Question

Problem 2. Suppose that demand for broccoli is given by QD = 1,000 – 5P, where QD 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 QS = 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 the consumer surplus at this equilibrium? What is the producer surplus? 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 that loss be shared if P = 140? 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 all results. Please be thorough.

Answer 1

Q_D= 1000-5P is the demand curve and Q_S= 4P-80 is the supply curve.

For demand curve , when P=0, Q=1000 and when Q=0,P=200. Thus, plotting price on y-axis and quantity on x-axis , vertical intercept will be 200 and horizontal intercept will be 1000.

For supply curve, when Q=0, P=20 and when P=200, Q=4*200-80 = 720.Both the curves are drawn below.

a.For equilibrium, Demand=Supply or, 1000-5P=4P-80 or, 9P=1080 or, P=120 and Q=1000-5*120 = 1000-600 = 400.

Total money spent on broccoli = P*Q = 400*120 = 48,000.

In the above diagram consumer surplus is denoted by area C.S = 1/2*400*(200-120)= 1/2*400*80=16000

and producer surplus is denoted by area P.S = 1/2*400*(120-20) = 20000.

b) If Q=300, then, Q_S = 1000 - 5P or, 300=1000-5P or, 5P=700 or, P=700/5 = 140

Then new consumer surplus = 1/2*300*(200-140) = 1/2*300*60 = 9000

and new producer surplus = 1/2*300*(140-20) = 1/2*300*120 = 17= 18000

c) Initially at P=120, total surplus=16000+20000 = 36000

But at P=140, total surplus = 9000+18000 = 27000

Thus, there is a loss in overall surplus or welfare as the price increases to 140. Here, loss in surplus shared by consumers = 16000-9000 = 7000 and loss in surplus shared by producers = 20000-18000 = 2000. Hence the consumers loses more.

d) When Q=450 instead of 400, equilibrium price changes.

From supply curve,Q=4P-80

or,4P=350+80=430

or, P=107.5

Then, consumer surplus = 1/2*450*(200-107.5) = 20,812.5

and producer surplus = 1/2*450*(107.5-20) = 19687.5

In this consumer surplus increases and its gain= 20812.5 - 16000 = 4812.5

but producer surplus decreases and its loss = 20000 - 19687.5 = 312.5

Thus, here consumers gain but producers lose. It is independent on the price of broccoli.