I have estimated the values of consumption equivalence for the 2 regimes as well as the values from the welfare equations, however i am not sure whether i need to compare the values of consumption equivalent or welfare values and determine the optimal policy.
I read in previous posts that the change of sign matters, but for example in my case both regimes have negative sign for consumption equivalent. Which is the best policy, the one with the lower negative value?
Morever, does the values of welfare or consumption equivalence have an interpretetion on their own?
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
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.
I am not aware of such papers, so I do not know the reason why they provided such and expression.
But start from the condition that \xi must satisfy: E[W^{p}_t(\xi)]=E[W^{ap}_t]. With log utility in consumption, you will have that E[W^{p}_t(\xi)]=\frac{1}{1-\beta}ln(1+\xi)+E[W^{p}_t].
Hence, he condition that \xi must satisfy is \frac{1}{1-\beta}ln(1+\xi)+E[W^{p}_t]=E[W^{ap}_t]
Re-arrange to have ln(1+\xi)=(1-\beta)(E[W^{ap}_t]-E[W^{p}_t])
The base of the natural log is e, so 1+\xi=exp((1-\beta)(E[W^{ap}_t]-E[W^{p}_t]))
And finally \xi=exp((1-\beta)(E[W^{ap}_t]-E[W^{p}_t]))-1
Hopefully having broken down the expression of the CE in such fashion will make it easier for you to see what is going on in those papers
@cmarch thank you very much for your help.
it is really illustrative and hopefully i will manage to solve it.
However, i would like to ask one more thing. In one of the simulations i make i get the value of consumption equivalence i.e 178.43. As discused above shouldn’t the value of ξ be less than 1 in order to multiply it with 100 and have the interpretation you where saying? Because i suppose that the 178.43 value do not have an interpretation.
I have calculated the Consumption Equivalent for different policies and i get i.e - 0,9810 as a value under a certain policy. Does this mean that the agent is willing to return in the benchmark economy p and will give up to 98, 2% of his consumption or give up (1-98,1%)=1,9% to return in the benchmark economy p vs the alternative economy?
How can I obtain a similar formula for \xi for models where the utility is not in logs and has leisure (or more complicated, where utility is not separable over time or arguments)?
For example, how to do this in the slightly more complicated case where U_t = \frac{C_t^{1-\sigma}}{1-\sigma} - \nu\frac{H_t^{1+\psi}}{1+\psi}.
The problem, in this case, is that the separation between (1+\xi) and the rest of the utility is not straightforward as in logs and hence it is not as easy to set everything in terms of E[W_1] - E[W_0] where the subindexes in welfare denote different models.
Another alternative, used by Paul Levine is to normalize the unconditional welfare wedge between 2 models by the increase in welfare obtained from a 1% increase in consumption. Then the utility units cancel and you get the Consumption Equiv. Variation too.
Hi Reuben, unfortunately, I haven’t been able to find the citation myself.
All I have are study notes from a course on DSGE designed by UofSurrey (P. Levine is one of the authors). That is where I took the screenshot from (not much more info than what I showed here is included there). You may want to use the image and ask yourself to the CIMS team (https://www.surrey.ac.uk/centre-international-macroeconomic-studies).
An alternative is to do it numerically as Johannes suggests above. The result should be similar (in an exercise with logs, I compared the analytical method (the first one mentioned in this post), the numerical method, and the normalization method (from screenshot)) to compare a battery of models, the results are very similar with each.
I realize that your answer is quite old, yet I have a question since I am currently trying to calculate the CEV coefficient.
you used a log utility, if one would use a CRRA utility u(c_t) = \frac{c_t^{1-\sigma}-1}{1-\sigma} where the coefficient is not equal to one. Would this be the correct expression? E[W_t^p(\xi)]= (1+ \xi)^{1-\sigma} * E[W_t^p]
My confusion stems from the \frac{1}{1-\beta} but as I understood it that’s specific to the log utility.
Thank you.
Best,
Fabian
That is the fundamental problem. The 1/(1-\beta) comes from the \sum_t^\infty in welfare. But only with log utility everything becomes additively separable. For more general utility functions, I recommend approaching it numerically. The only exception is when talking about the steady state welfare as the elements within the infinite sum are constant.
@jpfeifer Thanks . I spent quite some time reading through but I am still not sure. Do you know any material that will help me compute CE Welfare gain or Loss for non-log utility functions?
