Hi mikegouv,

Let superscript p denote the benchmark policy and ap denote the alternative state of policy, and E be the unconditional mathematical expectation sign and assume log utility.

Formally, \xi must satisfy

E\left\lbrace \sum_{i=0}^{\infty}\beta^{i}U\left(C_{t+i}^{p}\left(1+\xi\right),\cdot\right)\right\rbrace
\equiv E\left[W_{t}^{p}\left(\xi\right)\right]=E\left[W_{t}^{ap}\right]\equiv E \left\lbrace \sum_{i=0}^{\infty}\beta^{i}U\left(C_{t+i}^{ap},\cdot\right)\right\rbrace

Solving for \xi yields

\xi=\exp\left(\left(1-\beta\right)\left(E \left[W_{t}^{ap}\right]-E\left[W_{t}^{p}\right]\right)\right)-1

In order to derive a meaningful interpretation of welfare improvement, gains and losses of the agent are expressed in terms of consumption equivalent (CE) variation, that is, the maximum fraction of consumption \xi that the agent would be willing to forgo in an economy p to join the economy in which ap is active. Or, differently worded, the amount of consumption the agent would require to be indifferent between staying in the economy p and joining the economy ap.

Imagine the alternative economy is the one with highest welfare. If this is the case, you will have a positive CE. The implication is that you would require \xi \cdot 100 percent of consumption each period to be willing to remain in the p economy.

Imagine the benchmark economy is the one with highest welfare. If this is the case, you will have a negative CE. The implication is that you would be willing to give up \xi \cdot 100 percent of consumption each period to remain in the p economy.