Does a raise keep up with the cost of living?

A raise keeps up with the cost of living when the new salary is at least the old salary times one plus the CPI percent you are using. The gap is the new salary minus that cost-of-living salary.

The cost of living raise calculator does both multiplications from the percent you type.

Cost-of-living salary = salary × (1 + CPI percent)

Raised salary = salary × (1 + raise percent)

Gap = raised salary − cost-of-living salary

BLS explains escalation as a base payment that usually moves in proportion to the percent change in the CPI between two periods. The percent is the index-point change divided by the earlier index. You bring that percent from the series named in the letter.

How BLS turns two index levels into a percent

The escalation factsheet walks one pair of index levels.

3.130 divided by 229.815 is 0.01361965. Times 100, that is 1.362%. The factsheet shows 0.0136, then prints the percent change as 1.4%. One decimal on 1.36 is 1.4, so the printed line is that rounding.

BLS also asks you to name the series before you use a percent. CPI-U covers all urban consumers, which the factsheet puts at over 90% of the U.S. population. CPI-W covers urban wage earners and clerical workers, about 30%. Name the area too, and whether the basket is all items or something narrower, such as rent of primary residence.

The same page says seasonally adjusted indexes are a poor fit for escalation. Those factors get revised. Use the unadjusted index.

A cap or a floor in a contract sits outside this calculator. If the letter says the adjustment cannot exceed 3%, type 3 when the raw CPI percent is higher. The math will follow the number in the box.

Example 1. A 4% raise against a 3.2% change

Inputs: salary $72,000, cost-of-living change 3.2%, raise 4%, pay periods 26.

The 3.2% is the calculator’s starting value, so the arithmetic is easy to check before you swap in the percent from your letter.

Cost-of-living salary: 72,000 × 1.032 = $74,304

The 3.2% itself is 72,000 × 0.032 = $2,304

Raised salary: 72,000 × 1.04 = $74,880

The 4% itself is 72,000 × 0.04 = $2,880

Gap: 74,880 − 74,304 = $576

Per paycheck: 576 ÷ 26 = $22.15

Real change versus the cost-of-living salary: 74,880 ÷ 74,304 − 1 = 0.775%. The calculator rounds that line to 0.8%.

Four minus 3.2 is 0.8 percentage points. 0.8 points of the old $72,000 is $576, and that matches the dollar gap. The 0.8% on the result line is $576 divided by the higher cost-of-living salary, then rounded. The dollars and the rounded percent land on the same tenth. The unrounded share of the new base is 0.775%.

Example 2. The raise matches the percent

Same salary, same 3.2%, raise changed to 3.2%, still 26 periods.

Cost-of-living salary: 72,000 × 1.032 = $74,304

Raised salary: 72,000 × 1.032 = $74,304

Gap: $0

Real change: 0%

Per paycheck: $0

When the two percents match, the new salary sits on the cost-of-living line. BLS describes the usual agreement as a payment that moves in proportion to the CPI percent. A 3.2% raise is that proportion for a 3.2% index change. Taxes can still change the check you deposit. This page stops at gross salary. The raise take-home post applies a deduction rate you type to a raise.

Example 3. A 2% raise against the same 3.2%

Inputs: salary $72,000, cost-of-living change 3.2%, raise 2%, pay periods 26.

Raised salary: 72,000 × 1.02 = $73,440

Cost-of-living salary: $74,304

Gap: 73,440 − 74,304 = −$864

Per paycheck: −864 ÷ 26 = −$33.23

Real change: 73,440 ÷ 74,304 − 1 = −1.163%. The calculator rounds that line to −1.2%.

The raise is $1,440 (72,000 × 0.02). The cost-of-living line wanted $2,304. The shortfall is 2,304 − 1,440 = $864, the same $864 with the sign flipped.

Compared with example 1, the raise fell from 4% to 2%. The gap moved from +$576 to −$864. That swing is $1,440, which is two percentage points of $72,000. The annual gap moves by salary × (new raise − old raise).

Example 4. $125,000, paid monthly

Inputs: salary $125,000, cost-of-living change 2.5%, raise 3%, pay periods 12.

Cost-of-living salary: 125,000 × 1.025 = $128,125

The 2.5% is 125,000 × 0.025 = $3,125

Raised salary: 125,000 × 1.03 = $128,750

The 3% is 125,000 × 0.03 = $3,750

Gap: 128,750 − 128,125 = $625

Per month: 625 ÷ 12 = $52.08

Real change: 128,750 ÷ 128,125 − 1 = 0.488%. The calculator rounds that line to 0.5%.

Half a percentage point of $125,000 is $625. That is the dollar gap again. The paycheck is monthly here, so the $625 is split 12 ways. Example 1 split a different gap across 26 paychecks. The annual gap stays $625 either way. Only the per-paycheck line changes when the period count changes.

Example 5. $42,000, paid weekly

Inputs: salary $42,000, cost-of-living change 4%, raise 5%, pay periods 52.

Cost-of-living salary: 42,000 × 1.04 = $43,680

The 4% is 42,000 × 0.04 = $1,680

Raised salary: 42,000 × 1.05 = $44,100

The 5% is 42,000 × 0.05 = $2,100

Gap: 44,100 − 43,680 = $420

Per week: 420 ÷ 52 = $8.08

Real change: 44,100 ÷ 43,680 − 1 = 0.962%. The calculator rounds that line to 1%.

One percentage point of $42,000 is $420. On a weekly payroll the extra is about eight dollars. The same $420 on a monthly payroll would be 420 ÷ 12 = $35. The annual number is the one to compare with rent, groceries, or a benefit change. The rent increase post prices the lease side of that comparison.

Example 6. What if the raise is 6% instead of 4%

Hold example 1 still: salary $72,000, cost-of-living change 3.2%, pay periods 26. Change only the raise, from 4% to 6%.

Raised salary: 72,000 × 1.06 = $76,320

Cost-of-living salary: still $74,304

Gap: 76,320 − 74,304 = $2,016

Per paycheck: 2,016 ÷ 26 = $77.54

Real change: 76,320 ÷ 74,304 − 1 = 2.713%. The calculator rounds that line to 2.7%.

The raise dollars are 72,000 × 0.06 = $4,320. After the $2,304 cost-of-living piece, $2,016 is left. Example 1 left $576. The extra two percentage points (6 − 4) are 72,000 × 0.02 = $1,440, and 576 + 1,440 = 2,016.

Same salary. Same 3.2%. Only the raise moves.

Example 7. A 2% pay cut while the index is up 3.2%

Inputs: salary $72,000, cost-of-living change 3.2%, raise −2%, pay periods 26.

Raised salary: 72,000 × 0.98 = $70,560

The pay cut is 72,000 × 0.02 = $1,440

Cost-of-living salary: $74,304

Gap: 70,560 − 74,304 = −$3,744

Per paycheck: −3,744 ÷ 26 = −$144

Real change: 70,560 ÷ 74,304 − 1 = −5.039%. The calculator rounds that line to −5%.

The $3,744 has two parts. The paycheck lost $1,440. The cost-of-living line still asked for $2,304. 1,440 + 2,304 = 3,744. A pay cut and a rising index stack. They do not offset.

Example 8. The BLS factsheet percent, with a 2% raise

Use the percent the factsheet prints, 1.4%, on the same $72,000 salary. Raise 2%. Pay periods 26.

Cost-of-living salary: 72,000 × 1.014 = $73,008

The 1.4% is 72,000 × 0.014 = $1,008

Raised salary: 72,000 × 1.02 = $73,440

Gap: 73,440 − 73,008 = $432

Per paycheck: 432 ÷ 26 = $16.62

Real change: 73,440 ÷ 73,008 − 1 = 0.592%. The calculator rounds that line to 0.6%.

Type the unrounded quotient instead. 3.130 ÷ 229.815 × 100 = 1.362% (the full figure is 1.361965…).

Cost-of-living salary: 72,000 × 1.01362 = $72,980.64

Gap against the same $73,440 raise: 73,440 − 72,980.64 = $459.36

Per paycheck: 459.36 ÷ 26 = $17.67

The factsheet’s 1.4% and the unrounded 1.362% are $27.36 apart on this salary (73,008 − 72,980.64). I would type the percent the agreement names. If the agreement says “the percent change, one decimal,” that is the 1.4% line. If it says the index ratio without that rounding, the 1.362% line is the one in the box.

Example 9. The common mistake, typing the point change as a percent

The factsheet’s index-point change is 3.130. Someone in a hurry types 3.130 into the percent box, on the way to “about 3.1%,” and skips the division by 229.815.

Inputs: salary $72,000, cost-of-living change 3.13%, raise 2%, pay periods 26.

Cost-of-living salary: 72,000 × 1.0313 = $74,253.60

Raised salary: still 72,000 × 1.02 = $73,440

Gap: 73,440 − 74,253.60 = −$813.60

Per paycheck: −813.60 ÷ 26 = −$31.29

Real change: the calculator shows −1.1%.

Example 8, with the factsheet’s 1.4%, had the same 2% raise $432 ahead. Typing the point change as if it were already a percent puts the same raise $813.60 behind. The swing is 432 − (−813.60) = $1,245.60 a year, from one skipped division.

BLS states the order on the factsheet. Subtract the earlier index from the later index. Divide that point change by the earlier index. Multiply by 100. Then the percent is ready for the salary.

Same annual gap, three payroll calendars

Example 1’s annual gap is $576. Split it three ways.

The annual gap stays $576 in all three rows. If the offer letter quotes a biweekly bump and you compare it with a monthly rent increase, put both numbers on a year first. The salary to hourly calculator is a different split. It divides annual salary by weeks times hours. On its defaults, 40 hours and 48 weeks, $74,880 ÷ 1,920 hours = $39 an hour. The old $72,000 on those same defaults is $37.50. Paid time off at 20 days is $6,240 on the new hourly rate, because a day is one fifth of the 40-hour week: 39 × 8 × 20.

If the same two percents repeat

The next table runs the same formula again on each new salary. A later CPI release will carry its own percent. These rows keep 4% and 3.2% so the compounding is visible.

Start at $72,000. Each year, multiply by 1.04 for the raise and by 1.032 for the cost-of-living line.

Year 2 is year 1 times the same factors. 74,880 × 1.04 = 77,875.20. 74,304 × 1.032 = 76,681.728. The gap is 1,193.472, which rounds to $1,193.47. The later years are that step again. The salaries in the table from year 3 on are shown to the thousandth of a dollar so a reader can subtract and land on the printed gap.

The gap grows because each year’s percent applies to a larger salary, and because 4% keeps outrunning 3.2%. Swap in a raise that matches the CPI percent and the gap stays $0 every year. Swap in the 2% raise from example 3 and the shortfall gets wider on the same schedule.

The nine cases next to each other

Cases 1 through 7 and case 9 use percents you choose so the arithmetic is easy to check. Case 8 uses the 1.4% the BLS factsheet prints from index levels 232.945 and 229.815. Case 9 types that factsheet’s 3.130 point change into the percent box.

Which series, and what the gross gap leaves out

BLS wants the agreement to name CPI-U or CPI-W, the area, the items, and the index base (1982-84 = 100 is the usual one). A Chicago all-items CPI-U and the U.S. city average CPI-W can print two different percents for the same two months. The calculator multiplies the percent you type, so the series has to be the one the letter names.

Metropolitan indexes are noisier. The factsheet says BLS recommends the U.S. city average for escalator clauses, because the local samples are smaller.

The result here is the gross gap. A 4% raise can clear a 3.2% line by $576 and still land smaller in the bank account after withholding. Open the raise or pay-cut calculator when you want a deduction rate on the raise itself. The hourly post and the weekly-to-annual post are the conversions when the offer is not already a yearly salary. The employee true cost calculator is the employer’s stack on top of salary, with the taxes and benefits you type. That is a different question from whether the employee’s gross pay cleared the CPI line.

Open the cost of living raise calculator with the salary in the letter, the percent from the series the letter names, the raise on offer, and the number of paychecks in the year.