Unit vs. Ad Valorem tax Example

Suppose the demand for boxes of chocolate is described by the equation

Demand: P= 15 – 1/3 Q

And the supply for boxes of chocolate is described by the equation:

Supply: P= 3 + 1/3 Q

What would be the initial equilibrium price and quantity?

Answer: look for where D&S cross, i.e. where P and Q are the same for both D & S

For example substitute the supply equation P into the demand equation P and solve for Q.

3 + 1/3 Q = 15 – 1/3 Q

1/3 Q= 12 – 1/3 Q

2/3 Q = 12

Q = 18

Then to find P substitute into either the Demand or Supply equation the prices should match.

Demand: P = 15 – 1/3 (18) = 15 – 6 or P = 9

Supply: P= 3 + 1/3 Q = 3 + 1/3 (18) = 3 + 6 = 9 √

Unit tax example:

Now suppose chocolate is deemed a “demerit good” and a paternalistic government in need of revenue decides to use a unit tax of $1 to both discourage consumption and raise revenue.

How many boxes will be purchased?

Answer: The new supply with tax as perceived by the consumers will look like the original supply equation with the unit tax added to the price.

Supply with tax: P= 4 + 1/3 Q

We look for where this supply crosses demand to determine the new equilibrium quantity and price.

4 + 1/3 Q = 15 – 1/3 Q

1/3 Q= 11 – 1/3 Q

2/3 Q = 11

Q = 16.5

How much will consumers pay per box of chocolates with this tax in place?

Then to find the price consumers pay substitute into the Demand equation.

Demand: P = 15 – 1/3 (16.5) = 15 – 5.5 or P to consumers= 9.5

How much will suppliers receive per box of chocolates?

Answer: Substitute into the original supply equation (note it should also be equal to what the consumers pay minus the per unit tax or in this case $9.50 –1 = $8.50)

P=3 + 1/3 (16.5) = 3 + 5.5 or P to sellers = 8.5

How much tax revenue will be gathered?

Answer: Take the per unit tax and multiply by the new equilibrium quantity

$1 x 16.5 = $16.50

Ad Valorem tax example:

Now suppose chocolate is deemed a “demerit good” and a paternalistic government in need of revenue decides to use an ad valorem tax of 10% to both discourage consumption and raise revenue.

How many boxes will be purchased?

Supply with tax: P= (3 + 1/3 Q)(1+.1) or P= 3.3 + .3666 Q

3.3 + 11/30 Q = 15 – 1/3 Q

11/30 Q = 11.7 – 1/3 Q

21/30 Q = 11.7

Q = 11.7(30/21) = 16.714

How much will consumers pay per box of chocolates with this tax in place?

Then to find the price consumers pay substitute into the Demand equation.

Demand: P = 15 – 1/3 (16.714) = 15 – 5.57 = 9.43

How much will suppliers receive per box of chocolates?

Answer: Substitute into the original supply equation

P=3 + 1/3 (16.714) = 3 + 5.57 = 8.57

How much tax revenue will be gathered?

Answer: Find the amount of the tax and multiply by the new equilibrium quantity

Amount of the tax = 10% of the price to the seller = .1 (8.57) = .86

(Note it should also equal the difference between the price to the consumer and the price to the seller

9.43-8.57= .86 √)