Question

Suppose that 10 people live on a street and that each of them is willing to pay \(\$ 2\) for each extra streetlight, regardless of the number of streetlights provided. If the cost of providing \(x\) streetlights is given by \(c(x)=x^{2},\) what is the Pareto efficient number of streetlights to provide?

Short answer

10 streetlights provide a Pareto efficient solution.

Step-by-step solution

Understand the Demand

Each of the 10 people values an additional streetlight at $2. Therefore, the total willingness to pay for each additional streetlight is $20.

Recognize the Cost Function

The cost of providing \( x \) streetlights is given by the function \( c(x) = x^2 \). This means the cost of the first streetlight is \( 1^2 = 1 \), the cost of the second is \( 2^2 = 4 \), and so on.

Establish the Pareto Efficiency Condition

For streetlights to be Pareto efficient, the social marginal benefit of an additional streetlight must equal the social marginal cost. The social marginal benefit, in this case, is constant at $20 per streetlight.

Set Marginal Cost Equal to Marginal Benefit

Find the derivative of the cost function \( c(x) \) to determine marginal cost. The derivative is \( c'(x) = 2x \). Set this equal to the social marginal benefit: \( 2x = 20 \).

Solve for \( x \)

Solve the equation \( 2x = 20 \) to find the number of streetlights. Dividing both sides by 2 gives \( x = 10 \).

Key concepts

Marginal Cost

Marginal cost is the cost of producing one more unit of a good or service. It helps businesses and policymakers understand how much it costs to add more of something. In our streetlight problem, the cost function is represented by \( c(x) = x^2 \), which tells us how the total cost changes with the number of streetlights \( x \).
To find the marginal cost, we need to calculate the derivative of the cost function. This is done by finding \( c'(x) \), which gives us \( c'(x) = 2x \). This tells us that the cost of adding one more streetlight increases as the number of streetlights goes up.
Here, the marginal cost relates directly to how many streetlights are already present. For example:

• If there are 3 streetlights, the marginal cost of a fourth light is \( 2 \times 3 = 6 \).
• If there are 10 streetlights, adding one more costs \( 2 \times 10 = 20 \).

In our case, the Pareto efficient number of streetlights, where the marginal cost equals the marginal benefit, is found by equating \( 2x = 20 \), giving us \( x = 10 \).

Marginal Benefit

Marginal benefit shows how much value or satisfaction one gains from consuming an additional unit of a good or service. It's a crucial concept for understanding how much society values another unit of something.
In our example with streetlights, each resident is willing to pay \( \\(2 \) for an extra streetlight. With 10 residents, the social marginal benefit for each additional streetlight adds up to \( \\)20 \).
This value remains constant regardless of the number of streetlights because each resident values additional lighting equally. Thus, the social marginal benefit is crucial in determining the Pareto efficiency of a public good like streetlights. In simple terms:

• Every streetlight adds \( \\(20 \) of perceived value.
• The goal is to provide streetlights until the cost to add another (marginal cost) equals this \( \\)20 \) in value (marginal benefit).

Achieving Pareto efficiency means that you have the right number of streetlights - the point where the extra cost equals the extra value derived.

Cost Function

A cost function gives us a mathematical way to express how total costs change with different levels of production. It's like a recipe for understanding expenses.
In our example, the cost function for streetlights is \( c(x) = x^2 \), illustrating that costs rise exponentially as more streetlights are added. This tells us:

• The total cost for \( x \) streetlights is the square of \( x \). For example, for 5 streetlights, total cost \( = 5^2 = 25 \).
• The nature of this function reflects a quadratic increase. Each additional streetlight costs more than the last.

Understanding the cost function helps city planners grasp how increasing public goods, like streetlights, affects budget and resources. In decision-making, knowing that costs increase faster as you add more lights is critical. This is why equating marginal cost with marginal benefit leads to Pareto efficiency - it's about balancing costs with the societal value of those streetlights.