Most of the time, when an aircraft crashes or has an accident, the manufacturer also assists in the investigation. Is this a pro-bono kind of effort on part of the manufacturers, or do they charge the airline for it?
Investigating agencies are in charge of most air accident investigations not involving criminal acts. In the US this is the NTSB, which is required to investigate air accidents per US Code Title 49, Chapter 11. These agencies exist to provide independent investigation of accidents and provide recommendations to regulators.
The investigating agents will do their best to understand the circumstances and causes of an accident. Agents of the NTSB have authority under §1134 of the above mentioned law to gather information about an accident.
(a) Entry and Inspection.—An officer or employee of the National Transportation Safety Board—
(1) on display of appropriate credentials and written notice of inspection authority, may enter property where a transportation accident has occurred or wreckage from the accident is located and do anything necessary to conduct an investigation; and
(2) during reasonable hours, may inspect any record, process, control, or facility related to an accident investigation under this chapter.
Investigations can also benefit from working with people that are already familiar with the critical elements. As part of the investigation, the agency may invite various parties. This could include airlines, manufacturers, air traffic control, workers unions, other agencies, or anyone else with information or expertise that could be helpful. If these parties do not abide by terms for participation, this status can be revoked. The resources spent by parties during an investigation can in some cases be fairly intensive. For example, while supporting the investigation of the crash of British Airways Flight 38, Rolls-Royce and Boeing ran full-scale tests of the engine and the fuel system designs.
Investigating agencies are funded by their governments. The Aviation Safety department of the NTSB has an annual budget of about $50 million. Parties to the investigation represent their respective parties and are paid by their party to do so. Part of the party agreement acknowledges this:
Participation as a party to an NTSB investigation is a privilege and confers no rights or benefits.
They "pay" that parties get is good press for assisting in the investigation, and helping to both prove and improve the safety of their products and services.
Manufacturers are not necessarily involved in the investigations of an airplane crash while they may lend a hand they have no jurisdiction to cary out an investigation (generally speaking).
This depends on where the accident occurs as well as the aircraft involved. Aircraft Accident investigation is generally the responsibility of the nation and national organization that belongs to it, in which the accident occurs. However in practice not all nations have the facilities or resources to carry out an investigation completely so there is a lot of coordination. The NTSB will generally lend a hand if an American aircraft built or registered is involved or help is requested for something like a blackbox read.
If an accident or serious incident occurs in a foreign state involving a civil aircraft of U.S. Registry, a U.S. operator, or an aircraft of U.S. design or U.S. manufacture, where the foreign state is a signator to the ICAO Convention, that state is responsible for the investigation. In accord with the ICAO Annex 13 SARPS, upon receipt of a formal notification of the accident or serious incident that may involve significant issues, the NTSB may designate a U.S. Accredited Representative and appoint advisors to carry out the Obligations, receive the Entitlements, provide Consultation, and receive Safety Recommendations from the state of occurrence.
These agencies are government run/funded. The agencies can call on the manufacturer for advice/help such as requesting information on the aircraft or design. Although much of this data is submitted during the type certification process and already on hand.
Often times the manufacturer will offer to aid if at all possible. One thing to consider here is that his can be a conflict of interest. Most countries treat accident investigations as an open crime scene. Since the cause could potentially be a design flaw it is in the best interest of the investigating body to keep makers out of the investigation so as not to sway it.
Boeing does how ever have a team of investigators (that it presumably pays internally) that they send to accidents involving Boeing aircraft.
International protocol, defined by the International Civil Aviation Organization treaty, establishes that a government with jurisdiction over an accident site leads an investigation. Accredited representatives from the countries of the airplane’s manufacturer, designer, operator and registry are invited to join the investigative team. For international accidents involving Boeing airplanes, the accredited representative is the U.S. National Transportation Safety Board. The NTSB leads all domestic investigations. Boeing’s role is to support the NTSB as technical adviser.
In a legal sense the NTSB is actually Boeings seat at the international table when it comes to Investigations. While the NTSB can (and often might) call on Boeing directly they are ultimately responsible for the investigations as well as their cost.