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With overall law firm billing rates rising fast, firms that increased their rates more slowly are reaping rewards in the form of greater revenue growth, according to a new study.

The report, issued Tuesday by Thomson Reuters’ Peer Monitor, found that when firms increase their rates at a slower pace, they also see an increase in demand from clients, even as demand has been stagnant across the industry.

“We’ve been making the case that demand has been flat over the years,” said Leonard Lee, an analyst at Peer Monitor. But while demand for legal services has generally reached a plateau, some firms are getting much more work than others.

Lee said Peer Monitor wanted to know what separates the firms enjoying increased demand, measured as hours billed, from those that have seen a slowdown. “Earlier this year, we started to notice this trend with rates,” Lee said. “Firms that are willing to take less of a rate increase seem to be doing better with demand.”

The study found that in the first half of 2015, Am Law 100 firms increased their rates by 3 percent and demand grew by 1.3 percent. In the first half of this year, the largest firms pushed up rates more sharply, increasing them by 3.3 percent, and demand subsequently fell 0.2 percent.

The same holds true for Am Law Second Hundred firms, which have seen a very gradual increase in rate growth since 2014, according to the report. Demand went from growing 1.4 percent in the first half of 2014 to a drop of 1 percent in the first half of this year, Peer Monitor found.

Midsized firms, in contrast, seemed to benefit from a gentler approach to billing rates. They’ve gradually slowed their rate increases since 2014, and both demand and revenue have started to grow. In 2014, demand fell by 0.8 percent, but in the first half of this year, it grew by 1.6 percent.

The report cautioned that reducing rates or growing them at a slower pace is not a sure-fire way to increase revenue. There are other factors that may explain the slowdown in demand at Am Law Second Hundred firms, such as a drop in realization rates and changing client preferences.

“Over the last few years, a lot of work seems to be moving down market,” Lee said, noting that midsized firms are having a good year financially, compared to firms in the Second Hundred.

The figures reported in the Peer Monitor study are averages and every firm is different, but Lee said the findings show a clear correlation between rate growth and revenue.

“Firms that are willing to slow their rate increases are seeing higher average growth in demand and revenue than firms that are raising rates more aggressively,” the report concluded.

Contact Nell GLuckman at ngluckman@alm.com or on Twitter @NellGluckman.