This is a brief note about setting up warm-glow of bequests in an life-cycle model.
First, what will the warm-glow of bequests do? Mostly, it is going to increase the asset holdings of the elderly. Without a warm-glow agents typically aim to run assets down to zero in the final period, with a warm-glow they will no longer do so. Since in real-world data there is little to no running down of assets by the elderly this provides an easy way to hit this empirical fact. (Two main alternatives. First is modelling something like health expenses or nursing home care that comes near end of life. Second is more explicit modelling of bequests, such as caring about ‘child’s’ utility but this at a minimum means solving a dynastic OLG and can even mean dynamic games, so is a substantial complication to the model.)
Once we decide to implement warm-glow of bequests there are essentially two decisions: when you get the warm-glow, and what functional form to use?
When you get the warm-glow? There are two standard options. The first is that you get the warm-glow at the end of the terminal period (so period J+1). The second is that you get a warm-glow with probability 1-s_j each period, where s_j is the (age-dependent) conditional survival probability.
What functional form to use for warm-glow? There are again two standard options. The first is to use the same utility function as elsewhere; so if you utility of consumption is \frac{c^{1-\gamma}}{1-\gamma} then you can use \phi \frac{aprime^{1-\gamma}}{1-\gamma} as the warm-glow, where \phi is the relative importance of bequests and aprime is the assets you leave behind on death. The second is to use a functional form with a ‘target level’ of bequests, such as \phi \frac{(aprime-atarget)^{1-\gamma}}{1-\gamma}, where atarget is the target level.
Note that these two choices do somewhat interact. In particular if you use warm-glow based on the probability of dying (on 1-s_j) and a ‘target level’ of bequests, then you need to be careful as this target now applies at every age in which there is a non-zero probability of dying and so can influence asset holdings even at young ages pulling them towards the target level (a cheap solution/trick is only to give warm-glow of bequests on dying if you are, e.g., over 70yrs old).
Lastly, note that if you use \phi \frac{aprime^{1-\gamma}}{1-\gamma} then \phi is monotone in the strength of the bequest motive, but not linear. To see this think about first order conditions (aprime is being curved by -\gamma, but phi is not). If you instead use \phi^{\gamma} \frac{aprime^{1-\gamma}}{1-\gamma} then you have a linear effect from changing \phi, but this is unlikely to be important nor useful. Just be aware that doubling \phi is typically not going to result in doubling the bequest size.
Obviously in an OLG model warm-glow bequest will also importantly influence the size of the bequests being received.