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:
Given income, what are individuals' choices?
Based on preferences and income, what decisions will individuals make?
Page 3: Lecture Outline
Feasible Choices
Budget Constraints: Changes in Price and Income
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:
Illustrate feasible choices on diagrams.
Describe budget constraint adjustments due to price and income variations.
Analyze optimal choice bundles of goods.