Mathematically Solving for the Income and Substitution Effect of a Price Change

Economics in Many Lessons
21 Sept 201810:01
EducationalLearning
32 Likes 10 Comments

TLDRThis video script explores the income and substitution effects of a price change on consumer behavior. It uses a utility function to demonstrate graphically and mathematically how a price increase for good X from $5 to $10 impacts consumption. The total effect, substitution effect, and income effect are analyzed, showing a decrease in good X consumption and an increase in good Y, holding utility constant. The script concludes with a table summarizing the effects and their implications on consumer spending decisions.

Takeaways
  • 📚 The video discusses the income and substitution effects of a price change using a consumer's utility function.
  • 📈 The utility function is defined with both goods X and Y raised to the power of 0.5, representing their respective shares in the utility equation.
  • 💰 The initial prices for goods X and Y are given as $5 and $2, respectively, and the consumer's income and prices are taken into account.
  • 📊 The graphical analysis begins by maximizing utility with the initial prices, resulting in the consumer buying 2 units of X and 5 units of Y at Point A.
  • 🔄 The price of good X increases to $10, leading to a new budget constraint and a new optimal consumption point, Point B.
  • 🔍 The total effect of the price change is the movement from Point A to Point B, while the substitution effect is the movement from A to C, holding utility constant.
  • 🔢 The mathematical derivation involves taking partial derivatives to find the marginal utilities of goods X and Y and setting them equal to their respective prices.
  • ✂️ The substitution effect shows a decrease in the consumption of good X by 0.6 units and an increase in good Y by 2 units due to the price increase.
  • 💼 The income effect is calculated by comparing the consumption bundles after the price change and finding the difference in consumption of good X (a decrease of 0.4 units) and good Y (a decrease of 2 units).
  • 📉 The price increase results in a normal good behavior for both X and Y, where consumption decreases as the price increases, indicating a negative income effect.
  • 📋 The video concludes with a summary table that organizes the optimal consumption bundles, total effects, substitution effects, and income effects for both goods.
Q & A

  • What is the utility function used in the video?

    -The utility function used in the video is U(X, Y) = X^0.5 * Y^0.5, where X and Y are both raised to the power of 0.5.

  • What are the initial prices of good X and good Y in the video?

    -The initial prices are $5 for good X and $2 for good Y.

  • What is the point of tangency at the initial prices?

    -The point of tangency at the initial prices is point A, where the consumer buys 2 units of good X and 5 units of good Y.

  • What happens to the price of good X in the scenario presented in the video?

    -The price of good X increases to $10 in the scenario presented in the video.

  • How does the budget constraint change with the new price of good X?

    -The budget constraint becomes steeper with the new price of good X, leading to a new point of tangency at point B.

  • What is the total effect of the price change on good X?

    -The total effect of the price change on good X is a decrease in consumption from 2 units to 1 unit, while the consumption of good Y remains unchanged at 5 units.

  • What is the substitution effect of the price change on good X?

    -The substitution effect of the price change on good X is a decrease in consumption from 2 units to 1.4 units, while the consumption of good Y increases slightly.

  • How is the income effect calculated in the video?

    -The income effect is calculated by comparing the consumption at the new price (point B) with the consumption at the utility constant level (point C), resulting in a decrease in consumption of good X by 0.4 units and good Y by 2 units.

  • What utility level was used to find the substitution effect?

    -The utility level used to find the substitution effect was U = 3.16, which was the utility at the original consumption bundle of 2 units of good X and 5 units of good Y.

  • How does the consumer's spending decision change due to the price increase of good X?

    -Due to the price increase of good X, the consumer's spending decision changes by cutting back on good X by 0.6 units and consuming more of good Y by 2 units, considering the substitution effect.

  • What is the conclusion of the income effect for both goods X and Y?

    -The conclusion of the income effect is that the consumer would buy less of both goods X and Y, with a decrease of 0.4 units for good X and 2 units for good Y, indicating that both are normal goods.

Outlines
00:00
📈 Analyzing Price Change Effects on Consumer Behavior

This paragraph introduces a video on the income and substitution effects of a price change using a consumer's utility function. The utility function is given as a function of two goods, X and Y, both raised to the power of 0.5, with income and prices as variables. The video aims to demonstrate graphically and mathematically what happens when the price of good X increases to $10. It shows the initial maximization of utility with the original prices, leading to the consumption of 2 units of X and 5 units of Y at Point A. Then, it illustrates the new maximization with the increased price of X, resulting in consumption at Point B with 1 unit of X and 5 units of Y. The total effect of the price change is moving from A to B, while the substitution effect is moving from A to C, holding utility constant. The income effect is the remaining movement from C to B. The paragraph concludes with a promise to explain the graphical representation mathematically in the following content.

05:04
📉 Calculating the Total, Substitution, and Income Effects

The second paragraph delves into the numerical analysis of the effects of a price change on consumer behavior. It starts by reiterating the graphical results from the initial video segment, which showed a decrease in the consumption of good X from 2 to 1 unit and no change in the consumption of good Y, remaining at 5 units. The substitution effect is calculated by evaluating the utility function at the original consumption bundle, finding that utility equals approximately 3.16. The new consumption bundle, with the price of good X at $10, is then used to find the substitution effect, which shows a decrease of 0.6 units in good X and an increase of 2 units in good Y. The income effect is determined by comparing the new consumption bundle with the one found during the substitution effect analysis, revealing a decrease of 0.4 units in good X and 2 units in good Y. The paragraph concludes with a summary table that organizes the optimal consumption bundles, total effects, substitution effects, and income effects for both goods X and Y, highlighting the consumer's spending decisions in response to the price change.

Mindmap
Keywords
💡Utility Function
A utility function represents a consumer's preferences and measures the level of satisfaction or happiness derived from consuming goods and services. In the video, the utility function involves goods X and Y raised to the 0.5 power, illustrating how changes in prices impact consumer satisfaction and choices.
💡Price Change
Price change refers to the variation in the cost of a good or service over time. The video explores the effects of an increase in the price of good X from $5 to $10 on consumer behavior, examining how this change influences consumption patterns through the total, substitution, and income effects.
💡Substitution Effect
The substitution effect describes how consumers adjust their consumption of goods in response to a change in relative prices, while keeping utility constant. In the video, this effect is illustrated by the movement from point A to point C, where the consumer reduces consumption of good X and increases consumption of good Y.
💡Income Effect
The income effect explains how a change in the price of a good affects the consumer's purchasing power, and consequently, their consumption of goods. The video shows this effect as the movement from point C to point B, where a higher price of good X reduces the consumer's real income, leading to decreased consumption of good X.
💡Budget Constraint
A budget constraint represents the combinations of goods and services a consumer can purchase with their limited income, given the prices of those goods. The video demonstrates how the initial budget constraint changes when the price of good X increases, affecting the consumer's optimal consumption bundle.
💡Indifference Curve
An indifference curve represents all combinations of goods that provide the same level of utility to a consumer. The video uses indifference curves to graphically show the points of tangency where the consumer maximizes utility before and after the price change, illustrating the substitution and income effects.
💡Marginal Utility
Marginal utility is the additional satisfaction or utility a consumer gains from consuming one more unit of a good or service. The video explains how the marginal utility of goods X and Y is calculated and used to determine the consumer's optimal consumption choices before and after the price change.
💡Optimal Consumption Bundle
The optimal consumption bundle is the combination of goods that maximizes a consumer's utility given their budget constraint. In the video, the initial optimal bundle includes 2 units of good X and 5 units of good Y, which changes to 1 unit of good X and 5 units of good Y after the price increase.
💡Tangency Point
A tangency point is where an indifference curve is tangent to the budget line, indicating the optimal consumption bundle. The video shows points A and B as tangency points before and after the price increase, respectively, highlighting how the consumer's optimal choices shift due to the price change.
💡Total Effect
The total effect of a price change is the overall change in consumption of goods resulting from both the substitution effect and the income effect. The video illustrates the total effect as the movement from point A to point B, where the consumer decreases consumption of good X from 2 units to 1 unit.
Highlights

The video explores the income and substitution effects of a price change using a consumer's utility function.

The utility function is defined with both goods X and Y raised to the power of 0.5.

The initial prices for goods X and Y are $5 and $2, respectively.

The graphical approach is used to maximize utility and find the initial tangency point A.

At point A, the consumer buys 2 units of good X and 5 units of good Y.

The price of good X increases to $10, leading to a new budget constraint and tangency point B.

At point B, consumption shifts to 1 unit of good X and remains at 5 units of good Y.

The total effect of the price change is the movement from A to B.

The substitution effect is represented by the movement from A to C, holding utility constant.

At point C, the consumer would buy 1.4 units of X and 7 units of Y with the new budget constraint.

The income effect is the remaining movement from C to B after accounting for substitution.

Graphical representation helps to understand the mathematical calculations involved.

The marginal utility of good X and the marginal utility of good Y are derived through partial derivatives.

The utility-maximizing condition sets the marginal utility per dollar spent on each good equal to each other.

The consumption bundle at the new price point is found to be 1 unit of X and 5 units of Y.

The substitution effect leads to a decrease in good X consumption by 0.6 units and an increase in good Y consumption by 2 units.

The income effect shows a decrease in consumption of both goods, indicating normal goods behavior.

A summary table is provided to simplify the understanding of the total, substitution, and income effects.

Transcripts

Browse More Related Video

Perfect Substitutes Utility: Compensating Variation, Equivalent Variation, and Consumer Surplus

Consumer Welfare: Compensating Variation & Equivalent Variation

How to Calculate Compensating Variation and Equivalent Variation

Three Measures of Consumer Welfare: Compensating Variation, Equivalent Variation, Consumer Surplus

Managerial Economics 2.5: More Elasticity

16. Compensating Variation and Equivalent Variation

Rate This

5.0 / 5 (0 votes)

Thanks for rating:

Related Tags
Utility FunctionPrice ChangeIncome EffectSubstitution EffectConsumer BehaviorEconomic TheoryBudget ConstraintGraphical AnalysisMathematical ApproachOptimization