The Normal Case

The demand function for a market is the relationship between the price of the commodity and the quantity of it deamanded. Likewise the supply function is the relationship between the price of the commodity and the quantity of it supplied. The demand and supply functions can be represented as curves in a graph, such as is shown below.

Let p_eq and q_eq the price and quantity where the demand and supply curves intersect. Let D(p) be the demand function for the market and S(p) the supply function. If p > p_eq then the quantity demanded is less than the quantity supplied, D(p) < S(p), and the surplus results in the market price being bid down. On the other hand if p < p_eq then the quantity demanded is greater than the quantity supplied; D(p) > S(p); the resulting shortage causes the bprice to be bid up.

At p_eq the quantity demanded is exactly equal to the quantity supplied and there is no tendency for the price to change. It is, in fact, the equilibrium price. The quantity demanded and supplied at that price is the equilibrium output, q_eq.

The Impact of an Excise Tax

Suppose a tax of t is imposed upon the commodity and the tax is collected from the producers. One's first expectation would be that the market price would increase by the amount of the tax, to (p_eq+t). In this case the producers would still be gettiing p_eq and thus would supply the same amount q_eq. But the quantity demanded at (p_eq+t) will be less than q_eq and thus that could not be an equilibrium situation. This shortage would drive the price down. The new equilibrium would be somewhere between p_eq and (p_eq+t).

Algebraically the new equilibrium price for consumers p_c is the price such that

D(p_c) = S(p_c + t)
 

Graphically the determination of the new equilibrium price is shown below.

In the above graph P_t represents the price paid by consumers once the tax is imposed. P_t' is the price received by the producers once the tax is imposed. The difference between P_t and P_t is equal to the amount of the tax. The effect of the tax is to shift the supply curve, which is S without the tax, to St. The shift is an upward shift by the amount of the tax, but the upward shift is the same as a backward shift, a decrease in supply. As can be seen from the above graph, the impact of the tax is an increase in the price paid by consumers and a decrease in the price received by producers. Thus the consumers and producers share the burden of the tax. In this case the burden of the tax is equally shared by the consumers and producers, but that has to do with the relative slopes of the demand and supply curves. From this graph the loss in consumers' surplus and the loss in producers' surplus may be determined as shown below. The loss in consumers' surplus is shown in pink and the loss in producers' surplus in light blue (cyan).

The combined loss in consumers' and producers' surplus is offset in part by the gain to the government in tax revenue. But the offset is only partial; the loss to consumers and producers is greater than the tax revenue gained, as is shown below. The situation is analogous to a car burglar breaking into a car and doing $500 of damage to get a $50 item. The loss to the car owner is $550 whereas the burglar gets only $50.

The Impact of a Subsidy

Although the analysis of the impact of a tax is important the analysis of the impact of a subsidy is more interesting. The analysis is essentially the same, a subsidy merely being a negative tax. The effect of a subsidy is to shift the supply curve downward by the amount of the subsidy. Effectively this is an increase in supply. The graph below shows the results of a subsidy on a market.

In the above graph (and following graphs) P_s represents the price paid to consumers after the subsidy is created. P_s' represents the price received by the producers, which is the price paid by consumers plus the subsidy. The impact of the subsidy is to lower prices for consumers but to increase the price received by producers. The benefit of the subsidy is shared by the consumers and producers in a proportion that depends upon the relative slopes of the demand and supply functions.

The above graph shows the gains in consumers' and producers' surpluses as a result of the subsidy. Although the effect of the subsidy seems beneficial the important question is the cost of the subsidy relative to the benefits. In the graph shown below the cost of the subsidy to the government is the gray rectangle including the colored triangles. The graph shows the balance is negative; i.e., the cost of the subsidy is always greater than the benefits to consumers and producers. This is an important result of analysis.

The deadweight loss of the subsidy is the amount by which the cost of the subsidy exceeds the gains in consumers' and producers' surpluses, the triangles shown in pink and blue. The magnitude of the deadweight loss of a tax or subsidy depends upon the amount of the tax or subsidy and the change in production that results from the tax or subsidy. Specifically

Deadweight Loss = (1/2)(amount of tax or subsidy)*ΔQ
 

where ΔQ is the change in output.

Since the change in output ΔQ is proportional to the amount of the tax or subsidy the deadweight loss is proportional to the square of the tax or subsidy. This means that if the tax or subsidy is doubled the deadweight loss increases by a factor of four. If the tax or subsidy is tripled the deadweight loss increases by a factor of none.

The relationship between the tax rate and the amount of tax revenue collected is a parabola, a form popularized by Art Laffer. The Laffer Curve shows that beyond a certain point an increase in the tax rate results in a decrease in tax revenue rather than an increase. This relationship was popularized as part of the Supplyside Economics of the Reagan Administration.

The Economic Effect of a Tax (or Subsidy) is Independent of Which Party Makes (or Receives) the Payment to (or from) the Government

This is an important implication of the economic analysis of an excise tax that could easily be overlooked. In the previous analysis it was implicitly assumed that the producer makes the tax payment to the government. The price received by the producers was equal to the price paid by consumers less the amount of the tax. As a result if we look at the supply function as the relationship between the price paid by consumers and the quantity suppled this supply curve shifts vertically upward by the amount of the tax. The demand function as the relationship between the price paid by consumers and the quantity demanded remains unchanged.

Suppose the government required consumers to keep track of their purchases and send a tax payment to the government based on the number of units of taxable goods purchased. If the graph shows quantities demanded and quantities supplied as a function of the price received by producers the supply curve would stay fixed and the demand curve would shift vertically downward by the amount of the tax. This is equivalent to a leftward shift in the in demand resulting in a smaller quantity produced and bought. The price received by the producers would fall as a result of the tax. The price paid by consumers, which is the price received by producers plus the amount of the tax, would rise. It will rise by exactly the same amount as the it would rise if the same tax were collected from the producers rather than the consumers.

As a practical matter it is a lot easier for the government to collect taxes from the sellers (businesses), than from the buyers (consumers). Therefore the government usually imposes taxes are upon the producers rather than upon the consumers but the burden on the consumers is exactly the same.

The same conclusion applies to subsidies. The effect on prices is the same whether the subsidy payment is made to consumers or producers. Thus in a competitive market for medical services the effect of government reimbursement of medical costs to patients will have the same effect on medical service prices as an equal subsidy to the medical service providers.

The Case of a Backward Bending Supply Curve

A backward bending supply curve in the case of labor supply is a perfectly reasonable phenomenon. If wage rates in a particular occupation reach high enough levels the recipients of those wage rates may decided to work less rather than more. In effect, the recipients of the high wage rates may choose to take the benefit of the high pay in terms of more leisure time and hence less work.

An equilibrium of the wage rate in the backward bending portion of the supply curve is a completely legitimate economic equilibrium. But the effect of an excise tax or subsidy in such a market is quite surprising and contrary to what was intended.

First we should examine whether the intersection of the demand and supply curves for this type of market is a stable equilibium. That is to say, if the price is below the equilibrium does quantity demanded exceed quantity supplied and result in a shortage that drives up the price and if the price is above the equilibrium does the quantity supplied exceed the quantity demanded and result in a surplus on the market that drives the price down? As shown in the graph below it is shown that these conditions do prevail and therefore the equilibrium is stable.

When a tax is imposed in a market with a backward-bending supply curve the effect on the equilibrium prices for the consumers and producers is surprising, as is shown in the diagram below. The tax results in a vertical upward shift in the supply curve by the amount of the tax.

Likewise the effect of a subsidy in a market with a backward-bending supply curve is contrary to expectations, as is shown below. The subsidy results in a vertical downward shift in the supply curve by the amount of the subsidy.