Modern teaching of modern macroeconomics and modern monetary theory should reflect modern monetary policy -- what modern central banks actually do nowadays. That means the modern LM curve is vertical.
Modern central banks target inflation. More precisely, modern central banks target their own internal forecast of the near-future inflation rate. That means the modern LM curve is vertical, at that level of real GDP which the modern central bank believes is consistent with inflation remaining on target. Call it "estimated potential output", or Ype for short. Modern central banks make the money supply and interest rates adjust endogenously to keep money supply = money demand where Y=Ype. Any shift in the IS curve (or BP curve) will have no effect on Y (unless it also caused Ype to change), because the modern central bank targets inflation, and because the modern central bank believes that any change in Y, relative to Ype, would cause inflation to deviate from target.
Teaching an upward-sloping LM curve, which holds the money supply fixed, can at least be defended as a way to teach economic history. Or as a "what-if?" thought-experiment. Because targeting the money supply is a feasible monetary policy.
Teaching an upward-sloping LM curve could also be defended by assuming the central bank follows a Taylor Rule. (The slope of the LM curve would be the coefficient on Y in the Taylor Rule).
But inflation forecast targeting modern central banks do not follow Taylor Rules exactly. For example, if a central bank sees a loosening of fiscal policy, and believes it will cause the IS to shift up, and raise the natural rate of interest, the modern central bank will raise the market rate of interest accordingly, to prevent any increase of Y above Ype.
Teaching a horizontal LM curve, which holds the interest rate fixed, is indefensible. No central bank has ever targeted the rate of interest. And if a central bank did try to target a rate of interest, modern macroeconomic theory tells us it would fail. (Yes, I know the Bank of Canada targets a rate of interest, but only for 6 week periods, and 6 weeks is far too short a time for shifts in the IS curve or BP curve to have any significant effect on Y. And in those rare cases where the Bank of Canada thinks it would have an effect, it breaks its own 6-week rule and changes the overnight rate target between Fixed Announcement Dates.)
This post is a response to two more good posts, by Menzie Chinn and Simon Wren-Lewis, on teaching Mundell-Fleming. I apologise for all the "modern" stuff, where I was hamming it up a bit. And sorry for recycling this old post. And sorry it's so short; I gotta go teach my ECON1000 students on whether the national debt is a burden on future generations. 99% of them think it is a burden. First I've got to teach them that we owe it to ourselves, then I've got to teach them that's wrong too. Pray for me.