No, they borrow (uncollateralised) from depositors at 4% and borrow (collateralised) from the central bank at 1%.

They lend to each other, and to customers, at whatever the respective markets will bear.

And it's not my theory. It is a fact Competition for deposits in 2010-11 got insane and they were offering up to 4% for demand deposits (not time deposits!) which did squeeze margins as they were not increasing what they were charging for their lending in a commensurate way. And so the Central Bank got worried, especially as competition for deposits is a zero-sum game among banks and does not improve the overall health of the banking system.

However, what deposits (as liabilities) do is allow banks to book cash as assets. At a minimum they can park this cash as central bank reserves making a measly 0.25%. And, as assets, central bank reserves and cash don't count as risk weighted assets and therefore don't count in capital adequacy requirements (i.e., they don't imply regulatory or market needs for additional equity) whereas a loan which may collect 7% or upwards as interest income is a risky asset which does count as risk-weighted assets for capital adequacy purposes and does lead to additional equity requirements.

There are three stories about the euro crisis: the Republican story, the German story, and the truth. -- Paul Krugman