<p>If I understand well, the objective here is to illustrate the mathematical difference between centralized and decentralized bargaining (with a clear illustration of the model). I'll to my best to provide an answer. I'll base my modeling on Wallerstein (1990), and I greatly encourage you to read it (the references will be left after the text).</p>
<p><em>Decentralized wage bargaining</em></p>
<p>Wallerstein model of decentralized wage bargaining is basically based on the behavior of investors, and less on the behavior of firms. I appreciate it because it facilitate the link with macroeconomic considerations. So, the fundamental equation of the model is the following:</p>
<p><span class="math-container">\begin{equation*}
S\ =\ \frac{1}{y}\left( 1-\frac{p/v}{1-m_{t}}\right)
\end{equation*}</span></p>
<p>Where:</p>
<p><span class="math-container">\begin{gather*}
S\ =\ rate\ of\ investment\\
y\ =\ risk\ aversion\ constant\ ( sensibility\ to\ risk,\ is\ superior\ to\ 1)\\
v\ =\ productivity\ of\ capital\\
p\ =\ discount\ rate\\
m_{t} \ =\ aggregate\ wage\ share\ of\ worker
\end{gather*}</span></p>
<p>This equation tries to summarize the behavior of investors in the economy. According to Wallerstein investors will increase their investments with technological and managerial improvements, and will decrease it when risk or wages increase. There's then an implicit class conflict between financial capital holders and workers.
From this first formula, we can derive the optimal wage obtained from decentralized negotiations.</p>
<p><span class="math-container">\begin{equation*}
m^{*} \ =\ \frac{( p/v) +y-1}{1+n( y-1)}
\end{equation*}</span></p>
<p>Where</p>
<p><span class="math-container">\begin{gather*}
m^{*} \ =\ optimal\ wage\ share\ of\ workers\\
n\ =\ number\ of\ unions
\end{gather*}</span></p>
<p>According to this model, (1) wage decrease with improvement in technology and management, (2) wage decrease with the degree of wage decentralisation</p>
<p><em>Competitive centralization</em></p>
<p>Wallerstein will also modelize competitive centralization, where unions regroup in centrals, who in turn are in competition. In this scenario, the optimal solution for each central is:</p>
<p><span class="math-container">\begin{equation*}
m_{a} \ =\ \frac{M}{1+M}\left(\frac{( p/v) +y-1}{y}\right)
\end{equation*}</span></p>
<p>Where</p>
<p><span class="math-container">\begin{equation*}
\frac{M}{1+M} \ =\ wage\ proportion\ of\ central\ A
\end{equation*}</span></p>
<p>With a bit of algebraic manipulation, the author will show that this solution is not optimal.</p>
<p><em>Cooperative centralisation</em></p>
<p>This is not covered by Wallerstein, but it's worth exploring. What happen if each union central decide to cooperate?
In that case, there share of wage will be equal in the economy:</p>
<p><span class="math-container">\begin{equation*}
\frac{Ma}{1+Ma} =\ \frac{Mn}{1+Mn} =\ ....\ 1/n
\end{equation*}</span></p>
<p>In that case, we can easily derive the optimal wage share (for every individual firms/unions), which is equal to:</p>
<p><span class="math-container">\begin{equation*}
m^{*} \ =\ \frac{( p/v) +y-1}{ny}
\end{equation*}</span></p>
<p>Since
<span class="math-container">\begin{equation*}
ny\ \leqq \ 1+n( y-1)
\end{equation*}</span></p>
<p>We can assume (if every value other than n is constant across models), that cooperative central bargain is inferior to decentralized negotiation. However, the difference between the two models will be reduced with increased in population (at full employment).</p>
<p><em>Critiques and reflexions</em></p>
<p>Wallerstein model is simple to grasp and can be modified with further assumptions. It's nice and usefull, but not enough. Here's some limit that are transcended by contemporary models:</p>
<p>(1) There's a direct link between wages and labor productivity. Usually, when you increase wage, you increase productivity. This is lacking from the model. (see Kim et al. 2020)</p>
<p>(2) Theres other model of wage negotiation. For exemple, you could have a model where a third institutions (ex: the State) mediate and influence negotiation and tries to balance powers.</p>
<p>(3) This model assume that the capitalist firm - a model centered around the division of labor and capital - is the only thing in existence. Our economy is way more diversed. How can we modelise wage negociation in self-management? (the result may be way better, as shown by Vanek 1975)</p>
<p>(4) Finally, there's an omission between wages, investments, and savings/consumer's wealth.</p>
<p><strong>References</strong></p>
<p>Hyuncheol Bryant Kim, Seonghoon Kim, Thomas T. Kim; The Role of Career and Wage Incentives in Labor Productivity: Evidence from a Two-Stage Field Experiment in Malawi. The Review of Economics and Statistics 2020; 102 (5): 839–851. doi: <a href="https://doi.org/10.1162/rest_a_00854" rel="nofollow noreferrer">https://doi.org/10.1162/rest_a_00854</a></p>
<p>Wallerstein, Michael. “Centralized Bargaining and Wage Restraint.” American Journal of Political Science 34, no. 4 (1990): 982–1004. <a href="https://doi.org/10.2307/2111468" rel="nofollow noreferrer">https://doi.org/10.2307/2111468</a>.</p>
<p>Vanek, Jaroslav. 1975. Self-Management: Economic Liberation of Man. New York : Penguin Book</p>

CC BY-SA 4.0
2021-11-28T05:59:44.223Z
2021-11-28T05:59:44.223Z
Alexandre Michaud
<p>Consider the following setting:</p>

<ol>
<li><p>Profit maximizing firms with production functions $\Pi(w,L)$, where $w$ is the wage and $L$ is employment. </p></li>
<li><p>Unions who want to maximize the expected utility of their representative union members. To explicate, let $v(c)$ be the indirect utility function of a union member, where $c$ is consumption. If the union member is employed, he or she gets the wage $c=w$. Otherwise, he or she gets unemployment benefits $c=b$. Then the expected utility of a representative member is $$\nu(w)=lv(w)+(1-l)v(b)$$ where $l=\min(1,L/N)$ and where $N$ is the total amount of union members. (Note: In these problems, one usually assumes $L\leq N$ so that $l=L/N$.)</p></li>
<li><p>Firms and unions bargain over the wage $w$; i.e., this is a collective bargaining problem. The collective bargaining problem is modelled as the maximization of the Nash bargaining product w.r.t. $w$ (see below).</p></li>
</ol>

<p>Now, consider two outcomes of the bargaining process:</p>

<ol>
<li><p>Unions and firms <em>agrees</em> on some wage $w$. In this case, the expected utility of a representative member is $\nu(w)$. The profits to the firm are $\Pi(w,L)$.</p></li>
<li><p>Unions and firms <em>do not agree</em> on any wage $w$. In this case, the expected utility to union members is $v(b)$ and profits to the firm is $0$.</p></li>
</ol>

<p>In the right-to-manage model the collective bargaining is modelled as a symmetric Nash bargaining solution with $\gamma$ as the relative bargaining strength of the union, <em>given</em> that the firm maximizes its profits with respect to employment. I.e., it is the solution to $$\max_w\Omega(w)$$  such that $$\frac{\partial \Pi(w,L)}{\partial L}=0,$$ where $\Omega(w)=\big(\nu(w)-v(b)\big)^{\gamma}\Pi(w,L)^{1-\gamma}$ is the <em>Nash bargaining product</em>.</p>

<p>Now, when reading about this scenario/optimization problem I see two cases in the academic literature: The first one is called <strong>local (or firm-level) wage bargaining</strong> and the other is called <strong>central (or national) wage bargaining</strong>. Even though I have read about them, I do not understand the <em>mathematical</em> difference between them. </p>

<p><strong>So, what is the fundamental, mathematical difference between <em>local (or firm-level) wage bargaining</em> and <em>central (or national) wage bargaining</em> given that we apply the right-to-manage model (i.e., we let firms determine employment unilaterally)? How do I model the two situations?</strong></p>

<p><em>My guesses and thoughts so far (this will be updated as time goes by):</em></p>

<ul>
<li>Local wage bargaining is at the firm level. Central wage bargaining is not at the firm level; instead, the firms are organised into an national employers' federation.</li>
<li>In central wage bargaining, the firms takes the collective bargaining problem as an exogenous event. This would then mean that when they maximize their profits, they do not take into account the agreed on wage. However, in local wage bargaining, firms take the wage into account, meaning that when they maximize their profit, they take into account that the wage is a function of employment $w=w(L)$. Even though some authors seems to think about it this way, I do not understand <em>why</em>. Maybe it has to do with firms somewhow regarding the wage as exogenous and independent of their <em>own</em> investment decisions since they do not directly engage in the barganing process, but only indirectly through the employers' federation (?).</li>
<li>One idea I had was that under central wage bargaining, employment is fixed during the bargaining process, while in local wage bargaining, employment is a function of the wage $w$. This difference would reflect the fact that firms view the wage agreed on as exogenous when wage bargaining is centralised. According to this idea, local wage bargaining would be modelled as $\max_{w}\Omega(w)$ given that $L=L(w)$ is the solution to $\max_w\Pi(w,L)$; and central wage bargaining would be modelled as $\max_w\Omega(w)$ holding $L$ fixed, and the firms chooses $L$ so that it is the solution to $\max_L\Pi(w^*,L)$, where $w^*$ is the centrally determined wage.</li>
<li>The <em>timing of events</em> is a bit unclear in the articles I have read about local and central wage bargaining. But it seems to be this: <em>Firstly</em>, the wage is determined through wage bargaining. <em>Secondly</em>, production takes place as firms solve their profit maximization problem. However, since the model is solved by backwards induction, one often begins by solving the profit maximization problem before finding the Nash bargaining solutions.</li>
</ul>

<p><em>Examples of articles related to my question:</em></p>

<ol>
<li><p>Hoel, Michael. "Local versus central wage bargaining with endogenous investments." The Scandinavian Journal of Economics (1990): 453-469.</p></li>
<li><p>Holden, Steinar. "Local and central wage bargaining." The Scandinavian Journal of Economics 90.1 (1988): 93-99.</p></li>
<li><p>Holmlund, Bertil. "Centralized wage setting, wage drift and stabilization policies under trade unionism." Oxford Economic Papers 38.2 (1986): 243-258.</p></li>
</ol>

CC BY-SA 3.0
2017-02-14T08:59:44.110Z
To me, the beginning of an answer requires to clearly differentiate between employment at the firm-level and at the national level. Yet, you do not seem to do that. What is $L$? Aggregate employment? Also. who is unionised? All workers? Also, can firms coordinate in their bargaining?
luchonacho
CC BY-SA 3.0
2017-02-14T14:07:38.667Z
@luchonacho I agree. I will update the post when I have time. But note that this is the notation used in the articles mentioned; subscript $i$ is often ignored as they view the economy as symmetric. Thus, for firm $i$ we have employment level $L_i$. But since the economy is symmetric we may ingore the subscript. Aggregate employment is $\sum_iL_i\neq L_i$. I guess it does not matter too much who is unionised; the function of the union is to bargain over the wage. W.r.t. the workers, they are employed or unemployed. Firms cannot coordinate. It is unions and firms who wage bargain.
Elias
<microeconomics><mathematical-economics><labor-economics>
Local and Central Wage Bargaining: What Is the Difference?
Elias
# Local and Central Wage Bargaining: What Is the Difference?

(Asked by: username_0 on 2017-02-10)

Consider the following setting:

1. Profit maximizing firms with production functions $\Pi(w,L)$, where $w$ is the wage and $L$ is employment.
2. Unions who want to maximize the expected utility of their representative union members. To explicate, let $v(c)$ be the indirect utility function of a union member, where $c$ is consumption. If the union member is employed, he or she gets the wage $c=w$. Otherwise, he or she gets unemployment benefits $c=b$. Then the expected utility of a representative member is $$\nu(w)=lv(w)+(1-l)v(b)$$ where $l=\min(1,L/N)$ and where $N$ is the total amount of union members. (Note: In these problems, one usually assumes $L\leq N$ so that $l=L/N$.)
3. Firms and unions bargain over the wage $w$; i.e., this is a collective bargaining problem. The collective bargaining problem is modelled as the maximization of the Nash bargaining product w.r.t. $w$ (see below).

Now, consider two outcomes of the bargaining process:

1. Unions and firms *agrees* on some wage $w$. In this case, the expected utility of a representative member is $\nu(w)$. The profits to the firm are $\Pi(w,L)$.
2. Unions and firms *do not agree* on any wage $w$. In this case, the expected utility to union members is $v(b)$ and profits to the firm is $0$.

In the right-to-manage model the collective bargaining is modelled as a symmetric Nash bargaining solution with $\gamma$ as the relative bargaining strength of the union, *given* that the firm maximizes its profits with respect to employment. I.e., it is the solution to $$\max\_w\Omega(w)$$ such that $$\frac{\partial \Pi(w,L)}{\partial L}=0,$$ where $\Omega(w)=\big(\nu(w)-v(b)\big)^{\gamma}\Pi(w,L)^{1-\gamma}$ is the *Nash bargaining product*.

Now, when reading about this scenario/optimization problem I see two cases in the academic literature: The first one is called **local (or firm-level) wage bargaining** and the other is called **central (or national) wage bargaining**. Even though I have read about them, I do not understand the *mathematical* difference between them.

**So, what is the fundamental, mathematical difference between *local (or firm-level) wage bargaining* and *central (or national) wage bargaining* given that we apply the right-to-manage model (i.e., we let firms determine employment unilaterally)? How do I model the two situations?**

*My guesses and thoughts so far (this will be updated as time goes by):*

* Local wage bargaining is at the firm level. Central wage bargaining is not at the firm level; instead, the firms are organised into an national employers' federation.
* In central wage bargaining, the firms takes the collective bargaining problem as an exogenous event. This would then mean that when they maximize their profits, they do not take into account the agreed on wage. However, in local wage bargaining, firms take the wage into account, meaning that when they maximize their profit, they take into account that the wage is a function of employment $w=w(L)$. Even though some authors seems to think about it this way, I do not understand *why*. Maybe it has to do with firms somewhow regarding the wage as exogenous and independent of their *own* investment decisions since they do not directly engage in the barganing process, but only indirectly through the employers' federation (?).
* One idea I had was that under central wage bargaining, employment is fixed during the bargaining process, while in local wage bargaining, employment is a function of the wage $w$. This difference would reflect the fact that firms view the wage agreed on as exogenous when wage bargaining is centralised. According to this idea, local wage bargaining would be modelled as $\max\_{w}\Omega(w)$ given that $L=L(w)$ is the solution to $\max\_w\Pi(w,L)$; and central wage bargaining would be modelled as $\max\_w\Omega(w)$ holding $L$ fixed, and the firms chooses $L$ so that it is the solution to $\max\_L\Pi(w^\*,L)$, where $w^\*$ is the centrally determined wage.
* The *timing of events* is a bit unclear in the articles I have read about local and central wage bargaining. But it seems to be this: *Firstly*, the wage is determined through wage bargaining. *Secondly*, production takes place as firms solve their profit maximization problem. However, since the model is solved by backwards induction, one often begins by solving the profit maximization problem before finding the Nash bargaining solutions.

*Examples of articles related to my question:*

1. Hoel, Michael. "Local versus central wage bargaining with endogenous investments." The Scandinavian Journal of Economics (1990): 453-469.
2. Holden, Steinar. "Local and central wage bargaining." The Scandinavian Journal of Economics 90.1 (1988): 93-99.
3. Holmlund, Bertil. "Centralized wage setting, wage drift and stabilization policies under trade unionism." Oxford Economic Papers 38.2 (1986): 243-258.

**Tags:** microeconomics, mathematical-economics, labor-economics

> **Comments:**
> **username_1**: To me, the beginning of an answer requires to clearly differentiate between employment at the firm-level and at the national level. Yet, you do not seem to do that. What is $L$? Aggregate employment? Also. who is unionised? All workers? Also, can firms coordinate in their bargaining?  
> **username_0**: @username_1 I agree. I will update the post when I have time. But note that this is the notation used in the articles mentioned; subscript $i$ is often ignored as they view the economy as symmetric. Thus, for firm $i$ we have employment level $L_i$. But since the economy is symmetric we may ingore the subscript. Aggregate employment is $\sum_iL_i\neq L_i$. I guess it does not matter too much who is unionised; the function of the union is to bargain over the wage. W.r.t. the workers, they are employed or unemployed. Firms cannot coordinate. It is unions and firms who wage bargain.  

---

## 1 answers

### Answer by username_2 on 2021-11-28. Score: 2

If I understand well, the objective here is to illustrate the mathematical difference between centralized and decentralized bargaining (with a clear illustration of the model). I'll to my best to provide an answer. I'll base my modeling on Wallerstein (1990), and I greatly encourage you to read it (the references will be left after the text).

*Decentralized wage bargaining*

Wallerstein model of decentralized wage bargaining is basically based on the behavior of investors, and less on the behavior of firms. I appreciate it because it facilitate the link with macroeconomic considerations. So, the fundamental equation of the model is the following:

\begin{equation\*}
S\ =\ \frac{1}{y}\left( 1-\frac{p/v}{1-m\_{t}}\right)
\end{equation\*}

Where:

\begin{gather\*}
S\ =\ rate\ of\ investment\\
y\ =\ risk\ aversion\ constant\ ( sensibility\ to\ risk,\ is\ superior\ to\ 1)\\
v\ =\ productivity\ of\ capital\\
p\ =\ discount\ rate\\
m\_{t} \ =\ aggregate\ wage\ share\ of\ worker
\end{gather\*}

This equation tries to summarize the behavior of investors in the economy. According to Wallerstein investors will increase their investments with technological and managerial improvements, and will decrease it when risk or wages increase. There's then an implicit class conflict between financial capital holders and workers.
From this first formula, we can derive the optimal wage obtained from decentralized negotiations.

\begin{equation\*}
m^{\*} \ =\ \frac{( p/v) +y-1}{1+n( y-1)}
\end{equation\*}

Where

\begin{gather\*}
m^{\*} \ =\ optimal\ wage\ share\ of\ workers\\
n\ =\ number\ of\ unions
\end{gather\*}

According to this model, (1) wage decrease with improvement in technology and management, (2) wage decrease with the degree of wage decentralisation

*Competitive centralization*

Wallerstein will also modelize competitive centralization, where unions regroup in centrals, who in turn are in competition. In this scenario, the optimal solution for each central is:

\begin{equation\*}
m\_{a} \ =\ \frac{M}{1+M}\left(\frac{( p/v) +y-1}{y}\right)
\end{equation\*}

Where

\begin{equation\*}
\frac{M}{1+M} \ =\ wage\ proportion\ of\ central\ A
\end{equation\*}

With a bit of algebraic manipulation, the author will show that this solution is not optimal.

*Cooperative centralisation*

This is not covered by Wallerstein, but it's worth exploring. What happen if each union central decide to cooperate?
In that case, there share of wage will be equal in the economy:

\begin{equation\*}
\frac{Ma}{1+Ma} =\ \frac{Mn}{1+Mn} =\ ....\ 1/n
\end{equation\*}

In that case, we can easily derive the optimal wage share (for every individual firms/unions), which is equal to:

\begin{equation\*}
m^{\*} \ =\ \frac{( p/v) +y-1}{ny}
\end{equation\*}

Since
\begin{equation\*}
ny\ \leqq \ 1+n( y-1)
\end{equation\*}

We can assume (if every value other than n is constant across models), that cooperative central bargain is inferior to decentralized negotiation. However, the difference between the two models will be reduced with increased in population (at full employment).

*Critiques and reflexions*

Wallerstein model is simple to grasp and can be modified with further assumptions. It's nice and usefull, but not enough. Here's some limit that are transcended by contemporary models:

(1) There's a direct link between wages and labor productivity. Usually, when you increase wage, you increase productivity. This is lacking from the model. (see Kim et al. 2020)

(2) Theres other model of wage negotiation. For exemple, you could have a model where a third institutions (ex: the State) mediate and influence negotiation and tries to balance powers.

(3) This model assume that the capitalist firm - a model centered around the division of labor and capital - is the only thing in existence. Our economy is way more diversed. How can we modelise wage negociation in self-management? (the result may be way better, as shown by Vanek 1975)

(4) Finally, there's an omission between wages, investments, and savings/consumer's wealth.

**References**

Hyuncheol Bryant Kim, Seonghoon Kim, Thomas T. Kim; The Role of Career and Wage Incentives in Labor Productivity: Evidence from a Two-Stage Field Experiment in Malawi. The Review of Economics and Statistics 2020; 102 (5): 839–851. doi: <https://doi.org/10.1162/rest_a_00854>

Wallerstein, Michael. “Centralized Bargaining and Wage Restraint.” American Journal of Political Science 34, no. 4 (1990): 982–1004. <https://doi.org/10.2307/2111468>.

Vanek, Jaroslav. 1975. Self-Management: Economic Liberation of Man. New York : Penguin Book


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