To go by the pronouncements coming out of Germany over the last couple of weeks, you might naturally conclude that the euro is toast. Speaking before Parliament, Chancellor Angela Merkel broadly rejected “counterproductive” proposals to pool Europe’s resources to help floundering Mediterranean nations. Germany’s “strength is not infinite,” she stressed.


German voters are even more skeptical than their leaders about financing their “slothful” and “profligate” neighbors. Though most still tell pollsters they want to keep the single currency, almost four-fifths want Greece to leave — oblivious to the chain reaction that Greek departure would unleash against Portugal, Spain and even Italy.


Yet it would be wrong to kiss the euro goodbye just yet. For all of Berlin’s neins, shooting down every serious proposal to address its woes, the German government knows it must ultimately cave and open its wallet to save the single currency.


Berlin’s wall of hostility against bailouts of Europe’s south will be on display this week, when European leaders will again try to cobble together a plan to address their debt crisis. They will discuss Greece’s request to ease the terms of its $217 billion rescue package, as well as proposals to create a regional banking union and Spain’s request for $125 billion to shore up its failing banks.


Berlin will drag its feet as long as it can before offering help, as has been its wont throughout the crisis. It will demand assorted quid pro quos. Last week, the German finance minister, Wolfgang Schäuble, warned Greece not to expect much sympathy and demanded that Athens comply with the austerity measures “quickly and without delay.”


But Ms. Merkel knows that Germany must ultimately underwrite the euro’s rescue, pretty much regardless of whether its conditions are satisfied. There are three good reasons. First, the euro has been very good to Germany. Second, the bailout costs are likely to be much lower than most Germans believe. Third, and perhaps most important, the cost to Germany of euro dismemberment would be incalculably high — far more than that of keeping the currency together.


Let’s take the reasons in turn. Germany has had a fairly good crisis so far. Since 2009, when it fell into a deep recession, it has grown faster and suffered less unemployment than almost any other industrialized country. Wages are rising. And exports have rebounded sharply from the crisis to surpass their peak of 2008.


Germany owes much of this to the euro — which tethered its ultracompetitive manufacturing to the mediocre economies of its neighbors. Since the advent of the single currency, Germany’s labor costs have fallen more than 15 percent against the average labor costs of all the countries using the euro, and about 25 percent against those of the troubled nations on the periphery. If it dumped the euro for a new deutsche mark, its exchange rate would surge to make up for the difference, potentially crippling its exports, which have fed most of its economic growth over the last decade.


What about the cost of a bailout? German economists are pushing the story that Germany has already squandered enormous sums on the euro’s survival, going above and beyond the call of duty. Hans-Werner Sinn, who heads the Ifo Institute for Economic Research in Munich, argued in a commentary piece on the Op-Ed page of The New York Times that Germany had given Greece so far the equivalent of 29 times the aid given to West Germany under the Marshall Plan after World War II. His analysis omitted, however, that aid was just a small part of the Marshall Plan’s help to Germany. Most important, the plan also wiped out a majority of Germany’s debt.


While Germany has committed a few hundred billion euros to rescue the currency, if the rescue succeeds, it should recover all of it. And it can readily do more. William R. Cline, an economist at the Peterson Institute for International Economics, told me that covering the entire financing needs of Greece, Ireland, Portugal, Spain and Italy through 2015 would cost about $1.6 trillion.


If the International Monetary Fund contributed one-third, Germany and other rich euro area countries would be left to put up the rest. But even if Germany’s share reached $500 billion, it would not forfeit this money. After all, the goal of the bailout would be to prevent defaults. Germany could even turn a profit.


Most economists and policy makers outside Germany agree that keeping Europe’s common currency together over the long term will require a permanent mechanism to pool risk — transferring resources from the euro area’s powerful core to its weaker members. Germany, predictably, has balked at this prospect as way too expensive. Yet German estimates seem exaggerated.


One way to pool risk would be to allow countries to issue “euro bonds,” which would be jointly guaranteed by all the countries in the euro area and thus carry a much lower interest rate than markets are charging countries like Italy and Spain.


Kai Carstensen of the Ifo Institute estimated that euro bonds would end up raising Germany’s annual borrowing costs by 1.9 percent of the country’s gross domestic product — more than $60 billion — because it would pay a higher rate of interest than German bonds do now.


But an analysis by Mr. Cline of the interest rates paid by countries of different credit ratings since the late 1990s suggests a much lower price tag: the borrowing costs of triple-A countries like Germany and France would rise by 0.35 percent of G.D.P. per year. And the benefits would clearly outweigh the costs. Portugal, for instance, would save 1.9 percent of its G.D.P. in lower interest costs, giving it much-needed breathing room.