Question

3. Imagine there exist three consumers, each with their own demand curves for a Public Good. The equations below provide the demand curves for each consumer for this public good where P is the unit price of the public good and Q is the unit value of the public good. Consumer 1: P = 200 – Q Consumer 2: P= 40 – 30 Consumer 3: P = 50 - Q. The total cost (TC) producing the public good is given by: TC = 40Q, and the marginal cost is given by: MC = 40 a. Derive the aggregate demand curve for the public good. This can be accomplished by stacking four graphs vertically: the top graph should represent Consumer 1, the second graph just below the first one will represent Consumer 2, the third graph will represent Consumer 3, and the fourth graph will represent the aggregate or market demand for the public good. Make sure all your graphs are clearly and completely labeled. (Hint: Think of the price as the marginal benefit that the consumers get from consuming the public good). b. What is the Social Marginal Benefit of the Public Good? c. Find the socially optimal level of the public good. Also, find the total price of each unit of the public good. Show the steps you have taken to arrive at your result.

Answer 1

C1,P=200-Q

C2,P=40-3Q

C3,P=50-Q

TC=40Q

MC=40

a) AD

200-Q+40-3Q+50-Q

290-5Q

c)

TR=290Q-5Q^2

MR=290-10Q

MC=40

MC=MR

40=290-10Q

10Q=250

Q=250/10

=25

P=290-5*25

290-125

=165

TR1=200Q-Q^2

MR1=200-2Q

MC=MR1

40=200-2Q

2Q=160

Q=160/2

=80

P=200-80

=120

TR2=40Q-3Q^2

MR2=40-6Q

40=40-6Q

6Q=0

Q=0

P=40-3*0

=40

TR3=50Q-Q^2

MR3=50-2Q

40=50-2Q

2Q=10

Q=10/2

=5

P=50-5

=45

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b) The social marginal benefit curve (MBS) is the demand curve for a public good. It is derived by vertically summing the individual consumers' marginal benefit curves. The efficient level of a public good occurs where the social marginal benefit curve intersects the marginal cost curve: MBs = MC.

Prue MCZTB Social marginal Benefit AR me 0 5 10 15 20 25 30 &

c) The optimal quantity of the public good occurs where MB (society's marginal benefit) equals MC (provider's marginal cost), or where the two curves intersect. When MB = MC, resources have been allocated efficiently. Optimal Quantity of a public good.  The optimal quantity of public good occurs where MB = MC.

Optimal quantity is 20 units, where mc equals mb.