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 TC = 500 + 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

1. The regulators will charge a price of $1 if they want to avoid deadweight loss. The output, loss, and total consumer surplus will be 490 units, $500, and $12005, respectively.
2. The least price that LWE will charge to avoid losses is $2.04, and the quantity is 479.6 units. The total consumer surplus and deadweight loss will be $11,500.81 and $5.41, respectively.
3. Each household will have to pay $50 for Kristina’s plan to work. None of the households will refuse to pay because they will each earn a higher consumer surplus by paying the fixed cost.

Step-by-step solution

Computing the price, output, consumer surplus, and profit in the market

The individual demand of each household is Q = 50 - P.

You can find the market demand by the horizontal summation of all ten households as follows:

$$\begin{gathered} Q_M=\underset{i=1}{\overset{10}{\sum Q}}i \\ =10\left(50-P\right) \\ {Q}_M=500-10P \end{gathered}$$

Thus, the inverse demand function will be P = 50 - 0.1Q.

The total cost of producing electricity is TC = 500 + Q.

The marginal cost will be:

$MC=\frac{d}{dQ}\left(500+Q\right)=\$1$

If the regulators wish to avoid generating a deadweight loss, they will find the optimal quantity by setting the price equal to the marginal cost as follows:

$$\begin{gathered} 50-0.1Q=1 \\ 0.1Q=49 \\ Q=\frac{49}{0.1} \\ Q=490 \end{gathered}$$

The regulators will produce 490 units to avoid deadweight loss.

As the price will be $1 and the quantity is 490 units, the total revenue will be:

TR = 1 x 490 = $490

The total cost will be:

TC = 500 + 490 = $ 990

The total revenue and total costs are $490 and $990, respectively.

The profit will be as follows:

localid="1653918847609" $$\begin{gathered} \mathrm{\pi }=\mathrm{TR}-\mathrm{TC} \\ =490-990 \\ =-\$500 \end{gathered}$$

LWE will incur a loss worth $500.

You can compute the total consumer surplus as follows:

CS = (0.5)(490)(50-1)

= $ 12,005

The total consumer surplus will be $12,005, or $1200.50 per household.

Computing the lowest price the regulators can impose, along with other market conditions

If LWE does not wish to lose money, the regulators will set the price at the average cost of production.

The average cost is:

$AC=\frac{500+Q}{Q}=\frac{500}{Q}+1$

You can compute the required quantity as follows:

$$\begin{gathered} 50-0.1Q=\frac{500}{Q}+1 \\ 500+0.1Q^2=49Q \\ 0.1Q^2-49Q+500=0 \end{gathered}$$

The quantity is in the form of a quadratic equation. You can solve it as follows:

$$\begin{gathered} Q=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ =\frac{49\pm \sqrt{{\left(49\right)}^2-4\left(500\right)\left(0.1\right)}}{2\times 0.1} \\ =10.4,479.6 \end{gathered}$$

LWE can either produce 10.4 units or 479.6 units.

However, the quantity produced will be 479.6 units because it yields a greater consumer surplus than 10.4 units. Also, the profits earned at 479.6 units will be greater than 10.4 units.

The price at this quantity will be 50 - 0.1(479.6) = $ 2.04:

Thus, LWE will produce 479.6 units and charge a price of $2.04 per unit.

The consumer surplus will be:

CS = (0.5)(479.6)(50-2.04)

= $ 11,500.81

The total consumer surplus will be $11,500.81, or $1150.08 per household.

You can compute the deadweight loss as follows:

DWL = (0.5)(490 - 479.6)(2.04 - 1)

= $ 5.41

LWE generates a deadweight loss worth $5.41

Computing the fixed amount that households have to pay 

If the people pay a fixed charge, they can operate at the marginal cost pricing.

From the total cost function TC = 500 + Q, you can see that the fixed cost is $500. Thus, if all the households pay $500, they can operate at the marginal cost pricing and operate in the market without any deadweight loss.

As there are ten households, each household will have to pay $50.

None of the households will refuse to pay the fixed cost and deny electricity because on incurring the fixed cost, each household will obtain a consumer surplus worth $\$1200.05-\$50=\$1150.05$instead of not having consumer surplus if they don't pay the fixed cost and not use electricity. Thus, each household will be incentivized to incur the fixed cost.