Bargaining over productivity and wages when technical change is induced: implications for growth, distribution, and employment

Abstract

I study a model of growth and income distribution in which workers and firms bargain à la Nash (Econometrica 18(2):155–162, 1950) over wages and productivity gains, taking into account the trade-offs faced by firms in choosing factor-augmenting technologies. The aggregate environment resulting from self-interested, objective function-maximizing decision rules on wages, productivity gains, savings and investment, is described by a two-dimensional dynamical system in the employment rate and output/capital ratio. The economy converges cyclically to a long-run equilibrium involving a Harrod-neutral profile of technical change, a constant rate of employment of labor, and constant input shares. The type of oscillations predicted by the model is qualitatively consistent with the available data on the United States (1963–2003), replicates the dynamics found in earlier models of growth cycles such as Goodwin (A growth cycle, in C.H. Feinstein (ed). Socialism, Capitalism and Economic Growth. Cambridge University Press, Cambridge 1967. Cambridge University Press, Cambridge, 1967); Shah and Desai (Econ J 91:1006–1010, 1981); van der Ploeg (J Macroecon 9:1–12, 1987); Flaschel (J Econ: Zeitschrift für Nationalökonomie 44:63–69, 1984) and Sportelli (J Econ: Zeitschrift für Nationalökonomie 61(1):35–64, 1995), and can be verified numerically in simulations. Institutional change, as captured by variations in workers’ bargaining power, has a positive effect on the long-run rate of growth of output per worker but a negative effect on long-run employment. Economic policy can also affect the growth and distribution pattern through changes in the unemployment compensation, which also have a positive long-run impact on labor productivity growth but a negative long-run impact on employment. In both cases, employment can overshoot its new equilibrium value along the transitional dynamics.

Appendices

Appendix A: Solution of the bargaining problem

1.1 A.1 Proof of Proposition 1

Form the current-value Hamiltonian:

$$\begin{aligned} H_B=\eta \log (w-\rho V_U)+(1-\eta )\log \left(B\left(1-\frac{w}{A}\right)\right)+\gamma _1 \beta B+\gamma _2 g(\beta )A \end{aligned}$$

(22)

and let harmlessly $\gamma _1 \equiv p_1 e^{-\beta ^* t}, \gamma _2 \equiv p_1 e^{-g(\beta ^*)t}$, $\beta ^*$ being the solution value for $\beta$ to be determined. Necessary conditions for maximization are:

$$\begin{aligned} \frac{\partial H_B}{\partial w}&= \frac{\eta }{w- \rho V_U}-\frac{1-\eta }{A} \left(\frac{1}{1-\frac{w}{A}}\right)=0\end{aligned}$$

(23)

$$\begin{aligned} \frac{\partial H_B}{\partial \beta }&= p_1e^{-\beta ^* t} B+p_2 e^{-g(\beta ^*)t} g_\beta A=0 \end{aligned}$$

(24)

Solving (23) for $w$ gives (4). Also, a necessary condition for optimality of the solution is the existence of continuous function $p_1(t), p_2(t)$ satisfying:

$$\begin{aligned} (\rho +\lambda +\beta ^*)p_1 e^{-\beta ^*t}-e^{-\beta ^*t}\dot{p}_1&= \frac{1-\eta }{B}+\beta ^* p_1 e^{-\beta ^* t} \end{aligned}$$

(25)

$$\begin{aligned} (\rho {+}\lambda {+}g(\beta ^*))p_2 e^{-g(\beta ^*)t}{-}e^{-g(\beta ^*)t}\dot{p}_2&= \frac{1-\eta }{A}\left(\frac{\omega }{1-\omega }\right){+}g(\beta ^*)p_2 e^{-g(\beta ^*)t}\qquad \end{aligned}$$

(26)

where the LHS of the two equations are equal to the familiar $\rho \gamma _i-\dot{\gamma }_i, i=1,2$, and $\omega \equiv w/A$ is the share of wages in firm’s output. Also, the following transversality conditions must hold at an optimal control path:

$$\begin{aligned} \lim _{t\rightarrow \infty } e^{-\beta ^*t}p_1(t)=\lim _{t\rightarrow \infty } e^{-g(\beta ^*)t}p_2(t)=0 \end{aligned}$$

(27)

Because of concavity of the objective function and convexity of the constraint set, conditions (23)–(26) are also sufficient for a maximum of problem (3).

A stationary solution for (25) and (26) where $\dot{p}_1=\dot{p}_2=0$ satisfies:

$$\begin{aligned} e^{-\beta ^* t} p_1 B&= \frac{1-\eta }{\rho }\\ e^{-g(\beta ^*)t} p_2 A&= \frac{1-\eta }{\rho }\left(\frac{\omega }{1-\omega }\right) \end{aligned}$$

Divide the first equation by the second and use (24) to get (5).

1.2 A.2 Behavior of rates of factor-augmentations at the firm level

In order to get started in studying the behavior of factor-augmentations, rewrite Eq. (5) as:

$$\begin{aligned} G(\beta , \omega ;\rho ) \equiv \frac{\omega }{1-\omega } +g_\beta =0 \end{aligned}$$

Such an equation yields an implicit function $\beta (\omega , \rho )$ whose partial derivatives fulfill standard properties. We have that:

$$\begin{aligned} G_\beta&= g_{\beta \beta }<0 \end{aligned}$$

Clearly,

$$\begin{aligned} G_\omega&= -\frac{1}{\omega ^2} \end{aligned}$$

Therefore,

$$\begin{aligned} \text{ sign} \left(\beta _\omega \right) =\text{ sign} \left( - \frac{G_\omega }{G_\beta }\right) \end{aligned}$$

so that $\beta _\omega <0$. Our assumption about the IPF imply $\alpha _\omega >0$.

Appendix B: Proofs of various propositions

1.1 B.1 Proof of Proposition 2

We have:

$$\begin{aligned} \frac{\partial w}{\partial v}&= \eta (1-\eta )\frac{q_v(\rho +\lambda )}{\left[\rho +\lambda +\eta q(v)\right]^2}A>0\\ \frac{\partial w}{\partial \eta }&= \frac{\rho +\lambda +q(v)}{[\rho +\lambda +\eta q(v)]^2}(\rho +\lambda )A>0 \end{aligned}$$

Furthermore,

$$\begin{aligned} \frac{\partial w}{\partial \lambda }=\frac{\eta (\eta -1)}{[\rho +\lambda +\eta q(v)]^2}A<0 \end{aligned}$$

Finally,

$$\begin{aligned} \frac{\partial w}{\partial \rho }= \frac{\eta (\eta -1)q(v)}{[\rho +\lambda +\eta q(v)]^2}A<0 \end{aligned}$$

1.2 B.2 Proof of Proposition 3

Using the same approach as in Appendix A.2, we can construct a function $\Gamma (\beta , v; \rho , \lambda , \eta )=0$ such that:

$$\begin{aligned} \Gamma _\beta =g_{\beta \beta }<0 \end{aligned}$$

Since

$$\begin{aligned} \Gamma _v=-\frac{1-\eta }{\eta }\left[\frac{q_v(\rho +\lambda )}{(\rho +q(v)+\lambda )^2}\right]<0 \end{aligned}$$

we have that:

$$\begin{aligned} \text{ sign} \beta _v=\text{ sign} \left(-\frac{\Gamma _v}{\Gamma _\beta }\right)<0 \end{aligned}$$

and our assumptions about the IPF ensure that $\alpha _v>0$. Also,

$$\begin{aligned} \Gamma _\eta =-\frac{1}{\eta ^2}\left[\frac{\rho +\lambda }{\rho +q(v)+\lambda }\right]<0 \end{aligned}$$

Hence,

$$\begin{aligned} \text{ sign} \beta _\eta =\text{ sign} \left(-\frac{\Gamma _\eta }{\Gamma _\beta }\right)<0 \end{aligned}$$

and $\alpha _\eta >0$. Further,

$$\begin{aligned} \Gamma _\lambda =\frac{1-\eta }{\eta }\left[\frac{q(v)}{(\rho +q(v)+\lambda )^2}\right]>0 \end{aligned}$$

so that

$$\begin{aligned} \text{ sign} \beta _\lambda =\text{ sign} \left(-\frac{\Gamma _\lambda }{\Gamma _\beta }\right)>0 \end{aligned}$$

Therefore, $\alpha _\lambda <0$. Finally,

$$\begin{aligned} \Gamma _\rho =\frac{1-\eta }{\eta }\frac{q(v)}{\rho +q(v)+\lambda }>0 \end{aligned}$$

so that

$$\begin{aligned} \text{ sign} \beta _\rho =\text{ sign} \left(-\frac{\Gamma _\rho }{\Gamma _\beta }\right)>0 \end{aligned}$$

so that $\alpha _\rho <0$.

1.3 B.3 Proof of Proposition 4

Existence and uniqueness of a long-run equilibrium are easy to check after a glance at Fig. 3. A look at Eqs. (17) and (18) reveals that the long-run equilibrium is of the purely Harrod-neutral type, with zero capital-productivity growth and a growth rate of labor-augmenting technologies equal to $\alpha _{ss}=g(0; \cdot )$. The equilibrium employment rate is constant and equal to $\beta ^{-1}(0; \cdot )$. Finally, once the long-run employment rate of the model is achieved, the wage share will be constant, and given by $\eta \frac{\rho +\lambda +q(v_{ss})}{\rho +\lambda +\eta q(v_{ss})}$.

1.4 B.4 Proof of Proposition 5

Linearize the above system around its non-trivial rest point to obtain the Jacobian matrix:

$$\begin{aligned} J_{\{Bss,vss\}}=\left(\begin{array}{c@{\quad }c} 0&\beta _v\left(\frac{\rho +\alpha +n}{1-\eta }\right)\left[\frac{\rho +\lambda +\eta q(v_{ss})}{\rho +\lambda }\right]\\ (1-\eta )\frac{\rho +\lambda }{\rho +\lambda +\eta q(v_{ss})}&\left(\beta _v-\frac{\eta (1-\eta )}{\rho +\lambda +\eta q(v_{ss})}(\rho +\lambda )q_v -\alpha _v\right)v_{ss} \end{array}\right) \end{aligned}$$

This matrix has a negative trace, as $\beta _v<0,-q_v<0, -\alpha _v< 0$, and a positive determinant. Therefore, its two eigenvalues have real parts that are of the same sign and sum up to a negative number, and this proves the claim.

1.5 B.5 Proof of Proposition 7

We have that:

$$\begin{aligned} \frac{\partial B_{ss}}{\partial n}=\frac{1}{1-\eta }\left[\frac{\rho +\lambda +\eta q(v_{ss})}{\rho +\lambda }\right]>0 \end{aligned}$$

Further,

Also,

$$\begin{aligned} \frac{\partial B_{ss}}{\partial \eta }{=}\left[\frac{\alpha _\eta }{1-\eta }+\frac{\alpha (\cdot )+n+\rho }{(1-\eta )^2}\right]\left[\frac{\rho +\lambda +\eta q(v_{ss})}{\rho +\lambda }\right]+\frac{q(v_{ss})}{\rho +\lambda }\left(\frac{\rho +\alpha (\cdot )+n}{1-\eta }\right){<}0 \end{aligned}$$

Finally,

$$\begin{aligned} \frac{\partial B_{ss}}{\partial \lambda }=\frac{\alpha _\lambda }{1-\eta }\left[\frac{\rho +\lambda +\eta q(v_{ss})}{\rho +\lambda }\right]-\frac{\eta q(v_{ss})}{(\rho +\lambda )^2}\frac{\rho +\alpha (\cdot )+n}{1-\eta }<0 \end{aligned}$$

1.6 B.6 Proof of Proposition 8

Since $\beta (v_{ss}; \eta , \lambda , \rho )=0$, we must have that $\beta _{v_{ss}}dv_{ss}+\beta _\eta d\eta =0$, from which:

$$\begin{aligned} \frac{dv_{ss}}{d\eta }=-\frac{\beta _\eta }{\beta _{v_{ss}}} \end{aligned}$$

As both $\beta _\eta , \beta _{v_{ss}}<0$, the claim is proved.

1.7 B.7 Proof of Proposition 9

We have:

$$\begin{aligned}&\displaystyle \frac{\partial w}{\partial b}=(1-\eta )\frac{\rho +\lambda }{\rho +q(v)+\lambda }>0\\&\displaystyle \frac{\partial w}{\partial \eta }=(A-b)\frac{(\rho +\lambda )[\rho +\lambda +q(v)]}{[\rho +\lambda +q(v)]^2} >0 \iff A>b\\&\displaystyle \frac{\partial w}{\partial v}=\eta (1-\eta )\frac{(A-b)q_v(\rho +\lambda )}{[\rho +\lambda +q(v)]^2}>0 \iff A>b\\&\displaystyle \frac{\partial w}{\partial \lambda }=\eta ^{-1}\frac{\partial w}{\partial \rho }=\eta (\eta -1)\frac{(A-b)q(v)}{[\rho +\lambda +q(v)]^2}<0 \iff A>b \end{aligned}$$

1.8 B.8 Proof of Proposition 11

Since $\beta _{v_{ss}}dv_{ss}+\beta _b db=0$,

$$\begin{aligned} \frac{dv_{ss}}{db}=-\frac{\beta _b}{\beta _{v_{ss}}} \end{aligned}$$

As both $\beta _b, \beta _{v_{ss}}<0$, the claim is proved.

Appendix C: Bargaining in the model with substitution

The statement of the bargaining problem with substitution is similar to the non-substitution case, and therefore omitted. The current-value Hamiltonian $H_S$ associated with such problem is:

$$\begin{aligned} H_S=\eta \log (w-\rho V_U)+(1-\eta )\log \left[f(x)-\frac{w}{A}x\right]+\gamma _1\beta B+\gamma _2 g(\beta )A \end{aligned}$$

where $\gamma _1, \gamma _2$ are defined as in Appendix A. Setting $\partial H_S/\partial w=0$ yields:

$$\begin{aligned} w=\eta A f(x)/x+(1-\eta )\rho V_U \end{aligned}$$

(28)

and setting $\partial H_S/\partial \beta =0$ gives (24). Proceeding as in the non-substitution case, a stationary solution in which the adjoint variables $\gamma _1, \gamma _2$ don’t change over time satisfies:

$$\begin{aligned}&\displaystyle (\rho +\lambda ) B p_1 e^{-\beta ^*t}=(1-\eta )\left[\frac{f(x)-xf^{\prime }(x)}{f(x)-x w/A}\right]\end{aligned}$$

(29)

$$\begin{aligned}&\displaystyle (\rho +\lambda ) A p_2 e^{-g(\beta ^*)t}=(1-\eta )\left[\frac{xf^{\prime }(x)}{f(x)-x w/A}\right] \end{aligned}$$

(30)

Dividing both sides of (29) by (30) and using (24), one obtains $-g_\beta =\frac{f(x)-xf^{\prime }(x)}{xf^{\prime }(x)}$.

Once a bargaining deal is struck, profit-maximizing firms will demand labor up to the point where its marginal product $Af^{\prime }(x)$ is equal to the wage. The wage share in firm’s output is then $wL/Y=xf^{\prime }(x)/f(x)$, so that the firm-level optimal direction of technical change satisfies (5). Once the outside option $V_U$ is determined through (8), we can define a market equilibrium as in Sect. 2.5, at which real wage and profit rate are:

$$\begin{aligned} \begin{array}{c@{\quad }c}&\displaystyle w=\eta \left[\frac{\rho +q(v)+\lambda }{\rho +\lambda +\eta q(v)}\right]A\frac{f(x)}{x};\quad \displaystyle r=(1-\eta )Bf(x)\left[\frac{\rho +\lambda }{\rho +\lambda +\eta q(v)}\right] \end{array}\qquad \end{aligned}$$

(31)

Appendix D: Stability analysis in the model with substitution

The Jacobian matrix evaluated at the steady state is:

$$\begin{aligned} J^{subst}_{\{Bss,vss\}}= \left(\begin{array}{c@{\quad }c} 0&\beta _vB_{ss}\\ \frac{1}{1-\sigma \epsilon }f(x)(1-\omega )v_{ss}&\frac{1}{1-\sigma \epsilon }\left(\beta _v -\alpha _v\right)v_{ss} \end{array}\right) \end{aligned}$$

which has a negative trace and positive determinant if and only if $\sigma <1/\epsilon$, thus yielding the same results of the non-substitution case if this restriction on the elasticity of substitution is satisfied.

Appendix E: Numerical simulations

For this exercise, we assume the following parameter calibration: $n=.02$, $\alpha (v_{ss}, \cdot )=.02$, which match first moments in population growth and labor productivity growth in the US. As far as rates of job creation and job destruction are concerned, Davis et al. (1997) suggest to calibrate $\lambda =.113, q(v_{ss})=.092$. The plot in Fig. 2 points an average employment rate around 94 %, which we take as our benchmark value for $v_{ss}$. If we assume a linear specification for $q(\cdot )$, then we can internally calibrate the first derivative $q_v$. Finally, we need to set a value for the parameter describgin workers’ bargaining power. On the one hand, we could use the national US average for the unionization rate. The current value is 12.1 %, although this figure has been steadily decreasing in recent decades from 33 % in the 1960s (BLS 2009b). On the other hand, we could follow several authors (Abowd and Lemieux 1993; Blanchflower et al. 1996; van Reenen 1996) who estimated the percentage of rents produced by firms and captured by wages. These estimates provide values for $\eta$ ranging from 0.08 to 0.3. In line with these findings, I run simulations corresponding to several values of $\eta$, respectively $\eta =\{0.1, 0.2, 0.3\}$. The following table summarizes the parameter values used for the simulation runs displayed in Figs. 4, 5 and 6.