Short answer: If the goal is for minimum‑wage workers to maintain their purchasing power, then yes—the minimum wage should at least keep pace with the cost of living. But “keeping up” can mean different things, and design details matter.
What “keep up” could mean
•Index to inflation (CPI): preserves purchasing power nationally but ignores local price differences.
•Index to local costs: ties wages to regional prices or housing costs; better fit, more complexity.
•Tie to typical wages or productivity: e.g., a target like 50–66% of the median wage, or adding a small productivity factor so low wages don’t drift away from the broader economy.
Pros of automatic indexing
•Prevents silent pay cuts during inflation and reduces poverty risk.
•Makes changes predictable for workers and employers; less political delay.
•Low administrative burden once rules are set.
Cons and trade‑offs
•A single national index can miss big regional differences.
•Large inflation spikes can cause abrupt jumps; small firms in low‑margin sectors may struggle.
•Indexing preserves the status quo level; if the starting wage is too low to meet basic needs, indexing alone won’t fix adequacy.
Practical design options
•Annual CPI indexing with guardrails: use a 12–24 month average, caps/floors on yearly changes, and catch‑up adjustments.
•Regional indexing: tie to metro/state CPI or a cost‑of‑living basket, possibly with a national floor.
•Median‑wage target: set the minimum at a share of the local or national median, reviewed periodically.
•Phase‑ins for big changes, and small‑business offsets (e.g., tax credits) during transitions.
•Align or phase out sub‑minimums (like the tipped wage) to avoid loopholes.
•Strong enforcement so posted wages are actually paid.
Complements (so the wage doesn’t carry everything)
•Expand/streamline Earned Income Tax Credits, childcare and housing supports.
•Zoning and housing supply reforms where rent is the main cost driver.
•Transportation and training policies that raise productivity and earnings.
Bottom line: To avoid real wage erosion, automatic indexing to inflation is a sensible minimum. If the policy aim is a “living wage” that reflects local prices, pair indexing with a higher, locally calibrated starting level or a median‑wage target, plus smoothing rules to handle inflation spikes. If you tell me your state or city, I can sketch a concrete indexing rule with example numbers.