Editing Cobweb model (section)

== The model ==
[[File:Cobweb theory (convergent).svg|thumb|The ''convergent'' case: each new outcome is successively closer to the intersection of supply and demand.]]
[[File:Cobweb theory (divergent).svg|thumb|The ''divergent'' case: each new outcome is successively further from the intersection of supply and demand.]]
The cobweb model is generally based on a time lag between supply and demand decisions. Agricultural markets are a context where the cobweb model might apply, since there is a lag between planting and [[harvest]]ing (Kaldor, 1934, p.&nbsp;133–134 gives two agricultural examples: rubber and corn). Suppose for example that as a result of unexpectedly bad weather, farmers go to market with an unusually small crop of strawberries. This shortage, equivalent to a leftward shift in the market's [[supply curve]], results in high prices. If farmers expect these high price conditions to continue, then in the following year, they will raise their production of strawberries relative to other crops. Therefore, when they go to market the supply will be high, resulting in low prices. If they then expect low prices to continue, they will decrease their production of strawberries for the next year, resulting in high prices again.

This process is illustrated by the adjacent diagrams. The [[equilibrium price]] is at the intersection of the supply and demand curves. A poor harvest in period&nbsp;1 means supply falls to Q<sub>1</sub>, so that prices rise to P<sub>1</sub>. If producers plan their period&nbsp;2 production under the expectation that this high price will continue, then the period&nbsp;2 supply will be higher, at Q<sub>2</sub>. Prices therefore fall to P<sub>2</sub> when they try to sell all their output. As this process repeats itself, oscillating between periods of low supply with high prices and then high supply with low prices, the price and quantity trace out a spiral. They may spiral inwards, as in the top figure, in which case the economy converges to the equilibrium where supply and demand cross; or they may spiral outwards, with the fluctuations increasing in magnitude.

The cobweb model can have two types of outcomes:
* If the supply curve is steeper than the demand curve, then the fluctuations decrease in magnitude with each cycle, so a plot of the prices and quantities over time would look like an inward spiral, as shown in the first diagram. This is called the stable or ''convergent'' case.
* If the demand curve is steeper than the supply curve, then the fluctuations increase in magnitude with each cycle, so that prices and quantities spiral outwards. This is called the unstable or ''divergent'' case.

Two other possibilities are:
* Fluctuations may also maintain a constant magnitude, so a plot of the outcomes would produce a simple rectangle. This happens in the linear case if the supply and demand curves have exactly the same slope (in absolute value).
* If the supply curve is less steep than the demand curve near the point where the two curves cross, but more steep when we move sufficiently far away, then prices and quantities will spiral away from the equilibrium price but will not diverge indefinitely; instead, they may converge to a [[limit cycle]].

In either of the first two scenarios, the combination of the spiral and the supply and demand curves often looks like a [[Spider web|cobweb]], hence the name of the theory.

The Gori ''et al.'' group finds that cobwebs experience [[Hopf bifurcation]]s, in Gori ''et al.'' 2014, Gori ''et al.'' 2015a, and Gori ''et al.'' 2015b.<ref name="Hopf">
{{cite journal|year=2020|publisher=[[Elsevier BV]]|journal=[[Journal of Computational and Applied Mathematics]]|issn=0377-0427|volume=374|last1=Chen|first1=Churong|last2=Bohner|first2=Martin|last3=Jia|first3=Baoguo|title=Caputo fractional continuous cobweb models|doi=10.1016/j.cam.2020.112734|page=112734|s2cid=213816027 }}

{{cite journal|year=2014|publisher=[[Hindawi Limited]]|volume=2014|pages=1–8|journal=[[Discrete Dynamics in Nature and Society]]|issn=1026-0226|last1=Gori|first1=Luca|last2=Guerrini|first2=Luca|last3=Sodini|first3=Mauro|title=Hopf Bifurcation in a Cobweb Model with Discrete Time Delays|doi=10.1155/2014/137090|doi-access=free|hdl=11568/538470|hdl-access=free}}
</ref>