Question

The demand for mineral water is P=10 – (2/3)Q and supply function for mineral water is P=1+(1/3)Q

a) Find the equilibrium price and quantity and Price elasticities of demand and supply.

b) Suppose a unit tax (t) is imposed on suppliers (t= 3TL). Find the new equilbirium.

c) Find the price that consumers pay and the price that producers get after the tax.

d) What is the burden of the tax on producers and consumers and explain how the tax burden is related to elasticities?

Answer 1

Answer:

a.

• Equilibrium

We have:

+) Demand function: $P = 10 - \frac{2}{3} Q_{demand}$

=> $Q_{demand} = (10 - P)/\frac{2}{3} = (10-P).\frac{3}{2} = 15 - \frac{3}{2} P$

+) Supply function:  $P = 1 + \frac{1}{3} Q_{supply}$

=> $Q_{supply} = (P-1)/\frac{1}{3} = (P-1).3 = 3P - 3$

The market of mineral water is at equilibrium when the quantity of demand is equal to the quantity of supply at a price level.

We have: Qdemand = Qsupply

⇔$15 - \frac{3}{2} P = 3P - 3$

⇔ $15 + 3 = 3P + \frac{3}{2} P$

⇔$18 = \frac{9}{2} P$

⇔$P = 18/ \frac{9}{2} = 18 .\frac{2}{9} = 4$

When P = 4

⇒ $\left \{ {{Q_{demand} = 15 - \frac{3}{2}P = 15 -\frac{3}{2}.4 = 9 } \atop {Q_{supply} = 3P - 3 = 3.4 - 3 = 9}} \right.$

• Price elasticity:

At P = 4, Qd = Qs = 9

+) Demand function:

$Q_{demand} = 15 - \frac{3}{2} P$ => $dQ_{D} = -\frac{3}{2}$

Price elasticity of demand is:

$E_{D} = dQ_{D}.\frac{P}{Q_{D} } = -\frac{3}{2} . \frac{4}{9} = -\frac{2}{3}$

+) Supply function:

$Q_{supply} = 3P - 3$ => $dQ_{S} = 3$

Price elasticity of supply is:

$E_{S} = dQ_{S}.\frac{P}{Q_{S} } = 3. \frac{4}{9} = \frac{4}{3}$

So equilibrium price is 4, equilibrium quantity is 9.

Price elasticity of demand is -2/3; Price elasticity of supply is 4/3

b.

Initially, the market is at equilibrium at point A: P=4; Qd = Qs = 9

When a unit of tax is imposed on the suppliers, it will shift the supply curve to the left a distance = t = 3 TL. (We can see in the attached image).

The new supply curve intersects the demand curve at point B.

The market moves to the new equilibrium at point B: Q = Q2; Price producers get after tax = Price consumers pay - Tax

c.

When a unit of tax is imposed, it will result in the difference between the price consumers pay and price producers receive at the equilibrium , which is equal to the tax.

We have:

$P_{D} = 10 - \frac{2}{3} Q_{demand}$

$P_{S} = 1 + \frac{1}{3} Q_{supply}$

And at equilibrium, Q demand = Q supply, so we have the equation:

$P_{S} - P_{D} = t$

⇔ $10 - \frac{2}{3} Q - (1+ \frac{1}{3} Q) = 3$

⇔ 9 - Q = 3

⇔ Q = 9 - 3 = 6

⇒$P_{D} = 10 - \frac{2}{3} Q_{demand} = 10 - \frac{2}{3} .6 = 6$

$P_{S} = 1 + \frac{1}{3} Q_{supply} = 1 + \frac{1}{3} .6 = 3$

New equilibrium: Quantity = 6

+) Price consumers pay = 6

+) Price suppliers receive = 3

d.

When the tax is imposed on the producers, it will create the burden on both producers and consumers.

As we can see it the attached image, the blue part represents the total tax that the government would receive, it can be calculated as:

Tax revenue = Tax × Equilibrium Quantity = 3 x 6 = 18

The burden of tax on consumers is illustrated by the area of rectangle a

=> Burden tax on consumers = (4 - 3) x 6 = 6

=> Burden tax on producers = 18 - 6 = 12

When the government imposes the tax, if the price elasticity of demand is less than price elasticity of supply, the consumers will have heavier burden of tax.

Vice versa, if the price elasticity of demand is greater than price elasticity of supply, the consumers will have less burden of tax.

=> The more inelastic, the heavier burden of tax and vice - versa