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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 P1 = 400-Q1 and the off-peak demand is given by P2 = 380-Q2. The variable cost is 20 per unit (paid in both market) 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 condition for the problem.
b. Find the optimal output and capacity for this problem.
c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?
d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).?
a. Write out the lagrangian and Kuhn-Tucker condition for the problem.
b. Find the optimal output and capacity for this problem.
c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?
d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).?
Most Upvoted Answer
An electric company is setting up a power plant in a foreign country, ...
Step A: Lagrangian and Kuhn-Tucker Conditions
To set up the Lagrangian, denote output in peak and off-peak periods as \( Q_1 \) and \( Q_2 \), respectively. The Lagrangian is formulated as:
\[
\mathcal{L} = \left( 400 - Q_1 \right) Q_1 + \left( 380 - Q_2 \right) Q_2 - 20(Q_1 + Q_2) - 10C
\]
where \( C \) is the capacity.
Kuhn-Tucker conditions imply:
- \( \frac{\partial \mathcal{L}}{\partial Q_1} = 400 - 2Q_1 - 20 = 0 \)
- \( \frac{\partial \mathcal{L}}{\partial Q_2} = 380 - 2Q_2 - 20 = 0 \)
- Capacity constraints: \( C \geq Q_1, C \geq Q_2 \)
Step B: Optimal Output and Capacity
Solving the conditions:
1. From \( \frac{\partial \mathcal{L}}{\partial Q_1} = 0 \):
- \( 400 - 2Q_1 - 20 = 0 \) → \( Q_1 = 190 \)
2. From \( \frac{\partial \mathcal{L}}{\partial Q_2} = 0 \):
- \( 380 - 2Q_2 - 20 = 0 \) → \( Q_2 = 180 \)
Capacity is determined by the maximum of \( Q_1 \) and \( Q_2 \):
- \( C = \max(190, 180) = 190 \)
Step C: Capacity Allocation (λ1 and λ2)
The Lagrange multipliers \( \lambda_1 \) and \( \lambda_2 \) represent the shadow prices for capacity used in each market:
\[
\lambda_1 = \frac{P_1 - MC}{1} = \frac{(400 - Q_1) - 20}{1} = 190
\]
\[
\lambda_2 = \frac{P_2 - MC}{1} = \frac{(380 - Q_2) - 20}{1} = 180
\]
Step D: Capacity Cost Adjustment
If the capacity cost increases to 30 cents per unit, the new Lagrangian becomes:
\[
\mathcal{L} = (400 - Q_1)Q_1 + (380 - Q_2)Q_2 - 20(Q_1 + Q_2) - 30C
\]
Following similar steps, you would find new optimal outputs and capacity, taking into account the increased fixed cost.
Conclusion
- New quantities \( Q_1 \) and \( Q_2 \) will be recalculated.
- The respective values of \( \lambda_1 \) and \( \lambda_2 \) will reflect changes in capacity cost.
To set up the Lagrangian, denote output in peak and off-peak periods as \( Q_1 \) and \( Q_2 \), respectively. The Lagrangian is formulated as:
\[
\mathcal{L} = \left( 400 - Q_1 \right) Q_1 + \left( 380 - Q_2 \right) Q_2 - 20(Q_1 + Q_2) - 10C
\]
where \( C \) is the capacity.
Kuhn-Tucker conditions imply:
- \( \frac{\partial \mathcal{L}}{\partial Q_1} = 400 - 2Q_1 - 20 = 0 \)
- \( \frac{\partial \mathcal{L}}{\partial Q_2} = 380 - 2Q_2 - 20 = 0 \)
- Capacity constraints: \( C \geq Q_1, C \geq Q_2 \)
Step B: Optimal Output and Capacity
Solving the conditions:
1. From \( \frac{\partial \mathcal{L}}{\partial Q_1} = 0 \):
- \( 400 - 2Q_1 - 20 = 0 \) → \( Q_1 = 190 \)
2. From \( \frac{\partial \mathcal{L}}{\partial Q_2} = 0 \):
- \( 380 - 2Q_2 - 20 = 0 \) → \( Q_2 = 180 \)
Capacity is determined by the maximum of \( Q_1 \) and \( Q_2 \):
- \( C = \max(190, 180) = 190 \)
Step C: Capacity Allocation (λ1 and λ2)
The Lagrange multipliers \( \lambda_1 \) and \( \lambda_2 \) represent the shadow prices for capacity used in each market:
\[
\lambda_1 = \frac{P_1 - MC}{1} = \frac{(400 - Q_1) - 20}{1} = 190
\]
\[
\lambda_2 = \frac{P_2 - MC}{1} = \frac{(380 - Q_2) - 20}{1} = 180
\]
Step D: Capacity Cost Adjustment
If the capacity cost increases to 30 cents per unit, the new Lagrangian becomes:
\[
\mathcal{L} = (400 - Q_1)Q_1 + (380 - Q_2)Q_2 - 20(Q_1 + Q_2) - 30C
\]
Following similar steps, you would find new optimal outputs and capacity, taking into account the increased fixed cost.
Conclusion
- New quantities \( Q_1 \) and \( Q_2 \) will be recalculated.
- The respective values of \( \lambda_1 \) and \( \lambda_2 \) will reflect changes in capacity cost.
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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 P1 = 400-Q1 and the off-peak demand is given by P2 = 380-Q2. The variable cost is 20 per unit (paid in both market) 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 condition for the problem.b. Find the optimal output and capacity for this problem. c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).?
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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 P1 = 400-Q1 and the off-peak demand is given by P2 = 380-Q2. The variable cost is 20 per unit (paid in both market) 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 condition for the problem.b. Find the optimal output and capacity for this problem. c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about 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 P1 = 400-Q1 and the off-peak demand is given by P2 = 380-Q2. The variable cost is 20 per unit (paid in both market) 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 condition for the problem.b. Find the optimal output and capacity for this problem. c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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 P1 = 400-Q1 and the off-peak demand is given by P2 = 380-Q2. The variable cost is 20 per unit (paid in both market) 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 condition for the problem.b. Find the optimal output and capacity for this problem. c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).?.
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 P1 = 400-Q1 and the off-peak demand is given by P2 = 380-Q2. The variable cost is 20 per unit (paid in both market) 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 condition for the problem.b. Find the optimal output and capacity for this problem. c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about 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 P1 = 400-Q1 and the off-peak demand is given by P2 = 380-Q2. The variable cost is 20 per unit (paid in both market) 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 condition for the problem.b. Find the optimal output and capacity for this problem. c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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 P1 = 400-Q1 and the off-peak demand is given by P2 = 380-Q2. The variable cost is 20 per unit (paid in both market) 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 condition for the problem.b. Find the optimal output and capacity for this problem. c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).?.
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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 P1 = 400-Q1 and the off-peak demand is given by P2 = 380-Q2. The variable cost is 20 per unit (paid in both market) 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 condition for the problem.b. Find the optimal output and capacity for this problem. c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).?, a detailed solution for 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 P1 = 400-Q1 and the off-peak demand is given by P2 = 380-Q2. The variable cost is 20 per unit (paid in both market) 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 condition for the problem.b. Find the optimal output and capacity for this problem. c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).? has been provided alongside types of 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 P1 = 400-Q1 and the off-peak demand is given by P2 = 380-Q2. The variable cost is 20 per unit (paid in both market) 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 condition for the problem.b. Find the optimal output and capacity for this problem. c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).? theory, EduRev gives you an
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