Animal spirits and unemployment: a disequilibrium analysis

Abstract

The past decade has seen a number of advances in modelling disequilibrium dynamics. This paper draws on separate approaches to disequilibrium dynamics to demonstrate a Keynesian result concerning the formal relevance of “animal spirits” in production economies. Specifically, it is shown that a parameter that can be associated with the “animal spirits” of firms is crucial to the stability of full employment equilibrium in a production economy. This approach to “animal spirits” is different to that taken by recent New Keynesian DSGE-type models, but similar in spirit to “Old Keynesian” approaches, including that of the General Theory. The corollary of the main conclusion is that price flexibility is not a sufficient condition for convergence on full employment equilibrium.

Appendix

Determining the Walrasian equilibrium as per Sect. 2.1 is straightforward. Solving the firms’ problems gives relative prices (given CRTS and non-substitution), and solving the households’ problems gives the amounts of each commodity produced (given total labour endowment). Firm $j$’s problem is as follows:

$$\begin{aligned} \max p_{j} x_{j} - w l \text { s.t. } x_{j} = \alpha _{j} l \end{aligned}$$

Taking the wage as numeraire, this reduces to the following problem:

$$\begin{aligned} \max p_{j} \alpha _{j} l - l \end{aligned}$$

As the technology is CRTS, the maximisation problem does not yield a supply function. Instead, the F.O.C. gives $p_{j}$:

$$\begin{aligned} p_{j} \alpha _{j} - 1 = 0 \rightarrow p_{j} = \alpha _{j}^{-1} \end{aligned}$$

Hence, $p_{1} = \alpha _{1}^{-1}$ and $p_{2} = \alpha _{2}^{-1}$. Household $i$’s problem is as follows:

$$\begin{aligned} \max \beta _{i}\ln x_{1} + (1 - \beta _{i})\ln x_{2} \text { s.t. } p_{1}x_{1} + p_{2}x_{2} \le wl_{i} \end{aligned}$$

The F.O.C.s are then as follows:

$$\begin{aligned} x_{1}&= \beta _{i} \dfrac{w l_{i}}{p_{1}} \\ x_{2}&= (1 - \beta _{i}) \dfrac{w l_{i}}{p_{2}} \end{aligned}$$

Taking the wage as numeraire, noting that each household’s labour endowment is normalised to 1, and substituting for the relative prices calculated above, the above F.O.C.s reduce to:

$$\begin{aligned} x_{1}&= \dfrac{\beta _{i}}{p_{1}} = \alpha _{1} \beta _{i} \\ x_{2}&= \dfrac{(1 - \beta _{i})}{p_{2}} = \alpha _{2} (1 - \beta _{i}) \end{aligned}$$

Finally, as total demand for $x_{1}$ and $x_{2}$ is the demand of household 1 plus the demand of household 2, the total amounts produced of $x_{1}$ and $x_{2}$ are as follows:

$$\begin{aligned} x_{1}&= \alpha _{1} \beta _{1} + \alpha _{1} \beta _{2} = \alpha _{1} (\beta _{1} + \beta _{2}) \\ x_{2}&= \alpha _{2}(1 - \beta _{1}) + \alpha _{2} (1 - \beta _{2}) = \alpha _{2} (2 - \beta _{1} + \beta _{2}) \end{aligned}$$

This gives the Walrasian equilibrium as per Sect. 2.1.