Is the current proliferation of tax credits - or their
better spoken sibling, tax thresholds – inimical
to the welfare potential of the economy? The following well known proposition
of economic theory suggests that it is.

*For any structure of ‘threshold plus marginal rate’ there will always be some ‘flat rate scheme’that*

*for a given individual income tax payer*

*will increase the utility, consumption, labour supply and tax paid of that tax payer.*

If you like, for a given income tax payer, there will always
be some 'flat rate scheme' - one which abolishes the threshold and makes all income subject to a 'flat' rate of tax - that makes both the tax payer and the government
better offer.

What is going on here? How is this marvel achieved?

The answer does not lie in any strange grace of
‘simplicity’. Or in some abhorrence of nature of a vertical axis intercept. The
answer lies in the incentives that a flat rate scheme offer to a person to obtain utility from
consumption rather than leisure.

The explanation of this doesn't require any maths or diagrams. The logic can be conveyed solely through words.

It is easy to see the following 'trivial' proposition: there will always exist some flat
rate scheme that would yield exactly the same revenue as before IF the person
worked exactly the same hours as before. This is true merely by virtue of arithmetic: as any flat rate scheme involves abolishing
the tax threshold (and so increasing the tax
liability) and a cut in the marginal rate (decreasing the tax liability), there
will necessarily exist some marginal rate of tax such that the person would pay
exactly the same revenue IF they worked the same number of hours.

We now persue significance of this apparently hum drum proposition ... Is it not true that to work the same number of hours as before - while
paying the same tax - is to have same consumption as before? And is not working the same number
of hours as before to have the same leisure as before? Thus our proposition in the paragraph above says: 'there will necessarily
be a flat rate scheme such that the person could have the same leisure and the
same consumption as before by the working the same hours as before'. In short, we can construct a flat rate scheme such that the
person can obtain the same utility as before simply by working the same hours as they did before.

But - and this the key observation - under this scheme, the person
can now do BETTER than 'the same utility as before'; and they can do better by choosing to work MORE hours than they did before. This is because the reduction in the rate of tax
has increased the consumption reward for working an extra hour. Previously, the
consumption reward (in utility terms) from working an extra hour was just matched by the loss (in utility terms) from working an extra hour (that equality was why the person worked as long as they did, and no longer). Now that the consumption reward for working an extra hour has ncreased (on account of the cut in the marginal rate),
it is now utilty increasing to to work more than they did before. Further, by working more under the hypothesised scheme they must pay more tax, as a matter of arithmatic. And they must consume more.

The proposition of the dominance of a flat rate scheme over 'threshold plus marginal rate' is suggestive of the more general proposition:

*For any scheme of ‘threshold plus marginal rate’ there will always be some (i) reduction in the marginal rate, and (ii) reduction in the threshold that will, for any given income tax payer, increase the utility, consumption, labour supply and tax paid of that tax payer.*

The logic is just the same. The only difference is that the size of the effect of a mere reduction in the threshold is less than the size of the effect of outright abolition. Thus reducing the threshold to zero is better than reducing it merely to a positive number.

But if reducing the threshold to zero is better than reducing it merely to a positive number, would it not be better still to reduce the threshold to a

*negative*number? The answer is : yes, it would be better. Unlimited vistas of improvement through tax restructure seem to open up ...Why, then, do we resist this free lunch offered by schemes to simultaneously reduce thresholds and reduce marginal rates?

One answer is that the impacts of any given such restructure are heterogeneous and contrary across income.

The 'free lunch' proposition at stake is, recall, conditioned on a given individual income
tax payer, call them X.

*It is clear that the restructuring that increases the utility, consumption, and labour supply of tax payer X may have different effects on other payers.*To illustrate: any individual whose income was below the threshold obviously must now have lower utility and consumption (and pay more tax) on account of any abolition of the threshold. It is also easy to see that the person whose income was ‘significantly' above that of tax payer X, will win from the restructure that delivers 'lunch for free' to X. This high income person will certainly have higher utility, BUT (unlike X) they may pay less revenue. Thus what is a free lunch for X may be lunch, indeed, for those better off than X; but not, alas, a free one. And for those with incomes less than X, it may be no 'lunch' at all.
There is another reason that recommends caution against reducing thresholds and rates. Mancur Olson has suggested that Joseph Stalin used this logic to drive growth in the 1930s. That fact is an inauscipicous one. Public Choice theory can explain why what might seem like a tax reform can prove to a tax exploitation.

The next post elaborates.