Equity-efficiency dilemma and tax harmonization

Dalamagas, Basil A.; Leventides, John; Tantos, Stefanos Article Equity-efficiency dilemma and tax harmonization Central European Economic Journal (CEEJ) Provided in Cooperation with: Faculty of Economic Sciences, University of Warsaw Suggested Citation: Dalamagas, Basil A.; Leventides, John; Tantos, Stefanos (2022) : Equity-efficiency dilemma and tax harmonization, Central European Economic Journal (CEEJ), ISSN 2543-6821, Sciendo, Warsaw, Vol. 9, Iss. 56, pp. 342-353, https://doi.org/10.2478/ceej-2022-0020 This Version is available at: https://hdl.handle.net/10419/324578 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Central European Economic Journal, 9(56), 342-353. DOI: 10.2478/ceej-2022-0020 To link to this article: https://doi.org/10.2478/ceej-2022-0020 Equity-efficiency dilemma and tax harmonization Basil Dalamagas, John Leventides, Stefanos Tantos Open Access. © 2022 B. Dalamagas, J. Leventides, S. Tantos, published by Sciendo. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. Basil Dalamagas National and Kapodistrian University of Athens, Department of Economics, Sofokleous & Aristidou 1, P.C.10559, Athens, Greece corresponding author: [email protected] John Leventides National and Kapodistrian University of Athens, Department of Economics, Sofokleous & Aristidou 1, P.C.10559, Athens, Greece Stefanos Tantos National and Kapodistrian University of Athens, Department of Economics, Sofokleous & Aristidou 1, P.C.10559, Athens, Greece Equity-efficiency dilemma and tax harmonization Abstract The present paper attempts to demonstrate that finding an appropriate trade-off between direct and indirect taxes can help smooth policy makers’ way through reconciling the contradictory notions of equity and efficiency. Our theoretical and empirical analysis is based on the assumption that direct taxes discourage work effort, thus impinging on the incentives to supply labour, to save and to invest, and finally, to grow, whereas indirect taxes discourage consumption and bear more heavily on the poor. Central to our discussion is the argument that carefully designed adjustments in the tax mix can reduce distortions in the consumption-leisure decision, thus leading to an optimal allocation of resources between the equity and efficiency objectives. To derive a competitive equilibrium setting, a social welfare function is maximized and the first-order conditions are manipulated to trace out the optimal direct-indirect tax rates that pave the way for the equity-efficiency goals to be reconciled with each other. Keywords equity-efficiency trade off, direct-indirect tax rates, growth, income distribution, tax harmonization | JEL Codes H21, H30, E62 1. Introduction The main task of policy makers is to create a favourable economic environment for achieving high rates of growth at a low inflation level. A long-term growing GDP ensures that sufficient tax revenue would be raised to finance the provision of social services and attain optimal allocation of resources and welfare maximization. This in turn would contribute towards achieving the four desired fiscal goals: stabilization, growth, fair distribution of income, and efficient allocation of resources. Unfortunately, the above government targets do not seem easily attainable. For example, over the years we have had numerous debates about whether equity (income distribution) should or should not take priority over efficiency (growth). Researchers appear to have well-founded arguments over which of the two policy plans to support or to reject. The reason for the dispute is that growth without a properly designed policy of reducing income inequality may lead any country to a state of sociopolitical turmoil, whereas an income redistribution policy with low or zero growth tends to undermine the resilience of any social plan, given the scarcity of available resources. The heated controversy surrounding the equityefficiency trade-off, ever since the adoption of fiscal policy measures, has added a new dimension to politics via the emergence of parties that align with the notion of equity (socialists) or efficiency (liberals). CEEJ • 9(56) • 2022 • pp. 342-353 • ISSN 2543-6821 • DOI: 10.2478/ceej-2022-0020 344 For a detailed presentation and description of the equity-efficiency trade-off, as well as for an extensive argument in support of (or against) either of these principles, there is a good bibliography that deserves recognition for its inclusiveness and organization. See, for example, Saez and Stantcheva (2018), Chiappori and Mazzocco (2017), Gayle and Shephard (2019), Farhi and Gabaix (2020), Stevans (2012), Forbes (2000), Dawson (1998), Deininger and Squire (1996), Perotti (1993), Barro (1999), Saez (2001), Tillmann (2005), Slemrod (1990), Bird (1992), Alesina and Rodrik (1994), Okun (2015). A general conclusion that can be drawn from a review of these articles is that most of them argue over whether equity and efficiency are rivals or complementary factors in the design and implementation of economic policy. A number of researchers contend that equity and efficiency cannot be achieved simultaneously, as a greater equity comes at the cost of a loss of efficiency: income redistribution results in changes in work effort, in savings, and in physical and human capital, thus leading to less efficiency. This argument, however, is challenged by the incomplete markets constructive framework and sociopolitical theories of an inverse relationship between inequality and growth. Finally, other researchers claim that equity and efficiency complement each other. 2. The scope of the present study The aim of the present study is to move the discussion from the ongoing equity-efficiency tradeoff or conflict to the establishment of a well-founded relationship between the direct-indirect tax trade-off on the one hand and the equity-efficiency trade-off on the other, given that taxation is the main source of financing for the equity/efficiency policy goals. If each of these two categories of taxes could be assigned solely to one of the policy targets, then policy makers would be able to redirect resources to the most desired objective, simply by altering the size and the structure of the tax revenue, provided that the electorate would approve proposed fiscal policy changes. There are strands of literature on the effects of income/consumption taxes on equity (income redistribution) and efficiency (incentives and growth). For example, progressive labour taxes are shown to have a positive effect on income distribution (Journard et al., 2012). The corporate profits tax in open economies tends to fall on labour income and worsen income distribution, but in countries with a low mobility of capital, corporate taxes may discourage investment projects (Harberger, 1995). Consumption taxes tend to be shifted to consumers, with lowerincome households invariably paying a larger portion of their income on necessities. Therefore, indirect taxes are expected to result in income inequality (IMF, 2014). Theoretical discussion showing so many noticeable and opposing viewpoints on the economic effects of direct/indirect taxes implies that an appropriate mix of these taxes should be devised to attain the equity/ efficiency goals (Musgrave, 1959) and maximize social welfare. Atkinson and Stiglitz (1976) claim that a fair income distribution can be achieved by income taxation alone without employing consumption taxes. However, Cremer et al. (2001) hold the opposite view that, in a tax mix of direct/indirect taxes, the benefits of commodity taxes are of a strong redistributive nature. More of an interest should also be taken in the work of García‐Peñalosa and Turnovsky (2011), who examine how changes in the combinations of direct/indirect taxes affect income distribution. For example, an increase in income taxes ─ far in excess of consumption taxes ─ that would introduce disincentives to work effort and reduce labour supply are associated with lower output but also with lower income inequalities. A great contribution to the literature on the effects of direct/indirect taxes and their combinations on equity and/or efficiency has also been made by Chari et al. (2020), Heathcote et al. (2020), Stiglitz (2018), Aghion et al. (2016), Uchida and Ono (2021), Beffy et al. (2019), Jantti et al. (2015), Keane et al. (2016), Stiglitz (2015), Chang et al. (2021), and Reis (2020). The present study aims at contributing to the design of an optimal direct/indirect tax rate combination that would be perfectly in tune with the ideal of incorporating two opposing fiscal targets (equity-efficiency) into a single and manageable objective. To this end, a social welfare function is developed to amalgamate the widely used index of equity – that is, consumption or income – with the conventional index of efficiency – that is, the leisure or labour force supply. The utility function is then maximized with respect to the direct and indirect tax CEEJ • 9(56) • 2022 • pp. 342-353 • ISSN 2543-6821 • DOI: 10.2478/ceej-2022-0020 345 rate and the first-order conditions are manipulated to provide the optimal tax mix that can promote simultaneously both policy objectives. In carrying out the estimation process for a sample of six developed countries, a number of assumptions is adopted: 1) According to the preceding analysis, direct taxes – that is, progressive personal income taxes and proportional corporate taxes – are taken to discourage the creation of human and physical capital and hence to limit the prospects for growth, while ensuring a fair distribution of income. Conversely, indirect taxes, such as a VAT with a uniform proportional tax rate, favor income inequalities, since the indirect tax burden as a percentage of income rises as income falls. 2) It becomes evident that the present study sidesteps the question of whether the equity of efficiency principle is the most preferred choice, by focusing our attention on the inner equilibrium obtained from the first-order conditions. It is well known that the ratio of the marginal utilities derived from maximizing the utility function with respect to direct and indirect tax rates represents the marginal rate of substitution (MRS) in consumption between the above tax rates. By the same token, the ratio of the cost shares in terms of distortions that are caused by the introduction of taxation to equity or efficiency represents the marginal rate of transformation in production (MRT) between direct and indirect taxes. If there are no distorting factors arising from the level or the structure of direct-indirect tax rates, the manipulation of the first-order conditions leads to Pareto optimality, where MRS=MRT and no government intervention is required. 3) In the prevailing imperfect competition framework, however, private agents and fiscal authorities tend to escape the Pareto optimality rules, so that a scope is provided for policy makers to restore public confidence in the management of the economy. The present study extends the analysis beyond the conventional Pareto’s optimality conditions by postulating noncompetitive markets, which are characterized by the existence of monopoly power and distortions through price setting. Private agents are not taken to act as independent units and to interact via the price system, whereas externalities and public goods give rise to divergencies between the private valuation and the social valuation of a wide array of services provided by private and public agents. What should be stressed at this point is that the present study does not give any attention either to the description and the analysis of the nature of the distortions or to the establishment of the causes of their existence. Instead, we will concentrate on the lack of equality, MRS≠MRT, due to the existence of distortions, and the design of an optimal mix of direct/indirect tax rates that would help redress the balance, MRS=MRT. 4) Lastly, the present study further extends the traditional approach of evaluating the impact of taxation on equity (efficiency) in a way that restricts the analysis to include the effects of income taxes alone. The notion of mixing taxes requires that combinations of both direct and indirect tax-rate changes should be tested before finding a solution that is reasonable and acceptable to all agents. The underlying assumption is that consumers’ behaviour is significantly affected by adjustments in indirect tax rates, which are shown to “force wedges” between post-tax and pre-tax prices. The rest of the paper is organized as follows: In section 3, we outline some of the important considerations that have been largely ignored by the conventional analysis, by using econometric techniques, simulations, and mathematical tools to underline the practical implications of incorporating our theoretical work into the sphere of applied fiscal policy management. The empirical investigation of the model is presented in section 4, and section 5 concludes the discussion, laying out directions for further work. 3. The model 3.1. The linear optimization model The utility maximization model that will be built in the present study to determine the most suitable combination of direct/indirect taxes for attaining the optimal allocation of resources between equity and efficiency will be presented in two working forms, the linear and the logarithmic. The linear model has the property of putting the analysis into a form that is understandable to the reader by considering the circumstances under which distortions are generated and then eliminated. The logarithmic model facilitates developing computational techniques which lead to manageable solutions. CEEJ • 9(56) • 2022 • pp. 342-353 • ISSN 2543-6821 • DOI: 10.2478/ceej-2022-0020 346 Starting with the linear model, consider an economy consisting of H households indexed h=1…H. Each household has a utility function 𝑈𝑈𝑈𝑈 ℎ =𝑈𝑈𝑈𝑈 ℎ [𝑐𝑐𝑐𝑐 ℎ �𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖,𝑦𝑦𝑦𝑦 ℎ �,𝑙𝑙𝑙𝑙 ℎ �𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑤𝑤𝑤𝑤�] (1) =𝜇𝜇𝜇𝜇1𝑈𝑈𝑈𝑈1�𝑐𝑐𝑐𝑐1(𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖,𝑦𝑦𝑦𝑦1),𝑙𝑙𝑙𝑙1�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑤𝑤𝑤𝑤��+∑𝜇𝜇𝜇𝜇ℎ 𝐻𝐻𝐻𝐻 ℎ=2 �𝑈𝑈𝑈𝑈ℎ�𝑐𝑐𝑐𝑐ℎ�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑦𝑦𝑦𝑦ℎ�,𝑙𝑙𝑙𝑙ℎ�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑤𝑤𝑤𝑤��−𝑈𝑈𝑈𝑈ℎ�−−𝜆𝜆𝜆𝜆(𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝐶𝐶𝐶𝐶+𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤−𝐺𝐺𝐺𝐺) (2) (1) where ch is the consumption of household h of the vector of private goods and lh is the household’s h supply of labour. Private consumption is taken to be a function of disposable income, y, and the indirect tax rate, ti, whereas labour supply depends on an exogenously determined wage rate, w, and the direct tax rate, ty. To characterize the set of first-best or Paretoefficient allocations, each household chooses ch and lh, h=1…H, to maximize their utility level, constrained by the requirement that all the other households, 2 to H, obtain given utility levels, and by the condition that the government will raise sufficient revenue to finance the provision of public goods. Varying the given utility levels for households 2 to H traces out the set of Pareto-efficient allocations. The Lagrangian for this maximization problem is written 𝑈𝑈𝑈𝑈ℎ=𝑈𝑈𝑈𝑈ℎ[𝑐𝑐𝑐𝑐ℎ�𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖,𝑦𝑦𝑦𝑦ℎ�,𝑙𝑙𝑙𝑙ℎ�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑤𝑤𝑤𝑤�] (1) =𝜇𝜇𝜇𝜇1𝑈𝑈𝑈𝑈1�𝑐𝑐𝑐𝑐1(𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖,𝑦𝑦𝑦𝑦1),𝑙𝑙𝑙𝑙1�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑤𝑤𝑤𝑤��+∑𝜇𝜇𝜇𝜇ℎ 𝐻𝐻𝐻𝐻 ℎ=2 �𝑈𝑈𝑈𝑈ℎ�𝑐𝑐𝑐𝑐ℎ�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑦𝑦𝑦𝑦ℎ�,𝑙𝑙𝑙𝑙ℎ�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑤𝑤𝑤𝑤��−𝑈𝑈𝑈𝑈ℎ�−−𝜆𝜆𝜆𝜆(𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝐶𝐶𝐶𝐶+𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤−𝐺𝐺𝐺𝐺) (2) 𝑈𝑈𝑈𝑈ℎ=𝑈𝑈𝑈𝑈ℎ[𝑐𝑐𝑐𝑐ℎ�𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖,𝑦𝑦𝑦𝑦ℎ�,𝑙𝑙𝑙𝑙ℎ�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑤𝑤𝑤𝑤�] (1) =𝜇𝜇𝜇𝜇1𝑈𝑈𝑈𝑈1�𝑐𝑐𝑐𝑐1(𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖,𝑦𝑦𝑦𝑦1),𝑙𝑙𝑙𝑙1�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑤𝑤𝑤𝑤��+∑𝜇𝜇𝜇𝜇 ℎ 𝐻𝐻𝐻𝐻 ℎ =2 �𝑈𝑈𝑈𝑈 ℎ �𝑐𝑐𝑐𝑐 ℎ �𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑦𝑦𝑦𝑦 ℎ �,𝑙𝑙𝑙𝑙 ℎ �𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑤𝑤𝑤𝑤��−𝑈𝑈𝑈𝑈 ℎ �−−𝜆𝜆𝜆𝜆(𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝐶𝐶𝐶𝐶+𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤−𝐺𝐺𝐺𝐺) (2) 𝑈𝑈𝑈𝑈ℎ=𝑈𝑈𝑈𝑈ℎ[𝑐𝑐𝑐𝑐ℎ�𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖,𝑦𝑦𝑦𝑦ℎ�,𝑙𝑙𝑙𝑙ℎ�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑤𝑤𝑤𝑤�] (1) =𝜇𝜇𝜇𝜇1𝑈𝑈𝑈𝑈1�𝑐𝑐𝑐𝑐1(𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖,𝑦𝑦𝑦𝑦1),𝑙𝑙𝑙𝑙1�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑤𝑤𝑤𝑤��+∑𝜇𝜇𝜇𝜇ℎ 𝐻𝐻𝐻𝐻 ℎ=2 �𝑈𝑈𝑈𝑈ℎ�𝑐𝑐𝑐𝑐ℎ�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑦𝑦𝑦𝑦ℎ�,𝑙𝑙𝑙𝑙ℎ�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦,𝑤𝑤𝑤𝑤��−𝑈𝑈𝑈𝑈ℎ�−−𝜆𝜆𝜆𝜆(𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝐶𝐶𝐶𝐶+𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤−𝐺𝐺𝐺𝐺) (2) _ (2) where C=∑H h=1ch, L=∑H h=1lh, and Uh is the fixed utility level that is achieved by h=2…H. The term μh may be interpreted as the social welfare weight that is given to each household (μh=1 for h=1). Assuming that the specified utility levels can be reached simultaneously, the necessary conditions describing the optimal choice of consumption with respect to the indirect tax rate and the optimal choice of working time with respect to the direct tax rate are 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖=𝜇𝜇𝜇𝜇 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖−𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆+𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖�=0 (3) (3) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦=𝜇𝜇𝜇𝜇 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦−𝜆𝜆𝜆𝜆𝜆𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤+𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦�=0 (4) (4) dividing (3) by (4) and rearranging terms gives 𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈ℎ 𝜕𝜕𝜕𝜕𝑐𝑐𝑐𝑐ℎ 𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙ℎ = �𝑐𝑐𝑐𝑐+𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖�𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖 �� 𝜕𝜕𝜕𝜕𝑐𝑐𝑐𝑐ℎ 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖 [𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤+𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤�𝜕𝜕𝜕𝜕𝑤𝑤𝑤𝑤 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦�] 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 ℎ 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦 (5) (5) The left-hand term in (5), i.e. the ratio of the marginal utilities (∂Uh)/(∂ch)/(∂Uh)/(∂lh)=(∂lh)/(∂ch) represents the marginal rate of substitution in consumption between consumption and leisure (or labour), MRSc,l. The term MRSc,l measures the extent to which the utility (benefit) of the household from the consumption of the good will increase if they are willing to sacrifice an extra unit of leisure or to work an extra unit of time in the production of this good (demand function). It is presumed throughout that (i) the response of consumption spending to changes in indirect tax rates is equalized across households and to the population as a whole, i.e., 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖 =𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖 (6) (6) and (ii) the response of labour supply to changes in direct tax rates is equalized across households and to the population as a whole, i.e., 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦=𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦 (7) (7) Rearranging the terms in (5) gives 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦 =𝜕𝜕𝜕𝜕+𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑖𝑖𝑖𝑖 𝑤𝑤𝑤𝑤𝜕𝜕𝜕𝜕+𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤�𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦� (8) (8) Given that the entire disposable income is assumed to be consumed, i.e., wL=C, and following the simple mathematical formula 1+𝐴𝐴𝐴𝐴 1+𝐵𝐵𝐵𝐵=1+𝐴𝐴𝐴𝐴−𝐵𝐵𝐵𝐵 (9) (9) CEEJ • 9(56) • 2022 • pp. 342-353 • ISSN 2543-6821 • DOI: 10.2478/ceej-2022-0020 347 where 𝐴𝐴𝐴𝐴=𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖�𝜕𝜕𝜕𝜕𝑐𝑐𝑐𝑐ℎ 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖� and 𝐵𝐵𝐵𝐵=𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤(𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙ℎ 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦), , equation (9) takes the form 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦 =𝜕𝜕𝜕𝜕+𝐴𝐴𝐴𝐴 𝜕𝜕𝜕𝜕+𝐵𝐵𝐵𝐵 =1+𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶 1+𝐵𝐵𝐵𝐵 𝐶𝐶𝐶𝐶=1+𝐴𝐴𝐴𝐴 𝜕𝜕𝜕𝜕−𝐵𝐵𝐵𝐵 𝜕𝜕𝜕𝜕=𝜕𝜕𝜕𝜕+𝐴𝐴𝐴𝐴−𝐵𝐵𝐵𝐵 𝜕𝜕𝜕𝜕=𝜕𝜕𝜕𝜕+𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖�𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖�−𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤(𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦) 𝜕𝜕𝜕𝜕 (10) 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦 =𝜕𝜕𝜕𝜕+𝐴𝐴𝐴𝐴 𝜕𝜕𝜕𝜕+𝐵𝐵𝐵𝐵 =1+𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶 1+𝐵𝐵𝐵𝐵 𝐶𝐶𝐶𝐶=1+𝐴𝐴𝐴𝐴 𝜕𝜕𝜕𝜕−𝐵𝐵𝐵𝐵 𝜕𝜕𝜕𝜕=𝜕𝜕𝜕𝜕+𝐴𝐴𝐴𝐴−𝐵𝐵𝐵𝐵 𝜕𝜕𝜕𝜕=𝜕𝜕𝜕𝜕+𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖�𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖 �−𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤(𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦 ) 𝜕𝜕𝜕𝜕 (10) (10) The second term of the left-hand side of (10) is [(∂ch)/(∂ti)]/[(∂lh)/(∂ty)] = [(∂ch)/(∂lh)]×[(∂ty)/(∂ti)] and the term ∂ch/∂lh represents the marginal rate of transformation in production between consumption and labour. The term MRTc,l represents the hourly cost of employing an extra unit of labour in the production (and consumption) of the good (the marginal product of labour or the wage rate, w), i.e., the cost of transforming an extra unit of labour in the production (and consumption) of the good (supply function). Therefore, eq. (10) can be written 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ=𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑐𝑐𝑐𝑐,𝑙𝑙𝑙𝑙=𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑐𝑐𝑐𝑐,𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦 𝜕𝜕𝜕𝜕+𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖�𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖�−𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤�𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦� 𝜕𝜕𝜕𝜕 (11) (11) Thus, it becomes evident that, if taxation and other distorting factors are introduced into the analysis, Pareto efficiency can be achieved (MRSc,l=MRTc,l) only when the term 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦[𝑐𝑐𝑐𝑐+𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖�𝜕𝜕𝜕𝜕𝑐𝑐𝑐𝑐ℎ 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 �−𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤�𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙ℎ 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡 𝑦𝑦𝑦𝑦 � 𝑐𝑐𝑐𝑐] is equal to 1, or when 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖=1+𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖�𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑖𝑖𝑖𝑖� 𝑐𝑐𝑐𝑐−𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤�𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦� 𝑐𝑐𝑐𝑐 (12) (12) In eq. (12), we note that: 1) The term ∂ch/∂ti demonstrates the effect of a unit change in indirect tax rate on consumption. Since this relationship is expected to be negative, the term ∂ch/∂ti may be interpreted as the marginal loss of consumption (utility) resulting from a unit increase in indirect tax rate. 2) The term w((∂lh)/(∂ty)) represents the effect of a unit change in direct tax rate on labour supply (with the wage rate being treated as numeraire). Assuming that this relationship may bear a negative (positive) sign, the term w((∂lh)/(∂ty)) is taken to measure the loss (gain) in terms of hours worked (or income) that arises from a unit increase in the income tax rate. 3) In practice, the equilibrium condition MRSc,l =MRTc,l can rarely be satisfied. This occurs only if the effects of changes in indirect tax rates on consumption, as well as the effects of direct tax rate changes on labour supply, are exactly balanced out. In this case, which is an exception to the rule, (∂lh)/(∂ty)=-(∂ch)/(∂ti) and ti=ty, so that (∂ti)/(∂ty)=1. 4) When these two kinds of effects differ in magnitude, and given that reactions of both consumers and workers to taxation are not directly controlled by the government, it is only the fiscal instruments ty and ti that can be used by policy makers to ensure equilibrium between MRSc,l and MRTc,l . The extent to which direct and indirect tax rates can be re-combined to equalize MRSc,l and MRTc,l can be estimated from the solution of (12) for ty, in order to generate an equilibrium reaction function. Solving (12) for ty results in a complicated formula, as shown in Appendix 1, which however cannot be easily manipulated by policy makers to achieve fiscal objectives. A readily manageable form of the reaction function is given below by eq. (13a), 𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦=𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌−𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖 𝐶𝐶𝐶𝐶 (𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦 )+ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖+𝐶𝐶𝐶𝐶(𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡 𝑦𝑦𝑦𝑦 −𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 ) (𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦)2 (13a) (13a) where ρ stands for a constant. To simplify the analysis, we assume that labour supply is a linear function of the income tax rate, the wage rate, and a set of other explanatory variables (X), i.e., 𝑙𝑙𝑙𝑙ℎ=𝑏𝑏𝑏𝑏0+𝑏𝑏𝑏𝑏1𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦+𝑏𝑏𝑏𝑏2𝑤𝑤𝑤𝑤+𝑏𝑏𝑏𝑏3𝑋𝑋𝑋𝑋 𝑐𝑐𝑐𝑐ℎ=𝑎𝑎𝑎𝑎0+𝑎𝑎𝑎𝑎1𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖+𝑎𝑎𝑎𝑎2𝑤𝑤𝑤𝑤𝑙𝑙𝑙𝑙ℎ+𝑎𝑎𝑎𝑎3𝑍𝑍𝑍𝑍 while consumption spending is a linear function of the indirect tax rate, the wage income, wlh, and a set of its own explanatory variables (Z), i.e., 𝑙𝑙𝑙𝑙ℎ=𝑏𝑏𝑏𝑏0+𝑏𝑏𝑏𝑏1𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦+𝑏𝑏𝑏𝑏2𝑤𝑤𝑤𝑤+𝑏𝑏𝑏𝑏3𝑋𝑋𝑋𝑋 𝑐𝑐𝑐𝑐ℎ=𝑎𝑎𝑎𝑎0+𝑎𝑎𝑎𝑎1𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖+𝑎𝑎𝑎𝑎2𝑤𝑤𝑤𝑤𝑙𝑙𝑙𝑙ℎ+𝑎𝑎𝑎𝑎3𝑍𝑍𝑍𝑍 CEEJ • 9(56) • 2022 • pp. 342-353 • ISSN 2543-6821 • DOI: 10.2478/ceej-2022-0020 348 The partial derivatives of the above functions with respect to tax rates are the constant coefficients b1=(∂lh)/(∂ty), and a1=(∂ch)/(∂ti). Therefore, eq. (13a) can be re-written 𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦=𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌−𝑏𝑏𝑏𝑏1 𝐶𝐶𝐶𝐶𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖+𝑎𝑎𝑎𝑎1 𝑏𝑏𝑏𝑏1𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖+𝜕𝜕𝜕𝜕(𝑏𝑏𝑏𝑏1−𝑎𝑎𝑎𝑎1) (𝑏𝑏𝑏𝑏1)2 (13b) (13b) where the constant, ρ , describes the initial conditions. Following a similar procedure in solving (13a), a reaction function in which ti is expressed in term of ty may also be formulated. Eq. (13b) measures the required change in the direct tax rate following a one-percentage point increase (decrease) in the indirect tax rate. These tax-rate changes are introduced by fiscal authorities in order to eliminate distortions in demand and/ or in labour supply. However, eq. (13b), even in its simplified form, cannot be easily subjected to closer study and our discussion will fail to emphasize some crucial aspects of fiscal policy. These aspects are most relevant to elaborating the practical implications of employing tax-policy instruments, in order to handle distortions in the economy. This occurs because some technical features of the economic system that are strictly necessary for later analysis are obscured by the quite complicated relationship between direct and indirect tax rates. In Section 3.2, we provide an analytical foundation for tax-policy intervention in the economy, in order to cope with distortions in consumer’s (and/or worker’s) behaviour, by using a non-linear utility function. Such a function allows us to express the variables of interest in terms of elasticities rather than in terms of changes in absolute values. 3.2. The logarithmic model Keeping all the assumptions of Section 3.1 in the modified framework, the utility function takes the form Uh=Uh [lnch (ti), lnlh (ty)] while the Lagrangian for the maximization problem can be written =𝑈𝑈𝑈𝑈 ℎ �𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑐𝑐𝑐𝑐 ℎ (𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖),𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 ℎ �𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦��−ln[𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝐶𝐶𝐶𝐶(𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖)] −ln[𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤�𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦�] (14) (14) where lnch=a0+a1lnti+a2ln(wL)+a3ln(CPI), and lnlh= b0+b1ln(ty)+b2 lnw The necessary condition describing the choice of the indirect tax rate is 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖=𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑐𝑐𝑐𝑐 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑐𝑐𝑐𝑐 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖−�1+𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖�=0, or or 𝑎𝑎𝑎𝑎1𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑐𝑐𝑐𝑐 ℎ =1+𝑎𝑎𝑎𝑎1 (15a) (15a) For the choice of the level of the direct tax rate, optimizing with respect to lnty gives 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦=𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦−�1+𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕(𝜕𝜕𝜕𝜕) 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑦𝑦𝑦𝑦�=0, or or 𝑏𝑏𝑏𝑏1𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 ℎ =1+𝑏𝑏𝑏𝑏1 (15b) (15b) given that the wage rate is treated as an exogenous variable dividing (15b) by (15a) gives 𝑏𝑏𝑏𝑏1(𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 ℎ ) 𝑎𝑎𝑎𝑎1(𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑐𝑐𝑐𝑐 ℎ ) =1+𝑏𝑏𝑏𝑏1 1+𝑎𝑎𝑎𝑎1 (16) (16) It is well known from the preceding discussion that (i) 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑐𝑐𝑐𝑐 ℎ �=𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑐𝑐𝑐𝑐 ℎ 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 ℎ �=𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑐𝑐𝑐𝑐 ℎ ,𝑙𝑙𝑙𝑙 ℎ (ii) 𝑎𝑎𝑎𝑎1 𝑏𝑏𝑏𝑏1= 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝐿𝐿𝐿𝐿 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦=𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝐿𝐿𝐿𝐿𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖=𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑐𝑐𝑐𝑐,𝑙𝑙𝑙𝑙(𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖) Therefore, we can recast eq. (16) to make it consistent with eq. (11) of the linear model: 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑐𝑐𝑐𝑐 ℎ ,𝑙𝑙𝑙𝑙 ℎ =𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑐𝑐𝑐𝑐,𝑙𝑙𝑙𝑙�𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖�1 + 𝑏𝑏𝑏𝑏1 1 + 𝑎𝑎𝑎𝑎1��=𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑐𝑐𝑐𝑐,𝑙𝑙𝑙𝑙�𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖(1 + 𝑏𝑏𝑏𝑏1−𝑎𝑎𝑎𝑎1)� 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑐𝑐𝑐𝑐ℎ,𝑙𝑙𝑙𝑙ℎ=𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑐𝑐𝑐𝑐,𝑙𝑙𝑙𝑙�𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖�1 + 𝑏𝑏𝑏𝑏1 1 + 𝑎𝑎𝑎𝑎1��=𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑐𝑐𝑐𝑐,𝑙𝑙𝑙𝑙�𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖(1 + 𝑏𝑏𝑏𝑏1−𝑎𝑎𝑎𝑎1)� (17) Eq. (17) implies that market equilibrium exists (MRS=MRT) only if CEEJ • 9(56) • 2022 • pp. 342-353 • ISSN 2543-6821 • DOI: 10.2478/ceej-2022-0020 349 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖(1 + 𝑏𝑏𝑏𝑏1−𝑎𝑎𝑎𝑎1)= 1, that is, if , that is, if 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦= 1 + 𝑏𝑏𝑏𝑏1−𝑎𝑎𝑎𝑎1 (18) ∫𝑑𝑑𝑑𝑑𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖= (1 + 𝑏𝑏𝑏𝑏1−𝑎𝑎𝑎𝑎1)∫𝑑𝑑𝑑𝑑𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦, or (18) The interpretation of (18) is quite similar to that of the linear model (see eq.11), that is, a) Pareto efficiency results only if • the direct-tax elasticity of labour supply and the indirect-tax elasticity of consumption goods exactly offset each other, and • the percentage changes in both direct and indirect tax rates are exactly the same in size. b) When the direct-tax elasticity of labour supply differs from the indirect-tax elasticity of consumption, then the only way to achieve Pareto efficiency is to change the structure of the tax system (tax mix) by placing greater emphasis on direct or indirect taxation. In the usual case of asymmetric responsiveness of labour supply and consumption to changes in (in) direct tax rates, restructuring of the tax system is required to redress the balance. Policy makers have to reschedule the ratio of the proportional changes in the two categories of tax rates in a way that eliminates the distortions in demand and labour market. To establish a reasonable relationship between indirect and direct tax rates, eq. (18) is solved for ti to generate the following reaction function: 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦= 1 + 𝑏𝑏𝑏𝑏1−𝑎𝑎𝑎𝑎1 (18) ∫𝑑𝑑𝑑𝑑𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖= (1 + 𝑏𝑏𝑏𝑏1−𝑎𝑎𝑎𝑎1)∫𝑑𝑑𝑑𝑑𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦, or , or 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖=(1 + 𝑏𝑏𝑏𝑏1−𝑎𝑎𝑎𝑎1)𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦+𝑘𝑘𝑘𝑘 𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖=𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦 1+𝑏𝑏𝑏𝑏1−𝑎𝑎𝑎𝑎1 (19) where k is a constant that captures the initial conditions in the economy. From the last equation, we receive 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖=(1 + 𝑏𝑏𝑏𝑏1−𝑎𝑎𝑎𝑎1)𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦+𝑘𝑘𝑘𝑘 𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖=𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦 1+𝑏𝑏𝑏𝑏1−𝑎𝑎𝑎𝑎1 (19) (19) A reverse relationship between the above two tax rates can also be found by dividing (15a) by (15b) and then by replicating the procedure that led to eq. (19): 𝑡𝑡𝑡𝑡𝑦𝑦𝑦𝑦=𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖1+𝑎𝑎𝑎𝑎1−𝑏𝑏𝑏𝑏1 (20) 𝐺𝐺𝐺𝐺 𝑌𝑌𝑌𝑌= 𝑇𝑇𝑇𝑇 𝑌𝑌𝑌𝑌(=𝜏𝜏𝜏𝜏𝜏)=𝛵𝛵𝛵𝛵𝑦𝑦𝑦𝑦 𝑌𝑌𝑌𝑌+𝛵𝛵𝛵𝛵𝑖𝑖𝑖𝑖 𝑌𝑌𝑌𝑌(= 𝜏𝜏𝜏𝜏𝜏𝑦𝑦𝑦𝑦+𝜏𝜏𝜏𝜏𝜏𝑖𝑖𝑖𝑖) (21) (20) In order to measure the response of the indirect tax rate to changes in the direct tax rate in a practicable and manageable way that would help fiscal authorities to properly re-design the tax structure, we must turn to the empirical investigation of our model by using data from the national accounts of the countries considered. 4. Empirical evidence The motive of our analysis is to provide practical policy recommendations. This implies that existing tax rules must be capable of being adjusted to whatever an optimal tax mix would dictate for improving social welfare. All the data series used in estimating the parameters of the above relationships have been taken from Ameco Database (Eurostat) and OECD Statistics. To present estimates of optimal (in)direct tax rates, the first step is to maximize a social welfare function and manipulate the resulting first-order conditions with a view towards establishing a Pareto-efficient tax structure. The procedure used for this was discussed in Section 3. From now on, the focus of interest will be eq. (20), which describes an infinite number of combinations of optimal direct and indirect tax rates. It measures the extent to which the tax rates should change after distorting events in consumption or in labour markets come to the fore. Eq. (20) may be interpreted as providing a map of indifference curves that present the preferences of policy makers for direct (indirect) tax rates over indirect (direct) tax rates, as a means of eliminating distortions coming from the labour market and/or from the market for consumer goods. It can be shown that, in eq. (20), the direct tax rate is a convex function of the indirect tax rate, if 1+a1-b1>1, or a1>b1. In contrast, the indifference curves are concave to the origin if 1+a1-b1<1, or a1<b1. It is well understood that the design of a map per se, which includes an infinite number of indifference curves on the basis of (20), does not seem to comprehend the scale of the problem. It is clear that eq. (20) by itself cannot solve the problem of finding an equilibrium point, unless a constraint on the direct/indirect tax-rate structure is placed. A suitable constraint is considered to be the equality between the total tax revenue, T, and the sum of direct and indirect taxes (Ty+Ti), i.e. T=Ty+Ti or their ratio to GDP,