Bequest games with unbounded utility functions (Q907745)

The authors extend a result obtained in [\textit{D. Bernheim} and \textit{D. Ray}, Altruistic growth economies. I: Existence of bequest equilibria. Technical Report 419. Stanford University: Institute for Mathematical Studies in the Social Sciences (1983)] and \textit{W. Leininger} [Rev. Econ. Stud. 53, 349--367 (1986; Zbl 0602.90040)] about the existence of Markov perfect equilibria for a deterministic bequest game by proving that it holds for a wider class of utility functions. Let \(\mathbb N\) be the set of positive integers, and consider an infinite sequence of generations indexed by \(\mathbb N\). A single good is assumed which can be used for consumption or productive investment. Generation \(t\) lives for one period and inherits an amount of good \(s_{t} \in [0,+\infty)\) from generation \(t-1\). Generation \(t\) consumes \(a_{t} \in [0,s_{t}]\) and saves \(y_{t}=s_{t} - a_{t}\). The inheritance of generation \(t+1\) is \(s_{t+1}=p(y_{t})\) where \(p : [0,+\infty) \rightarrow [0,+\infty)\) is a production function. The utility of generation \(t\) is \(u(a_{t}, a_{t+1})\) where \(u : [0,+\infty) \times [0,+\infty) \rightarrow [-\infty,+\infty)\). A stationary Markov perfect equilibrium is a function \(g : [0,+\infty) \rightarrow [0,+\infty)\) such that \(g(s) \in [0,s]\) for all \(s \in [0,+\infty)\), and \[ g(s) \in \arg \max_{a\in[0,s]}u(a,g(p(s-a)))~(s \in [0,+\infty)). \] For \(b >0\), a function \(\delta : [b, +\infty) \rightarrow (-\infty,+\infty)\) is said to have the strict single crossing property on \([b, +\infty)\) if the following holds: if there exists \(x \geq b\) such that \(\delta(x)\geq 0\), then for each \(x'>x\) one has \(\delta(x') >0\). The main result of the paper is the following theorem. Assume {\parindent=0.7cm\begin{itemize}\item[(A1)] \(u(0, 0) >-\infty\), \(u\) is continuous on \([0,+\infty) \times [0,+\infty)\) and increasing in each variable; or \(\lim_{x \rightarrow 0+}u(x, y) =u(0, y) =-\infty\) for each \(y \geq 0\), \(u(x, 0) >-\infty\) for each \(x >0\), \(u\) is continuous on \( (0,+\infty) \times [0,+\infty)\) and increasing in each variable on \((0,+\infty) \times [0,+\infty)\). \item[(A2)] For every \(y_1 > y_2\) in \( [0,+\infty)\) and \(h >0\), the function \(\Delta _{h} u(x) =u(x, y_1)-u(x +h, y_2)\) has the strict single crossing property on \([b, +\infty)\) for each \(b >0\). \item[(A3)] \(p : [0,+\infty) \rightarrow [0,+\infty)\) is continuous and increasing, and \(p(0) =0\). \end{itemize}} Then there exists a stationary Markov perfect equilibrium \(g : [0,+\infty) \rightarrow [0,+\infty)\) of the form \(g(s) = s-f(s)\) where \(f: [0,+\infty) \rightarrow (-\infty,+\infty)\) is a non-decreasing lower semi-continuous function satisfying \(f(s) \in [0,s]\) for all \(s \in [0,+\infty)\). The authors point out that this formulation of the result allows to consider unbounded from below (isoelastic) utility functions; moreover, the proof provided is relatively short and proceeds for the unbounded state space without the truncation of the production function applied in both papers referenced above.