Question

1. Suppose that the demand and supply functions for good X are
   

Q_d = 56 – 2P_X + 0.01M +7P_R

Q_s = -600 + 10P_X

Where P_X is the sales price of good X, M is average consumer income, P_R is the price of a related good.

1. Is good X a normal or inferior good?
2. Are good X and R complements of substitutes? Explain?
   
   Suppose M = $50,000 and P_R = $20
   
3. What is the direct demand function for good X?
4. What is the equilibrium price and quantity?
5. What is the market outcome if M rises to $60,000, other things being equal?
6. What is the market outcome if price is of good P_R is decreases to $14 other things being equal (M= $50,000?
7. What happens to equilibrium price and quantity if the supply function becomes Q_S = - 360 + 10P_X, for the original demand function and the original values for M and P_R?

Answer 1

(a) $\Delta$Qd/$\Delta$M = 0.01 >0 . This implies that good X is a normal good.

(b) $\Delta$Qd/$\Delta$PR =7 >0 . This implies that good X and R are substitute goods. Because partial derivative w.r.t PR is positive.

(c) M=$50,000 and PR= $20

Qd= 56- 2PX+ 0.01M + 7PR

= 56- 2PX + 0.01(50000)+ 7(20)

= 56- 2PX+500+140

Qd = 696 - 2PX (Demand function for good X).

(d) At equilibrium Qd=Qs

696- 2PX= -600+10PX

1296 = 12PX

PX = $ 108 (Equilibrium price)

Q = -600+10(108)

= -600+ 1080 = 480 (Equilibrium quantity)

(e) If M=$60,000

Then, Qd = 56-2PX + 0.01(60000) + 7(20)

= 56- 2PX+ 600 +140

Qd = 796 - 2PX

Now, at equilibrium Qd =Qs

796-2Px = -600 +10PX

1396 =12PX

PX =$ 116.33 (Equilibrium price)

Q= -600+10(116.33)

= -600 + 1163.33

Q = 563.33 (Equilibrium quantity)

(f) If PR=$14

Qd = 56- 2PX +0.01(50000) +7(14)

= 56-2PX +500+ 98

Qd = 654 -2PX

Now at equilibrium Qd=Qs

654-2PX = -600+10PX

1254 =12PX

PX = $ 104.5 (Equilibrium price)

Q= -600+10(104.5)

= -600+ 1045

Q = 445 (Equilibrium quantity)

(g) Qs = -360+10PX

Qd = 696 -2PX

At equilibrium Qd=Qs

696-2PX = -360+10PX

1056 +12PX

PX = $ 88 (Equilibrium price)

Q= -360+10(88)

= -360+880

Q =520 (Equilibrium quantity)