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Published in: Dynamic Games and Applications 4/2019

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

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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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Metadata
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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