# RDP 2019-05: Cost-benefit Analysis of Leaning against the Wind Appendix A: Decomposition of Dynamic Estimates

This appendix provides details of how the change in social loss is decomposed into separate elements interpretable as benefits and costs. We assume the change in welfare can be represented by the following quadratic loss function.

(A1) $L t =( 1− p t ) ( u t ) 2 + p t ( c t ) 2 Δ L t = L t law − L t base$

where L is expected loss, $\text{Δ}L$ is the expected welfare change as a result of leaning against the wind, u is the expected change in the non-crisis unemployment gap, c is the expected unemployment gap if a crisis occurs, and p is the probability of being in a crisis. The superscripts base and law represent the baseline and leaning against the wind scenarios, respectively. In other words, expected loss is a probability-weighted average of the crisis and non-crisis losses.

To simplify, assume that the non-crisis unemployment gap is zero in the baseline scenario, that is, ubase = 0. Then add and subtract ${p}_{t}^{base}{\left({u}_{t}^{law}\right)}^{\text{2}}+{p}_{t}^{base}{\left({c}_{t}^{law}\right)}^{\text{2}}$ and rearrange:

(A2) $Δ L t =( 1− p t base ) ( u t law ) 2 −( p t base − p t law )[ ( c t law ) 2 − ( u t law ) 2 ]+ p t base [ ( c t law ) 2 − ( c base ) 2 ]$

Crises under leaning against the wind $\left({c}_{t}^{law}\right)$ are variable, so have a time subscript, whereas crises in the baseline are not. Equation (A2) can be rewritten in terms of costs and benefits.

(A3) $Δ L t =COS T t −BENEFI T t +OTHE R t$

where:

(A4) $COS T t =( 1− p t base ) ( u t law ) 2$
(A5) $BENEFI T t =( p t base − p t law )[ ( c t law ) 2 − ( u t law ) 2 ]$
(A6) $OTHE R t = p t base [ ( c t law ) 2 − ( c base ) 2 ]$

In addition to weighting factors, the expression COST represents the unemployment gap arising from leaning against the wind, BENEFIT represents the reduced probability of a crisis multiplied by the cost of the crisis (abstracting from the effect of policy on the crisis severity), and OTHER represents the difference in the size of a crisis that might arise under the two policies (this could be classified as either a cost or a benefit, depending on whether ${c}_{t}^{law}$ is greater or less than cbase).

In Section 3, we assume that ${c}_{t}^{law}={c}^{base}$, so that the unemployment gap in a crisis is the same regardless of previous policy. Then the term OTHER becomes zero and welfare loss is expressible as a simple difference between COST and BENEFIT, as shown in Figure 3. In Section 5.2 we relax this assumption. Implications and details are discussed in Appendix B.

Equations (A4) and (A5) are used to estimate the flow of costs and benefits. To make an assessment of whether leaning against the wind is desirable, we compare the cumulative sum of costs and benefits over the nine years following the initial change in interest rates.