Why do companies pay differently based on location?

Many companies pay employees differently based on location. The most common claim is that companies pay employees based on cost of living differences. This means that a New York City or San Francisco employee gets paid often significantly higher than an employee working in Chicago or Boston for example.
There are differences based on city, for example New York City vs Boston. There are also differences based on country, for example United States vs Mexico.

Cost of Living Explained

Cost of living is the cost of maintaining a certain standard of living. Changes in the cost of living over time are often operationalized in a cost-of-living index. Cost of living calculations are also used to compare the cost of maintaining a certain standard of living in different geographic areas. Differences in cost of living between locations can also be measured in terms of purchasing power parity rates.

Cost of Living Adjustments

Employment contracts and pension benefits can be tied to a cost-of-living index, typically to the consumer price index (CPI). A COLA adjusts salaries based on changes in a cost-of-living index. Salaries are typically adjusted annually. They may also be tied to a cost-of-living index that varies by geographic location if the employee moves. In this later case, the expatriate employee will likely see only the discretionary income part of their salary indexed by a differential CPI between the new and old employment locations, leaving the non-discretionary part of the salary (e.g., mortgage payments, insurance, car payments) unmodified.
Annual escalation clauses in employment contracts can specify retroactive or future percentage increases in worker pay which are not tied to any index. These negotiated increases in pay are colloquially referred to as cost-of-living adjustments or cost-of-living increases because of their similarity to increases tied to externally determined indexes. Cost-of-living allowance is equal to the nominal interest minus the real interest rate.

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