Question

1. Market demand and supply for a commodity are given by the following equations:

Demand: X = 30 – (1/3) P Supply: X = -2.5 + (1/2) P where X= quantity (units), and P=price per unit ($)

Suppose that the government is planning to impose a tax on this commodity and considering the following two options:

Option 1: A unit tax of $15

Option 2: An ad valorem tax of 20%

1. a) Find the tax incidence on buyers and producers, and the tax revenue of the government under each the two options (4 marks)

1. b) Compare the two options in terms of their welfare costs (2 marks)

Answer 1

Market demand: X = 30 - 1/3P

Market supply: X = -2.5 + 1/2 P

In order to find the equilibrium price and quantity, we need to equate the two,

30 - 1/3P = -2.5 + 1/2P ,

32.5 = 1/2P + 1/3P = 5/6 P ,

P* = 32.5 * 6/5 = 39

X* = 30 - 1/3 * 39 = 30 - 13 = 17

These are the equilibrium price and quantities.

i) Now, if an unit tax of $15 is levied, the tafter-tax price and quantity will be,

Rewriting the demand and supply functions as their inverse functions:

P = 90 - 3X , P = 2X + 5 ,

with $15 tax on producers, the supply curve after tax becomes: P = 2X + 5 + 15 = 2X + 20

So, equating the new after-tax supply curve and the previous demnd curve,

90 - 3X = 2X + 20, 5X = 70, X_T = 14 This is the after-tax equilibrium quantity,

Price producers recieve is from the pre-tax supply equation = 2X_T + 5 = $33

Price consumers have to pay is determined from the demand equation = 90 - 3 * X_T = 90 - 42 = $48

Governemt revenue is calculated by tax times the quantity transacted = 15* 14 = $210

Consumer's tax incidence = after tax price - pre-tax equilibrium price * quantity = (48-39) * 14 = $126

Producer's tax incidence = pre-tax eq. price - after tax price = (39-33)* 14 = $84

Welfare cost = deadweight loss = 1/2 * 15 * (17-14 i.e. the diff between eq quantity and after-tax quantity)

= 1/2 * 5 * 3 = $7.5

ii) Now, when an ad-valorem tax of 20% is imposed :

This tax is a charge based on the value of the good or asset transacted and is a percentage of the price and is a kind of sales/property tax. So, as demand eqn = X = 30 - 1/3P  and supply =  X = -2.5 + 1/2 P

So, when the 20% tax is levied on consumer, 30-1/3 (1+1/5)P = -2.5 + 1/2P .

30 - 2/5P = -2.5 + 1/2P , 32.5 = 9/10P, P = $36 = price recieved by seller

Quantity supplied = -2.5 + 1/2 * 36 = 15.5 16 (as quantity cant be a fraction)

Price paid by consumer = from inverse DD function = P = 90 - 3*16 = $42

So, tax incidence on buyer = (42 - 39) * 16 = $48

Tax incidence on sellers = (39-36) * 16 = $48

Tax revenue recieved by government = 16* 6 = 96

Welfare costs = 1/2* 6 * (17-16) = $3

So, the welfare cost is much less (3<7.5) in the option option as compared to the first option. So, the option of levying an ad valorem tax is better for the economy.