Optimal tax enforcement and the income tax rate: the role of taxable income inequality

Abstract

This paper considers the problem of jointly determining the optimal income tax rate and the optimal degree of enforcement by the authorities. It extends the analysis of Keen and Slemrod, who combine the well-known elasticity of taxable income with an enforcement elasticity of taxable income. The model is extended to the many-person context with both differing wages and preferences, and with a tax and transfer system along with the desire for redistribution on the part of the government. The effect of inequality, defined in terms of an equally-distributed-equivalent taxable income measure, is explored. There is also government expenditure on a pure public good. The model is also extended to allow for a direct effect on labour supply of tax enforcement, and the implications for the optimal enforcement gap.

KEYWORDS:

• Tax enforcement
• elasticity of taxable income
• optimal tax

Acknowledgments

I should particularly like to thank Norman Gemmell for discussions and very helpful comments on earlier versions of this paper. I have also benefited from discussions with Matt Benge, Richard Braae, Phil Whittington, Hamish Slack and Josh Teng from the New Zealand Inland Revenue Department.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. For example, see Saez (2001) and Mirrlees (2011). For details and applications to New Zealand, see Creedy (2015). On the elasticity of taxable income, see Saez, Slemrod, and Giertz (2012).

2. For a detailed exposition of the model, with a diagrammatic illustration, and an explanation of the solution process using explicit functional forms and numerical examples, see Creedy (2016).

3. Some of the KS notation has been changed here. In particular, the symbol, h, replaces ℓ. This can too easily be confused with e when writing by hand. Also, KS use φ to denote the elasticity of taxable income with respect to the enforcement parameter, α, but here φ replaces KS's ϕ to denote the disutility from work. In writing elasticities, KS use, say, E(a, b) to denote the elasticity of a with respect to b: here this is denoted η _a,b .

4. Later KS consider a many person economy where individuals face different wage rates, so that the elasticity of taxable income is a suitably weighted aggregate.

5. This first-order condition is expressed in terms of the enforcement parameter which influences the individual's cost of concealing income. Keen and Slemrod do not consider the cost of imposing or achieving any given value of α and, hence, the implications for the government budget constraint. It would be possible to rewrite the model in terms of expenditure, but this is not considered here. However, (17(17) $${\eta }_{z,\alpha }=\alpha \left(\frac{\frac{c_{\alpha }}{v\prime }+a_{\alpha }}{tz}\right)$$(17) ) can be expressed in terms of the government expenditure on enforcement, q say. Write α = α(q) to express the way in which α depends on expenditure. Then η _z,q = η _z,αη_α,q and (17(17) $${\eta }_{z,\alpha }=\alpha \left(\frac{\frac{c_{\alpha }}{v\prime }+a_{\alpha }}{tz}\right)$$(17) ) can be rewritten as: ${\eta }_{z,q}=\alpha {\eta }_{\alpha ,q}\left(\frac{c_{\alpha }}{v\prime }+a_{\alpha }\right)/tz$.

6. If the assumption of quasi-linearity is dropped, the elasticity of taxable income (with respect to one minus the marginal income tax rate) depends on the change in the average as well as the marginal tax rate. However, in empirical work, such income effects have been found to be weak: see Creedy, Gemmell, and Teng (in press).