## Abstract

This paper examines the Laffer effect in the Ramsey tax-model with linear consumption taxes and a representative consumer. It is assumed that the private goods and the public good are weakly separable. It is demonstrated that if all of the private goods are weak gross complements to each other, then the Laffer effect does not exist, in other words, higher tax rates can always achieve more tax revenue. In contrast, if all of the private goods are strict gross substitutes, then the Laffer effect does exist. Moreover, if all of the private goods are weak gross substitutes, then the government cannot fully acquire the leisure endowment through taxes on consumption goods. We also show that gross substitution works to raise the marginal cost of public funds.

### Similar content being viewed by others

## Notes

It is known in microeconomic theory that one kind of gross substitution is a sufficient condition for the uniqueness of Walrasian equilibrium for a general exchange economy. For example, please refer to Definition 17.F.2 and Proposition 17.F.3 of Mas-Colell et al. (1995).

The budget constraint of the consumer is \(\sum_{i=1}^{n}p_{i}x_{i}+wz=0\) where

*w*denotes the wage rate which is normalized to unity. Therefore,*Z*is homogeneous of degree zero in all prices, implying that \(\sum_{i=1}^{n}p_{i}\partial Z/\partial p_{i}+w\partial Z/\partial w=0\). As a result,*∂Z*/*∂p*_{ i }⪋0, ∀*i*⇒*∂Z*/*∂w*⪌0. If*n*=1, then*∂Z*/*∂p*_{1}⪋0⇔*∂Z*/*∂w*⪌0. We are indebted to a referee for pointing out these properties.In a Ramsey setting, Corlett and Hague (1953) first show that it is socially optimal to tax complements to leisure and to subsidize substitutes which serves to offset the distorting effect of taxation on labor supply.

The tax revenue function may not have an interior maximum (Malcomson 1986, p. 263), and hence the maximum tax revenue, max{

*R*(*p*_{1},…,*p*_{ n },*y*)∣*p*_{ i }≥0,*i*=1,…,*n*}, may not be defined. Example 3.1 below is a case in point: the maximum tax revenue does not exist when*σ*≤1. In contrast, because*y*is an upper bound for {*R*(*p*_{1},…,*p*_{ n },*y*)∣*p*_{ i }≥0,*i*=1,…,*n*}, this set should have a least upper bound (supremum).For Example 3.1, the private goods are gross complements (substitutes) if

*σ*falls below (exceeds) unity.We can show that in Example 3.1, (3.3) is indeed satisfied if and only if the goods are strictly gross substitutes.

Let

*L*(*w*) denote the Marshallian labor supply function where*w*is the posttax wage rate. The tax revenue*R*(*t*) is equal to \(tL(\hat{w}-t)\) where \(\hat{w}\) is the pretax wage rate, and \(t=\hat{w}-w\). Therefore,*L*(*w*)/*R*′(*t*) is equal to*L*(*w*)/[*L*(*w*)−*tL*′(*w*)], implying that*L*/*R*′⪋1 if*L*′⪋0.\({\partial \sum_{j=1}^{n}p_{j}X_{j}/ \partial p_{i}}<0\) can be rewritten as \(X_{i}+\sum_{j=1}^{n}p_{j}{\partial X_{j}/ \partial p_{i}}<0\), which, in turn, can be rewritten as \(s_{i}-\sum_{j=1}^{n} s_{j}\varepsilon_{ji}<0\).

## References

Aronsson, T. (2008). Social accounting and the public sector.

*International Economic Review*,*49*(1), 349–375.Atkinson, A. B., & Stern, N. H. (1974). Pigou, taxation and public goods.

*Review of Economic Studies*,*41*(1), 119–128.Atkinson, A. B., & Stiglitz, J. E. (1980).

*Lectures on public economics*. New York: McGraw-Hill.Chang, M. C. (2000). Rules and levels in the provision of public goods: the role of complementarities between the public good and taxed commodities.

*International Tax and Public Finance*,*7*, 83–91.Chang, M. C., & Chang, T.-H. (2001). A note on Ramsey pricing—do Ramsey prices exceed marginal costs?

*Academia Economic Papers*,*29*(3), 365–381.Chang, M. C., & Peng, H.-P. (2009). Structure regulation, price structure, cross subsidization and marginal cost of public funds.

*The Manchester School*,*77*(5), 618–641.Christiansen, V. (2007). Two approaches to determine public good provision under distortionary taxation.

*National Tax Journal*,*60*(1), 25–43.Corlett, W. J., & Hague, D. C. (1953). Complementarity and the excess burden of taxation.

*Review of Economic Studies*,*21*(1), 21–30.Fullerton, D. (1982). On the possibility of an inverse relationship between tax rates and government revenues.

*Journal of Public Economics*,*19*, 3–22.Gaube, T. (2000). When do distortionary taxes reduce the optimal supply of public goods?

*Journal of Public Economics*,*76*(2), 151–180.Gronberg, T., & Liu, L. (2001). The second-best level of a public good: an approach based on the marginal excess burden.

*Journal of Public Economic Theory*,*3*(4), 431–453.Iorwerth, A. A., & Whalley, J. (2002). Efficiency considerations and the exemption of food from sales and value added taxes.

*Canadian Journal of Economics*,*35*(1), 166–182.Kaplow, L. (2010). Taxing leisure complements.

*Economic Inquiry*,*48*(4), 1065–1071.Malcomson, J. M. (1986). Some analytics of the Laffer curve.

*Journal of Public Economics*,*29*, 263–279.Mas-Colell, A., Whinston, M. D., & Green, J. R. (1995).

*Microeconomic theory*. New York: Oxford University Press.Pigou, A. C. (1947).

*Public finance*. London: Macmillan.Slemrod, J., & Yitzhaki, S. (2001). Integrating expenditure and tax decisions: the marginal cost of funds and the marginal benefit of projects.

*National Tax Journal*,*54*(2), 189–201.Smith, A. (1776).

*An inquiry into the nature and causes of the wealth of nations*.West, S. E., & Parry, I. W. H. (2009).

*Alcohol/leisure complementarity: estimates and implications for tax policy*(RFF Discussion Paper No. 09-09).West, S. E., & Williams, R. C. III (2007). Optimal taxation and cross-price effects on labor supply: estimates of the optimal gas tax.

*Journal of Public Economics*,*91*, 593–617.Wilson, J. D. (1991a). Optimal public good provision with limited lump-sum taxation.

*American Economic Review*,*81*(1), 153–166.Wilson, J. D. (1991b). Optimal public good provision in the Ramsey tax model—a generalization.

*Economics Letters*,*35*, 57–61.

## Acknowledgements

The authors are grateful to the editor Professor E. Janeba and two anonymous referees for their very helpful and detailed comments.

## Author information

### Authors and Affiliations

### Corresponding author

## Appendix

### Appendix

### Proof of Fact 3.1

Because *R*(*p*,*y*) is strictly concave with respect to *p*, if the Laffer effect exists, then there exists a finite number *p*
^{∗} such that

implying that the supreme tax revenue is equal to *R*(*p*
^{∗},*y*). If *X*(*p*
^{∗},*y*)=0, then *R*(*p*
^{∗},*y*)=0<*y*. Because *q*>0, if *X*(*p*
^{∗},*y*)>0, then *R*(*p*
^{∗},*y*)<*p*
^{∗}
*X*(*p*
^{∗},*y*), implying that *R*(*p*
^{∗},*y*)<*y* since *p*
^{∗}
*X*(*p*
^{∗},*y*)≤*y*. □

### Proof of Lemma 3.1

It is next to apply the same technique used by Chang and Chang (2001, Proposition 2.2) to conduct the proof. Suppose that *p*
_{
i
}<*q*
_{
i
}, *i*=1,…,*m* (*m*≥1), and *p*
_{
j
}≥*q*
_{
j
}, *j*=*m*+1,…,*n*. If we raise *p*
_{
i
} to *q*
_{
i
}, *i*=1,…,*m*, then \(\sum_{i=1}^{m}(p_{i}-q_{i})X_{i}\) rises from a negative number to zero. Moreover, \(\sum_{j=m+1}^{n}(p_{j}-q_{j})X_{j}\) also increases since *p*
_{
j
}−*q*
_{
j
}≥0, *j*=*m*+1,…,*n*, and *∂X*
_{
j
}/*∂p*
_{
i
}≥0 for \(j\not{=}i\). Therefore, it is without loss of generality to exclude negative tax rates when we want to find out the supreme tax revenue. □

### Proof of Proposition 3.3

Equation (3.3) can be rewritten as^{Footnote 10}

where *ε*
_{
ji
}≡−*∂*ln*X*
_{
j
}/*∂*ln*p*
_{
i
}, and *s*
_{
i
}≡*p*
_{
i
}
*X*
_{
i
}/*R*. Partially differentiating *R* with respect to *p*
_{
i
} yields

Therefore,

where *t*
_{
j
}=(*p*
_{
j
}−*q*
_{
j
})/*p*
_{
j
}. In order to establish that the Laffer effect exists, it suffices to consider the case with uniform tax rates where *t*
_{1}=⋯=*t*
_{
n
}=*t*. Equation (A.3) implies that *∂R*/*∂p*
_{
i
} is positive (*i*=1,…,*n*) if *t* is small enough. Equation (A.2) implies that *∂R*/*∂p*
_{
i
} is negative (*i*=1,…,*n*) if *t* is large enough (i.e., if *t* approaches 1). □

## Rights and permissions

## About this article

### Cite this article

Chang, M.C., Peng, HP. Laffer effect, gross substitution, marginal cost of public funds and the level property of public good provision.
*Int Tax Public Finance* **19**, 650–659 (2012). https://doi.org/10.1007/s10797-011-9200-1

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10797-011-9200-1

### Keywords

- Laffer effect
- Gross substitution
- Gross complementarity
- Marginal cost of public funds
- Leisure complements