Optimal consumption decisions of family networks with similar felicity functions

Abstract

Family networks have increased at an accelerated pace through social networking through social networking services, online platforms, and social media clubs. This paper develops a model of an economy populated by networks of families with similar preferences (the same functional form of the felicity or utility function) but family members differ in their deep parameters (compulsion for consumption and risk tolerance), which, in turn, leads to some kind o heterogeneity among the families. In this context, the problem of felicity (utility) maximization of the average family in the network is solved. The optimal path of consumption is obtained for the average family and graphical comparative static exercises are performed. Finally, the economic welfare of the typical family in a network is calculated and comparative static experiments are carried out. This approach improves the understanding of the consumption decision making of families interacting in social networking services.

Appendices

Appendix A

Let $k_{t} = k_{H\,t} + k_{M\,t} ,$ so, then, the allocation of resources is obtained by solving the equation:

$$ \dot{k}_{H\,t} + \dot{k}_{M\,t} = rk_{H\,t} + rk_{M\,t} - \dot{c}_{H\,t} - \dot{c}_{M\,t} , $$

which solution is given by (integration by parts)

$$ k_{H0} + k_{M0} = \int\limits_{0}^{\infty } {c_{H0} e^{ - rt} {\text{d}}\,t} + \int\limits_{0}^{\infty } {c_{M0} e^{ - rt} {\text{d}}\,t} . $$

Appendix B

The optimal consumption paths are given by

$$ c_{j} = \sqrt {\frac{{\lambda \,\mu_{j} }}{{\beta_{j} }}\frac{{\prod\nolimits_{i \ne j} {\mu_{i} e^{rt} } }}{{\left( {t + \lambda } \right)\sum\nolimits_{i \ne j} {\left( {\mu_{i} + c_{i} } \right)} }}} - \sum\nolimits_{i \ne j} {\mu_{i} } ,\quad \forall i,j \in \left\{ {1,2, \ldots ,N} \right\}. $$

Therefore,

$$ \sqrt {\frac{{\lambda \mu_{j} }}{{\beta_{j} }}} = e^{{ - \frac{rt}{2}}} \left( {c_{j} + \sum\limits_{i \ne j} {\mu_{i} } } \right)\left\{ {\sqrt {\left( {\left( {t + \lambda } \right)\sum\limits_{i \ne j} {\left( {\mu_{i} + c_{i} } \right)} } \right)\left( {\prod\limits_{i \ne j} {\mu_{i} } } \right)^{ - 1} } } \right\}, $$

which is substituted into

$$ c_{j} = \sqrt {\frac{{\lambda \,\mu_{j} }}{{\beta_{j} }}\frac{{\sum\nolimits_{i \ne j} {\mu_{i} e^{rt} } }}{{\left( {t + \lambda } \right)\prod\nolimits_{i \ne j} {\left( {\mu_{i} + c_{i} } \right)} }}} - \mu_{j} ,\quad \forall j \in \left\{ {1,2, \ldots ,N} \right\} \, $$

to obtain

$$ c_{j} = \frac{{e^{{\frac{A}{2}\left( {t - \lambda } \right)}} }}{{\sqrt {t + \lambda } }}\left( {\frac{{Ak_{j0} + \mu_{j} }}{{\sqrt {8\pi A} \left( {1 - \varPhi \left( {\sqrt {A\lambda } } \right)} \right)}}} \right) - \mu_{j} $$

which readily provides Eq. (8).