So from the outset, police union officials felt that the city was more sincere and serious this time around. Indeed, Patrick J. Lynch, the president of the union, said he saw more flexibility on the city’s part than in previous rounds of negotiations. For instance, the city agreed to make a required annual day on the shooting range a work day, rather than a vacation day. Far more important, the city’s opening offer in the negotiations was a raise of 4 percent a year.  “It was a real negotiation starting with a real offer, and there was give and take on both sides,” Mr. Lynch said. “When we went into our other rounds of recent negotiations, the city was proposing zeros for some years,” meaning a pay freeze for a year or two.  The contract that Mr. Lynch and Mr. Bloomberg announced on Thursday calls for a 4 percent increase for each of four years. By the time the contract expires, the starting salary will be nearly $42,000, and it raises the base salary after five and a half years on the job to $76,488, up from $65,382. (The contract calls for two years of retroactive raises because it runs from Aug. 1, 2006, through July 31, 2010.)  For Mr. Bloomberg, the motivation to achieve a deal was enhanced, city officials said, by a sense that the city had not fared well under the arbitration panel’s decision in May. That panel awarded police officers a raise of 9.7 percent over two years, considerably more than the city had anticipated.  Moreover, heading into his last year in office, the mayor was clearly eager to eliminate one more potential problem. And if he is seriously considering making it possible to run for a third term, it would certainly help to reach a contract with the Patrolmen’s Benevolent Association because that would go far to neutralize a union that, probably more than any other, has rarely shrunk from public denunciations of Mr. Bloomberg.  “The mayor’s take was, arbitration is an unknown,” Mr. Skyler said. “Why would we leave the future to someone else? A bad arbitration can really hurt the city. When you add Commissioner Kelly’s concerns about the starting salaries, we had all the motivation we needed to make a deal.”  And the union had motivation, too, fearing that the city’s financial problems might grow worse.  Joshua B. Freeman, a history professor at City University Graduate Center and author of “Working-Class New York,” said: “The P.B.A. isn’t unique here. The sense is there’s a difficult economic stretch ahead with budgetary implications, and that has led a number of unions to settle quicker than usual.”  In this instance, the negotiations had been carefully watched because New York City faces serious budget shortfalls over the next several years. Mr. Bloomberg estimates a $2 billion shortfall in 2010.  City officials said Thursday’s agreement would increase payroll costs by a total of $540 million over the four years. The 4 percent raises were based on a pattern set last year in a contract with the police sergeants’ union. City officials said they had already budgeted for the Patrolmen’s Benevolent Association’s 4 percent increases.  One reason the patrolmen’s union was reluctant to go to arbitration was that its leaders recognized that a pattern had been set by the police sergeants’ contract and that it might be very hard to persuade arbitrators to award more than that 4 percent a year.  But even as the union accepted the 4 percent raises, negotiators for the two sides still needed to thrash out dozens of other issues, large and small, including the city’s persuading the union to drop a lawsuit against taking police officers’ hair for drug tests.  “It came together through a lot of hard work,” said James F. Hanley, the city’s labor commissioner.  A version of this news analysis appears in print on , on Page A31 of the New York edition with the headline: A Police Contract Without Tears Or Arbitration. Order Reprints| Today's Paper|Subscribe
Considering the history of bad blood, news that the city had struck a tentative four-year contract with the city’s main police union came as a shock.