Health insurance costs and market competition - a clarification

In a recent piece in which I wrote in favor of a national single-payer health care plan, I stated:

By eliminating market competition and a for-profit business structure for essential health services, we can keep costs down.

I received a reply that alerted me to the need for some clarification.

That reply, from someone who identifies himself only as "Capitalist PigDog" from Wyoming, started out as follows:

"Market competition is the single most effective means of lowering costs."

I see his point. And I see that I should have elaborated.

Below is my response to Capitalist PigDog's comment:

In many cases, perhaps most cases, market competition does indeed keep costs down. With health insurance, however, it has had the opposite effect. In order to remain competitive, insurers take on a lot of unnecessary overhead, including administrative costs and marketing, which drives costs up. And those costs are subsidized via the pervasiveness of employer-sponsored medical benefits, so the market will bear the high costs.

And so the poor and the unemployed remain uninsured.

Many thanks to Capitalist PigDog for keeping me on my toes.