User:MiloKing/SalesTaxEdit

Effect of sales tax on a single good with a monopolist-controlled market

We consider comparative statics between two situations:

A world where there are no sales taxes
A world where sales taxes are introduced on a single good (or class of goods) for which we are drawing the supply and demand curves.

Note that the same analysis also works for comparative statics where we change the sales tax on only one class of goods.

Unlike in the previous section, here we assume that the market for the single good is supplied solely by one firm which is seeking to maximize its profits. Here, the demand curve the firm faces for its product is the same as the demand curve for the industry as a whole, so the price they receive for their goods declines as the quantity sold increases. This means that the marginal revenue from selling a unit is lower than the price of that unit. Since maximizing profits involves setting marginal cost equal to marginal revenue, the monopolist will produce goods at a level where price exceeds the marginal cost of producing a good.

The marginal revenue from the sale of a good equals $d(PQ)/dQ$ .

Analytical tools

There are three kinds of diagrams that we draw to study the situation:

Consider the world without sales tax. We can draw the usual supply, marginal revenue, and demand curves and do the usual analysis to find the market price and equilibrium quantity traded.
Consider the world with sales tax. In this world, consider supply, marginal revenue, and demand curves drawn with respect to pre-tax prices.
Consider the world with sales tax. In this world, consider supply, marginal revenue, and demand curves with respect to post-tax prices.

Analysis with pre-tax prices

We want to perform comparative statics between the world without sales tax and the world with sales tax. We consider the supply and demand curves for the latter in terms of the pre-tax prices.

Supply curve: The supply curve using pre-tax prices is expected to remain the same as the supply curve in a world without taxes, because the price that the seller sees is the pre-tax price.

Demand curve: The demand curve changes. In general, it moves downward. Assuming the law of demand, that is the same as moving inward, i.e., a contraction of the demand curve. Arithmetically, the new demand curve is related to the old demand curve as follows:

Case	What happens to the demand curve algebraically	Pictorial depiction
price-propotional sales tax	For a price-proportional sales tax with a factor of $x$ , the new demand function at a price of $p$ equals the old demand function at a price of $p(1+x)$ . The reason is that, as far as buyers are concerned, the effective price is $p(1+x)$ . The upshot is that the demand curve shrinks downward by a factor of $1+x$ .	An example picture is below, comparing the no-tax demand curve (solid blue) with the pre-tax demand curve for a 50% sales tax (dashed purple). The original curve shrinks downward to 2/3 of its original level.
quantity-proportional sales tax	For a quantity-proportional sales tax with a tax of $b$ per unit quantity, the new demand function at a price of $p$ is the old demand function at a price of $p+b$ . The reason is that, as far as buyers are concerned, the effective price is $p+b$ . The upshot is that the demand curve shifts downward by a vertical distance of $b$ .	An example picture is below, comparing the no-tax demand curve (solid blue) with the pre-tax demand curve for a 0.4 price units/unit quanity sales tax (dashed purple). The original curve shrinks downward by 0.4 price units from its original level.

Marginal revenue curve: The marginal revenue curve changes. In general, it moves downward. Assuming the law of demand, the marginal revenue curve slopes downwards, so that is the same as moving inward, i.e., a contraction of the demand curve. Arithmetically, the new marginal revenue curve is related to the old marginal revenue curve as follows:

Case	What happens to the marginal revenue curve algebraically	Pictorial depiction
price-propotional sales tax	For a price-proportional sales tax with a factor of $x$ , the new marginal revenue function equals $(1/(1+x))(Q(dP/dQ)+P)$ . The reason is that the demand curve is multiplied by 1/(1+x), so instead of marginal revenue equaling $d(PQ)/dQ$ , it now equals $d(PQ/(1+x))/dQ$ .	An example picture is below, comparing the no-tax marginal revenue curve (solid blue) with the pre-tax marginal revenue curve for a 50% sales tax (dashed purple). The original curve shrinks downward to 2/3 of its original level. INSERT IMAGE
quantity-proportional sales tax	For a quantity-proportional sales tax with a tax of $b$ per unit quantity, the new marginal revenue function equals $Q*(dP/dQ)+P-b$ . The reason is that the demand curve shifts downward by b, so instead of marginal revenue equaling $d(PQ)/dQ$ , it now equals $d((P-b)Q)/dQ$ .	An example picture is below, comparing the no-tax marginal revenue curve (solid blue) with the pre-tax marginal revenue curve for a 0.4 price units/unit quanity sales tax (dashed purple). The original curve shrinks downward by 0.4 price units from its original level. INSERT IMAGE

The upshot is that the supply curve remains the same and the demand and marginal revenue curves move inward. Thus, as with the general analysis of comparative statics for demand and supply, we obtain that:

The market price drops. In other words, the new pre-tax market price is lower than the market price in the world without taxes.
The equilibrium quantity traded falls.

The general picture of what happens, showing the contraction of demand curve and consequent move to a lower equilibrium price and lower quantity traded, is below.

INSERT IMAGE

Analysis with post-tax prices

We want to perform comparative statics between the world without sales tax and the world with sales tax. We consider the supply, marginal revenue, and demand curves for the latter in terms of the post-tax prices.

Demand curve: The demand curve using post-tax prices is expected to remain the same as the demand curve in a world without taxes, because the price that the buyer sees is the post-tax price.

Marginal revenue curve: The marginal revenue curve using post-tax prices is expected to remain the same as the demand curve in a world without taxes, because it is determined by the demand curve.

Supply curve: The supply curve changes. In general, it moves upward. Assuming the law of supply, that is the same as moving inward, i.e., a contraction of the supply curve. Arithmetically, the new supply curve is related to the old supply curve as follows:

For a price-proportional sales tax with a factor of $x$ , the new supply function at a price of $p$ equals the old supply function at a price of $p/(1+x)$ . The reason is that, as far as sellers are concerned, the effective price is the pre-tax price $p/(1+x)$ . The upshot is that the supply curve moves upward by a factor of $1+x$ .
For a quantity-proportional sales tax with a tax of $b$ per unit quantity, the new supply function at a price of $p$ is the old supply function at a price of $p-b$ . The reason is that, as far as sellers are concerned, the effective price is $p-b$ . The upshot is that the supply curve shifts upward by a vertical distance of $b$ .

The upshot is that the supply curve contracts and the demand curve remains the same. Thus, as with the general analysis of comparative statics for demand and supply, we obtain that:

The market price rises. In other words, the new post-tax market price is higher than the market price in the world without taxes.
The equilibrium quantity traded falls.

Combined analysis and conclusions

Combining both these analyses, we obtain the three conclusions:

The pre-tax market price is lower than the market price in a world without the tax (this can be seen via the comparative statics between the no-tax and the pre-tax curves)
The post-tax market price is higher than the market price in a world without the tax (this can be seen via the comparative statics between the no-tax and the post-tax curves)
The equilibrium quantity traded is less than the equilibrium quantity traded in a world without the tax (this can be seen using either of the two comparative statics methods employed above)
The sales tax effectively adds additional deadweight loss on top of the existing deadweight loss caused by the monopoly

Extreme cases of elastic and inelastic supply and demand

We consider some extreme cases. The first row describes the standard case, and subsequent rows describe extreme cases:

Assumption for price-elasticity of demand	Assumption for price-elasticity of supply	Conclusion about pre-tax market price (relative to market price in a world without the tax)	Conclusion about post-tax market price (relative to market price in a world without the tax)	Conclusion about equilibrium quantity traded (relative to a world without the tax)
negative (satisfies the law of demand)	positive (satisfies the law of supply)	falls	rises	falls
infinite, i.e., a horizontal demand curve (e.g., when the good has a perfect substitute)	positive (satisfies the law of supply)	falls	stays the same	falls
zero, i.e., a vertical demand curve. We also say that the demand is perfectly price-inelastic	positive (satisfies the law of supply)	stays the same	rises	stays the same
negative (satisfies the law of demand)	infinite, i.e., a horizontal supply curve, e.g., a constant cost industry. Alternatively, this also applies if the jurisdiction where sales tax is imposed is a small subjurisdiction of the economy and the pre-tax prices of the goods are determined by the world economy.	stays the same	rises	falls
negative (satisfies the law of demand)	zero, i.e., a vertical supply curve. We say that the supply is perfectly price-inelastic	falls	stays the same	stays the same

How price-elasticity affects the nature of the effect of sales tax

The extent to which the sales tax affects the pre-tax price, post-tax price, and equilibrium quantity traded depends upon the price-elasticity of demand and price-elasticity of supply. The following are two general principles:

Between demand and supply, the relatively more price-inelastic side absorbs more of the price burden of the sales tax. In other words, if demand is more inelastic, then the effect of the sales tax is largely seen in terms of an increase in the post-tax price. If supply is more inelastic, then the effect of the sales tax is largely seen in terms of a decrease in the pre-tax price. This is in keeping with the general principle attributed to Ricardo that rents are captured by the most inelastic side. Here, the rents are reversed in sign, but the principle stays the same.
In general, the extent to which the equlibrium quantity traded is affected is negatively related to the price-elasticities of both demand and supply. In other words, if we reduce the price-elasticity of either demand or supply, the sensitivity of the equilibrium quantity traded to the sales tax reduces. In particular, if either demand or supply is perfectly price-inelastic, the equilibrium quantity traded is independent of the sales tax.