Question

Suppose that Spongebob and Patrick have the following marginal benefit schedules for the public good G:

MB_S=50-2G        MB_P=40-3G.

1. Suppose there are 4 units of the public good. What is the marginal benefit to Patrick of an additional unit of the public good?

2. What is the total benefit to Patrick from consuming 4 units of the public good? Draw a graph to illustrate this total benefit.

Show your calculations in the space provided.

3. If the marginal cost of providing an additional unit of public good is $40 (i.e. MC = 40). What is the efficient quantity of the public good?

Draw a graph to illustrate the social marginal benefit schedule, the marginal cost and the efficient level of the public good.

Show your calculations in the space provided.

Answer 1

SOLUTION:

The solution of this particular question can be explained very easily with the help of diagrams and graphical Analysis.

Consider the Diagram below.

PART (i)

Thus, we can see from the above diagram that the Marginal Benefits of the two consumers have been indicated very clearly. The Green Curve is for Patrick, and the Blue Curve is for SpongeBob.

Thus, we can say that at Quantity of Public Good G = 4, we will have,

MB_P = 40 - 3G = 40 - 3*4 = 40 - 12 = 28 utils, as shown in the diagram.

Similarly, MB_S = 50 - 2G = 50 - 2*4 = 42 utils, as shown in the diagram.

Thus, at G = 4 units, for every additional unit of G, Patrick has a Marginal Benefit of 28 utils.

PART (ii)

The Total Benefit (TB) for Patrick can be found by Integration of the MB curve for Patrick, just as the MB is found by differenting the TB.

Thus, TB_P = Integration (MB_P) = Integration (40 - 3G) = 40G - 1.5G²

Thus, at G =4, we will have TB_P = 40*4 - 1.5*4² = 136 utils, as shown in the diagram.

PART (iii)

The last part actually requires us to perform a VERTICAL Summation of the Individual MB curves for the two consumers. Thus, The Vertical Summation actually gives us, the Marginal Social Benefit (MSB) curve as indicated in the diagram by the Red curve.

Thus, the Red Curve indicates the Marginal Social Benefit for SpongeBob and Patrick.

It is also given that the MC curve is MC = 40, as shown in the diagram.

The Efficient outcome will be the point where the Red Curve (MSB) intersects with the Marginal Cost curve of the society, MC = 40.

Remeber that the Marginal Social Benefit curve is actually found by the Vertical Summation of the MBs of the two consumers derived by joining the added points from the MB curves (adding their corner points; i.e. 40 + 50 = 90), and the Value by taking a particular level of Quantity say G = 10 (giving us for G = 10 (say) from the MB curves, as MB_P = 10, and MB_S = 30, and adding them to get a point 10+30 = 40, as shown).

Thus, the MSB curve is the line joining the points 40 and 90, extending up to the Blue curve and then coinciding with it.

Thus, we see that the Intersection point, i.e. (10,40) gives us the Equilibrium Amount of Public Goods G = 10 units, and Equilibrium MB = MC = 40 utils.

Equilibrium G = 10 units.

Equilibrium MC=MB = 40 utils, which are the Efficient Levels.