Proof please! (economics & elasticity)
(this has been re-posted from the business math section)
I'm trying to follow the working from a paper* that looks at the revenue maximising tariff for a country.
It gives the expression for a government's tariff revenue as:
R = t.p.M
where R is revenue, t is the average ad valorem tariff rate, p is the average price of imports and M is the volume of imports.
The author totally differentiates this exression with respect to the tariff to get:
dR/dt = p.M + t.p.dM/d(p(1+t)) + t.M.dp/dt
The last expression drops out as we assume dp/dt = 0. Finally, the author sets dR/dt = 0 so as to find the optimal (i.e. revenue maximing) tariff.
So far so good. But what I don't get is how he then arrives at the next expression for the optimal tariff:
t* = -1/(1+n)
where t* is the optimal tariff and n is the price elasticity of import demand.
I'm assuming that
n = [dM/M]/[dP(1+t)/p(1+t)]
though this isn't made explicit in the text. Assuming this expression for n is correct, can you show me how to get to the last expression above, i.e. t* = -1/(1+n) ?
Thanks for your time
* Douglas Irwin, 'Higher Tariffs, Lower Revenue? Analyzing the Fiscal Aspects of "The Great Tariff Debate of 1888"', The Journal of Economic History, Mar. 1998.