Reeves, who became an MP in 2010, has been pushed forward by Ed Miliband himself as a star of his front bench. When the Labour leader and Ed Balls went to Gregg's flagship store to exploit George Osborne's "pastygate" fumbles, they were joined by Reeves, eating sausage rolls for the cameras. But many in the Labour Party believe she can also be their first female leader, whenever that time comes. A year ago, when Miliband's leadership was running into difficulty, MPs and members looked around for an election-winner – and Rachel Reeves's name came up every time.
Today, Ed Miliband is on safer ground as he begins his second conference as leader. But the party's 2011 reaction shows that, despite his current good run, many are still looking to the future for a leader with the same election-winning formula as Tony Blair.
Sitting in Bramley Baths, an Edwardian swimming pool in her Leeds West constituency which local people saved from closure by the (Labour-run) council, the Shadow Chief Secretary to the Treasury is calmly dismissive of the leadership question.
"That's not my ambition," she says. "My ambition is for Labour to win the next election, and to be part of the team that's leading the change that Ed Miliband's been speaking about.
"I just really want the Labour Party to win the next election – not for me or for Ed Miliband, but for people in my constituency and for people around the country who are being so hard hit by what the Government are doing."
The swimming pool was saved when Reeves held public meetings to bring together a local partnership, the Friends of Bramley Baths, who in December will take ownership of the pool from Leeds City Council, which says it was forced to threaten closure because of government cuts. Despite wearing three-inch heels, the MP walks around the edge of the pool to point out the original features from 1904: the stained-glass windows and white tiled walls, where her constituents, their parents and grandparents have spent a century coming together to meet, swim and play. We then go to a room next door, painted the same colour purple as her top, where there is proper Yorkshire hospitality: mugs of tea and a gigantic cream sponge cake from Morrison's.
Reeves explains Ed Miliband's idea of "predistribution" – ensuring "income and wealth are distributed before government gets involved, through the benefits system and taxes", using minimum and living wages to spread income more fairly and boost skills-based education.
"When the economy grows again, the proceeds of that growth must be fairly distributed. If people at the top of the banks get bigger bonuses, that does nothing for ordinary people who want to see an improvement in their living standards. That's what the next Labour manifesto is going to be about."
Reeves admits that the policy is quite hard for voters to understand: "We're not going to have a pledge card that says: 'No 1, we want a better predistribution of income'. But we need to have a narrative to be able to develop policies."
She insists Miliband will be PM after the next election: "I find him hugely inspirational and I know that he cares passionately about the challenges that Britain faces, that ordinary people in Britain face."
But even if Reeves can see Ed Miliband in No 10 after May 2015, can voters really see it in their minds?
"It's difficult being leader of the Opposition because you're aspiring to do a job that you don't have any chance to prove you can do until you're in it. Conference is an opportunity for Ed Miliband to be centre-stage.
"People are calling out for change; they are desperate for change. In a way which we haven't had at many conferences for a few years, people are looking to Labour for that alternative vision for the country. They want some hope, which they're not getting from this government."
Reeves leaves the door open to Labour signing up to Osborne's spending plans, an issue of heated debate inside the Shadow Cabinet. "At the moment, it's very difficult to know what the economy's going to look like in three months' time, let alone in two and a half years' time.
"Labour have to be cautious: we can't promise to reverse any of the cuts or the tax increases that the Government has put in place. So until closer to the election, we're not going to be able to set out our spending plans and commitments because, if we did that now, the likelihood is we would have to revise them every time a new lot of data came in."
When she discusses the economy, she talks quickly, words spilling out, which suggests her brain is working faster than her mouth can keep pace with. She has been criticised for having an estuarine twang in her voice – she was brought up in Lewisham, south-east London – which could hamper her leadership chances, but I wonder whether it is this that makes some of her words sound strained. When the tape recorder is off and she is more relaxed, the twang seems to be smoothed out.
A year on from joining the Shadow Cabinet as Shadow Chief Secretary to the Treasury, she seems to have grown in political stature, but she can still sound as if she's reading from a script. Maybe this is the cautious behaviour of someone who wants to keep a clean slate to prepare for leadership. And maybe becoming a mother will make her sound less like a career politician.
Reeves, the daughter of Labour-supporting teachers, went to a state school in Bromley and was a junior chess champion. After studying politics, philosophy and economics at Oxford, her first job was as an economist at the Bank of England. On her first day, she met Matt Hancock, who would later be Osborne's adviser before also becoming an MP in 2010 – he is now Business minister. Was Hancock right to compare himself, as he did in an interview recently, to Churchill and Disraeli? "I didn't see it when I worked with him."
If Labour is the largest party in a hung parliament in 2015, could it work with Clegg?
"I think it would be very hard to work with Nick Clegg. He is fully signed up to what David Cameron and George Osborne are doing. He's lost the trust of people."
But asked whether Labour could work with a different Lib Dem leader, such as Vince Cable, Reeves says: "I think the Labour Party can get an overall majority at the next election. But we've also got two and a half years until the next election and, during that time, we're happy to work with politicians from all parties."
Did she have any sympathy for Osborne when he was booed at the Paralympics? "I have no sympathy for George Osborne. It was the Tories' biggest-ever focus group, and 80,000 people booed him. The people I feel sorry for are the people who come to my surgeries every week who are just facing desperate situations. Those are the people my heart goes out to, not George Osborne, who should take his medicine."
Reeves is planning to take some maternity leave, and her husband, Nick Joicey, a Cabinet Office civil servant, will take paternity leave.
How will motherhood affect her views on politics? "Ask me this in six months' time or a year's time! In my constituency I meet lots of mums and dads who balance having a job or career and bringing up a family. Having children is one of the greatest joys, but it's a big challenge as well. I'm sure it will make me more in touch with some of the challenges my constituents face."
Her response sounds a little rehearsed, so it is refreshing to hear her joke about names. "We don't know whether it's a boy or a girl yet – but, if it's a girl, it definitely won't be called Ed." Will the baby be called Ed if it's a boy? "Probably not. But I'll have to discuss it with Nick."
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