Tax evasion in a Cournot oligopoly with endogenous entry

See, for example, Cremer and Gahvari (1993) and Besfamille et al. (2009, 2013) for similar approaches. In contrast to these contributions, we do not assume \(S_{1}(0, \pi _j) = 0\) to also allow for the possibility that no evasion constitutes optimal behaviour.

See Slemrod (2001) and Grubert and Slemrod (1998) for models of personal or corporate income tax avoidance which are based on assumptions which are tantamount to \(S_{12} < 0\).

Note that the costs of evasion could also be specified as increasing with the tax rate, \(\tau \). For a given tax rate, such modification would not alter the main findings because the alteration would primarily affect the impact of the tax rate on evasion activities (see “Impact of higher tax rate” section of Appendix).

It could be argued that tax evasion is easily detectable because all firms are ex-post identical and evade the same amount. Accordingly, authorities could eradicate evasion by auditing all firms. Such an outcome will not occur if either proving evasion activities is required and sufficiently costly to tax authorities such that they cannot establish illegal activities for all firms. Alternatively, one could assume that operating profits are given by \((P(X) - c)x - k\), where k varies across firms. If authorities do not know the value of k, a given tax payment does no longer indicate illegal behaviour. Since the inclusion of such firm-specific operational costs would considerably extend the notation without yielding additional insights, we refrain from doing so.

Besley (1989) considers a unit tax on output in a Cournot oligopoly in the absence of tax evasion. He shows for a setting with constant unit costs that output per firm rises with the tax rate if \(\eta > 0\), while the number of firms declines with the tax rate if \(2 + \eta > 0\), while aggregate output unambiguously falls. Delipalla and Keen (1992) provide comparable predictions and, furthermore, show that the consequences of a higher ad valorem tax are more ambiguous because this tax also reduces marginal revenues and, thus, mitigates the product market imperfection (see also Stern 1987; Hamilton 1999).

The positive impact of the tax rate on tax evasion is a common prediction in models of tax evasion by firms which maximise (expected) profits (see, for example, Cremer and Gahvari 1993). If the objective is defined in terms of utility, risk aversion also plays a role (cf. Yaniv 1995, inter alia), as it is the case with regard to income taxes paid by individuals.

von Weizsäcker (1980) and Suzumura and Kiyono (1987), for example, also compare the market outcome to the first-best.

Combining (13a) and (16), the tax rate which ensures the second-best optimal number of firms, \(n^{0,\hbox {opt}}\), can be calculated. It is implicitly defined by \(\tau - (n^{0,\hbox {opt}}(\tau )+\eta )/(1 + n^{0,\hbox {opt}}(\tau )+\eta ) = 0\).

As in a framework without evasion, the tax rate which ensures the constrained-optimal number of firms can be calculated on the basis of Eqs. (7) and (17). For \(\beta = 1\), it is possible to show that the tax rate must satisfy \(\tau (1 - \alpha )- (n^{+,\hbox {opt}}(\tau )+\eta )/(1 + n^{+,\hbox {opt}}(\tau )+\eta ) < 0\), where \(n^{0,\hbox {opt}}(\tau ) = n^{+,\hbox {opt}}(\tau )\) for any given tax rate and \(\hbox {d}n/\hbox {d}\tau < 0\). Hence, the constrained-optimal effective tax rate \(\tau (1- \alpha )\) in the presence of evasion activities must be lower than the second-best optimal tax rate \(\tau \) in a world without evasion. This is the case because the costs of evasion deter entry, ceteris paribus. For \(\beta < 1\), the optimal number of firms generally depends on the extent of tax evasion.

A lower detection probability combines a fall in the absolute costs of tax evasion with a decrease in marginal costs. An increase in tax evasion activities would also come about if either only the marginal costs of evasion decline, while the cost level is held constant, or alternatively if the absolute costs of evasion rise, while marginal costs remain unchanged. In terms of the model, the first type of change would be equivalent to a reduction in the detection probability, q, and a rise in the fixed costs, \(c_\mathrm{fix}\), such that total costs \(T = q\pi (\alpha , \alpha ) + c_\mathrm{fix}\) remain the same. The second type of change would be tantamount to a rise in the fixed costs. We do not explicitly consider the consequences of such alterations in the costs of tax evasion because they are more difficult to implement than a combined decrease in marginal and absolute costs. The relevant derivations can be obtained from the author upon request and are used to some extent to provide intuition for the results relating to a change in q.

If \(E_{\pi } > 0\), higher profits imply that less evasion is needed to cover its costs, since the tax base rises. Accordingly, a fall in the absolute costs of evasion T, holding constant the marginal costs, \(T'\), results in less evasion.At the same time, higher profits induce more firms to enter the market. This implies that a fall solely in the cost level reduces evasion and enhances entry.

I am very grateful to an anonymous referee for suggesting this illustration of the main results of the paper.

As profits exceed unity, as shown below, costs of evasion \(T(\alpha , 0) = q\alpha ^{2}\) are substantially lower than in the case of \(T(\alpha , \pi ) = q(\alpha \pi )^{2}\). Consequently, the fraction of taxes evaded will be noticeably higher in the former case, given the same detection probability, q.