Nested/Recursive Dynastic Utility Functions

Question

I want to find a way of representing a dynastic utility function in which not only the head of the dynasty's utility is dependent on its descendants' utility, but all members of the family tree gain utility from the utility of some of their offspring. In particular, I would like for the total utility of a person $U_{x_{i}}=u(x_i)+u(x_{i+1}) + u(x_{i+2})$ for all $i$ in the dynasty, where '$u(x_{i})$' denotes the utility of $x_i$ which is not dependant on the utility of children and grand-children ('non-parental utility'). So I want a model which allows us to say that the utility of the dynastic head is partly a function of the utility of his children and grandchildren's utility, but where his children's utility is again partly a function of his grandchildren's and great-grandchildren's utility, and so on.

In Becker and Barro's model, we have that \begin{equation} U_{x_0}=\sum_{i=0}^{n}N_i \beta_i(u(x_i)). \end{equation},

where $N$ is the number of descendants in a generation $i$ and $\beta$ is an intergenerational altruistic discount rate. The problem with this model is it lacks the 'nested' or 'recursive' feature explained above--it simply states that the dynastic head derives utility from his descendants' utility without taking into the consideration that his descendants utility might in turn depend on their descendants' utility.

Adding in the discounting element, let me give a concrete example: suppose I, as the dynastic head ($x_0$), directly receive 50% of the utility of all of my children (i=1) and 25% of the utility of my grandchildren (i=2). Let us say I do not directly receive any utility form the utility of my great-grandchildren. I will still indirectly receive utility from my great-grandchildren since my children receive 25% of their utility and my grandchildren receive 50% of their utility. If we assume for simplicity that my child had one child, which in turn had one child, which in turn had one child $u(x_3)=1$, then I indirectly gain $50\%*25\%*1 + 25\%*50\%*1$= 0.25 utils from my great-grandchild. How do I get a general expression for this phenomenon which holds for $n$ generations?

Any suggestions or pointers are hugely appreciated!

Answer 1

Let $\delta_i=N_i\beta_i$. I think what you want is for generation $i$'s utility to be something like: \begin{equation} U_i=u(x_i)+\delta_{i+1}U_{i+1}+\cdots+\delta_{i+n}U_{i+n}, \end{equation} where $i$ derives utility from his own consumption $u(x_i)$ as well as from his descendants' utilities, $U_{i+t}$ for $t=1,\dots,n$, over their own consumption and their descendants utilities. You can try to expand the above expression recursively to get a form that only involves $u$ and $\delta_i$'s. However, that's going to be messy, mainly because of the discount factors.

If you are willing to assume a constant discount factor, i.e. $\delta_{i+t}=\delta^t$, so that \begin{equation} U_i=u(x_i)+\delta U_{i+1}+\cdots+\delta^n U_{i+n}, \end{equation} then things become more manageable. In particular, with $n=2$ (i.e. one only cares about two generations down the line) and letting $v_i=u(x_i)$ for notational simplicity, you get \begin{align} U_i=v_i+\delta v_{i+1}+2\delta^2 v_{i+2}+3\delta^3v_{i+3}+5\delta^4v_{i+4}+8\delta^5v_5+\cdots = \sum_{t=0}^\infty F_{t+1}\delta^t v_{i+t}, \end{align} where the $F_{t+1}$'s are the Fibonacci numbers.

Answer 2

Here are two nice papers on recursive utility functions, a generalization of additive utility functions and compatible with "utility of the dynastic head to be partly a function of the utility of his children and grandchildren's utility, but where his children's utility is again partly a function of his grandchildren's and great-grandchildren's utility, and so on":

Blackorby, C., Nissen, D., Primont, D., & Russell, R. (1973). Consistent Intertemporal Decision Making. The Review of Economic Studies, 40(2), 239-248.
Blackorby, C., Nissen, D., Primont, D., & Russell, R. (1974). Recursively Decentralized Decision Making. Econometrica, 42(3), 487-496.