Every day, it seems, more damaging details emerge about the rollout of the federal online insurance exchange at the heart of the Affordable Care Act. Today’s revelation, courtesy of the Washington Post: days before the launch, officials and government contractors conducted a test of the new Web site, during which it crashed when just a few hundred people tried to log in simultaneously. But the Obama Administration went ahead with the rollout anyway, only for the site to seize up just hours into October 1st.

Until the Administration gets the site working properly, this story will dominate the news and overshadow the underlying reality about Obamacare: judged on its own terms, the new health-care system is likely to work. In the coming decade, tens of millions of Americans will end up using the new health-insurance marketplaces—both the federal one and the state ones—and the number of uninsured will drop quite dramatically. Not everybody will end up being covered, but, excluding unauthorized immigrants, who won’t be eligible to use the new system, it seems likely that, at a minimum, the proportion of people who are uninsured will be cut in half.

What is the basis of these statements? Not any particular affection on my behalf for the new system, which largely preserves the private-insurance model that has proved so costly and inefficient. As somebody who grew up using the British National Health Service, I’ve always been more attracted to a single-payer system that guarantees coverage. (No, my Republican friends: introducing such a system wouldn’t amount to imposing an alien European-style socialism on the American public. Under the rubric of Medicare, almost fifty million Americans already enjoy, and value greatly, precisely this type of system.)

My confidence that the Affordable Care Act will meet its coverage objectives is based on a belief, shared by most economists, that financial incentives, if they are big enough, tend to work. Obamacare, once it gets up and running, will provide very large incentives for people to get coverage—subsidies of as much as ten thousand dollars a year for some low-to-middle-income families. Over time, the presence of these incentives, which include substantial fines for individuals who refuse to buy coverage, will almost certainly overcome any glitches or difficulties in the system of enrolling. When people think they are getting a good deal, they are willing to put up with a bit of hassle. And for many of the uninsured, Obamacare is a very good deal.

To make the new system work, the Administration is spending a great deal of money—about $1.4 trillion over the next ten years, according to the latest analysis by the Congressional Budget Office. About half of this will go toward subsidies for people who buy insurance policies on the new exchanges. The rest will go toward expanding Medicaid, the health-care system for the poor and indigent, which is jointly funded by the federal government and the states.

(Note, though, that despite the substantial expenditures it involves, the Affordable Care Act won’t increase the budget deficit by $1.4 trillion, or anything near it. That’s because it also features hefty cuts to the Medicare budget and a number of tax increases that will raise substantial amounts of new revenue. Indeed, the C.B.O. says that, over all, the Act will reduce the deficit slightly during the next ten years.)

The price of obtaining coverage on the new exchanges, and the precise subsidies on offer, vary from city to city and state to state. But with the aid of a very helpful online “subsidy calculator” from the Kaiser Family Foundation, I obtained a few representative figures. Take, for example, a family consisting of two parents (both age thirty-five) and three children that has an annual income of forty thousand dollars and lives in Nashville, Tennessee. According to the calculator, a silver insurance plan for such a family—i.e., one that provides more coverage than a bronze plan but less than a gold one—would cost $10,148 a year. The federal government would cover eighty-five per cent of that sum, which is $8,662. The actual cost to the family would be $1,485, or about a hundred and twenty-five dollars a month.

Generally speaking, smaller families will get less help from the government, but many of them will still receive sizeable subsidies. Consider a single mother of one living in Denver, Colorado, and earning thirty thousand dollars a year. A silver plan for her and her child would cost $4,366 a year, according to the calculator, but a federal subsidy of $2,567 would bring her actual cost down to $1,799, which is a hundred and fifty dollars a month. With these sorts of subsidies in place, it is hardly surprising that the new online exchanges have received a lot of visitors—in some cases, too many for them to handle.

Many single people and folks who earn a decent salary won’t get any subsidies at all. (If you are a single person living in New York or Los Angeles and earning fifty thousand dollars a year or more, that’s you.) But thanks to the individual mandate, such people will face substantial penalties if they don’t buy insurance. For 2014, the Obamacare “fine” has been set at a lowly introductory figure of ninety-five dollars per adult. Thereafter, though, it will climb steeply. From 2016 onwards, it will be six hundred and ninety-five dollars per person or two and a half per cent of family income—whichever is greater. For a single person earning eighty thousand dollars a year, that’s two thousand dollars. In some parts of the country, such a person would find it almost as cheap to buy coverage as pay the fine.

Maybe I’m wrong, but my feeling is these carrots and sticks will be sufficient to persuade most Americans to sign up for health insurance. Indeed, one of my concerns is that the new exchanges will prove too popular, and end up costing taxpayers a lot more than planned. As municipalities and private companies gain more knowledge about how the Affordable Care Act works, they will have an incentive to shift some of their employees and retirees from their existing plans onto the exchanges, where they may be eligible for federal subsidies. Chicago and some other cities have already indicated that they are thinking along these lines.

And let’s not forget Medicaid, which frequently gets ignored in the debate, and which is the cheapest and most effective way to provide low-income people with health care. Under the Affordable Care Act, Washington will provide states with enough money to cover almost anybody whose family earns less than a hundred and thirty-eight per cent of the poverty line. (For a family of three, that’s about twenty-six thousand dollars a year.) Last year, the Supreme Court decided that states could opt out of this expansion, if they wished to, and more than twenty G.O.P.-controlled states have done so. But the federal government is offering the states such a good deal (for newly eligible Medicaid recipients, it will cover all the costs until 2016 and at least ninety per cent of them thereafter) that Republican resistance is weakening.

On Monday, a panel of state lawmakers in Ohio approved the decision of Governor John Kasich, a Republican, to accept the additional money from Washington, which, according to a report by the Kaiser Family Foundation, will enable the state to cover an additional two hundred and seventy-five thousand residents, and cut the proportion of Ohioans who are uninsured by almost two-thirds. As time goes on, and voters in Republican states realize that their elected officials are leaving a lot of cash on the table that could be used to help their neighbors, it seems likely that other states will follow Ohio’s lead. Just how many more Americans across the country will end up being covered by Medicaid is anybody’s guess, but it’s sure to be a large number. (For what it’s worth, the C.B.O.’s estimate is thirteen million by 2023.)

If you put together the subsidies, the fines, and the expansion of Medicaid, you have a formula for substantially reducing the number of uninsured. Here, too, attempting to make precise predictions is perhaps silly. But the C.B.O., which is legally obligated to provide lawmakers with forecasts, reckons that between now and 2023 the number of uninsured Americans will fall from fifty-five million to thirty-one million. Excluding unauthorized immigrants, the agency says, the proportion of the population with coverage will rise from eighty-two per cent to ninety-two per cent.

Would such a jump justify all the cost and effort that has gone into the Affordable Care Act? Could it be accomplished more cheaply in other ways? And will the reforms slow down the over-all growth in health-care spending? These are questions that can be debated. But as far as I can see, there is little doubt that Obamacare will succeed in its primary aim of expanding health-care coverage.

Photograph by Brian Shumway/Redux.