<p>Here are two nice papers on recursive utility functions, a generalization of additive utility functions and compatible with &quot;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&quot;:</p>
<p>Blackorby, C., Nissen, D., Primont, D., &amp; Russell, R. (1973). Consistent Intertemporal Decision Making. The Review of Economic Studies, 40(2), 239-248.<br />
Blackorby, C., Nissen, D., Primont, D., &amp; Russell, R. (1974). Recursively Decentralized Decision Making. Econometrica, 42(3), 487-496.</p>

CC BY-SA 4.0
2021-02-16T16:10:24.837Z
2021-02-16T16:10:24.837Z
Bertrand
<p>Let <span class="math-container">$\delta_i=N_i\beta_i$</span>. I think what you want is for generation <span class="math-container">$i$</span>'s utility to be something like:
<span class="math-container">\begin{equation}
U_i=u(x_i)+\delta_{i+1}U_{i+1}+\cdots+\delta_{i+n}U_{i+n},
\end{equation}</span>
where <span class="math-container">$i$</span> derives utility from his own consumption <span class="math-container">$u(x_i)$</span> as well as from his descendants' utilities, <span class="math-container">$U_{i+t}$</span> for <span class="math-container">$t=1,\dots,n$</span>, over their own consumption and their descendants utilities. You can try to expand the above expression recursively to get a form that only involves <span class="math-container">$u$</span> and <span class="math-container">$\delta_i$</span>'s. However, that's going to be messy, mainly because of the discount factors.</p>
<p>If you are willing to assume a constant discount factor, i.e. <span class="math-container">$\delta_{i+t}=\delta^t$</span>, so that
<span class="math-container">\begin{equation}
U_i=u(x_i)+\delta U_{i+1}+\cdots+\delta^n U_{i+n},
\end{equation}</span>
then things become more manageable. In particular, with <span class="math-container">$n=2$</span> (i.e. one only cares about two generations down the line) and letting <span class="math-container">$v_i=u(x_i)$</span> for notational simplicity, you get
<span class="math-container">\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}</span>
where the <span class="math-container">$F_{t+1}$</span>'s are the <a href="https://en.wikipedia.org/wiki/Fibonacci_number" rel="nofollow noreferrer">Fibonacci numbers</a>.</p>

CC BY-SA 4.0
2021-02-17T08:29:29.530Z
Brilliant, @HerrK ! Would you mind explaining some more how $U_i=u(x_i)+\delta U_{i+1}+...+\delta^nU_{i+n}$ gets transformed into the last expression? Where does the Fibonacci sequence come from?
1.618
CC BY-SA 4.0
2021-02-17T13:19:38.617Z
@prebbon: It's just successively substituting out the $U_{i}$'s. With $n=2$:
\begin{align}
U_i&=v_i+\delta U_{i+1}+\delta^2 U_{i+2}\\
&=v_i+\delta[v_{i+1}+\delta U_{i+2}+\delta^2 U_{i+3}]+\delta^2 U_{i+2}\\
&=v_i+\delta v_{i+1}+2\delta^2 U_{i+2}+\delta^3U_{i+3}\\
&\;\;\vdots
\end{align}
Herr K.
CC BY-SA 4.0
2021-02-16T17:49:36.220Z
2021-02-16T17:49:36.220Z
Herr K.
<p>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 <span class="math-container">$U_{x_{i}}=u(x_i)+u(x_{i+1}) + u(x_{i+2})$</span> for all <span class="math-container">$i$</span> in the dynasty, where '<span class="math-container">$u(x_{i})$</span>' denotes the utility of <span class="math-container">$x_i$</span> 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.</p>
<p>In Becker and Barro's model, we have that <span class="math-container">\begin{equation}
    U_{x_0}=\sum_{i=0}^{n}N_i   \beta_i(u(x_i)).
\end{equation}</span>,</p>
<p>where <span class="math-container">$N$</span> is the number of descendants in a generation <span class="math-container">$i$</span> and <span class="math-container">$\beta$</span> 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 <em>their</em> descendants' utility.</p>
<p>Adding in the discounting element, let me give a concrete example: suppose I, as the dynastic head (<span class="math-container">$x_0$</span>), 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 <em>indirectly</em> 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 <span class="math-container">$u(x_3)=1$</span>, then I indirectly gain <span class="math-container">$50\%*25\%*1 + 25\%*50\%*1$</span>= 0.25 utils from my great-grandchild. How do I get a general expression for this phenomenon which holds for <span class="math-container">$n$</span> generations?</p>
<p>Any suggestions or pointers are hugely appreciated!</p>

<microeconomics><mathematical-economics><utility><welfare-economics><ces-function>
Nested/Recursive Dynastic Utility Functions
1.618
# Nested/Recursive Dynastic Utility Functions

(Asked by: username_0 on 2021-02-16)

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!

**Tags:** microeconomics, mathematical-economics, utility, welfare-economics, ces-function


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## 2 answers

### Answer by username_2 on 2021-02-16. Score: 2 | Accepted answer

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](https://en.wikipedia.org/wiki/Fibonacci_number).

> **Comments:**
> **username_0**: Brilliant, @HerrK ! Would you mind explaining some more how $U_i=u(x_i)+\delta U_{i+1}+...+\delta^nU_{i+n}$ gets transformed into the last expression? Where does the Fibonacci sequence come from?  
> **username_2**: @prebbon: It's just successively substituting out the $U_{i}$'s. With $n=2$: \begin{align} U_i&=v_i+\delta U_{i+1}+\delta^2 U_{i+2}\\ &=v_i+\delta[v_{i+1}+\delta U_{i+2}+\delta^2 U_{i+3}]+\delta^2 U_{i+2}\\ &=v_i+\delta v_{i+1}+2\delta^2 U_{i+2}+\delta^3U_{i+3}\\ &\;\;\vdots \end{align}  

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### Answer by username_1 on 2021-02-16. Score: 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.


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