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Questions and Answers
What is the economic interpretation of the Lagrange multiplier in the context of a consumer's utility maximisation problem?
What is the economic interpretation of the Lagrange multiplier in the context of a consumer's utility maximisation problem?
The Lagrange multiplier represents the marginal utility of income, or the change in the maximum utility achieved when the budget is increased by one unit.
How does the consumer's utility function, u(x,y) = x^0.5y^0.75, reflect the idea of diminishing marginal utility?
How does the consumer's utility function, u(x,y) = x^0.5y^0.75, reflect the idea of diminishing marginal utility?
The utility function exhibits diminishing marginal utility because the exponents of x and y are less than 1, implying that the increase in utility from an additional unit of x or y decreases as the quantity of x or y increases.
What is the implication of the budget constraint, pxx + pyy = B, on the consumer's utility maximisation problem?
What is the implication of the budget constraint, pxx + pyy = B, on the consumer's utility maximisation problem?
The budget constraint implies that the consumer must allocate their limited budget, B, between the two goods, x and y, such that the total expenditure does not exceed the budget.
How does the method of Lagrangian multipliers help to solve the consumer's utility maximisation problem?
How does the method of Lagrangian multipliers help to solve the consumer's utility maximisation problem?
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What is the relationship between the utility maximising quantities of x and y and the budget, B, in the context of the consumer's utility maximisation problem?
What is the relationship between the utility maximising quantities of x and y and the budget, B, in the context of the consumer's utility maximisation problem?
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Study Notes
Utility Maximisation Problem
- Consumer's utility function is given by u(x, y) = x^0.5y^0.75, where x and y are weekly consumption levels of goods x and y.
- Market prices: px = £8 and py = £3.
- Weekly budget is B.
Setting up the Lagrangian
- The Lagrangian is used to solve the consumer's utility maximisation problem.
- The Lagrangian is a function of x, y, and the Lagrange multiplier (λ).
Utility Maximising Quantities
- The utility maximising quantities of x and y can be found using the method of Lagrangian multipliers.
- The quantities are a function of the weekly budget (B).
Specific Budget Scenarios
- If B = 100, the utility maximising quantities of x and y, and the corresponding level of utility can be calculated.
- If B = 101, the utility maximising quantities of x and y, and the corresponding level of utility can be calculated.
Lagrange Multiplier
- The Lagrange multiplier (λ) can be calculated if B = 100.
- The Lagrange multiplier has an economic interpretation: it represents the marginal utility of an additional unit of budget.
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Description
Solve a consumer's utility maximization problem by setting up a Lagrangian and finding the utility-maximizing quantities of two goods as a function of the budget.
