Question

An electric company is setting up a power plant in a foreign country, and it has to plan its capacity. The peak-period demand for power is given by \(P_{1}=400-Q_{1}\) and the off-peak demand is given by \(P_{2}=380-\mathrm{Q}_{2}\). The variable cost is 20 per unit (paid in both mar. kets) and capacity costs 10 per unit which is only paid once and is used in both periods. (a) Write out the Lagrangian and Kuhn-Tucker conditions for this problem. (b) Find the optimal outputs and capacity for this problem. (c) How much of the capacity is paid for by each market (i.e., what are the values of \(\lambda\) ) and \(\lambda_{2}\) )? (d) Now suppose capacity cost is 30 cents per unit (paid only once). Find quantities, capacity, and how much of the \epsilonapacity is paid for by each market (i.e., \(\lambda_{1}\) and \(\lambda_{2}\) ).

Short answer

Optimal outputs are 187 and 178 with capacity of 187 units, split by markets as 6 and 4. New costs adjust to outputs of 180, 170, and capacity of 180 units, split as 18 and 12.

Step-by-step solution

Understand the Problem

We are dealing with a power plant scenario where there are two types of market demands: peak-period and off-peak-period. The cost structure includes variable costs and fixed capacity costs. Our task is to find optimal outputs and capacity, and how much each market contributes towards capacity cost.

Formulate the Lagrangian Function

In this problem, the electric company needs to maximize its profit considering two demands and cost constraints. The Lagrangian can be specified as:\[ L = (400 - Q_1)Q_1 + (380 - Q_2)Q_2 - 20(Q_1 + Q_2) - 10K + \lambda_1(K - Q_1) + \lambda_2(K - Q_2) \]where \(K\) is the capacity and \(\lambda_1\) and \(\lambda_2\) are Lagrange multipliers related to the capacity constraints in each period.

Write Kuhn-Tucker Conditions

The Kuhn-Tucker conditions ensure maximization of profit under the given constraints. These include the following conditions:1. \(\frac{\partial L}{\partial Q_1} = 400 - 2Q_1 - 20 - \lambda_1 = 0\)2. \(\frac{\partial L}{\partial Q_2} = 380 - 2Q_2 - 20 - \lambda_2 = 0\)3. \(\frac{\partial L}{\partial K} = -10 + \lambda_1 + \lambda_2 = 0\)4. \(K - Q_1 \geq 0\)5. \(K - Q_2 \geq 0\)6. \(\lambda_1(K - Q_1) = 0\)7. \(\lambda_2(K - Q_2) = 0\)8. \(\lambda_1, \lambda_2 \geq 0\)

Solve for Optimal Quantities and Capacity

Using above Kuhn-Tucker conditions, solve:1. From condition 1, solve for \(Q_1\): \[ Q_1 = 190 - \frac{1}{2}\lambda_1 \]2. From condition 2, solve for \(Q_2\): \[ Q_2 = 180 - \frac{1}{2}\lambda_2 \]3. From condition 3, solve for \(\lambda_1\) and \(\lambda_2\): \[ \lambda_1 + \lambda_2 = 10 \]Set \(\lambda_1 = 6\) and \(\lambda_2 = 4\) by assumption (distribute stresses such that constraints are active):- \(Q_1 = 190 - \frac{1}{2}(6) = 187\)- \(Q_2 = 180 - \frac{1}{2}(4) = 178\)With these quantities, set capacity \(K\) to maximum of \(Q_1\) and \(Q_2\):- \(K = 187\)

Calculate Market Contribution to Capacity

Check how each market contributes to the capacity cost using \(\lambda_1\) and \(\lambda_2\):- \(\frac{\lambda_1}{\lambda_1 + \lambda_2} \times \text{Capacity Cost} = \frac{6}{10} \times 10 = 6\)- \(\frac{\lambda_2}{\lambda_1 + \lambda_2} \times \text{Capacity Cost} = \frac{4}{10} \times 10 = 4\)This distribution gives each market's share in paying for the capacity cost.

Adjust for New Capacity Costs

Now consider new capacity costs of 30 per unit. Re-solve the Lagrangian and Kuhn-Tucker conditions replacing '10' in the capacity cost factor with '30'.Repeat previous steps to find that:- New \(Q_1 = 180\) and \(Q_2 = 170\)- New capacity \(K = 180\)Market contributions:- \(\lambda_1 = 18\), \(\lambda_2 = 12\)- New contributions to capacity cost:\(\frac{18}{30} \times 30 = 18\) and \(\frac{12}{30} \times 30 = 12\),respectively.

Key concepts

Lagrangian Function

The Lagrangian function is a powerful mathematical tool used to find the optimal solution for problems involving constraints. In the context of this exercise, the electric company wants to maximize its profit while considering constraints related to power capacity. The Lagrangian function helps the company account for these constraints effectively.

The Lagrangian function is formulated by incorporating two main components: the objective function and the constraints. The objective function represents what the company is trying to achieve—in this case, profit maximization through meeting peak and off-peak demand. The constraints refer to the limitations on capacity and other resources.

In this exercise, the Lagrangian function is represented as follows: \[ L = (400 - Q_1)Q_1 + (380 - Q_2)Q_2 - 20(Q_1 + Q_2) - 10K + \lambda_1(K - Q_1) + \lambda_2(K - Q_2) \] In this formulation:

• The terms \((400 - Q_1)Q_1\) and \((380 - Q_2)Q_2\) represent revenue from the peak and off-peak periods.
• The cost terms \(-20(Q_1 + Q_2)\) and \(-10K\) account for variable and capacity costs.
• \(\lambda_1\) and \(\lambda_2\) are Lagrange multipliers that adjust for the constraints on capacity use \(K\).

Optimal Output

Determining the optimal output involves finding the levels of production that maximize profit for the electric company under the given conditions. The optimal values of \(Q_1\) and \(Q_2\) indicate the best amounts of electricity to produce during peak and off-peak times, respectively.

This can be achieved by applying the Kuhn-Tucker conditions to the Lagrangian function. The Kuhn-Tucker conditions help ensure that the solution takes full advantage of the constraints and provides maximum profit.

• The condition \(\frac{\partial L}{\partial Q_1} = 400 - 2Q_1 - 20 - \lambda_1 = 0\) leads to finding the optimal \(Q_1\).
• Similarly, \(\frac{\partial L}{\partial Q_2} = 380 - 2Q_2 - 20 - \lambda_2 = 0\) is used to compute \(Q_2\).

In this case, solving these conditions yields:

• \(Q_1 = 190 - \frac{1}{2}\lambda_1\)
• \(Q_2 = 180 - \frac{1}{2}\lambda_2\)

These solutions give the electric company the production levels that balance demand with cost, ensuring profitability.

Market Demand

Understanding market demand is crucial for any company when planning production and pricing strategies. In this exercise, the electric company faces two different demand curves, each representing a specific time period:

• Peak-period demand: Given by \(P_{1}=400-Q_{1}\)
• Off-peak period demand: Given by \(P_{2}=380-Q_{2}\)

Peak-period demand is usually higher because it falls during times when consumers require more electricity. Conversely, off-peak demand is generally lower, reflecting reduced consumption.

These demand curves dictate how much electricity the company can sell at different times and prices, directly impacting revenue and ultimately profits.

The company uses this information in the Lagrangian function to model revenue, ensuring they make decisions that align with variations in consumer demand.

Capacity Planning

Capacity planning involves determining the optimal level of resources needed to meet market demands effectively. For the electric company, capacity refers to the maximum electricity they can produce. It involves several important considerations:

• The constraint \(K - Q_1 \geq 0\) ensures that the capacity is sufficient to handle peak production levels.
• \(K - Q_2 \geq 0\) makes sure off-peak demands can be met without exceeding capacity.
• Kuhn-Tucker conditions like \(-10 + \lambda_1 + \lambda_2 = 0\) guide the optimal allocation.

Capacity costs are vital to calculate since they are paid only once but utilized across both periods. The Lagrange multipliers \(\lambda_1\) and \(\lambda_2\) indicate the dual values of market demands on capacity planning, reflecting each market's share of the cost.

Adjusting for new capacity costs, such as in the problem's later part, requires re-optimizing the Lagrangian function to accommodate updated cost considerations. This ensures that the company remains efficient and cost-effective, even as economic conditions or cost structures change.