Suppose that the demand for steel in Japan is given by the equation Qd S = 1200 – 4PS + PA + PT, where QS is the quantity of steel purchased (millions of tons per year), PS is the price of steel (yen per ton), PA is the price of aluminum (yen per ton), and PT is the price of titanium (yen per ton). The supply curve for steel is given by QS S = 4PS. Similarly, the demand and supply curves for aluminum and for titanium are given by QdA = 1200 – 4PA + PS + PT (demand curve for aluminum), QSA = 4PA (supply curve for aluminum), QdT = 1200 – 4PT + PS + PA (demand curve for titanium), and QST = 4PT (supply curve for aluminum). a) Find the equilibrium prices of steel, aluminum, and titanium in Japan. b) Suppose that a strike in the Japanese steel industry shifts the supply curve for steel to QS S = PS. What does this do to the prices of steel, aluminum, and titanium? c) Suppose that growth in the Japanese beer industry, a big buyer of aluminum cans, fuels an increase in the demand for aluminum so that the demand curve for aluminum becomes QdA = 1500 – 4PA + PS + PT. How does this affect the prices of steel, aluminum, and titanium?

See explanation below for answer.

Explanation:

a) In equilibrium the quantity supplied will equal the quantity demanded in all three markets. Algebraically this implies:

Qd S = Qs S

Qd A = Qs A

Qd T = Qs T

Substituting in the given curves implies:

1200 - 4PS + PA + PT = 4PS

1200 - 4PA + PS + PT = 4PA

1200 - 4PT + PS + PA = 4PT

Solving the first equation for PT and substituting into the second equation implies:

1200 - 4PA + PS + (8PS - PA - 1200) = 4PA

9PS = 9PA

Cancel 9 on both sides, we have:

=> PS = PA

Substituting these results into the third equation implies:

1200 - 4(8PA - PA - 1200) + PA + PA = 4(8PA - PA - 1200)

=> 10800 = 54PA

=> PA = 200

At:

PA = 200

PS = 200

PT = 200

The equilibrium quantities are:

QA = 800

QS = 800

QT = 800

b) Substituting the new supply curve for steel into the equilibrium condition, we have:

1200 - 4PS + PA + PT = PS

1200 - 4PA + PS + PT = 4PA

1200 - 4PT + PS + PA = 4PT

Again solving for PT in the first equation and substituting into the second equation, we have:

1200 - 4PA + PS + (5PS - PA - 1200) = 4PA

=> 6PS = 9PA

=> PS = 1.5PA

Substituting these results into the third equation, we have:

1200 - 4(5(1.5PA) - PA - 1200) + 1.5PA + PA = 4(5(1.5PA) - PA - 1200)

=> 10800 = 49.5PA

=> PA = 218.18

At:

PA = 218.18

PS = 327.27

PT = 218.18

At these prices, the equilibrium quantities are:

QA = 872.72

QS = 327.27

QT = 872.72

The shift in the supply of steel raises the equilibrium price for all three goods, lowering the equilibrium quantity of steel and raising the equilibrium quantities of aluminum and titanium. This last effect comes as a result of the demand curves for aluminum and titanium increasing in response to the shift in the steel supply curve.

c) Returning to the original equilibrium, this shift in the demand for aluminum implies:

1200 - 4PS + PA + PT = 4PS

1500 - 4PA + PS + PT = 4PA

1200 - 4PT + PS + PA = 4PT

Solving the first equation for PT and substituting into the second equation, we have:

1500 - 4PA + PS + (8PS - PA - 1200) = 4PA

=> 9PS + 300 = 9PA

=> PS = PA - 33.33

Substituting these results into the third equation, we have:

1200 - 4(8(PA - 33.33) - PA - 1200) + (PA - 33.33) + PA = 4(8(PA - 33.33) - PA - 1200)

=> 12900 = 54PA

=> PA = 238.89

At:

PA = 238.89

PS = 205.56

PT = 205.56

At these prices, the equilibrium quantities are:

QA = 955.56

QS = 822.24

QT = 822.24

An increase in the demand for aluminum will raise the equilibrium prices and quantities in all three markets. The price and quantity in the steel and aluminum industries increase because as the price of aluminum rises, the demand for steel and titanium increases.