Expenditure Minimization in Economics
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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 $UÌ„$ (B)</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 (A)</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. (A), The consumer spends less money without changing their utility level. (D)</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. (C)</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 UÌ„ = 4 utility. (D)</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. (B)</p> Signup and view all the answers

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

<p>Reaching above UÌ„ indicates inefficiency in spending. (D)</p> Signup and view all the answers

What represents the slope of the expenditure line?

<p>Negative price ratio (A)</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) (D)</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 (B)</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) (B)</p> Signup and view all the answers

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

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

What is the fixed income value indicated in the context?

<p>16 (B)</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. (C)</p> Signup and view all the answers

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

<p>16 (D)</p> Signup and view all the answers

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

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

Which of the following best describes Hicksian demand?

<p>Utility-level driven demand analysis (C)</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. (B)</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. (A)</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. (C)</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. (C)</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. (D)</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. (A)</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. (B)</p> Signup and view all the answers

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

<p>The consumer must find the cheapest combination to achieve that utility. (D)</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. (B)</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. (A)</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 (C)</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) (C)</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 (A)</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 (C)</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 (B)</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 (D)</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 (A)</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 (A)</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 (D)</p> Signup and view all the answers

Flashcards

Expenditure Minimization Problem

Finding the least expensive way to achieve a specific level of utility.

Utility Level (UÌ„)

A specific level of satisfaction that a consumer aims to reach.

Lagrangian (L)

A mathematical function that combines the objective function (expenditure) and the constraint (utility level) into a single expression.

First-Order Conditions

Equations obtained by setting the derivatives of the Lagrangian with respect to each variable to zero.

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MRS = Price Ratio

The condition where the marginal rate of substitution (MRS) between two goods is equal to their price ratio.

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Expenditure Minimization

The process of finding the cheapest bundle of goods that meets a given level of utility. Consumers aim to reach a specific utility level (UÌ„) at the lowest possible cost.

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Binding Constraint

In expenditure minimization, the consumer always reaches the specified utility level (UÌ„). They will never exceed it because any additional spending would not increase utility, only cost more.

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Utility Function

A mathematical representation that describes a consumer's preferences for different combinations of goods. It assigns a utility level to each consumption bundle. For example, U = x^(1/2)y^(1/2)

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Feasible Set

The set of all combinations of goods a consumer can afford given their budget and the prices of the goods.

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Expenditure Line

A line representing all possible combinations of goods 'x' and 'y' that can be purchased with a given level of expenditure.

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Expenditure Line Equation

The mathematical representation of the expenditure line, showing the relationship between the quantities of two goods 'x' and 'y' and the total expenditure. It's derived from the expenditure equation.

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Expenditure Line Slope

The slope of the expenditure line is the negative price ratio (-px/py), which represents the rate at which you can substitute one good for another while maintaining the same expenditure.

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Optimal Point (Expenditure Minimization)

The point on the indifference curve that intersects the lowest possible expenditure line, representing the most affordable bundle of goods that provides the desired level of utility.

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Relationship to Utility Maximization

The optimal point in expenditure minimization is the same as the optimal point in utility maximization, given the same level of income and utility.

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Hicksian Demand

The demand for a good when utility is held constant and price changes occur. It shows how the quantity demanded changes as the price changes while maintaining the same level of utility.

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Hicksian Demand vs. Marshallian Demand

Hicksian demand holds utility constant, while Marshallian demand holds income constant. Both show the relationship between price and quantity demanded, but differ in their underlying constraints.

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Parameters in Hicksian Demand

The key parameters influencing Hicksian demand are the price of the goods and the desired utility level.

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Hicksian Demand Equation

The mathematical expression that shows the relationship between the quantity demanded of a good and its price, given a fixed utility level (UÌ„).

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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.

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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.

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