- How does a tax affect the gains from trade?

The presence of the deadweight loss implies that raising $1 in taxes costs society more than $1. But how much more? This idea—that the cost of taxation exceeds the taxes raised—is known as the excess burden of taxationThe amount by which the cost of taxation exceeds the taxes raised., or just the excess burden. We can quantify the excess burden with a remarkably sharp formula.

To start, we will denote the marginal cost of the quantity *q* by *c*(*q*) and the marginal value by *v*(*q*). The elasticities of demand and supply are given by the standard formulae

and

$$\eta =\frac{\raisebox{1ex}{$dq$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}{\raisebox{1ex}{$dc$}\!\left/ \!\raisebox{-1ex}{$c$}\right.}=\frac{c(q)}{q{c}^{\prime}(q)}\text{.}$$Consider an ad valoremAt value. (at value) tax that will be denoted by *t*, meaning a tax on the value, as opposed to a tax on the quantity. If sellers are charging *c*(*q*), the *ad valorem* tax is *tc*(*q*), and the quantity *q** will satisfy *v*(*q**) = (1 + *t*)*c*(*q**).

From this equation, we immediately deduce

$$\frac{dq*}{dt}=\frac{c(q*)}{{v}^{\prime}(q*)-(1+t){c}^{\prime}(q*)}=\frac{c(q*)}{-\frac{v(q*)}{\epsilon q*}-(1+t)\frac{c(q*)}{\eta q*}}=-\frac{q*}{(1+t)\left(\frac{1}{\epsilon}+\frac{1}{\eta}\right)}=-\frac{q*\epsilon \eta}{(1+t)\left(\epsilon +\eta \right)}.$$Tax revenue is given by
*Tax* = *tc(q*)q**.

The effect on taxes collected, *Tax*, of an increase in the tax rate *t* is

Thus, tax revenue is maximized when the tax rate is *t*_{max}, given by

The value $\frac{\epsilon}{\epsilon -1}$ is the monopoly markup rate, which we will meet when we discuss monopoly. Here it is applied to the sum of the inverse elasticities.

The gains from trade (including the tax) is the difference between value and cost for the traded units, and thus is

$$GFT={\displaystyle \underset{0}{\overset{q*}{\int}}v(q)-c(q)\text{\hspace{0.17em}}\text{\hspace{0.17em}}dq}\text{.}$$Thus, the change in the gains from trade as taxes increase is given by

$$\frac{dGFT}{dTax}=\frac{\raisebox{1ex}{$\partial GFT$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}{\raisebox{1ex}{$\partial Tax$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}=\frac{\left(v(q*)-c(q*)\right)\frac{dq*}{dt}}{\frac{c(q*)q*}{(1+t)\left(\epsilon +\eta \right)}\left(\epsilon +\eta -t\eta (\epsilon -1)\right)}=-\frac{\left(v(q*)-c(q*)\right)\frac{q*\epsilon \eta}{(1+t)\left(\epsilon +\eta \right)}}{\frac{c(q*)q*}{(1+t)\left(\epsilon +\eta \right)}\left(\epsilon +\eta -t\eta (\epsilon -1)\right)}$$ $=-\frac{\left(tc(q*)\right)\epsilon \eta}{c(q*)\left(\epsilon +\eta -t\eta (\epsilon -1)\right)}=-\frac{\epsilon \eta t}{\epsilon +\eta -t\eta (\epsilon -1)}=-\frac{\epsilon}{\epsilon -1}\text{\hspace{0.17em}}\frac{t}{{t}_{\mathrm{max}}-t}\text{.}$The value *t*_{max} is the value of the tax rate *t* that maximizes the total tax taken. This remarkable formula permits the quantification of the cost of taxation. The minus sign indicates that it is a loss—the deadweight loss of monopoly, as taxes are raised, and it is composed of two components. First, there is the term $\frac{\epsilon}{\epsilon -1}\text{,}$
which arises from the change in revenue as quantity is changed, thus measuring the responsiveness of revenue to a quantity change. The second term provides for the change in the size of the welfare loss triangle. The formula can readily be applied in practice to assess the social cost of taxation, knowing only the tax rate and the elasticities of supply and demand.

The formula for the excess burden is a local formula—it calculates the increase in the deadweight loss associated with raising an extra dollar of tax revenue. All elasticities, including those in *t*_{max}, are evaluated locally around the quantity associated with the current level of taxation. The calculated value of *t*_{max} is value given the local elasticities; if elasticities are not constant, this value will not necessarily be the actual value that maximizes the tax revenue. One can think of *t*_{max} as the projected value. It is sometimes more useful to express the formula directly in terms of elasticities rather than in terms of the projected value of *t*_{max}, in order to avoid the potential confusion between the projected (at current elasticities) and actual (at the elasticities relevant to *t*_{max}) value of *t*_{max}. This level can be read directly from the derivation shown below:

- The cost of taxation that exceeds the taxes raised is known as the excess burden of taxation, or just the excess burden.
- Tax revenue is maximized when the tax rate is ${t}_{\mathrm{max}}=\frac{\epsilon}{\epsilon -1}\left(\frac{1}{\eta}+\frac{1}{\epsilon}\right)\text{.}$
- The change in the gains from trade as taxes increase is given by $\frac{dGFT}{dTax}=-\frac{\epsilon}{\epsilon -1}\text{\hspace{0.17em}}\frac{t}{{t}_{\mathrm{max}}-t}\text{.}$

- Suppose both demand and supply are linear,
*qD*= (*a*–*b*)*p*and*qS*= (*c*+*d*)*p*. A quantity tax is a tax that has a constant value for every unit bought or sold. Determine the new equilibrium supply price*pS*and demand price*pD*when a quantity tax of amount*t*is applied. - An
*ad valorem*tax is a proportional tax on value, like a sales tax. Repeat the previous exercise for an*ad valorem*tax*t*. -
Let supply be given by

*p = q*and demand by*p =*1 –*q*. Suppose that a per-unit tax of 0.10 is applied.- What is the change in quantity traded?
- Compute the tax revenue and deadweight loss.