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 for no deadweight loss is $25, with an output of 25 units, a consumer surplus of $312.5 and a profit of $0 for LWE. To avoid losses, the lowest price they can charge is $21, leading to an output of 29 units, a consumer surplus of $420.5, a profit of $0 and a deadweight loss of $8. With Kristina's plan, a fixed amount of $52.5 per household can ensure there's no deadweight loss.

Step-by-step solution

Calculate the equilibrium price for no deadweight loss

A deadweight loss is avoided when the market is in equilibrium. This happens when the quantity demanded equals the quantity supplied. Thus, set \(Q = 50 - P\) equal to \(Q = TC - 500\), giving \(50 - P = 500 + P\). Solving for \(P\) gives the equilibrium price as \(P = 25\).

Find the equilibrium quantity

The equilibrium quantity is found by substituting the equilibrium price into the demand equation: \(Q = 50 - P\), yielding \(Q = 50 - 25 = 25\).

Calculate the consumer surplus and LWE's profit

The consumer surplus is the area under the demand curve and above the price line, represented by the formula: \(0.5 * (max price - equilibrium price) * quantity \), which gives: \(0.5 * (50 - 25) * 25 = 312.5\). LWE's profit is the total revenue minus total costs, given by \( (P * Q) - TC\), which gives: \( (25 * 25) - (500 + 25) = 0\).

Find the lowest price to avoid loss

The lowest price for LWE to avoid losses occurs when price is equal to the average total cost. Since the total cost function is linear (TC = 500 + Q) and the number of households are 10, we need to find the price where the total cost equals the total revenues. Setting these two equal we get, \( (P * Q) = 500 + Q \). Solving for \(P\), we get \(P = 500/Q + 1 = 21\).

Calculate quantities, consumer surplus, profit and deadweight loss

Substituting \(P = 21\) into the demand equation gives us \(Q = 50 - 21 = 29\). The consumer surplus is: \(0.5 * (50 - 21) * 29 = 420.5\), and LWE's profit is \( (21 * 29) - (500 + 29) = 0\). Since price is above the equilibrium price, there is a deadweight loss given by \(0.5 * (25 - 21) * (29 - 25) = 8\).

Determine the fixed amount under Kristina's plan

With Kristina's plan, households must pay a fixed amount and a per-unit charge for electricity. Since the per-unit charge is the equilibrium price to break even, which we calculated in Step 1 as $25, the fixed amount can be determined by calculating the difference between the total cost and total revenue at the equilibrium quantity of 25. So, the fixed amount per household will be \( (500 + 25)/10 = 52.5\). Since the households are paying less than their maximum willingness to pay for electricity (which is $50), we can be confident they will choose to pay rather than go without electricity.

Key concepts

Consumer Surplus

Consumer surplus is a vital concept in understanding how consumers benefit from market transactions. It represents the difference between what consumers are willing to pay for a good or service and what they actually pay. It is the area under the demand curve and above the price line in a graph.
For example, in the electricity market of Lake Wobegon, Minnesota, the equilibrium price was set at \(25. The maximum price consumers were willing to pay was \)50. This results in a consumer surplus calculated as:

• Consumer Surplus Formula: \( ext{Consumer Surplus} = 0.5 \times ( ext{Maximum Price} - ext{Equilibrium Price}) \times ext{Quantity} \)
• Using the numbers provided, we find the surplus is \(0.5 \times (50 - 25) \times 25 = 312.5\)

This consumer surplus shows the total additional benefit the households receive from purchasing electricity at the equilibrium price.

Deadweight Loss

Deadweight loss occurs in a market when the allocation of resources is not efficient, leading to a loss of total welfare. In a perfectly competitive market in equilibrium, there is no deadweight loss because resources are optimally allocated between producers and consumers. Any deviation from this equilibrium, such as taxation or price floors/ceilings, may lead to this inefficiency.
In the provided exercise, if the electricity price is set higher than the equilibrium price, the producers might still cover their costs, but some consumer demand is unmet. This is evident when the price was set at \(21 to avoid company losses, leading to a deadweight loss because the equilibrium price was originally \)25. The exact loss is computed as:

• Deadweight Loss Formula: \(0.5 \times ( ext{Equilibrium Price} - ext{New Price}) \times ( ext{New Quantity} - ext{Equilibrium Quantity}) = 8\)

Thus, this $8 loss represents the inefficiency from not operating at true equilibrium.

Price Equilibrium

Price equilibrium is the state in an economic market where the quantity demanded matches the quantity supplied at a particular price. It is the moving force which balances supply and demand, creating market stability. At this point, neither shortages nor surpluses occur.
In Lake Wobegon's electricity market, equilibrium is achieved when the quantity demanded by all households equals the amount supplied by Lake Wobegon Electric. The demand function was given as \(Q = 50 - P\), and to find equilibrium, this was set equal to the supply, described by the cost function adjusted for units \(P = 500 + Q - 500\). After computing, the equilibrium price was calculated to be $25, creating an equilibrium quantity of 25 units.

Profit Calculation

In business, profit calculation is fundamental to determining how much gain a company receives from its operations. It is computed as total revenue minus total costs. Managers and regulators often use this to decide pricing and strategize financial health.
For Lake Wobegon Electric, several computations were necessary to understand the profit outcomes at different prices:

• At equilibrium price $25: Profit = Revenue - Total Cost = (25 * 25) - (500 + 25) = 0
• With a price at $21 to cover costs but below equilibrium, the profit was still calculated as zero because total revenue matched the adjusted cost.

These calculations ensure that Lake Wobegon Electric can break even and maintain operations while providing fair value to customers.