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 to be aired is 112 hours. However, a competitive private market would only provide up to 80 hours.

Step-by-step solution

Aggregate Demand

To aggregate the demand curves, simply add together the willingness to pay equations of all three groups. The aggregate demand curve is found by adding the quantity demanded by three groups at each price. This gives us \[ W = W_{1}+ W_{2}+ W_{3}\] Which simplifies to \( W = \$760 - 5T \)

Determine the efficient number of hours

The efficient number of hours of public television is found by balancing the cost and the aggregate willingness to pay. We equate the marginal cost (the cost of an extra unit of good) with aggregate willingness to pay: \[ \$200 = \$760 -5T \]. Solving this for T will yield the efficient number of hours of public television.

Calculate the quantity

To solve the equation derived in step 2 for T, first subtract $200 from both sides to get \[ -5T = \$560 \]. Next, divide each side by -5 to solve for T yielding \[ T = 112 hours \]

Determine the market-provided television hours

The competitive private market would provide the amount of public television that a group would be willing to buy at the price of $200 per hour. Looking at the individual demand curves and setting the price to $200, we find that the largest amount of hours that a group is willing to buy is 80 hours, derived from \( \$ 200 = \$320 - 2T \)

Key concepts

Aggregate Demand

When studying microeconomics, understanding aggregate demand is crucial, especially when dealing with public goods. Aggregate demand in microeconomics refers to the total demand for a good or service within a market. This is the sum of all individual demands from different buyers or consumer groups. In the context of public television, which is a pure public good, the aggregate demand represents the total amount of television programming that the community collectively values and wishes to consume.

For a public good like public television, the demand curves of different groups in the community are paramount. Each curve shows the amount of the good the group is willing to consume at different levels of provision. To find the aggregate demand, these individual willingness-to-pay curves are summed up. The exercise presents three linear demand curves for different groups, and by adding them together, we establish the community’s overall valuation of each additional hour of television programming.

However, it’s not just about totaling up these individual valuations. The balance is struck when this aggregate demand intersects with the marginal cost of providing the good. By setting the marginal cost equal to the overall willingness to pay, the efficient amount of public television programming can be determined—ensuring that the community’s resources are used to their fullest potential without wastage.

Marginal Cost

Marginal cost is a term that might seem intricate at first, but with a bit of explanation, it becomes a simple yet powerful concept in microeconomic analysis. Marginal cost represents the expense of producing one additional unit of a good or service. It’s fundamental in determining how many goods should be produced to achieve maximum economic efficiency. In our example with public television, the marginal cost remains constant at $200 per hour of programming.

This constant marginal cost simplifies the calculation of efficiency. When comparing this to the aggregate demand, we can pinpoint the exact number of hours where producing one more hour of public television is no longer justifiable by the community’s aggregate willingness to pay. Hence, the efficient production point is where the last hour of television provided costs just as much as the community’s valuation of that hour.

In a perfectly competitive market, producers would continue to supply more public television hours up until the point where the marginal cost equals the market price—which is also the aggregate willingness to pay. If the marginal cost were to exceed this amount, the production would lead to losses, as it would not be valued as much by the consumers.

Efficiency in Public Goods Allocation

Attaining efficiency in public goods allocation is like fine-tuning an instrument to reach the perfect harmony; it requires balancing the scales of cost and benefit to the community. In the realm of microeconomics, this efficiency occurs when the social marginal cost of providing the good equals the social marginal benefit, which is reflected in the aggregate willingness to pay.

In our textbook scenario, we seek the efficient number of hours for public television, where the total benefit to the community (measured by the sum of individual group demands) matches the cost of providing that service. By solving the equation where the marginal cost equals the aggregate demand, we derive the efficient number of hours that maximizes societal benefit without incurring unnecessary costs.

However, this ideal state is often difficult to achieve in reality, particularly with public goods. A competitive private market may underprovide a public good like television programming due to its non-excludable and non-rivalrous nature. This means that individuals cannot be effectively excluded from its use, and one person’s consumption does not reduce the amount available for others—leading to potential free-riding issues. The competitive market would typically provide the quantity where the group with the highest demand is willing to pay the marginal cost, which is often less than the efficient quantity. Through understanding these concepts, students can gain better insights into the challenges of public goods allocation and the role of economic analysis in addressing those challenges.