Question

In Example 9.1 (page 332), we calculated the gains and losses from price controls on natural gas and found that there was a deadweight loss of \(5.68 billion. This calculation was based on a price of oil of \)50 per barrel.

a. If the price of oil were \(60 per barrel, what would be the free-market price of gas? How large a deadweight loss would result if the maximum allowable price of natural gas were \)3.00 per thousand cubic feet?

b. What price of oil would yield a free-market price of natural gas of $3?

Short answer

1. The free-market price of gas would be $8.94. The deadweight loss will be $63.45.

2. The price of natural gas would be $3 at a price of oil =$19.75

Step-by-step solution

Step 1. Determining the free-market price of gas and deadweight loss

The supply and demand equations for natural gas are:

The supply equation of gas (Q^S) is: QS = 15.90 + 0.72P_G+ 0.05P_O

The demand equation of gas, (Q^D) is: QS = 0.02 - 1.8P_G+ 0.69P_O

(P_G is the price of natural gas in dollars per thousand cubic feet, and P_O is the price of oil in dollars per barrel.)

The free-market price of the gas will be calculated by equating the supply equation (Q^S) of gas with the demand equation of gas (Q^D). The price of oil (P_O) is $60 per barrel. The free-market price of the gas is calculated below:

$$\begin{gathered} Q_s=Q_D \\ 15.90+0.72P_G++0.05\left(60\right)=0.02-1.8P_G+0.69\left(60\right) \\ 41.4-15.90-3+0.02=1.8P-0.72P \\ P=8.94 \end{gathered}$$

The price of natural gas is $8.94 per thousand cubic feet.

Deadweight loss:Deadweight loss is the loss to society because of restrictive policies of the government. It is calculated by multiplying the difference between the prices (P2- P1) and quantities (Q1– Q2) of the free-market situation and after the price ceiling.

The quantity demanded by the society during the free market situation is:

$$\begin{gathered} Q1=0.02-1.8\left(8.94\right)+0.69\left(60\right) \\ =0.02-16.092+41.4 \\ =25.33 \end{gathered}$$

The quantity demanded by the society after the maximum allowable price policy:

$$\begin{gathered} Q2=0.02-1.8\left(3\right)+0.69\left(60\right) \\ =0.02-5.4+41.4 \\ =36.02 \end{gathered}$$

The value of the deadweight loss is calculated by putting the values of P1 (Free market equilibrium price), P2 (maximum allowable price policy price), Q1 (free-market quantity), and Q2 (quantity after maximum allowable price policy).

$$\begin{gathered} DWL=\left(P_2-P_1\right)\left(Q_1-Q_2\right) \\ =\left(8.94-3\right)\left(36.02-25.33\right) \\ =5.94\times 10.69 \\ =63.45 \end{gathered}$$

The deadweight loss to society because of the government’s maximum allowable price policy is $63.45 billion.

Step 2. Determining the price of oil when the natural gas is $3

The price of oil can be calculated by putting the price of natural gas (PG = $3) in the free market demand-supply equations. It is calculated below:

$$\begin{gathered} 15.90+0.72\left(3\right)+0.05P_O=0.02-1.8\left(3\right)+0.69P_O \\ 12.64=0.69P-0.05P \\ P=\frac{12.64}{0.64} \\ =19.75 \end{gathered}$$

The price of oil would be $19.75 per barrel.