Employee share ownership in a unionised duopoly

Abstract

Profit sharing schemes have been analysed assuming Cournot competition and decentralised wage negotiations, and it has been found that firms share profits in equilibrium. This paper analyses a different remuneration system: employee share ownership. We find that whether firms choose to share ownership or not depends on both the type of competition in the product market and the way in which workers organise to negotiate wages. If wage setting is decentralised, under duopolistic Cournot competition both firms share ownership. If wage setting is centralised, only one firm shares ownership if the degree to which goods are substitutes takes an intermediate value; otherwise, the two firms share ownership. In this case, if the union sets the same wage for all workers neither firm shares ownership. Therefore, centralised wage setting discourages share ownership. Finally, under Bertrand competition neither firm shares ownership regardless of how workers are organised to negotiate wages.

Appendix

1.1 Second order conditions under Cournot competition

Decentralised wage setting. In the second stage, firm i chooses the value of α _i that maximises B _i  = (1–α _i )π _i . Solving this problem we obtain first order conditions:

$$ \frac{\partial {B}_i}{\partial {\alpha}_i}=-\frac{4{a}^2\left({b}^4-4{b}^2\left(3+{\alpha}_i-{\alpha}_j\right)-16{\alpha}_i\left(-2+{\alpha}_j\right)\right){\left(-8+2b+{b}^2+4{\alpha}_j\right)}^2}{{\left({b}^4+16\left(-2+{\alpha}_i\right)\left(-2+{\alpha}_j\right)+4{b}^2\left(-5+{\alpha}_i+{\alpha}_j\right)\right)}^3}=0,i\ne j;\ i,\ j=1,2. $$

Thus,

$$ \frac{\partial^2{B}_i}{\partial {\alpha_i}^2}=\frac{64{a}^2\left({b}^4-2{b}^2\left(7+{\alpha}_i-2{\alpha}_j\right)+8\left(1+{\alpha}_i\right)\left(2-{\alpha}_j\right)\right)\left(-8+{b}^2+4{\alpha}_j\right){\left(8-2b-{b}^2-4{\alpha}_j\right)}^2}{{\left({b}^4-16\left(2-{\alpha}_i\right)\left(2-{\alpha}_j\right)+4{b}^2\left(-5+{\alpha}_i+{\alpha}_j\right)\right)}^4}. $$

As α _i <1/2, α _j <1/2 and b is such that 0 < b < 1, we find that b ⁴ − 2b ²(7 + α _i  − 2α _j ) + 8(1 + α _i )(2 − α _j ) = (16 − 14b ² + b ⁴ − 8α _j  + 4b ² α _j ) + (16α _i  − 2b ² α _i  − 8α _i α _j ) > 0, (−8 + b ² + 4α _j ) < 0 and (8 − 2b − b ² − 4α _j )² > 0. As a result ∂² B _i /∂α _i ² < 0.

Centralised wage setting. In the second stage, firm i chooses the value of α _i that maximises B _i  = (1–α _i )π _i . Solving this problem we obtain first order conditions:

$$ \frac{\partial {B}_i}{\partial {\alpha}_i}=-\frac{a^2\left({b}^2+{\alpha}_i\left(-2+{\alpha}_j\right)\right){\left(2-b-{\alpha}_j\right)}^2}{4{\left({b}^2-\left(2-{\alpha}_i\right)\left(2-{\alpha}_j\right)\right)}^3}=0,i\ne j;\ i,\ j = 1,\ 2. $$

Thus,

$$ \frac{\partial^2{B}_i}{\partial {\alpha_i}^2}=-\frac{a^2\left(2{b}^2-\left(1+{\alpha}_i\right)\left(2-{\alpha}_j\right)\right)\left(-2+{\alpha}_j\right){\left(2-b-{\alpha}_j\right)}^2}{2{\left({b}^2-\left(2-{\alpha}_i\right)\left(2-{\alpha}_j\right)\right)}^4}, $$

which is negative since α _i and α _j are lower than 1/2 and b is such that 0 < b < 1.

1.2 Bertrand competition and decentralised wage setting

From (13) the profit of firm i can be expressed as:

$$ \begin{array}{c}\hfill {\pi}_i = {a}^2\left(1-b\right){\left(2-{b}^2\right)}^2{\left(8+6b-3{b}^2-2{b}^3-2\left(2+2b-{b}^2-{b}^3\right){\alpha}_j\right)}^2/\hfill \\ {}\hfill \left(\left(1+b\right)\ \left(64-84{b}^2+33{b}^4-4{b}^6-\left(32-52{b}^2+26{b}^4-4{b}^6\right){\alpha}_j+\right.\right.\hfill \\ {}\hfill \left.{\left.2\left(2-{b}^2\right){\alpha}_i\left(-8+9{b}^2-2{b}^4+2\left(2-3{b}^2+{b}^4\right){\alpha}_j\right)\right)}^2\right).\hfill \end{array} $$

It can be shown that ∂πi/∂α i > 0. In the second stage, firm i chooses the value of α _i that maximises B _i  = (1–α _i )π _i , i = 1, 2. Solving these problems the following is obtained:

$$ \begin{array}{l}\frac{\partial {B}_i}{\partial {\alpha}_i}=\Big({a}^2\left(1-b\right){\left(2-{b}^2\right)}^2{\left(8+6b-3{b}^2-2{b}^3+2\left(-2-2b+{b}^2+{b}^3\right){\alpha}_j\right)}^2\\ {}\left(2\left(-2+{b}^2\right){\alpha}_i\left(-8+9{b}^2-2{b}^4+2\left(2-3{b}^2+{b}^4\right){\alpha}_j\right)+\right.\\ {}\left.\left.{b}^2\left(20-19{b}^2+4{b}^4-2\left(6-7{b}^2+2{b}^4\right){\alpha}_j\right)\right)\right)/\ \left(\left(1+b\right)\right.\\ {}\left(84{b}^2-33{b}^4+4{b}^6-64+\left(32-52{b}^2+26{b}^4-4{b}^6\right){\alpha}_j-\right.\\ {}\left.{\left.2\left(2-{b}^2\right){\alpha}_i\left(-8+9{b}^2-2{b}^4+2\left(2-3{b}^2+{b}^4\right){\alpha}_j\right)\right)}^3\right).\end{array} $$

The numerator of the above expression is positive and the denominator is negative since b < 1, α _i  < 1/2 and α _j <1/2. This means that B _i strictly decreases with α _i , and thus α _i  = 0, i ≠ j; i, j = 1, 2.

1.3 Bertrand competition and centralised wage setting

In the fourth stage, firm i chooses the price, p _i , that maximises B _i  = (1–α _i )π _i . Solving these problems gives expression (11). In the third stage, the union chooses the wages w _i and w _j that maximise its total income: I _i  = α _i π _i  + w _i q _i  + α _j π _j  + w _j q _j . Solving these problems it is obtained that the wage paid by firm i (i ≠ j; i, j = 1, 2) is:

$$ {w}_i=\frac{a\left(b{\alpha}_i+2\left(1-{\alpha}_i\right)\left(2-{\alpha}_j\right)-{b}^2\left(1-2{\alpha}_i\right)\left(1-{\alpha}_j\right)\right)}{2\left(\left(2-{\alpha}_i\right)\left(2-{\alpha}_j\right)+{b}^2\left(1-{\alpha}_i\right)\left(1-{\alpha}_j\right)\right)}. $$

Thus, the profit of firm i can be expressed as:

$$ {\pi}_i = \frac{a^2\left(1-b\right){\left(2+b\left(1-{\alpha}_j\right)-{\alpha}_j\right)}^2}{4\left(1+b\right){\left(\left(2-{\alpha}_i\right)\left(2-{\alpha}_j\right)+{b}^2\left(1-{\alpha}_i-{\alpha}_j+{\alpha}_i{\alpha}_j\right)\right)}^2}. $$

In the second stage, firm i chooses the value of α _i that maximises B _i  = (1–α _i ) π _i . Solving these problems the following is obtained:

$$ \frac{\partial {B}_i}{\partial {\alpha}_i}=\frac{a^2\left(1-b\right)\left({\alpha}_i\left(2-{\alpha}_j\right)+{b}^2\left(1-{\alpha}_i\right)\left(1-{\alpha}_j\right)\right){\left(2+b-\left(1+b\right){\alpha}_j\right)}^2}{4\left(1+b\right){\left(-\left(2-{\alpha}_i\right)\left(2-{\alpha}_j\right)+{b}^2\left(1-{\alpha}_i\right)\left(1-{\alpha}_j\right)\right)}^3} $$

The numerator of the above expression is positive and the denominator is negative since b < 1, α _i  < 1/2 and α _j <1/2. This means that B _i strictly decreases with α _i , and thus α _i  = 0, i ≠ j; i, j = 1, 2.

1.4 Cut-off $\overline{\alpha}$ lower than 1/2

Decentralised wage-setting. It can be shown that when $\alpha =\overline{\alpha}$ the income of each owner, in the different cases, is:

$$ {\overline{B}}^{YY}=\frac{4{a}^2\left(1-\overline{\alpha}\right)}{{\left(8+2b-{b}^2-4\overline{\alpha}\right)}^2},\ {\overline{B}}^{YN}=\frac{4{a}^2{\left(2-b\right)}^2{\left(4+b\right)}^2\left(1-\overline{\alpha}\right)}{{\left({b}^4-4{b}^2\left(5-\overline{\alpha}\right)+32\left(2-\overline{\alpha}\right)\right)}^2},\ {\overline{B}}^{NY}=\frac{4{a}^2{\left(8-2b-{b}^2-4\overline{\alpha}\right)}^2}{{\left({b}^4-4{b}^2\left(5-\overline{\alpha}\right)+32\left(2-\overline{\alpha}\right)\right)}^2} $$

For $\overline{\alpha}=0.01$ it is obtained that ${\alpha}^{YY}\le \overline{\alpha}$ for b = 0.1631, and ${\alpha}^{YN}\le \overline{\alpha}$ for b = 0.1632. Therefore, if b < 0.1631 we obtain the same result as in Section 3 (since ${\alpha}^{YY}<\overline{\alpha}$ and ${\alpha}^{YN}<\overline{\alpha}$). If 0.1630 ≤ b < 0.1632, we find that α ^YY = 0.01, α ^YN < 0.01, ${\overline{B}}^{YY}>{B}^{NY}$ and ${\overline{B}}^{YN}>{B}^{NN}$. Finally, if b ≥ 0.1632 we find that α ^YY = 0.01, α ^YN = 0.01, ${\overline{B}}^{YY}>{\overline{B}}^{NY}$ and ${\overline{B}}^{YN}>{B}^{NN}$. Therefore, the best response of each firm in all three cases is to share ownership, so in equilibrium that is what the firms do. A similar result is obtained for other values of $\overline{\alpha}$ (for example, for $\overline{\alpha}$ =0.1 and $\overline{\alpha}=0.05$).

Centralised wage-setting. It can be shown that when $a=\overline{\alpha}$ the income of each owner, in the different cases, is:

$$ {\overline{B}}^{YY}=\frac{a^2\left(1-\overline{\alpha}\right)}{4{\left(2+b-\overline{\alpha}\right)}^2},\ {\overline{B}}^{YN}=\frac{a^2{\left(2-b\right)}^2\left(1-\overline{\alpha}\right)}{4{\left(4-{b}^2-2\overline{\alpha}\right)}^2},\kern0.37em {\overline{B}}^{NY}=\frac{a^2{\left(2-b-\overline{\alpha}\right)}^2}{4{\left(4-{b}^2-2\overline{\alpha}\right)}^2}. $$

For $\overline{\alpha}$ =0.05 it is obtained that ${\alpha}^{YY}\le \overline{\alpha}$ for b = 0.3122, and ${\alpha}^{YN}\le \overline{\alpha}$ for b = 0.3162. Therefore, if b < 0.3122 we obtain the same result as in Section 3 (since ${\alpha}^{YY}<\overline{\alpha}$ and ${\alpha}^{YN}<\overline{\alpha}$). If 0.3122 ≤ b < 0.3162 we find that α ^YY = 0.05, α ^YN < 0.05, ${\overline{B}}^{YY}>{B}^{NY}$ and ${\overline{B}}^{YN}>{B}^{NN}$. Finally, if b ≥ 0.1632 we find that α ^YY = 0.05, α ^YN = 0.05, ${\overline{B}}^{YY}>{\overline{B}}^{NY}$ and ${\overline{B}}^{YN}>{B}^{NN}$. Therefore, the best response of each firm in all three cases is to share ownership, so in equilibrium that is what the firms do. Therefore, for a low cut-off the asymmetric equilibrium disappears. There is an asymmetric equilibrium as long as b ₃ > b ₂. Therefore, when b ₃ ≤ b ₂ the only equilibrium is that in which both firms share ownership. It is shown in Section 4 that B ^YY = B ^NY for b = b ₂ = 0.7691 (note that it does not depend on $\overline{\alpha}$). Moreover, b ₃ is such that ${\overline{B}}^{YY}={B}^{NY}$. It can be shown that b ₃ = b ₂ = 0.7691 for $\overline{\alpha}$ = 0.3610. Therefore, if $\overline{\alpha}$ ≤ 0.3610 we obtain that in equilibrium both firms share ownership for all values of parameter b.

1.5 Welfare analysis under Cournot competition

Decentralised wage setting. In this case we obtain the following:

$$ \begin{array}{l}C{S}^{NN}=\frac{4{a}^2\left(1+b\right)}{{\left(4-b\right)}^2{\left(2+b\right)}^2},\;P{S}^{NN}=\frac{8{a}^2}{{\left(4-b\right)}^2{\left(2+b\right)}^2},\;2\ {I}^{NN}=\frac{4{a}^2\left(2-b\right)}{{\left(4-b\right)}^2\left(2+b\right)},{W}^{NN}=\frac{4{a}^2\left(7+b-{b}^2\right)}{{\left(4-b\right)}^2{\left(2+b\right)}^2}.\\ {}C{S}^{YY} = \frac{4{a}^2\left(1+b\right)}{{\left(4+2b-{b}^2+\sqrt{16-12{b}^2+{b}^4}\right)}^2},\;P{S}^{YY}, = \frac{a^2\sqrt{16-12{b}^2+{b}^4}}{{\left(4+2b-{b}^2+\sqrt{16-12{b}^2+{b}^4}\right)}^2},\\ {}2{I}^{YY} = \frac{a^2\left(4-2{b}^2+\sqrt{16-12{b}^2+{b}^4}\right)}{{\left(4+2b-{b}^2+\sqrt{16-12{b}^2+{b}^4}\right)}^2},\;{W}^{YY} = \frac{4{a}^2\left(3+b-{b}^2+\sqrt{16-12{b}^2+{b}^4}\right)}{{\left(4+2b-{b}^2+\sqrt{16-12{b}^2+{b}^4}\right)}^2}.\\ {}C{S}^{YN} = \frac{a^2\left(8192+4096b-9216{b}^2-3200{b}^3+3216{b}^4+880{b}^5-408{b}^6-100{b}^7+17{b}^8+4{b}^9\right)}{2{\left(8-{b}^2\right)}^2{\left(32-16{b}^2+{b}^4\right)}^2},\\ {}P{S}^{YN} = \frac{a^2\left({\left(2-b\right)}^2{\left(4+b\right)}^2\left(8-{b}^2\right)\left(32-16{b}^2+{b}^4\right)+16{\left(32-8b-14{b}^2+{b}^3+{b}^4\right)}^2\right)}{4{\left(8-{b}^2\right)}^2{\left(32-16{b}^2+{b}^4\right)}^2},\\ {}{I}^{YN}+{I}^{NY} = \frac{a^2\left(2-b\right)\left(32768-31744{b}^2-2560{b}^3+10240{b}^4+1600{b}^5-1240{b}^6-244{b}^7+50{b}^8+11{b}^9\right)}{4{\left(8-{b}^2\right)}^2{\left(32-16{b}^2+{b}^4\right)}^2},\\ {}{W}^{YN}=\frac{a^2\left(57344-20480b-54272{b}^2+15488{b}^3+18544{b}^4-3696{b}^5-2920{b}^6+356{b}^7+215{b}^8-12{b}^9-6{b}^{10}\right)}{2{\left(8-{b}^2\right)}^2{\left(32-16{b}^2+{b}^4\right)}^2}.\\ {}\end{array} $$

From the above expressions we obtain the following:

$$ \begin{array}{l}{W}^{YY}-{W}^{YN}={a}^2\Big(-131072+131072b+245760{b}^2-190464{b}^3-170752{b}^4+\\ {}98176{b}^5+60032{b}^6-23072{b}^7-12064{b}^8+2760{b}^9+1384{b}^{10}-166{b}^{11}-83{b}^{12}+\\ {}4{b}^{13}+2{b}^{14}-\left(\sqrt{16-12{b}^2+{b}^4}\right)\Big(-32768+32768b+12288{b}^2-26112{b}^3+\\ {}7872{b}^4+6816{b}^5-4848{b}^6-720{b}^7+908{b}^8+26{b}^9-71{b}^{10}+2{b}^{12}\left)\right)/\\ {}\left({\left(8-{b}^2\right)}^2{\left(32-16{b}^2+{b}^4\right)}^2{\left(4+2b-{b}^2+\sqrt{16-12{b}^2+{b}^4}\right)}^2\right)>0\ \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\;b;\\ {}{W}^{YN}-{W}^{NN}={a}^2{b}^2\left(12-{b}^2\right)\Big(24576-6144b-23296{b}^2+4864{b}^3+7888{b}^4-\\ {}1216{b}^5-1192{b}^6+120{b}^7+81{b}^8-4{b}^9-2{b}^{10}\Big)/\\ {}\left(2{\left(4-b\right)}^2{\left(2+b\right)}^2{\left(8-{b}^2\right)}^2{\left(32-16{b}^2+{b}^4\right)}^2\right)>0\ \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\;b\end{array} $$

Therefore: W ^YY > W ^YN > W ^NN for all b. Operating in a similar way we obtain the following: CS ^YN > CS ^YY > CS ^NN >, PS ^NN > PS ^YN > PS ^YY, and 2I ^YY > I ^YN + I ^NY > 2I ^NN.

Centralised wage setting. In this case we obtain the following:

$$ \begin{array}{l}{W}^{NN}=\frac{a^2\left(7+3b\right)}{4{\left(2+b\right)}^2},\;C{S}^{NN}=\frac{a^2\left(1+b\right)}{4{\left(2+b\right)}^2},\ P{S}^{NN}=\frac{a^2}{2{\left(2+b\right)}^2},2\ {I}^{NN}=\frac{2{a}^2}{8+2b-{b}^2}\\ {}{W}^{YY}=\frac{a^2\left(3+3b+4\sqrt{1-{b}^2}\right)}{4{\left(1+b+\sqrt{1-{b}^2}\right)}^2},\;P{S}^{YY}=\frac{a^2\sqrt{1-{b}^2}}{2{\left(1+b+\sqrt{1-{b}^2}\right)}^2},\ C{S}^{YY}=\frac{a^2}{8+8\sqrt{1-{b}^2}}\\ {}\ 2{I}^{YY}=\frac{a^2}{3+2b}\ \mathrm{if}b\ge {b}_1;\;{W}^{YN}=\frac{a^2\left(224-128b-128{b}^2+60{b}^3+11{b}^4\right)}{128{\left(2-{b}^2\right)}^2},\;P{S}^{YN}=\\ {}\frac{a^2\left(32-32b-8{b}^2+12{b}^3-{b}^4\right)}{64{\left(2-{b}^2\right)}^2},\ C{S}^{YY}=\frac{a^2\left(32-32{b}^2+4{b}^3+5{b}^4\right)}{128{\left(2-{b}^2\right)}^2},\ {I}^{YN}+{I}^{NY}=\frac{a^2\left(8-4b-{b}^2\right)}{16\left(2-{b}^2\right)}\end{array} $$

From the above expressions we obtain the following:

$$ \begin{array}{l}{W}^{YY}-{W}^{YN} = {a}^2\left(1+b\right)\left(128b-64{b}^2-60{b}^3+37{b}^4-32\right)+\sqrt{1-{b}^2}\Big(32-96b+\\ {}68{b}^3-7{b}^4-11{b}^5\Big)/\left(64{\left(-2+{b}^2\right)}^2{\left(1+b+\sqrt{1-{b}^2}\right)}^2\right)>0\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\;b,{b}_1>b>0;\\ {}{W}^{YY}-{W}^{YN}=-\frac{a^2\left(-544+768{b}^2+28{b}^3-333{b}^4-12{b}^5+44{b}^6\right)}{128{\left(3+2b\right)}^2{\left(2-{b}^2\right)}^2}>0\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\;b,\ 1>b\ge {b}_1;\\ {}{W}^{YN}-{W}^{NN}=\frac{a^2{b}^2\left(96-16b-68{b}^2+8{b}^3+11{b}^4\right)}{128{\left(2+b\right)}^2{\left(-2+{b}^2\right)}^2}>0\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\;b.\end{array} $$

Therefore: W ^YY > W ^YN > W ^NN for all b. Operating in a similar way we obtain the following: CS ^YY > CS ^YN > CS ^NN, PS ^NN > PS ^YN > PS ^YY and 2I ^NN > 2I ^YY > I ^YN + I ^NY for all b.