Multiplier–Accelerator Interaction in the Presence of an Underground Economy and Taxation

1. Introduction

The theory of multiplier–accelerator interaction has a long history (Puu, 2006). The basic idea has come to be that increased government spending, for instance, will raise national income as specified by the multiplier; and the higher income will in turn increase consumption expenditure, enhancing next investment as stipulated by the accelerator and raising national income even further. The interaction between the multiplier and the accelerator can lead to cyclical fluctuations, which nevertheless are predicted by the model to be transitory. Moreover, if the parameters of the model were taking on real-world values, business cycles would be unstable. The overall picture is one allowing one to think that cycles would be temporary because they would lead immediately to chaos. And since reality is one of cycles, chaos should have already set in.

Two extensions of the basic model that address this feature of it are the incorporation of nonlinear progressive income taxation by Fanti and Manfredi (2003) and nonlinear investor expectations by Westerhoff (2006). Both extensions conclude that once long-run equilibrium is disturbed, complex potential dynamics may arise depending on the interplay of the forces upon which each study concentrates. This article adds underground economy as one more factor that may be held responsible for complex dynamics. It does so within an economy containing a progressive income tax, since this tax is elastic with respect to income by definition and hence, stabilising (Creedy & Gemmell, 2007). Therefore, if the informal sector is found to have a destabilising effect within such an economy, this effect should be deemed to be a powerful one. It is for the same methodological reason that linear rather than the more likely destabilising nonlinear progression is assumed.

An extension of the multiplier-accelerator framework, acknowledging the presence of an informal sector in the economy is nowhere to be found although the operation of this sector is a major source of frustration in the system (Williams, 2014). Excluding the illegal or criminal economy, the terms informal, underground, hidden and unofficial economy are used interchangeably to denote the non-observed economy as defined, for instance, by Orsi, Raggi and Francesco (2012). The next section claims that if the multiplier-accelerator model is seen as an inherently unstable one, the introduction of the underground economy in this model leaves no doubt about it. Also, although the matter of the emergence of such an economy is of no concern to the macro-economic nature of the analysis, we do find out that the accelerator is the element that may be held responsible for the empirical documentation of a positive relationship between official economy and tax rate (Williams, 2014).

Since, microeconomics holds the opposite view due to increased tax evasion incentives in response to tax increases, the macroeconomic ‘accelerator effect’ is in effect found to neutralise this ‘tax evasion effect’ of higher taxation. It is certainly an important finding, identifying a link between official and unofficial economies, which is missing from the relevant literature. Tax policy comes up thus as a critical factor influencing stability, and its design as a particularly demanding task because it has to be addressing at the same time the matter of the desirable formal/informal economy mix. This article concludes with still another section where the destabilising role of the underground economy is evaluated in the light of the theories on formal and informal business cycle interaction.

2. Formal Considerations

Let Y, C, I, G and T denote national economy, consumption, investment, government expenditure and tax revenue, respectively. A version of the standard model with a linear progressive income tax (Turnovsky, 1977) has equations as follows:

Y = C + I + G

(1)

C = θ ( Y − 1 − T − 1 )

(2)

I = I 0 + ε ( C − C − 1 )

(3)

T = − ​ τ 0 + τ Y

(4)

where the subscript “–1” denotes one period lag. Coefficients θ and ε are positive while I₀ and τ₀ are the autonomous parts of (3) and (4), respectively. Inserting (2) in (3) and the result along with (2) and (4) in (1), yields the linear second-order difference equation:

Y = I 0 + G 0 + θ τ 0 + [ θ ( 1 + ε ) ( 1 − τ ) ] Y − 1 − θ ε ( 1 − τ ) Y − 2

(5)

for some G = G₀, where subscript ‘–2’ designates second period lag. The fixed point of this equation is as follows:

Y ^ = I 0 + G 0 + θ τ 0 1 − θ ( 1 − τ )

(6)

Appendix 1 shows that the stability condition is clear-cut when the homogeneous equation has only one general solution. The condition in this case is as follows:

0 ≤ 1 − 2 θ ( 1 + ε ) ≤ τ ≤ 1

(7)

which is satisfied for the values of τ in the area below the curves of Figure 1 where y ≡ τ and x ≡ ε. These curves depict the case of the equality sign for three different θ’ s: θ = 0.75, θ = 0.85 (dotted line in the middle) and θ = 0.95 (upper line).

If there are two real solutions, the stability condition applying to one of them is exactly the opposite of the condition related to the other solution. The critical value of τ distinguishing stability by solution is as follows:

0 ≤ τ = 1 − ε + 1 + 3 ε θ ( 1 + 2 ε ) ≤ 1

(8)

OPEN IN VIEWER

Figure 1

Stability in the Absence of an Underground Economy with One Real Root

with one of the conditions pertaining to τ’ s below the lines depicted in Figure 2 and the second condition applying to τ’s in the area above these lines. Again y ≡ τ and x ≡ ε in this figure, and there are three different θ’ s: θ = 0.75, θ = 0.85 (dotted line in the middle) and θ = 0.95 (upper line). The equality sign is the case of motion along the uniform cycle either in Figure 1 or in Figure 2.

These are results confirming the stabilising role of linear progressive taxation.

Next, let (1) be decomposed according to official and underground economy quantities, indexed by superscripts f’ and ‘u’, respectively, as follows:

h Y = Y f = C f + I f + G = a C + β I + G

(9)

( 1 − h ) Y = Y u = C u + I u = ( 1 − a ) C + ( 1 − β ) ​ I

(10)

where coefficients h, a and β belong to the interval (0,1). Moreover, let:

C = θ ( Y − 1 f − T − 1 ) + γ Y − 1 u = [ θ h + γ ( 1 − h ) ] Y − 1 − θ T − 1

(11)

where 0 < θ ≤ γ ≤ 1, θh + γ(1 – h) < 1 and

T = − τ 0 + τ Y f = − ​ τ 0 + τ h Y

(12)

OPEN IN VIEWER

Figure 2

Stability in the Absence of an Underground Economy with Two Real Roots

That is, total consumption is treated as the sum of two separate consumption functions; consumption behaviours in the formal and informal sectors are presumed to be unrelated. Finally, let

I = I 0 + I 0 u ​ + ​ ε ​ ( C − C − 1 )

(13)

Given that consumption is dealt with as it does through (11), if total investment was modelled (apart from its autonomous part) as the sum of two separate investment functions too, two fully distinct economies would be postulated in effect, which is not the case. We chose to link the two economies through investment rather than through consumption or both, acknowledging the possibility of smooth (official) consumption as in Blanchard (1981).

Now, inserting (11) in (13) and the result along with (11) and (12) in (1), yields the linear second-order difference equation:

Y = I 0 + I 0 u + G 0 + θ τ 0 + { [ θ h + γ ​ ( 1 − h ) ] − θ τ h } ( 1 + ε ) Y − 1 − ε { [ θ h + γ ( 1 − h ) ] − θ τ h } Y − 2

(14)

with a fixed point at:

Y ˜ = I 0 + I 0 u + G 0 + θ τ 0 1 − θ h ​ ( 1 − τ ) − γ ( 1 − h )

(15)

Appendix 1 shows that stability requires that:

0 ≤ τ ≤ φ ( 1 + ε ) 2 − 4 ε θ h ​ ( 1 + ε ) 2 ≤ 1

(16)

when the solution of the homogeneous equation is only one, as depicted by the τ’s on and below the curves in Figure 3 for x ≡ ϵ > 1.9, where φ = 0.85 and θ = 0.80 for h = 0.70, h = 0.80 (middle dotted line) and h = 0.90 (upper line). Motion along the uniform cycle will be the case when the equality sign holds.

The stability condition for one of the two real solutions is as follows:

τ ≥ 1 − 1 − γ ( 1 − h ) θ h

(17)

but as Figure 4 illustrates, it is of no real-world relevance: y ≡ τ and x ≡ h with γ = 0.9 and γ = 0.95, and θ = 0.80 and θ = 0.85. The other real solution is unstable as shown in Appendix 1. Underground economy activities may turn out to be quite destabilising under an inappropriate tax policy. In the presence of underground economy, tax policy is quite significant in maintaining stability and fighting against instability; much more so when our results point to instability leading to crisis, and not just to persistence of some cyclical behaviour of the economy.

OPEN IN VIEWER

Figure 3

Stability in the Presence of an Underground Economy with One Real Root

Also, although we have not assumed anything about the sources of this economy, the results appear to confirm the traditional viewpoint that tax evasion is one major such source: Solving (16) for h:

h ≤ γ ​ ( 1 + ε ) 2 − 4 ε ( 1 + ε ) 2 [ γ − θ ( 1 − τ ) ]

(18)

with ^h/^τ < 0. Appendix 2 illustrates the interaction among h ≡ z, ϵ ≡ x, and τ ≡ y, for θ = 0.80 and γ = 0.90 when the equality sign holds in (18), that is, when the maximum percentage of the official economy is considered. The slightly positive correlation between h and τ in the figure of this Appendix is confirmed empirically by Williams (2014) and suggests that the viewpoint that ^h/^τ < 0 may be true only within the context of micro-economics. It is a viewpoint that does not address the circumstances in the overall economy that can justify an increase in taxes. Taxes may be raised when macro-economic performance is good and the motive thereby to go underground is weak. Consequently, the effect from those who do choose to do so for tax evasion purposes is more than offset by the operation of the accelerator.

Finally, note that once stability prevails, equilibrium output is higher in the presence of underground economy:

Y ˜ > Y ^ ⇒ I 0 + I 0 u + G 0 − θ τ 0 I 0 + G 0 − θ τ 0 > 1 − θ h ​ ( 1 − τ ) − γ ​ ( 1 − h ) 1 − θ ​ ( 1 − τ )

(19)

OPEN IN VIEWER

Figure 4

Stability in the Presence of an Underground Economy with Two Real Root

which is true because the left-hand side of this inequality is greater than 1, whereas the right-hand side is less than 1: 1 – θh(1 – τ) – γ(1 – h) < 1 – θ(1 – τ) ⇒ θ(1 – τ) < γ, which is true since θ ≤ γ. Therefore, (19) would still hold even if I ^u₀ = 0 in the left-hand side because it is this term that makes this side exceed 1. This result confirms the viewpoint that the underground economy does not crowd out the official one (Onoshchenko, 2012). It is as if the underground economy rewards/avenges the entire economy once tax policy is such that it can/cannot or does not want to sustain the minimal at least equilibrium output, which is found by inserting (16) with the equality sign in (15):

min Y ˜ = ( I 0 + I 0 u + G 0 + θ τ 0 ) ( 1 + ε ) 2 ( 1 + ε ) 2 − 4 ε

It is not that taxes only are to blame for the emergence of underground activities; this is of micro-economic mostly concern. It is that taxes (and money) are what matter for macro-economic performance regardless the micro-economics surrounding informal sector operations.

The destabilising more likely role of the underground economy and the ‘accelerator effect’ countering the ‘tax evasion effect’ are the two results of this investigation. The latter is more or less common sense but it has escaped the attention of the literature on tax evasion because of the microeconomic character of this literature. Therefore, let us focus here on the former result. It was more or less anticipated and the claim that the option of entering underground economy during a recession exercises a stabilising effect on the overall economy (Soldatos, 2016) can be valid only when the multiplier–accelerator interaction allows it. It is an interaction which may be further assessed in the light of studies distinguishing between formal and informal business cycles, implying in effect two separate but interrelated multiplier–accelerator interactions. Most of these studies are empirical without a theoretical foundation and with mixed evidence, because if not anything else there is always the problem of the definition and measurement of the underground economy. Are both cycles symmetric or one only of them or both of them? Are underground economy cycles pro-cyclical or countercyclical? Based exclusively on empirical research, no definite answer can be given to these questions (Granda-Carvajal, 2010) and hence, such studies cannot be useful to consult within the context of this article.

Nevertheless, there do exist a few empirical studies based on dynamic general equilibrium models and documenting that (a) at the phase level, in the ‘medium-run’ so to speak, ‘a rise in the level the informal economy size is associated with a larger and more frequent formal expansions and deeper formal recessions’ (Birinci & Elgin, 2010, p. 4) while (b) in the log-run, the existence of a ‘double cycle’ with opposite peaks and troughs appears to be the case (Russo, 2008). One thus is inclined to conclude that the destabilising role of the underground economy identified herein is a ‘temporary’ matter of the medium-run while the stabilising argument appears to hold in the long-run. The underground economy acts much like a countercyclical policy shock which is initially destabilising but beneficial with the passage of time. Instability and then order like in a seismogram after an earthquake, ‘double cycle’ order for us here, inducing one to think by extension that the inherent instability of the multiplier-accelerator model as such relates to the short- and medium-run rather than to the long-run.

These, of course, are remarks made with every caution as to their validity before the dynamic properties of the multiplier-accelerator model per se, and the relation between the unofficial and the official sector from a business cycle perspective are studied thoroughly. In any case, what is made clear in this article is that long-run stability under an operative multiplier presupposes the manipulation of taxation rather than of public spending as Guest and Makin (2011) have already cautioned. Judging from our result here concerning the interaction between accelerator and taxation and from Westerhoff’s (2006) instability result induced by nonlinear investor expectations, such a tax policy may prove to be a complex, delicate task.