Budget Constraints and Consumer Optimization

Jul 14, 2024

Lecture on Budget Constraints and Consumer Optimization

Introduction to Budget Constraints

  • Objective of the Consumer: Maximize utility
    • Reach the indifference curve farthest from the origin
  • Constraint: Limited income

Assumptions

  • Fixed Prices: Prices of goods (X and Y) do not change during the decision period
  • Fixed Income: Consumer’s income is constant for the short term
  • No Savings/Borrowing: Consumer spends all income on goods X and Y

Budget Constraint Formulation

  • Equation: Income (I) = PX * QX + PY * QY
    • PX: Price of good X
    • QX: Quantity of good X
    • PY: Price of good Y
    • QY: Quantity of good Y

Graphical Representation

  1. Solve for QY: QY = (Income / PY) - (PX / PY) * QX
  2. Intercepts:
    • Vertical Intercept: Income / PY
    • Horizontal Intercept: Income / PX
  3. Slope: PX / PY (Negative slope)
  4. Example:
    • Income = $100
    • PX = $10, PY = $5
    • Budget constraint: QY = 20 - 2QX
    • Graph intercepts: (10, 0) and (0, 20)

Changes in Budget Constraint

Change in Income

  • Income Decreases: Budget constraint shifts inward (parallel shift)
  • Income Increases: Budget constraint shifts outward (parallel shift)

Change in Prices

  • Increase in PY: Budget constraint pivots downward
  • Decrease in PY: Budget constraint pivots upward
  • Increase in PX: Budget constraint pivots inward
  • Decrease in PX: Budget constraint pivots outward

Non-Standard Budget Constraints

Quantity Discounts

  • Example:
    • Two goods: Pizza and Tickets
    • Tickets: First 10 cost $10, additional cost $5
    • Budget: $200
  • Graph: Initial portion before discount is a standard budget constraint; after discount, line shifts indicating cheaper additional units

Quantity Limits

  • Example: Limit on good Y
  • Graph: Budget constraint has a flat segment at the limit, cannot exceed the limit

Consumer Optimization Problem

Maximizing Utility

  • Objective: Choose bundle on budget constraint that maximizes utility
  • Indifference Curves: Higher curves represent higher utility
  • Optimal Point: Point where budget constraint is tangent to the highest indifference curve
    • Example: Move from bundle A to B to C (C is optimal)

Tangency Condition

  • At Optimal Point: Slope of indifference curve = Slope of budget constraint
  • Formula: Marginal Rate of Substitution = Price ratio of two goods
    • \frac{MU_X}{MU_Y} = \frac{P_X}{P_Y}

Implications

  • All Consumers: Face the same prices, hence same slope of budget constraint
  • Differences: Preferences and income levels lead to different choices

Corner Solutions

  • Definition: All income spent on one good (occurs with unusual indifference curves)
  • Typical Scenario: Indifference curves are unusually steep or perfect substitutes

Graphical Explanation

  • Indifference Curves: Lead to buying all of one good if they prefer it significantly more
    • Example using pizza and burritos: Strong preference for pizza leads to no burritos

Expenditure Minimization

  • New Problem Formulation: Minimize spending to achieve a specific utility level
  • Dual Problem: Maximization of utility vs. minimization of expenditure for same utility
  • Objective Function and Constraint Swap: Minimization problem's constraint is the horizontal line; Objective is the curved indifference curves

Graphical Explanation

  • Example: Similar to utility maximization but flipped in terms of minimization

Conclusion

  • Future Directions: Work with utility maximization in more detail and apply concepts to cost minimization for firms in future chapters.