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I'm trying to derive the tax incidence for an excise $t$ so that producers receive $p$ and consumers pay $q=p+t$ per unit. The formula I am seeing everywhere is some version of

$$ \frac{\mathrm{d}p}{\mathrm{d}t} = \frac{\epsilon_D}{\epsilon_S - \epsilon_D}, $$

but I keep ending up (derivation below) with:

$$ \frac{\mathrm{d}p}{\mathrm{d}t} = \frac{\epsilon_D}{(q/p) \epsilon_S - \epsilon_D}. $$

I use definitions of elasticity $\epsilon_D = \frac{q}{D}\frac{\partial D}{\partial q}$ and $\epsilon_S = \frac{p}{S}\frac{\partial S}{\partial p}$. I can see that I would get the expected result if I used $\epsilon_D = \frac{p}{D}\frac{\partial D}{\partial q}$ instead, but surely this would not be sound, since the consumer adjusts their demand based on $q$, not $p$?

What is my mistake here?

My derivation

Starting from the equilibrium condition

$$ D(q) = S(p), $$

I differentiate by $t$ to get

$$ \frac{\partial D}{\partial q} \frac{\mathrm{d} q}{\mathrm{d} t} + \frac{\partial D}{\partial q} \frac{\partial q}{\partial p} \frac{\mathrm{d} p}{\mathrm{d} t} = \frac{\partial S}{\partial p} \frac{\mathrm{d} p}{\mathrm{d} t}, $$

Which rearranges to

$$ \frac{\mathrm{d} p}{\mathrm{d} t} = \frac{ \frac{\partial D}{\partial q} }{ \frac{\partial S}{\partial p} - \frac{\partial D}{\partial q} }, $$

Noting that $\frac{\partial q}{\partial p} = \frac{\partial q}{\partial t} = 1$.

Now I apply the definitions of the elasticities $\epsilon_D = \frac{q}{D}\frac{\partial D}{\partial q}$ and $\epsilon_S = \frac{p}{S}\frac{\partial S}{\partial p}$, giving

$$ \frac{\mathrm{d} p}{\mathrm{d} t} = \frac{ \frac{D}{q}\epsilon_D }{ \frac{S}{p} \epsilon_S - \frac{D}{q} \epsilon_D } = \frac{\epsilon_D}{\frac{q}{p} \epsilon_S - \epsilon_D}, $$

where I simplify by dividing through with $\frac{D}{q}$ (recalling that $S = D$).

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  • $\begingroup$ Looking at these slides it seems that the tax incidence is the derivative of price with respect to tax evaluated for a tax $t = 0$. In this case $q = p$ so the ratio becomes 1. $\endgroup$
    – tdm
    Dec 6, 2023 at 15:30

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