Short answer: If the goal is that a minimum-wage job preserves a stable standard of living, then yes—the minimum wage should at least keep up with the cost of living. How you design that link matters.
Ways to link it (and the trade‑offs):
•Index to inflation (CPI): Preserves purchasing power automatically and predictably; simplest to administer. Con: Doesn’t reflect local housing/childcare spikes or productivity trends.
•Index to local prices: Adjust by region or city so wages track local living costs; fairer geographically. Con: More complex; burdens can fall unevenly on small firms in high‑cost areas.
•Link to median wages (e.g., target 50–60% of the median): Lets the floor rise with overall pay and productivity, not just prices; helps reduce inequality. Con: Can raise costs faster than prices in weak labor markets.
•Hybrid rules: CPI indexing each year plus periodic resets toward a median‑wage target; guardrails that pause hikes in severe recessions.
Pros of automatic indexing:
•Prevents erosion between legislative updates
•Gives workers and businesses predictability
•Reduces politicized, lumpy jumps
Common concerns and mitigations:
•Employment/small‑business pressure: Phase‑ins, temporary small‑employer credits, or lower payroll taxes can smooth adjustments.
•Price pass‑through: Pair with competition policy and targeted supports (e.g., childcare or housing vouchers) to ease household budgets.
•One‑size‑fits‑all: Use regional floors or allow higher local minima.
Complements (often more precise against poverty than wages alone):
•Earned Income Tax Credits or wage subsidies
•Housing, childcare, and healthcare supports
•Transit access and skills training
Bottom line: To avoid real wage erosion, a minimum wage should at least be indexed to inflation. If the objective is to let low‑wage workers share in broader income growth and reflect local living costs, consider a hybrid: annual CPI indexing, periodic calibration to a share of the local median wage, regional variation, and supportive tax/benefit policies.
If you have a specific place or policy proposal in mind, I can tailor a concrete design with numbers.