Sales tax

Definition

A sales tax is a tax imposed by a government on final sales or purchases.

A sales tax is a kind of consumption tax, and it differs from a value-added tax in that the tax is collected only at the point of final purchase rather than at intermediate steps on the value added at each step.

A sales tax may be of either of these two forms:

• A price-proportional sales tax: Here, the sales tax is specified as a fraction of the total pre-tax price and added to it. The buyer pays a post-tax price which is the sum of the pre-tax price and the sales tax. The seller keeps the pre-tax price and the sales tax is remitted to the taxing authority.
• A quantity-proportional sales tax: Here, the sales tax is specified per unit of the quantity being sold. For instance, a sales tax on wheat flour might specify a tax in money units per unit mass of wheat flour.

Interpretation of percentages for price-proportional sales tax

Note that unlike the income tax, the sales tax is computed as a percentage of the pre-tax price and added to the pre-tax price to determine how much the buyer will pay. Thus, the interpretation of percentages for the sales tax is somewhat different from that for income taxes.

For instance, for a sales tax of 25%, the post-tax price is $1.25$ times the pre-tax price. The taxing authority collects $0.25$ times the pre-tax price, which is $0.20$ times, or $20\%$ of, the post-tax price. Note that the percentage 20% is different from the sales tax percentage. In general, the percentage of post-tax price collected by the taxing authority is lower than the percentage of sales tax. However, for small values of price-proportional sales tax, these two percentages are almost the same.

The explicit formula is this. In fractional term, if the sales tax fraction of pre-tax price is $x$, the fraction of post-tax price is:

$\!{\frac {x}{1+x}}$

Effects of sales tax

The effect of a sales tax is understood to mean the effect of introducing the sales tax on a good where there was none before. The analysis of such effects is an example of comparative statics.

Effect on market price and quantity traded

Further information: effect of sales tax on market price and quantity traded

Consider the simplifying assumptions of a competitive market, and assume that the sales tax is levied only on a particular good and not on the various substitute goods and complementary goods. Also, we assume that the law of demand and law of supply hold (these assumptions are usually valid in the short run). Under these assumptions, the following are the effects of introducing or increasing a sales tax:

Item of interest	Direction of effect (non-edge case) relative to a world without sales tax	Would the same effect be observed if an existing sales tax percentage is increased?	Edge case	Conditions for edge case
pre-tax price (i.e., the effective price for the seller)	goes down	yes	pre-tax price stays the same	One of these: The demand curve is vertical, i.e., the price-elasticity of demand is zero. Intuitively, what this means is that the quantity demanded is unaffected by price, so the entire burden of the sales tax is passed on to buyers (consumers). The supply curve is horizontal, i.e., the price-elasticity of supply is infinite.
post-tax price (i.e., the effective price for the buyer)	goes up	yes	post-tax price stays the same	One of these: The supply curve is vertical, i.e., the price-elasticity of supply is zero. Intuitively, what this means is that the quantity supplied is unaffected by price, so the entire burden of the sales tax is absorbed by sellers. The demand curve is horizontal, i.e., the price-elasticity of demand is infinite.
quantity traded	goes down	yes	quantity traded stays the same	Either the demand or the supply curve is vertical, i.e., either the price-elasticity of demand or the price-elasticity of supply is zero.

Effect on social surplus and its distribution

Further information: effect of sales tax on social surplus

In the absence of a sales tax, the social surplus arising from the sale of a good is captured by two types of parties -- sellers/producers (in the form of producer surplus) and buyers/consumers (in the form of consumer surplus).

Once a sales tax is introduced, there are generally two important effects:

1. Now, the social surplus is captured by three types of parties -- producers (producer surplus), consumers (consumer surplus), and the taxing authority (government surplus).
2. The total social surplus goes down: The new total social surplus, even including the portion captured by the taxing authority, is less than the original social surplus. Both the producer surplus and the consumer surplus go down. While part of the reduction in their surpluses is captured by the taxing authority, part of it is lost. The loss can be attributed to the trades that do not occur because of the decline in quantity traded, and is an example of a deadweight loss due to taxation. Geometrically, it can be measured as the area of a Harberger triangle.

Here are some of the relationships:

Producer surplus in a world without sales tax [corresponds to C + D + F] = (Producer surplus in a world with sales tax [corresponds to D]) + (Part of government surplus whose incidence falls on the producers [corresponds to C]) + (Part of deadweight loss whose incidence falls on the producers [corresponds to F])

and:

Consumer surplus in a world without sales tax [corresponds to A + B + E] = (Consumer surplus in a world with sales tax [corresponds to A]) + (Part of government surplus whose incidence falls on the consumers [corresponds to B]) + (Part of deadweight loss whose incidence falls on the consumers [corresponds to E])

Overall:

Social surplus in a world without sales tax [corresponds to A + B + C + D + E + F]= (Social surplus in a world with sales tax [corresponds to A + B + C + D]) + (Deadweight loss due to taxation [corresponds to E + F])

Note that this analysis includes government surplus in the social surplus. The bad effect of taxation, as per this analysis, is the deadweight loss represented by the Harberger triangle.

The tax incidence question is: how is the burden of taxation (both in terms of the part captured through government surplus and the deadweight loss) distributed between producers and consumers? The answer depends on the relative price-elasticities of the demand and supply curves.