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 (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 (1956) and Goodwin (1967) that is also compatible with the evolutionary perspective on coordination (or the lack thereof) within markets by Metcalfe et al. (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.

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Fig. 5

Notes

  1. 1.

    As argued in Section 8, our approach is also compatible with an evolutionary-like approach that emphasizes coordination (or the lack thereof) through market mechanisms (see Metcalfe et al.2006).

  2. 2.

    However, Foley (1985) notes that reasons for these lags differ between Marx and Keynes. For Keynes, spending lags are due to speculation arising from changing expectations in conditions of fundamental uncertainty. For Marx, these spending lags—and thus the possibility of inadequate aggregate demand—result from liquidity and spending constraints inherent to the nature of capitalist production.

  3. 3.

    Crotty (2019) similarly emphasizes the General Theory’s contribution to microeconomic analysis: “[T]o understand Keynes’s thinking about the nature of financial markets, we must first understand the revolution he created in micro theory or the theory of agent choice, a revolution not recognized by or incorporated in Mainstream Keynesian theory or in neoclassical micro theory” (p.13).

  4. 4.

    Similar reasoning applies to the assumption of competitive firms: the results of the model do not depend on slow price adjustments, as Skott (2017) has argued to be the case in neo-Kaleckian growth models.

  5. 5.

    Such overshooting behavior is typical in models with forward-looking optimizing agents. Classic papers on this topic from a mainstream viewpoint are Dornbush (1976) and Pissarides (1985).

  6. 6.

    Interestingly, the growth rate of utilization obtained log-differentiating (13) is identical to the corresponding equilibrium growth rate as per equation (8)

  7. 7.

    Setting τ = 0 amount to impose a deficit-financed user cost subsidy.

  8. 8.

    We leave the evaluation of growth effects with fully endogenous technical change to future research.

  9. 9.

    In the conventional AK-growth literature, the stock of labor-augmenting technologies depends on the aggregate capital stock, A = ζKϕ, ζ > 0, ϕ ∈ (0, 1), through a learning-by-doing externality. Log-differentiation of this expression gives equation (17). See Romer (1987).

  10. 10.

    The figure uses the same parameter calibration as the baseline model, plus a value of ϕ that ensures a long-run growth rate of 3%. It is displayed for illustrative purposes, and it is not meant to argue that demand policies can boost labor productivity growth to over 12% as a literal interpretation of the figure would suggest.

  11. 11.

    The proof goes along the same lines as Proposition 4.

  12. 12.

    Erixon (2005) argues that Verdoorn’s law, which plays a central role in our analysis here, should more properly be referred to as Svennilson’s law—after the findings by Svennilson (1954).

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Acknowledgments

We thank participants at the 2018 Analytical Political Economy Workshop, Renzo Cavalieri, and Luca Zamparelli for helpful comments and suggestions on earlier drafts. Addressing comments by two anonymous Referees greatly improved the paper. All errors are ours.

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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 (gc = gK).

Appendix C: Stability analysis

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.

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 Dω 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

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

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Tavani, D., Petach, L. Firm beliefs and long-run demand effects in a labor-constrained model of growth and distribution. J Evol Econ (2020). https://doi.org/10.1007/s00191-020-00680-w

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Keywords

  • Beliefs
  • Capacity utilization
  • Factor shares
  • Growth cycles

JEL Classification

  • D25
  • E12
  • E22
  • E25
  • E62