Answer to Question #204019 in Financial Math for Thabiso

Question

The inverse demand for table salt is 𝒑 = 𝟐𝟎𝟎 , while the inverse supply of table salt is

𝒒𝒅 +𝟏

𝒑 = 𝟏𝟎 + 𝟐𝒒𝒔.

a. Find the equilibrium price of table salt before AND after the imposition of a 40% ad valorem tax on the

consumers of table salt. [2]

b. Describe the distribution of the burden (incidence) of this ad valorem tax between consumers and

producers. [2]

c. Find and interpret the price elasticity of supply (𝒆𝒔) at the after-tax equilibrium price and quantity. [2]

Answer 1

a.

Before tax

$Qd=Qs$

$\frac{p-1}{200}=\frac{p-10}{2}$

$2(p-1)=200(p-10)$

$2p-2=200p-2000\\200p-2p=2000-2\\198p=1998\\p=\frac{1998}{198}=10.091$

Therefore pe=10.091

$qd=\frac{p-1}{200}=\frac{10.091-1}{200}$

$=0.04545$

$qs=\frac{p-10}{2}=\frac{10.091-10}{2}$

$=0.04545$

After tax imposition

$qd=qs$

$\frac{p-1}{200}=\frac{p-t-10}{10}$

$2(p-1)=200p-200t-2000\\198p=2000-2+2t\\198p-198+200t\\p=10.0909+\frac{200t}{198}$

$qd=\frac{p-1}{200}$

$=\frac{(10.0909+\frac{200t}{198})-1}{200}$

$=\frac{9.0909}{200}+\frac{\frac{200t}{198}}{200}$

$=0.045445+\frac{1t}{198}$

$qs=\frac{p-10}{2}$

$=\frac{(10.0909+\frac{200t}{198})-10}{2}$

$=\frac{0.0909}{2}+\frac{\frac{200t}{198}}{2}$

$=0.04545+\frac{100t}{198}$

As the price increase by 200/198 of the tax, the consumer pays 200/198 of the tax and the supplier absorbs -2/198 of the tax. The equilibrium quantity increases by 100/198 of the tax.

$p=10.0909+\frac{200(0.4)}{198}\\=10.0909+0.4040\\=10.4949$

$qs=0.04545+\frac{100(0.4)}{198}\\=0.04545+0.2020\\=0.2475$

$p=10.0909=\frac{2(0.4)}{198}\\=10.0909+0.004040\\=10.0949$

If the consumer pays 10.4949 after-tax, the producer pays 10.0949. The equilibrium quantity rises by 0.24745.

 

Result

1. The equilibrium price before the tax is equal to 10.091, and after the tax is equal to 10.4949.
2. The distribution of the tax between the consumer and producer is equal to the consumer pays 200/198 of the tax and the supplier absorbs -2/198 of the tax. Hence, the consumer pays 10.4949 but the producer receives 10.0949.

c.

Price elasticity of supply (PES) refers to the change in quantity supplied when there is a change in the market price of the good.

When the PES is equal to 1, the supply is unitary.

When the PES is less than 1, the supply is inelastic.

When the PES is greater than 1, the supply is elastic.

When the tax is imposed, the price of the good would rise.

$\%change \space in\space qs=\frac{New\space qs-Initial\space qs}{Initial\space qs}\times100$

$=\frac{0.2475-0.0455}{0.0455}\times100$

$=\frac{0.20195}{0.0455}\times100$

$=443.846$

$\%change\space in\space price=\frac{New \space price-Initial\space price}{Initia\space price}\times 100$

$=\frac{10.4949-10.091}{10.091}\times100$

$\frac{0.4039}{10.091}\times100$

$=4.00258$

Price Elasticity of Supply $=\frac{\%change\space in\space quantity\space Supplied}{\%change\space in\space price}$

$=\frac{443.846}{4.00258}\\=110.88998$

Therefore, The price elasticity of supply is equal to 110.88998.