Question

The market demand function for a good is given by Q = D(p) = 800 − 50p. For each firm that produces the good the total cost function is TC(Q) = 4Q+( Q2/2) . Recall that this means that the marginal cost is MC(Q) = 4 + Q. Assume that firms are price takers. In the short run (with 100 firms), and assume that the government imposes a tax of $3 per unit.

(a) What would be the new equilibrium quantity supplied after the tax is imposed?

(b) What would be the price consumers pay and the price sellers receive with the tax? Explain how the burden of the tax is shared between consumers and producers.

c) Compute consumer and producer surplus before and after the tax. How much government revenue is generated by the tax? How large is the deadweight loss?

(d) What would be the long-run equilibrium quantity in this market with the tax? What are the prices that consumers pay and sellers receive? Compare this to the long-run equilibrium without the tax and determine how much of the burden of the tax is borne by consumers and producers.

Answer 1

The demand function is given by . The total cost is given as and marginal cost is . There are a 100 firms. The marginal cost of a firm is basically the inverse supply curve of the firm, ie or . The supply function of one firm is hence , and of 100 firms is , ie the supply curve of the industry is . The equilibrium price and quantities can be found by equating the quantity demanded and supplied, ie or or , and the equilibrium quantity is or .

Further, the government has imposed a tax of $3 per unit, assuming, on suppliers.

(a) After the tax is imposed, the supply of the producers would be or , as for every unit, producer will have to pay $3 to government, receiving only $P-3. Hence, the new equilibrium price can be found as or or dollars. The equilibrium quantity is or .

(b) The consumer would pay the equilibrium price dollars, but the sellers actually receives dollars, paying $3 to government. But the burden can better be understood by the graph.

The tax increase is shown as ME', length of which is ($)3. The revenue of the goverment by the tax is area of MNKE', as ME' is the tax, and NM is the quantity sold. The revenue of the government is or dollars. Among this, the burden on consumer is represented by the area LRKE', which is ; as LR is equal to MN which is 300 untis, and LE' is price difference between old and new price equilibrium, hence or dollars. Also, the burden on producer is represented by the area LMNR, which is ; as MN is 300 units and LM is difference between pre tax equilibrium price and pre tax price at the new equilibrium price, hence or dollars.

Hence, the burden on producer is $300, and burden on consumer is $600. The burden on consumer is more, as the demand curve is relatively more elastic than the supply curve. Also, it is a general phenomena that the burden of tax is more on consumer if the demand curve is relatively less elastic than the supply curve, and is more on producer if demand curve is relatively more elastic than the supply curve, and will be equal on both if demand and supply curve's elasticity is the same.

(c) In pre-tax scenario: The consumer surplus, CS is area of triangle AER, which is or or dollars. The producer surplus, PS is the area of triangle BER, which is or or dollars.

In post-tax scenario: The consumer surplus, CS is area of triangle AKE', which is or or dollars. The producer surplus, PS is the area of triangle NKE', which is or or dollars.

The government revenue generated is calculated as area of MNKE', which is 900 dollars. The dead weight loss is the area of triangle EME', which is , and since ME' is the tax amount of $3, and LE difference between pre tax and post tax equilibrium quantity, hence or dollars.

(d) In the long run, the supply curve will be completely elastic at the pre-tax equilibrium price. Hence the long run supply curve is , but due to the reflection of tax, the long run supply curve with $3 tax will be . Hence, the long run equilibrium quantity with tax, since the demand is , is thus or units.

Price paid by consumer is $11, and price received by seller is $8, as $3 received by seller goes to government.

In case there wouldn't be a tax, the long run supply curve would be , and equilibrium quantity is or . As the supply curve is prefectly elastic, and the demand curve is less elastic than the supply curve, the entire burden of the tax falls on the consumers, which is equal to the government revenue, by the way. The burden of tax on consumers would be dollars, as 250 is bought and sold, and $3 tax is paid. The burden of tax on producers would be $0, as entire burden falls on consumer.