Summer ends this week with the long-promised economic recovery still not underway. Chastened forecasters continue to maintain that the U.S. economy will soon begin climbing out of the black hole of recession into which it has tumbled, but they now believe the recovery from the worst recession in 50 years will be painfully slow.

President Reagan claimed recently, "The recovery has been sighted," speaking largely on the basis of four consecutive increases in the index of leading indicators, which often foreshadow changes in economic activity.

But all the latest figures -- for retail sales, industrial production, unemployment insurance claims, housing starts and business investment plans, among others -- indicate that August was a bummer. The earliest weekly numbers show no pickup for September, either.

As a result, the odds are high the nation's unemployment rate, which was 9.8 percent in July and August, will hit 10 percent for September.

On the other hand, interest rates did fall sharply in July and August, probably setting the stage for the beginning of the recovery in the fourth quarter, most forecasters say. Similarly, on the price side the news was encouraging, with inflation remaining modest and the rate of increase in wages continuing to slow.

Welcome as it was, the interest rate decline was more the result of the economy's continuing weakness drying up business loan demand than of any other factor, including Federal Reserve policy. And the sharp runup in initial claims for unemployment benefits since late July to the highest level of the recession is hardly what one would expect on the eve of a recovery.

However, the precise timing of a turn from recession to recovery is not nearly as important as the nature of the recovery once it begins. This one, after all, will begin with the economy extraordinarily depressed.

After more than three full years of economic ups and downs, the level of the nation's output of goods and services probably is no higher today than it was in the first quarter of 1979. A postwar record for idle production capacity was set last month.

Corporate profits are down sharply and bankruptcies are piling up. Many thrift institutions are in desperate straits and are being kept afloat only through mergers greased with federal aid and questionable accounting techniques. Some commercial banks are going under, too, often as a result of uncollectable loans, and some uninsured depositors have lost their money along with stockholders.

The output being lost in the recession will plague the economy for years to come, particularly that portion which would have been saved and invested. The personal saving rate has risen from about 6 percent to about 8 percent of disposable income in recent months. But that is a higher proportion of a figure much diminished by the recession.

Strains such as these argue for a fast enough recovery to put idle employes and machines back to work quickly and close the yawning gap between what is actually being produced and what might be produced if the economy were close to full employment.

Yet most forecasters expect that gap to be closed slowly over the next several years, particularly if the Federal Reserve, as it says it intends, continues to slow the rate of growth of the money supply to reduce inflation. Some economists believe inflation remains such a serious threat that any attempt to achieve something approaching a normal recovery would be unwise.

In any event, only a handful of economists, including some in the Reagan administration, expect the gross national product, after adjustment for inflation, to rise even by as much as 4 percent in 1983. The more common forecast is for growth in real GNP of between 3 percent and 3 1/2 percent.

That's distinctly slower than in most other postwar recoveries and could easily leave unemployment above 9 percent throughout all of next year. The forecasts that show larger drops in unemployment assume there will be little rebound in productivity in 1983.

One or two forecasters, such as H. Erich Heinemann and Charles Lieberman of Morgan Stanley & Co., say there will be a renewed recession after a brief revival this fall.

Townsend-Greenspan & Co., which is headed by economist Alan Greenspan, believes the recovery will be so sluggish -- with real output rising only 2.3 percent next year and 3.2 percent in 1984 -- that inflation will continue to drop. Greenspan expects consumer prices to go up 5.6 percent in 1983 and 4.5 percent in 1984. With such slow growth, unemployment averages 9.7 percent next year.

But Greenspan also has an "alternate scenario" in which interest rates stay high and the economy continues to decline until the middle of 1983. In that case, unemployment rises nearly to 11 percent in the second quarter. Inflation drops to 4.6 percent next year and 3.3 percent in 1984.

A striking thing about most of the forecasts at this point is that they show little further improvement in inflation. The predictions have inflation, as measured by the consumer price index, hovering around 6 percent.

Only a handful show either the improvement seen by Greenspan with his slow recovery or the kind of reacceleration of inflation the Reagan administration--pushed by the need to show higher federal revenues to hold down budget deficits--included in its midyear economic review. The administration predicted the consumer price index would rise 5.9 percent this year, jumping to 6.7 percent in 1983 and 6.9 percent in 1984.

The critical question for inflation, along with the pace of the recovery, is whether wage gains continue to grow at a lower rate. The change in hourly earnings over a 12-month period peaked at 9.9 percent in February 1981. In the last 12 months, the increase was only 6.5 percent as the recession pushed unemployment up and profits down, curbing workers' demands and their employers' ability to pay.

Most forecasters expect there will be little further downward movement in the rate of wage gains once the recovery is underway and the unemployment rate begins to fall. That could be a too pessimistic an assessment, however, for several reasons.

For one thing, in a number of major industries individual companies have fared so differently during the last three years that industry-wide bargaining has all but gone by the boards. Trucking and autos are two examples.

In addition, collective bargaining experts say that agreements at one firm or in an industry that used to set patterns for other firms or industries may not do so in the future. In many cases, weaker companies simply cannot afford to accept the same terms as stronger firms, and workers know it.

But if the majority of forecasters are correct, and inflation does not fall below roughly 6 percent in 1983, then the rate of increase in consumer prices would have been reduced by more than half from 1980's 13.5 percent rate.

However, in the opinion of many economists, the consumer price index overstated inflation in 1980 because of the large weight given to mortgage interest rates. (That portion of the index will be replaced in January by figures based on what it would cost to rent a house rather than buy it.)

Some estimates of the underlying rate of inflation, which seek to abstract from temporary price surges not built into wages, show a 1980 peak of between 9 percent and 9 1/2 percent. Such an estimate for that year by Data Resources, Inc., for example, was 9.3 percent, and DRI predicts it will be down to 6.6 percent in 1983--a much more modest improvement in inflation than implied by the consumer price index.

The improvement in inflation has, of course, been achieved at a high cost. The Federal Reserve has slowed the growth of the money supply over the last two years only by letting interest rates reach record levels. High rates initially clobbered the homebuilding industry and other interest-sensitive sectors of the economy. But they have stayed high so long--mortgage rates still are generally above 15 percent--that the damage in those industries has spilled throughout the economy.

There are several reasons that only a modest recovery now is expected.

First, the Federal Reserve is not seeking such a rapid recovery that the gains on the inflation front are jeopardized. The Fed fully intends to provide enough money that there will be a recovery, but even the upper limit of its present tentative money targets for next year probably is consistent with no more than a 9 percent increase in GNP, conceivably 10 percent. If inflation runs 6 percent or so, then there is room for only about a 3 percent to 4 percent increase in real output.

Second, the recession has lasted so long and has become so deep that many businesses will not have the resources to respond quickly to any increase in the demand for goods and services. In extreme cases, of course, the businesses no longer exist.

Third, there is so much idle productive capacity available, it will be many months if not years before there is a general move toward investment intended to expand that capacity. Most forecasters expect business investment in new plants and equipment to continue to decline until the middle of next year, and perhaps until the beginning of 1984.

Fourth, given the Fed's likely policy, and the need for the federal government to borrow unprecedented amounts of money to cover equally unprecedented budget deficits, interest rates will stay high enough that there will be no housing boom, and very likely no big rebound for auto sales, either.

Finally, given the cuts in federal spending programs and the recession's impact on state and local government revenues, there will be no surge in government spending at any level -- other than for national defense.

The key difference this time around will be that monetary policy is not geared to promote a swift recovery. Thus, the massive federal deficits will continue to add to total demand even as monetary policy seeks to dampen the total rise in national income.

As private sector credit demands weakened this summer as the recession dragged on and on, interest rates did fall. The Federal Reserve, seriously worried about spreading business bankruptcies, including those of financial institutions, moved as aggressively as possible to add funds to the banking system whenever it could do so without throwing over its basic monetary targets.

Most analysts now think the drop in rates has run its course, and that they could rise again at the first clear signs of a recovery. At the moment, the question for the Fed is what it will do if there are no such signs and money growth gets well above target once more. In the short run, the higher money growth likely will be tolerated, but it should be a bar to any further easing of policy.

The nation has paid an enormous price in lost output and lost jobs to reduce inflation. Even if the recovery does get underway before the year is out, the total price still will be mounting at a rapid clip. There is a long way to go before the U.S. economy can climb out of its hole.