A municipal water utility employs quasi-fixed capital inputs-the water treatment plant and distribution lines to homes-to supply water to 20,000 households in the community it serves.The figure below shows the cost structure of this utility for various levels of water service.Quantity of water consumption is measured in 1,000-gallon units per month.AQFC is the average quasi-fixed cost curve,and LAC is long-run average cost.Long-run marginal cost,LMC,is constant and equal to $4 per 1,000-gallon unit.The inverse demand equation is $$P = 24 - 0.0004 Q _ { d }$$ . If the Public Service Commission implements an optimal two-part pricing plan,the usage fee is set at $_____ per 1,000-gallon units per month and the fixed monthly access charge must then be $_____ per household.

Question

A municipal water utility employs quasi-fixed capital inputs-the water treatment plant and distribution lines to homes-to supply water to 20,000 households in the community it serves.The figure below shows the cost structure of this utility for various levels of water service.Quantity of water consumption is measured in 1,000-gallon units per month.AQFC is the average quasi-fixed cost curve,and LAC is long-run average cost.Long-run marginal cost,LMC,is constant and equal to $4 per 1,000-gallon unit.The inverse demand equation is $$P = 24 - 0.0004 Q _ { d }$$ .  If the Public Service Commission implements an optimal two-part pricing plan,the usage fee is set at $_____ per 1,000-gallon units per month and the fixed monthly access charge must then be $_____ per household.

Quizplus · Accepted Answer

In order to find the optimal two-part pricing plan, we need to take the following steps:

1. Find the profit-maximizing level of output.
To do this, we need to find the quantity at which marginal revenue equals marginal cost. Since the inverse demand equation is given, we can find the total revenue function as follows:
$$TR = P \times Q = (24 - 0.0004 Q) Q = 24Q - 0.0004Q^2$$
Taking the derivative of TR with respect to Q, we get:
$$MR = \frac{dTR}{dQ} = 24 - 0.0008Q$$
Setting MR equal to LMC, we get:
$$4 = 24 - 0.0008Q$$
Solving for Q, we get:
$$Q^* = \frac{20000\times20\times4}{200\times4+0.8} = 17,778 \text{ thousand gallons/month}$$

2. Find the optimal usage fee.
The optimal usage fee should be equal to LMC at the profit-maximizing level of output, in order to ensure that marginal cost is covered. Since LMC is given to be $4 per 1,000-gallon unit, the optimal usage fee is $4.00 per 1,000-gallon unit.

3. Find the optimal access charge.
The optimal access charge should be set such that the utility earns zero economic profit, i.e., covers all its costs including quasi-fixed costs. To find this, we need to determine the total quasi-fixed costs of the utility for 20,000 households. From the AQFC curve, we know that at a output level of 17,778 thousand gallons/month, the average quasi-fixed cost is $9.00 per household. Therefore, the total quasi-fixed costs for 20,000 households is $180,000 (i.e., $9.00 per household times 20,000 households). Adding this to the total variable costs at the profit-maximizing level of output (which can be obtained by multiplying LMC by Q*), we get the total cost at the profit-maximizing level of output:
$$TC = VC + FC = LMC \times Q^* + AQFC \times 20,000 = 4 \times 17,778 + 180,000 = 250,712$$
Now, since the utility has 20,000 households, the optimal access charge per household is given by:
$$A = \frac{TC}{20,000} = \frac{250,712}{20,000} = $12.54$$
Rounding this to the nearest cent, we get $12.50 per household.

Therefore, the optimal two-part pricing plan is:
Usage fee: $4.00 per 1,000-gallon unit
Access charge: $12.50 per household per month.

A Municipal Water Utility Employs Quasi-Fixed Capital Inputs-The Water Treatment P=24−0.0004QdP = 24 - 0.0004 Q _ { d }P=24−0.0004Qd​

A Municipal Water Utility Employs Quasi-Fixed Capital Inputs-The Water Treatment $P = 24 - 0.0004 Q _ { d }$