A cartoon in the Polish weekly Polityka recently showed two arm wrestlers resting their arms on a table, the legs of which had been broken out fragilely repaired with string. For either wrestler to win, it was evident he would have to lean on the table. Common sense would dictate that the table be fixed first.

Common sense would dictate the same to opposing powers here about Poland's damaged economy.

Following yesterday's Supreme Court decision that registered the independent trade union federation Solidarity, the Warsaw government called on the new organization to help repair the nation's economy, crippled by months of labor unrest and years of economic mismanagement.

Much of the chance for future social peace in Poland depends on this recovery and reform effort. But the dimensions of the problem -- ever more urgent now with the onset of winter -- are staggering.

Nearly everything in Poland is in desperately short supply. The lines people commonly stand in to buy anything from meat to chocolates to newspapers have grown longer since the summer's strikes and the quality of goods for sale is notably poorer.

Bad harvests and an early frost have resulted in a potato crop only two-thirds the size of last year's and a crop of sugar beets about two-thirds the amount expected. Industrial plants have fallen far short of targets. Coal production is considerably under the demand for it. The housing shortage is acute.

Communist Party chief Stanislaw Kania last week disclosed in a speech to steelworkers in Krakow that Poland would experience a drop in national income in 1980 for the second consecutive year -- and only the second time in the country's postwar history. About the only thing flourishing around Warsaw these days is the black market.

One paradox of the gains won by workers last summer is that they have aggravated the economic difficulties that sparked the strikes. The economic concessions made by the government will have the twin effect of reducing the work week and injecting the equivalent of about $4 billion of new purchasing power into the economy in the next year.

The short-term consequences of these concessions can only be greater shortages, higher prices and even greater discontent, unless accompanied by miraculous increases in productivity or massive increases in imports of goods and other consumer items from Poland's hard-pressed socialist allies, or additional credit from the West, to which Warsaw already owes $21 billion.

The Polish government has announced a sharp curtailment in industrial investment next year. Rationing of meat, fats and butter is expected to be introduced before the year's end.

Most of the goods on the market have been placed under strict price control. Improved taxation schemes and incentives to get people to save more are under consideration as ways of soaking up the excess cash resulting from recent wage increases.

But these short-term options are desperately limited.Under unprecedented pressure by the new union movement, Polish authorities have finally been pushed to consider basic long-term reform of the country's economic system.

The recent labor upheaval fully discredited the heavy borrowing, production-investment strategy that guided Poland for 10 years under former party chief Edward Gierek. But the outlines of a new general plan to replace it have only begun to take shape in the meetings of a special commission appointed for this purpose following the summer revolt.

Preliminary reports of the commission's deliberations suggest that the group -- a mix of 380 government, professional and worker representatives -- will recommend a program of changes likely to move Poland away from the Soviet model and toward the less centralized systems already practiced in the socialist states of Hungary and Yugoslavia.

The basic aim of the reform is said to be to dismantle much of Poland's stultified economic planning bureaucracy and give more responsibility and flexibility to individual business enterprises. Central authorities would then be left principally with responsibility for long-term economic strategy.

Specifics are still lacking and resistance to such a broadscale reform is inevitable from a deeply entrenched bureaucracy and ideological hard-liners.

But advocates of change feel they now have political momentum. "The summer events added something," said Jozef Pajestka, chairman of the Polish Economic Society and a member of Poland's Central Committee. "The reform has now become an absolute political and economic necessity."

The urgency to act is one factor that distinguishes the Polish situation from the Hungarian reform which led to the Soviet Bloc's most progressive economic system. In Hungary, the reforms had the luxury of careful planning and gradual introduction.

Proponents of change in Poland say there is little time to wait both economically, because of the severeness of the current situation, and politically, because the favorable climate for reform may disppear.

While representatives from Solidarity last week for the first time attended a meeting of the special commission on economic reform, union leaders say they do not have time now to consider the reform issue since they are preoccupied still with organizing.

For now, this quiet Polish revolution is being left to the authorities and economic professionals who nonetheless will have to answer again to Solidarity.