Find the incidence tax imposer by demander

Question

The demand for a good XD=1500-5P and the supply is XS=1000 where X is the number of units and P is price per unit. In equilibrium without taxes the price is 100 and the quantity of X that is traded is 1000.

A) Find the incidence of $100 sales tax imposed by law on the demander.

B) What sales tax rate imposed on the demander will produce the same outcome as this per unit rate?

Answer 1

So first step, we need to calculate the equilibrium with taxes.

1500-5P = 1000, P =100, Q = 1000. With a per-unit sales tax of $100 imposed on demand. With this tax, the consumer’s paying an effective price of P+100, while the supplier receives a price P.

This means the demand function becomes:

X_{D} = 1500 - 5(P+100)

= 1500 - 5P - 500

= 1000 - 5P.

Because supply is fixed at 1000 units in your example (i.e., it’s supply is perfectly inelastic), we get the following:

1000 - 5P = 1000, meaning the new price is 0.

So, supply receives $0 per unit, and the consumer’s total price is 100. This means the entire tax burden falls on supply here, while the consumer’s sticker price doesn’t change at all.

Now for part B!

What Ad Valorem sales tax imposed on demand would lead to the same outcome?

With a percentage (ad valorem) tax rate r, consumers pay P and remitting a tax of rP. Equivalently, suppliers receive a net price of P(1-r). Demand is based on the consumer’s final price, so:

X_{D}=1500 - 5P

= 1000

meaning P=100.

Thus, with an ad valorem tax, the supplier’s net revenue becomes: 100(1-r).

To replicate the per-unit tax outcome, the supplier’s net revenue needs to be zero, so:

100(1 - r) = 0

Which means r = 1.

A 100% ad valorem sales tax imposed on demand would result in the same outcome as a $100 per unit tax in the market you’ve described.