Constrained Utility Maximization — Budget Constraint Fundamentals

Introduction to Constrained Utility Maximization

• Constrained maximization = optimizing choices subject to limits (money, time, energy, etc.)
• This lecture focuses exclusively on the money limitation → the budget constraint
• Goal: understand how consumers choose when limited by income

Why Model the Budget Constraint?

• Real–world decision (e.g., buying steak) illustrates two fundamental determinants of choice:
  • Budget: limited cash → price matters
  • Preferences: taste/desire for a good (you may dislike steak even if affordable)
• Key observations:
  • Unlimited money ⇒ prices irrelevant; real life ≠ this
  • Strong desire ≠ ability to purchase
• Analytical strategy in economics:
  1. Isolate the budget component
  2. Analyze preferences separately
  3. Re‐combine for full constrained utility maximization

Simplest Analytical Setting: The Two-Goods Model

• To build intuition start with only two products (goods X and Y)
  • Here: Cakes (C) and Movies (M)
• Widely used because:
  • Easy to visualize graphically (2-D plane)
  • Extends later to labor–leisure choice, policy analysis, etc.

Mathematical Form of a Budget Constraint

• Generic form for two goods: $Y = P<em>C \cdot C + P</em>M \cdot M$ where
  • $Y$ = total money available ("income" or spending budget)
  • $P<em>C, P</em>M$ = prices of cakes & movies
  • $C, M$ = quantities purchased
• Interpretation
  • Equality sign assumes the consumer ultimately spends the entire budget (nothing else to do with leftover money in this simple world)
  • More generally: $P<em>C C + P</em>M M \le Y$ (cannot exceed income)
• Can be generalized to $n$ goods: $Y = \sum<em>{i=1}^{n} P</em>i Q_i$

Graphical Construction (Cakes on Y-axis, Movies on X-axis)

1. Y-intercept (all cakes, 0 movies)
   • $C = \dfrac{Y}{P_C}$
2. X-intercept (all movies, 0 cakes)
   • $M = \dfrac{Y}{P_M}$
3. Connect the two intercepts → straight line because prices are constant

Numerical Example

• Income $Y = 96$
• $P<em>C = 16$ per cake, $P</em>M = 8$ per movie
• Intercepts
  • Cakes: $C = \dfrac{96}{16} = 6$ → point $A(0,6)$
  • Movies: $M = \dfrac{96}{8} = 12$ → point $B(12,0)$
• Budget line = line through A and B

Economic Meaning of the Budget Line

• Points on the line: exhaust entire budget (feasible & efficient use of $Y$)
• Points below/inside: affordable but leave unspent money
• Points above/outside: unattainable with current $Y$ and prices

Slope & Opportunity Cost

• Slope of budget line (rise/run):
  $\text{slope} = -\frac{C\text{-intercept}}{M\text{-intercept}} = -\frac{6}{12} = -\frac{1}{2}$
• Interpretation (good on X-axis = movies):
  • Buying 1 additional movie costs $1/2$ cake in foregone consumption
  • Opportunity cost of 1 movie = $0.5$ cakes
• For the good on Y-axis (cakes), take the inverse:
  • Opportunity cost of 1 cake = $\dfrac{1}{|\text{slope}|} = 2$ movies

Conceptual Takeaways

• Budget & preferences jointly determine choice; analytically helpful to master each alone first
• Budget constraint summarizes all feasible consumption bundles given $Y, P<em>C, P</em>M$
• Opportunity cost emerges from the slope: choosing more of one good automatically means less of the other
• Two-goods simplification builds intuition transferable to richer settings (many goods, labor–leisure, policy studies)

Key Formulas & Numbers Recap

• General two-good equation: $Y = P<em>C C + P</em>M M$
• Intercepts: $C = Y/P<em>C$ and $M = Y/P</em>M$
• Slope (opportunity cost of X-good): $\text{slope} = -\frac{P<em>M}{P</em>C}$ (since intercepts ratio simplifies to price ratio)
• Example values: $Y=96,\; P<em>C=16,\; P</em>M=8\; \Rightarrow\; \text{slope} = -1/2$

Looking Ahead

• Next step (future lecture): model preferences via utility functions and indifference curves
• Final objective: merge budget constraint with preferences to solve the constrained utility-maximization problem and predict actual consumer choices