Question

who is more important in influencing americas economy ... the president or the head chairman of the federal reserve board & why?

Answer 1

It shall be noted that head chairman of Federal Reserve Board is the most important one influencing American economy vis-a-vis US President.

US President does not control the economy. Presidential economic records are highly dependent on the dumb luck of where the nation is in the economic cycle. And the White House has no control over the demographic and technological forces that influence the economy.

The chairman of the Federal Reserve Board is the public face of the Federal Reserve Bank. Officially, it is the active executive officer of the Federal Reserve Board. The main responsibility of the chairman is to promote maximum employment, stable prices, and moderate long-term interest rates.

The chairman of  Federal Reserve Board is also the chairman of the Federal Open Market Committee and is responsible for determining short-term US monetary policy.

During normal economic situations when mild fluctuations ripple around the economy, the task of keeping a stable growth path lies with the Chairman of Federal Reserves. The chair of the Fed, who’s chosen by the president, has a large impact on how monetary policy is conducted.With members of the board of governors (which includes the chair), who are also appointed by the president, they can set the policy agenda. The US President has limited role here.
In severe crisis, President has a large impact on fiscal policy, but it alone can’t determine fiscal policy. That requires the cooperation of Congress. It is rightly said, that US President does not control the economy

Thus, overall, Chairman of Federal Reserves is important figure influencing American Economy