On Tax Over-Shifting in Wage Bargaining Models

Abstract

It has frequently been noted in the wage bargaining literature that increasing average labour taxes may in fact be over-shifted in the pre-tax wage that is negotiated between unions and firms, raising workers post-tax wages. In this paper, we study the precise conditions for such tax over-shifting to occur under both Nash and Right-To-Manage bargaining structures, and considering both competitive and imperfectly competitive output market conditions. In the case of competitive output markets, we derive and interpret the conditions for over-shifting to occur and show that they hold for an entire class of commonly used production functions. Moreover, under monopolistically competitive output markets we show that tax over-shifting will occur when the firm has sufficient market power. The conditions on the production function, that were necessary and sufficient for tax over-shifting to occur under perfect competition, are shown to be no longer necessary. These findings hold for all bargaining structures considered.

Appendices

Appendix A

We here derive the condition for tax over-shifting in the case of Nash bargaining over wages and employment with imperfectly competitive output markets. First, differentiating the system of first-order conditions (18a–18b) and solving by Cramer’s rule yields the effect of a tax increase on the pre-tax wage:

$$ \frac{{dw}} {{dz}} = - \frac{{g_{{1z}} g_{{2L}} }} {{\rm H}} $$

where ${\rm H} = g_{{1w}} g_{{2L}} - g_{{2w}} g_{{1L}}$. The various terms are given by:

$$ \begin{array}{*{20}c} {g_{{1w}} = - \lambda \prime {\left( {1 - T_{w} } \right)}^{2} {\left[ {w - pf\prime {\left( {1 + \frac{1} {\beta }} \right)}} \right]}} \\ {g_{{1L}} = \lambda {\left( {1 - T_{w} } \right)}{\left[ {p{\left( {1 + \frac{1} {\beta }} \right)}f + pf\prime {\left( {1 + \frac{1} {\beta }} \right)}\frac{{f\prime }} {{\beta f}}} \right]}} \\ {g_{{2w}} = 1} \\ {g_{{2L}} = \frac{\mu } {{L^{2} }}{\left[ {pf - pLf\prime {\left( {1 + \frac{1} {\beta }} \right)}} \right]} - {\left( {1 - \mu } \right)}{\left[ {p{\left( {1 + \frac{1} {\beta }} \right)}f + pf\prime {\left( {1 + \frac{1} {\beta }} \right)}\frac{{f\prime }} {{\beta f}}} \right]}} \\ {g_{{1z}} = T_{z} {\left\{ {\lambda \prime {\left( {1 - T_{w} } \right)}{\left[ {w - pf\prime {\left( {1 + \frac{1} {\beta }} \right)}} \right]} - \lambda } \right\}}} \\ \end{array} $$

Using λ′ < 0, assuming positive profit, and noting that system (18a and 18b) implies:

$$ w > pf\prime {\left( {1 + \frac{1} {\beta }} \right)} $$

the following signs can be shown to hold: g _1w  > 0, g _1L  < 0, g _2L  > 0, g _1z  < 0, H > 0. It then immediately follows that a tax increase raises negotiated wages:$\frac{{dw}} {{dz}} > 0$.

The effect of a tax increase on the net of tax wage is given by:

$$ \frac{{d\widetilde{W}}} {{dz}} = \frac{{dw}} {{dz}}{\left( {1 - T_{w} } \right)} - T_{z} $$

which leads after standard manipulations, using the above expressions, to:

$$ \frac{{{\text{d}}\widetilde{W}}} {{dz}} = p{\left\{ {\frac{{T_{z} }} {{\rm H}}\lambda {\left[ {1 - T_{w} } \right]}\frac{\mu } {L}} \right\}}{\left\{ {\frac{f} {L} - f\prime {\left( {1 + \frac{1} {\beta }} \right)} + {\left( {1 + \frac{1} {\beta }} \right)}{\left[ {fL + \frac{{{\left( {f\prime } \right)}^{2} L}} {{\beta f}}} \right]}} \right\}} $$

(A.1)

Next, note that simple algebra allows us to write:

$$ \frac{f} {L} - f\prime {\left( {1 + \frac{1} {\beta }} \right)} + {\left( {1 + \frac{1} {\beta }} \right)}{\left[ {fL + \frac{{{\left( {f\prime } \right)}^{2} L}} {{\beta f}}} \right]} = {\left( {fL + \frac{f} {L} - f\prime } \right)} + K $$

where

$$ K = \frac{1} {\beta }{\left\{ {fL - f\prime + \frac{{{\left( {f\prime } \right)}^{2} L}} {f}{\left( {1 + \frac{1} {\beta }} \right)}} \right\}} $$

(A.2)

Substitution of Eq. A.2 in Eq. A.1 then yields:

$$ \frac{{{\text{d}}\widetilde{W}}} {{{\text{d}}z}} = p{\left\{ {\frac{{T_{z} }} {{\rm H}}\lambda {\left[ {1 - T_{w} } \right]}\frac{\mu } {L}} \right\}}{\left\{ {{\left( {fL + \frac{f} {L} - f\prime } \right)} + K} \right\}} $$

so that ${\left( {fL + \frac{f} {L} - f\prime } \right)} + K > 0$ is a sufficient condition for tax over-shifting. This is expression (20) in the main body of the paper.

The term K captures the effects of imperfect competition. If we assumed perfect competition, K would be zero and we reproduce the results for perfect competition, see Eq. 10. Note that K can be shown to be positive as follows. First, use the Nash curve (18b) to obtain:

$$ f\prime {\left( {1 + \frac{1} {\beta }} \right)} = \frac{w} {{{\left( {1 - \mu } \right)}p}} - \frac{\mu } {{1 - \mu }}\frac{f} {L} $$

Substitute this into Eq. A.2, use the definition of profit $\pi = pf - wL$, and rearrange to find:

$$ K = \frac{1} {\beta }{\left[ {\frac{{ - f\prime \pi }} {{{\left( {1 - \mu } \right)}pf}} + fL} \right]} $$

This is positive as long as profit is positive.

Finally, interpretation of Eq. 20 is the same as under perfect competition. One easily shows that:

$$ sgn{\left\{ {\frac{{\partial {\left[ {\frac{{pf}} {L} - pf\prime {\left( {1 + \frac{1} {\beta }} \right)}} \right]}}} {{\partial L}}} \right\}} = \operatorname{s} gn{\left\{ {{\left( {fL + \frac{f} {L} - f\prime } \right)} + K} \right\}} $$

so that tax over-shifting occurs when the difference between average and marginal revenue product curves is declining in employment. Alternatively, the condition can again also be interpreted in terms of the difference in the slope of the Nash curve and the firm’s labour demand curve. Using Eq. 18b, the slope of the Nash curve is easily shown to be given by:

$$ \frac{{\partial L^{{{\text{Nash}}}} }} {{\partial w}} = \frac{1} {{p{\left( {1 + \frac{1} {\beta }} \right)}{\left[ {f + \frac{{{\left( {f\prime } \right)}^{2} }} {{f\beta }}} \right]} - \frac{\mu } {L}{\left\{ {{\left( {fL + \frac{f} {L} - f\prime } \right)} + K} \right\}}}} $$

(A.3)

Similarly, the slope of the labour demand curve is easily derived as:

$$ \frac{{\partial L^{{\text{d}}} }} {{\partial w}} = \frac{1} {{p{\left( {1 + \frac{1} {\beta }} \right)}{\left[ {f + \frac{{{\left( {f\prime } \right)}^{2} }} {{f\beta }}} \right]}}} $$

(A.4)

Comparison of the slopes immediately produces the desired result. Finally, the expression for the slope of the Nash curve also shows that tax over-shifting depends on the effect of higher union power on the wage sensitivity of labour demand.

For purposes of further interpretation, note that, at given employment, K can be written as a function of the price elasticity only, i.e., K(). Moreover, simple differentiation then shows that, provided that K() > 0, we have $\frac{{\partial K{\left( \beta \right)}}} {{\partial \beta }} > 0$. This observation is used in the main body of the paper.

Appendix B

For the right to manage model, the first order condition of the bargaining problem can be written as:

$$ g_{3} {\left( {w,z} \right)} = \mu {\left[ {\frac{{\lambda {\left( {1 - T_{w} } \right)}}} {{U{\left( . \right)} - D}} + \frac{1} {L}\frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}}} \right]} - {\left( {1 - \mu } \right)}\frac{{L^{{\text{d}}} {\left( w \right)}}} {\pi } = 0 $$

It immediately follows that an increase in the average tax rate raises the negotiated wage:

$$ \frac{{{\text{d}}w}} {{{\text{d}}z}} = - \frac{{g_{{3z}} }} {{g_{{3w}} }} = \frac{1} {{g_{{3w}} }}{\left\{ {\frac{{\mu T_{z} {\left( {1 - T_{w} } \right)}}} {{U - D}}{\left[ {\lambda \prime - \frac{{{\left( \lambda \right)}^{2} }} {{U - D}}} \right]}} \right\}} > 0 $$

since $g_{{3w}} = \mu {\left[ {\frac{{{\left( {1 - T_{w} } \right)}^{2} }} {{U - D}}{\left( {\lambda \prime - \frac{{\lambda ^{2} }} {{U - D}}} \right)} + \frac{1} {L}{\left( {\frac{{\partial ^{2} L^{{\text{d}}} {\left( w \right)}}} {{\partial ^{2} w}} - \frac{1} {L}{\left( {\frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}}} \right)}^{2} } \right)}} \right]} - {\left( {1 - \mu } \right)}{\left[ {\frac{1} {\pi }\frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}} + \frac{{L^{2} }} {{\pi ^{2} }}} \right]}$ is negative by the second order condition.

Furthermore, the impact on the net of tax wage is given by:

$$ \frac{{{\text\rm{d}}\widetilde{W}}} {{{\text{\rm{d}}}z}} = \frac{{{\text\rm{d}}w}} {{{\text\rm{d}}z}}{\left( {1 - T_{w} } \right)} - T_{z} $$

This can easily be shown to be:

$$ \frac{{{\text{d}}\widetilde{W}}} {{{\text{d}}z}} = \frac{{T_{z} }} {{g_{{3w}} }}{\left\{ { - \frac{\mu } {{L^{2} }}{\left( {L\frac{{\partial ^{2} L^{{\text{d}}} {\left( w \right)}}} {{\partial ^{2} w}} - {\left( {\frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}}} \right)}^{2} } \right)} + \frac{{1 - \mu }} {{\pi ^{2} }}{\left[ {\pi \frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}} + L^{2} } \right]}} \right\}} $$

(B.1)

An interesting issue is whether imperfect competition raises or reduces the likelihood of tax over-shifting under right to manage bargaining. By working out the various terms appearing in Eq. B.1, this question can be answered.

First, differentiating the slope of the labour demand curve derived before, see Eq. A.4, and rearranging we have:

$$ \frac{{\partial ^{2} L^{{\text{d}}} {\left( w \right)}}} {{\partial ^{2} w}} = - {\left( {\frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}}} \right)}^{3} {\left[ {p{\left( {1 + \frac{1} {\beta }} \right)}} \right]}{\left\{ {f\prime + \frac{{3f\prime f}} {{f\beta }} + \frac{{{\left( {f\prime } \right)}^{3} {\left( {1 - \beta } \right)}}} {{{\left( {f\beta } \right)}^{2} }}} \right\}} $$

(B.2)

Combining Eqs. A.4 and B.2, we further derive:

$$ L\frac{{\partial ^{2} L^{{\text{d}}} {\left( w \right)}}} {{\partial ^{2} w}} - {\left( {\frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}}} \right)}^{2} = - {\left( {\frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}}} \right)}^{3} {\left[ {p{\left( {1 + \frac{1} {\beta }} \right)}} \right]}{\left\{ {Lf\prime + f + R} \right\}} $$

(B.3)

where

$$ R = \frac{{f\prime }} {{f\beta }}{\left\{ {3Lf + f\prime {\left[ {1 + \frac{{Lf\prime }} {{f\beta }}{\left( {1 - \beta } \right)}} \right]}} \right\}} $$

(B.4)

It follows from Eq. B.3 that a sufficient condition for tax over-shifting under the monopoly union model is given by $Lf\prime + f + R > 0$, see Eq. 23 in the main body of the paper.

To see the determinants of the sign of R, note that a sufficient condition for it to be positive is that

$$ {\left[ {1 + \frac{{Lf\prime }} {{f\beta }}{\left( {1 - \beta } \right)}} \right]} $$

(B.5)

be negative. Now note from the condition for profit maximising behaviour

$$ w = pf\prime {\left( {1 + \frac{1} {\beta }} \right)} $$

that

$$ f\prime = \frac{w} {p}\frac{\beta } {{\beta + 1}} $$

Substituting in Eq. B.5 implies using the definition of profit and some simple rearrangement:

$$ {\left[ {1 + \frac{{Lf\prime }} {{f\beta }}{\left( {1 - \beta } \right)}} \right]} = \frac{1} {{\beta + 1}}{\left[ {\frac{{{\left( {\beta - 1} \right)}\pi }} {{pf}} + 2} \right]} $$

(B.6)

Since the price elasticity was assumed to strictly exceed unity in absolute value and profit relative to revenues is smaller than one, it follows from Eqs. B.4 to B.6 that R will be positive unless the price elasticity is very large, i.e., unless the firm has very limited market power.

Note that at given employment, the term R can be written directly as a function of the price elasticity only: R = R(). As before, simple differentiation then shows that, provided R() > 0, we have $\frac{{\partial R{\left( \beta \right)}}} {{\partial \beta }} > 0$. This observation is used to arrive at Proposition 4 in the text.

Second, when the firm has some negotiation power, the term $\pi \frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}} + L^{2}$ also matters. Rewrite this term as:

$$ \pi \frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}} + L^{2} = L\frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}}{\left[ {\frac{\pi } {L} + L\frac{1} {{\frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}}}}} \right]} $$

Then substitute the profit maximising price into the definition of profit, use Eq. A.4, and rearrange to find:

$$ \pi \frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}} + L^{2} = pL\frac{{\partial L^{{\text{d}}} {\left( w \right)}}} {{\partial w}}{\left[ {{\left( {fL + \frac{f} {L} - f\prime } \right)} + K} \right]} $$

(B.7)

where K is defined as in Eq. A.2:

$$ K = \frac{1} {\beta }{\left\{ {fL - f\prime + \frac{{{\left( {f\prime } \right)}^{2} L}} {f}{\left( {1 + \frac{1} {\beta }} \right)}} \right\}} $$

For the right to manage model, using the first order condition for profit maximising behaviour, K can again be shown to be positive.