f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} - t_{d} \hat{w}^{\prime }_{t_{s}} \int \limits _{\kappa (\hat{w})}^{\overline{n}^{d}} n^{d} f_{s\mid d}(\hat{w} \mid n^{d}) f_{d}(n^{d}) dn^{d} \\&- t_{d} \int \limits _{w^{*}}^{\hat{w}} \kappa ^{\prime }_{t_{s}} \kappa (n^{s}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} - \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} n^{s} \mathrm{{d}}F(n^{d},n^{s}) - t_{s} w^{*\prime }_{t_{s}} \int \limits _{\underline{n}^{d}}^{\kappa (w^{*})} w^{*} f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&- t_{s} \int \limits _{\underline{n}^{s}}^{w^{*}} \kappa ^{\prime }_{t_{s}} n^{s} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} - \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*} dF(n^{d},n^{s}) - t_{s} \int \limits _{w^{*}}^{\hat{w}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*\prime }_{t_{s}} \mathrm{{d}}F(n^{d},n^{s}) \\&- t_{s} (-w^{*\prime }_{t_{s}}) \int \limits _{\underline{n}^{d}}^{\kappa (w^{*})} w^{*} f_{s\mid d}(w^{*}\mid n^{d}) f_{d}(n^{d}) dn^{d} - t_{s} \hat{w}^{\prime }_{t_{s}} \int \limits _{\underline{n}^{d}}^{\hat{w}} w^{*} f_{s\mid d}(\hat{w}\mid n^{d}) f_{d}(n^{d}) dn^{d} \\&- t_{s} \int \limits _{w^{*}}^{\hat{w}} \kappa ^{\prime }_{t_{s}} w^{*} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} - \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*} \mathrm{{d}}F(n^{d},n^{s}) - t_{s} \int \limits _{\hat{w}}^{\overline{n}^{s}} \int \limits _{\underline{n}^{d}}^{\kappa (n^{s})} w^{*\prime }_{t_{s}} dF(n^{d},n^{s}) \\&\left. - t_{s} (-\hat{w}^{\prime }_{t_{s}}) \int \limits _{\underline{n}^{d}}^{\kappa (\hat{w})} w^{*} f_{s\mid d}(\hat{w} \mid n^{d}) f_{d}(n^{d}) dn^{d} - t_{s} \int \limits _{\hat{w}}^{\overline{n}^{s}} \kappa ^{\prime }_{t_{s}} w^{*} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \right\} = 0 \end{aligned}$$

The same simplification steps used before are also used now. Applying those steps and doing algebra yields

$$\begin{aligned} t_{s} =&\dfrac{\overbrace{ \left[ \int \limits _{\underline{n}^{s}}^{w^{*}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} (1-g(n^{s})) n^{s} \mathrm{{d}}F(n^{d},n^{s}) + \int \limits _{w^{*}}^{\overline{n}^{s}} \int \limits ^{\kappa (n^{s})}_{\underline{n}^{d}} (1- g(n^{s})) (w^{*} + t_{s} w^{*\prime }_{t_{s}}) \mathrm{{d}}F(n^{d},n^{s}) \right] }^{E} (1- t_{d})}{\underbrace{\int \limits _{\underline{n}^{s}}^{w^{*}} (n^{s})^{2} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} + \int \limits ^{\hat{w}}_{w^{*}} w^{*} (w^{*} + t_{s} w^{*\prime }_{t_{s}}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s}}_{F}} \\&+ \dfrac{t_{d} \overbrace{ \left( \int \limits _{\underline{n}^{s}}^{w^{*}} \kappa (n^{s}) n^{s} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} + \int \limits ^{\hat{w}}_{w^{*}} \kappa (n^{s}) (w^{*} + t_{s} w^{*\prime }_{t_{s}}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} \right) }^{C}}{\underbrace{\int \limits _{\underline{n}^{s}}^{w^{*}} (n^{s})^{2} f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s} + \int \limits ^{\hat{w}}_{w^{*}} w^{*} (w^{*} + t_{s} w^{*\prime }_{t_{s}}) f_{s\mid d}(n^{s}\mid \kappa (n^{s})) f_{d}(\kappa (n^{s})) dn^{s}}_{F}} \end{aligned}$$

As before, some parts are denoted with letters to simplify the explanation. Note that the term C is the same as in the derivation of $t_{d}$. Rearranging terms

$$\begin{aligned} t_{s} = \dfrac{E}{F} - t_{d} \dfrac{(E - C)}{F} \end{aligned}$$

In this case, $t_{s} \le 1$ is needed to fulfill the requirement for a direct incentive-compatible mechanism. Applying this condition to the above equation yields

$$\begin{aligned} \dfrac{E - F}{E - C} \le t_{d} \le 1 \Rightarrow F \ge C \end{aligned}$$

Let us analyze each possible combination of the marginal tax rates and see if the above conditions hold. First, if $t_{s} < t_{d}$, we have $\kappa (n^{s}) > n^{s}$ for $n^{s} \le \hat{w}$, hence $C > F$, producing a contradiction. Therefore, $t_{s} < t_{d}$ is not a solution. Second, if $t_{s} = t_{d}$, it produces that $\kappa (n^{s}) = n^{s}$ for $n^{s} \le w^{*}$ and $\kappa (n^{s}) > n^{s}$ for $w^{*} \le n^{s} \le \hat{w}$, resulting in $C > F$. Therefore, $t_{s} = t_{d}$ cannot be a solution either. Third, if $t_{s} > t_{d}$, it makes $\kappa (n^{s}) < n^{s}$ for $n^{s} \le w^{*}$ and $\kappa (n^{s}) \lesseqgtr n^{s}$ for $w^{*} \le n^{s} \le \hat{w}$, resulting in $F \ge C$ for some cases. Therefore, the only possible solution for differential taxation is $t_{s} > t_{d}$.