Just Amazon’s or everyone’s? The mandate to deliver to everyone everywhere can’t be cheap. You have to wonder if they ever should have privatized them at all since they’re in a kind of limbo.
There are already private options, USPS just isn't correctly funded and government keeps trying to gut it. The requirement to pre-fund pensions and retiree healthcare so far in the future doesn't allow them to invest money into new research and development.
They are likely getting around 50 cents a pound (based on FBA inbound averages that they charge sellers) from UPS and FEDEX, which means in cases where they are shipping via USPS, they are getting an even lower rate.
Try printing your own label and ship a 2 lbs package an hour away for $1.