L8 Constrained optimisation

Principles of Microeconomics: Consumer Theory Lecture Notes

Page 1: Overview

• Lecture Topic: Consumer Theory

• Focus: Constrained Optimization

• Instructor: Tien-Der Jerry Han

• Lecture Number: 8

Page 2: Aims of the Lecture

• Understanding Constrained Optimization in consumer theory:

  • Examines how individuals make decisions under conditions of scarcity.

  • Key questions addressed:

    1. Given income, what are individuals' choices?

    2. Based on preferences and income, what decisions will individuals make?

Page 3: Lecture Outline

• 1. Feasible Choices

• 2. Budget Constraints: Changes in Price and Income

• 3. Optimal Solutions: Interior and Corner

Page 4: Required Readings

• Textbooks:

  • Lipsey and Chrystal (14th ed.), Chapter 4, pp. 83-89

  • Lipsey and Chrystal (13th ed.), Chapter 4, pp. 81-85

  • Sloman, Wride, & Garratt (11th ed.), Chapter 4, pp. 104-119

  • Varian, Intermediate Microeconomics: A Modern Approach (8th ed.), Chapter 2, pp. 20-26 and pp. 48-52

Page 5: Feasible Set of Choices

• Choice Set: Determined by income and prices of goods.

  • Example:

    • Income = £100 per week

    • Price of beer = £2; Price of pizza = £5

  • Budget Constraint Equation:

    • Total Expenditure ≤ Income

    • £100 ≥ 2X + 5Y

    • X = amount of beer bought; Y = amount of pizza bought

Page 6: Budget Constraint

• Illustrates which combinations of goods are affordable.

• Feasible Sets:

  • If income is entirely spent on pizza:

    • 100/5 = 20 units of pizza (Y)

  • If entirely on beer:

    • 100/2 = 50 units of beer (X)

  • Slope indicates trade-off: for every additional 25 beers, 10 pizzas must be foregone.

Page 7: Impact of Income Increase

• Scenario: Income rises to £120.

• Budget constraint shifts right parallel to existing line:

  • New Maximum Pizza = 120/5 = 24 units

  • New Maximum Beer = 120/2 = 60 units

Page 8: Impact of Price Increase

• Scenario: Price of beer increases to £2.50.

• Budget Constraint Shapes:

• New Slope: Price change leads to pivoting of the budget constraint solving for affordable combinations.

  • If spending only on beer = 100/2.5 = 40 units.

Page 9: Mathematical Representation of Budget Constraints

• General Formula:

  • M = pxX + pyY

  • Slope: ∆Y/∆X = –(px/py)

  • The relationship illustrates trade-offs between goods upon changing prices or income.

Page 10: Non-Linear Budget Constraints: Quantity Discounts

• Bulk purchase implications change relative price: degrade slope of the budget curve if buying over a threshold.

  • Consumers adapt choices in response to pricing changes orchestrated by quantity purchased.

Page 11: Non-Linear Budget Constraints: Rationing

• Government-imposed limits on goods.

  • Functional representation shifts feasible sets despite income levels.

  • Conceptualize as infinite slope when maximum quantities are reached.

Page 12: Interior Solutions

• Indifference Curves: Represent preferences.

  • Budget Constraint Intersects with Indifference Curve at optimal choice.

  • Tangency point between constraints indicates maximum utility.

Page 13: Marginal Rate of Substitution (MRS)

• Measures consumer's willingness to trade one good for another at varying utility levels.

  • Diminishing marginal returns depicted within the trade-off curves.

Page 14: Continued Marginal Rate of Substitution Exploration

• Emphasis on concept continuity in trade-off willingness as consumption units vary.

  • Linear margin adjustments critically influence consumer choice mechanics.

Page 15: Properties of the Interior Solution

• At optimal consumption:

  • MRSyx = px/py signifies balance between personal trade rate and market exchange rate for goods.

Page 16: Corner Solutions

• Occurs when consumers disregard certain products due to preference or rationing limitations.

  • Optimal choice situated at corners within feasible sets highlighting constraints.

Page 17: Summary of Budget Constraints

• Key terms:

  • Budget Constraint: Set of affordable options based on income and goods' prices.

  • Interior Solutions and Corners defined by underlying preferences vs. market limits.

Page 18: Key Learning Outcomes

• Ability to:

  1. Illustrate feasible choices on diagrams.

  2. Describe budget constraint adjustments due to price and income variations.

  3. Analyze optimal choice bundles of goods.