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A consumer live on an island where she produces two goods X and Y according to the production possibility frontier X^2+Y^2≤200 and she consumers all the goods herself. Her utility function is U=XY^3. The consumer also faces an environmental constraints on her total output of both goods. The environmental constraints is given by X+Y≤20. a. Write out the kuhn-tucke first order condition. b. Find the consumer's optimal X and Y. Identify which constraints are binding.?
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a) Kuhn-Tucker First Order Condition:
The Kuhn-Tucker first order condition is used to find the optimal solution for a constrained optimization problem. In this case, the consumer's problem is to maximize her utility function subject to the production possibility frontier constraint and the environmental constraint.
The Kuhn-Tucker first order condition can be written as follows:
∂U/∂X - λ(∂(XY^3)/∂X - 20) = 0
∂U/∂Y - λ(∂(XY^3)/∂Y - 20) = 0
Where:
∂U/∂X and ∂U/∂Y are the partial derivatives of the utility function with respect to X and Y, respectively.
λ is the Lagrange multiplier associated with the environmental constraint.
∂(XY^3)/∂X and ∂(XY^3)/∂Y are the partial derivatives of the production function with respect to X and Y, respectively.
b) Finding the Consumer's Optimal X and Y:
To find the consumer's optimal X and Y, we need to solve the Kuhn-Tucker first order condition equations.
Differentiating the utility function with respect to X and Y, we get:
∂U/∂X = Y^3
∂U/∂Y = 3XY^2
Differentiating the production function with respect to X and Y, we get:
∂(XY^3)/∂X = Y^3
∂(XY^3)/∂Y = 3XY^2
Substituting these values into the Kuhn-Tucker first order condition equations, we have:
Y^3 - λ(Y^3 - 20) = 0
3XY^2 - λ(3XY^2 - 20) = 0
Simplifying these equations, we get:
(1 - λ)Y^3 = 20λ
(1 - λ)XY^2 = 20λ
Dividing these equations, we get:
X = Y
Substituting this back into the production possibility frontier constraint, we have:
X^2 Y^2 ≤ 200
(Y^2)^3 ≤ 200
Y^6 ≤ 200
Y ≤ (200)^(1/6)
Therefore, the consumer's optimal X and Y are X = Y = (200)^(1/6).
Identifying Binding Constraints:
To identify which constraints are binding, we need to compare the equality and inequality constraints.
Equality Constraint:
X = Y = (200)^(1/6)
Inequality Constraint:
XY ≤ 20
Comparing the equality constraint with the inequality constraint, we can see that the inequality constraint is binding. This means that the consumer is producing at her maximum capacity given the environmental constraint.
Therefore, the binding constraint is XY ≤ 20.
The Kuhn-Tucker first order condition is used to find the optimal solution for a constrained optimization problem. In this case, the consumer's problem is to maximize her utility function subject to the production possibility frontier constraint and the environmental constraint.
The Kuhn-Tucker first order condition can be written as follows:
∂U/∂X - λ(∂(XY^3)/∂X - 20) = 0
∂U/∂Y - λ(∂(XY^3)/∂Y - 20) = 0
Where:
∂U/∂X and ∂U/∂Y are the partial derivatives of the utility function with respect to X and Y, respectively.
λ is the Lagrange multiplier associated with the environmental constraint.
∂(XY^3)/∂X and ∂(XY^3)/∂Y are the partial derivatives of the production function with respect to X and Y, respectively.
b) Finding the Consumer's Optimal X and Y:
To find the consumer's optimal X and Y, we need to solve the Kuhn-Tucker first order condition equations.
Differentiating the utility function with respect to X and Y, we get:
∂U/∂X = Y^3
∂U/∂Y = 3XY^2
Differentiating the production function with respect to X and Y, we get:
∂(XY^3)/∂X = Y^3
∂(XY^3)/∂Y = 3XY^2
Substituting these values into the Kuhn-Tucker first order condition equations, we have:
Y^3 - λ(Y^3 - 20) = 0
3XY^2 - λ(3XY^2 - 20) = 0
Simplifying these equations, we get:
(1 - λ)Y^3 = 20λ
(1 - λ)XY^2 = 20λ
Dividing these equations, we get:
X = Y
Substituting this back into the production possibility frontier constraint, we have:
X^2 Y^2 ≤ 200
(Y^2)^3 ≤ 200
Y^6 ≤ 200
Y ≤ (200)^(1/6)
Therefore, the consumer's optimal X and Y are X = Y = (200)^(1/6).
Identifying Binding Constraints:
To identify which constraints are binding, we need to compare the equality and inequality constraints.
Equality Constraint:
X = Y = (200)^(1/6)
Inequality Constraint:
XY ≤ 20
Comparing the equality constraint with the inequality constraint, we can see that the inequality constraint is binding. This means that the consumer is producing at her maximum capacity given the environmental constraint.
Therefore, the binding constraint is XY ≤ 20.
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A consumer live on an island where she produces two goods X and Y according to the production possibility frontier X^2+Y^2≤200 and she consumers all the goods herself. Her utility function is U=XY^3. The consumer also faces an environmental constraints on her total output of both goods. The environmental constraints is given by X+Y≤20. a. Write out the kuhn-tucke first order condition. b. Find the consumer's optimal X and Y. Identify which constraints are binding.?
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A consumer live on an island where she produces two goods X and Y according to the production possibility frontier X^2+Y^2≤200 and she consumers all the goods herself. Her utility function is U=XY^3. The consumer also faces an environmental constraints on her total output of both goods. The environmental constraints is given by X+Y≤20. a. Write out the kuhn-tucke first order condition. b. Find the consumer's optimal X and Y. Identify which constraints are binding.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A consumer live on an island where she produces two goods X and Y according to the production possibility frontier X^2+Y^2≤200 and she consumers all the goods herself. Her utility function is U=XY^3. The consumer also faces an environmental constraints on her total output of both goods. The environmental constraints is given by X+Y≤20. a. Write out the kuhn-tucke first order condition. b. Find the consumer's optimal X and Y. Identify which constraints are binding.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A consumer live on an island where she produces two goods X and Y according to the production possibility frontier X^2+Y^2≤200 and she consumers all the goods herself. Her utility function is U=XY^3. The consumer also faces an environmental constraints on her total output of both goods. The environmental constraints is given by X+Y≤20. a. Write out the kuhn-tucke first order condition. b. Find the consumer's optimal X and Y. Identify which constraints are binding.?.
A consumer live on an island where she produces two goods X and Y according to the production possibility frontier X^2+Y^2≤200 and she consumers all the goods herself. Her utility function is U=XY^3. The consumer also faces an environmental constraints on her total output of both goods. The environmental constraints is given by X+Y≤20. a. Write out the kuhn-tucke first order condition. b. Find the consumer's optimal X and Y. Identify which constraints are binding.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A consumer live on an island where she produces two goods X and Y according to the production possibility frontier X^2+Y^2≤200 and she consumers all the goods herself. Her utility function is U=XY^3. The consumer also faces an environmental constraints on her total output of both goods. The environmental constraints is given by X+Y≤20. a. Write out the kuhn-tucke first order condition. b. Find the consumer's optimal X and Y. Identify which constraints are binding.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A consumer live on an island where she produces two goods X and Y according to the production possibility frontier X^2+Y^2≤200 and she consumers all the goods herself. Her utility function is U=XY^3. The consumer also faces an environmental constraints on her total output of both goods. The environmental constraints is given by X+Y≤20. a. Write out the kuhn-tucke first order condition. b. Find the consumer's optimal X and Y. Identify which constraints are binding.?.
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