Competitive wages and tax evasion in a Cournot duopoly

Abstract

In a Cournot duopoly with indirect tax evasion, this paper counter-intuitively shows that, in the presence of positive competitive wages, a higher indirect taxation may increase expected profits. This result is likely to occur if the market size (or alternatively, if the cost pressure exerted by wages) is adequately large and the detection probability is not too high, and it is equivalent irrespective of firms optimally choosing either the tax base to disclose to the tax authority or the amount of evasion tout-court.

1 Introduction

Within public economics, the issue of tax evasion has never been more prominent than it currently is. In the last decades, at least since the pioneering work of Allingham and Sandmo (1972), the economic literature on tax evasion has developed as a branch of the public finance. However, scholars have mainly focused on individuals’ direct taxes, while the research on indirect taxes lags behind¹ despite their increasing relevance in the public tax revenue as well as in tax evasion. Merely limiting to the sales tax examined in this paper, data reveal that the revenues raised from general consumption taxes are about 30% of total tax revenues (with the VAT representing 20.2% of them) in the OECD countries (OECD, 2022) and 26.8% for European Union member countries in 2020 (European Commission, 2022a).

VAT evasion has impressive dimensions; therefore, it represents a huge public finance worry (Keen & Smith, 2006). The European Commission (2022b) has pointed out that—mostly because of tax evasion—the VAT gap of the EU27 in 2019 and 2020 is, as a percentage of the VAT total tax liability, on average 9.1% and 6.9% for the median country; however, those figures range from 35.7% for Romania to 1.3% of Finland.

Moreover, the literature investigating the incentives for tax-evading firms under different market structures has been prevalently concentrated on perfect competition (Cremer & Gahvari, 1992, 1993, 1999; Hashimzade et al., 2010; Panteghini, 2000; Virmani, 1989), and on monopoly (Kreutzer & Lee, 1986, 1988; Lee, 1998; Marrelli, 1984; Wang, 1990; Wang & Conant, 1988; Yaniv, 1996). Oligopolistic markets, like those considered in this work, have been investigated by Marrelli and Martina (1988), Goerke and Runkel (2006, 2011), Bayer and Cowell (2009), Besfamille et al., (2009a, 2009b) and Fanti and Buccella (2021, 2022).

In particular, Marrelli and Martina (1988) build a model in which firms choose tax evasion in terms of taxes not paid. In case of detection, firms incur a fine that includes the tax plus the tax evaded which is multiplied for a penalty rate larger than one. The main result is that the more markets are competitive, the lower is the amount of the tax evaded, both in symmetric and asymmetric duopolies with non-excessive differences in production costs.

Instead, Goerke and Runkel (2011) assume that firms evade in terms of undeclared sales; the fine that tax authorities impose is an increasing, convex function of revenues evaded. The authors conclude that no clear-cut relation exists between market competition and evasion activities: positive if demand is inelastic, negative if elastic.

In line with the methodology of Goerke and Runkel (2011), Fanti and Buccella (2021, 2022) further investigate the nexus between market competition and the tax evasion of firms in a duopoly with differentiated goods under both Cournot and Bertrand competition. Their key result is to show that a negative or a positive relation between competition and tax activities depends on the source of the competitive pressure (that is, a marginal cost increase, higher product substitutability or a change in the competition mode) and the pre-existing level of competition.²

This paper focuses on the interaction between the labour market (labour as input), in our context assumed to be perfectly competitive (wages as given), and product market with quantity competition. Previous contributions have also analyzed the interaction between the inputs markets and the Cournot equilibrium, see inter alias Szidarovszky and Yakowitz (1977), Chang and Tremblay (1991), Okuguchi (1998), Chen and Zhao (2014), among others.

Because the link between tax evasion—particularly as regards the indirect taxation—and labour markets has not been so far explored in depth in oligopolistic contexts, the aim of this paper is to contribute to this discussion in this precise field of the public finance literature. We present a stylized model in which firms compete in output under the burden of a sales tax, taking as given labour wages, extending the economic analysis of tax compliance in a simple one-stage game. The main findings are as follows.

First, provided that certain conditions are satisfied, in the presence of strictly positive competitive wage rates, high indirect tax rates may, rather counter-intuitively, increase profits. Second, the interesting finding of the relation “more taxation-higher profits” is more likely obtained if the market size is sufficiently large and the likelihood of the detection probability is not too high. The driving force of this results is that, when the cost pressure exerted by wages is adequately high, higher tax rates intuitively reduce the firms’ sales declaration.

The remainder of the paper is organized as follows. Section 2 presents the model and characterizes the market equilibrium and the tax effects in the case of competitive labour market. Section 3 shows an equivalence result as a robustness check. Section 4 closes the paper summarizing the findings and outlining future research.

2 The model

A standard Cournot duopoly with homogeneous goods is considered in which firms must pay an ad valorem sales tax that, however, firms may partially evade. The (inverse) demand function is assumed linear:

$$p = z - Q$$

(1)

where $p$ is the price of goods and $Q = q_{i} + q_{j}$ denotes the industry output. The parameter $t \in (0,1)$ defines the sales tax rate. Firm i’s authentic tax base is $pq_{i}$. To evade indirect taxes, firms undervalue their sales volume: firm i discloses as tax base to the tax authority $a_{i} \in [0,p\,q_{i} ]$. Therefore, the amount $pq_{i} - a_{i}$ is firm i’s unreported revenues, and its tax bill equals $t\,a_{i}$. The tax authority detects evasion with a probability $y \in (0,1)$. If evasion is detected, in addition to taxes on the entire sales revenues, $pq_{i}$, firm i must pay a penalty function $P(pq_{i}-a_{i})$ which depends on evaded revenues,³ and whose analytical expression is

$$P(q_{i} ,a_{i} ) = \frac{{\left( {pq_{i} - a_{i} } \right)^{2} }}{2}$$

(2)

The expected penalty,$yP(pq_{i} - a_{i} )$, is a measure of the expected cost of tax avoidance. In this model, the detection probability, $y$, is assumed constant; on the other hand, the penalty function, P, is quadratic, therefore strictly increasing and convex in evaded revenues. Consequently, given the convexity of P and the constant value of $y$, the expected penalty, $yP$, is increasing and convex in evaded sales as well.

In general, tax authorities design four forms of penalty: automatic financial, automatic nonfinancial, criminal financial, and criminal nonfinancial (Tait, 1988). The form of the penalty function P can be justified as follows (Goerke & Runkel, 2011, p. 716, F in their terminology): “The penalties generally increase with the severity and extent of insufficient tax payments, supporting our assumption that F is increasing in evaded revenues. Moreover, many penalty schemes involve prison sentences for severe tax evasion activities. If F reflects not only monetary but also non-monetary penalties, such prison sentences suggest that F will be convex.”

Numerous countries, in fact, contemplate in their legislations the presence of penalties whose properties are in line with the penalty function the model proposes. Furthermore, countries such as Denmark and Spain (and Ireland as regards interests to be paid for late tax payments) have financial penalties increasing in the amount of evaded taxes (see OECD, 2009, 2011, 2013).⁴

Let us assume that firms use only labour as input for production, and workers are hired in a competitive labour market, in which the exogenous, uniform wage is given by $w > 0$, the per-worker wage paid by firm i. Firms exhibit a constant returns technology,

$$q_{i} = l_{i}$$

(3)

which represents the number of workers employed by the firm i to produce q_i output units. Firm i’s cost function is $wq_{i}$. Given the constant returns technology, marginal costs are constant. Firm i’s expected net profits are given by.

$$\pi_{i} = y\left\{ {(1 - t)pq_{i} - wq_{i} - \frac{{(pq_{i} - a_{i} )^{2} }}{2}} \right\} + (1 - y)\left\{ {pq_{i} - wq_{i} - ta_{i} } \right\}$$

(4)

The first term in brackets in Eq. (4) is firm i’s profits if tax evasion is detected, while the second term represents profits if such an evasion remains undetected. Firm i maximizes $\pi_{i}$, simultaneously⁵ choosing output $q_{i}$ and declared revenues $a_{i}$, taking as given the rival firm’s output. The first-order conditions for an interior solution are, as regards declared revenues

$$\frac{{\partial \pi_{i} }}{{\partial a_{i} }} = 0\; \Leftrightarrow a_{i} = \frac{{y[q_{i} z - (q_{i} q_{j} + q_{i}^{2} - t)] - t}}{y}$$

(5)

and, exploiting (5), as regards output

$$\frac{{\partial \pi_{i} }}{{\partial q_{i} }} = 0\; \Leftrightarrow q_{i} = \frac{{(z - q_{j} )(1 - t) - w}}{2(1 - t)}$$

(6)

From (6), by substituting its counterpart for firm j, we get the equilibrium output and declared sales revenue, respectively, by firm i,

$$q_{i} (w) = \frac{z}{3} - \frac{w}{3(1 - t)}$$

(7)

$$p_{i} (w) = \frac{z}{3} + \frac{2w}{{3(1 - t)}}$$

(8)

$$a_{i} = \frac{{(z - w)(2w + z)y - 9(1 - y)t^{3} + (yz^{2} - 18y + 18)t^{2} + [(9 - wz - 2z^{2} )y - 9]t}}{{9y(1 - t)^{2} }}$$

(9)

From (8), it is easy to see that, with a linear demand, the possibility and the degree of tax-shifting would depend on the level of costs.⁶ Then, making use of (7), (8) and (9), profits are:

$$\pi_{i} = \frac{{9t^{2} \left[ {\left( {1 + y^{2} } \right) - 2y} \right] + 2y\left\{ {\left[ {z^{2} (1 + t)} \right] - 2wz} \right\}}}{18y} + \frac{{2w^{2} }}{18(1 - t)}$$

(10)

The condition ensuring that an interior solution for a does exist, i.e., $a \in (0,pq)$, which guarantees positive declared revenue, is

$$y > y^\circ = \frac{9t}{{(z^{2} + 9t) - \frac{[2w + z(t - 1)]w}{{(1 - t)^{2} }}}}.$$

(11)

This means that a market size (i.e., a value of z) sufficiently large, as generally assumed in Cournot duopoly models, always ensures an economically meaningful value of the declared tax base. The next proposition shows the effect of the taxation on profits and evasion with competitive labour markets.

Proposition 1.

(a) Expected net profits can be increasing with an increasing tax rate; (b) the declared tax base decreases with an increasing tax rate if and only if the competitive wage is adequately high.

Proof:

See the Appendix.

The rationale for the profit increasing effect of higher taxes occurring only when y is sufficiently low is as follows. First, notice that, when the cost pressure exerted by wages is adequately high, higher tax rates intuitively reduce the firms’ sales declaration (see the Appendix), while they do not affect price and output (see Eqs. (7) and (8)). Second, it is easy to observe from that the tax induced reduction of the declared sales has a twofold effect on profits. On the one hand, it reduces profits through the penalty effect, and this reductive effect is higher the higher is the probability (y) of being detected. On the other hand, it increases the profit through the reduction of the tax burden in the case in which tax evasion is undetected, whose probability is inversely related to y.

3 An equivalence result

In the previous section, it has been assumed that duopolistic firms’ decision variable is the declared revenue, $a$. Let us now consider instead that the firms’ decision variable is unreported sales, defined as $e$. In the present framework, the firms’ quantity and evasion decisions are taken on their own. To see this fact clearly, Eq. (4) can be re-arranged as follows to expresses the objective of firm i in terms of unreported sales and quantity

$$\pi_{i} = \pi_{i}^{f} + \pi_{i}^{c} = [(1 - t)p - w_{i} ]q_{i} + (1 - y)te_{i} - y\frac{{e_{i}^{2} }}{2}$$

(12)

where $\pi_{i}^{f} = [(1 - t)p - w_{i} ]q_{i}$ are the profits when the firm behaves fairly term, and $\pi_{i}^{c} = (1 - y)te_{i} - y\frac{{e_{i}^{2} }}{2}$ are the extra profits (or loss) due to the cheating activity, with $e_{i}$ representing the unreported sales. This denotes that the choices relative to $\pi_{i}^{f}$ and $\pi_{i}^{c}$ are independent: the latter term indicates that the deliberate evasion actions mean audit activities by tax administration should be directed to put in place strategies and structures to ensure that non-compliance with tax law is kept to a minimum (OECD, 2004). Moreover, the two firms’ evasion decisions are independent of each other. Irrespective of the strategic quantity/price choices, the optimal unreported sales of firm i leads to $e_{i} = e^{*} = \frac{(1 - y)t}{y} \ge 0$, which implies $\pi_{i}^{c} = \frac{{t_{{}}^{2} }}{2}(1 - y)^{2}$, with $\frac{{\partial \pi_{i}^{c} }}{\partial t} = \frac{t}{y}(1 - y)^{2} \ge 0$, i.e., an increase in the tax rate increases cheating profits. Provided that the condition $pq_{i} \ge e^{*}$ is satisfied (to guarantee that the declared revenue is positive), the price and quantities prevailing are identical to those when firms do not misreport their revenues. Firms independently set their unreported sales to maximize the extra profit. As a consequence, maximization of the fair profits lead to the equilibrium quantity and price as in (7) and (8), profits $\pi_{i}^{f} = \frac{(1 - t)}{9}\left( {z - \frac{w}{1 - t}} \right)^{2}$, and $\frac{{\partial \pi_{i}^{f} }}{\partial t} = \frac{{w^{2} - z^{2} (1 - t)^{2} }}{{9(1 - t)^{2} }} \le 0$ since $t \le 1 - \frac{w}{z}$. The key point is to know how and when an increase of the tax rate can increase the overall profit of the firms. One can easily check that differentiation of the overall profits with respect to tax rate exactly replicate the result in Proposition 1.

Because $\frac{{\partial \pi_{i}^{f} }}{\partial w} \le 0$ and $\frac{{\partial^{2} \pi_{i}^{f} }}{\partial t\partial w} > 0$, the higher is the average wage, the lower is the fair profit, and the less it decreases with respect to the tax rate; while the extra profit is unaffected by wage rates. More generally, if the input is provided by an upstream supplier (firm, or union in the case of labour) which exerts market power, this may lead to tax evasion based on cheating profit maximization, looking like an aboveground activity of the firms which tends to dominate the output fixing for high values of the wages.⁷

4 Conclusions

This paper investigates a unionized Cournot duopoly model with evasion of indirect taxes. The main finding is that, rather counter-intuitively, a higher indirect taxation may increase profits in the presence of competitive labour markets. In particular, the result of “more taxation and higher profits” can be obtained if the cost pressure of wages (alternatively, if the market size) is adequately large and the likelihood of the detection probability is not too high. Moreover, we have shown that our findings are equivalent irrespective of firms optimally choosing either the tax base to disclose to the tax authority or the amount of evasion tout-court.

One must be aware of the extremely simplified nature of the model employed, based on a set of specific assumptions such as a convex penalty rate function, and linear demand schedule. In future research, those caveats need to be relaxed. For instance, different penalty functions, the presence of differentiated goods, and a general demand function call for additional robustness check of our results.

Appendix

Proof of Proposition 1

Part (a)

(i) $\frac{{\partial \pi_{i} }}{\partial t} = \frac{{w^{2} - z^{2} (1 - t)^{2} }}{{9(1 - t)^{2} }} + \frac{{t(1 - y)^{2} }}{y}$;

(ii) $\frac{{\partial \pi_{i} }}{\partial t} \ge 0\; \Leftrightarrow \;y < y_{1}^{^\circ } \;or\;y > y_{2}^{^\circ }$,

where $y_{1}^{^\circ } = 1 + \frac{{z^{2} }}{18t} - \frac{{\sqrt {\{ [(1-t)z]^{2} - w^{2} \} [(z^{2} + 36t)(1-t)^{2} + w^{2} ]} }}{18t}$,$y_{{_{2} }}^{^\circ } = 1 + \frac{{z^{2} }}{18t} + \frac{{\sqrt {\{ [(1-t)z]^{2} - w^{2} \} [(z^{2} + 36t)(1-t)^{2} + w^{2} ]} }}{18t} > 1$;

(iii) $y_{2}^{^\circ } > 1$; however, since $y_{1}^{^\circ } < y^\circ \;if\;\frac{w}{1 - t} \le z < \frac{9t(1 - t)}{{2w}} - \frac{{3\sqrt {9t^{6} - 36t^{5} + 54t^{4} + (2w^{2} - 36)t^{3} + (9 - 4w^{2} )t^{2} + 2tw + w^{4} } - w^{2} }}{w(1 - t)}$ but $y^\circ < y_{1}^{^\circ } \;if\;\frac{9t(1 - t)}{{2w}} - \frac{{3\sqrt {9t^{6} - 36t^{5} + 54t^{4} + (2w^{2} - 36)t^{3} + (9 - 4w^{2} )t^{2} + 2tw + w^{4} } - w^{2} }}{w(1 - t)} \le z$, there are pairs $(t,w)$ such that the condition $(1 - t)^{2} z^{2} \ge w^{2} \ge 9\left[ {\frac{{z^{2} }}{9} - \frac{{t(1 - y)^{2} }}{y}} \right](1 - t)^{2}$ is satisfied. Then, part a) of Proposition 1 is proved.

Part (b)

i) $\frac{\partial a}{{\partial t}} = \overbrace {{ - \frac{1 - y}{y}}}^{audit\;effect} - \overbrace {{\frac{w[4w - (1 - t)z]}{{9(1 - t)^{3} }}}}^{market\;effect}$. The first effect, the audit effect, is always negative: the higher the probability of evasion detection is, the lower the undeclared sales are. The second effect, the market effect, is negative if and only if $w \ge \frac{(1 - t)z}{4}$, that is, if the cost pressure of wages is high. Q.E.D.