Study Notes for Slutsky Equation and Price Changes in Microeconomic Theory

Overview of Microeconomic Theory

• Course: ECON 360 Microeconomic Theory
• Instructor: Lucas de Lara
• Institution: Binghamton University, State University of New York

Slutsky Equation

• Alternate Name: “Substitution Effects and Income Effects”

Price Changes

General Remarks

• Price changes can result in ambiguous effects that may yield unexpected results.
• Example of Giffen Goods:
  • Experiment by Jensen & Miller focusing on rice consumption in China.
  • Observation: Consumption decreased when the price decreased.

Reaction to Wage Changes

• Increase in hourly wages leads to:
  • More income earned, thus increasing spending power.
  • Increased opportunity cost of leisure time (not working).

General Price Increase Effect

• When the price of a good increases:
  • Consumers have more money but it becomes more expensive to purchase the same quantity.
  • Higher opportunity cost for holding onto the good instead of selling it.

Price Change Implications

• Understanding price changes aids in:
  • Grasping demand within the basic model.
  • Facilitating model extensions in future analysis.

Effects of a Price Change

Price Decrease Effects (1)

• Substitution Effect:
  • When a commodity's price decreases, it becomes relatively cheaper. Consumers tend to substitute this commodity for more expensive ones.

Price Decrease Effects (2)

• Income Effect:
  • Consumer’s budget allows purchasing more than before due to the price drop, akin to an increase in income.
  • This leads to changes in quantities demanded across different goods.

Graphical Representation

• Consumer's budget denoted as $m$.
• Price changes illustrated graphically indicating shifts in budget constraints and optimal consumption choices.

Slutsky's Findings

• Slutsky demonstrated:
  • Changes in demand from a price change can be quantified as the sum of:
  • Pure substitution effect.
  • Income effect.

Pure Substitution Effect

Definition and Concept

• Identifies the change in demand attributing solely to relative price changes without adjusting the overall budget.
• Formulated by the question:
  • "What change in demand occurs if income is adjusted so the consumer can afford the original bundle at new prices?"

Graphical Representation

• Graphical analyses show:
  • Original budget line and new budgets, maintaining tangency to the original bundle.

Sign of Substitution Effect

• Negative Reaction:
  • When the price rises, the quantity demanded decreases.
  • Conversely, when the price falls, the quantity demanded increases.
• Assumes monotone preferences are at play.

Income Effect Introduction

• Post adjusting for substitution effects, the new situation appears to consumers as an increased budget in light of reduced prices.
• For normal goods, this typically results in increased demand in reaction to the income effect.

Slutsky Identity

• Total change in demand when the price of good 1 changes:
• $ext{Total Change} = ext{Substitution Effect} + ext{Income Effect}$
  • Mathematically illustrated as:
  • $riangle x<em>1 = riangle x</em>1^s + riangle x_1^n$

Rate of Change Formulation

• Slutsky’s identity can be rewritten in terms of rates of change for practical application:
  • $\frac{ riangle x<em>1}{ riangle p</em>1} = \frac{ riangle x<em>1^s}{ riangle p</em>1} - \frac{ riangle x_1^m}{ riangle m}$

Application of Slutsky’s Effects

Normal Goods

• Characterization: Demand increases with income.
• Both income and substitution effects act in alignment, boosting demand when pricing shifts.

Inferior Goods

• Defined by demand decreasing with increased income.
• When price changes, the substitution effect countermands the income effect, potentially leading to less demand.

Giffen Goods

• Rare cases arise whereby extreme income inferiority leads to:
  • Income effect outweighing the substitution effect, resulting in a decrease in quantity demanded as the price drops.

Example Problems

Example Problem 4: Todd's Consumption

Overview

• Budgets, prices for pens (x) and pencils (y) set.
• Utility function defined as $u(x, y) = xy$ with total budget of $48.

Optimal Bundle Calculation

• Using Cobb-Douglas model:
  • Derived optimal consumption as $x^* = 24, y^* = 24$

Utility Level Calculation

• Utility attained at optimal bundle:
  • $u(x^<em>, y^</em>) = 24 imes 24 = 576$

Price Adjustment Scenario

• Impact of a change in pen price ($1 to $4) analyzed for new optimal bundle.
  • Results in new consumption pattern found as $x^{<strong>} = 6, y^{</strong>} = 24$

Compensated Bundles

• Calculation of a compensated bundle maintaining a utility of 576 under new price constraints.
• Final outcomes yielded: $x<em>c = 12, y</em>c = 48$

Substitution and Income Effects

• Calculated substitution effect (SE) and income effect (IE) from changes in pen demand:
  • SE = $x_c - x^* = 12 - 24 = -12$
  • IE = $x^{**} - x_c = 6 - 12 = -6$

Classification of Substitution Effect

• Analyzed the type of substitution effect, determining it to be Hicksian due to constant utility.

Example Problem 5: Coco's Consumption

Overview

• Coco's income and utility function for food and clothing defined.

Optimal Bundle under Initial Prices

• With food at $1 per unit:
• Derived optimal bundle findings of $C^* = 2, F^* = 12$

Price Variation and Response

• Price of food increases to $4 per unit, creating a new optimal bundle through recalibrated calculations. Final consumption yielded as $C^{<strong>} = 2, F^{</strong>} = 3$

Effect Attribution

• Total consumption variance analyzed from initial to modified prices,
  • Substitution effect and income effects articulated clearly with combined results maintained throughout the practical examples.

Conclusion

• Mastery of these microeconomic principles formulates a clearer understanding of consumer behavior in response to price changes, leading to accurate predictions of market dynamics.