A few months ago, Matt Bedsole got a call from two real estate developers asking for his help. Their plan to build a four-story apartment complex in Chattanooga, Tennessee, had a financial hole that no backer seemed eager to fill. The developers needed $8 million. Would Bedsole be interested in stepping in? Bedsole is not a normal investor. He is the CEO of Invest Chattanooga, a fund set up by the city of 200,000 to invest in local apartment projects. Unlike private equity firms — the main backers of new construction — he judges deals not solely on their financial return but also on how much housing they can deliver the city. The apartment complex cleared that hurdle. It called for 170 units that would replace a self-storage center ringed by barbed wire in a gentrifying part of the city. But Bedsole had terms. In exchange for the $8 million investment, he got a 51% stake in the building and an agreement that 30% of its units be priced below the market rate. The developers said yes.…