Question

There are 10 households in Lake Wobegon, Minnesota, each with a demand for electricity of \(Q=50-P\). Lake Wobegon Electric's (LWE) cost of producing electricity is \(\mathrm{TC}=500+\mathrm{Q}\) a. If the regulators of LWE want to make sure that there is no deadweight loss in this market, what price will they force LWE to charge? What will output be in that case? Calculate consumer surplus and LWE's profit with that price. b. If regulators want to ensure that LWE doesn't lose money, what is the lowest price they can impose? Calculate output, consumer surplus, and profit. Is there any deadweight loss? c. Kristina knows that deadweight loss is something that this small town can do without. She suggests that each household be required to pay a fixed amount just to receive any electricity at all, and then a per-unit charge for electricity. Then LWE can break even while charging the price calculated in part (a). What fixed amount would each household have to pay for Kristina's plan to work? Why can you be sure that no household will choose instead to refuse the payment and go without electricity?

Short answer

The price to avoid deadweight loss is 1, and the output is 49 units. The consumer surplus is 1204.5 and LWE's profit is -500. To ensure no loss for LWE, the price should be 500, however, it gives an impractical output of -450 units. Therefore, no practical price can ensure no loss for LWE in this regulation. In the third part, each household has to pay a fixed amount of 50 units for LWE to break even while pricing per unit as calculated before.

Step-by-step solution

Calculation of price and output for avoiding dead weight loss

To avoid deadweight loss, price should be equal to marginal cost (P=MC). As in this case, total cost is \( TC=500+Q \), the marginal cost (MC) is first derivative of total cost which is 1. So, price P=MC=1. Given the demand equation as \( Q=50-P \), by substituting P=1, we get \( Q=50-1=49 \). So, the price is 1 and output is 49 units.

Calculation of consumer surplus and LWE's profit

Consumer Surplus (CS) is given by \( \frac{1}{2}*(Highest price - Market price)*Quantity \) = \( \frac{1}{2}*(50-1)*49=1204.5 \). Total cost (TC) is \( 500+49=549 \). So, the profit is \( (49*1)-549=-500 \) as the revenue (R) is \( P*Q \) which is \( 49*1=49 \)

Calculation of minimum price to ensure no loss for LWE

To ensure no loss for LWE, price should be such that the Revenue meets the Total Cost. So, there is no profit and no loss which means total revenue is equal to total cost, so \( P*Q=TC=500+Q \). Solving this equation, we get \( P=500 \). By substituting P=500 in the demand function, we get \( Q=50-500=-450 \). Thus, the price has to be 500 and output is -450 units (which is not practical). So, in this case the price cannot be determined using this method. Hence, regulators cannot ensure no loss for LWE with this regulation.

Finding fixed amount each household has to pay to break even

In the third part of the question. The price per unit is already calculated as 1 in part (a). Now, to make LWE break even, the fixed cost of 500 has to be distributed among all 10 households. So each household has to pay a fixed amount of \( \frac{500}{10}=50 \) units in addition to a per unit cost of electricity consumption.

Key concepts

Marginal Cost Pricing

Marginal Cost Pricing is an economic concept that involves setting the price of a good or service equal to the additional cost to produce one more unit, known as the marginal cost. In the context of our Lake Wobegon Electric (LWE) example, the total cost (TC) of producing electricity is given by the equation \( TC=500+Q \), where \( Q \) represents the quantity of electricity produced. To find the marginal cost, we calculate the derivative of the total cost with respect to quantity, which yields a marginal cost (MC) of 1.

Thus, regulators aiming to avoid deadweight loss, a term that refers to the lost economic efficiency when the optimal quantity of goods is not produced, would set the price equal to the marginal cost (\(P=MC=1\)). At this price, output would be maximized without incurring any deadweight loss, ensuring that resources are allocated efficiently in the market for electricity in Lake Wobegon.

Understanding Deadweight Loss

Deadweight loss occurs when market inefficiencies prevent a market from reaching its full potential. By adhering to marginal cost pricing, LWE would be producing at a point where the price consumers pay equals the cost of producing an additional unit, maximizing total welfare in the market.

Consumer Surplus Calculation

Consumer surplus is a measure of the economic benefit that consumers receive when they are able to purchase a product for less than the maximum price they would be willing to pay. It is essentially the area between the demand curve and the market price, extending up to the quantity consumed.

In our example, when regulators set the price equal to the marginal cost of 1, we can calculate consumer surplus using the formula \( CS = 0.5 \times (Highest price - Market price) \times Quantity \) which gives us \( CS = 0.5 \times (50 - 1) \times 49 = 1204.5 \). This number represents the sum of the differences between what each household is willing to pay for each unit of electricity (up to 50 units) and what they actually pay (1 unit), summed across all units sold (49 in total).

Maximizing Consumer Benefit

By pricing goods at marginal cost, we ensure that consumer surplus is maximized because consumers are paying exactly what it costs to produce one more unit. Any price set above this would create deadweight loss, as the area representing consumer benefit would shrink. The maximization of consumer surplus, together with the minimization of deadweight loss, ensures a more efficient and welfare-maximizing market.

Break-Even Analysis

Break-even analysis is a financial assessment that determines when a business will be able to cover all its expenses and begin to make a profit. It’s the point where the total revenue equals total costs, leading to a situation where the business is neither making a loss nor a profit. In the Lake Wobegon Electric example, regulators want LWE to at least break even, that is, to ensure that LWE's total revenue from selling electricity equals its total costs.

Calculating the Break-Even Point

Initially, we see that simply raising the price to cover costs would lead to impractical outcomes, as the demand would drop below zero. Therefore, Kristina proposes a fixed fee which allows LWE to break even while maintaining the marginal cost pricing of 1. By assessing LWE’s fixed costs of 500 units and dividing this amount by the 10 households, it has been determined that each household would need to pay a fixed fee of 50 units. This amount, when combined with the per-unit cost of electricity, ensures LWE's financial stability without leading to any deadweight loss.

The Benefit of the Fixed Fee

The fixed fee guarantees LWE recovers its fixed costs upfront, circumventing the potential for a loss even when it charges the optimal price per unit of electricity. This financial security encourages the company to continue producing, while households enjoy the benefits of marginal cost pricing and the corresponding consumer surplus. It is a systemic adjustment to maintain the balance of cost recovery and market efficiency.