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As the student protesters have pointed out, the vouchers fail to cover an estimated 20 percent of school costs, and many families must pitch in to fill the gap.

Student leaders have denounced the system as a cornerstone of social inequality in a country with one of the widest income gaps in Latin America, and they are demanding a return to the old system of free public universities.

That would mean raising taxes on the rich to subsidize widespread free education, the students say.

Perhaps the biggest losers in the battle are the public high schools, routinely of poor quality because their government funding does not begin to meet their needs, the protesters say.

Mr. Pinera has said that an ideal structure “is a mixed system, in which there is private education and public education.”

“What we have said is that [a university] education will be free for all young people who can’t pay for it, but those who can pay for it should make an effort to do so,” Mr. Pinera said last month.

Mr. Pinera has proposed ensuring a free university education to 40 percent of Chile’s poorest families, who earn less than about $500 a month, an offer that eventually would increase to 60 percent.

Chilean education specialist Patricio Meller argues that doing away with private universities won’t halt the soaring cost of tuition, which is hitting all institutions as enrollment rises.

As long as schools set their own rates, costs will continue to grow, Mr. Meller said in a recent newspaper interview.

“All the problems, in their origin, lie in the market,” he said. “If for-profit universities are closed, this isn’t going to stop the accelerated increase in tuition. This isn’t going to increase student credit. This isn’t going to decrease the cost of credit. … The university students have confused profit with the market.”

After playing defense for months, Mr. Pinera recently has taken a harder stand toward the protesters, including pushing a law to penalize anyone who incites rioting or acts of violence such as the occupation of schools or university campuses.

Chile has seen more than 40 student marches, many of them resulting in vandalism and violence, since the movement began.

Still, support for the students remains high among Chileans: 67 percent last month compared to 79 percent in September.

While admitting disappointment, the students point out the modest gains they have made.

Interest rates on university student loans have been reduced from 6 percent to 2 percent; Congress has discussed refinancing debt for 110,000 people behind in their payments; and opposition lawmakers have promised they won’t reach any backroom deals with the government, Miss Vallejo said.

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