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Math Help - Economics - import elasticity question

  1. #1
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    Economics - import elasticity question

    Hi, 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) ?? Ive been struggling with with!

    Thanks for your time

    Steve

    * Douglas Irwin, 'Higher Tariffs, Lower Revenue? Analyzing the Fiscal Aspects of "The Great Tariff Debate of 1888"', The Journal of Economic History, Mar. 1998.
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  2. #2
    Member garymarkhov's Avatar
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    Quote Originally Posted by slosy View Post
    Hi, 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) ?? Ive been struggling with with!

    Thanks for your time

    Steve

    * Douglas Irwin, 'Higher Tariffs, Lower Revenue? Analyzing the Fiscal Aspects of "The Great Tariff Debate of 1888"', The Journal of Economic History, Mar. 1998.
    Since you haven't got any help on this yet, I suggest you post it to the main calculus section of the forum.
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  3. #3
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    Thanks, will do
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