Question

There are three groups in a community. Their demand curves for public television in hours of programming, T, are given respectively by

W1 = \(200 -T

W2 = \)240 - 2T

W3 = \(320 - 2T

Suppose public television is a pure public good that can be produced at a constant marginal cost of \)200per hour.

a. What is the efficient number of hours of public television?

b. How much public television would a competitive private market provide?

Short answer

1. The efficient number of hours of public televisions is 112 hours.

2. The competitive market will provide 80 hours of public television.

Step-by-step solution

The efficient number of public television

To determine an efficient number of hours of public television, set the marginal benefit equal to the marginal cost. Since the demand curve reflects the marginal benefit,adding all the demand functions(willingness to pay) will give the total marginal benefit as:

$$\begin{gathered} {\text{MSB =W}}_{\text{1}}{\text{+W}}_{\text{2}}{\text{+W}}_{\text{3}} \\ \text{= 200 - T + 240 - 2T + 320 - 2T} \\ \text{= 760 - 5T} \end{gathered}$$

Setting it equal to marginal cost:

$$\begin{gathered} \text{MSB = MSC} \\ \text{760 - 5T = 200} \\ \text{T = 112} \end{gathered}$$

Therefore, 112 hours will be an efficient number of hours.

Public television by competitive market

Assume that the private marginal cost is $200. Find the efficient levels of all the groups by setting the MPC=MPB (demand function/willingness to pay).Add them together to find the hours provided by the competitive market.Therefore;

$$\begin{gathered} \text{200 - T = 200} \\ \text{T = 0} \\ \text{240 - 2T = 200} \\ \text{T = 20} \\ 32\text{0 - 2T = 200} \\ \text{T = 60} \\ \text{0 + 20 + 60 = 80} \end{gathered}$$

Hence, the competitive market provides 80 hours.