FORTUNE — There are very few surprised faces from either side of the Verizon picket lines. When a labor contract signed in buoyant times expires in a bleaker era – as is the case with the company’s current 45,000-worker strike — the stage is set for conflict.

Both Verizon’s (VZ) management and the unions are responding to clear, immediate issues and changes, but they are both acting somewhat myopically. Each side could have pursued a stronger strategy. And there still exists the possibility of a grand cooperative outcome.

Verizon’s management has asked the union workers to take fewer sick days, make larger contributions to their health plans, and accept a freeze on pension benefits. (The pension plan demands mirror changes Verizon made six years ago to 50,000 of its non-union employees.)

The Communications Workers of America and the Brotherhood of Electrical Workers argue that Verizon is a hugely profitable enterprise, with a net income (fully consolidated) of $10 billion in 2010. Union leaders also claim that management is trying to push profits even higher by “destroying middle-class lives,” a policy of pure “greed.”

Although Verizon is making decent profits, only 0.5% of its income is attributable to the slowly shrinking landline business where its unionized employees are concentrated. The difference between the wireless and wireline sides of Verizon’s house is driven by its network architecture. In the wireline business, there are hundreds of thousands of miles of individual telephone and data lines, most sagging from aging telephone poles, exposed to the elements and accidents. After a major storm in southern California it may take 10,000 truck rolls to repair the damage. That is why the wireline business needs 111,000 employees — about one-third more than the wireless business — to support revenues only two-thirds as large.

In the huge, mass-market portion of the wireline business, Verizon competes with cable companies in offering triple-pay (video, Internet, and telephone) services. In that competition, cable companies have simpler networks and many fewer unionized workers, paying installers an average of about $19 per hour compared with Verizon’s $26 per hour.

Given the current economic climate, where austerity and benefit reductions are discussed daily, it seems a propitious time to bargain. But, Verizon management’s position is weakened because of the company’s overall success. While the company may argue that wireline profits are only $3,064 per wireline employee, the union can point to the company’s large overall profits.

In cases like these, where the pie is shrinking, management may have to legally separate the two businesses in order to avoid cross subsidization. When the business in question has no surplus, the union’s position is greatly weakened.

Turning to the union’s strategy, it is smart to push back on management before such a split is engineered, exploiting the company’s overall profitability. And, it is smart to push back at a moment when the unions have the luxury of having sympathetic ears in the White House.

But winning concessions from management in this round of contract negotiations does nothing to halt the gradual shrinkage of the wireline business. This shrinkage is a problem for more than the workers at Verizon; it affects many of the estimated 500,000 union workers in the U.S. wireline industry.

The Communication Workers union would have a much stronger position were it to use its clout across the industry to work with Congress and the administration to promote a national fiber-to-the-home initiative. Such a program would build infrastructure, create jobs, and, like Eisenhower’s Interstate highway program, would have benefits to the economy that many conservative Republicans could embrace.

Because such an initiative would benefit wireline businesses more than the cable companies, it offers a way for both Verizon shareholders and the union to gain. Even if it does not surface in this round of negotiations, don’t be surprised when wireline operators, unions, and the national government devise a fiber initiative that jointly advances their separate agendas.

Richard Rumelt is the Harry & Elsa Professor at the UCLA Anderson School of Management. He is the author of Good Strategy/Bad Strategy. Visit his website at strategyland.com.

Editor’s note: A previous version of this story incorrectly stated the number of employees working in Verizon’s wireline business and misstated the profits Verizon earned per wireline employee. These figures have been adjusted.