Among households with annual incomes of less than $250,000 per year, 41.7 percent of respondents said that they would certainly relocate during retirement, and only 0.4 percent said they were not likely to relocate.

For higher income households in the $750,000–$1 million annual income bracket, a similar 40.8 percent of respondents said they would certainly relocate for retirement, and 0.5 percent said they were not likely to relocate.

And while many of these folks certainly don't appear to be on the bottom end of the income scale, this study isn't the only one to indicate that retirees might get itchy feet.

A Money report points out that location and money aren't the only questions that have to be answered.

The report highlights a Merrill Lynch retirement study conducted with Age Wave, a research and consulting company, that finds that close to two thirds of retirees have either moved or plan to move.

But that brings up additional questions—such as whether to buy a retirement home in the location of their choice while they're still working, so that they can spend vacations there and pay the bills on the house while they have a job to do so.

They often fail to take into account how big the expenses of a second home can be, or how little they may make on rent should they rent it out till they actually plan to move in.

Or should they simply rent, planning on moving again later?

They also may not know quite when to relocate, although if they leave it too late they may find themselves unable to move at all, for health or other reasons.

There's also the consideration of how a surviving spouse would manage if the other spouse dies early in retirement; will there be a community to support them, or will they be isolated in a new and unfamiliar location?

But with money looming so large in most people's minds—GoBankingRates.com points out that more than half of Americans have less than $10,000 saved toward retirement—a cheaper, rather than a more expensive, place to live can help make what money people do have last a lot longer.

So GoBankingRates.com rounded up data from a range of places, starting with cost-of-living indices from Sperling's Best Places.

Once they were finished comparing the cost of housing, groceries, transportation, utilities and health in 150 U.S. cities, it then used that data, as well as U.S. Bureau of Labor Statistics data on average annual expenditures for adults 65 and older, to arrive at a formula to predict annual expenditures in each city.

Here are the 10 cheapest cities they found:

10. Montgomery, Alabama

Annual expenditures: $36,971

Annual amount spent on housing: $7,149

Annual amount spent on health: $5,335

With a cost of living about 19 percent lower than the national average, Montgomery also has the fifth lowest annual amount spent on health and the 10^th lowest annual amount spent on housing.

Also, its mild winters might not just be pleasant to retirees, but will also save them money on heating costs.

9. Akron, Ohio

Annual expenditures: $36,147

Annual amount spent on housing: $5,401

Annual amount spent on health: $5,514

Akron's cost of living is even lower—coming in 21 percent below the national average.

Akron has the fifth lowest annual amount spent on housing—but there are a couple of caveats: it has the fourth highest grocery costs and third highest transportation costs.

8. Cleveland, Ohio

Annual expenditures: $36,056

Annual amount spent on housing: $4,448

Annual amount spent on health: $6,474

Even though it's bigger than neighboring Akron (and on Lake Erie) Cleveland is a tad cheaper for retirement.

It has the second lowest annual amount spent on housing, but unfortunately has the second highest annual amount spent on health among the most affordable places to retire.

But back on the plus side, Cleveland, known as an industrial city, has undergone a revitalization, according to Sperling's Best Places. It has nationally recognized health facilities, a world-renowned orchestra, three professional sports teams and the Rock and Roll Hall of Fame.

7. Augusta, Georgia

Annual expenditures: $35,781

Annual amount spent on housing: $6,672

Annual amount spent on health: $5,574

Augusta, best known for hosting the Masters Golf Tournament, is also one of the best places to retire on a budget. The cost of living is about 22 percent lower than the national average; it also has the ninth lowest annual amount spent on housing.

And the annual amount spent on health is lower than in most of the cheapest cities to retire. Another financial benefit is the cost of senior care services, which is lower in Georgia than the national median.

6. Brownsville, Texas

Annual expenditures: $35,461

Annual amount spent on housing: $6,513

Annual amount spent on health: $5,694

Brownsville is one of the cheapest places to retire thanks to low housing, grocery and health costs.

The annual amount spent on housing is the eighth lowest on the list. It's also close to beaches and boasts a warm climate.

5. Toledo, Ohio

Annual expenditures: $35,095

Annual amount spent on housing: $4,925

Annual amount spent on health: $6,174

Toledo has a cost of living about 23 percent lower than the national average, and ties with Jackson, Mississippi for the third lowest housing costs on the list.

It's an industrial city, but its location on Lake Erie offers access to plenty of water and outdoor recreation.

4. Memphis, Tennessee

Annual expenditures: $33,859

Annual amount spent on housing: $6,354

Annual amount spent on health: $5,694

The cost of living in Memphis is 26 percent lower than the national average.

Memphis has the fourth lowest annual amount spent on transportation, fourth lowest utility costs and seventh lowest annual amount spent on housing. It also has mild winters and highly ranked health care facilities.

3. Jackson, Mississippi

Annual expenditures: $33,676

Annual amount spent on housing: $4,925

Annual amount spent on health: $5,514

Jackson is one of the cheapest places to retire thanks to a cost of living 26.4 percent below the national average; it ties with Toledo for the third lowest annual amount spent on housing, and has the fourth lowest grocery costs and seventh lowest annual amount spent on health.

In addition to a low cost of living, it has four seasons, museums, festivals and a strong network of health facilities.

2. Detroit, Michigan

Annual expenditures: $33,356

Annual amount spent on housing: $3,177

Annual amount spent on health: $5,994

Although Detroit is the largest city to file for bankruptcy, it's reinventing itself as "America's Great Comeback City," creating a 50-block sports and entertainment district. Its new streetcar and bikesharing system will save money on transportation to the new restaurants, galleries, markets and community gardens.

But the big draw is its incredibly low cost of living, 27 percent lower than the national average.

1. Birmingham, Alabama

Annual expenditures: $33,219

Annual amount spent on housing: $5,242

Annual amount spent on health: $4,915

The cost of living in Birmingham is 27.4 percent lower than the national average, with the annual amount spent on health the lowest of any city on the list. 

It also has the third lowest transportation costs and fourth lowest annual amount spent on housing.

But it has goodies, too, like local parks, 15 golf courses, nearby lakes, festivals, and schools such as the University of Alabama at Birmingham and Samford University that provide an abundance of concerts and arts programming.