Question 3.11: Find the equilibrium price and quantity for two substitute g......

The equilibrium condition for this two-goods market is

Q_{dX} = Q_{sX}  \text {and}  Q_{dY} = Q_{sY}

Therefore, the equilibrium prices and quantities are calculated as follows:

82 − 3P_X + P_Y = −5 + 15P_X   equating equations (3.11) and (3.12)
−18P_X + P_Y = −87              simplifying                             (3.15)

and

92 + 2P_X − 4P_Y = −6 + 32P_Y  equating equations (3.13) and (3.14)
2P_X − 36P_Y = −98          simplifying                                  (3.16)

Equations (3.15) and (3.16) are two equations in two unknowns, P_X  \text {and}  P_Y.

Therefore, solve these simultaneous equations for the equilibrium prices, P_X  \text {and}  P_Y

−18P_X + P_Y = −87            equation (3.15)

18P_X  −  324P_Y = −882   equation (3.16) multiplied by 9

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−323P_Y = −969

P_Y = 3

Solve for P_X by substituting P_Y = 3 into either equation (3.15) or equation (3.16):

−18P_X + 3 = −87   substituting P_Y = 3 into equation (3.15)
−18P_X = −90
P_X = 5

Now, solve for Q_X \text  {and}  Q_Y

Solve for Q_X by substituting P_X = 5 and P_Y= 3 into either equation (3.11) or equation (3.12) as appropriate:

Q_X = −5 + 15P_X      using equation (3.12)
Q_X = −5 + 15(5)         substituting PX = 5
Q_X = 70

Solve for Q_Y by substituting P_Y = 3 and P_X = 5 into either equation (3.13) or equation (3.14) as appropriate:

Q_Y = −6 + 32P_Y      using equation (3.14)
Q_Y = −6 + 32(3)         substituting P_Y = 3
Q_Y = 90

The equilibrium prices and quantities in this two-goods market are

P_X = 5 ,      Q_X = 70,       P_Y = 3,         Q_Y = 90