Question

b) The demand for baseball tickets is Q = 360 − 10P and the supply of baseball tickets is Q = 20P. Calculate the equilibrium quantity and price before tax. Now assume a per-ticket tax of $4. Calculate the price paid by consumers and price received by seller (allow prices to go up to 5 decimal places) after a per-ticket tax of $4. Calculate the new equilibrium quantity and the dead-weight loss, as a result of tax. Interpret your results. Graph the old and new prices, quantities and the dead-weight loss [3 marks].

Answer 1

Below are the Pre-Tax Conditions:

Let the given Demand Curve be: Q =360 – 10P and the Supply curve: Q = 20P

Equilibrium is at that point where Quantity Demanded = Quantity Supplied

Q (Demanded) = Q (Supplied)

360 – 10P = 20P

360 = 20P + 10P

360 = 30P

360/30 = P

P = 12

Thus, at equilibrium price of $12 , the equilibrium quantity demanded and supplied be :

Q (Demanded) =360 – 10P

Q (Demanded) =360 – 10(12)

Q (Demanded) =360 – 120

Q (Demanded) = 240

Q (Supplied) = 20P

Q (Supplied)= 20(12)

Q (Supplied)= 240

Q (Demanded) = Q (Supplied )= 240

Below are the Post -Tax Conditions:

If per ticket tax of $4 is imposed, it shifts the supply upwards by $4 units. This is the supply curve actually faced by the consumer as they have to pay a higher price now as compared to before.

We write the inverse demand and supply equation:

Inverse Demand :

10P = 360 – Q

P = 36 – 0.1Q

Inverse Supply Pre-Tax

Q = 20P

P = Q/20

P = 0.05Q

Inverse Supply Post-Tax

P = 0.05Q + 4

Thus the new equilibrium can be calculated by equating the inverse demand function with the new post-tax supply function:

36 – 0.1Q = 0.05Q + 4

36 – 4 = 0.05Q + 0.1Q

32 = 0.15Q

Q = 32/0.15

Q = 213.33

Price that consumer pay is given by the inverse demand function:

P = 36 – 0.1Q

P = 36 – 0.1(213.33)

P = 36 – 21.3

P (C) = 14.66

Price that producer receives is given by the pre-tax supply function:

P = 0.05Q

P = 0.05(213.33)

P(S) = 10.66

Deadweight loss is given by the area of triangle AE1E2

= ½ x Base x Height

= ½ x (Pre Tax – Post Tax Quantity) x Tax

= ½ x (240 – 213.33) x 4

= ½ x 26.67 x 4

= 53.34

In the below figure, the X-axis shows the market quantity and Y-axis shows the Price Level.

o Before tax the demand curve Q = 360 − 10P is downward sloping curve : When P = 0, Q = 360 which gives the Y-axis intercept and when Q = 0, P = 36 which gives the X-axis intercept. For the supply curve, when Q = 0, then P= 0 which means the supplu curve starts from the origin and slopes upwards towards the equilibrium price and quantity combination. The pre-tax supply and demand curve intersect at E1 with equilibrium quantity 240 and equilibrium price $12.

o After tax, the supply curve shifts upwards by the tax amount of $4. For the new supply curve P = 0.05Q + 4, when Q = 0, we get the Y –axis intercept as $4. This verifies the difference of shift between the old and new supply curve (S +Tax). The new supply curve intersects with the unchanged demand curve at point E2. Here the equilibrium quantity has fallen to 213.33. The price paid by the consumer has increased to 14.66 (shown on the demand curve), whereas the price received by the producer has decreased to 10.66 (shown by the old supply curve).

o Thus the incidence of tax burden has fallen on both the consumers as well as sellers. The shaded triangular area AE1E2 represents the deadweight loss in the market.