## 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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## Notes

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

- 3.
We do not focus on perpetual growth. We could obtain endogenous growth by assuming a Romer [13] like technology.

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

- 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.
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.
That the objective is well defined is established in Appendix.

- 9.
- 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.
Recall that Strulik uses a continuous time setting. We translate his modeling assumptions in a discrete time setting.

- 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.
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.
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.
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.
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.

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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

where

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*

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

It follows that for all nonnegative \(c_{t}\), we have

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

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

so that

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

Now, notice that

We thus have

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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### Cite this article

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