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Income distribution, productivity growth, and workers’ bargaining power in an agent-based macroeconomic model

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Abstract

We investigate the effect of labor productivity growth, workers’ bargaining power, and legal minimum wage revision rules on income distribution in a novel agent-based macroeconomic model mostly inspired by the post-Keynesian literature. Its main novelties are a wage bargaining process and a mark-up adjustment rule featuring a broader set of dimensions and coupled channels of interaction. The former allows nominal wages to be endogenously determined by interactions involving firms and workers, which are mediated by workers’ bargaining power. The latter assumes that firms also consider their position relative to workers (through their unit costs) to set their mark-up rates, thus linking the evolution of nominal wages in the bargaining process and labor productivity growth to the functional income distribution. This has implications for the personal income distribution through a three-class structure for households. The model reproduces numerous stylized facts, including those concerning the income distribution dynamics. By capturing the inherent social conflict over the distribution of income, our results show the importance of the coevolutionary interaction between workers’ bargaining power and productivity growth to the dynamics of income inequality and to its relationship with output. This leads to a policy dilemma between promoting productivity growth and improving income equality which can, nonetheless, be attenuated by combining policies and institutions that protect workers with policies that stimulate technological innovation and productivity growth.

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Data Availability

the artificial datasets generated through computational simulations undertaken during the current study are available from the corresponding author on reasonable request.

Notes

  1. While productivity growth is also interpreted as an opportunity to increase profit margins in Caiani et al. (2019a), this results from an adaptive behavior of firms, rather than being determined strategically by firms following a mark-up pricing rule as proposed in our model.

  2. In the agent-based literature, Caiani et al. (2019b) also divide households into different classes and Ciarli et al. (2010) also assumes that workers have different functions in the production process.

  3. The following subscripts are used throughout this article: h for households, c for consumption goods firms, m for machines, k for the capital goods firm, f for both firms, b for the bank, and g for the public sector. The superscripts res, man, ind, dir, and cap refer to researchers, managers, indirect workers, direct workers, and capitalists, respectively, while j refers to households from all classes. The superscripts $, D, d , and e identify nominal, demand, desired, and expected variables, respectively. Finally, the subscript t identifies the time period, which encompasses the production, commercialization, and investment periods. Parameters that do not change in a given simulation are referred to as fixed parameters and, as such, are not accompanied by t.

  4. The assumption of monopolist agents is adopted in other models in the literature (Cardaci and Saraceno 2018; Carvalho and Di Guilmi 2020; Dweck et al. 2020; Oliveira et al. 2020) and is consistent with the idea of keeping the model as simple as possible in the elements that are less essential to the main relations under analysis.

  5. A detailed matrix presenting the transaction flows between the agents is reported in Appendix A. The model is implemented and simulated in the Laboratory for Simulation Development (LSD) software with the assistance of the Purpurea program (https://github.com/lirolim/purpurea). The data analysis was undertaken in R and greatly benefited from R scripts from model examples available in LSD.

  6. These values are respectively rounded to the closest integer and rounded down. The reason for this treatment in the first case is to achieve a stable average relation between \(L^{D,dir}_{k,t}\) and \(L^{D,man}_{k,t}\), while in the second case it is to guarantee that the cost with researchers does not exceed the R&D budget.

  7. Hired indirect workers are split between researchers and managers proportionally to their participation in \(L^{D,ind}_{k,t}\).

  8. The use of mark-up rules is largely supported by the empirical literature based on surveys in different countries (Correa et al. 2018; Fabiani et al. 2006). Note that since ρ3 is the number of managers per direct workers in the capital goods firm, \(\rho _{3}w^{ind,\$}_{k,t}/{y^{k}_{t}}\) is equal to costs with indirect workers per unit of output. Since \(w^{dir,\$}_{k,t}/{y^{k}_{t}}\) is the cost with direct workers per unit of output, the price level depends on total unit costs.

  9. In each period, the most productive machines are used first. When fully utilized, each machine can produce \(Q^{fc}_{m}\) consumption goods units.

  10. Given the possibility of measuring the consumption goods firms’ size, for this sector we incorporate the idea that managers perform office administration activities (that become more complex and require more managers as the firms grow in size) as well as directly supervise the direct workers’ production activities. In this sense, managers’ income has an overhead nature and their income tends to fluctuate less during the cycle than direct workers’ income (in aggregate terms), as noted by Kalecki (1971). For a theoretical model with overhead labor, see Lavoie (2014).

  11. Available resources are split between direct and indirect workers following the same proportion as the relative labor demand for each type of worker. Thus, if firms are financially constrained, they hire as many workers as they can given the current wage level up to this labor demand. Wage levels only affect labor demand in a dynamic way: if firms are paying a relatively higher wage level, their costs and prices may be higher, and they may become less competitive, which may reduce the demand for their output production, leading to downward adjustments of the production level in the following periods.

  12. Note that firms achieve their desired production level only if they are able to hire all indirect and direct workers demanded.

  13. A key implication of the normal-cost pricing procedure in the presence of fixed costs is that net profit margins become endogenous to the business cycle and the wage share becomes countercyclical, as shown by the stylized facts in Section 5.

  14. In case firms wish to reduce their productive capacity (\(Q^{fc,d}_{c,t} < Q^{fc}_{t}\)), replacement investment is reduced by the number of machines corresponding to the desired reduction.

  15. The latter condition simply means that firms pay back the entire sum of interest and principle even if the R ratio is achieved. In other words, firms always have the possibility of rolling over their previous loans and loan defaults are a possibility only when firms exit the market.

  16. Inventories are evaluated at the current unit cost of production, in accordance with accounting rules, and the FIFO (first in, first out) criterion is adopted. This means that the change in the value of inventories is composed of the change in its volume (evaluated at the current unit cost of production) and of the change in the unit production cost (known as inventory appreciation). As only the former is related to a production flow, profits in national accounts consider the value from Eq. 21. See Godley and Lavoie (2012) for more details.

  17. The amount of profit dividends may be limited by the amount of resources available at this stage (that is, bank deposits minus resources necessary for loan repayment in the beginning of the next period, which are kept as deposits at the bank from one period to the other).

  18. For simplicity, we do not input a real value for the financial sector’s services when computing real output. In order to guarantee consistency, we do not consider the bank’s profits when calculating nominal output and the wage share. As the interest rate on bonds, loans, and deposits is the same, the bank’s profits are almost negligible. This is also the reason why it is assumed that the bank has no owners.

  19. More specifically, workers consider the growth rate of private aggregate demand, since they are bargaining wages with the consumption and capital goods firms. It is assumed that they have more information on the output growth rate than on the productivity growth rate. Since both variables are positively correlated, a specification in which wages were sensitive to productivity growth would present similar qualitative properties.

  20. Workers’ desired wage will always be at least equal to the minimum wage set by the government.

  21. Once again, the reservation wage is never lower than the minimum wage. Since the reservation wage depends on the desired wage, it is also affected by the macroeconomic context. Indeed, through Eq. 24 and Eq. 25, the relation between unemployment rates and wages emerges endogenously, similarly to other approaches in the agent-based literature (Caiani et al. 2016, 2019b; Dosi et al. 2017, 2018, 2020).

  22. This is also related to the interplay between household debt, inequality and consumption (Cynamon and Fazzari 2008; Kim et al. 2014; 2015). As we assume that households have no access to the credit market, this relation is beyond the scope of this article. For agent-based models that deal with the relation between inequality and household debt, see Cardaci and Saraceno (2018) and Carvalho and Di Guilmi (2020).

  23. This revision rule represents a neutral minimum wage policy that does not target a specific redistributive outcome. In the second set of experiments we introduce a more active minimum wage policy, which targets an exogenous Gini coefficient (see Section 6.2).

  24. Government expenditure in real terms is calculated assuming that the productivity of direct workers in the public sector grows in line with the average productivity in the private sector.

  25. In other words, the government never lays off its employees and the public servants never quit their job. This simplifying assumption is adopted in order to reduce the influence of the government’s demand for labor in the labor market.

  26. However important, our model does not intend to explore the effect of labor market reforms that have facilitated flexibility and created the so-called dual labor market, wherein primary sector workers are relatively more insulated from labor market fluctuations than secondary sector workers (Caju et al. 2015). In this sense, we aim to capture a more collective dimension of workers’ bargaining power, rather than individual dimensions that nonetheless may have macro implications. For a complementary discussion on labor market flexibility in an agent-based framework, see Dosi et al. (2017, 2018).

  27. This also means that negative inflation rates are not considered when workers adjust their desired wages or in the minimum wage adjustment.

  28. The firm may also have to fire additional workers in case it does not have the financial resources to hire them in the current period. Workers to be fired are randomly selected among current employees.

  29. By randomly selecting workers among the employed and unemployed workers, firms survey, on average, a set of workers that reflects the participation of unemployed and employed workers in the economy. Therefore, if the unemployment rate is low (high), more (less) employed workers are surveyed. The unemployment rate is also key to the wage setting process through Eq. 27.

  30. Fagiolo et al. (2004) adopt a similar equation, but in their model firms’ bargaining power is exogenous. Our model captures two important determinants of workers’ bargaining power dynamics: institutions that support workers’ bargaining power (captured by the ϕ parameter) and changes in the employment rate. The effect of workers’ bargaining power on the nominal wage can be interpreted as representing direct collective negotiations or the indirect effect of labor market conditions on firms’ offered wage. This allows for a richer and more encompassing analysis of income distribution dynamics, as the same wage share can result from different combinations of ϕ and ηt and, consequently, can be related to markedly different macroeconomic contexts, as discussed more broadly and with respect to the experience of the U.S. economy in recent times by Setterfield (2021). The inclusion of productivity growth adds a third determinant to the story, as shown in our results in Section 6.

  31. Unless mentioned otherwise, all figures, tables, and statistics refer to the time span from period 201 to 500.

  32. Stock and Watson (1999) show that total fixed investment tends to be coincident with output, while investment in equipment lags the cycle. Indeed, in accordance with the dynamics of investment in equipment, some theoretical models assume that investment lags the cycle (following endogenously the dynamics of aggregate demand). For a discussion on induced investment, see, for instance, Cesaratto (2015). In our model, a leading investment probably results from the effect of replacement investment and to productivity growth being the main driver of output growth.

  33. Note that the same initial conditions are used for all firms in the consumption goods sector.

  34. For a discussion on the cyclical behavior of the real wage, see Basu and House (2016), Bils (1985), and Solon et al. (1994). For empirical evidence on the countercyclicality of the wage share, see Giovannoni (2010) and Schneider (2011).

  35. This is the case because in the expansion phase of the business cycle, firms do not need to hire a proportionately higher number of managers. Consequently, their share of income in output falls and productivity, measured by total output divided by total number of workers employed (including managers) increases. For a theoretical discussion on productivity and functional income distribution during the business cycle see Lavoie (2014, 2017). See also Cauvel (2019) and Rolim (2019) for empirical evidence.

  36. In the model, consumption inequality is measured by the distribution of the number of goods acquired by households (in real terms), while wealth inequality refers to the distribution of deposits held by the households as this is the only asset held by households the value of which is measured.

  37. Note that there is an intrinsic asymmetry in this conflict over income distribution: firms strength will be manifested both in the labor market when they negotiate with workers and in their pricing decisions, while workers’ strength only affects nominal wages.

  38. Average values for the main variables are reported in Table 5 in Appendix C.

  39. The key role of workers’ bargaining power in shaping the income distribution dynamics is also supported by empirical studies, such as Guschanski and Onaran (2021).

  40. While suggestive that the demand regime (i.e. relation between the wage share and output) depends on the policy shock, our results are not directly comparable with the Kaleckian theoretical models, since they often treat the wage share as an exogenous variable. Our results indicate the effect of specific parameters on both output and distribution and the resulting relation between both variables, but the output dynamics can be partially due to a direct effect of the parameters on output (and not entirely through demand mechanisms linking distribution and growth). Nevertheless, our results do offer an important insight for empirical studies investigating the relation between income distribution and economic activity, as these studies may capture a negative correlation between the wage share and output that is mostly due to the productivity dynamics and, consequently, has limitations as a counterfactual (e.g., how the economy would behave in case workers’ bargaining power was enhanced by institutional changes) or as an explanation for the performance of an economy during a specific period (e.g., what was the contribution of specific policies aiming to increase the nominal wage). Given this endogeneity of the wage share, an alternative strategy for empirical studies would be to use the components of the wage share. This is the case of Cauvel (2019), for instance, who replaces the wage share with the real wage rate and labor productivity.

  41. Technological unemployment arises when productivity growth reduces the demand for workers and aggregate demand does not grow enough to counteract this effect.

  42. Note that even when workers’ bargaining power is high (ϕ = 0.9), productivity gains are not entirely shared with workers, since this depends on the parameters associated with the mark-up deviation (17). If productivity growth is higher, firms will still be able to maintain relatively higher mark-up rates, even if partially limited by the higher workers’ bargaining power. This explains why for each level of ϕ the unemployment rate is higher in the scenarios with higher ζ (which are associated with more productivity growth).

  43. Inflation rates are modest and some scenarios experience negative average inflation rates due to the predominant effect of technological progress, as suggested by Fig. 9. This indicates that when technological progress is too strong (relative to workers’ bargaining power) there is a significant downward pressure on nominal unit labor costs (both directly through more productivity and indirectly through the increase in unemployment rates) which leads to lower prices. Therefore, the inflation dynamics reported in Fig. 10 reflects an important implication of the coevolutionary interplay between productivity growth and workers’ bargaining power, which is intrinsic to the social conflict over the income distribution. It also bespeaks the necessary (and possibly sufficient) conditions for positive inflation rates in models with a similar structure. See Santoro (2020) for an agent-based model also inspired by the conflicting-claims inflation model.

  44. Arguably, a more active minimum wage policy (as well as other policies protecting workers) might result from workers obtaining more bargaining power also in the political sphere. In the experiments carried out in the preceding section, however, the scope of influence of the bargaining power of workers is assumed to be confined to wage negotiations.

  45. Note that this is an asymmetrical rule: while the government increases the minimum wage to reduce inequality, it does not reduce it when the Gini coefficient is below the targeted level.

  46. Note that the experiments presented in Section 6.1 could be characterized by the same revision rule for the minimum wage with λ = 0. As in the previous cases, the nominal minimum wage cannot be adjusted downward.

  47. See Table 6 in Appendix C.

  48. Since the unemployment benefit is equal to the minimum wage, the adjustment of the nominal minimum wage also has implications for the unemployed workers’ income level, which is captured by the net income distribution but not by the gross income distribution (unemployment benefits are transfers).

  49. See Table 6 in Appendix C for more details.

  50. See also Saltelli et al. (2008). For examples of agent-based models using a similar methodology, see Dosi et al. (2018) and Pedrosa and Lang (2021), and Possas et al. (2020).

  51. For instance, if a machine is assumed to belong to a vintage from t = −Tk/2, its productivity will be equivalent to a ratio \((1-{\varrho }_3)^{T^k/2}\) of the productivity of a machine produced in t = 0.

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Acknowledgements

The authors thank Federico Bassi and João Santoro for helpful comments, as well as Esther Dweck and Marcelo Pereira for suggestions on earlier stages of the research. We also thank two anonymous reviewers for comments and suggestions that helped to improve the paper.

Funding

LNR gratefully acknowledges funding from the São Paulo Research Foundation (FAPESP) under grants #2018/21762-0 and #2019/22413-1 and from the Brazilian National Council for Scientific and Technological Development (CNPq) under grant #140426/2018-3. GTL gratefully acknowledges research support from CNPq under grant #311811/2018-3. This study was also financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

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Authors and Affiliations

  1. Institute of Economics, University of Campinas, Campinas, Brazil

    Lilian N. Rolim & Carolina Troncoso Baltar

  2. Department of Economics, University of São Paulo, São Paulo, Brazil

    Gilberto Tadeu Lima

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  1. Lilian N. Rolim
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  2. Carolina Troncoso Baltar
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Correspondence to Lilian N. Rolim.

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Appendices

Appendix A: Transaction flows matrix

The interactions between the agents are represented by the transaction flows matrix in Table 3. Aggregate values are calculated before the entry and exit of consumption goods firms takes place. These aggregate values are used to evaluate the consistency of the model by checking if the relations expressed by Table 3 hold. We also check for the consistency of real and nominal output from the income, expenditure, and production approaches.

Table 3 Transaction flows matrix
Full size table

Appendix B: Model initialization and parameters

The proportion of capitalists, indirect and direct workers reflect the Brazilian social structure reported by Baltar and Rolim (2018). Accordingly, the number of indirect workers and capitalists depends on the exogenous number of direct workers, as follows:

$$ N^{ind} = \left \lceil \frac{N^{dir}}{n^{dir}} n^{ind} \right \rceil $$
(29)
$$ N^{cap} = \left \lceil \frac{N^{dir} (1-n^{dir}-n^{ind})/n^{dir}}{N^c + N^k} \right \rceil (N^c + N^k) $$
(30)

where Ndir is the number of direct workers, ndir and nind are the proportion of direct and indirect workers respectively. The number of capitalists per firm is equal to ρ1 = Ncap/(Nc + Nk). The number of direct workers as public servants (\(L_g^{dir}\)) is given by a multiple ng of the number of direct workers employed by the private sector in the model’s initialization, while the number of indirect workers as public servants is given by \(L_g^{ind} = L_g^{dir} \lceil N^{ind}/N^{dir} \rfloor \).

Workers’ initial wages are set according to their class, as follows:

$$ w^{dir,\$} = {\varrho}_1 w^{\min,\$}_{0} $$
(31)
$$ w^{ind,\$} = {\varrho}_2 w^{dir,\$} $$
(32)

where ϱ1,2 > 1 are parameters.

Consumption goods firms start with the same full capacity production level (\(Q^{fc}_{c,0}\)). It is assumed that the machines were produced between t = −Tk/2 and t = 0 and that their productivity rate differs by a factor ϱ3 > 0 per period.Footnote 51 Thus, each firm has a heterogeneous set of machines, but, as this heterogeneity is equal for all firms, there is a homogeneous composition of the capital stock across firms. It is also assumed that firms’ initial production and sales are equal to their desired capacity utilization level and that inventories are at the desired level. The capital goods firm’s initial production is proportional to the number of capital goods owned by the consumption goods firms and their lifetime and it is assumed that there is no increase in productivity in the first time step.

The parameters used in the baseline scenario are reported Table 4

Table 4 Parameters and initial values in baseline scenario
Full size table

Appendix C: Performance comparisons

Table 5 Average values for alternative values of ϕ and ζ
Full size table
Table 6 Average values for alternative values of λ
Full size table

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Rolim, L.N., Baltar, C.T. & Lima, G.T. Income distribution, productivity growth, and workers’ bargaining power in an agent-based macroeconomic model. J Evol Econ 33, 473–516 (2023). https://doi.org/10.1007/s00191-022-00805-3

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