A big part of the pitch for states to participate in the Medicaid expansion through President Obama's health care law is rooted in the fallacy that it represents “free” money from Washington.

Republican governors who have fought conservatives to expand Medicaid in their states have relied on the argument that it would be cost-free to residents. Gov. John Kasich, R-Ohio, presented it as “a chance to bring Ohio money back to Ohio.” On her official website, Gov. Jan Brewer has a page dedicated to selling the Medicaid expansion as “The Conservative Choice for Arizona,” which boasts that the plan carries “no state expense” and comes at “no cost.”

So far, the Medicaid expansion has now been enacted in eight states with Republican governors. But after 2016, GOP executives seeking to expand Medicaid will have to find a new pitch.

The reason is that the way the law was designed, the federal government only agreed to cover the full cost of the expansion until 2016. Starting in 2017, states will have to start kicking in money, and by 2020, will be expected to cover 10 percent of the cost of the expansion.

According to the Congressional Budget Office, state spending on Medicaid will be $46 billion higher in the coming decade than it would have been before Obamacare passed (which is on top of the $792 billion cost to federal taxpayers). The Center on Budget and Policy Priorities has noted that this still represents a mere 1.6 percent increase over the $2.9 trillion that states would be spending on the existing Medicaid program anyway.

But this understates the cost of the Medicaid expansion in several ways. To start, the $46 billion figure is for 2015 through 2024, meaning that for half of the period being measured (2015 through 2019), the state share of Medicaid won’t have reached its full level. Additionally, the CBO analysis is being made at a time at which two dozen states have not agreed to expand Medicaid. If every state were to expand Medicaid, the total cost to the states would be higher than $46 billion. This also assumes the federal government keeps paying the same share in perpetuity -- which is far from assured.

It’s no doubt true that even accounting for these factors, the ultimate state share of the Medicaid expansion is going to be a fraction of what states already spend on the program and of what the federal government will be spending. But that says more about the staggering cost of Medicaid than it does about the meagerness of state contributions.

When it comes to expanding Medicaid, Republican governors have found themselves torn between conservative activists who want no part of Obamacare and hospitals that are aggressively lobbying for more government money, especially given other provisions in the law that cut hospital payments. Liberal activists are pursuing a public shaming campaign to attack governors who have thus far rejected the expansion of Medicaid.

After 2016, it’s going to be a lot more difficult for Republican governors – especially those seeking higher office – to explain their decision to expand Medicaid to conservative supporters.

Even if costs being imposed by the Medicaid expansion are small relative to the massive size of the joint state-federal program, they will still be objectively real. Republican governors will no longer be able to use the argument that expanding the program through Obamacare is cost-free. This will change the dynamic of the battlefield.

This reality makes the next few years that much more crucial for advocates of limited government aiming to limit the scope of Obamacare.