Expenditure Minimization in Economics
39 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the primary goal of the consumer’s expenditure minimization problem?

  • Reach a certain level of utility in the cheapest possible way (correct)
  • Determine the maximum quantity of goods consumed
  • Achieve a specific utility level at the highest cost
  • Maximize total expenditure

    • Which equation represents the consumer's total expenditure in the minimization problem?

  • $p_x x + p_y y$ (correct)
  • $p_x x - p_y y$
  • $p_x/y + p_y x$
  • $p_x x + p_y/y$

    • What condition must be satisfied for the solution to the expenditure minimization problem based on MRS?

  • Price ratio = $U(x, y)$
  • MRS = price ratio (correct)
  • MRS < price ratio
  • MRS > price ratio

    • Which of the following is a requirement in the formulation of the expenditure minimization problem?

    <p>Minimize $px x + py y$ subject to achieving a utility level of $Ū$</p> Signup and view all the answers

    What does the Lagrangian represent in the context of expenditure minimization?

    <p>A tool for constrained optimization with a utility constraint</p> Signup and view all the answers

    What happens when the marginal rate of substitution (MRS) is greater than the price ratio?

    <p>The consumer achieves the highest utility by consuming more x compared to y.</p> Signup and view all the answers

    In the context of expenditure minimization, what is the fixed constraint?

    <p>The indifference curve representing a specific utility level.</p> Signup and view all the answers

    What is the significance of the shaded area in the graphical representation of utility?

    <p>It represents all combinations of x and y that yield at least Ū = 4 utility.</p> Signup and view all the answers

    What is a key aspect of the expenditure minimization problem compared to the utility maximization problem?

    <p>Expenditure minimization requires a fixed utility level as a constraint.</p> Signup and view all the answers

    Why would a consumer not want to exceed a utility level of Ū = 4 in expenditure minimization?

    <p>Reaching above Ū indicates inefficiency in spending.</p> Signup and view all the answers

    What represents the slope of the expenditure line?

    <p>Negative price ratio</p> Signup and view all the answers

    At which point does the expenditure line achieve optimality in relation to the indifference curve?

    <p>(8,2)</p> Signup and view all the answers

    What does fixing an arbitrary level of expenditure allow us to analyze?

    <p>Minimization of expenditure at varying utility levels</p> Signup and view all the answers

    When varying expenditure while keeping utility fixed, what is the key relationship that must hold?

    <p>Price ratio must equal marginal rate of substitution (MRS)</p> Signup and view all the answers

    In the expenditure minimization problem, what is varied to achieve optimization?

    <p>Expenditure level</p> Signup and view all the answers

    What is the fixed income value indicated in the context?

    <p>16</p> Signup and view all the answers

    How does the expenditure line compare to the budget constraint?

    <p>Their slope is identical but allows expenditure to vary.</p> Signup and view all the answers

    What is the value of expenditure represented by the point (8,2)?

    <p>16</p> Signup and view all the answers

    What do we hold constant while varying expenditure in related graphs?

    <p>Utility level</p> Signup and view all the answers

    Which of the following best describes Hicksian demand?

    <p>Utility-level driven demand analysis</p> Signup and view all the answers

    Which condition demonstrates that the solution to both the expenditure minimization and utility maximization problems occurs at the same point?

    <p>Marginal rate of substitution equals the price ratio.</p> Signup and view all the answers

    In the context of the expenditure minimization problem, what is the function of the Lagrangian?

    <p>To identify the combination of x and y that minimizes expenditure.</p> Signup and view all the answers

    What represents the main objective for the consumer in the expenditure minimization problem?

    <p>To minimize the total expenditure while maintaining a fixed level of utility.</p> Signup and view all the answers

    Which of the following is a key component of setting up the consumer’s expenditure minimization problem?

    <p>Establishing a specific utility level that must be achieved.</p> Signup and view all the answers

    In the first order conditions of the Lagrangian for the expenditure minimization problem, what do the equations px = λ and py = λ indicate?

    <p>The marginal utility per dollar spent on each good is equal.</p> Signup and view all the answers

    What does the shaded area in the graphical representation signify?

    <p>All combinations of x and y that provide at least 4 utility.</p> Signup and view all the answers

    What is the relationship between the marginal rate of substitution (MRS) and the price ratio when maximizing utility?

    <p>MRS equals the price ratio, indicating optimal consumption.</p> Signup and view all the answers

    In the context of expenditure minimization, what does maintaining a fixed utility level like Ū = 4 imply?

    <p>The consumer must find the cheapest combination to achieve that utility.</p> Signup and view all the answers

    What does the expenditure line represent in the context of the given graphs?

    <p>It indicates the relationship between fixed utility levels and costs.</p> Signup and view all the answers

    Why is it said that there will always be a binding constraint in the expenditure minimization problem?

    <p>Exceeding the utility level is not cost-effective.</p> Signup and view all the answers

    What does the optimal point of tangency between the expenditure line and the indifference curve represent?

    <p>Minimum expenditure for the given utility level</p> Signup and view all the answers

    When solving for y in the expenditure equation, which variable is treated as a function of the others?

    <p>y (quantity of good y)</p> Signup and view all the answers

    In comparing the expenditure minimization problem to the utility maximization problem, which variable is held constant in the former?

    <p>Utility level</p> Signup and view all the answers

    What is the significance of the negative price ratio in both the expenditure line and the budget constraint?

    <p>Represents the trade-off between goods x and y</p> Signup and view all the answers

    What does the coordinate (8,2) indicate in the context of the expenditure minimization problem?

    <p>The lowest expenditure for given prices</p> Signup and view all the answers

    What is the role of the expenditure line in relation to price ratios?

    <p>It illustrates the price ratio for given expenditure levels</p> Signup and view all the answers

    What would be an incorrect assumption about the expenditure line compared to the budget constraint?

    <p>The expenditure line is always higher than the budget constraint</p> Signup and view all the answers

    What does varying expenditure while keeping utility fixed aim to achieve?

    <p>To discover the optimal consumption bundle for less expense</p> Signup and view all the answers

    Which component is critical in determining whether the point of tangency represents the lowest expenditure line?

    <p>The marginal rate of substitution equals the price ratio</p> Signup and view all the answers

    Study Notes

    Expenditure Minimization

    • Consumers aim to achieve a specific utility level at the lowest possible cost.
    • This is a constrained optimization problem, like utility maximization, but the constraint is a specific utility level, not a budget constraint.
    • The expenditure minimization problem involves finding the least expensive combination of goods (x and y) that yields a certain level of utility (Ū).
    • This problem is solved using a Lagrangian, similar to utility maximization, with the utility level as a constraint.
    • The Lagrangian includes the expenditure function (Pxx + Pyy) and the utility constraint (Ū - U(x,y)).

    First Order Conditions

    • Setting the partial derivatives of the Lagrangian with respect to x, y, and λ to zero yields first-order conditions.
    • These conditions state that the ratio of the marginal rate of substitution (MRS) between goods x and y must equal the ratio of their prices (Px/Py).
    • These conditions are identical to the conditions found in utility maximization, but with the utility level fixed rather than the budget.

    Graphical Representation

    • The expenditure minimization problem can be visualized graphically, using indifference curves (representing different utility levels) and expenditure lines (representing different total expenditure levels).
    • The optimal bundle occurs where the lowest expenditure line is tangent to the indifference curve representing the desired utility level.
    • The point of tangency signifies that the MRS between the two goods equals the price ratio.
    • The shaded area on the graph represents all bundles that offer at least the desired level of utility.
    • Expenditure lines are similar to budget lines, but with expenditure varying instead of income.

    Hicksian Demand

    • The optimal quantities of x and y that minimize expenditure for a given utility level are the Hicksian demands.
    • These are functions that describe the optimal consumption levels of goods x and y as a function of prices and desired utility level.
    • Hicksian demands differ from Marshallian demands in that they are derived for a fixed level of utility rather than a fixed budget.
    • The Hicksian demands are derived by substituting the first-order conditions into the Lagrangian function.

    Special Utility Functions

    • The analysis in expenditure minimization problems applies to different utility functions, including perfect substitutes and perfect complements.
    • Corner solutions occur when the optimal bundle involves consuming only one of the goods (e.g., in the case of perfect substitutes).
    • Perfect substitutes result in consumption of only one good if that good offers a better price-utility ratio than the other.
    • Expenditure minimization and utility maximization, especially with corner solutions, yields strikingly similar results, with the only difference being that expenditure is fixed, while utility is maintained.
    • For perfect complements, the optimal consumption bundle satisfies the condition Ax = By which means that the amounts consumed by complements are proportional.

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Related Documents

    Expenditure Minimization PDF

    Description

    This quiz explores expenditure minimization, focusing on how consumers achieve a specific utility level at the lowest cost. It covers the constrained optimization problem, first-order conditions, and graphical representations. Test your understanding of these key concepts in economic theory.

    More Like This

    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser