State Conditions to Guarantee the Value Function is Continuous Monotone and Concave

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The concavity of the value function of the extended Barro–Becker model

Abstract

In this paper, the model of endogenous fertility proposed by Becker and Barro (1988), and extended by Benhabib and Nishimura (1989) is considered. However, in their model, the uniqueness of the optimal path may fail to hold. Furthermore, the value function may not be concave, because of the variable discount factor with respect to the choice made about fertility. In this paper, we show that under a set of mild conditions, based on the assumption that the cost of raising a child is non-constant, there exists a unique optimal path and the value function is concave and continuously differentiable. We also show the existence of the unique steady state and a monotonically optimal path, and confirm that the steady state is saddle point stable.

Introduction

Becker and Barro (1988) established an endogenous fertility model, the so-called dynasty model, in which the number of children and intergenerational allocation are chosen by an agent, whose utility takes into account the satisfaction gained from own consumption and the discounted utility of his or her descendents. They provided an analysis of fertility and the factors that affect the fertility in an open economy model. Barro and Becker (1989) used the framework in a closed economy model to analyze the effects of child-raising costs, the tax system, technical progress, and altruism on the fertility. However, with the discount factor defined as an exponential function of the number of children, the optimal path reaches a steady state in one period. Benhabib and Nishimura (1989) gave a more precise formulation and defined the discount factor as a concave function of the number of children. With the extension, they improved the results of Barro and Becker (1989), and proved that the optimal capital path is monotonic or oscillates depending on the elasticity of the ratio of the discount factor function to the differentiation of the discount factor function. Nishimura and Raut (1999) considered an endogenous fertility model in which an altruistic agent lives for three periods: young, adult and old. The agent determines the number of children and her own consumption and the amount of savings in her adult life, while in her childhood, she depends on her parents, and in old age, she lives on her saving and the inheritance from her deceased parents. They obtained similar results as those of Benhabib and Nishimura (1989). In order to analyze the transition of the optimal path, the uniqueness of the optimal path and the concavity of the value function are assumed in above papers. However, the properties may not necessarily hold without the concavity of the reduced utility function with respect to the number of children and the capital of each child. Kanaya (2001) gave a graphical analysis on the possibility of multiple solutions to the dynamic optimal problem in the original model in which it is assumed that the cost of raising a child is constant. In order to rule out the existence of the multiple paths, he proposed modifying the cost of raising a child such that it became an increasing and convex function with respect to the number of the children. The present paper provides a set of conditions on the elasticity of the discount factor function and that of return on capital under a variable cost framework to guarantee the uniqueness of the optimal path. We also show that under our mild conditions, the value function is concave and continuously differentiable. Like the problems of endogenous time preference, in the variable discount factor case, we face the difficulty of characterizing the optimal path: the value function cannot be concave because of the choice of the number of children. This paper provides a solution to this problem. Moreover, under our conditions, it is shown that the optimal capital path is monotone and the optimal fertility is increasing in the initial capital. We also show that there exists a unique steady state and it is saddle point stable.

In Section 2, we formalize the problem to be analyzed. It is shown that the correspondence feasible set is a continuous correspondence and that the operator is a contraction mapping. These two lemmas together immediately imply the existence of the value function. In Section 3, we show by Theorem 2 that under Assumption 1, the value function V is concave. Next, we show by Theorem 3 that the optimal path is unique. Moreover, V is shown by Theorem 4 to be continuously differentiable. It is also shown that V is almost twice differentiable by Theorem 5. In Section 4, by Theorem 6, monotone of the optimal path is shown. By Theorem 7, a unique steady state is shown in to exist and is characterized by the modified golden rule. A dynamic analysis is conducted by Theorem 8 to show that the steady state is saddle point stable.

Section snippets

The model

In this section, we formalize the model studied by Barro and Becker (1989), and later generalized by Benhabib and Nishimura (1989). In this model, the optimization problem of the t-th generation is $V(k_t)=\underset{c_t,n_t,k_{t+1}}{\max }\{u(c_t)+\beta (n_t)V(k_{t+1})\}\text{,}$ $\mathrm{s}.\mathrm{t}.f(k_t)=c_t+n_t{k}_{t+1}+B(n_t)\text{,}$ where $c_t$ is consumption of the t-th generation; $k_t$ capital; $n_t$ the number of children; $\gamma (n)={\gamma }_1(n)+B_0$ the cost for raising a child with ${\gamma }_1^{\prime }(n)>0$ , ${\gamma }_1^{″}(n)>0$ , and $B_0>0$ ; $B(n)=n\gamma (n)$ the cost for raising n children; and the function $\beta (n)$

Uniqueness of the optimal path

In this section, we show the uniqueness of the solution to problem (I). Since the differentiability of the value function V is not shown, we cannot characterize the solution using the first-order conditions. To address this difficulty, we first consider the maximization problem for a finite-horizon (problem IV) and set a sequence of value functions of finite-horizon maximization problems, $v_m$ , each of which is associated with m periods of consumption remaining, respectively. Then, we show the

Characterization of the optimal path

In this section, we characterize the optimal path using the first-order conditions and the Euler equation. From Theorem 4, the value function V is continuously differentiable in the interior point of X. The optimal path satisfies the following equations: $-[k_1+B^{\prime }(n)]u^{\prime }[f(k)-n{k}_1-B(n)]+{\beta }^{\prime }(n)V(k_1)=0\text{,}$ and $-n{u}^{\prime }[f(k)-n{k}_1-B(n)]+\beta (n)V^{\prime }(k_1)=0\text{.}$

The optimal path must satisfy the Euler equations: $-n_t{u}^{\prime }(c_t)+\beta (n_t)u^{\prime }(c_{t+1})f^{\prime }(k_{t+1})=0,t=1,2,\dots \text{.}$

Theorem 6

Under Assumption 1, the optimal path $(H(k),N(k))$ of problem (I) is

Concluding remarks

In this paper, we have provided a set of reasonable conditions to guarantee uniqueness of the optimal path and the concavity of the value function. Under these conditions, the optimal path is monotone and the steady state is saddle point stable. Our elasticity conditions are mild ones because they guarantee neither concavity of $\beta (n)V(k_1)$ , nor quasi-concavity of $\beta (n)V(k_1)$ in $(n,k_1)$ . Becker and Mulligan (1997) considered an endogenous time preference model, in which the discount rate $\beta (S)$ is a

Acknowledgment

Ling Qi thanks Professors Kazuo Nishimura, Koji Shimomura and Makoto Tawada for their constant encouragement and guidance on this paper. Ling Qi's research was supported by the Humanities and Social Science Foundation Grant 08JA790138 of Ministry of Education of the People's Republic of China, and "211 Project" for Central University of Finance and Economics (the 3rd phase), China.

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