The "can't push on a string" metaphor appears in many intro-level economics texts. It has also gotten a heavy work-out these last few years as people have sought to understand why either economic output or inflation wasn't stimulated more greatly by having the Federal Reserve's target interest rate (the "federal funds" rate) near zero percent for going on seven years now, especially when combined with "forward guidance" promises that this policy would continue into the future and a couple trillion dollars of direct Federal Reserve purchases of Treasury debt and mortgage-backed securities.
The first use of "pushing on a string" in a monetary policy context may have occurred in hearings before House Committee on Banking and Currency on March 18, 1935, concerning the proposed Banking Act of 1935. Marriner Eccles, who was appointed Chairman of the Fed in 1934 and served on the Board of Governors until 1951, was taking questions from Rep. Thomas Alan Goldsborough (D-MD) and Prentiss M. Brown (D-MI). The hearings are here; the relevant exchange is on p. 377, during a discussion of what the Fed might be able to do to end deflation.
Governor Eccles: Under present circumstances there is very little, if anything, that can be done.Later in the hearings, several other speakers refer back to the "push on a string" comment, which clearly had some resonance. Although I have seen the "can't push on a string" metaphor attributed to John Maynard Keynes in a number of places, I haven't seen an actual primary source where Keynes used the phrase.
Mr. Goldsborough: You mean you cannot push a string.
Governor Eccles: That is a good way to put it, one cannot push a string. We are in the depths of a depression and, as I have said several times before this committee, beyond creating an easy money situation through reduction of discount rates and through the creation of excess reserves, there is very little, if anything that the reserve organization can do toward bringing about recovery. I believe that in a condition of great business activity that is developing to a point of credit inflation monetary action can very effectively curb undue expansion.
Mr. Brown: That is a case of pulling the string.
Governor Eccles: Yes. Through reduction of discount rates, making cheap money and creating excess reserves, there is also a possibility of stopping deflation, particularly if that power is used combined with this broadening of eligibility requirement.
For those who like digging into monetary policy metaphors, here's a post about the speech where William McChesney Martin coined the metaphor that the job of a central bank is, when the party is just starting to heat up, to take away the punch bowl.