Growth and Insecure Private Property of Capital

Abstract

This paper revisits Strulik’s model of growth with insecure property rights. In this model, different social groups devote some effort to control a share of the capital stock. We show that a slight variation in the modeling of strategic interactions results in the coexistence of savings and efforts to control a share of the capital stock. We also study the effects of a change in the number of social groups on growth. We also show that an increase in social fractionalization may lead to less effort devoted to control capital and to a higher growth rate.

A Definition of the Objective

In this section, we check that the sum used to define the value function is meaningful. We recall that the value function $V: \mathbb {R}^+ \rightarrow \mathbb {R} \cup \lbrace \infty \rbrace$ is defined by

$$\begin{aligned} V(k_{0})&= \sup _{(c_{t}, \tau _{t}, k_{t})_{t}} \sum _{t=0}^{\infty }\beta ^t \ln c_{t}, \end{aligned}$$

(81)

where

$$\begin{aligned} c_{t}&\le \left( \frac{g(\tau _{t})}{g(\tau _{t})+(n-1)g(\tau )} k_{t} \right) ^{\alpha } (1-\tau _{t})^{1-\alpha }, \end{aligned}$$

(82)

$$\begin{aligned} k_{t+1}&= (n-1) \psi k_{t}^{\alpha } + \left( \frac{g(\tau _{t})}{g(\tau _{t})+(n-1)g(\tau )} k_{t} \right) ^{\alpha } (1-\tau _{t})^{1-\alpha } - c_{t}, \end{aligned}$$

(83)

and $\psi = (1/n)^{\alpha }(1-\tau )^{1-\alpha }$.

It will be useful to bound all feasible sequences of capital stocks. To do this, observe that for all t

$$\begin{aligned} k_{t+1}&\le \bar{k}_{t+1} = n \bar{k}_{t}^{\alpha }. \end{aligned}$$

(84)

One can check that $\bar{k}_{t} = n^{\frac{1-\alpha ^t}{1-\alpha }} k_{0}^{\alpha ^t}.$

Now, observe that since $\ln c_{t}$ is concave, for all positive $\hat{c}$ and for all positive $c_{t}$ one has

$$\begin{aligned} \ln c_{t} - \ln \hat{c}&\le \frac{1}{\hat{c}} (c_{t}-\hat{c}). \end{aligned}$$

(85)

It follows that for all nonnegative $c_{t}$, we have

$$\begin{aligned} U^+(c_{t}) = \max \lbrace 0, \ln c_{t} \rbrace&\le A c_{t} + B, \end{aligned}$$

(86)

where $A = 1/\hat{c}$ and $B = \max \lbrace 0, \ln \hat{c} -1 \rbrace$. Therefore, we have

$$\begin{aligned} \sum _{t=0}^T U^+(c_{t})&\le \sum _{t=0}^T \left( A c_{t} + B \right) , \end{aligned}$$

(87)

$$\begin{aligned}&\le \sum _{t=0}^{T} \left( A \bar{k}_t^{\alpha } + B \right) . \end{aligned}$$

(88)

Since $\bar{k}_{t}^{\alpha }$ is a concave function, we also have for all positive $\hat{k}$

$$\begin{aligned} \bar{k}_{t}^{\alpha } -\hat{k}^{\alpha }&\le \alpha \hat{k}^{\alpha -1} (\bar{k}_{t} - \hat{k}), \end{aligned}$$

(89)

so that

$$\begin{aligned} \bar{k}_{t}^{\alpha }&\le \gamma \bar{k}_{t} + \gamma ', \end{aligned}$$

(90)

where $\gamma = \alpha \hat{k}^{\alpha -1}$ and $\gamma ' = (1-\alpha ) \hat{k}^{\alpha }$.

Now, notice that

$$\begin{aligned} \bar{k}_{t+1}&= n \bar{k}_{t}^{\alpha } \end{aligned}$$

(91)

$$\begin{aligned}&\le n( \gamma \bar{k}_{t}+\gamma ')\end{aligned}$$

(92)

$$\begin{aligned}&\le \left( n \gamma \right) ^2 \bar{k}_{t-1} +n \gamma '\left( 1+ n \gamma \right) \end{aligned}$$

(93)

$$\begin{aligned}&\le \left( n \gamma \right) ^{t+1} k_{0} + n \gamma ' \frac{1- \left( n \gamma \right) ^{t+1}}{1-\left( n \gamma \right) }\end{aligned}$$

(94)

$$\begin{aligned}&= \left( n \gamma \right) ^{t+1} \left( k_{0} - \frac{n \gamma '}{1-\left( n \gamma \right) }\right) + \frac{n \gamma '}{1-\left( n \gamma \right) } . \end{aligned}$$

(95)

We thus have

$$\begin{aligned} \sum _{t=0}^T \beta ^t U^+(c_{t})&\le \sum _{t=0}^T \beta ^t \left( A \left( n^t \gamma ^{t+1} k_{0}+ \frac{ \gamma '}{1-\left( n\gamma \right) } \right) + B \right) . \end{aligned}$$

(96)

Assuming $n \beta \gamma <1$, we see that $\lim _{T \rightarrow + \infty } \sum _{t=0}^T \beta ^t U^+(c_{t})$ is well defined, and so is $\lim _{T \rightarrow + \infty } \sum _{t=0}^T \beta ^t U(c_{t})$.