1. Introduction
The optimal tax–debt mix to finance public expenditure provision has generated a lively debate in the literature. Public debt may create fiscal illusion and ultimately undermine growth (Khalid and Guan Reference Khalid and Guan1999); taxes reduce fiscal illusion by making agents aware of the costs of public provision, but they may create dead-weight losses (Tresch Reference Tresch2002). Furthermore, an increment in taxes may also generate or worsen tax evasion, which is a wide spread phenomenon and whose magnitude is quite relevant (Cebula and Feige, Reference Cebula and Feige2012). Recent estimates show that intentional under-reporting of income is about 18–19% of total reported income in the US (i.e. a tax gap of about 500 billion dollars) which may increase to about one trillion dollars if we take into account tax avoidance (Davison Reference Davison2021). In Europe, the level of tax evasion is about 20% of GDP, with a loss of about 1 trillion Euros (Buehn and Schneider Reference Buehn and Schneider2012; Murphy Reference Murphy2014; Albarea et al. Reference Albarea, Bernasconi, Marenzi and Rizzi2020).
In some countries (Greece and Italy for example) tax evasion is associated with a high public debt, but the relationship between these variables and growth is not clear, in spite of a growing literature (Argentiero and Cerqueti Reference Argentiero and Cerqueti2021; Halkos et al. Reference Halkos, Papageorgiou, Halkos and Papageorgiou2020; Schilirò Reference Schilirò2019). Public debt allows to increase public expenditure without raising the tax rate or reducing tax evasion; however, this process may undermine economic growth if debt becomes unsustainable.
In this note we study the effects of tax evasion on growth and debt sustainability by using an approach in the spirit of Blanchard et al. (Reference Blanchard, Chouraqui, Hagemann and Sartor1990). We define debt to be sustainable if the dynamics of the debt/GDP ratio follows a mean reverting process that converges toward a stable equilibrium (which is endogenous to the model; see Debrun et al., Reference Debrun, Ostry, Willems and Wyplosz2020) instead of in terms of an outside threshold for the debt/GDP ratio (Alloza et al. Reference Alloza, Andrés, Pérez and Rojas2020; Fournier and Fall Reference Fournier and Fall2017; Caner et al. Reference Caner, Grennes, Koehler-Geib and Koehler-Geib2010).
A representative consumer optimally chooses the intertemporal consumption and the evasion. If the agent is caught evading, a fine must be paid on the evaded yield. We show that, although the level of public expenditure which maximizes growth may not depend on these parameters, the latter reduces the range of parameters for which the debt/GPD ratio is mean reverting. This implies that the level of expenditure that maximizes growth and welfare may not be compatible with debt sustainability because of the burden that interests and debt create on the economy. The adverse impact of tax evasion is even more pronounced in economies with slower growth or those heavily reliant on public expenditure (where private capital productivity is relatively low). In such case, the debt-to-GDP ratio remains mean reverting only if the total factor productivity is sufficiently strong. From a policy perspective, our model highlights a trade-off: while tolerating tax evasion can boost expected optimal economic growth, it comes at the cost of undermining government debt stability. As a result, minimizing tax evasion is particularly crucial for economies that rely heavily on public expenditure to sustain economic growth.
2. Related literature
Debt sustainability is a controversial issue: the literature which has investigated sovereign debt sustainability (see Lindgren Reference Lindgren2021; Debrun et al., Reference Debrun, Ostry, Willems and Wyplosz2020; Willems and Zettelmeyer Reference Willems and Zettelmeyer2022 for a review) has not reached a definite conclusion.
Fournier and Fall (Reference Fournier and Fall2017) use OECD data to show that Governments are vulnerable to changes in macroeconomic conditions and interest rates.
In a controversial paper, Reinhart and Rogoff (Reference Reinhart and Rogoff2010) showed that debt/GDP above 90% may undermine growth, but according to Herndon et al. (Reference Herndon, Ash and Pollin2014) this is due to errors in the coding and the dataset, while, in actual fact, a high debt/GDP ratio could even slightly improve growth, as suggested by most of the theoretical papers that do not find any endogenous “dangerous” threshold for debt/GDP ratio (see Willems and Zettelmeyer (Reference Willems and Zettelmeyer2022) for a review.).
From an empirical point of view, the World Bank (Caner et al., Reference Caner, Grennes, Koehler-Geib and Koehler-Geib2010) argues that an alert threshold for the debt/GDP ratio is about 77% for mature economies and 64% for emerging ones, while Eberhardt and Presbitero (Reference Eberhardt and Presbitero2015) argue that a negative relationship may exist, but a common debt threshold does not exist when observed and unobserved heterogeneity across countries is taken into account.
The relationship between tax evasion, public debt, and growth has been rather unexplored from a theoretical point of viewFootnote 1 while from an empirical point of view (see Loayza, Reference Loayza2016) there seems to be only a negative relationship between the size of the informal sector and growth. Our model adds to the present literature by studying under which conditions the debt-to-GDP ratio follows a mean reverting path also in the presence of tax evasion without setting a specific threshold for the debt/GDP ratio.
3. The model
We model a stylized general equilibrium where Government sets the level of public expenditure that maximizes welfare. The agent is endowed with a production technology that allows to transform capital into yield through a constant return to scale production function whose arguments are capital and public expenditure as in Barro (Reference Barro1990); Futagami et al. (Reference Futagami, Iwaisako and Ohdoi2008); Minea and Villieu (Reference Minea and Villieu2013); Mirrlees et al. (Reference Mirrlees, Adam, Besley, Blundell, Bond, Chote, Gammie, Johnson, Myles and Poterba2011).
The representative agent maximizes the expected present value of their future utility by controlling their intertemporal consumption and tax evasion. Tax evasion adds uncertainty to these lifetime decisions because of random audits: if the agent is caught evading, they have to pay a fine on top of evaded taxes. Tax revenues are a source of uncertainty also for the government which may have to issue debt if the tax revenue is lower than expected. In this environment, government has to set the level of public expenditure that maximizes the agent’s value function, under the constraint that debt is sustainable; we assume that this condition is satisfied if the debt/GDP stochastic process is mean reverting, that is, the debt-to-GDP ratio converges to a stable equilibrium.
We first solve the problem for the agent in order to find their optimal consumption and tax evasion; we then compute the dynamics of the optimal debt/GDP ratio and define under which conditions it is mean reverting. Finally, we compute the public expenditure that maximizes growth and met the mean reverting conditions.
As in Barro (Reference Barro1990), we model an economy with a representative individual. GDP
$y_{t}$
is produced by using both private capital
$k_{t}$
and public expenditure
$G_{t}$
through a constant return to scale technology:
\begin{equation*} y_{t}=AG_{t}^{\beta }k_{t}^{1-\beta }, \end{equation*}
where
$A$
is the constant total factor productivity and
$\beta$
is the elasticity of the product w.r.t. public expenditure. We assume that public expenditure is set to be a constant proportion of private capital, that is,
\begin{equation*} G_{t}=gk_{t}, \end{equation*}
and, accordingly, the production function can be written as:
\begin{equation} y_{t}=Ag^{\beta }k_{t}. \end{equation}
Contrary to Barro (Reference Barro1990), our representative consumer takes
$g$
(rather than
$G_{t}$
) as a fixed parameter, that is, they can predict that a change in
$k_{t}$
will cause
$G_{t}$
to adjust, that is, we rule out fiscal illusionFootnote
2
.
Public expenditure is financed through a proportional tax
$\tau$
on income and through public debt. However, a fraction
$\phi$
of the total tax revenue
$\tau y_{t}$
is lost in tax collection activities: the higher the
$\phi$
, the more inefficient is Government. The agent is fully rational, that is, they can see the effect that public expenditure has on economic growth and can also observe collection cost.
$B_{t}$
is the total amount of public debt which: (i) increases because of its service at the rate
$r$
, (ii) increases because of the public expenditure
$G_{t}$
, (iii) decreases because of taxes collected on declared yield, and (iv) decreases when evasion (
$e_{t}$
) is caught and the agent must pay a fine
$\eta \left (\tau \right )$
on the evaded income
$e_{t}y_{t}$
.
As in Levaggi and Menoncin (Reference Levaggi and Menoncin2013); Bernasconi et al. (Reference Bernasconi, Levaggi and Menoncin2015), we assume that the audit happens according to a Poisson jump process. If we call
$d\Pi _{t}$
this process, it may have value either
$1$
(if an audit happens) with probability
$\lambda dt$
or
$0$
(if no audit happens). The constant parameter
$\lambda$
is the jump intensity.Footnote
3
Accordingly, the dynamics of the public debt is:
\begin{equation} dB_{t}=\left (B_{t}r+gk_{t}-\left (1-\phi \right )\tau \!\left (1-e_{t}\right )y_{t}\right )dt-\left (1-\phi \right )\eta \left (\tau \right )e_{t}y_{t}d\Pi _{t}, \end{equation}
where we assume that the same inefficiency
$\phi$
applies also when collecting fines following an audit. We immediately see that if an audit occurs (i.e.
$d\Pi _{t}=1$
) the debt reduces by the fee paid by the caught evader. Instead, if no audit happens, the last term of (2) is zero. On average the expected fine cashed by the government is
\begin{equation*} \mathbb {E}_{t}\left [\!\left (1-\phi \right )\eta \left (\tau \right )e_{t}y_{t}d\Pi _{t}\right ]=\left (1-\phi \right )\eta \left (\tau \right )e_{t}y_{t}\lambda dt. \end{equation*}
The audit regimes can be modeled through the shape of the function
$\eta \left (\tau \right )$
which allows to take into account several actual forms of fines.Footnote
4
The private capital
$k_{t}$
: (i) increases because of production
$y_{t}$
, (ii) decreases because of taxes on the non-evaded income
$\tau \!\left (1-e_{t}\right )y_{t}$
, (iii) decreases because of consumption
$c_{t}$
, (iv) increases because of the interest rate paid by Government to the bond holders
$B_{t}r$
, (v) decreases because of the loans to Government
$dB_{t}$
, and (vi) decreases because of the fee that must be paid on the audited income. Thus, we can write
\begin{equation} dk_{t}=\left (y_{t}-\tau \!\left (1-e_{t}\right )y_{t}-c_{t}+B_{t}r\right )dt-dB_{t}-\eta \left (\tau \right )e_{t}y_{t}d\Pi _{t}. \end{equation}
After substituting
$dB_{t}$
into the capital dynamics we get:
\begin{equation} \frac {dk_{t}}{k_{t}}=\left [\!\left (1-\phi \tau \!\left (1-e_{t}\right )\right )Ag^{\beta }-g-\frac {c_{t}}{k_{t}}\right ]dt-\phi \eta \left (\tau \right )Ag^{\beta }e_{t}d\Pi _{t}. \end{equation}
If there are no collection costs (i.e.
$\phi =0$
) evasion does not play any role: it is just a reallocation of fund between the agent and Government. The expected value of (4) is
\begin{equation*} \mathbb {E}_{t}\left [\frac {dk_{t}}{k_{t}}\right ]=\left (\left (1-\phi \tau \right )Ag^{\beta }-g-\frac {c_{t}}{k_{t}}+\phi \!\left (\tau -\lambda \eta \left (\tau \right )\right )Ag^{\beta }e_{t}\right )dt, \end{equation*}
and tax evasion is profitable, on average, if
$\tau \gt \lambda \eta \left (\tau \right )$
, that is, if the tax that must be paid is higher than the fee weighted by the intensity of being caught. Finally, the debt/GDP ratio has the following dynamics:
\begin{align} d\!\left (\frac {B_{t}}{y_{t}}\right ) & =\left (\frac {g^{1-\beta }}{A}-\left (1-\phi \right )\tau \!\left (1-e_{t}\right )-\frac {B_{t}}{y_{t}}\left [\!\left (1-\phi \tau \!\left (1-e_{t}\right )\right )Ag^{\beta }-g-\frac {c_{t}}{k_{t}}-r\right ]\right )dt\nonumber \\ & -\left (\frac {1-\phi }{\phi }-\frac {B_{t}}{y_{t}}Ag^{\beta }\right )\frac {\phi \eta \left (\tau \right )e_{t}}{1-\phi \eta \left (\tau \right )e_{t}Ag^{\beta }}d\Pi _{t}. \end{align}
3.1 The agent’s optimization problem
Over an infinite time horizon, a representative consumer maximizes his/her intertemporal utility which depends on the consumption of a private good (
$c_{t}$
). We assume that agent’s utility has a constant relative risk Aversion (
$\delta$
) and agent’s (constant) subjective discount is
$\rho$
. Thus, the optimization problem is
\begin{equation} \max _{\left \{ c_{t},e_{t}\right \} _{t\in \left [t_{0},\infty \right [}}\mathbb {E}_{t_{0}}\left [\int _{t_{0}}^{\infty }\frac {c_{t}^{1-\delta }}{1-\delta }e^{-\rho \!\left (t-t_{0}\right )}dt\right ], \end{equation}
with
$k_{t}$
following the dynamics in (4).
Proposition 1. Given the capital dynamics ( 4 ), the optimal consumption and evasion that solve Problem ( 6 ) are
\begin{align} \frac {c_{t}^{*}}{k_{t}} & =\frac {\rho }{\delta }+\frac {\delta -1}{\delta }\!\left (\left (1-\phi \tau \right )g^{\beta }A-g\right )\nonumber \\ & +\frac {\tau }{\eta \left (\tau \right )}\!\left (1-\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1}{\delta }}-\frac {1}{\delta }\!\left (1-\frac {\eta \left (\tau \right )\lambda }{\tau }\right )\right ),\end{align}
\begin{equation} e_{t}^{*}=\frac {1}{\phi \eta \left (\tau \right )g^{\beta }A}\!\left (1-\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1}{\delta }}\right ). \end{equation}
Proof. See Appendix A.
Corollary 2.
Given the capital dynamics (
4
), if the optimal evasion is zero (i.e.
$\tau =\eta \left (\tau \right )\lambda$
) the optimal consumption that solves Problem (
6
) is
\begin{equation} \left .\frac {c_{t}^{*}}{k_{t}}\right |_{\lambda =\frac {\tau }{\eta \left (\tau \right )}}=\frac {\rho }{\delta }+\frac {\delta -1}{\delta }\!\left (\left (1-\phi \tau \right )g^{\beta }A-g\right ). \end{equation}
Proof.
It is sufficient to substitute
$\lambda =\frac {\tau }{\eta \left (\tau \right )}$
in the result of Proposition 1.
From (7) and (8) we note that both the optimal consumption and the optimal evasion are constant fractions of income (we recall that, in this model, income is a linear transformation of capital as in (1)). This result is perfectly in line with the so-called “consumption smoothing” behavior that is often observed in the market.
It is interesting to note that if evasion is expedient (i.e.
$e^{*}\gt 0$
) and
$\delta \gt 1$
, the optimal consumption with evasion is always higher than that without evasion. To show this, let us consider the term in (7) that depends on the fiscal parameters
$\eta$
and
$\lambda$
, and set
$m:=\frac {\tau }{\eta \left (\tau \right )\lambda }$
. Consumption is higher with tax evasion if:
\begin{equation*} 1-m^{-\frac {1}{\delta }}\gt \frac {1}{\delta }\!\left (1-\frac {1}{m}\right ). \end{equation*}
When
$m=1$
the two sides of the inequality are zero, but for evasion to be expedient
$m\gt 1$
. In this case, the derivative on the left-hand side w.r.t.
$m$
(i.e.
$\frac {1}{\delta }m^{-\frac {1}{\delta }-1}$
) is higher than the same derivative on the right-hand side (i.e.
$\frac {1}{\delta }m^{-2}$
) if
$\delta \gt 1$
, and, accordingly,
$c$
is higher.
The optimal capital dynamics is
\begin{equation} \frac {dk_{t}^{*}}{k_{t}^{*}}=\frac {\!\left (1-\phi \tau \right )g^{\beta }A-g-\rho +\frac {\tau }{\eta \left (\tau \right )}-\lambda }{\delta }dt-\left (1-\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1}{\delta }}\right )d\Pi _{t}. \end{equation}
Should Government set the audit parameter to erase evasion (i.e.
$\lambda =\tau /\eta \left (\tau \right )$
), the optimal growth with
$e_{t}^{*}=0$
would be
\begin{equation} \left .\frac {dk_{t}^{*}}{k_{t}^{*}}\right |_{\lambda =\frac {\tau }{\eta \left (\tau \right )}}=\frac {\!\left (1-\phi \tau \right )g^{\beta }A-g-\rho }{\delta }dt. \end{equation}
From these results we see that the optimal consumption is an affine transformation of the capital growth rate
\begin{equation*} \frac {c_{t}^{*}}{k_{t}}=\rho +\!\left (\delta -1\right )\frac {1}{dt}\mathbb {E}_{t}\left [\frac {dk_{t}^{*}}{k_{t}^{*}}\right ]+\lambda \!\left (\delta -\left (\delta -1\right )\!\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1}{\delta }}-\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1-\delta }{\delta }}\right ). \end{equation*}
In particular, we can conclude that the optimal consumption is proportional to the expected capital growth. In other words, any policy that maximizes the expected growth of capital, will also maximize optimal consumption. Furthermore, since utility is a monotonic function of consumption, maximizing consumption also coincides with utility maximization.
Proposition 3. The government expenditure levels that maximize the agent’s welfare, the expected optimal economic growth, and the consumer’s consumption ratio, all coincide:
\begin{align*} \arg \max _{g}\frac {1}{dt}\mathbb {E}_{t}\left [\frac {dk_{t}^{*}}{k_{t}^{*}}\right ] & =\arg \max _{g}\mathbb {E}_{t}\left [\int _{t}^{\infty }U\!\left (c_{s}^{*}\right )e^{-\rho \!\left (s-t\right )}dt\right ]\\ & =\arg \max _{g}\frac {c_{t}^{*}}{k_{t}}. \end{align*}
3.2 Mean reverting conditions
The optimal debt/GDP ratio has a dynamics whose form is like in the following stochastic differential equation:
\begin{equation*} d\!\left (\frac {B_{t}}{y_{t}}\right )=p_{0}\!\left (p_{1}-\frac {B_{t}}{y_{t}}\right )dt+\text {stoch. var.}, \end{equation*}
where
$p_{0}$
and
$p_{1}$
are highly nonlinear combinations of all model parameters (see Appendix B). If the parameters
$p_{0}$
is positive, then the process converges toward a long-term equilibrium given by
$p_{1}$
. Instead, if
$p_{0}\lt 0$
, the process is divergent (independently of the value of
$p_{1}$
). The higher
$p_{0}$
, the faster the convergence toward the long-term equilibrium value. Thus, we can conclude what follows.
Proposition 4. The optimal debt/GDP is mean reverting if and only if
\begin{equation} \frac {\!\left (1-\phi \tau -\delta \!\left (1-\tau \right )\right )g^{\beta }A-g-\rho }{\delta }+\lambda \underset {=:\Lambda \left (m,\delta, \phi \right )}{\underbrace {\left (-\frac {m-1}{\phi }\!\left (1-m^{-\frac {1}{\delta }}\right )+\frac {\delta -1}{\delta }+\frac {m}{\delta }-m^{\frac {1}{\delta }}\right )}}\gt 0, \end{equation}
in which
$m:=\frac {\tau }{\eta \left (\tau \right )\lambda }$
. In this case, the long-term equilibrium of debt/GDP is
\begin{equation} \frac {\delta }{g^{\beta }A}\frac {g-\left (1-\phi \right )\tau g^{\beta }A+\frac {1-\phi }{\phi }\lambda \!\left (m^{1-\frac {1}{\delta }}-1\right )\!\left (m^{\frac {1}{\delta }}-1\right )}{\!\left (\left (\delta -\phi \right )\tau -\left (\delta -1\right )\right )Ag^{\beta }-g-\rho +\delta \lambda \!\left (\frac {1-m}{\phi }\!\left (1-m^{-\frac {1}{\delta }}\right )+\frac {\delta -1}{\delta }+\frac {m}{\delta }-m^{\frac {1}{\delta }}\right )}. \end{equation}
Proof. See Appendix B.
We recall that tax evasion is expedient if
$m\gt 1$
, while for
$m=1$
the optimal tax evasion is zero. Since
$\Lambda \!\left (1,\delta, \phi \right )=0$
, while the function
$\Lambda$
is always negative for any
$m\gt 1$
(as shown in Appendix B), then the second term of (12) undermines the mean reversion property of the debt/GDP ratio.
Public expenditure
$g$
may contribute to both convergence and growth, but in a different way so that both objectives may not be simultaneously pursued.
We can provide a strong interpretation of the debt/GDP equilibrium value in (13) if we take into account the case without evasion (i.e.
$m=1$
). In this case it is easy to show that (13) becomes
\begin{equation*} \frac {\frac {g-\left (1-\phi \right )\tau g^{\beta }A}{g^{\beta }A}}{\frac {\!\left (1-\phi \tau \right )Ag^{\beta }-g-\rho }{\delta }+\!\left (\tau -1\right )Ag^{\beta }}. \end{equation*}
The first fraction at the denominator coincides with the optimal capital growth (without evasion) as shown in (10). Let us call
$\gamma ^{*}$
this optimal growth rate. If we multiply and divide by
$k_{t}$
the fraction in the numerator we get
\begin{equation*} \frac {\frac {G_{t}-\left (1-\phi \right )\tau y_{t}}{y_{t}}}{\gamma ^{*}-\left (1-\tau \right )Ag^{\beta }}. \end{equation*}
This ratio can be seen as the present value of a perpetual annuity as follows:
\begin{equation*} \int _{t}^{\infty }\frac {G_{s}-\left (1-\phi \right )\tau y_{s}}{y_{s}}e^{-\left (\gamma ^{*}-\left (1-\tau \right )Ag^{\beta }\right )\left (s-t\right )}ds, \end{equation*}
if the optimal growth rate
$\gamma ^{*}$
is higher than the (net of tax) total factor productivity, which is the only convergence condition in this framework with optimal zero tax evasion.
Thus, we can conclude that the debt/GDP ratio converges to the discounted value of all the future deficit/GDP ratios and the discount rate is given by the optimal capital growth, reduced by the (net of tax) total factor productivity.Footnote
5
In this formula we see that the risk aversion parameter
$\delta$
affects only the discount rate in the growth of capital
$\gamma ^{*}$
. Instead, if tax evasion is expedient, risk aversion enters both the cash flows and the rate of discount and, thus, affects in a nontrivial way government debt sustainability (as shown in the numerical simulations).
3.3 Social welfare, tax evasion, and debt dynamics
In the previous sections we have shown that there exists a level of public expenditure that maximizes consumption, growth, and agent’s welfare simultaneously, but such level may not be compatible with debt sustainability, that is, with the convergence of the debt/GDP ratio toward an equilibrium value.
Thus, in order to guarantee that the debt/GDP is convergent over time, the government should set
$g$
to the level that maximizes one of the welfare measures, under the constraint that the mean reversion strength of the debt/GDP dynamics is positive. If we call
$J\!\left (t,k_{t};\ g\right )$
the value function of the agent (as defined in (17)), the Government problem can be written as follows:
\begin{align} & \max _{g}J\!\left (t,k_{t};\ g\right )\\ & \text {s.t.}\nonumber \\ & \frac {\!\left (1-\phi \tau \right )g^{\beta }A-g-\rho }{\delta }-\left (1-\tau \right )Ag^{\beta }+\lambda \!\left (\frac {1-m}{\phi }\!\left (1-m^{-\frac {1}{\delta }}\right )+\frac {\delta -1}{\delta }+\frac {m}{\delta }-m^{\frac {1}{\delta }}\right )\gt 0.\nonumber \end{align}
This problem makes sense because public debt allows Government to disentangle the tax rate
$\tau$
from the public expenditure
$g$
. In fact, without debt, the amount
$g$
should satisfy a budget constraint. If we call
$T_{t}$
the net income of the government, its dynamics without debt would be
\begin{equation*} dT_{t}=\left (\left (1-\phi \right )\tau \!\left (1-e_{t}^{*}\right )y_{t}-gk_{t}\right )dt+\!\left (1-\phi \right )\eta \left (\tau \right )e_{t}^{*}y_{t}d\Pi _{t}, \end{equation*}
whose expected value is
\begin{equation*} \mathbb {E}_{t}\left [dT_{t}\right ]=\left (\left (1-\phi \right )\!\left (\tau -\tau e_{t}^{*}+\eta \left (\tau \right )e_{t}^{*}\lambda \right )y_{t}-gk_{t}\right )dt, \end{equation*}
and, accordingly, the level
$g$
should satisfy the condition
$\mathbb {E}_{t}\left [dT_{t}\right ]=0$
.
Furthermore, in our framework the Government may be tempted to tolerate evasion for allowing a higher expected economic growth, according to the following result.
Proposition 5. The expected economic growth with evasion is always higher than that achieved without evasion:
\begin{equation*} \frac {1}{dt}\mathbb {E}_{t}\left [\frac {dk_{t}^{*}}{k_{t}^{*}}\right ]\gt \frac {1}{dt}\mathbb {E}_{t}\left [\left .\frac {dk_{t}^{*}}{k_{t}^{*}}\right |_{e^{*}=0}\right ]. \end{equation*}
Proof. If we substitute the optimal economic growth from (10), we get
\begin{equation*} \frac {\!\left (1-\phi \tau \right )g^{\beta }A-g-\rho +\frac {\tau }{\eta \left (\tau \right )}-\lambda }{\delta }-\lambda \!\left (1-\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1}{\delta }}\right )\gt \frac {\!\left (1-\phi \tau \right )g^{\beta }A-g-\rho }{\delta }, \end{equation*}
which becomes
\begin{equation*} \frac {1}{\delta }\frac {\tau }{\eta \left (\tau \right )\lambda }+\!\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1}{\delta }}\gt \frac {1}{\delta }+1, \end{equation*}
and it is easy to show that this inequality always holds for any
$\tau \gt \eta \left (\tau \right )\lambda$
.
However, while evasion may increase the economic growth, it also reduces government revenue; the optimal debt/GDP ratio is likely to increase with tax evasion and its dynamics may explode.
If the Government is not constrained by the mean reverting condition on debt/GDP, the public expenditure rate that maximizes growth/value function/consumption is
\begin{equation} g_{growth}^{*}=\left (A\beta \!\left (1-\phi \tau \right )\right )^{\frac {1}{1-\beta }}, \end{equation}
which is easily obtained from (10). If this is not the case the level of public expenditure compatible with mean reversion is the one for which the constraint in (14) is still greater than zero.
The algebraic solution to Problem (14) does not give any true policy insight since it is a highly nonlinear combination of the model parameters. Thus, in the following section we propose a numerical simulation.
3.4 Numerical simulation
Some further insights into the relationship between growth, tax evasion and debt sustainability may be gained through some numerical simulations that allow to better understand the role of the different structural parameters in this relationship.
Table 1 shows the values of the parameters used for the benchmark case.
Table 1. Values of the parameters for the baseline simulation
*For an emerging economy see https://www.imf.org/external/datamapper/NGDP_RPCH@WEO/OEMDC
As a measure of tax inefficiency we have used some recent estimates of tax collection activities (OECD 2011) which are somehow a lower bound estimate of these costs. The marginal productivity of public expenditure is derived from recent OECD estimates while the fiscal and total productivity parameters are in line with Bernasconi et al. Reference Bernasconi, Levaggi and Menoncin2020 while A in the benchmark has been chosen to replicate the growth of emerging economies, which rely more than mature ones on public expenditure as a drive for growth. Figure 1 shows the value of
$J\!\left (t,k_{t};\ g\right )$
as a function of
$g$
, the ratio of public expenditure to income/production. The curve is green if the value of
$g$
satisfied the mean reverting condition and red when this condition is not met.
Figure 1. Value function
$J(t,k_{t};\ g)$
as a function of public expenditure
$g$
. In the upper left graph the base case is drawn using the parameters in Table 1. In the upper right graph
$A=0.42$
. In the lower left graph,
$\beta =0.15$
. In the lower right graph,
$\delta =1.07$
. In the green (red) section of the curve, the debt/GDP ratio is convergent (divergent).
In the baseline case (upper left) the economic system is not strong enough to support tax evasion. The highest level of public expenditure for which debt is sustainable is much lower than the level that maximizes the value function (about 40% less). On the contrary, evasion is over 6 point percent higher (0.504 against 0.478) than the level that would allow to get a sustainable debt. Growth as well as welfare is lower as expected. It would be necessary to increase the productivity
$A$
in order to get a level of public expenditure that both maximizes growth and secures debt sustainability. In our example the value of 0.42 (almost double the baseline) would reduce the gap between optimal and sustainable public expenditure, even if the mean reverting constraint is still binding.
The bottom left graph shows the role of the private capital productivity (
$1-\beta$
). A value of
$\beta =0.15$
(i.e. an economy that relies less on public expenditure to grow, as it would be the case of a more mature economy) would allow to set expenditure to its optimal level and the debt to be mean reverting.
Finally, in the bottom right graph we show the role of risk aversion: for lower levels of risk aversion, the debt/GDP is not mean reverting.
If debt/GDP dynamic is not mean reverting, Government will have to reduce public expenditure to get back on track. From (12) we see that, if
$\delta \lt \frac {1-\phi \tau }{1-\tau }$
, the public expenditure rate that maximizes convergence, that is, the public expenditure that makes debt/GDP to converge as quick as possible, is:
\begin{equation} g_{conv}^{*}=\left (A\beta \!\left (1-\phi \tau -\delta \!\left (1-\tau \right )\right )\right )^{\frac {1}{1-\beta }}, \end{equation}
and we can immediately conclude that
\begin{equation*} g_{conv}^{*}\lt g_{growth}^{*}, \end{equation*}
which means that the level of public expenditure that maximizes growth may not be compatible with debt/GDP convergence.
Optimal government expenditure
$g_{growth}^{*}$
depends neither on the audit parameters, nor on tax evasion. This is an interesting result of our model: the presence of the debt allows Government to set public expenditure independently on the level of tax evasion, but this does not prevent evasion from having undesired effects.
3.5 The effect of evasion on an equilibrium budget
In a model without evasion the public expenditure which maximizes the objectives in Proposition 3 is given byFootnote 6
\begin{equation*} g_{E}=\left (\left (1-\phi \tau \right )\beta A\right )^{\frac {1}{1-\beta }}, \end{equation*}
and the tax rate
$\tau _{E}$
which guarantees the equilibrium of the Government budget balance (without debt) solves
\begin{equation*} \tau _{E}y_{t}=g_{E}k_{t}, \end{equation*}
which becomes
\begin{equation*} \tau _{E}A\!\left (g_{E}\right )^{\beta }k_{t}=g_{E}k_{t}, \end{equation*}
and whose solution is
\begin{equation*} \tau _{E}=\frac {\beta }{1+\phi \beta }. \end{equation*}
This is the tax rate that we will use in the numerical simulations as a basic framework. In Figure 2 we show the same simulations we performed in the previous subsection, but with the tax defined in this way. The tax rate resulting from applying the values presented in Table 1 is lower than the basis value used in the previous simulations. Thus, we decided to increase the value of
$\lambda$
(and take
$\lambda =0.097$
) in order to keep a reasonable value for the optimal evasion.
Figure 2. Value function
$J(t,k_{t};g)$
as a function of public expenditure
$g$
. In the upper left graph the base case is drawn with parameters in Table 1, but with
$\tau =\frac {\beta }{1+\phi \beta }$
, and
$\lambda =0.097$
. In the upper right graph
$A=0.42$
. In the lower left graph,
$\beta =0.255$
and
$A=0.42$
. In the lower right graph,
$\delta =1.5$
. In the green (red) section of the curve, the debt/GDP ratio is convergent (divergent).
All the graphs of the figure show a narrower range of public expenditure
$g$
that allows for debt/GDP convergence. In particular, the converge happens at a cost of a smaller value function or growth. Thus, we can conclude that the presence of the public debt allows the Government to increase the public expenditure by tolerating a level of evasion without preventing the debt/GDP to be stable over time.
4. Conclusions
Debt sustainability, its effect of debt on economic growth, and its relationship with tax evasion have become one of the most well-researched topics recently. We show that debt allows Governments to set the level of expenditure that maximizes growth also in the presence of tax evasion. However, in a general economic framework (where the equilibrium interest rate depends on the factor productivity), we show that tax evasion always reduces the range of parameters for which the debt/GDP ratio is mean reverting. Furthermore, mean reversion can be achieved with a sufficiently high total factor productivity whose increment also increases welfare (production is more efficient), reduces tax evasion and increases investments (consumption increases at a lower rate than income). Thus, evasion shows to be a problem especially for low-growth economies, which should try to reduce tax evasion as much as possible. This result is quite interesting in the light of recent estimates (Loayza Reference Loayza2016) showing that the size of the informal sector (a proxy for tax evasion) dampens economic growth. In this respect our model could be either interpreted as theoretical support for those findings or as the signal of a more structural problem. We have shown that, in order to preserve the debt/GDP sustainability, a relatively big informal sector leads to less growth. Furthermore, it is possible to use public expenditure and other fiscal policies to increase growth only if Government reduces its size.
From a policy perspective, our model reveals that while tolerating tax evasion may enhance expected optimal economic growth, it simultaneously undermines the stability of government debt. Therefore, tax evasion should be minimized as much as possible, particularly in economies that depend heavily on public expenditure to drive economic growth.
For future research, we plan to take into account the opportunity for the Government to sell its debt abroad in order to reduce the importance of the link between evasion and debt stability.
Acknowledgements
No external funds received; no conflict of interest to report.
Appendix A. Proof of Proposition 1
Given the optimization problem, we can define the value function
$J\!\left (t,k_{t}\right )$
as follows:
\begin{equation} J\!\left (t,k_{t}\right )=\max _{\left \{ c_{s},e_{s}\right \} _{s\in \left [t,\infty \right [}}\mathbb {E}_{t}\left [\int _{t}^{\infty }U\!\left (c_{s}\right )e^{-\rho \!\left (s-t\right )}ds\right ], \end{equation}
which must solve the following partial differential equation (so-called Hamilton–Jacobi–Bellman (HJB) equation):
\begin{align*} 0 & =J_{t}-\left (\rho +\lambda \right )J+J_{k}\!\left (\left (1-\phi \tau \right )g^{\beta }Ak_{t}-gk_{t}\right )\\ & +\max _{c_{t}}\left \{ \frac {c_{t}^{1-\delta }}{1-\delta }-J_{k}c_{t}\right \} \\ & +\max _{e_{t}}\left \{ J_{k}\phi \tau e_{t}g^{\beta }Ak_{t}+\lambda J\!\left (k_{t}-\phi \eta (\tau )e_{t}y_{t}\right )\right \}, \end{align*}
in which the lower scripts on
$J$
indicate partial derivatives. We take the guess function for
$J$
in the following form:
\begin{equation*} J=F^{\delta }\frac {k_{t}^{1-\delta }}{1-\delta }, \end{equation*}
where
$F$
is assumed to be constant such that the following equations is solved:
\begin{align*} 0 & =-\left (\rho +\lambda \right )F^{\delta }\frac {k_{t}^{1-\delta }}{1-\delta }+F^{\delta }k_{t}^{-\delta }\!\left (\left (1-\phi \tau \right )g^{\beta }Ak_{t}-gk_{t}\right )\\ & +\max _{c_{t}}\left \{ \frac {c_{t}^{1-\delta }}{1-\delta }-F^{\delta }k_{t}^{-\delta }c_{t}\right \} \\ & +\max _{e_{t}}\left \{ F^{\delta }k_{t}^{-\delta }\phi \tau e_{t}g^{\beta }Ak_{t}+\lambda F^{\delta }\frac {\!\left (k_{t}-\phi \eta (\tau )e_{t}y_{t}\right )^{1-\delta }}{1-\delta }\right \} . \end{align*}
The first-order condition on consumption is
\begin{equation*} c_{t}^{*}=\frac {k_{t}}{F}, \end{equation*}
while the first-order condition on evasion is
\begin{equation*} e_{t}^{*}=\frac {1}{\phi \eta \left (\tau \right )g^{\beta }A}\!\left (1-\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1}{\delta }}\right ). \end{equation*}
Once these optimal values are substituted into the HJB equation, we get
\begin{align*} 0= & -\left (\rho +\lambda \right )F^{\delta }\frac {k_{t}^{1-\delta }}{1-\delta }+F^{\delta -1}\frac {k_{t}^{1-\delta }}{1-\delta }+F^{\delta }k_{t}^{1-\delta }\!\left (\left (1-\phi \tau \right )g^{\beta }A-g\right )\\ & -F^{\delta -1}k_{t}^{1-\delta }+F^{\delta }k_{t}^{1-\delta }\frac {\tau }{\eta \left (\tau \right )}\!\left (1-\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1}{\delta }}\right )+\lambda F^{\delta }k_{t}^{1-\delta }\frac {\!\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1-\delta }{\delta }}}{1-\delta }, \end{align*}
which gives
\begin{equation*} \frac {1}{F}=\left (\rho +\lambda \right )\frac {1}{\delta }+\frac {\delta -1}{\delta }\!\left (\left (1-\phi \tau \right )g^{\beta }A-g+\frac {\tau }{\eta \left (\tau \right )}\right )-\lambda \!\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1-\delta }{\delta }}. \end{equation*}
When
$F$
is substituted into the optimal consumption the result of the proposition is obtained.
Appendix B. Proof of Proposition 4
After plugging the optimal consumption and evasion in the debt/GDP dynamics we obtain the following expected value:
\begin{align*} \mathbb {E}_{t}\left [d\!\left (\frac {B_{t}}{y_{t}}\right )\right ] & =\left (\frac {g^{1-\beta }}{A}-\left (1-\phi \right )\tau +\frac {1-\phi }{\phi }\frac {\lambda }{g^{\beta }A}\!\left (\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1}{\delta }\!\!-1}-1\right )\!\left (\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{-\frac {1}{\delta }}-1\right )\right )dt\\ & -\frac {B_{t}}{y_{t}}\left [\frac {\!\left (1-\phi \tau \right )g^{\beta }A-g+\frac {\tau }{\eta \left (\tau \right )}-\left (\rho +\lambda \right )}{\delta }+\lambda \!\left (1-\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{-\frac {1}{\delta }}\right )-r\right ]dt, \end{align*}
which is a mean reverting process if the coefficient of
$\frac {B_{t}}{y_{t}}$
in the square brackets on the right-hand side is positive. Let us now investigate the general equilibrium solution by substituting the value of the interest rate
$r$
with its equilibrium value:
\begin{equation} r^{*}=\left (1-\tau \right )Ag^{\beta }+\frac {\tau -\eta \left (\tau \right )\lambda }{\phi \eta \left (\tau \right )}\!\left (1-\left (\frac {\eta \left (\tau \right )\lambda }{\tau }\right )^{\frac {1}{\delta }}\right ), \end{equation}
which is obtained by setting the interest rate to the level of the expected marginal product of private capital.
We recall that the expected production net of taxes is given by
\begin{equation*} y_{t}-\tau y_{t}\!\left (1-e_{t}^{*}\right )-\lambda \eta \left (\tau \right )e_{t}^{*}y_{t}, \end{equation*}
and after substituting the optimal evasion and computing the derivative w.r.t.
$k_{t}$
, we get the result shown in the text.
After substituting for the equilibrium value of the interest rate (18), the debt/GDP dynamics becomes
\begin{align*} \mathbb {E}_{t}\left [d\!\left (\frac {B_{t}}{y_{t}}\right )\right ] & =\left (\frac {g^{1-\beta }}{A}-\left (1-\phi \right )\tau +\frac {1-\phi }{\phi }\frac {\lambda }{g^{\beta }A}\!\left (m^{1-\frac {1}{\delta }}-1\right )\!\left (m^{\frac {1}{\delta }}-1\right )\right )dt\\ & -\frac {B_{t}}{y_{t}}\left [\frac {\!\left (1-\phi \tau \right )g^{\beta }A-g-\rho }{\delta }-\left (1-\tau \right )Ag^{\beta } \right. \\ &\left.+\lambda\, \underset {=:\Lambda \left (m,\delta, \phi \right )}{\underbrace {\!\left (\frac {1-m}{\phi }\!\left (1-m^{-\frac {1}{\delta }}\right )+\frac {\delta -1}{\delta }+\frac {m}{\delta }-m^{\frac {1}{\delta }}\right )}}\right ]dt, \end{align*}
in which, for the sake of simplicity, we set
$m:=\frac {\tau }{\eta \left (\tau \right )\lambda }$
, and the sustainability condition is the one written in the proposition.
If this process is mean reverting, then it converges to the ratio of the first term and the coefficient of
$\frac {B_{t}}{y_{t}}$
. Furthermore, it is easy to show that
\begin{equation*} \Lambda \!\left (1,\delta, \phi \right )=0, \end{equation*}
\begin{equation*} \left .\frac {\partial \Lambda }{\partial m}\right |_{m=1}=\left .-\frac {1}{\phi }\!\left (1-m^{-\frac {1}{\delta }}\right )-\frac {1}{\delta }\frac {m-1}{\phi }m^{-\frac {1}{\delta }-1}+\frac {1}{\delta }-\frac {1}{\delta }m^{\frac {1}{\delta }-1}\right |_{m=1}=0, \end{equation*}
\begin{align*} \left .\frac {\partial ^{2}\Lambda }{\partial m^{2}}\right |_{m=1} & =\left .-\frac {1}{\delta }\frac {1}{\phi }m^{-\frac {1}{\delta }-1}-\frac {1}{\delta }\frac {1}{\phi }m^{-\frac {1}{\delta }-1}-\frac {1}{\delta }\frac {m-1}{\phi }\!\left (-\frac {1}{\delta }-1\right )m^{-\frac {1}{\delta }-2}-\frac {1}{\delta }\!\left (\frac {1}{\delta }-1\right )m^{\frac {1}{\delta }-2}\right |_{m=1}\\ & =\frac {1}{\delta ^{2}}\frac {1}{\phi }\!\left (\left (\phi -2\right )\delta -\phi \right )\lt 0. \end{align*}
The sign of the first derivative is negative for
$\phi =1$
and, thus, it is also negative for any lower value of
$\phi$
. Finally, we can conclude that
$\Lambda \!\left (m,\delta, \phi \right )$
is always negative in the domain, decreasing and concave.
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