32  An expression equivalent to the IS curve above can then be used to define  the natural interest rate, 𝑟𝑡∗ , which is the interest rate that would prevail in the absence of any nominal rigidities: ∗ 𝑦𝑡∗ = 𝐸𝑡 𝑦𝑡+1 −  1 ∗ 𝑟 𝜎 𝑡  Using this, the IS curve can be re-written in output gap form, using the output gap definition 𝑥𝑡 ≡ 𝑦𝑡 − 𝑦𝑡∗ and the definition of the natural interest rate to give:  (2)  1 𝜎  𝑥𝑡 = 𝐸𝑡 𝑥𝑡+1 − [(𝑖𝑡 − 𝐸𝑡 𝜋𝑡+1 ) − 𝑟𝑡∗ ]  31  In other words, the simple model here abstracts from all channels of monetary transmission other than the inter-temporal substitution channel. Of course, building in addition frictions and model features, like borrowing constraints and open economy features, can restore the roles of the other channels. 32 In particular, in the simple model, one can show that 𝑦𝑡∗ = 𝜓𝑎𝑡 , where 𝑎𝑡 is the log-level of TFP, assumed to follow an exogenous process. 18  All speeches are available online at www.bankofengland.co.uk/publications/Pages/speeches  18  This says that the output gap is negatively related to the sequence of “real rate gaps” (𝑖𝑡 − 𝐸𝑡 𝜋𝑡+1 ) − 𝑟𝑡∗ delivered by monetary policy. For example, when monetary policy delivers real interest rates equal to the natural rate, so 𝑖𝑡 − 𝐸𝑡 𝜋𝑡+1 = 𝑟𝑡∗ , the output gap is zero at all times.
In other words, some trade-off between inflation and output has become possible and lambda has become relevant. As I mentioned earlier, the period of inflation targeting can be divided into pre- and post-crisis periods. During the Great Moderation, the economy’s supply capacity grew at a steady pace, and, for the most part, 15  In New Keynesian models, the slope of the Phillips Curve is related to the deep parameters governing the frequency with which firms are assumed to be able to re-optimise their prices, following the formalisation of Calvo (1983). See Calvo, G (1983), “Staggered prices in a utility-maximising framework”, Journal of Monetary Economics, 12. 9  All speeches are available online at www.bankofengland.co.uk/publications/Pages/speeches  9  deviations of inflation from target reflected temporary disturbances to demand that were expected to drive output and inflation in the same direction (Chart 1). 16  Generally, monetary policy faced few trade-offs; divine  coincidence largely ruled. The crisis brought shocks both to demand and supply, higher inflation and large shortfalls in output and employment. The era of divine coincidence was over and the MPC had to balance a series of challenging trade-offs (Chart 2). What can be learned about lambda from this era? Taking these trade-offs at face value suggests a value of lambda of between 0.1 and 0.2 for the inflation targeting period as a whole, on the basis of data outturns 17  (Chart 6) and on the basis of MPC forecasts (Chart 7).