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In his State of the Nation address in Auckland this afternoon, Mr Cunliffe also set out a raft of other measures for parents of young children, including free antenatal classes for all first time mothers, and extending early childhood education subsidies from 20 free hours a week to 25 hours.
Paid parental leave will also be extended from 14 to 26 weeks - a feature which was already Labour Party policy.
The total package is expected to cost $147 million in its first year, 2015-2016, rising to $528 million by 2018/19. The child payments have been costed at $151 million in the first full year they apply of 2016/17, rising to $272 million by 2018/19.
Dubbed Best Start, the first year payments will apply for those with household incomes of less than $150,000 a year, which Labour estimates will be about 59,000 households or 95 per cent of children aged under one. The longer term payments for those on lower incomes would cover about 56 per cent of one and two year olds. It will apply to children born after April1 2016, if Labour forms the next Government.
Mr Cunliffe said the Childrens' Commissioner's expert panel report on Solutions to Child Poverty made it clear it was important to provide support to those with young children to help reduce poverty.
"Sixty dollars will make a real difference to the lives of struggling parents. For example, it will pay for a weekly supply of nappies and baby food."
The Children's Commissioner report had recommended a universal child payment until the child turned six with ongoing support for lower income families.
The payments replace the party's 2011 policy to extend the Working for Families in-work tax credit to beneficiary families.
Labour last week ditched its policies for GST-free fruit and vegetables and an income tax free threshold, which it said freed up about $1.5 billion in its policy platform.
Mr Cunliffe spoke to a packed house of about 400 at the Kelston Girls' College hall today.
He spoke about his desire to build an economy that delivered for all New Zealanders.
"They are told things are getting better, but in their own lives they see prices going up while wages stay still. Too many people feel nervous about the monthly rates, power and insurance bills. Too many families are struggling even to cover the basics."
He said any Government had to put its resources where they would do most good, and confirmed Labour would "unashamedly" tax the wealthiest more to help pay for it.
"It's not good enough for a quarter of out kids to be growing up in poverty, or for many to lack access to support and education in their early years. We will fix this."
He said Labour would set out its wider programme of reforms in economic and social areas as the election year carried on.
Economic Development Minister Steven Joyce reacted swiftly on Twitter.
"Labour advises their spend-a-thon would resume immediately. Labour & Greens already an extra 3/4 a billion a year & it's not even end of Jan."
National-aligned blogger and pollster David Farrar quickly posted that the polices amounted to "turning families on $140,000 into beneficiaries".
"Welfare should be targeted at those most in need. A family on $140,000 with one child do not need our taxes. At the other end of the scale, this is a huge incentive to have more children if you are already on welfare."
He also said it appeared that backbench MPs would be eligible for the new payment.
"Yep, if a backbench MP gets pregnant (or their wife gets pregnant), then taxpayers will be paying them $60 a week welfare because they're in such dire need. Sickening."
But the policies met with approval from the Greens.
Co-leader Metiria Turei said, like her party's schools policies announced yesterday, Labour's Best Start plan drew from the Expert Advisory Group on Solutions to Child Poverty report prepared for the Childrens' Commissioner in late 2012, and " show a real commitment to children".
"The Greens are fully behind them," said Mrs Turei.
"If we give our kids the best start in life, they will grow up happier and healthier, they will learn more, and contribute more to our communities and our economy."
By ignoring the report to the Children's Commissioner in its own policies, "John Key's Government has ignored the evidence and failed to address the issues that are holding our kids back".
United Future Leader Peter Dunne was lukewarm, saying the Best Start policy was "good in some parts, but not others".
He supported the extension of Paid Parental Leave to 26 weeks which went some way toward his own policy to extend it to 12 months.
Mr Dunne also said the plan to provide free antenatal services, was " a positive step towards ensuring children get the best start in life".
However, he described the $60 a week "baby bonus" as "just a crass bribe".
"Using the funding committed to this policy could have been better used to cut tax rates, or to fund the further extension of Paid Parental Leave," he said.
Labour's 'Best Start' package
* $60 a week payment for the first year of a child's life, if parents earn less than $150,000. Will apply from April 2016.
* Ongoing $60 weekly payments for middle/lower income earners until the child turns three.
* Extend free early childhood education from 20 hours to 25 hours a week for over-3s.
* Extend paid parental leave to 26 weeks.
* free antenatal classes for all first time mothers.
Total cost: $147 million in 2015/16 rising to $528 million in $2018/19
- By Claire Trevett of the New Zealand Herald