Heterogeneity in the Harrodian sentiment dynamics, entailing also some scope for stability

Abstract

Drawing on recent ideas in the literature to model ‘animal spirits’, the paper considers the investment of heterogeneous firms that probabilistically switch between optimism and pessimism. Combining the herding component in the endogenous transition probabilities with a Harrodian feedback, a one-dimensional macroeconomic adjustment equation is set up. Assuming a ‘neutral’ herding coefficient and considering the local dynamics around the steady state with normal utilization, the equation shows a one-to-one correspondence with the neo-Kaleckian baseline model of Harrodian instability. The paper thus provides a rigorous specification of its sentiment adjustment story. In a second and innovative step, the firms are additionally allowed to be neutral. In this variant, up to three ‘fully-adjusted’ steady states come into being, which, however, cannot be distinguished from a macroeconomic point of view. While two of them exhibit the usual instability, it turns out that the equilibrium with the highest share of neutral agents can easily be locally stable. The result shows that the common macroeconomic view of Harrodian instability that essentially treats all firms alike may be too simple. Also, the economic significance of the paper’s findings goes beyond the present very limited framework.

Appendix

1.1 Derivation of Eq. 11

Using the identities for the hyperbolic sine and cosine and setting ν = 1 for brevity, we have $\dot {a} = (1-a) \exp (s) - (1+a) \exp (-s) = 2 \{ [\exp (s) - \exp (-s)]/2 - \ a [\exp$$(s) + \exp (-s)]/2 \} = 2 [\sinh (s) - a \cosh (s)]$. Equation 11 results from dividing the square bracket by $\cosh (s)$ and then using the identity $\tanh = \sinh /\cosh$.

1.2 Derivation of Eq. 17

Using the identities $\exp (x) = \cosh (x) + \sinh (x)$, $\cosh (x) = [\exp (x) + \exp (-x)]/2$ and again $\tanh = \sinh /\cosh$, the first equation of Eq. 13 can be rewritten as follows:²¹

$$ \begin{array}{@{}rcl@{}} \dot{b}/\nu & = & (1-b-c) \exp(s_{b}) \ - \ b \exp(-s_{b})\\ &=& (1-c) \exp(s_{b}) \ - \ 2b [\exp(s_{b}) + \exp(-s_{b})]/2\\ &=& (1-c) \cosh(s_{b}) \ + \ (1-c) \sinh(s_{b}) \ - \ 2b \cosh(s_{b})\\ &=& (1-c) \sinh(s_{b}) \ - \ (2b+c-1) \cosh(s_{b})\\ &=& [ (1-c) \sinh(s_{b})/\cosh(s_{b}) \ - \ (2b+c-1)] \cosh(s_{b}) \qquad \qquad\\ &=& [ (1-c) \tanh(s_{b}) \ - \ (2b+c-1)] \cosh(s_{b}) \end{array} $$

By the same token,

$$ \begin{array}{@{}rcl@{}} \dot{b}/\nu & = & [ (1-b) \tanh(s_{c}) \ - \ (2c+b-1)] \cosh(s_{c}) \hspace{3.3cm} \end{array} $$

1.3 The Jacobian matrix for an equilibrium with b = c

Set up the partial derivatives of the switching indices and, using b = c, abbreviate:

$$ \begin{array}{@{}rcl@{}} \begin{array}{rrrcl} \partial\! s_{b}/\partial\! b &=& \partial\! s_{c}/\partial\! c &=& 2\phi_{h} + \widetilde{\phi}_{y}\\ \partial\! s_{b}/\partial\! c &=& \partial\! s_{c}/\partial\! b &=& \phi_{h} - \widetilde{\phi}_{y}\\ \partial\! s_{b}/\partial\! v &=& -\partial\! s_{c}/\partial\! v &=& -\phi_{v}\\ C &:=& \multicolumn{3}{l}{\nu \cosh[\phi_{h} (3b-1)]}\\ D &:=& \multicolumn{3}{l}{\tanh^{\prime}[\phi_{h} (3b-1)] \ \ = \ \ 1 / \cosh^{2}[\phi_{h} (3b-1)]}\\ H &:=& \multicolumn{3}{l}{\tanh[\phi_{h} (3b-1)]} \end{array} \end{array} $$

The entries of the Jacobian are then given by

$$ \begin{array}{@{}rcl@{}} \begin{array}{lclclcrcr} \hspace{0.7cm} j_{11} &=& \partial\! \dot{b}/\partial\! b &=& C [(1-b) D (2\phi_{h} + \widetilde{\phi}_{y}) - 2] &=& \partial\! \dot{c}/\partial\! c &=& j_{22}\\ \hspace{0.7cm} j_{12} &=& \partial\! \dot{b}/\partial\! c &=& C [(1-b) D (\phi_{h} - \widetilde{\phi}_{y}) - H - 1] &=& \partial\! \dot{c}/\partial\! b &=& j_{21} \end{array} \end{array} $$