WASHINGTON — In a well-known auto insurance commercial, a garrulous gecko promises you can save 15 percent if you switch insurers.

It turns out something roughly similar is happening with the Obama administration’s health insurance overhaul:

A study out today finds that this year’s most popular health law plans are raising premiums an average of 15 percent in 2016. You’ll have to switch if you don’t want to pay more.

The study by the nonpartisan Kaiser Family Foundation comes as sign-up season for subsidized private coverage under President Barack Obama’s law is in its third week. Due to rising premiums, high out-of-pocket costs, and skeptical consumers, the administration faces challenges trying to grow the pool of people covered through the law’s online markets.

The study looked at a type of coverage called the “lowest cost silver plan,” which is the health law’s most popular. Silver is the middle tier, and the lowest cost silver plan can be different in every community. The catch is, the lowest cost silver plan can also change every year. If your plan holds that distinction this year, odds are it won’t have it in 2016.

Looking at premium data across 2,635 counties in 36 states, the analysis found that a hypothetical 40-year-old faces an average premium of $264 for the lowest cost silver plan in 2015. If the consumer stays in the same plan for 2016, his premium would be $304, or 15 percent more. That’s before taxpayer-financed subsidies, which lower the cost to consumers by about 70 percent on average.

“The bottom line is that insurance companies are increasing premiums for what had been the lowest cost silver plan,” said Cynthia Cox, lead author of the study. “In many cases, people could save money on their premiums by switching to a new lowest cost silver plan.”

The study found that’s the case in 3 out of 4 counties examined. Among all consumers, those who switch could save $322 over the course of the year.

The Obama administration didn’t quarrel with the study. “Our message to returning marketplace customers is simple,” said spokesman Aaron Albright. “Shopping may save you money.”

The drafters of the health care law intended to create a competitive market in which insurers would feel pressure to keep premiums low. But shopping for a new plan can be daunting because of variables like the deductible, co-payments, and the provider network.