THE ANALYSIS OF AN ECONOMIC GROWTH MODEL WITH TAX EVASION AND DELAY

Transcription

1 Key words: delayed differential equations, economic growth, tax evasion In this paper we formulate an economic model with tax evasion, corruption and taxes In the first part the static model is considered, there are a representative agent and a public institution he public institution by its tax collectors detects the tax evasion and enacts a system of tax on capital and fines he representative agent is endowed with a capital k, k > and he has to pay a tax on this capital at a rate t, t (,] he agent can try to evades the tax on capital by e of the total capital concealing a capital he optimal tax evasion level which maximizes expected net profit is determined In the second part, the dynamic model of tax evasion is presented, the representative agent chooses at each moment in time the level of tax evasion so as maximize expected net profit on infinite horizon, taking into account of the motion equation for the capital () that depends on ( ) and ( ) Using the delay as bifurcation parameter we have shown that a Hopf bifurcation occurs when this parameter passes through the critical value he direction of the Hopf bifurcation, the stability and the period of bifurcating period solution are also discussed and characterized JEL Classification: A, C6, D, O49 HE ANALYSIS OF AN ECONOMIC GROWH MODEL WIH AX EVASION AND DELAY Olivia BUNDĂU Mihaela NEAMŢU Introduction In this paper, we construct a new economic growth model with tax evasion, taxes and delay he main feature of our model is that the control variable and the state variable enter with a delay in the motion equation of the state variable he introduction of the time delay yields a system of mixed functional differential equations We determine the steady state of this system and we investigate the local stability of the steady state by analyzing the corresponding transcendental characteristic equation of its linearized system In the following, by choosing of the delay as a bifurcation parameter, we show that this model with a delay exhibits the Hopf bifurcation herefore, the dynamics are oscillatory and this is entirely due to time-to-build technology hen, we discuss the direction and stability of the bifurcating periodic solutions by applying the normal form theory and the center manifold theorem he outline of this paper is as follows In Section, we present the static model of tax evasion and we determine the optimal tax evasion level which maximizes expected net profit In Section, we formulate the dynamic model of tax evasion Section 4, by choosing of the delay as a bifurcation parameter, some sufficient conditions for the existence of Hopf bifurcation are derived In Section 5, the direction of Hopf bifurcation is analyzed by the normal form theory and the center manifold theorem introduced by Hassard and some criteria for the stability of the bifurcating periodic solutions are obtained In Section 6, the conclusions are discussed he Model We will consider an economic model with tax evasion, corruption and taxes In economy of this model there are a representative agent and a public institution he public institution by its tax collectors detects the tax evasion and enacts a system of tax on capital and fines he representative agent is endowed with a capital k, k > and he has to pay a tax on this capital at a rate t, t (,] he agent can try to evades the tax on capital by Assistant, PhD, Politehnica University of imişoara, Department of Mathematics, Romania Associate Professor, PhD, West University of imişoara, Faculty of Economics and Business Administration, imişoara, Romania

2 Olivia BUNDĂU, Mihaela NEAMŢU concealing a capital e of the total capital k, e [, k] We assume that the transaction cost for tax evasion is he, with h > A smaller h captures the situation when an economy is more corrupt (Chen, B-L,, 8-4) In making this decision, the agent takes its tax evasion costs, the tax rate and the auditing probability into account If the agent evades and the tax collectors do not detect the tax evasion then the result capital is given by: ( ) k t k te he () If the agent evades and the tax evasion gets detected then the agent has to pay the tax on capital plus a fine at a penalty rate s, s > he disposable capital is given by Allingham and Sandmo (Allingham & Sandmo, 97): k ( t ) k se h e () A or Yitzhaki (raxler, C, 6): k ( t ) k st e h e () Y In the following, we consider the disposable capital is given by: k ( a) k ak, a, (4) [ ] A Y From (), () and (4) result: b a at he profit of a risk averse agent are represented by the following von Neuman-Morgenstern profit function namely the expected net profit (raxler, C, 6) (5) (, ) ( ) ( )( ) (6) p [,] is the probability that the agent is discovered and convicted by the tax collectors he functions u satisfy the properties u'( x ) >, u"( x ) < he agent chooses e, so as to maximize expected net profit Proposition a) If the utility function is ux ln x, x >, the optimal tax evasion level which maximizes expected net profit is the positive solution from [, k ] of the equation: a ax ax ax a (7) h, a h ( pt ( p) bs), a h ( t ) k bst, k ( t ) k bse h e a ( t) ( p) t pbs k (8) b) If the utility function is u( x) x σ, x >, σ (,) then the optimal tax evasion level which maximizes expected net profit is the positive solution from [, k ] of the equation: σ ( ) ( ) σ ( t k t x h x ) t h x p ( t) k bsx hx bs hx ( p) σ x c) If the utility function is ux, x >, σ (,) σ then the optimal tax evasion level which maximizes expected net profit is the positive solution from [, k ] of the equation () p ( t) k bsx hx ( bs hx) ( p) If ( ) ; ( e is optimal tax evasion level, the net profit is Examples: a) If t 6, p 8, a 4, k, s from (7) and () result: ) i h, then k 6, k 54, e 5599 and V k, e, 965; ( ) ) if h, then k 5, k 59, e 8 and V k, e, 94; ( ) b) If t 6, p, a 4, k, s, σ from (9) and () result: ) if h, then k 474, k 979, V k, e 59; and ( ) 4) if h, then k 7966, k 77, and V k, e 5585 (9) () () he dynamic model of tax evasion with delay Consider an economy that is inhabited by infinitely-lived households In this economy, a representative agent chooses at each moment in time the level of tax evasion V k(), t e () t e () t so as maximize expected net profit given by: σ σ ( t ) k t x h x ( t h x) pu( t k te ) he Vke (, ) pu( t) k bse he ( ) ( ) ( )

3 HE ANALYSIS OF AN ECONOMIC GROWH MODEL WIH AX EVASION AND DELAY (, ) ( ) ( )( ) () p [,] he infinite planning problem for this economy is given by: subject to () ( ) ( ) ( ) k( θ ) ϕ ( θ ); θ [ τ,]; e ( t) [, k( t)]; e () e ; () (4) kt ( τ ) is the productive capital stock at time t τ, δ [,] is the rate at which capital depreciates; ρ > is the discount rate, f : R R is the production function with f '( k ) >, f ''( k ) <, lim f '( k), lim f '( k), and k k ϕ :(,] R is the initial capital function, it need to be specified in order to identify the relevant history of the state variable he coefficient d is given by d pbsh ( p) t (5) o solve this optimization problem, we apply the generalized Maximum Principle for time-lagged optimal control problems he Hamiltonian associated to this problem is: ρ t Η kt (), e() t V kt (), e() t e p( t τ ) () () () (6) p denotes the costate variable or the shadow price of the capital kt () Using (5) result: Proposition he equations of motion for the capital kt () and evasion e () t is given by: e () t max V k( t), e ( t) e dt e () t g ( k(), t e ()) t ( f ( k( t τ)) δ k( t τ) de ( t τ ) ) g( kt, e( t)) ( f '( k( t)) δ ρ ) dg ( k ( t ), e( t)) (7) V (, k e ) V (, k e ) g ( ke, ), g( ke, ), V (, k e ) V (, k e ) V(, k e) g (, k e ) V (, k e ), Vke (, ) V(, k e), V k ρt kt f kt ( τ) δ kt ( τ) de( t τ) V ( k, e), e Vke (, ) (, k e ) e V 4 Local stability analysis and Hopf bifurcation (8) he steady states ( k, e ) of the functional differential equation system (7) are determined by setting e, k From (7) we have: Proposition a) he steady states is ( k, e ), equation and e mf ( k ) nk k is the root of () () b) If f ( x) x σ, x >, σ (,), ux ln x, then the steady states is ( k, e ), k is the root of equation σ σ p d( t ) ( bs hk( mk n)( σk δ ρ)) ( σ σ σ pk pk pk pk 4 pk 5 ) ( p) ( qk pk qk σ pk σ pk σ 5 ) σ σ [ d( t ) ( t h k( mk n)( σk δ ρ))] () σ e and mk nk c) If f ( x) x σ, x >, σ (,), ux ln xand h then the steady states ( k, e ) is given by k σ a b Vke (, ), e k (, k e ), k e V δ k d σ Vke (, ) ( ( ( )( ' δ ρ)) ) p d t bs h mf k nk f k u qk p k q f k p kf k p f k '( 5 ) ( δ ρ ) ( p) d( t) ( t h( mf( k) nk)( f '( k) )) u' p k p k p f ( k) p kf( k) p f( k) (9) ( 4 5 ) δ,,,, m n p t nt p h n p t m, d d p mnh, p h m, q 4 5 q t bsn, q mbs ()

4 Olivia BUNDĂU, Mihaela NEAMŢU With respect to the transformation x () t k () t k, x() t e() t e system (7) becomes x () t F( x( t τ), x( t τ)) x () t F ( x (), t x (), t x ( t τ), x ( t τ)) ( ) a p p t t bs p ( pt ) ( p) ( t) q F( u, u, u, u4) g( u k, u e) ( fu ( k) δ( u k) du ( 4 e) ) g ( u k, u e ) b t mp( p)( t ) mbst ( p) ( t ) F ( u, u, u, u ) f( u k ) δ ( u k ) 4 du ( e) 4 ( f '( u k ) δ ρ) dg ( u k, u e ) (4) (5) (6) Expanding F, F, given above in aylor series around O (,, ) and neglect the terms of higher order than the third order, we can rewrite system (5) in the form: x () t ax( t τ) ax( t τ) ax ( t τ) ax ( t τ)!! x () t bx( t τ) bx( t τ) cx () t cx() t [ lx () t! l x () t l x ( t τ) lx ( t) x( t) lx ( t) x ( t τ ) lx () t x ( t τ)] [ () () ()! l x t l x t x t lx ( t τ) l x ( t) x ( t) l x ( t) l l x () t x ( t τ) l x () t x ( t τ)] F F F a (); a (); a (); u u4 u F F F a (); b b u u u 4 F F F c (); c (); l (); u u u (); (); (7) F F F (); (); (); u u u u F F F l u u l u u u l l l l ω ( a b ) c (( a b) c ) 4( ac ac) c ( a c a c ) ω ( a b ) τ arctg ω( a c a c c b ) dλ M Re () d λ ωτ τ τ i, (8) o investigate the local stability of steady state we linearize system (7) hen, the liniarized system is given by: () () ( ) (9) a a A, B c c b b () he characteristic equation corresponding to system (9) is given by λ λ λ c ( a b ) a c a c e λτ () From () using the standard Hopf bifurcation theory (Hassard, BD etal, 98) we get: Propoziţia 4 i) Ifτ, and a c a c <, c a b < then the steady states ( k, e ) is asymptotically stable ii) Let λ λ( τ) be a solution of () If τ, ω are given by then a Hopf bifurcation occurs at the steady state given by k, e as τ passes through τ ( ) (); (); (); F F F (); (); (); l l l u u u u4 F F l (); l () u u u u 5 Direction and local stability of the Hopf bifurcation for system (7) From section 4, ifτ τ given by (), then all the roots of equation () other than ± iω have negative real part For notational convenience, let τ τ μ, μ ( ε, ε) with ε > hen μ is the Hopf bifurcation value of system (7) In the study of the Hopf bifurcation problem, first we transform system (8) into an operator equation of the form

5 HE ANALYSIS OF AN ECONOMIC GROWH MODEL WIH AX EVASION AND DELAY () () () x ( x, x), xt x( t θ ), θ [ τ,] he operators and and Ρ are defined as ( μφθ ) dφθ, θ [ τ,) dθ (4) Aφ() Bφ( τ), θ C (,,C ) and A, B are given by () and (5): ( μφθ ) (, ), θ [ τ, ) (5) ( F( μφ, ), F( μφ, )), θ For ψ C (,,C ),the adjunct operator of is defined as: dψ ( s) ( μψ ) ( s), s [, τ ) ds (6) A ψ () B ψ ( τ ), s τ For C (,,C ) and ψ C (,,C ) we defined the bilinear form θ ψ, φ ψ () φ() ψ ( ξ θ ) Bφξ dξdθ τ Proposition 5 i) he eigenvector φ of associated with eigenvalue λ is given by φθ ve λθ, θ [ τ,] v ( v, v) and v ae λτ, v ( λ a) e λτ ii) he eigenvector φ of associated with eigenvalue λ λ is given by s φ s we λ, s [, τ], λ η λ λτ ( v av ) ( ) e ( a ab ) v ( a ab ) v λ a e a c λτ λτ be (7) We calculate the eigenvector φ of associated with eigenvalue λ iω and the eigenvector φ of associated with eigenvalue λ λ λτ λ a e w ( w, w) and w η, w c iii) With respect to (7) we have: φ φ φ φ,,, λτ λτ b e φ φ φ φ,, η, (8) Next, we construct the coordinates of the center of the manifold Ω at μ (Lorenz, HW, 997; Mircea, G et al, ) We consider zt () φ, xt, wt (, θ ) xt Re{ zt () φ( θ )} On the center manifold Ω, wt (, θ ) wzt ( (), zt (), θ ), (9) and z, z are the local coordinates of the center manifold Ω in the direction of φ and φ respectively For the solution xt Ω of (), notice that for μ we have zt () λ zt () φ, ( wt (, θ) Re{() ztφ( θ)} ) (4) We rewrite (4) zt () λ zt () gzz (, ) with gzz (, ) φ () ( wzz (,, θ) Re{ zφ( θ)}) (4) We expand the function g( z, z ) on the center manifold Ω in powers z and z z z z z gzz (, ) g g zz g g For the system (7) we have: (4) g wf wf, g wf wf, (4) g w F w F g wf wf,, vv ) l vve l ( v ve vv e ), F and w ( θ) ( w ( θ), w ( θ)), z z wzz (,, θ) w ( θ) w( θ) zz w ( θ) λτ λτ F a v e, F a v e, F F a ( w ( τ) v e λτ av v, F l v l v l v e λτ λτ τ w ( ) ve ) a v v e, λτ λτ λτ l vv e l v e l vv e, F l v v l v v l v v l ( v v λτ λτ λτ F l ( v w () w () v ) l ( v w () v w ()) λτ l ( ve w ( τ) v w ( τ) e λτ ) l ( vw () v w ()) l v w w w w are given by g g w ( θ) ve ve Ee λθ λθ λθ λ λ λθ λθ λ λ (44) F, ( θ) ( ( θ), ( θ)) g g w ( θ) ve ve E, θ [ τ,] λτ (45)

6 Olivia BUNDĂU, Mihaela NEAMŢU (, ) E D F F, D ( A e λτ λ I) E D ( F, F ), D ( A B) (46) herefore, we can compute the following parameters: i g C() g g g g ω Re ( C() ) μ, β Re ( C () ) M ( C ) Im () μn ω N d λ Im dτ λ iω, τ τ (47) Proposition 6 In formulas (47), μ determines the direction of the Hopf bifurcation: if μ > ( < ) the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exist for τ > τ( < τ); β determines the stability of the bifurcation periodic solutions: the solutions are orbitally stable (unstable) if β < ( > ) determines the period of the bifurcating periodic solutions: the period increases (decreases) if > ( < ) 6 Conclusion In this paper, we formulate a growth model with delay for capital and tax evasion Using the delay τ as a bifurcation parameter we have shown that a Hopf bifurcation occurs when this parameter passes through a critical value τ he direction of the Hopf bifurcation, the stability and period of the bifurcating periodic solutions are also discussed and characterized Bibliography Allingham MG, Sandmo A (97) - Income tax evasion a theoretical analysis, Journal of Public Economics, vol, Issue -4:-8 Asea, PK, Zac, PJ (999) - ime-to-build and cycles, Journal of Economic Dynamics and Control, Elsevier, vol (8), pp Chen, B-L () - ax evasion in a model of endogenous growth, Journal of Economic Dynamics, vol 6:8-4 4 Hale, JK, Lunel, SM (99) - Introduction to functional diferential equations, Applied Mathematical Sciences, Springer-Verlang 5 Hassard, BD, Kazarino, ND, and Wan, HY (98) - heory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge 6 Lorenz, HW (997) - Nonlinear Dynamical Economics and Chaotic Motion, Springer-Verlang 7 Mircea, G, Neamtu, M, Opris, D () - Dynamical system from economy, mechanic and biology described by diferential equations with time delay, Ed Mirton, imisoara 8 raxler, C (6) - Social Norms and Conditional Cooperative axpayers, Springer-Verlang 9 Valencia Arana, MO (5) - Economic Growth and the Household Optimal Income ax Evasion, Archivos de Economia Winkler, R, Brand-Pollmann, U, Moslener, U, Schlsoder, J (4) - ime lags in capital Accumulation, Operations Research Proceedings Springer, Heidelberg