No effect. But as everyone knows from undergrad economics, the answer is always "it depends."
Generally, FP compensation depends on your negotiated contract (salary vs. "eat-what-you-kill"). Salaried contracts are more "sticky" while production contracts will vary depending on your business. Secondly, it'll depend on your payer mix: government vs. commercial insurance vs. self/cash pay. Thirdly, it'll depend on your payer contract, i.e. discounted-fee-for-service (PPO) or capitation (which are rare in some practice areas).
If the economy stagflates, self/cash pay FP's will take a hit. These are your community health centers (like some FP residency programs), your walk-in clinics, your urgent cares as well as your luxury clinics like concierge medicine, botox, and med-spas. Luxuries will take a hit, but the CHC's and walk-ins can go either way.
If the economy stagflates, the unemployment will push more people out of employer-based insurance into the government and self-pay category. Even if you got tax credits to buy your own insurance outside of employment, there's no risk-pooling or "community-rating" to make insurance cheaper. Healthy people will opt out of buying insurance, thus increasing insurance for everyone else.
Commercial insurance follows Medicare anyways. Expect Medicare to grow. Expect Medicaid to grow. And remember both Medicare/Medicaid compete with Afghanistan and Iraq for Congressional appropriations.
If the economy contracts, everyone will suffer, although doctors will suffer less (as usual). The Treasury bailout hopefully extend credit to businesses (large/small) so that unemployment won't be so sharp. The Fed's signaling a rate cut, which should help with some of that liquidity. But both are incredibly pro-inflationary in the long run.
So... I expect some FP's will do well, some worse. Average effect is no change. Long run, I think it'll appear as if salaries are going up... but because of inflation, it may not mean anything.
(But to answer your question, technically, Presidents have no control over the economy. I don't think it matters who's president. It depends on Congress, which controls fiscal policy. The Fed Reserve can lessen the short-term blow by controlling the monetary policy, but it'll depend on how the WORLD market receives it.)