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Published in:

01-01-2019

# Growth and Insecure Private Property of Capital

Authors: Bertrand Crettez, Naila Hayek, Lisa Morhaim

Published in: Dynamic Games and Applications | Issue 4/2019

## 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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Appendix
Available only for authorised users
Footnotes
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.

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

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Title
Growth and Insecure Private Property of Capital
Authors
Bertrand Crettez
Naila Hayek
Lisa Morhaim
Publication date
01-01-2019
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 4/2019
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-018-00294-9

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