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M13.L01 ā€” Budget Constraints: The Consumer's Choice Set

Module: Module 13 ā€” Budget Constraints, Preferences, and Utility Lesson: L01 of 05 Duration: ~30 minutes Level: Intermediate Provenance: Intermediate Microeconomics with Excel (Barreto) | MIT OCW 14.01 Principles of Microeconomics

Learning Objective

Key Diagram

Figure 9: Consumer Optimum. The optimal bundle occurs where the budget line is tangent to the highest attainable indifference curve ā€” the MRS equals the price ratio.
Figure 9: Consumer Optimum. The optimal bundle occurs where the budget line is tangent to the highest attainable indifference curve ā€” the MRS equals the price ratio.

Derive and interpret the budget constraint equation for consumer choice problems.

The Budget Constraint

A consumer with income m faces prices Pā‚ and Pā‚‚ for goods xā‚ and xā‚‚ respectively. The budget constraint is:

Pā‚xā‚ + Pā‚‚xā‚‚ ā‰¤ m

The budget line (all bundles where the consumer spends exactly their income) is:

Pā‚xā‚ + Pā‚‚xā‚‚ = m

Rewriting to show intercepts:

xā‚‚ = (m/Pā‚‚) - (Pā‚/Pā‚‚)xā‚

  • Vertical intercept (xā‚=0): m/Pā‚‚
  • Horizontal intercept (xā‚‚=0): m/Pā‚
  • Slope: -Pā‚/Pā‚‚ (opportunity cost of xā‚ in terms of xā‚‚)

Australian Example: Consider a student with \(100 weekly budget for textbooks (Pā‚=\)50) and coffee (Pā‚‚=$5). Maximum textbooks = 2, maximum coffees = 20.

Worked Example

Australian Budget Adjustment

  1. Original budget: 50xā‚ + 5xā‚‚ = 100
  2. If textbook price rises to $60:
  3. New intercepts: xā‚ = 100/60 ā‰ˆ 1.67, xā‚‚ = 20
  4. New slope: -60/5 = -12 (steeper)
  5. If income increases to $120:
  6. New intercepts: xā‚ = 120/50 = 2.4, xā‚‚ = 24
  7. Parallel shift outward

Common Misconception

"A price increase always reduces the maximum quantity of both goods"

Correction: Only the good whose price increased sees reduced maximum quantity. The other good's maximum remains unchanged (though affordable combinations change).

Key Takeaways

  • The budget line shows all combinations of goods that exhaust income
  • Slope represents the market's tradeoff rate between goods
  • Income changes cause parallel shifts
  • Price changes rotate the budget line
  • The numeraire good simplifies analysis by setting one price to 1

Practice

  1. Derive the budget line equation when Pā‚=10, Pā‚‚=4, m=200
  2. How does the budget set change if Pā‚‚ increases to 5?
  3. Calculate the new intercepts if income rises by 20% with original prices

Further Resources

Provenance: Intermediate Microeconomics with Excel (Barreto) | MIT OCW 14.01 Principles of Microeconomics