Republicans are doing their best to persuade Americans that the Affordable Care Act will drive health insurance premiums so high that consumers will experience “rate shock.” That fear-provoking tale seems much less credible now that the rates proposed by health insurers for individual coverage on the new health care exchanges have been made public in several states. For the most part, the premiums will increase only slightly or even decrease for individuals and family coverage on the exchanges, electronic marketplaces in which consumers choose among a variety of plans with differing benefits and costs.

The reported rates are in states that have elected to run their own exchanges, which are set to start enrolling people in October for coverage that begins in January 2014. Many states with Republican governors have refused to set up exchanges and have left that task to the Obama administration. In 19 states where the exchanges will be entirely run by the federal government, more health insurance companies are entering the markets and injecting competition that should help hold down costs, the administration said on Thursday. Premiums are subject to many competing pressures, but, in several states, competition appears to be offsetting factors that might push rates up.

In California, officials announced last week that 13 insurers were awarded contracts to sell policies on the state’s health care exchanges next year. The companies submitted bids, which came in much lower than virtually all experts and actuaries had forecast.

Blue Shield of California said that its premiums for individual customers would increase about 13 percent on average next year over current premiums; about 8 percent of that increase would the cover rising cost of medical care, which would have happened anyway. The other 5 percent pays for better benefits, guaranteed coverage for pre-existing conditions, and taxes and fees imposed on insurers by the reform law.