Rent seeking games with tax evasion

In [3] a model with tax evasion is presented. The authors consider n firms which enter the market with a homogenous good. These firms have to pay an ad valorem sales tax, but may evade a certain amount of their tax duty. The aims of the firms are to maximize their profits. The equilibrium point is determined and an economic analysis is made.

Based on [1], [3], [7], [8], [10], in our paper we present three economic models with tax evasion: the static model of Cournot duopoly with tax evasion in Section 2, the dynamic model of Cournot duopoly with tax evasion in Section 3 and the dynamic model with tax evasion and time delay in Section 4.

In Section 2, in the static model the purpose of the firms is to maximize their profits. We determine the firms’ outputs and the declared revenues which maximize the profits, as well as the conditions for the model’s parameters in which the maxim profits are obtained. Using Maple 11, the variables orbits are displayed.

In Section 3, the dynamic model describes the variation in time of the firms’ outputs and the declared revenues. We study the local stability for the stationary state and the conditions under which it is asymptotically stable.

In Section 4, we formulate a new dynamic model, based on the model from Section 3, in which the time delay is introduced. That means, the two firms do not enter the market at the same time. One of them is the leader firm and the other is the follower firm. The second one knows the leader’s output in the previous moment tτ, τ ≥ 0.

Using classical methods [6], [9] we investigate the local stability of the stationary state by analyzing the corresponding transcendental characteristic equation of the linearized system. By choosing the delay as a bifurcation parameter we show that this model exhibits a limit cycle.

Finally numerical simulations, some conclusions and future research possibilities are offered.

The static model of Cournot duopoly is described by an economic game, where two firms enter the market with a homogenous consumption product. The elements which describe the model are: the quantities which enter the market from the two firms x i ≥ 0, i = 1, 2; the declared revenues z i , i = 1, 2; the inverse demand function p :

). The government levies an ad valorem tax on each firm’s sales at the rate t 1 ∈ (0, 1) and q ∈ [0, 1] is the probability with which the tax evasion is detected.

The true tax base of firm i is x i p (x 1 + x 2 ) . Firm i declares z i ≤ x i p (x 1 + x 2 ) as tax base to the tax authority. Accordingly, evaded revenues of firm i are given by x i p (x 1 + x 2 )z i . With probability 1q tax evasion remains undetected and the tax bill of firm i amounts to t 1 z i . The tax authority detects tax evasion of firm i with probability q. In case of detection, firm i has to pay taxes on the full amount of revenues, x i p (x 1 + x 2 ) , and, in addition, a penalty F (x i p (x 1 + x 2 )z i ). The penalty is increasing and convex in evaded revenues x i p (x 1 + x 2 )z i . Moreover, it is assumed that F (0) = 0, namely law-abiding firms go unpunished.

The profit functions of the two firms are: P i : R 2 + → R + , i = 1, 2, given by:

The first bracketed term in (1) equals the profit of firm i if evasion activities remain undetected. The second term in (1) represents the profit of firm i in case tax evasion is detected. The firm’s aim is to maximize (1) with respect to output x i and declared revenues z i . This aim represents a mathematical optimization problem which is given by: max {xi,zi}

(2)

From the hypothesis about the functions p, F, C i , i = 1, 2 , we have:

Proposition 1 The solution of problem (2) is given by the solution of the following system:

In what follows, we will consider the penalty function

From (3) we can deduce:

given by :

For the parameters c 1 = 0.3, c 2 = 0.6, q = 0.12, t 1 = 0.16, the variations of the variables z 1 , z 2 , and the profits P 1 , P 2 are given in the following figures:

The dynamic model describes the variation in time of output x i (t) , i = 1, 2 taking into account the marginal profits ∂P i ∂x i , i = 1, 2. Assume that each agent adjusts its declared revenue z i (t), i = 1, 2 proportionally to the marginal profits ∂P i ∂z i , i = 1, 2. Then, the dynamic model is given by the following differential system of equations:

By expanding (6) in a Taylor series around the stationary state (x * 1 , x * 2 , z * 1 , z * 2 ) and neglecting the terms of higher order than the first order, we have the following linear approximation of system (6) :

where:

The characteristic equation associated to ( 7) is given by:

where: A necessary and sufficient condition as equation ( 9) has all

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