Piecewise Pricing Models

Business Calculus · Axiom Academy

REAL WORLD Piecewise Pricing Models How businesses use tiered pricing in the real world Piecewise functions in action! Many businesses use tiered or graduated pricing where rates change based on quantity, time, or usage level. These create piecewise functions that may or may not be continuous at the boundaries. A utility company charges residential customers: This creates the cost function: This pricing function is continuous — there are no sudden jumps at tier boundaries. The cost at 500 kWh is 40 whether calculated from the first tier or as the starting point of the second tier. This is by design to avoid customer complaints about arbitrary bill increases. A mobile carrier charges 50/month for 10GB of data, then 10 per GB over the limit: This creates a jump discontinuity at g = 10. A customer using 9.9 GB pays 50, but using 10.1 GB pays 51. The discontinuity is intentional! It encourages customers to upgrade to higher-tier plans rather than risk overage charges. The psychological impact of "going over" is greater than the actual 1 difference. Progressive tax rates that increase for higher income levels (continuous) Weight-based tiers with potential jumps at boundaries Hourly rates with daily maximums (creates ceiling) Lower per-unit price for larger orders (discontinuous) Multiplier that can change abruptly with demand Zero cost until deductible met, then coverage kicks in

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