In OLG Model 3,it’s only sj in calculating V(j) in the formula, while Params.mewj(jj)=Params.sj(jj-1)*Params.mewj(jj-1)/(1+Params.n) in the code. why? how to understand it?
Why does the conditional death probability sj of a representative household change with the population growth rate n? I think an existing representative family may gradually die out, but it should not change with the population growth rate?
Roughly:
The household problem is the perspective of one household, they care about s_j (which is the probability they survive/die). But they do not care about the population growth rate.
By constrast mewj relates to the entire population of agents. It is from the perpective of the whole economy/country. Hence we need sj and n.
PS. This is not a representative agent (there are no idiosyncratic shocks, but there is still age; you could think of it as representative conditional on age but I am not sure this is a useful way to think of it). In any case, rep agent or not, there is a continuum of mass one of households (so an infinite number of households), so it might help to think of this as ‘just one household’. [‘Complete markets’ is a more useful way to think of the model]
PPS. ‘mewj’ is the age-distribution of the population, and we could set it to anything we want (as long as they sum to one). This example assumes that the age-distribution should be that which will result from the sj and the n, if they lasted forever; it is the stationary age-distribution implied by sj and n.
Another way to think about it that might help. Everything the household does can be thought of as already being per-capita (per household, but in this model there is no distinction between people and households). So you are not going to divide things that are already per-capita by the population (and n is the population growth rate, so is closely related to population). Hence n is not going to appear in the household problem.
I am grateful for your response. I would like to ascertain whether the formula and the code are consistent. Is there a more optimal consistency option? I would like to enquire as to whether 1/(1+Params.n) shouldn’t be placed in mewj, or whether the relevant n variable should be added to Vj.
n should not be in V (it would be putting ‘population’ into the household problem which is already ‘per-capita’)
You can set mewj to anything, as long as they sum to 1. The code will solve just fine.
That code sets mewj to the population weights that would be implied as the stationary population arising from n and sj, with n being the growth rate of the population (or more accurately the growth rate of the population of age 1, but they are the same thing in this example due to considering a stationary population) and sj being the age-conditional survival probabilities. So in period t+1, mu_{1,t+1} (mass of age one in t+1), would be (1+n)mu_{t,1} (mass of age one at t, but grow by rate n). Since mu_{2,t} is the mass of age one from t-1 that have now become age 2, it follows that mu_{2,t}=m_{1,t}/(1+n). We then add that only some survive (namely fraction s_1; the conditional survival probability at age 1) and we now have mu_{2,t}=s_1*m_{1,t}/(1+n). This is formula mu_j=s_{j-1} mu_{j-1} /(1+n) which the population weights are iteratively calculated from in the code (and then normalized to one).
