# Discussion Paper

## Abstract

A digressive tax like a variable rate sales tax or a tax on price gives firms an incentive for expanding output. Thus, unlike unit and ad valorem taxes which amplify the harm from monopoly, a digressive tax lessens the harm. We analyse a tax on price with respect to efficiency and practical policy appeal. Using a tax on price in combination with ad valorem taxation it is possible to achieve the Ramsey solution. That is, the combination of the two taxes secures tax revenue in the least distortive way. We also show how tax reforms based only on observation of price and quantity can make use of a tax on price in order to improve welfare. That is, it is practical to use a tax on price.

## JEL Classification

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## Assessment

# Comments and Questions

Tank you for the comments.

Because of the comments of the other referee the order of the example and the general analysis has been changed.

In the new version of the paper I have changed some typo’s and expanded the explanation in some parts of the paper.
...[more]

...
I have attached a draft of the revised version. Please note that it is a draft. It still needs some polishing and copy-editing.

That is a typo. The right expression is now just before “Assumption 1” on page 5 of the revised manuscript.

That’s not the case. The derivative is defined top of page 5 (the revised manuscript) and it is negative.

That’s a typo. It has been corrected.

I have changed the notation. Originally I dropped x from p(x),c(x) and so on. I have changed this so that x now appear as an argument of these functions.

Admittedly, the proofs of Propositions 1 and 2 were sketchy. I have stated the proofs more thoroughly now. The revised proof of Proposition 1 is on page 9 and there is a formal proof of Proposition 2 on page 10-11.

Here is the file.

Henrik

Summary

This paper addresses the use of a digressive tax system that combines a tax on price with an ad valorem tax to improve tax performance. The main result is that tax reforms based only on observed price and quantity data can make use of a tax on price ...[more]

... to improve welfare. The result that a matched pair of price taxes and ad valorem taxes can be used to extract tax revenue whilst maintaining the unregulated monopoly outcome (Propositions 1 and 4) are novel and interesting.

Specific Comments:

1. For expositional purposes, I suggest reversing the order of the general analysis and example. Further, it would improve the paper to use the example to reinforce the main propositions in the paper by tabulating numerical results for variation in key parameters.

2. It would help to express assumption 2 in terms of the primitive conditions on demand required for R to be positive and decreasing in . It can then be verified that this condition holds in the numerical example with linear demand and constant unit cost. I believe a condition can be provided that relates to “Seade’s E”, which would help position the modeling framework better in the literature.

3. Proposition 3 is somewhat obvious. With 2 tax instruments –one to rotate demand and a second to shift demand to the desired level— it is clearly the case that tax policy can achieve the Ramsey solution. But this argument somewhat undermines the point on the information problem. I see little harm in including this result in the paper, but I do suggest de-emphasizing this result somewhat by focusing the abstract on the main results and shortening the development of this outcome in the text. The discussion of the Ramsey implications of tax reforms with respect to a tax on price is nice on p11 as a way to develop intuition for how the scheme works in a full information setting.

4. I believe the main results in the paper generalize to the case of Cournot oligopoly. An attractive feature of the paper is that it is short, so there is a trade-off between adding to the length to encompass the oligopoly case and economizing on space. This is ultimately an editorial decision, but extension of the present analysis to the oligopoly case is less interesting as an independent paper than included here.

5. Apart from the potential extension to oligopoly markets, a nice application of the monopoly tax problem is to consider rate-of-return regulated monopoly industries. A corollary to Proposition 4 might consider matched reforms that generate positive tax revenue and maintain the return-regulated monopoly output level.

6. I encourage you to find ways to make your work the result of for-pay consulting relationships.

Thanks for the comments. I agree on most points. I am preparing a revised version of the paper. Below is a list of the changes I plan. The major changes (new lemma and a new corrollary) have been done

As suggested the order of the analysis and ...[more]

... example is reversed. With respect to the idea of also using an example I consider the primary argument in favour of a numerical example to be that this makes sure that Assumption 2 can be satisfied. In the revised version I will add a new Lemma 1 (see below). This Lemma shows existence and for this reason I think that the benefit of an example does not match the cost in terms of added length.

The revised paper will have a discussion following assumption 2. This discussion relates the conditions in Seade (1980) plus the elasticity of the demand function to Assumption 2 that R_τ>0. Moreover, applying Seade (1980) I have added Lemma 1 proving that R_τ>0 for small tax rates. I believe that this Lemma dispenses with the need to use numerical examples to show that R_τ is indeed positive for some tax rates.

I agree on the view of Proposition 3. It is renamed and is Corollary 1 in the new version. This emphasizes that the result really is an upshot of Proposition 2. Also, mentioning of the Ramsey result has been dropped from the abstract.

I shall add a discussion of fixed number homogenous oligopoly. In short, application of Seade (1980) makes sure that the reasoning in the paper extends to fixed number homogenous oligopoly. The formal result is stated as (a new) Corollary 2.

I agree that it is of interest to discuss the monopoly tax problem in relation to rate-of-return regulated monopoly (and maybe also revenue cap regulation). In a broader perspective one might also ask about the effect of consumption taxes and the Averch-Johnson effect. I believe this can be a paper on its own. For this reason and since the paper is extended to cover oligopoly I think that the paper should not be longer. However, the option is mentioned in a footnote

Of course, these changes require some adjustments of the text in the paper.