<p>The answer by @NickCHK is very good and the derivation is correct for quantity taxes. Nevertheless, let me clarify a few things. In the comments you give a counterexample to this formula and state that the formula only holds for linear S or D. That is not true.</p>

<p>It also holds for non-linear S/D. Your counterexample about the supply shape is not only non-linear, it is also non-continuous. That means the derivative does not exist over the whole domain, which complicates things. Nevertheless, the formula would hold on either side of the kink points. Furthermore, your comment seems to argue that the formula implies constant incidence shares. This is also not true as elasticities are not necessarily constant with regards to quantity.</p>

<p>There is, however, a common case where this formula does not hold exactly. For ad valorem taxes (e.g. the VAT) you would have slightly different expressions. Nevertheless, the incidence formula for quantity taxes is still a good proxy. This is discussed a bit in <a href="https://www.sciencedirect.com/science/article/pii/S0047272706001708" rel="nofollow noreferrer">Carbonnier (2007)</a>, who analyzes VAT, but uses the formula in your OP as a proxy for the consumer share of the tax.</p>

<p>It is also worth noting that the formula is derived for marginal tax rate changes. With very large changes, the incidence will not be exactly as described above, but that is a caveat of all such analyses using (non-discrete) derivatives.</p>

CC BY-SA 4.0
2018-10-15T17:59:42.817Z
2018-10-16T18:31:41.647Z
2018-10-16T18:31:41.647Z
BB King
<p>Ah, never mind, got there.</p>

<p>Both supply and demand share the same change in quantity as a result of the tax, so</p>

<p>$$\Delta Q_S = \Delta Q_D$$</p>

<p>The amount by which the quantities changed is the derivative of that quantity wrt price, times the change in price that side experienced</p>

<p>$$\frac{\partial Q_S}{\partial P_S}\Delta P_S = \frac{\partial Q_D}{\partial P_D}\Delta P_D$$</p>

<p>Multiply both sides by the equilibrium P/Q</p>

<p>$$\frac{\partial Q_S}{\partial P_S}(\frac{P^*}{Q^*})\Delta P_S = \frac{\partial Q_D}{\partial P_D}(\frac{P^*}{Q^*})\Delta P_D$$</p>

<p>which of course is the formula for elasticity at the equilibrium</p>

<p>$$\varepsilon_S \Delta P_S = \varepsilon_D \Delta P_D$$
$$\frac{\varepsilon_S}{\varepsilon_D}\Delta P_S = \Delta P_D$$</p>

<p>So then the producer's share is </p>

<p>$$Incidence_S=\frac{P^*-P_S}{T}=\frac{|\Delta P_S|}{|\Delta P_S|+|\Delta P_D|} = \frac{|\Delta P_S|}{|\Delta P_S|+|\frac{\varepsilon_S}{\varepsilon_D}\Delta P_S|}$$</p>

<p>the price change cancels out and you get producer's share as</p>

<p>$$Incidence_S=\frac{1}{1+|\frac{\varepsilon_S}{\varepsilon_D}|}=\frac{|\varepsilon_D|}{|\varepsilon_D|+\varepsilon_S}$$</p>

<p>So it all comes from the fact that this is the unique formula that produces the price-change ratios that generate identical quantity changes in both sides of the market. Probably not any intuition that can come down to a principles course level but makes a lot more sense than it did before!</p>

CC BY-SA 3.0
2017-04-12T09:21:38.687Z
Although thinking about it some more, the standard equation necessarily has to break down for nonlinear S/D. Imagine a linear demand with supply shaped like _/. Incidence shares would necessarily be nonconstant. The second step in the proof won't hold. So yeah it's a linear-S/D thing only, although a decent enough approximation for nonlinear S/D with small taxes that all my counterexamples failed.
NickCHK
CC BY-SA 3.0
2017-04-12T21:46:03.323Z
Specifically, these equations will apply whenever the ratio of the slopes of supply and demand is constant. This works for both linear S/D and also, for example, S/D where Q enters only as a square in both, or only as a cube in both, or as ln(Q) in both, which just happens to contain all the counterexamples I tried!
NickCHK
CC BY-SA 3.0
2017-04-12T06:04:57.850Z
2017-04-12T06:04:57.850Z
NickCHK
<p>You're probably familiar with the formula for tax incidence (in a standard Principles framework) from elasticity. Specifically, that the consumer share is</p>

<p>$$\frac{\varepsilon_S}{\varepsilon_S+|\varepsilon_D|}$$
and the producer share is
$$\frac{|\varepsilon_D|}{\varepsilon_S+|\varepsilon_D|}$$</p>

<p>I had always assumed that this was just a byproduct of linear supply/demand and therefore not of much interest, but just now have proven to myself that it's not with a few nonlinear supply/demand forms that also produce this result. </p>

<p><strong>Is there an intuitive reason why the tax incidence takes this particular functional form, or is it just a common mathematical happenstance?</strong> This isn't the only functional form that satisfies what I think of as the intuitive tenets of the tax incidence solution (consumer + producer shares add to 1, increase in your side's elasticity increases your side's burden). Another form that satisfies this, but is not in fact a calculation of incidence share, is $$\frac{(|\varepsilon_D|/\varepsilon_S)}{(|\varepsilon_D|/\varepsilon_S)+(\varepsilon_S/|\varepsilon_D|)}$$</p>

<p>I suppose an alternative explanation is that the counterexamples I happened to test this with satisfy some condition necessary to have those be the shares, and it's not actually universal.</p>

<microeconomics><supply-and-demand><taxation>
Is there an intuitive explanation for the tax incidence formula from elasticity?
NickCHK
# Is there an intuitive explanation for the tax incidence formula from elasticity?

(Asked by: username_0 on 2017-04-12)

You're probably familiar with the formula for tax incidence (in a standard Principles framework) from elasticity. Specifically, that the consumer share is

$$\frac{\varepsilon\_S}{\varepsilon\_S+|\varepsilon\_D|}$$
and the producer share is
$$\frac{|\varepsilon\_D|}{\varepsilon\_S+|\varepsilon\_D|}$$

I had always assumed that this was just a byproduct of linear supply/demand and therefore not of much interest, but just now have proven to myself that it's not with a few nonlinear supply/demand forms that also produce this result.

**Is there an intuitive reason why the tax incidence takes this particular functional form, or is it just a common mathematical happenstance?** This isn't the only functional form that satisfies what I think of as the intuitive tenets of the tax incidence solution (consumer + producer shares add to 1, increase in your side's elasticity increases your side's burden). Another form that satisfies this, but is not in fact a calculation of incidence share, is $$\frac{(|\varepsilon\_D|/\varepsilon\_S)}{(|\varepsilon\_D|/\varepsilon\_S)+(\varepsilon\_S/|\varepsilon\_D|)}$$

I suppose an alternative explanation is that the counterexamples I happened to test this with satisfy some condition necessary to have those be the shares, and it's not actually universal.

**Tags:** microeconomics, supply-and-demand, taxation

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## 2 answers

### Answer by username_0 on 2017-04-12. Score: 3 | Accepted answer

Ah, never mind, got there.

Both supply and demand share the same change in quantity as a result of the tax, so

$$\Delta Q\_S = \Delta Q\_D$$

The amount by which the quantities changed is the derivative of that quantity wrt price, times the change in price that side experienced

$$\frac{\partial Q\_S}{\partial P\_S}\Delta P\_S = \frac{\partial Q\_D}{\partial P\_D}\Delta P\_D$$

Multiply both sides by the equilibrium P/Q

$$\frac{\partial Q\_S}{\partial P\_S}(\frac{P^\*}{Q^\*})\Delta P\_S = \frac{\partial Q\_D}{\partial P\_D}(\frac{P^\*}{Q^\*})\Delta P\_D$$

which of course is the formula for elasticity at the equilibrium

$$\varepsilon\_S \Delta P\_S = \varepsilon\_D \Delta P\_D$$
$$\frac{\varepsilon\_S}{\varepsilon\_D}\Delta P\_S = \Delta P\_D$$

So then the producer's share is

$$Incidence\_S=\frac{P^\*-P\_S}{T}=\frac{|\Delta P\_S|}{|\Delta P\_S|+|\Delta P\_D|} = \frac{|\Delta P\_S|}{|\Delta P\_S|+|\frac{\varepsilon\_S}{\varepsilon\_D}\Delta P\_S|}$$

the price change cancels out and you get producer's share as

$$Incidence\_S=\frac{1}{1+|\frac{\varepsilon\_S}{\varepsilon\_D}|}=\frac{|\varepsilon\_D|}{|\varepsilon\_D|+\varepsilon\_S}$$

So it all comes from the fact that this is the unique formula that produces the price-change ratios that generate identical quantity changes in both sides of the market. Probably not any intuition that can come down to a principles course level but makes a lot more sense than it did before!

> **Comments:**
> **username_0**: Although thinking about it some more, the standard equation necessarily has to break down for nonlinear S/D. Imagine a linear demand with supply shaped like _/. Incidence shares would necessarily be nonconstant. The second step in the proof won't hold. So yeah it's a linear-S/D thing only, although a decent enough approximation for nonlinear S/D with small taxes that all my counterexamples failed.  
> **username_0**: Specifically, these equations will apply whenever the ratio of the slopes of supply and demand is constant. This works for both linear S/D and also, for example, S/D where Q enters only as a square in both, or only as a cube in both, or as ln(Q) in both, which just happens to contain all the counterexamples I tried!  

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### Answer by username_1 on 2018-10-15. Score: 1

The answer by @NickCHK is very good and the derivation is correct for quantity taxes. Nevertheless, let me clarify a few things. In the comments you give a counterexample to this formula and state that the formula only holds for linear S or D. That is not true.

It also holds for non-linear S/D. Your counterexample about the supply shape is not only non-linear, it is also non-continuous. That means the derivative does not exist over the whole domain, which complicates things. Nevertheless, the formula would hold on either side of the kink points. Furthermore, your comment seems to argue that the formula implies constant incidence shares. This is also not true as elasticities are not necessarily constant with regards to quantity.

There is, however, a common case where this formula does not hold exactly. For ad valorem taxes (e.g. the VAT) you would have slightly different expressions. Nevertheless, the incidence formula for quantity taxes is still a good proxy. This is discussed a bit in [Carbonnier (2007)](https://www.sciencedirect.com/science/article/pii/S0047272706001708), who analyzes VAT, but uses the formula in your OP as a proxy for the consumer share of the tax.

It is also worth noting that the formula is derived for marginal tax rate changes. With very large changes, the incidence will not be exactly as described above, but that is a caveat of all such analyses using (non-discrete) derivatives.


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