User:MiloKing/SalesTaxEdit CompSubEdit

Effect of sales tax that also affects complementary and substitute goods

The analysis of the preceding section was based on the assumption that the sales tax is levied only on that particular good for which the analysis is being performed. This assumption is necessary to ensure that the other determinants of demand and determinants of supply are unaffected.

However, in real world situations, sales taxes are levied on large classes of goods, and changes to sales taxes are made simultaneously on large classes of goods. In particular, the sales tax may also affect the market prices of complementary goods and substitute goods. This means that we either need a more complicated partial equilibrium analysis (that somehow accounts for the prices of all the complementary and substitute goods) or an even more complicated general equilibrium analysis. This is extremely tricky. We consider some special cases to illustrate the kinds of effects that may be operational.

Sales tax on two mutually substitute goods in perfect competition

Consider two goods $A,B$ that are partial substitutes for each other as far as buyers are concerned. Starting with a world with no sales tax, a sales tax is then levied on both $A$ and $B$ . Also assume that the suppliers of $A$ and $B$ are disjoint, so there is no supply-side substitution. Finally, we assume that the markets for both $A$ and $B$ are perfectly competitive. We can draw the following conclusions:

The post-tax market prices for both $A$ and $B$ are higher than the market price in a world without taxes.
The pre-tax market price for at least one of $A$ and $B$ must fall (or stay the same) relative to its market price. In other words, it cannot happen that both pre-tax market prices are higher than the respective market prices in a world without taxes.
The equilibrium quantity traded for at least one of $A$ and $B$ must fall (or stay the same) relative to the original equilibrium quantity traded. In other words, it cannot happen that the quantity traded for both goods rises relative to the world without taxes.

The justification of these using a combined partial equilibrium analysis is hard, but we can try to do a seriatim analysis of the two goods:

In the first approximation, ignore the substitution effects entirely. Then, the analysis proceeds as in the preceding section, so we expect that for both $A$ and $B$ , in the first approximation, that the pre-tax price is lower than the original market price, the post-tax price is higher than the original market price, and the equilibrium quantity traded is lower than the original.
We now consider the substitution effects. Since the post-tax price for $B$ has risen, this is a change to one of the determinants of demand for $A$ , and the effect of the change is an expansion in the demand curve for $A$ . This expansion applies to both the pre-tax and post-tax demand curves. The upshot is that the pre-tax price goes up and the post-tax price goes up too, while the equilibrium quantity traded rises.
The combined effect on the pre-tax price of $A$ is ambiguous (in the first approximation, it goes down, but the substitution effect pushes it back up, and it is unclear which effect dominates). The combined effect on the equilibrium quantity traded is also ambiguous. However, the combined effect on the post-tax price is unambiguous: both reasons cause it to go up.
Of course, these are just two approximations. We can continue the process ad infinitum -- we can use the changes to the price of $A$ to shift the demand curve for $B$ via the substitution effect, and then again back to $A$ , and so on. The hope is that the process will eventually converge. To avoid this kind of infinite process, a combined partial equilibrium analysis is useful.

The crude seriatim analysis does show that the post-tax price goes up. However, it does not clearly show the conclusions mentioned about pre-tax price and quantity traded. Those conclusions can be better derived by thinking of the combined market for both goods and applying a single analysis to that.

Sales tax on two mutually complementary goods in perfect competition

Consider two goods $A,B$ that are complements for each other as far as buyers are concerned. Starting with a world with no sales tax, a sales tax is then levied on both $A$ and $B$ . Also assume that the suppliers of $A$ and $B$ are disjoint, so there is no supply-side complementation or substitution. Finally, we assume that the markets for both $A$ and $B$ are perfectly competitive. We can draw the following conclusions:

The pre-tax market prices for both $A$ and $B$ are lower than the market price in a world without taxes.
The post-tax market price for at least one of $A$ and $B$ must rise (or stay the same) relative to its market price. In other words, it cannot happen that both post-tax market prices are lower than the respective market prices in a world without taxes.
The equilibrium quantity traded for both $A$ and $B$ must fall (or stay the same) relative to the original equilibrium quantity traded.

The justification of these using a combined partial equilibrium analysis is hard, but we can try to do a seriatim analysis of the two goods, similar to that done for substitution effects.

In the first approximation, ignore the complementarity effects entirely. Then, the analysis proceeds as in the preceding section, so we expect that for both $A$ and $B$ , in the first approximation, pre-tax price is lower than the original market price, the post-tax price is higher than the original market price, and the equilibrium quantity traded is lower than the original.
We now consider the complementarity effects. Since the post-tax price for $B$ has risen, this is a change to one of the determinants of demand for $A$ , and the effect of the change is an contraction in the demand curve for $A$ . This contraction applies to both the pre-tax and post-tax demand curves. The upshot is that the pre-tax price goes down and the post-tax price goes down too, while the equilibrium quantity traded falls.
The combined effect on the post-tax price of $A$ is ambiguous (in the first approximation, it goes up, but the complementarity effect pushes it back down, and it is unclear which effect dominates). The combined effects on the pre-tax price and equilibrium quantity traded are unambiguous, however: both reasons cause them to go down.
Of course, these are just two approximations. We can continue the process ad infinitum -- we can use the changes to the price of $A$ to shift the demand curve for $B$ via the complementarity effect, and then again back to $A$ , and so on. The hope is that the process will eventually converge. To avoid this kind of infinite process, a combined partial equilibrium analysis is useful.

The crude seriatim analysis does show that the pre-tax price and quantity traded both go down. However, it does not clearly show the conclusions mentioned about the post-tax price. Those conclusions can be better derived by thinking of the combined market for both goods and applying a single analysis to that.

Sales tax on two mutually substitute goods with one supplied by a monopolist

Consider two goods $A,B$ that are partial substitutes for each other as far as buyers are concerned. Starting with a world with no sales tax, a sales tax is then levied on both $A$ and $B$ . Also assume that the suppliers of $A$ and $B$ are disjoint, so there is no supply-side substitution. Also assume that the market for good A is supplied by only one firm which seeks to maximize its profits. The market for good B is perfectly competitive.

We can draw the following conclusions:

The post-tax market prices for $A$ and $B$ change ambiguously.
The pre-tax market prices of $A$ and $B$ change ambiguously; they can either rise or fall for each good. This is different from the perfectly competitive case, in which it is impossible for the pre-tax price to rise for both goods.
The equilibrium quantity traded for at least one of $A$ and $B$ must fall (or stay the same) relative to the original equilibrium quantity traded. In other words, it cannot happen that the quantity traded for both goods rises relative to the world without taxes.

The justification of these using a combined partial equilibrium analysis is hard, but we can try to do a seriatim analysis of the two goods:

In the first approximation, ignore the substitution effects entirely. Then, the analysis proceeds as in the preceding section, so we expect that for both $A$ and $B$ , in the first approximation, the post-tax price is higher than the original market price and the equilibrium quantity traded is lower than the original. As for the pre-tax price, for $B$ we expect it will fall, but for $A$ it may not fall for reasons described in the section on sales tax in a monopoly.
We now consider the substitution effects. Since the post-tax price for $B$ has risen, this is a change to one of the determinants of demand for $A$ , and the effect of the change is an expansion in the demand curve for $A$ . This expansion applies to both the pre-tax and post-tax demand curves. However, as we pointed out in the section on monopoly for a single good, it is possible for an increase in demand to actually reduce the price of a good supplied by a monopolist (by encouraging the monopolist to sell the good to a much larger market). So while we might normally expect that the expansion in the demand curve for $A$ would cause an increase in the pre-tax and post-tax prices (relative to the baseline established after the previous bullet point), it could actually cause a decrease in the pre-tax price, or even the post-tax price as well.
The combined effect on the pre-tax price is ambiguous (in the first approximation, it may either go down or up, and the substitution effect also has an ambiguous effect). The combined effect on the equilibrium quantity traded is also ambiguous. Finally, the effect on the post-tax price is ambiguous for the same reason listed in the second bullet point.
Of course, these are just two approximations. We can continue the process ad infinitum -- we can use the changes to the price of $A$ to shift the demand curve for $B$ via the substitution effect, and then again back to $A$ , and so on. (In particular, an observation made above is that the post-tax price of $B$ might not rise. We could justify this if the post-tax price of $A$ fell in the stage described in the second bullet point. If the price of $A$ falls, this will reduce demand for $B$ to the point where the eventual post-tax price of $B$ might be lower.) The hope is that the process will eventually converge. To avoid this kind of infinite process, a combined partial equilibrium analysis is useful.

The crude seriatim analysis does show that the pre-tax and post-tax prices change ambiguously for both goods. However, it does not clearly demonstrate the effects on equilibrium quantity traded. Those conclusions can be better derived by thinking of the combined market for both goods and applying a single analysis to that.

Sales tax on two mutually complementary goods with one supplied by a monopolist

Consider two goods $A,B$ that are complements for each other as far as buyers are concerned. Starting with a world with no sales tax, a sales tax is then levied on both $A$ and $B$ . Also assume that the suppliers of $A$ and $B$ are disjoint, so there is no supply-side complementation or substitution. Also assume that the market for good A is supplied by only one firm which seeks to maximize its profits. The market for good B is perfectly competitive.

We can draw the following conclusions:

The pre-tax market price for $A$ is ambiguous; it can either rise or fall. The pre-tax market price for $B$ falls.
The post-tax market price for at least one of $A$ and $B$ must rise (or stay the same) relative to its market price. In other words, it cannot happen that both post-tax market prices are lower than the respective market prices in a world without taxes.
The equilibrium quantity traded for both $A$ and $B$ must fall (or stay the same) relative to the original equilibrium quantity traded.

The justification of these using a combined partial equilibrium analysis is hard, but we can try to do a seriatim analysis of the two goods:

In the first approximation, ignore the complementarity effects entirely. We expect that for both $A$ and $B$ , in the first approximation, the post-tax price is higher than the original market price and the equilibrium quantity traded is lower than the original. For $B$ , we expect that the pre-tax price is lower than the original market price, but for $A$ , the pre-tax price may be higher or lower than the original market price for reasons described in the section on sales tax in a monopoly.
We now consider the complementarity effects. Since the post-tax price for $B$ has risen, this is a change to one of the determinants of demand for $A$ , and the effect of the change is an contraction in the demand curve for $A$ . This contraction applies to both the pre-tax and post-tax demand curves. The equilibrium quantity traded of $A$ falls, but the pre-tax and post-tax prices could either rise or fall. Normally, they would be expected to fall, but as explained in the section on sales taxes on a monopolist, they could rise if the monopolist decides to shrink the market and only sell the good to higher-end consumers. So the effects on the pre-tax and post-tax prices are ambiguous.
The combined effect on the post-tax price of $A$ is ambiguous (in the first approximation, it goes up, and the complementarity effects are ambiguous). The combined effect on the pre-tax price of $A$ is ambiguous (in the first approximation, it is ambiguous, and the complementarity effect is also ambiguous). The combined effect on equilibrium quantity traded are unambiguous, however: both reasons cause it to go down.
Of course, these are just two approximations. We can continue the process ad infinitum -- we can use the changes to the price of $A$ to shift the demand curve for $B$ via the complementarity effect, and then again back to $A$ , and so on. The hope is that the process will eventually converge. To avoid this kind of infinite process, a combined partial equilibrium analysis is useful.

The crude seriatim analysis does show that the pre-tax price of $B$ falls, and that the quantity traded for both goods falls. However, it does not clearly show the conclusions mentioned about post-tax price. Those conclusions can be better derived by thinking of the combined market for both goods and applying a single analysis to that.