Question

Question 2.(20 points): A furniture store in North Carolina is selling a new model of couch. It can sell 210 couches per month if the price is $240, but it can only sell 130 couches per month if the price is $260. The suppliers of this store can provide 100 couches per month if the price is $240, but they can increase their supply by 100 couches per month more if the price is at $260. (a) Find two linear functions that express the demand and supply, respectively. b) Compute the equilibrium price. Interpret the shortage and the surplus in this case. (c) If the actual price is $230 for each couch, estimate the projected shortage or surplus at that price (that is the absolute difference between demand and supply) (d) Assume that the store sells couches according to the demand. What is the revenue function as a function of the price in this case? Compute the total revenue at the price of $262 per bed

Answer 1

(a)

(i) Linear demand function: P = a - bQ

When P = 240, Q = 210

240 = a - 210b.......(1)

When P = 260, Q = 130.

260 = a - 130b.......(2)

(2) - (1) yields: 80b = 20

b = 0.25

a = 240 + 210b [From (1)] = 240 + (210 x 0.25) = 240 + 52.5 = 292.5

Demand equation: P = 292.5 - 0.25Q

(ii) Linear supply equation: Q = c + dQ

When P = 240, Q = 100

240 = c + 100d.........(3)

When p = 260, Q = 100 + 100 = 200

260 = c + 200d.........(4)

(4) - (3) yields: 100d = 20

d = 0.2

c = 240 - 100d [From (3)] = 240 - (100 x 0.2) = 240 - 20 = 220

Supply equation: P = 220 + 0.2Q

(b) In equilibrium, demand equals supply.

292.5 - 0.25Q = 220 + 0.2Q

0.45Q = 72.5

Q = 161

P = 220 + (0.2 x 161) = 220 + 32.2 = $252.2

In equilibrium, quantity demanded equals quantity supplied, so the market clears and there is no surplus or shortage.

(c) When P = $230,

From demand function,

230 = 292.5 - 0.25Q

0.25Q = 62.5

Q = 250

From supply function,

230 = 220 + 0.2Q

0.2Q = 10

Q = 50

Since quantity demanded is higher than quantity supplied, there is a shortage equal to (250 - 50) = 200.

(d) Revenue = P x Q

From demand function,

0.25Q = 292.5 - P

Q = 1170 - 4P

R = Q x P = 1170P - 4P²

When p = $262,

Revenue = (1170 x 262) - (4 x 262 x 262) = 306540 - 274576 = $31964