Chapter 10: Problem 16

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

Expert verified

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.