Before beginning the hunt for their first house, Tennessee residents Brittany and Craig Murphy pared their student debt, saved for a down payment and got an income boost from her new job. The major hurdle was what came next.

In the last month, the couple lost two bidding wars on Nashville homes to competitors willing to pay more than 10 percent above the asking price.

“I was not expecting the actual finding of the house to be the difficult part,” said Brittany Murphy, a 26-year-old Web designer whose husband, 27, is a software developer.

Steady job growth, low mortgage rates and record apartment rents are turning millennials like the Murphys into homebuyers — if they can find a house. As the key U.S. spring sales season gets under way, robust real estate demand is being outweighed by a persistent lack of lower-priced supply that’s poised to limit transactions and worsen an affordability crunch for young people. They’re faring worse than purchasers at the higher end of the market, where inventory is piling up.

Rising interest in home tours indicates prospective buyers are coming out in droves. An index by Redfin that measures requests for property visits rose in the first two months of the year to the highest level since at least 2012, when the data began.

“As soon as a house hits the market, it will be eaten by the huge demand appetite,” said Nela Richardson, Redfin’s chief economist.

Limited Inventory

Surging homebuying interest won’t necessarily translate into a big jump in sales. Prices will rise while limited inventory will put a cap on transactions, said Doug Duncan, chief economist of Fannie Mae. He estimates that U.S. single- family home prices will climb 5 percent this year, about the same as in 2015, while sales will increase 3 percent. That’s a slowdown from 2015, when existing-home purchases jumped 7 percent.

“Affordability is a challenge this spring,” Duncan said. Prospective buyers “would have gotten their credit in shape and they’ll have a job. But they will be frustrated because, in their market, there simply won’t be affordable homes.”

The shortage is most acute for starter homes. Across the country, inventory was down 8.2 percent in January from a year earlier for properties priced below $250,000, data from the National Association of Realtors show.

While first-time buyers aren’t necessarily millennials, people age 35 or younger have become housing’s largest segment, accounting for 35 percent of buyers in the 12 months through June 2015, up from 28 percent three years earlier, according to a survey from the trade group released last week.

They’re facing weaker affordability as home prices climb faster than incomes, said Ralph McLaughlin, chief economist of Trulia, a unit of Zillow Group Inc. Starter-home buyers on average now need to devote 38 percent of income to housing costs, up from 32 percent four years ago, he said.

Nashville Demand

In the Nashville area, the median single-family home price in February jumped 14 percent from a year earlier to $235,000, according to the Greater Nashville Association of Realtors.

The Murphys amassed $20,000 for a down payment by moving in with a friend so they could save money, and from the added income of the better-paying job Brittany Murphy got in late September.

So far, they’ve come up short, even though they’ve offered well over the asking price twice. The couple is now the back-up buyer for a home if the winning bidder, who agreed to pay more than $250,000, falls through. The asking price was $225,000, said Scott Cornett, the Murphys’ real estate agent.

“I’m concerned about the current ability of the millennial generation to buy a home,” Trulia’s McLaughlin said. “Right now, inventory pressures are keeping them from doing that.”

Construction Needed

There’s only one way to boost inventory — by building more homes. But homebuilders, seeking larger profit margins, have focused on pricier offerings rather than smaller starter homes. And many homeowners aren’t selling because they owe more than their properties are worth or they can’t afford to trade up.

Starter homes made up 27.7 percent of the market’s inventory through February this year, down from 28.7 percent a year earlier, McLaughlin said. That’s the smallest share for the start of the year since 2012, he said.

Buyers with bigger budgets — or homeowners trading up — have more to choose from. The inventory of properties priced above $250,000 was up 10.9 percent in January, according to the National Association of Realtors. Economic instability from China to South America and the stronger U.S. dollar is dulling demand from foreigners, who tend to purchase higher-end homes.

“Foreign buyers are facing some headwinds,” Scholastica Cororaton, a research economist with the Realtors group, wrote in a blog post on March 7. “On a positive note, the slowdown in demand from foreign buyers may help ease the tightness of supply in the market.”

That’s of little comfort to Jennifer Lan and her husband, Jared Tompkins, who are both physicians finishing residencies in Memphis, Tennessee. They started hunting for their first house near her parents in Potomac, Maryland, last month and have already had to increase their budget to more than $900,000 after seeing what they would get for $800,000.

“I don’t think I’m going to find a bargain,” said Lan, 30. “I’m just hoping to find a home that’s not terribly overpriced.”