Why did the East German government decide that they should join the West? Was it their economical decline, political turmoil or anything else?
Well, I would say that the main reason for this reunification (apart from the 40 years of Cold War, Ostpolitik, "Star Wars", Oil price boom etc) was that the Bundesrepublik Deutschland (German Federal Republic, a.k.a. West Germany) had written in its Constitutional Law (Germany has no Constitution) that it understood itself as the Republic of the German People.
So it was written in the constitutional law that any of the former Länder of Germany (Mecklenburg Vorpommern, Brandenburg, Saxe, Saxe-Anhalt et Thuringe) could apply anytime to membership in the German Federal Republic (even if these Länder had been replaced by the Bezirke in the German Democratic Republic in 1952).
That's why it was so easy to achieve reunification after the Wall fell.
I'll look at the economics.
Despite its history of being a Soviet economic powerhouse, the economic state of the DDR was poor. The East German mark was performing badly even in the 1980s. East Germany had a GDP per capita of 6064 DM compared to West Germany's 19 864 DM, p131. Clearly the subsidy the DDR could expect as well as the competent administration they would receive would have been seen as hugely beneficial,as it turned out to be. So East Germany needed needed West Germany's help, but why did West Germany need East Germany? As you can see from the link at the top of the page it wasn't because they expected it to deliver short term benefits to the West Germany economy.
In that case it seems clear that the benefits to West Germany were instead political benefits, both at home and in terms of international prestige, as mentioned in the answer above. In addition, West Germany had been incorporated into NATO in 1955, and it was probably strategically desirable for its members that East Germany be incorporated into a stable, friendly NATO power (although perhaps this was balanced out be fear of German militarism) (speculative as hell).
I've noticed Cyprus has similar issues. Turkey has changed its stance to largely pro-reunification (the analogue of the USSR losing influence), the two sides have few cultural differences, the militants have grown old, etc. The opposition for reunification now comes from the (Greek) Republic of Cyprus. It would seem prosperity makes all the difference when it comes to public support for reunification in the "senior" partner.
The question could be is there any reason not to reunite GDR and FRG? These are trivial reasons why they united:
You can watch many documentaries even on youtube, the communist party power was only supported by military and police, the politicians were all old and clueless what to do, they didn't even follow what happened in USSR where Gorbachev started reforms.