Suppose that the demand for broccoli is given by: Demand: Q= 1,000 - 5P Where Q
ID: 2495449 • Letter: S
Question
Suppose that the demand for broccoli is given by:
Demand: Q= 1,000 - 5P
Where Q is quantity per year measured in hundreds of bushels and P is price in dollars per hundred bushels. The long-run supply curve for broccoli is given by
Supply: Q= 4p - 80
A) Show that the equilibrium quantity here is Q=400. At this output, what is the equilibrium price? How much in total is spent on broccoli? What is consumer surplus at this equilibrium? What is producer surplus at this equilibrium?
B) How much in total consumer and producer surplus would be lost if Q=300 instead of Q=400?
C) Show how the allocation of the loss of total consumer and producer surplus between suppliers and demanders described in part b depends on the price at which broccoli is sold. How would the loss be shared if P=140? How about if P=95?
D) What would the total loss of consumer and producer surplus be if Q=450 rather than Q=400? Show that the size of this total loss also is independent of the price at which the broccoli is sold.
E) Graph your results.
Explanation / Answer
Demand: Q = 1000 - 5P
Supply: Q = 4P - 80
(A) In equilibrium, Demand = Supply
1000 - 5P = 4P - 80
9P = 1080
P = 120
Q = 4P - 80 = (4 x 120) - 80 = 480 - 80 = 400
Total spending = P x Q = 120 x 400 = 48,000
Consumer surplus (CS) = Area between demand curve & equilibrium price
From demand curve: When Q = 0, P = 1000/5 = 200 [Reservation price]
CS = (1/2) x (200 - 120) x 400 = (1/2) x 80 x 400 = 16,000
Producer surplus (PS) = Area between supply curve & price
From supply curve: When Q = 0, P = 80/4 = 20 [Minimum acceptable price]
PS = (1/2) x (120 - 20) x 400 = (1/2) x 100 x 400 = 20,000
(B) If Q = 300, [Holding equilibrium price unchanged]:
New CS = (1/2) x (200 - 120) x 300 = (1/2) x 80 x 300 = 12,000
Loss in CS = 16,000 - 12,000 = 4,000
New PS = (1/2) x (120 - 20) x 300 = (1/2) x 100 x 300 = 15,000
Loss in PS = 20,000 - 15,000 = 5,000
(C)
(i) P = 140
Quantiity demanded, QD = 1000 - (5 x 140) = 1000 - 700 = 300
CS = (1/2) x (200 - 140) x 300 = (1/2) x 60 x 300 = 9,000
Loss in CS = 16,000 - 9,000 = 7,000
Quantity supplied, QS = 4P - 80 = (4 x 140) - 80 = 560 - 80 = 480
PS = (1/2) x (140 - 20) x 480 = (1/2) x 120 x 480 = 28,800
Gain in PS = 28,800 - 20,000 = 8,800
Therefore, while there is a loss in CS, there is a gain in PS by higher amount.
(ii) P = 95
QD = 1000 - (5 x 95) = 1000 - 475 = 525
CS = (1/2) x (200 - 95) x 525 = (1/2) x 105 x 525 = 27,562.5
Gain in CS = 27,562.5 - 20,000 = 7,562.5
QS = (4 x 95) - 80 = 380 - 80 = 300
PS = (1/2) x (95 - 20) x 300 = (1/2) x 75 x 300 = 11,250
Loss in PS = 20,000 - 11,250 = 8,750
So, gain in CS is higher than loss in CS.
So it is shown that relative loss in CS and PS is dependent on how high or low the price is (relative to equilibrium price).
(D) Q = 450 [Assuming equilibrium price is unchanged at 120]
CS = (1/2) x (200 - 120) x 450 = (1/2) x 80 x 450 = 18,000
Gain in CS = 18,000 - 16,000 = 2,000
PS = (1/2) x (120 - 20) x 450 = (1/2) x 100 x 450 = 22,500
Gain in PS = 22,500 - 20,000 = 2,500
Again, there is a total gain of 4,500 which is dependent on quantity and resultant price.
NOTE: First 4 sub-parts are answered.
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