Question

In 2011, Americans smoked 16 billion packs of cigarettes. They paid an average retail price of \(5.00 per pack.

a. Given that the elasticity of supply is 0.5 and the elasticity of demand is -0.4, derive linear demand and supply curves for cigarettes.

b. Cigarettes are subject to a federal tax, which was about \)1.00 per pack in 2011. What does this tax do to the market-clearing price and quantity?

c. How much of the federal tax will consumers pay? What part will producers pay?

Short answer

1. The linear demand curve is (Q = 22.4 – 1.28P) and supply curve is (Q = 8 + 1.6P).

2. The market-clearing price will increase and the quantity will decrease.

3. The consumers will pay 45 percent of the tax and the producers will pay 55 percent of the tax.

Step-by-step solution

Step 1. Deriving the linear demand and supply curves.

• Demand equation

The elasticity of demand is given by:

$E_d=-b\left(\frac{P^{*}}{Q^{*}}\right)$

, where P* is the equilibrium price, Q* is the equilibrium quantity and ‘b’ represents the change in quantity demanded by the change in price.

The value of ‘b’ is calculated by putting the values of E_{d ,} P*, and Q*.

$$\begin{gathered} -0.4=-b\left(\frac{5}{16}\right) \\ -5b=6.4 \\ b=\frac{6.4}{5} \\ =1.28 \end{gathered}$$

The demand equation is given by: Q = a –b*P,

The value of ‘a’ is determined by putting the values of Q, P, and ‘b’.

The demand equation for cigarettes is shown below by putting the values of Q, ‘a’, ‘b’ and P in the demand equation.

Demand: Q = 22.4 -1.28P

• Supply equation

The elasticity of supply is given by:

$E_S=-d\left(\frac{P*}{Q*}\right)$, where P* is equilibrium price, Q* is equilibrium quantity and ‘d’ represents the change in quantity supplied by change in price.

The value of ‘d’ is calculated by putting the values of Es, P* and Q*.

$$\begin{gathered} 0.5=d\left(\frac{5}{16}\right) \\ 5d=8 \\ d=\frac{8}{5} \\ =1.6 \end{gathered}$$

The supply equation is given by: Q = c + dP

The value of ‘c’ is determined by putting the values of Q, P and ‘d’.

$$\begin{gathered} Q=c+dP \\ 16=c+1.6\left(5\right) \\ c=16-8 \\ c=8 \end{gathered}$$

The supply equation for cigarettes is shown below by putting the values of Q, ‘c’, ‘d’ and P in the supply equation.

The supply equation is given by: Q = 8 + 1.6P.

Step 2. Impact of tax on equilibrium quantity and price.

The taxation of $1 will raise the cost of producing of cigarettes. The increased production cost will discourage producers, resulting in a decrease in the supplied quantity. Hence, the supply curve will shift leftward causing a new equilibrium point on demand curve.

The leftward shift in the supply curve would increase the price level and decrease the quantity produced in the market.

These changes in equilibrium position must have been included in the taxed price of cigarettes. Thus, it concludes that the taxation has caused the price to rise and quantity to decline.

The demand equation for cigarettes is shown below by putting the values of Q, ‘a’, ‘b’ and P in the demand equation.

Demand: Q = 22.4 -1.28P

• Supply equation

The elasticity of supply is given by:

, where P* is equilibrium price, Q* is equilibrium quantity and ‘d’ represents the change in quantity supplied by change in price.

The value of ‘d’ is calculated by putting the values of Es, P* and Q*.

The supply equation is given by: Q = c + dP

The value of ‘c’ is determined by putting the values of Q, P and ‘d’.

The supply equation for cigarettes is shown below by putting the values of Q, ‘c’, ‘d’ and P in the supply equation.

The supply equation is given by: Q = 8 + 1.6P.

Step 3. Calculating the part tax paid by consumers and producers.

The federal tax of $1 will increase the price level of cigarettes by $1. The new price will be $6.

The part of tax paid by producers: The increase in tax has caused a leftward shift in the supply curve. The supply curve equation will change. The new supply curve equation will be:

$$\begin{gathered} Q=8+1.6\left(P-1\right) \\ =8+1.6P-1.6 \\ =6.4-1.6P \end{gathered}$$

With the help of new supply equation, onc can find the price that the producer should at the new supply equation. The price that producer should receive:

$$\begin{gathered} Demand=NewSupply \\ 22.4-1.28=6.4-1.6 \\ 2.88P=16 \\ P=5.55 \end{gathered}$$

The producer should get $5.55 of after the increase in costs due to the tax, but it is receiving only $5 (6 -1). The extra $0.55 is taken by the government as a tax revenue from producers side. Hence, the producers are paying 55 percent of the tax (0.55×100).

The part of tax paid by consumers:Since producers are paying 55 percent of the tax, the consumers will be paying the remaining percent of federal tax. Thus, the consumers will pay 45 percent (100 – 55) of the tax.