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.

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Fig. 1

Notes

  1. 1.

    In the literature on the depletion of common resources, it is possible to accumulate some privately and safely held capital, but there is no costly competition to access the resources.

  2. 2.

    They also write that: “ethnicity is possibly a marker for organizing similar individuals along opposing lines.”

  3. 3.

    We do not focus on perpetual growth. We could obtain endogenous growth by assuming a Romer [13] like technology.

  4. 4.

    Strulik shows that conflict and growth can coexist in an economy populated by two groups of unequal sizes. Here, we do not need to assume a difference in size to obtain the coexistence result.

  5. 5.

    Physical feasibility implies that \(\sum _{i=1}^{n} \phi (\tau ^i_{t}, \varvec{\tau }^{-i}_{t}) = 1\).

  6. 6.

    We thank a referee for noticing that Strulik’s production function, namely \(f(k,l) = kl\), is not a special case of our production function when \(\alpha = 1\). To obtain Strulik’s specification from a Cobb–Douglas function, we could set: \(y_t = (A_t k_t)^{\alpha }(B_t l_t)^{1-\alpha }\) where, following Frankel [5] and Romer [13], \(A_t = l_t\), and \(B_t = k_t\). Generally, it is assumed that the decision maker is unaware of the specifications of \(A_{t}\) and \(B_{t}\), except if he is a social planner. That a social group behaves as a social planner is not strictly impossible, but this is probably a strong assumption.

  7. 7.

    In “Appendix A,” we show that under our specific assumptions the values of the agents’ objectives involved in the feedback Nash are always well defined.

  8. 8.

    That the objective is well defined is established in Appendix.

  9. 9.

    A working paper is also available at: https://www.iza.org/publications/dp/4762/does-conflict-disrupt-growth-evidence-of-the-relationship-between-political-instability-and-national-economic-performance.

  10. 10.

    The data are available in Feenstra et al. [4] available for download at www.ggdc.net/pwt The capital stock is evaluated at constant 2011 national prices (in mil. 2011 US$).

  11. 11.

    Recall that Strulik uses a continuous time setting. We translate his modeling assumptions in a discrete time setting.

  12. 12.

    The crucial hypothesis in this theorem is an invertibility condition which is satisfied in our case since \((1-\delta )+\sum _{j=1}^{n} f_1^j \frac{g(\tau ^j_{t})}{\sum _{h=1}^{n} g(\tau ^h_{t})} \ne 0\). Moreover, it is clear that the multiplier \(\lambda _0 \) in the cited Theorem 2.2 is different from zero in our case, so we have set it equal to one.

  13. 13.

    In the following expression, \(f^j_{1}\) stands for \(f'_{1}\left( \frac{ g(\tau ^j_{t}) }{ \left( \sum _{h=1}^{n} g(\tau ^h_{t}) \right) } k_{t}, 1-\tau ^j_{t} \right) \).

  14. 14.

    The fractionalization index gives the probability that two people drawn at random from the society will belong to different groups [3]. That is, if \(n_{i}\) is the population share of group i, the fractionalization index is \( \sum _{i=1}^{m} n_i (1-n_i)\). Another measure of social fragmentation is the polarization index P introduced by Esteban and Ray [7], where \(P = \sum _{i=1}^m \sum _{j=1}^m n_i^2 n_j d_{ij}\), with \(d_{ij}\) being the intergroup perceived distance. The R index is a special case of the P index where \(d_{ij} = 1\) when \(i \not = j\) [9]. Theses indexes are briefly discussed in Ray and Esteban [12], Sect. 5.1. In our symmetric society, the fractionalization and the R indexes have the same value.

  15. 15.

    Moreover, recall that our setting slightly differs from Strulik’s: we use a Cobb–Douglas function, whereas Strulik uses the production function \(A k_{i} (1-\tau ^{i})\), and we consider a general g(.) function, whereas Strulik concentrates on the case \(g(\tau ) = \alpha +\tau \) (Strulik, ibid, page footnote 3, however, indicates that most of his results can be obtained with more general contest success functions).

  16. 16.

    Our analysis has been cast in a neo-classical setting preventing perpetual growth. But we can recover our results (regarding predation time and savings) in a perpetual growth setting by assuming that \(y_t = (k_t)^{\alpha }(B_t l_t)^{1-\alpha }\) and that each social group takes \(B_t = k_{t}\) as given. Everything would be as if \(y_t = C_t k_t^{\alpha }l_t^{1-\alpha }\) (with \(C_t \equiv k_t^{1- \alpha }\)). In particular, the predation game would remain unchanged. Only the savings game would be different (although we could use the same methods to find the equilibrium savings). People would still save a part of individual production, but in equilibrium, production would be a linear function of capital.

  17. 17.

    In this vein, Vahabi [18, 19] considers that this quality is linked to the characteristics of the goods.

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Correspondence to Bertrand Crettez.

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We thank two anonymous referees as well as the Editor, Georges Zaccour, for useful comments on a previous version of this paper. We also thank Susan Crettez for very helpful remarks on this version.

A Definition of the Objective

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 \(\lim _{T \rightarrow + \infty } \sum _{t=0}^T \beta ^t U(c_{t})\).

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Crettez, B., Hayek, N. & Morhaim, L. Growth and Insecure Private Property of Capital. Dyn Games Appl 9, 1042–1060 (2019). https://doi.org/10.1007/s13235-018-00294-9

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Keywords

  • Dynamic games
  • Insecure private property of capital
  • Growth

JEL Classification

  • C 73
  • D 02
  • D72
  • E 22
  • O 11
  • O 43