Refer to Example 2.10 (page 81), which analyzes the effects of price controls on natural gas.

1. Using the data in the example, show that the following supply and demand curves describe the market for natural gas in 2005–2007:

Supply: Q = 15.90 + 0.72P_G+ 0.05P_O

Demand: Q = 0.02 - 1.8P_G+ 0.69P_O

Also, verify that if the price of oil is \(50, these curves imply a free-market price of \)6.40 for natural gas.

1. Suppose the regulated price of gas was \(4.50 per thousand cubic feet instead of \)3.00. How much excess demand would there have been?

2. Suppose that the market for natural gas remained unregulated. If the price of oil had increased from \(50 to \)100, what would have happened to the free market price of natural gas?

Let the demand curve be Q = a + bP_G + eP_O, where a is the intercept value, b is the change in the quantity of gas by change in the price of gas, e is the change in the quantity of gas by change in the price of oil.

The price elasticity of demand will be -0.5, cross-price elasticity of demand will be 1.5, gas price is $6.40, the quantity of gas 23 Tcf, and oil price is $50, given in Example 2.10 (Page 81).

The cross-price elasticity of demand is calculated below:

$$\begin{gathered} E_{GO}=\frac{△Q}{△P_O}\times \frac{P_O}{Q} \\ 1.5=\frac{△Q}{△P_O}\times \frac{50}{23} \\ \frac{△Q}{△P_O}=\frac{1.5\times 23}{50} \\ e=0.69 \end{gathered}$$

The price elasticity of demand is calculated below:

$$\begin{gathered} E_D=\frac{△Q}{△P_G}\times \frac{P_G}{Q} \\ -0.5=\frac{△Q}{△P_G}\times \frac{6.40}{23} \\ \frac{△Q}{△P_G}=\frac{-0.5\times 23}{6.40} \\ e=-1.8 \end{gathered}$$

The intercept value, i.e., a, is calculated below:

Q = a - 1.8P_G +0.69P_O

23 = a - 1.8(6.40) + 0.69(50)

23 = a - 11.52 + 34.5

a = 23 + 11.52 -34.5

a = 0.02

The demand curve will be Q = 0.02 – 1.8P_G+ 0.69P_o.

Let the supply curve be Q = c + dP_G + fP_O, where c is the intercept value, d is the change in the quantity of gas by change in the price of gas, f is the change in the quantity of gas by change in the price of oil.

The price elasticity of supply will be 0.2, cross-price elasticity of supply will be 0.1, gas price is $6.40, the quantity of gas 23 Tcf, and oil price is $50, given in Example 2.10 (Page 81).

The cross-price elasticity of supply is calculated below:

$$\begin{gathered} E_{GO}=\frac{△Q}{△P_O}\times \frac{P_O}{Q} \\ 0.1=\frac{△Q}{△P_O}\times \frac{50}{23} \\ \frac{△Q}{△P_O}=\frac{0.1\times 23}{50} \\ f=0.05 \end{gathered}$$

The price elasticity of supply is calculated below:

$$\begin{gathered} E_s=\frac{△Q}{△P_G}\times \frac{P_G}{Q} \\ 0.2=\frac{△Q}{△P_G}\times \frac{6.40}{23} \\ \frac{△Q}{△P_G}=\frac{0.2\times 23}{6.40} \\ d=0.72 \end{gathered}$$

The intercept value, i.e., c, is calculated below:

Q = c - 0.05P_G +0.72P_O

23 = c + 0.72(6.40) + 0.05(50)

23 = c + 4.608 + 2.5

c = 23 - 4.608 - 2.5

c = 15.9

The supply curve will be Q = 15.9 + 0.72P_G + 0.05P_o.

Thus, the supply curve is Q = 15.90 + 0.72P_G + 0.05P_O, and demand curve Q = 0.02 - 1.8P_G + 0.69P_O; hence, proved.

The market price of gas when the price of oil is $50 is calculated below:

At equilibrium, demand is equal to supply.

D = S

0.02 - 1.8P_G + 0.69(50) = 15.90 + 0.72P_G + 0.05(50)

34.52 - 1.8P_G = 18.4 + 0.72P_G

1.8P_G + 0.72P_G = 34.52 - 18.4

2.52P_G = 16.12

P_G = $6.40

Thus, the price is $6.40.

The demand and supply at $4.50 are calculated below:

Q_D = 0.02 - 1.8(4.50) + 0.69(50)

= 0.02 - 8.1 + 34.5

= 26.38

Q_S = 15.9 + 0.72(4.50) + 0.05(50)

= 15.9 + 3.24 + 2.5

= 21.64

The demand is greater than supply; thus,the excess demand will be 4.74 Tcf(=26.38 – 21.64).

The market price of gas when the price of oil is $100 is calculated below:

At equilibrium, demand is equal to supply.

D = S

0.02 - 1.8P_G + 0.69(100) = 15.90 + 0.72P_G + 0.05(100)

69.02 - 1.8P_G = 20.9 + 0.72P_G

1.8P_G + 0.72P_G = 69.02 - 20.9

2.52P_G = 48.12

P_G = $19.10

The price will be $19.10.