# Tax revenue formula maximization- whats the name of this formula?

I have been teaching intermediate microeconomics and have come across a formula a number of times relating the tax rate at which revenue is maximized if elasticities are constant. This formula is defined as follows:

$$T_{max}=\frac{\varepsilon}{\varepsilon-1}\left(\frac{1}{\eta}+\frac{1}{\varepsilon}\right)$$

where $$\varepsilon$$ is our price elasticity of demand and $$\eta$$ is our price elasticity of supply.

This seems to be just a spin on Ramsey's rule. By Ramsey's rule (derived through Lagrangian method assuming zero cross-elasticities) the optimal tax for a market where supply is not explicitly modeled is given by:

$$\frac{t}{1+t}=\frac{1}{\epsilon}$$

Well technically it is $$\frac{t}{1+t}=\frac{1-b}{\epsilon}$$ but $$b$$ which is parameter that depends on marginal cost of public funds is often omitted/assumed to be zero in introductory problems for students.

The Ramsey rule above can be solved for $$t$$ as:

$$t = \frac{\epsilon}{\epsilon-1} \left( \frac{1}{\epsilon}\right)$$

The version you showcase is just derived with also explicitly taking into account supply as opposed to just from consumer problem where supply is not explicitly modeled showed above.

Given that I think it would still be called 'optimal Ramsey taxation'. To my best knowledge this kind of tweaking of the model would not have it's own special name, but I also tried to do literature search on Ramsey models with supply elasticity which did not yielded any results, but absence of evidence is not necessarily evidence of absence.

• in the derivation of conventional rule, is the constraint in the langrangian that of demand curve? – Dayne Oct 30 at 2:59
• @Dayne no in fact the lagrangian is modeled from the point of view of the government where benevolent social planner tries to optimize person’s indirect utility $v(t_x,t_y)$ subject to government budget constraint $t_xx+t_yy=G$ where $G$ is gov spending. Although there are various ways of deriving it. – 1muflon1 Oct 30 at 3:05
• Ok. Actually I couldn't find the formal definition or derivation on wiki, so asked. Thanks for the reply. – Dayne Oct 30 at 3:19