Effect of sales tax on economic surplus

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This article attempts to discuss the effects of a sales tax on the social surplus. The reference point for studying these effects is a world without the sales tax, where the price is the market price and the quantity traded is the equilibrium quantity traded at that market price.

In a world without the sales tax, we have (assuming away external costs and external benefits):

social surplus in absence of sales tax = (producer surplus in absence of sales tax)+ (consumer surplus in absence of sales tax)

The introduction of the sales tax introduces a third party into the equation: the taxing authority, which we call the government.

social surplus in presence of sales tax = (producer surplus in presence of sales tax)+ (consumer surplus in presence of sales tax)

The goal is to ask the questions:

  • How does the producer surplus in the presence of a sales tax compare with the producer surplus in the absence of a sales tax? If the values differ, what accounts for this difference? How much of this difference is captured by the government through tax, nd how much of it takes the form of deadweight loss?
  • How does the consumer surplus in the presence of a sales tax compare with the consumer surplus in the absence of a sales tax? If the values differ, what accounts for this difference? How much of this difference is captured by the government through tax, nd how much of it takes the form of deadweight loss?
  • How does the social surplus in the presence of a sales tax compare with the social surplus in the absence of a sales tax? If the values differ, what accounts for this difference? How much of this difference is captured by the government through tax, nd how much of it takes the form of deadweight loss?

Assumptions

Prior to beginning the analysis, we note the following:

  1. A sales tax may be price-proportional (proportional to the price of the trade) or quantity-proportional (proportional to the quantity being traded). The quantitative analysis differs somewhat in both these cases. However, the qualitative analysis largely does not.
  2. In this article, we largely focus on the effect of the introduction of a sales tax, by performing comparative statics between a world without sales tax and a world with sales tax. Much of this analysis can also be applied to increases in sales tax. Conversely, a decrease in, or elimination of, a sales tax should have the opposite effect.
  3. For the most part, we focus on short run effects. In particular, this means that we assume the law of demand and law of supply.
  4. We assume away the costs of compliance with the tax laws, and do not deal with issues of tax evasion.
  5. For the most part, we assume competitive markets (though we also discuss other cases). Hence, the law of one price is assumed to hold, so that we can talk of the market price.

Effect of sales tax on a single good with a competitive market

Effect on the pre-tax and post-tax price and quantity traded

This analysis builds on a previous analysis, namely effect of sales tax on market price and quantity traded. The conclusions of that analysis are that the introduction of a sales tax has the following general effects:

  • The pre-tax market price is lower than the market price in a world without the sales tax.
  • The post-tax market price is higher than the market price in a world without the sales tax.
  • The equilibrium quantity traded is lower than the equilibrium quantity traded in a world without the sales tax.

Identifying the old and new surpluses

We list all the surpluses and what they mean:

Quantity Area that computes it
Producer surplus in a world without sales tax The area bounded by the price axis, the supply curve, and the horizontal line at the price level (for the market price without the sales tax). In other words, this is the area between the supply curve and the price level. It aggregates the individual surpluses of all producers.
Consumer surplus in a world without sales tax The area bounded by the price axis, the demand curve, and the horizontal line at the price level (for the market price without the sales tax). In other words, this is the area between the demand curve and the price level. It aggregates the individual surpluses of all consumer.
Government surplus in a world without sales tax Zero
Social surplus in a world without sales tax The region bounded by the price axis, demand curve, and supply curve.
Producer surplus in a world with sales tax The area bounded by the price axis, the supply curve, and the horizontal line at the price level (for the pre-tax market price in a world with sales tax). In other words, this is the area between the supply curve and the price level. It aggregates the individual surpluses of all producers.
Consumer surplus in a world with sales tax The area bounded by the price axis, the demand curve, and the horizontal line at the price level (for the post-tax market price in a world with the price level). In other words, this is the area between the demand curve and the price level. It aggregates the individual surpluses of all consumers.
Government surplus in a world with sales tax The area of the rectangle bounded by the horizontal lines for pre-tax and post-tax prices from below and above, by the price axis on the left, and by a vertical line for equilibrium quantity traded (with the sales tax) on the right.
Social surplus in a world with sales tax The region bounded by the price axis, the demand curve, the vertical linefor equilibrium quantity traded, and the supply curve.
Deadweight loss due to taxation The area of the Harberger triangle. The sides of the triangle are the vertical line for equilibrium quantity traded with the sales tax, the supply curve, and the demand curve.
Portion of original producer surplus that is transfered to government surplus The area of the rectangle bounded by the horizontal lines for pre-tax price from below, market price without sales tax from above, the price axis on the left, and a vertical line for equilibrium quantity traded (with the sales tax) on the right.
Portion of original consumer surplus that is transfered to government surplus The area of the rectangle bounded by the horizontal lines for market price without sales tax from below, post-tax price from above, the price axis on the left, and a vertical line for equilibrium quantity traded (with the sales tax) on the right.
Portion of original producer surplus that is lost to deadweight loss The area of the half-Harberger triangle: the sides are the vertical line for equilibrium quantity traded with the sales tax, the supply curve, and the horizontal line for market price without sales tax.
Portion of original consumer surplus that is lost to deadweight loss The area of the half-Harberger triangle: the sides are the vertical line for equilibrium quantity traded with the sales tax, the demand curve, and the horizontal line for market price without sales tax.

The upshot is that:

Producer surplus in a world without sales tax = (Producer surplus in a world with sales tax) + (Part of government surplus whose incidence falls on the producers) + (Part of deadweight loss whose incidence falls on the producers)

and:

Consumer surplus in a world without sales tax = (Consumer surplus in a world with sales tax) + (Part of government surplus whose incidence falls on the consumers) + (Part of deadweight loss whose incidence falls on the consumers)

Overall:

Social surplus in a world without sales tax = (Social surplus in a world with sales tax) + (Deadweight loss due to taxation)

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.

Note that this analysis is quite similar to that of effect of quantity ceiling on social surplus. The deadweight loss is qualitatively similar in both cases. The key difference is that in the case of a quantity ceiling, there is no government surplus, but rather, the producer and consumer surplus (after taking away deadweight loss) get redistributed based on the bargaining powers of buyers and sellers).

Extreme case 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 consumer surplus relative to a world without sales tax Conclusion about producer surplus relative to a world without sales tax Conclusion about deadweight loss
negative (satisfies the law of demand) positive (satisfies the law of supply) falls, captured by both government surplus and deadweight loss falls, captured by both government surplus and deadweight loss positive
infinite, i.e., a horizontal demand curve (e.g., when the good has a perfect substitute) positive (satisfies the law of supply) it was zero to begin with, and stays zero falls, captured by both government surplus and deadweight loss positive
zero, i.e., a vertical demand curve. We also say that the demand is perfectly price-inelastic positive (satisfies the law of supply) falls, but it was infinite to begin with, remains infinite stays the same zero, because there is no decline in quantity traded
negative (satisfies the law of demand) infinite, i.e., a horizontal supply curve, e.g., a constant cost industry falls, captured by both government surplus and deadweight loss it was zero to begin with, and stays zero positive
negative (satisfies the law of demand) zero, i.e., a vertical supply curve. We say that the supply is perfectly price-inelastic stays the same falls zero, because there is no decline in quantity traded

The question of tax incidence

The tax incidence question for any tax asks the question: how are the effects of the tax distributed among the various parties affected by the tax? The question is typically not just a question of who pays the tax, but rather a question of how a world with the tax differs from a world without the tax.

When we talk of tax incidence, we could mean two related things. The first is simply the decline in producer surplus and consumer surplus that specifically goes to tax revenue, i.e., the part that is converted to government surplus. The second is the decline in producer surplus and consumer surplus overall, including both the part that is converted to government surplus and the deadweight loss due to taxation.

The general rule is that tax incidence is borne by the side with lower price-elasticity. If demand is perfectly price-inelastic, then the entire burden of the tax is passed on to consumers. On the other hand, is supply is perfectly price-inelastic, then the entire burden of the tax is passed on to producers.

Effects of increasing and decreasing the tax rate on the social surplus

In general, increasing the sales tax from an already positive amount to a higher amount has the following effects:

  • The pre-tax market price drops further
  • The post-tax market price rises further
  • The equilibrium quantity traded falls further

As a result, the effects on surpluses are as follows:

Measure What happens to it? Conceptual explanation Pictorial explanation
Producer surplus falls both the equilibrium quantity traded and the pre-tax price fall, so producers are squeezed from both sides. The triangle whose area measures producer surplus becomes strictly smaller.
Consumer surplus falls the equilibrium quantity traded falls and the post-tax price rises, so consumers are squeezed from both sides (they make fewer trades, and the surplus per trade falls). The triangle whose area measures consumer surplus becomes strictly smaller.
Government surplus ambiguous two opposite effects: on the one hand, a higher sales tax means a larger difference between post-tax and pre-tax price (so a larger revenue per unit quantity traded). On the other hand, the quantity traded is lower. This is a sales tax analogue of the Laffer curve effect. The vertical thickness (height) of the rectangle representing government surplus increases, while the horizontal length (the equilibrium quantity traded) falls. It is unclear which effect dominates.
Deadweight loss due to taxation rises the equilibrium quantity traded falls the triangle whose area measures deadweight loss becomes strictly larger.
Social surplus (includes producer surplus, consumer surplus, government surplus) falls follows from deadweight loss becoming larger the area we are trying to measure becomes strictly smaller

In other words, raising taxes has clear negative effects on everything except the government surplus (the tax revenue collected), where the effect is ambiguous.