The manner in which the MPC optimises these trade-offs can be represented in “linear-quadratic” form – a set of linear constraints describing the behaviour of the economy, and quadratic preferences that penalise deviations of inflation from its target and output from its potential. The relative weight the policymaker places on output stabilisation, relative to inflation stabilisation, is often denoted 𝜆 – or ‘lambda’. The policymaker’s objective function can be described as the following “loss function”: 𝐻 ∗ )2 ]} min ℒ𝑡 ≡ 𝐸𝑡 {∑ 𝛽 𝑖 [(𝜋𝑡+𝑖 − 𝜋 ∗ )2 + λ(𝑦𝑡+𝑖 − 𝑦𝑡+𝑖  {𝑟𝑡+𝑖 }𝑇 𝑖=0  𝑖=𝑜  where 𝜋𝑡 is inflation, 𝜋 ∗ is the inflation target, 𝑦𝑡 is output and 𝑦𝑡∗ is ‘trend’ output, so that 𝑦𝑡 − 𝑦𝑡∗ is the output gap, 𝑟𝑡 is the nominal interest rate – the policy instrument, 𝛽 is the discount factor and 𝑡 subscripts denote time periods. 𝐻 is 3-5 years after which the monetary policy transmission mechanism fades. The policymaker’s objective is to minimise the discounted sum of losses over time. In this formulation, a lambda of zero would imply no weight on the stabilisation of real activity – so-called “inflation nutter” preferences. When lambda is positive, the policymaker is willing to strike some trade-off between output and inflation stabilisation, as directed by the MPC remit. This objective function is optimised subject to the constraints implied by the aggregate behaviour of the economy, including the relationship between interest rates and activity; and the relationship between activity and inflation – the Phillips curve.
On balance, the Bank’s analysis and empirical studies suggest there are no strong reasons to think that the housing tools have any significant negative long-run impact on the growth of supply capacity. The housing tools will bind more when housing activity is stronger and lenders are loosening underwriting standards, increasing both the gross benefits (i.e. reducing GDP-at-risk in a downturn) and gross costs (i.e. reducing demand by excluding some borrowers from the market). To test if net benefits are positive across different states of the world the FPC reviews central, upside and boom scenarios.20 In each scenario we first estimate the distribution of borrower indebtedness absent the tools (shown in blue shading, Chart 13, for the boom scenario). This includes an assumption about how lenders’ underwriting standards would have evolved without the tools in place. A second simulation then estimates how the distribution of indebtedness evolves differently with the tools in place (shown in pink shading, Chart 13, for the boom scenario). The gross benefits of the tools arise from a reduction in GDP-at-risk associated with the less risky debt distribution with the tools in place, compared to the risker scenario without the tools. To weigh these benefits off against the costs (the temporary reduction in housing activity) requires assumptions about the persistence of each, society’s discount rate, the probability of a downturn occurring – as the benefits are only realised in a downturn, and the preferences of the policymaker (as encapsulated in parameter 𝜙).