Social conflict,property rights,and the capital

Abstract

I examine some political economy aspects of the capital–labor split. In a game-theoretic general equilibrium model, rent-seekers attempt to appropriate the output of manufacturing firms on the offensive end, while such firms safeguard their income on the defensive end. Then the presence of imperfect property rights is a breeding ground for social conflict and has two indirect consequences. First, it evens out changes in the gross (before-social-conflict) labor and capital shares of total output that are caused by changes in the aggregate factor endowments, leading to more rigid equilibrium factor shares. Second, if social conflict is more labor-intensive than manufacturing, weaker property rights lead to a larger equilibrium labor share.

Keywords

Capital factor shares labor property rights rent-seeking safeguarding

1. Introduction

A well-known observation is that, at least since 1900, the labor and capital shares of gross domestic product (GDP)―which correspond to the allocation of GDP between laborers and owners of capital―have tended to remain relatively rigid in several countries over long periods of time despite substantial changes in the aggregate endowments of capital and labor (e.g., Jones, 2016; Kaldor, 1955; Piketty, 2014; Piketty and Saez, 2014). For example, as Kaldor (1955: 83–84) points out, ‘In fact no hypothesis as regards the forces determining distributive [labor and capital] shares could be intellectually satisfying unless it succeeds in accounting for the relative stability of these shares in the advanced capitalist economies over the last 100 years or so, despite the phenomenal changes in the techniques of production … and in real income per head’. A second important stylized fact is that there appears to be a gradual decrease in the labor share of GDP around the world in the last several decades (e.g., Feenstra et al., 2015; Jones, 2016; Karabarbounis and Neiman, 2014; Piketty, 2014). Overall, factor shares are slow-moving variables in a rapidly changing world that also exhibit a gradual reallocation of GDP away from labor (and toward capital) in recent decades.¹

A standard explanation for the relative rigidity of the capital–labor split is that economic activity may be approximated by Cobb–Douglas production functions, which have a unit elasticity of substitution between labor and capital (e.g., Cobb and Douglas, 1928; Jones, 2016; Piketty, 2014). Then the labor and capital shares of GDP are constant, regardless of the economy’s endowments of capital and labor. However, empirical research casts doubt on this view; actual production functions may not be Cobb–Douglas since the elasticity of substitution between capital and labor may be significantly different from one.² Furthermore, such a view of factor-share rigidity does not account for the gradual decline in the labor share in recent decades.

Unlike most of the literature that points to economic parameters, such as production functions and technology, as the main determinants of the capital–labor split, some anecdotal evidence brings out the important role of law and politics (e.g., Piketty, 2014). For example, Krugman (2009) stresses that political conflict has often been the principal driver of the capital–labor split in the USA in the 20th and the 21st centuries; labor market regulation and unionization, minimum wage laws, and the degree of progressivity of the tax system have been of paramount importance.³ However, despite the related anecdotal evidence, the political economy aspects of the capital–labor split are not fully understood in formal economic or formal political science theory.

In this paper, I present a simple game-theoretic general equilibrium analysis of some political economy aspects of the capital–labor split. As is standard in a large part of the general equilibrium literature, the modeling approach places emphasis on the examination of possible ubiquitous elements, abstracting from local features tuned to specific case studies. I show that the presence of imperfect property rights and social conflict may constitute a possible explanation both for the relative rigidity (or the slow-moving nature) of factor shares―i.e., for the relative unresponsiveness of factor shares to changes in the aggregate endowments of capital and labor―and for the recent gradual decline in the labor share. Although, as is well known (e.g., Acemoglu et al. 2005; North and Thomas, 1973), stronger property rights tend to be associated with a higher level of GDP per capita, I suggest that they may also have the indirect (and not necessarily intended) consequence, first, of decreasing the labor share and, second, of increasing its responsiveness to changes in aggregate factor endowments.

As in Segal and Whinston (2013), property rights enable the owner of an asset―or of manufacturing output in the case of our model―to use the benefits of the asset and to exclude others from it; a society may allocate property rights in various ways. Then, regarding our terminology (and in the spirit of North and Thomas (1973) and Acemoglu et al. (2005)), among others), ‘stronger’ property rights amount to manufacturing firms having effective legal rights to a larger fraction of their own gross output and thus, ceteris paribus, passing on ensuing benefits to employed factors of production (productive labor, capital) in competitive factor markets. The remaining manufacturing output (to which manufacturing firms have no effective legal rights) is vulnerable to rent-seeking, which can be a breeding ground for social conflict. Such imperfection of property rights implies that the gross income of laborers and owners of capital―which is their marginal product if markets are perfectly competitive―may be effectively redistributed through social conflict so that their equilibrium income is different from their gross income.

As is well known, rent-seeking may take several forms, such as theft, brute force, coercive encroachment, lawsuits, litigation, or lobbying the State for extractive policies (Hirshleifer, 2000). For example, rent-seeking may occur through special interest groups (and lobbyists or lawyers) that benefit from government policies, such as taxes and subsidies, or manipulate the legal system (Baumol, 1990; Dal Bo and Di Tella, 2003). Rent-seeking may also occur through corrupt bureaucratic organizations or agencies within the State that extract payments and bribes from productive projects (Acemoglu, 1995; Dal Bo et al., 2006).

In the model each laborer may choose to be employed either in the manufacturing sector as a producer or in the social conflict sector―i.e., in rent-seeking on the offensive end or in safeguarding on the defensive end―as a redistributor. Manufacturing firms generate their gross (before-social-conflict) output by hiring laborers as producers and buying capital from owners of capital. In social conflict, however, rent-seeking firms hire laborers as redistributors to prey on the appropriable fraction of the gross output of manufacturing firms. Manufacturing firms may contest such rent-seeking attempts against them by also hiring laborers as redistributors to safeguard the appropriable fraction of their output. Thus, a feature of the model is that the labor-intensity of social conflict is greater than the labor-intensity of manufacturing, a standard assumption in the literature (e.g., Dal Bo and Dal Bo, 2011; Grossman, 2001; Skaperdas and Syropoulos, 2001); such an assumption appears to be especially appropriate for modern economies.⁴

The analysis shows that, first, weaker property rights lead to a larger labor share of total output in equilibrium. Weaker property rights increase the relative importance of social conflict activities in the economy. Then, since social conflict is more labor-intensive than manufacturing, the increased weight of social conflict boosts the equilibrium share of labor relative to that of capital. Second, weaker property rights decrease the responsiveness of equilibrium factor shares to changes in the aggregate endowments of labor or capital. Specifically, in equilibrium, a proportion of the gross (i.e., before-social-conflict) capital income is effectively captured by (redistributors’) labor that preys on and safeguards gross manufacturing output. Such automatic (partial) gross-income-pooling implies that the presence of social conflict evens out any changes in the gross factor (labor or capital) shares of total output that are triggered by changes in the economy’s aggregate labor or capital endowments. The smoothing of factor shares is more pronounced when property rights are weaker and social conflict is more intense; then, a larger proportion of the gross capital income is eventually captured by labor in equilibrium (i.e., there is more extensive gross-income-pooling).⁵

In addition, the smoothing effect of social conflict may be further reinforced when (similar to Polinsky and Shavell, 1992) the quality of legal enforcement is sensitive to (or negatively related to) the overall extent of social conflict as society’s legal infrastructure becomes more overextended. An increase in capital’s gross share of total output―which may be caused by a relatively larger (smaller) aggregate capital endowment if the elasticity of substitution between capital and productive labor in manufacturing is larger (smaller) than one―reduces a laborer’s opportunity cost of becoming a redistributor, rather than a producer, since productive labor’s gross share is now smaller; more laborers are encouraged to become redistributors. Then a larger proportion of redistributors―or a larger extent of social conflict―may overstretch the legal infrastructure and lead to weaker enforcement of property rights, which makes more pronounced the evening-out of the equilibrium factor shares that is brought about by social conflict. The proportion of gross capital income that is eventually captured by laborers in equilibrium increases (decreases) when the gross capital share of total output increases (decreases). Thus, the evening-out of factor shares―or the relative unresponsiveness of such shares to changes in factor endowments―is more pronounced when legal enforcement is more sensitive to the extent of social conflict.⁶

In practice, the analysis implies that the gradual decline in the labor share in recent decades may, to some extent, be attributed to the overall improvement of property rights; as is well known, higher levels of GDP per capita (such as the ones that were attained in recent decades) tend to be linked to stronger property rights (e.g., Acemoglu et al., 2005; North and Thomas, 1973). Thus, given such a link, the analysis is generally in line with the empirical finding of Feenstra et al. (2015) that, in the period from 1950 to 2014, after including country fixed effects, a country’s ‘best estimate’ labor share (which also accounts for self-employed professionals) is negatively (and significantly) correlated with GDP per capita.⁷

Furthermore, two incidents of sudden institutional change appear to be generally consistent with the result on the link between property rights and the labor share. In particular, the sudden collapse of communism at the beginning of the 1990s led to a rather abrupt improvement in property rights in several countries. As our analysis implies, ex-Soviet republics and ex-Eastern-bloc European countries experienced a pronounced decrease in the labor share of GDP after 1990. For example, the data set of Karabarbounis and Neiman (2014) measures the corporate labor share (which may be subject to fewer measurement errors than the overall labor share) in countries for which at least 15 years of data between 1975 and 2012 are available. After 1990, the percentage point decline per decade of the corporate labor share is substantially greater in ex-Soviet and ex-Eastern-bloc European countries than in the rest of the world.

The two world wars and the Great Depression in the first half of the 20th century may constitute another incident of sudden institutional change. Government policies for tackling major wars and extreme depressions―such as active government intervention, taxation, capital controls, nationalizations, and financial repression―often infringe on individual economic liberties and on free markets, leading to a larger divergence of equilibrium agent income from gross before-social-conflict income and, in this regard, to weaker property rights to produced output. Thus, our model may point to such an emergence of a new legal or political status quo as a possible factor in the especially large labor share that many countries exhibited for a few decades after World War II. For example, the empirical analysis of Piketty and Saez (2014) supports the possibly important role of such government policies in the increase in labor shares.⁸ In any case, our model indicates that more formal empirical analysis is warranted to follow up such casual observations and to examine the link between social conflict and the capital–labor split.

1.1. Review of the theoretical literature

The paper is most closely related to the theoretical political economy literature on the role of factor endowments. In particular, Dal Bo and Dal Bo (2011) examine the effects of an economy’s capital and labor endowments on rent-seeking; changes in such endowments impact the degree of rent-seeking. Furthermore, several articles explore the effects of natural endowments or natural resources (e.g., Baland and Robinson, 2012; Robinson et al., 2006) and of comparative technological endowments (e.g., Levchenko, 2013) on rent-seeking and institutions. I supplement the literature by examining a different issue, the capital–labor split. I bring out the potentially important role of social conflict and property rights in the distribution of aggregate output between laborers and owners of capital; changes in an economy’s capital and labor endowments may affect the equilibrium capital–labor split not only directly (through the production function) but also indirectly, by shaping social conflict.

In the macroeconomics and labor economics literature Blanchard and Giavazzi (2003) and Caballero and Hammour (1998) show that exogenous shocks, which affect bargaining between workers and firms in the labor market, may impact the factor shares of total output. In this paper, I explore a different issue, namely, the rigidity, or the slow-moving nature, of factor shares, which is not examined by this literature (which focuses on the size, rather than on the rigidity, of factor shares). The paper also incorporates economy-wide social conflict, instead of focusing on negotiations in the labor market. Furthermore, the result on long-term trends is often different from that found in the literature; in the model, more social conflict leads to a larger labor share, while in the literature more bargaining power for workers (which allows for more ‘forceful’ redistribution away from purely economic outcomes) may often lead to a smaller labor share through eventual overcompensation in the form of decreased firm entry and decreased investment.

In the economic growth literature it is well known that along a steady-state balanced-growth path the economy may exhibit a constant growth rate of total output along with constant labor and capital shares of total output (since the ratio of factor endowments may remain constant). This is a standard feature of the benchmark neoclassical model, as well as of models that incorporate labor-augmenting technological change (e.g., Mankiw, 2016).⁹ However, there are indications that, in practice, economies may often operate for long periods outside a steady-state balanced-growth path (e.g., Jones, 2002). For example, it may take an economy a long time to reach a steady state, and since the steady state itself often changes or evolves (rather than constituting a permanent condition), an economy may remain outside (and in the process of reaching) a steady-state path for especially long periods.¹⁰ Then our analysis brings out a novel explanation for the relative rigidity of factor shares that is not restricted to the steady-state balanced-growth path. I also examine the recent decline in the labor share, which is not the focus of the growth literature. I complement the literature by introducing imperfect property rights and social conflict; agents choose between production and conflict, rather than between different production techniques and inputs.

Finally, there is a large body of literature in economics that discusses several economic or technological parameters that may have contributed to the recent decline in the labor share of GDP. For example, Karabarbounis and Neiman (2014) attribute such a decline to a decrease in the relative price of investment goods, which may have led to greater capital accumulation, while Acemoglu and Restrepo (2018) suggest that the automation of tasks previously performed by labor may have contributed to the decline in the labor share. I complement this literature by focusing on imperfect property rights and the political economy, rather than the technological parameters, of the capital–labor split.

2. The model

The economy is populated by $N^{L}$ laborers and $N^{K}$ owners of capital. Each laborer is endowed with $L / N^{L}$ units of labor so that the total amount of labor in the economy is $L$ . For example, an agent’s labor endowment may be impacted by his level of education or training. Each owner of capital, conversely, is endowed with $K / N^{K}$ units of (physical) capital so that the total amount of capital is $K$ . There are a large number of perfectly competitive firms that each manufactures a homogeneous output. Manufacturing entails two inputs, namely productive labor and capital. Each manufacturing firm has a standard constant-elasticity-of-substitution (CES) production function, $f (l, k) = (l^{r} + k^{r})^{1 / r}$ , where $r \in (- \infty, 1)$ and $l$ and $k$ are the amounts of productive labor and capital, respectively, that the firm employs. As in the standard framework of perfect competition, the distribution of firms’ ownership shares among agents is historically given; such ownership shares, however, have no effect on agent payoffs, since firms earn zero profits.

Property rights to manufacturing output are imperfect. As in Segal and Whinston (2013), property rights give the owner of an asset―or of manufacturing output in our model―the right to use the benefits of the asset and to exclude others from it; societies may allocate property rights in various ways. In the model a manufacturing firm has effective legal rights only to a fraction $1 - ρ$ ( $ρ \in (0, 1]$ ) of its output, which is thus insulated from rent-seeking attempts. Then, like Murphy et al. (1991, 1993), a proportion $ρ$ ( $ρ \in (0, 1]$ ) of the output of each manufacturing firm is appropriable and vulnerable to rent-seeking. As I explain in the introduction, rent-seeking may take several forms, such as theft, brute force, coercive encroachment, lawsuits, litigation, manipulation of the legal system, corruption, and lobbying the State for extractive policies (Hirshleifer, 2000).

Regarding the terminology, in the spirit of North and Thomas (1973) and Acemoglu et al. (2005), among others, I consider a more favorable allocation of property rights (to their own output) to manufacturers―i.e., a smaller $ρ$ ―to be ‘stronger’ property rights. As we will see, in competitive factor markets, the benefits of a smaller $ρ$ are, ceteris paribus, passed on to factors of production (productive labor, capital) that are employed in manufacturing.¹¹ Furthermore, since my objective is to focus on the impact of property rights on the capital–labor split, rather than on the causes of property rights, I take $ρ$ to be exogenous, perhaps the outcome of a historical accident (in the spirit of Acemoglu and Robinson (2006) and Acemoglu et al. (2005), among others).

In the model there are rent-seeking firms that prey on and attempt to appropriate the appropriable fraction $ρ$ of the gross (before-social-conflict) output of manufacturing firms. Manufacturing firms, conversely, have the opportunity to engage in safeguarding activities to protect the appropriable fraction $ρ$ of their output from rent-seeking. As in Dal Bo and Dal Bo (2011), Grossman (2001), and Skaperdas and Syropoulos (2001), among others, social conflict is a labor-intensive activity relative to manufacturing, which may be an especially appropriate assumption for modern economies, where social conflict tends to take rather subtle forms (see Note 4). I thus assume that rent-seeking (on the offensive end) or safeguarding against rent-seeking (on the defensive end) merely utilizes labor, rather than capital. Capital is only utilized in manufacturing. In Section 3.2, I will discuss how the results may be applied to capital-intensive conflict.

Like Acemoglu (1995), Dal Bo and Dal Bo (2011), Grossman (2001), Murphy et al. (1991, 1993), and Stefanadis (2010), among others, a laborer has the opportunity to be employed either in manufacturing as a producer or in social conflict (rent-seeking, safeguarding) as a redistributor, obtaining a wage for selling his $L / N^{L}$ units of labor. The proportions of producers and redistributors in the population of laborers are $θ^{P}$ ( $θ^{P} \in [0, 1]$ ) and $1 - θ^{P}$ , respectively, where $θ^{P}$ will be determined in equilibrium. There is free entry of rent-seeking firms. Putting together a rent-seeking firm entails a fixed investment of $α$ units of labor. Appropriation opportunities are allocated symmetrically among rent-seeking firms (as in Murphy et al. (1993), for example). Thus, in the presence of $n^{R}$ rent-seeking firms, each firm has the opportunity to prey on a proportion $1 / n^{R}$ of the gross output of manufacturing firms. As is standard in the theory of conflict, I adopt a Tullock contest success function; each side’s success in a conflict is determined by the ratio of respective inputs (Tullock, 1980).¹² If a rent-seeking firm utilizes $l^{R}$ units of redistributors’ labor in its appropriation campaign against a manufacturing firm, while such a manufacturing firm applies $l^{S}$ units of redistributors’ labor to its safeguarding, the rent-seeking firm succeeds in appropriating a proportion $ρ l^{R} / (l^{R} + l^{S})$ of its gross output (or a proportion $l^{R} / (l^{R} + l^{S})$ of the appropriable fraction $ρ$ of its gross output).¹³

2.1. Equilibrium social conflict

As explained in the appendix in the proof of Lemma 1, if a proportion $θ^{P}$ of laborers are employed as producers in manufacturing, the wages of each unit of producers’ and redistributors’ labor are $\hat{y_{P}^{L}} (θ^{P})$ and $\hat{y_{W}^{L}} (θ^{P})$ , respectively (see equations (14) and (15)). In a Nash equilibrium the wages of all laborers―i.e., of producers and redistributors―are equal since laborers can switch freely between the manufacturing and the social conflict sectors, i.e., $\hat{y^{W}} (θ^{P}) = \hat{y^{P}} (θ^{P}) = \hat{y^{L}} (θ^{P})$ . It follows that in a Nash equilibrium

{(\frac{K}{L})}^{r} {(θ^{P})}^{1 - r} = \frac{4 (1 - θ^{P})}{3 ρ} - 1

(1)

We can see that there exists a unique proportion, $θ^{P} * \in (0, 1)$ , of producers in the population of laborers so that equation (1) is met; $\hat{y_{P}^{L}} (θ^{P}) > \hat{y_{W}^{L}} (θ^{P})$ , $\forall θ^{P} \in [0, θ^{P} *)$ and $\hat{y_{P}^{L}} (θ^{P}) < \hat{y_{W}^{L}} (θ^{P})$ , $\forall θ^{P} \in (θ^{P} *, 1]$ . Lemma 1 follows.

Lemma 1. There exists a unique Nash equilibrium, which entails a proportion $θ^{P} * \in (0, 1)$ of producers in the population of laborers.

Proof. The proof is in the appendix.

The way in which the two endowments, aggregate capital and aggregate labor, shape the proportion of producers in the population of laborers depends on the elasticity of substitution, $σ_{KL} = 1 / (1 - r)$ , between capital and productive labor in the production function, $f (l, k) = (l^{r} + k^{r})^{1 / r}$ .¹⁴ When the ratio, $K / L$ , of aggregate capital to aggregate labor strictly increases (i.e., when $K$ increases or $L$ decreases), the equilibrium proportion, $θ^{P} *$ , of producers in the population of laborers strictly decreases if $σ_{KL} > 1$ ―i.e., if capital and productive labor are substitutes―and strictly increases if $σ_{KL} < 1$ ―i.e., if capital and productive labor are complements. If $σ_{KL} = 1$ , i.e., if the production function, $(l^{r} + k^{r})^{1 / r}$ , is Cobb–Douglas, $θ^{P} *$ is unaffected by changes in $K / L$ . This is summarized in Lemma 2.¹⁵

Lemma 2. When capital and productive labor are substitutes (complements), the equilibrium proportion of producers in the population of laborers is strictly decreasing (strictly increasing) in the ratio of aggregate capital to aggregate labor, i.e., when $σ_{KL} > 1$ ( $< 1$ ), $\partial θ^{P} * / \partial (K / L) < 0$ ( $> 0$ ).

Proof. The proof is in the appendix.

Intuitively, for a given $θ^{P}$ , the gross capital share constitutes the ratio of capital’s gross income, $M P^{K} (θ^{P} L, K) K$ ―the income that owners of capital would earn in the absence of rent-seeking and safeguarding (i.e., the marginal product, $M P^{K} (θ^{P} L, K)$ , of capital multiplied by the aggregate capital endowment, $K$ )―to the economy’s total output, $[(θ^{P} L)^{r} + K^{r}]^{1 / r}$ . As is well known, an increase in the ratio, $K / L$ , of the economy’s aggregate capital to aggregate labor endowments leads to an increase in capital’s gross share of output (and a corresponding decrease in productive labor’s gross share), ceteris paribus, if the elasticity of substitution, $σ_{KL}$ , between capital and productive labor is greater than one.¹⁶ Then, ceteris paribus, a redistributor’s wage relative to a producer’s wage increases since the gross capital share (part of which is eventually passed on to redistributors that prey on and safeguard manufacturing output) is larger. As a result, the equilibrium proportion of redistributors (producers) in the population of laborers increases (decreases). The opposite reasoning applies when the elasticity of substitution, $σ_{KL}$ , between capital and productive labor is less than one.

We can also see that stronger property rights lead to a strictly larger equilibrium proportion, $θ^{P} *$ , of producers in the population of laborers, i.e., $\partial θ^{P} * / \partial ρ < 0$ . Lemma 3 follows.

Lemma 3. The equilibrium proportion of producers in the population of laborers is strictly increasing in the strength of property rights, i.e., $\partial θ^{P} * / \partial ρ < 0$ .

Proof. The proof is in the appendix.

In particular, weaker property rights―i.e., a larger $ρ$ ― increase the impact of social conflict activities, i.e., of rent-seeking on the offensive end and of safeguarding on the defensive end (see equations (7) and (9)), leading to larger wages for redistributors, ceteris paribus (see equation (14)). Furthermore, since weaker property rights facilitate the appropriation of gross output and increase the safeguarding expenses of manufacturing firms, they also lead to smaller wages for producers, ceteris paribus (see equation (15)). As a result, weaker property rights encourage more laborers to work as redistributors, rather than as producers. Apparently, weaker property rights also strictly reduce the economy’s total equilibrium output, $[(θ^{P} * L)^{r} + K^{r}]^{1 / r}$ , since the equilibrium amount of productive labor, $θ^{P} * L$ , becomes strictly smaller.

3. The capital–labor split

I will now examine how the economy’s equilibrium total output, $Y * = [(θ^{P} * L)^{r} + K^{r}]^{1 / r}$ , is allocated to laborers and owners of capital. Furthermore, I will explore the degree of stability of such labor and capital shares of total output when the aggregate endowments, $L$ and $K$ , of labor and capital change. The analysis will place special emphasis on the role of social conflict and of related property rights in the capital–labor split.

In a Nash equilibrium the left-hand side of equation (1), $(K / L)^{r} (θ^{P} *)^{1 - r}$ , stands for the ratio of the aggregate equilibrium income of owners of capital to the aggregate equilibrium income of laborers (both producers and redistributors). In particular, equation (15) (and the fact that $\hat{y^{P}} (θ^{P} *) = \hat{y^{W}} (θ^{P} *) = \hat{y^{L}} (θ^{P} *)$ ) implies that the aggregate equilibrium income of labor is equal to $L (1 - 0.75 ρ) [(θ^{P} * L)^{r} + K^{r}]^{(1 / r) - 1} (θ^{P} * L)^{r - 1}$ ; the equilibrium labor share, $s_{L} *$ , of total output is $L (1 - 0.75 ρ) [(θ^{P} * L)^{r} + K^{r}]^{(1 / r) - 1} (θ^{P} * L)^{r - 1} / Y *$ . Equation (16) implies that the aggregate equilibrium income of owners of capital is equal to $K (1 - 0.75 ρ) [(θ^{P} * L)^{r} + K^{r}]^{(1 / r) - 1} K^{r - 1}$ ; the equilibrium capital share, $s_{K} * = 1 - s_{L} *$ , of total output is $K (1 - 0.75 ρ) [(θ^{P} * L)^{r} + K^{r}]^{(1 / r) - 1} K^{r - 1} / Y *$ . Thus, according to equation (1), $(1 - s_{L} *) / s_{L} * = 4 (1 - θ^{P} *) / (3 ρ) - 1$ , or

s_{L} * = \frac{3 ρ}{4 (1 - θ^{P} *)}

(2)

We can see that stronger property rights lead to a strictly smaller equilibrium labor share of total output, i.e., $\partial s_{L} * / \partial ρ > 0$ (or $\partial s_{K} * / \partial ρ < 0$ ). Proposition 1 follows.¹⁷

Proposition 1. The equilibrium labor share of total output is strictly decreasing in the strength of property rights, i.e., $\partial s_{L} * / \partial ρ > 0$ .

Proof. The proof is in the appendix.

Intuitively, as explained in Lemma 3, weaker property rights―i.e., a larger $ρ$ ―increase the relative importance of social conflict activities in the economy. Then, the increased weight of social conflict boosts the equilibrium labor share since social conflict is labor-intensive. In particular, some of capital’s gross share of output is effectively passed on to (redistributors’) labor, which preys on and safeguards the gross output of manufacturing firms (equation (16)). The presence of weaker property rights leads to the transfer of a larger proportion of the gross capital share to labor, increasing the equilibrium labor share of total output.

I define the elasticity, $ε_{SL} *$ , of the equilibrium labor share to be the percentage change in the equilibrium labor share, $s_{L} *$ , divided by the percentage change in the ratio, $K / L$ , of the aggregate capital and labor endowments, i.e., $ε_{SL} * = [(K / L) / s_{L} *] \partial s_{L} * / \partial (K / L)$ . Thus, such elasticity, $ε_{SL} *$ , constitutes a measure of how responsive the equilibrium labor share is to relative changes in aggregate endowments. A smaller absolute value $| ε_{SL} * |$ of the elasticity implies that the equilibrium labor share is more stable when the ratio, $K / L$ , of the aggregate capital and labor endowments changes. It follows from equation (2) (see the proof of Proposition 2) that

| ε_{SL} * | = \frac{| r |}{\frac{(1 - θ^{P} *) (1 - r)}{θ^{P} *} + 1 + {(\frac{K}{L})}^{- r} {(θ^{P} *)}^{r - 1}}

(3)

The absolute value, $| ε_{SL} * |$ , of the elasticity of the equilibrium labor share is impacted by social conflict and, specifically, by parameter $ρ$ . We can see that the equilibrium labor share is less elastic to changes in $K / L$ when property rights are weaker, i.e., $\partial | ε_{SL} * | / \partial ρ \leq 0$ (the inequality is strict ( $< 0$ ), as long as $| r | \neq 0$ ). This is summarized in Proposition 2.

Proposition 2. The absolute value of the elasticity of the equilibrium labor share is weakly increasing in the strength of property rights, i.e., $\partial | ε_{SL} * | / \partial ρ \leq 0$ (strictly increasing, as long as $| r | \neq 0$ ).

Proof. The proof is in the appendix.

Intuitively, in the presence of social conflict a proportion of the economy’s gross capital income is effectively passed on to (redistributors’) labor that attempts to appropriate and safeguards the gross output of manufacturing firms (equation (16)). Thus, because of such automatic (partial) gross-income-pooling, social conflict evens out any changes in the gross labor, or in the gross capital, share of total output that are brought about by changes in the ratio $K / L$ of aggregate endowments; changes in the equilibrium labor and capital shares are smaller than in the corresponding gross shares. The smoothing is more pronounced when property rights are weaker so that a larger proportion of the gross capital share is eventually passed on to labor.

We can see that equation (3) also accommodates the literature’s standard explanation for the relative stability of the labor share, which focuses on the characteristics of the manufacturing production function. Specifically, when $r = 0$ ―i.e., when the manufacturing function, $(l^{r} + k^{r})^{1 / r}$ , is Cobb–Douglas―we have $| ε_{SL} * | = 0$ .

3.1. Sensitivity of legal enforcement to social conflict

In the base setup property rights, as well as legal enforcement, corresponded to a single constant parameter, $ρ$ , which allowed the link between social conflict and the capital–labor split to be clearly shown. In practice, however, the enforcement of property rights may often depend on the extent of social conflict, rather than being constant; the quality of enforcement may be lower when the proportion of agents that engage in social conflict is larger (e.g., Polinsky and Shavell, 1992).¹⁸ Thus in this section I introduce function $z (1 - θ^{P})$ , which stands for the quality of legal enforcement of society’s property rights, to complement the strength, $ρ$ , of property rights (which, as in the base model, is an exogenous parameter). As we will see, incorporating legal enforcement that is sensitive to the extent of social conflict may strengthen and enrich the results about the smoothing effects of social conflict on factor shares.

In the model a smaller $z (1 - θ^{P})$ implies better enforcement; a rent-seeking firm succeeds in appropriating a fraction $ρ z (1 - θ^{P}) l^{R} / (l^{R} + l^{S})$ of the gross output of a manufacturing firm on which it preys. I assume that $\partial z (1 - θ^{P}) / \partial (1 - θ^{P}) \geq 0$ and $0 < z (0) < z (1) \leq 1$ . Thus, for a given $ρ$ , the quality of legal enforcement is weakly lower when the extent of aggregate social conflict in the economy is larger―i.e., when the proportion, $1 - θ^{P}$ , of redistributors in the population of laborers is larger―so that society’s legal infrastructure is more overextended. Since $1 - θ^{P}$ corresponds to a burden, or to a cost, for the legal infrastructure, I also make the standard assumption that $\partial^{2} z (1 - θ^{P}) / \partial (1 - θ^{P})^{2} \geq 0$ (i.e., legal enforcement deteriorates in a convex manner as related burdens rise). Furthermore, to ensure the existence of interior equilibria (in which strictly positive proportions of producers and redistributors coexist), I assume that the quality of legal enforcement is not overly sensitive to the extent of aggregate conflict. The sensitivity, $ω_{z}$ , of enforcement with respect to the proportion, $1 - θ^{P}$ , of redistributors is weakly less than one, i.e., $ω_{z} = [(1 - θ^{P}) / z (1 - θ^{P})] z' (1 - θ^{P}) \leq 1$ ; a percentage change in the proportion of redistributors translates into a weakly smaller percent change in $z$ .¹⁹

Then, equation (1) becomes

{(\frac{K}{L})}^{r} {(θ^{P})}^{1 - r} = \frac{4 (1 - θ^{P})}{3 ρ z (1 - θ^{P})} - 1

(4)

By following the same procedure as in the base model, we can see that, as in Lemma 1, there exists a unique proportion, $θ^{P} * * \in (0, 1)$ , of producers in the population of laborers so that equation (4) holds.²⁰ Furthermore, Lemmas 2 and 3 carry through. The equilibrium labor share is

s_{L} * * = \frac{3 ρ z (1 - θ^{P} * *)}{4 (1 - θ^{P} * *)}

(5)

Proposition 1 directly carries through.

Furthermore, equation (3) becomes

| ε_{SL} * * | = \frac{| r |}{\frac{(1 - θ^{P} * *) (1 - r)}{θ^{P} * * (1 - ω_{z})} + 1 + {(\frac{K}{L})}^{- r} {(θ^{P} * *)}^{r - 1}}

(6)

We can see that Proposition 2 carries through; the equilibrium labor share is less elastic to changes in $K / L$ when property rights are weaker, i.e., $\partial | ε_{SL} * * | / \partial ρ \leq 0$ (the inequality is strictly less than zero, as long as $| r | \neq 0$ and $ω_{z} \neq 1$ ). Furthermore, the equilibrium labor share is less elastic to changes in $K / L$ when the sensitivity of legal enforcement to the proportion of redistributors is larger, i.e., $\partial | ε_{SL} * * | / \partial ω_{z} \leq 0$ (the inequality is strictly less than zero, as long as $| r | \neq 0$ ). This is summarized in Proposition 3.

Proposition 3. When the quality of legal enforcement is sensitive to the extent of social conflict:

Proposition 2 carries through.

The absolute value of the elasticity of the equilibrium labor share is weakly decreasing in the sensitivity of legal enforcement to the extent of aggregate social conflict, i.e., $\partial | ε_{SL} * * | / \partial ω_{z} \leq 0$ (strictly decreasing, as long as $| r | \neq 0$ ).

Proof. The proof is in the appendix.

Intuitively, as in Proposition 2, because of the automatic (partial) gross-income-pooling that it entails, social conflict evens out any changes in the gross factor shares of total output that result from changes in the ratio $K / L$ of aggregate endowments. Such a smoothing effect is more pronounced when property rights are weaker. In addition, as explained in Lemma 2, a ceteris paribus increase (decrease) in the gross capital share that is brought about by changes in $K / L$ leads to a larger proportion of redistributors in the population and thus to a decrease (increase) in equilibrium legal enforcement―i.e., to an increase (decrease) in $z (1 - θ^{P} * *)$ ―which reinforces the social-conflict-related smoothing of the equilibrium labor and capital shares. In particular, the proportion of gross capital income that is eventually passed on to laborers increases (decreases) when the gross capital share of total output increases (decreases). Such smoothing is more pronounced when the sensitivity, $ω_{z}$ , of the quality, $z (1 - θ^{P} * *)$ , of legal enforcement to the extent of social conflict is larger.

We can see that if the forces that are in effect in Proposition 3(ii) take extreme forms, we have $| ε_{SL} * * | = 0$ , i.e., the equilibrium labor share, $s_{L} * *$ , of total output is constant, or totally insensitive to changes in the ratio, $K / L$ , of factor endowments. In particular, as equation (6) implies, $| ε_{SL} * * | = 0$ when the sensitivity of legal enforcement to the extent of social conflict is one, i.e., $ω_{z} = 1$ .

3.2. Capital-intensive social conflict

In the base model social conflict is a more labor-intensive activity than manufacturing or production. This is a standard assumption in the literature (e.g., Dal Bo and Dal Bo, 2011; Grossman, 2001; Skaperdas and Syropoulos, 2001), which is especially suitable for modern economies (see Note 4). However, we can see that even if social conflict is capital-intensive, rather than labor-intensive, the framework of the analysis remains relevant. In particular, assume that social conflict is capital-intensive; each owner of capital decides whether to channel his capital to production―becoming a producer―or to social conflict―becoming a redistributor. Labor can be used only in manufacturing. Then, Propositions 2 and 3 about the relative stability, or the slow-moving nature, of the capital–labor split directly carry through, and their intuition remains similar (of course, it is now the gross labor income that may, to some extent, be passed on to redistributors’ capital). Proposition 1, conversely, is inverted; stronger property rights cause an increase in the equilibrium labor share.²¹

3.3. Social conflict for the legal system

In the base model social conflict takes place in a given legal system (which establishes a given strength $ρ$ of property rights). We can use a similar approach to model social conflict for, rather than in, the legal system; the manufacturing and rent-seeking sectors may employ redistributors as lobbyists, lobbying the State for the establishment of stronger and weaker property rights, respectively. In such a framework, the emergence of imperfect, rather than perfect, property rights may then be an equilibrium outcome (shaped by contest success functions). Since there are vested interests, i.e., the rent-seeking sector, which compete politically with the manufacturing sector and lobby for weak property rights, the emergence of property rights imperfection is completely normal and unsurprising. This reinforces the empirical importance of examining possible links between imperfect property rights and the capital–labor split, which is the main question of the analysis.

4. Conclusion

I present a simple game-theoretic general equilibrium analysis of some political economy aspects of the capital–labor split. As is well known in economics and political science, stronger property rights tend to be associated with a higher level of GDP. However, our analysis suggests that stronger property rights may also have indirect (and not necessarily intended) consequences; they may lead, first, to a decline in the labor share and, second, to an increase in its responsiveness to changes in the aggregate endowments of labor and capital. Overall, the political economy effects that are brought out by the analysis may complement (without being mutually exclusive with) other views about the capital–labor split that are based on purely economic parameters, such as the characteristics of the production function in manufacturing. More empirical research is warranted to assess the relevance of the various views from a comparative perspective.

Footnotes

Appendix

Acknowledgements

I thank Andres Neumeyer, Geraldo Della Paolera, John Patty, Laura Veldkamp, and two anonymous referees for helpful suggestions. I am also grateful to seminar participants at Universidad de San Andres and Universidad Torcuato di Tella. Part of this work was completed while I was visiting Harvard University (The Institute for Quantitative Social Science), whose hospitality is gratefully acknowledged.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Christodoulos Stefanadis

Notes

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Social conflict,property rights,and the capital–labor split

Abstract

Keywords

1. Introduction

1.1. Review of the theoretical literature

2. The model

2.1. Equilibrium social conflict

3. The capital–labor split

3.1. Sensitivity of legal enforcement to social conflict

3.2. Capital-intensive social conflict

3.3. Social conflict for the legal system

4. Conclusion

Footnotes

Appendix

Acknowledgements

Declaration of Conflicting Interests

Funding

ORCID iD

Notes

References