Firm beliefs and long-run demand effects in a labor-constrained model of growth and distribution

Abstract

One of the most debated questions in alternative macroeconomics regards whether demand policies have permanent or merely transitory effects. While demand matters in the long run in (neo-) Kaleckian economics, both economists operating within other Keynesian traditions (e.g. Skott 1989) as well as Classical economists Duménil and Levy (Manchester School 67(6):684–716, 1999) argue that in the long-run output growth is constrained by an exogenous natural growth rate. This paper attempts to bridge the gap by analyzing the role of firm beliefs about the state of the economy in a labor-constrained growth and distribution model based on Kaldor (Review of Economic Studies 23(2):83–100, 1956) and Goodwin (Journal of Evolutionary Economics 1(1):29–47, 1991) that is also compatible with the evolutionary perspective on coordination (or the lack thereof) within markets by Metcalfe et al. (Cambridge Journal of Economics 30:7–32, 2006). The main innovation is the inclusion of beliefs about economic activity in an explicitly dynamic choice of capacity utilization at the firm level. We show that: (i) the relevance of such beliefs generates an inefficiently low utilization rate and labor share in equilibrium, but (ii) the efficient utilization rate can be implemented through fiscal policy. Under exogenous technical change, (iii) the inefficiency does not affect the equilibrium employment rate and growth rate, but expansionary fiscal policy has positive level effects on both GDP and the labor share. However, (iv) with endogenous technical change à la Verdoorn (1949), fiscal policy has also temporary growth effects. Finally, (v) the fact that the choice of utilization responds to income shares has a stabilizing effect on growth cycles, even under exogenous technical change, that is analogous to factor substitution.

Appendices

Appendix A: The capitalists’ optimization problem

Suppose that the representative capitalist household has logarithmic preferences over consumption streams, and discounts the future at a constant rate ρ > 0. Then, the household solves:

$$ \begin{array}{@{}rcl@{}} \text{Given } \{\tilde u(t), \omega(t)\}_{\forall t},\\ \text {Choose } \{c(t),u(t)\}_{t\in[s,\infty)} & \text{to max} & \displaystyle {\int}_{s}^{\infty} \exp\{-\rho(t-s)\} \ln c(t)dt\\ & \text{s. t. } & \dot K(t) = [1-\omega(t)]u(t)K(t) -c(t) \\ &&\qquad\quad -\lambda[u(t);\tilde u(t) ]K(t)\\ && K(s)\equiv K_{s}>0, \text{ given}\\&& \displaystyle \lim_{t\to \infty} \exp\{-\rho (t-s)\}K(t)\ge 0 \end{array} $$

(22)

Observe first that this problem involves a strictly concave objective function to be maximized over a convex set. Thus, with co-state variable μ(t), the standard first-order conditions on the associated current-value Hamiltonian

$$\mathcal H =\ln c +\mu[u(1-\omega)K -c- \lambda(u;\tilde u )K]$$

will be necessary and sufficient for an optimal control. They are:

$$ \begin{array}{@{}rcl@{}} c^{-1} &=& \mu \end{array} $$

(23)

$$ \begin{array}{@{}rcl@{}} 1-\omega&=&\lambda_{u}(u, \tilde u ) \end{array} $$

(24)

$$ \begin{array}{@{}rcl@{}} \rho \mu -\dot \mu &= &\mu[(1-\omega)u-\lambda(u;\tilde u )] \end{array} $$

(25)

$$ \begin{array}{@{}rcl@{}} \lim_{t\to\infty}\exp\{-\rho t\}\mu(t)k(t)&=&0 \end{array} $$

(26)

Solving (24) for the rate of utilization under the specific functional form (1) gives equation (3). To obtain the Euler equation for consumption, differentiate (23) with respect to time and use (23) and (25) to get:

$$g_{c} \equiv \frac{\dot c}{c} = (1-\omega)u(\omega;\tilde u )-\{\lambda[u(\omega; \tilde u )]+\rho\}.$$

Using both equations (3) and (1) while imposing a balanced growth path where consumption and capital stock grow at the same rate gives equation (4).

Appendix B: The efficient solution

A benevolent planner solves the accumulation problem under the additional constraint that $u=\tilde u$ at all times. Accordingly, the control problem (22) is solved under the modified accumulation constraint, omitting the time-dependence for notational simplicity:

$$ \dot K = u(1-\omega)K -c -\beta u^{\frac{1-\gamma}{\beta}}K $$

(27)

The first-order condition on consumption is the same as (23) above. On the other hand, the choice of utilization and the co-state equation satisfy the first-order conditions which, once again, are necessary and sufficient for an optimal control:

$$ \begin{array}{@{}rcl@{}} 1-\omega & =& (1-\gamma)u^{\frac{1-\beta-\gamma}{\beta}} \end{array} $$

(28)

$$ \begin{array}{@{}rcl@{}} \displaystyle \rho -\frac{\dot \mu}{\mu} & = & u(1-\omega)-\beta u^{\frac{1-\gamma}{\beta}} \end{array} $$

(29)

Solving equation (28) for utilization gives (13). To obtain the efficient accumulation rate (14), simply impose balanced growth (g_c = g_K).

Appendix C: Stability analysis

1.1 C.1 Equilibrium path: Proof of Proposition 2

The Jacobian Matrix evaluated at a steady state has the following structure:

$$ J(e_{ss}, \omega_{ss}) =\left[\begin{array}{ll} -\frac{\beta}{1-\beta-\gamma}\frac{\omega_{ss}}{1-\omega_{ss}}f^{\prime}(e_{ss})e_{ss} & -\frac{(1-\beta)(1-\gamma)}{1-\beta-\gamma}(1-\omega_{ss})^{\frac{\beta}{1-\beta-\gamma}}e_{ss}\\ (-) & (-) \\ f^{\prime}(e_{ss})\omega_{ss} & 0 \\ (+) & (0) \end{array}\right] $$

Thus, it has a negative trace and a positive determinant. It follows that its eigenvalues are of the same sign and sum to a negative number, which can only occur if they both have uniformly negative real parts. We conclude that the steady state is locally stable.

1.2 C.2 Efficient path

The Jacobian Matrix evaluated at the efficient steady state is:

$$ J(e_{ss},\omega^{*}_{ss}) =\left[\begin{array}{ll} -\frac{\beta}{1-\beta-\gamma}\frac{\omega^{*}_{ss}}{1-\omega^{*}_{ss}}f^{\prime}(e_{ss})e_{ss} & -(1-\gamma)\left( \frac{1-\omega^{*}_{ss}}{1-\gamma}\right)^{\frac{\beta}{1-\beta-\gamma}}e_{ss}\\ (-) & (-) \\ f^{\prime}(e_{ss})\omega^{*}_{ss} & 0\\ (+) & (0) \end{array}\right] $$

again, with negative trace and positive determinant, so that the efficient steady state is locally stable, too.

Appendix D: Proofs

Proposition 3

Consider that that, using (5) and (13),

$$\frac{u^{*}}{u} = \left( \frac{1}{1-\gamma}\right)^{\frac{\beta}{1-\beta-\gamma}}>1$$

since 0 < γ < 1 − β by assumption.

Proposition 4

Showing that ω^∗ > ω is tantamount to showing that $\ln (1-\omega )-\ln (1-\omega ^{*})>0$. We have that

$$ \begin{array}{lll} D_{\omega} & \equiv & \ln (1-\omega)-\ln(1-\omega^{*})\\ & =& \displaystyle \frac{1-\beta-\gamma}{1-\gamma}\left[\ln(1-\beta-\gamma) - \ln(1-\beta)\right] -\ln(1-\gamma) \end{array}$$

and

$$\frac{\partial D_{\omega}}{\partial \gamma} =-\frac{\beta}{(1-\gamma)^{2}}\left[\ln(1-\beta-\gamma)-\ln(1-\beta)\right] $$

Hence, the difference D_ω increases in γ provided that the term in brackets is negative. This is certainly true under 0 < γ < 1 − β, since $\partial \ln (1-\beta -\gamma )/\partial \gamma <0$.

Proposition 5

First, observe that the first-order necessary condition for the choice of utilization with the tax and subsidy solves for the firm-level utilization as

$$ u =\left( \frac{1-\omega}{1-s}\right)^{\frac{\beta}{1-\beta}}\tilde u^{\frac{\gamma}{1-\beta}} $$

(30)

Imposing the equilibrium condition $u=\tilde u$, we find

$$ u^{subs} = \left( \frac{1-\omega}{1-s}\right)^{\frac{\beta}{1-\beta-\gamma}} $$

(31)

The comparison with Equation (13) makes it clear that s = γ achieves the efficient utilization rate.

To prove the second claim, differentiate Equations (31) and (30) (after taking logs for simplicity) with respect to the subsidy s to see that

$$ \frac{\partial \ln u^{subs}}{\partial s} =\frac{\beta}{1-\beta-\gamma}\frac{1}{1-s}>\frac{\partial \ln u}{\partial s} = \frac{\beta}{1-\beta}\frac{1}{1-s} \iff \gamma \in (0,1-\beta). $$

The size of the fiscal multiplier m can be recovered by dividing the aggregate response by the individual response. We have that

$$ m = \frac{1-\beta}{1-\beta-\gamma} =\frac{1}{1-\frac{\gamma}{1-\beta}} $$