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M14.L02 ā€” Interior and Corner Solutions

Module: Module 14 ā€” Consumer Choice and Demand Lesson: L02 of 05 Duration: ~30 minutes Level: Intermediate Provenance: Principles of Microeconomics (Curtis and Irvine) | University of Utah Intermediate Microeconomics

Learning Objective

Identify when consumer optimization yields interior versus corner solutions.

Types of Solutions

Interior Solution: Occurs when consuming positive quantities of all goods (MRS = pā‚/pā‚‚). Common with normal goods like Australian households buying both electricity and gas.

Corner Solution: Occurs when consuming zero of one good. Three main cases: 1. Perfect substitutes (e.g., Coles vs Woolworths brand) 2. Strong preferences (vegetarians buying no meat) 3. Extreme price ratios (no private school at high tuition)

Mathematically, corner solutions violate the tangency condition:

MRS > pā‚/pā‚‚ ā‡’ consume only good 1 MRS < pā‚/pā‚‚ ā‡’ consume only good 2

Worked Example

Australian Energy Choice

Consider a consumer choosing between electricity (xā‚, $0.30/kWh) and gas (xā‚‚, $0.10/MJ) with utility U = 2xā‚ + xā‚‚ (perfect substitutes):

  1. MRS = 2 (constant)
  2. Price ratio = 0.30/0.10 = 3
  3. Since 2 < 3, consumer chooses only gas (xā‚‚* = m/0.10, xā‚* = 0)

Common Misconception

"Corner solutions mean the consumer is irrational"

Corner solutions often reflect rational specialization - like Australian miners using only diesel vehicles (no electric) in remote sites.

Key Takeaways

  • Interior solutions satisfy MRS = price ratio
  • Corner solutions occur when preferences or technology prevent substitution
  • Perfect substitutes always yield corner solutions
  • Real-world examples include: medication choices, fuel selection, education modes
  • Kinked budget constraints (like Medicare levy) can create corner solutions

Practice

  1. Find the optimal bundle for U = min{xā‚, 2xā‚‚} with pā‚=4, pā‚‚=2, m=40
  2. Show graphically why U = xā‚ + xā‚‚ always gives corner solutions
  3. Explain why Australian retirees might choose corner solutions in leisure/work choices

Further Resources

Provenance: Principles of Microeconomics (Curtis and Irvine) | University of Utah Intermediate Microeconomics