To finance public expenditure a government needs to raise revenue, which mainly comes from taxes and borrowings. During a financial crisis, however, financing of budget deficit is particularly difficult because of a rise in debt servicing costs that crowd out other expenses and raise the concern for government solvency. In extreme cases, governments are constrained to tax, as borrowing opportunities are strictly limited or unavailable. Still, governments can choose from tax menu options (income and consumption taxes), given the flexibility of the tax mix. This article presents a long-term dynamic model of fiscal solvency that shows the equilibrium the revenue maximising government can obtain with reasonable tax rates when capital income can be shifted and there are constraints on the consumption tax. Specifically, the solution predicts a positive level of bonds in the long-term equilibrium and the tax rates dependent positively on the abundance of the tax bases.

#### Keywords

- dynamic modelling
- capital tax
- consumption tax
- fiscal solvency

#### JEL Classification

- C61
- H21
- H26

To finance public expenditure a government needs to raise revenue, which mainly comes from taxes and borrowings. During a financial crisis, however, financing of budget deficit is particularly difficult because of a rise in debt servicing costs that crowd out other expenses and raise the concern for government solvency. In extreme cases, governments are constrained to tax, as borrowing opportunities are strictly limited or unavailable. Still, governments can choose from tax menu options (income and consumption taxes), given the flexibility of the tax mix, that is, the substitutability of one tax for another as a revenue source. However, any two taxes are not perfectly substitutable with each other, but they do differ in their impact on capital and labour supply, output, prices, risk aversion, income distribution and so on. Moreover, taxation is often motivated by non-fiscal reasons, such as redistributive objectives or curbing production and consumption of good with negative external effects. All of these make designing the optimal tax system a challenging task.

Although the theoretical literature has presented a number of arguments for differential taxation, we are still far

The model presented hereafter takes into account two different taxes, on capital and consumption but not on labour, and concentrates on the fiscal solvency without detailed specification of the labour market. This implies one production factor (capital) and two sources of budgetary revenues: capital income and consumption. Consumption is only indirectly linked to capital by its net remuneration. The government budget constraint is complemented with bonds revenue. We analyse the problem of determining the capital-consumption tax mix with the possibility of public debt issuance. It means that the optimisation problem has to be set by the government, not by economic agents. The government’s choice is limited by the fiscal solvency rule and possible income shifting hindering tax collection. We strive to determine some features of the long-term equilibrium, solving dynamic optimisation problem with continuous time. In other words, we are looking for optimal long-term tax policy of the government aiming to sustain fiscal solvency, contingent on the values of chosen parameters. We assume capital income to be partially or fully shifted abroad as a consequence of tax avoidance strategies of economic agents. We believe that this better reflects the limited mobility of physical capital and the treat of possible income shifting. For example, one can expect the latter to intensify during an insolvency crisis when fast liquidation of physical investments is not feasible but the possibility of capital income withdrawal is not affected. It should be noted that taxation of consumption depends greatly on the values of external parameters because only part of consumption is explicit and can be taxed.

The proposed approach exhibits several derogations from the standard analyses. First, we assume maximisation of budget revenues instead of utility (or welfare) maximisation, that is, the government is of the Leviathan type. Therefore, the objective function does not require the determination of a specific utility form and better fits to the governmental behaviour during financial distress when constraints on borrowing are binding. Government revenues come from the two tax instruments (capital and consumption taxes) and bond issuance. Second, the analysis concentrates on the values of capital, consumption, bonds and the respective tax rates in equilibrium. Next, the effects of parameter deviations from equilibrium are calculated with the procedure originally proposed by Boadway (1979).

Moreover, we propose three more extensions to the typical multi-tax dynamic model. First, the bond issuance is positive in the steady state but its level is limited by the ratio of tax revenues to the product of labour and capital. Fulfilling the constraint ensures fiscal solvency of the country. This is a simplifying assumption that allows for introducing the solvency rule into the model without detailed specification of the financial market. The idea stems from creditworthiness and measures the capacity of the market to accept the given level of governmental bond issue. Hence, the bonds are not a transitory phenomenon of budgetary financing, and they remain positive in the equilibrium simply because they provide less distortion to the economy than taxes (on capital and consumption). At the same time, there is an upper limit on the tax burden and debt financing that does not violate the fiscal solvency postulate. This assumption better fits to the persistent nature of debt and prevents the debt to be zero in the long term. Second, taxing of capital income can be partially avoided by shifting it abroad. The shifting possibility is exogenous because it depends on the existing ‘tax shifting technology’ and the level of foreign ownership of capital. The tax shifting technology includes the existence of tax havens and a group of controlled enterprises that may generate tax savings. At a given moment, it is determined by local tax regulations and tax enforcement efficiency. Third, the consumption tax can be raised but there are some external factors limiting its maximum rate as consumption tax burden may be partly avoided. They are not directly modelled but stay linked with legislative or competitive reasons such as cross-border purchases or shadow economy expansion. It rules out the possibility of fully taxing consumption as an equivalent of imposing distortionary taxes on capital.

There exists equilibrium in the model for the sufficiently small shifting parameter and time preference parameter. The obtained equilibrium has one internal (non-boundary) solution with positive bond issuance and two positive tax rates. Despite the fact that the model cannot be solved analytically, it allows for testing the impact of some parameter changes on the tax rates, bonds and the value of state variables (capital and consumption) in the equilibrium. The numerical simulation includes the impact of four important parameters on the state variables and two tax rates. The parameters cover capital income shifting, the market reaction on the indebtedness of a given country to the proxy of production, the explicit consumption rate (the part of consumption which is taxed) and intertemporal preferences.

The article is structured as follows. First, a dynamic model of government revenue maximization in continuous time is described. In the next step, we present the sketch of dynamic optimisation procedure involving the description of differential equations of control variables. This lets us carry out numerical analyses curbed with control variables (tax rates) bounded to the range 0–1. The analysis concentrates on the impact of the four parameters affecting consumption, capital, the capital tax, the consumption tax and the issuance of bonds. Then the time-varying rates of change are provided according to the procedure originally proposed by Boadway (1979). The article ends with conclusions.

Government maximises the revenues consisting of tax receipts of consumption _{t}

The control variables in the model are the tax rates on consumption _{c}_{k}

This excludes the possibility of consumption or capital subsidisation sometimes postulated in optimal taxation models.

The first two elements of equation (1) describe the tax revenues. Consumption is divided into two sets: explicit (taxable) and implicit (untaxed). The share of these two sets of consumption is specified by the parameter _{c}_{c}_{k}

Apart from the taxes, the government can issue bonds. By and large, debt financing is superior to tax revenues because it does not generate distortions to an economy such as taxes levied on the factors of production or consumption. However, the size of bond issuance is not arbitrary. We assume that the level of new bonds is determined on the financial market and requires meeting a pre-specified ratio of tax revenues to the output. The fulfilment of this ratio means fiscal solvency. As the production function in the equilibrium is not known (it depends, amongst others, on the relation between capital and technological progress), we assume that it could be substituted by the product of capital as a proxy of the output. Accordingly, the bond issuance is the greater, the higher is the tax revenues in relation to the product of capital. In other words, the new debt level depends positively on the relation similar to the rate of fiscalism, that is, the ratio of tax revenues to the output. Nevertheless in the contrast to the rate of fiscalism, we augment the formula with non-negative parameter

where

There are two state variables: consumption and capital. Each of them is dependent on taxation. Specifically, changes in consumption over time are driven by two tax rates, and changes in capital depend on capital tax rate. Differential equations for the state variables are written as follows:

where

Consumption is determined by disposable income (the expression in the square brackets of equation (3), the proportion of taxed and untaxed consumption,

Capital is created from capital income, net of tax and gross consumption. The government does not create capital in the economy. The depreciation of capital (similar to that in Wildasin, 2011) is omitted in the model merely because we find it less important for modelling the decision taken by the government striving for the long-term solvency.

Taking the objective function and the differential equations for the state variables, we can formulate the dynamic maximisation problem:

where the constraints on the state variables are given by equations (3) and (4).

The necessary condition for the problem (5) under the constraints (3) and (4) can be expressed using the current-value Hamiltonian function _{1} and _{2} denote the right-hand sides of equations (3) and (4), respectively. Then,

By maximising

Then, the equations of motion for the co-state variables can be written as follows:

Equation (7) defines the control variables as functions of the state and co-state

and co-state variables as functions of the state and control variables:

Taking the time derivatives of equation (9) and substituting equations (3), (4), (8) and (10), we get the differential equations that describe the dynamics of the control variables (see Appendix for discussion):

Finally, we should find the solutions of the four-dimensional system of nonlinear differential equations describing the dynamics of state and control variables (equations (3), (4) and (11)). Unfortunately, this problem cannot be solved analytically, but we can formulate the following proposition:

such that 0 < _{c}_{k}

where

and

The described model cannot be solved analytically, so we need to run numerical simulation to discover some features of the obtained solution. The value of parameters used in the numerical simulation should help to describe the behaviour of the two control variables: the level of bond issues and state variables. We found

The first scenario examines the impact of a parameter

ensure that the consumption will not be tax avoided. The consumption increases at first and then decreases at the equilibrium as the a becomes larger. The increasing dynamic of capital tax is probably responsible for the consumption drop. Despite the rising taxation, the capital is growing all the time together with explicit consumption share (Fig. 1–3).

The second scenario describes the impact of parameters referring to shifting (or hiding) capital income. The high

The third scenario involves the behaviour of intertemporal preferences. Higher preferences for the future require the increase in capital and consumption but the proportion of these two state variables is changing differently with

Let _{c}_{k}

where α =

it is necessary to derive the ‘variational equations’ (see: Boadway 1979, Wildasin 2011). If

where _{x}^{*}_{α}(^{*}^{*}

To see how changes in the chosen parameter affect the solution to equation (12), we assume that the dynamic system (12) is in the equilibrium. Then we increase the chosen parameter (initial value of the parameter that is perturbed is indicated in braces) and we observe how this perturbation evolves over time. This allows for numerical simulation. The parameters of the simulations are set to

The time-varying rate of change in the solution to the system (12) with respect to the parameter

One can observe that change in

The time-varying rate of change in the solution to the system (12) with respect to the parameter

The change in

The change in

There are some interesting numerical results referring to the equilibrium. First, the higher the explicit part of consumption (the taxed consumption), the higher are the taxation of capital, taxation of consumption and the level of debt. The latter is a consequence of dynamic constraint on bond issuance. The level of bond indebtedness is positively affected by the level of tax revenues. In other words, higher revenue allows for higher level of debt. But for very high level of capital taxation, we can expect rapid fall of the consumption.

If consumption tax is not efficient in generation of revenue (because of high level of untaxed consumption), then there is no policy providing for the high level of revenues. The consumption achieves the maximum value for some level of explicit consumption, and for higher values of explicit, consumption is decreasing. The latter effect is triggered by very high capital taxation. However, the tax rates on capital and consumption are independent in many cases (for

The changes in parameters at the equilibrium trigger the adjustment over time. In most cases, this adjustment was monotonic but sometimes (for capital or capital tax) the adjustment process involves the changes in both directions. It can indicate that taxation of capital is very vulnerable to the changes in parameter, and it is difficult to obtain sustainable solution after such a change.

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