Question

2. Consider the U.S. market for cigarettes. Suppose an econometric analysis estimates that the equation of the market demand curve is P=98−9Q (where P is measured in dollars per pack, and Q is measured in billions of packs per year), and at first the equation of the market supply curve is P=−2+Q. a. Solve the supply and demand equations simultaneously to find the initial market equilibrium quantity (Q∗) and price (P∗). Show your work. b. Now suppose the government imposes a tax of $t on each pack of cigarettes produced. - After the per-unit tax of $t is imposed, what is the new equation of the market supply curve? - Solve the new supply equation and the demand equation simultaneously to find equations for the new market equilibrium quantity (Q∗tax​) and price (P∗tax​) as functions of t. Show your work. c. The government's revenue from the cigarette tax is equal to tQ. - Using your equation for Q∗tax from the last part, write down an equation for the government's tax revenue as a function of the size of the tax, - If the government wants to earn $19.6 (billions) from the cigarette tax, how high should it set the tax per pack? Show your work. d. Suppose the government sets the cigarette tax at t=$1 per pack. - Using your results above, calculate the new market equilibrium quantity and price after the tax(Q∗tax and P∗tax). Show your work. - In this case, what percentage of the $1 tax is passed on to buyers as a price increase? Show your work.

Answer 1

a. To find the initial market equilibrium quantity (Q∗) and price (P∗), we need to solve the supply and demand equations simultaneously.

Demand equation: P = 98 - 9Q

Supply equation: P = -2 + Q

Setting the two equations equal to each other:

98 - 9Q = -2 + Q

Combining like terms:

10Q = 100

Dividing both sides by 10:

Q = 10

Substituting the value of Q back into either equation (let's use the demand equation):

P = 98 - 9(10)

P = 98 - 90

P = 8

Therefore, the initial market equilibrium quantity (Q∗) is 10 billion packs per year, and the price (P∗) is $8 per pack.

b. After the per-unit tax of $t is imposed, the new equation of the market supply curve can be derived by adding the tax to the original supply equation:

New supply equation: P = -2 + Q + t

To find the new market equilibrium quantity (Q∗tax) and price (P∗tax), we need to solve the new supply equation and the demand equation simultaneously:

Demand equation: P = 98 - 9Q

New supply equation: P = -2 + Q + t

Setting the two equations equal to each other:

98 - 9Q = -2 + Q + t

Combining like terms:

10Q = 100 + t

Dividing both sides by 10:

Q = 10 + 0.1t

Substituting the value of Q back into either equation (let's use the demand equation):

P = 98 - 9(10 + 0.1t)

P = 98 - 90 - 0.9t

P = 8 - 0.9t

Therefore, the equations for the new market equilibrium quantity (Q∗tax) and price (P∗tax) as functions of the tax (t) are:

Q∗tax = 10 + 0.1t (market equilibrium quantity)

P∗tax = 8 - 0.9t (market equilibrium price)

c. The government's tax revenue from the cigarette tax is equal to tQ. Using the equation for Q∗tax from the previous part, we can write down an equation for the government's tax revenue as a function of the size of the tax:

Government's tax revenue = t * Q∗tax

Government's tax revenue = t * (10 + 0.1t)

Government's tax revenue = 10t + 0.1t^2

d. If the government sets the cigarette tax at t = $1 per pack, we can calculate the new market equilibrium quantity and price after the tax (Q∗tax and P∗tax):

Q∗tax = 10 + 0.1t

Q∗tax = 10 + 0.1(1)

Q∗tax = 10.1 billion packs per year

P∗tax = 8 - 0.9t

P∗tax = 8 - 0.9(1)

P∗tax = 7.1 dollars per pack

Therefore, after the tax of $1 per pack, the new market equilibrium quantity (Q∗tax) is 10.1 billion packs per year, and the price (P∗tax) is $7.1 per pack.

To calculate the percentage of the $1 tax passed on to buyers as a price increase, we can use the following formula:

Percentage passed on = (