Question

There are three groups in a community. Their demand curves for public television in hours of programming, \(T,\) are given respectively by \\[ \begin{array}{l} W_{1}=\$ 200-T \\ W_{2}=\$ 240-2 T \\ W_{3}=\$ 320-2 T \end{array} \\] Suppose public television is a pure public good that can be produced at a constant marginal cost of \(\$ 200\) per 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

The efficient number of hours of public television is 112 hours, while a competitive private market would provide 280 hours.

Step-by-step solution

Summing the demand curves

In case of public goods, the social benefit is equal to the sum of the individual benefits. Thus, it is necessary to first sum the three demand curves in order to find the community's overall demand function for public television. This function is given by \( W=\$ 760-5T \).

Find the efficient number of hours

Next, for finding the efficient number of hours of public television, equate marginal cost (MC) with the marginal benefit (MB). The demand function derived in step 1 gives the relation for marginal benefit. In this case, equate the marginal cost of \(\$200\) per hour to aggregated demand: \(\$200=\$760-5T\). Thus, solving for \(T\), the efficient number of public television hours is \(112\) hours.

Determine television hours provided by private market

On the other hand, a competitive private market would equate the marginal cost with each demand curve separately as they perceive benefits individually. Doing so yields \( T=280, 120, 60\) hours respectively for \( W_1, W_2, W_3 \). The quantity of TV programming supplied by the private market would be the quantity corresponding to the highest willingness to pay, i.e., the highest \( T \) value, which is \( 280 \) hours.

Key concepts

Demand Curves

Understanding demand curves is essential for analyzing how consumers respond to changes in price and quantity. In the context of public goods like public television, each group within a community has a different valuation for the service, represented by their individual demand curves. For instance, given by the equations \( W_1 = \$200 - T \), \( W_2 = \$240 - 2T \), and \( W_3 = \$320 - 2T \), where \( W \) represents willingness to pay and \( T \) stands for hours of programming.

These demand curves slope downward, reflecting the economic principle that willingness to pay decreases as the quantity consumed increases. For public goods, each person's benefit does not diminish others' ability to enjoy it, which is why the total demand in such a scenario is the vertical sum of individual demand curves. This concept is crucial in determining the community's overall demand, and ultimately, the efficient level of provision for the public good.

Marginal Cost

Marginal cost is the cost of producing one additional unit of a good or service. In microeconomics, it's a fundamental concept as it helps determine the most efficient point of production. When it comes to public goods, the notion remains the same, although the social nature of these goods presents unique challenges. For public television, the exercise assumes a constant marginal cost of \( \$200 \) per hour, signifying that producing each additional hour of programming incurs the same expense.

Efficient provision occurs where the marginal cost of providing the service equals the collective marginal benefit, the point at which the additional cost to provide the service is just equal to the additional benefit as valued by the consumers. This balance is crucial in ensuring resources are not wasted and maximum social welfare is achieved.

Efficiency in Public Goods Provision

Efficiency in providing public goods like television programming is achieved when the sum of the community's marginal benefit equals the marginal cost of production. To obtain this social efficiency, the total willingness to pay (or aggregated demand) by all groups is considered rather than the individual demand curves.

Illustrating Efficiency

For the given problem, the summed demand curves reveal the community's overall valuation of public television, which, when set equal to the marginal cost, helps us find the efficient number of hours. The efficiency calculation resulted in \( 112 \) hours of programming, indicating the level where the last hour provided costs just as much as the value it delivers to the community as a whole.

Private Market Provision

When dealing with public goods, private market provision can lead to inefficiency due to the nature of these goods being non-rivalrous and non-excludable. In a private market, providers are motivated by profits and thus equate their marginal cost with the willingness to pay of individual groups, not the community as a whole.

As illustrated, each group's demand results in a different level of desired television programming. The private market tends to cater to those with the highest willingness to pay; however, this does not lead to the socially efficient outcome as it overlooks the combined benefit to all groups. In our exercise, the private market would supply \( 280 \) hours, far exceeding the efficient quantity, and leading to potential overproduction and misallocation of resources.