The paper is about a generalization of the EGM to more than one dimensions. Besides this, the model used in the paper is relatively simple and I think it could be solved rather easily with the VFI toolkit, given today’s computing power on a GPU.
So I wanted to double check if indeed the VFI toolkit can handle the model in the paper
I can summarize here the relevant equations.
There are two state variables, financial assets a and human capital h, and of course age j. There is no shock to earnings, only mortality risk. The choice variables are consumption c and investment in human capital i. The Bellman equation is
where f(i) is a strictly increasing, strinctly concave production function for human capital and prob_surv(h) is the per-period survival probability that is increasing in h. Note the h can be interpreted as human capital or health capital.
So the model is pretty simple, the only difficulty is that there are two continuous state variables and two choice variables, and h’ requires interpolation.
VFI Toolkit would easily solve the finite horizon version of this model (using vfoptions.experienceasset; see the Attanasio, Low & Sanchez-Marcos 2008 codes for example). But I have not implemented experienceasset for infinite horizon so currently this is not something the toolkit can solve.
They say they solve a finite horizon version with 100 periods, but I cannot see anywhere what they set for the terminal period? (V=0?)
Actually there is a more problematic issue. Specifically, the discount factor depends on human capital. VFI Toolkit cannot presently handle this. [Eqn is in in middle of Section 2.1.2 on pg 469]
Dear Aledinola, I came across your question while browsing the forum and found this paper interesting as well. I’m also curious—have the authors made their code publicly available? Additionally, regarding the example model you summarized here, do you know where I can find a complete solution code?
Thanks for your question. The Ludwig and Schon paper is quite interesting. It is published on Computational Economics, so I would start looking there for the replication code.
@robertdkirkby It seems that the toolkit cannot replicate this particular model, right?
More in general, there is a literature on Bewley model with health risk and health investment that is very interesting, see e.g. this paper by Jung and Tran:
This is a quantitative life-cycle model with income and health risk (uninsurable) used to study optimal tax progressivity. I am pretty sure that the toolkit can handle this because I am working on a similar model myself using the toolkit and so far I am happy about the toolkit’s performance
You will find in this forum that one of my posts is about this “health risk” lifecycle model. Robert was kind enough to explain how to handle this type of models with the toolkit. There is a slight complication with the discount factor/probability of survival but it can be dealt with.
Ignoring the part about the discount factor depending on human capital, and switching to finite instead of infinite horizon, the model essentially becomes the same core setup as used in Life Cycle Model 42 (in the Intro to Life Cycle Models).
Beyond this the toolkit cannot presently solve the Ludwig & Schon (2018) model. [It cannot do the experienceasset in infinite horizon, and cannot do that the discount factor is a function of state h.]
Link to health risk post Alessandro mentioned: Age-dependent health shocks
As he said, it is a loosely similar setup, and fits with what toolkit can do.
Hi Aledinola，thanks for the helpful pointers! I’ll check Computational Economics for the code of the Ludwig and Schön paper. The literature on Bewley models with health risk and health investment is indeed fascinating. The Jung and Tran paper looks particularly relevant. I’ll also take a look at your post on the “health risk” lifecycle model. Thanks again.
Thanks, Robert! That clarification is really helpful. I’ll take a closer look at Life Cycle Model 42 and the health risk post Alessandro mentioned. It’s good to know where the toolkit’s limitations are. Appreciate your insights!
A comment that might help clarify the difference between Ludwig & Schon (LS) vs Jung & Tran (JT) (from a purely mathematical/computational perspective).
LS is infinite horizon, while JT is finite horizon
Both have a stochastic discount factor.
LS has health as ‘human capital’, which in VFI Toolkit is done with experienceasset [which is when aprime(d,a), which in the LS notation is that h’(i,h)]. Note that there are no shocks to h, you have total deterministic control over it’s evolution.
JT has health as a state that transitions with semi-exogenous probabilities (you can influence the probabilities with a decision variable). In VFI Toolkit this is done with a semi-exogenous state.
VFI Toolkit does not have a way to directly handle stochastic discount factors. But in a semi-exogenous state you can include a ‘dead’ state, and transitions to this state can be used to implement the stochastic discount factor. It is slightly computationally wasteful to do things this way (why are you keeping track of all the ‘dead’ people and what they are up to?), but it typically works fine in practice. The main downside is just that you have to be careful when computing model statistics that you don’t include all the dead people (you can always use simoptions.conditionalrestrictions to just report all statistics conditional on being alive).
Note that in the LS setup, you can transition from alive to dead by chance (by the prob_surv(h)), but you cannot transition from health to unhealthy by chance (you have total control via h’=(1-delta)(h+f(i)) and choosing i). By contrast, in the JT setup, you can still transition from alive to dead by chance, but you can now also transition from healthy to unhealthy by chance. To my mind this makes JT the better setup—this comparison should of course come down to which is more empirically plausible, but my strong prior is that JT will be as I suspect you need the ability to go from healthy to unhealthy by chance.
experienceasset is demonstrated in Life-Cycle Model 42, where it is used to model human capital. semi-exogenous state is demonstrated in Life-Cycle Model 29, where it is used to model endogenous fertility/children.
Thanks for further comments, Robert. So this means that when studying the impact of negative health shock, JT’s exogenous health framework is kind of better than LS’s endogenous health framework. Of course, this answer is not absolute; it depends on the specific research question.
Yes, JT research question is: by how much would optimal tax progressivity change once you take into account health risk? It turns out that it is also very important to model public health insurance in a realistoc way. In the US they have low public health insurance, which means that you need a lot of redistribution via taxes and transfers.
JT do a counterfactual in which they introduce Medicare for all and find that in this case the desired progressivity of taxes is even lower than in the benchmark
I think what they miss is that health risk is fully exogenous, so they cannot study how public policies distort investment in health
Just to mention, there is probably also a third option with VFI Toolkit. You could model health as experienceassetu (note the u). This would allow you to have the mix of decisions-shocks combining to change health like in the semi-exogenous state setup (you have some control, e.g., quit smoking, but some is largely out of your control, e.g., car accident).
So what are the differences between modelling health as experienceassetu and semi-exogenous shock, since both have health evolve by a mix of decisions and shocks. The experienceassetu is better if health is continuous (say a health index from 0 to 1), and semi-exo is better if it is discrete (say there are just two states, health and unhealthy). My feeling is that setting up the semi-exo is probably also a bit easier from users perspective.
