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We develop a cobweb model with discrete time delays that characterise the length of production cycle. We assume a market comprised of homogeneous producers that operate as adapters by taking the (expected) profit-maximising quantity as a target to adjust production and consumers with a marginal willingness to pay captured by an isoelastic demand. The dynamics of the economy is characterised by a one-dimensional delay differential equation. In this context, we show that (1) if the elasticity of market demand is sufficiently high, the steady-state equilibrium is locally asymptotically stable and (2) if the elasticity of market demand is sufficiently low, quasiperiodic oscillations emerge when the time lag (that represents the length of production cycle) is high enough.

Time series of prices of nonstorable goods are observed on a daily basis and are subject to strong fluctuations, while production of such commodities requires a longer time period (for instance, from sowing to harvest with regard to agricultural ones). The cobweb model, originally developed by Kaldor [

The interest in the study of price dynamics has led Hommes [

By considering quantity instead of price as the main variable, Onozaki et al. [

To capture in a better way the functioning of a market whose price fluctuations are observed on a daily basis, this paper extends Onozaki et al.’s [

The rest of the paper is organised as follows. Section

We consider a continuous time version with

Let firms be (expected) profit maximisers and use the quantity that corresponds to maximum expected profits,

Since there are

With regard to consumers’ side, by following Fanti et al. [

Market equilibrium, therefore, implies that aggregate demand equals aggregate supply, that is,

The steady-state equilibrium of (

Assume now that

Let

Let

If

The following transversality condition:

Let

The previous lemma implies that the root of characteristic equation (

If

Summing up, we can state the following results (see Figure

Stability and instability regions in

Let

If

If

If

In the previous section we have obtained conditions for Hopf bifurcation to occur when

For

Using the same notations as in Hassard et al. [

Since there are

Based on the above analysis, we can see that each

Let

We now illustrate the theoretical results stated in previous sections by performing some numerical simulations. For this purpose, we fix the following parameter set:

Trajectory that converges to the steady-state equilibrium.

Three different closed invariant curves for three values of

Trajectory that converges to the limit cycle.

This paper extended the discrete time cobweb model by Onozaki et al. [

This model could be extended especially by considering heterogeneous producers, to stress how small perturbations can generate important changes in the structure of the dynamic system and different dynamic behaviours than when firms are homogeneous.

The authors declare that there is no conflict of interests regarding the publication of this paper.