Cobweb model

The convergent case: each new outcome is successively closer to the intersection of supply and demand.

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 harvesting (Kaldor, 1934, p. 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 1 means supply falls to Q₁, so that prices rise to P₁. If producers plan their period 2 production under the expectation that this high price will continue, then the period 2 supply will be higher, at Q₂. Prices therefore fall to P₂ 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 cobweb, hence the name of the theory.

The Gori et al. group finds that cobwebs experience Hopf bifurcations, in Gori et al. 2014, Gori et al. 2015a, and Gori et al. 2015b.[1]

When supply and demand are linear functions the outcomes of the cobweb model are stated above in terms of slopes, but they are more commonly described in terms of elasticities. The convergent case requires that the slope of the (inverse) supply curve be greater than the absolute value of the slope of the (inverse) demand curve:

${\displaystyle {\frac {dP^{S}}{dQ^{S}}}>\left|{\frac {dP^{D}}{dQ^{D}}}\right|.}$

In standard microeconomics terminology, define the elasticity of supply as ${\displaystyle {\frac {dQ^{S}/Q^{S}}{dP^{S}/P^{S}}}}$, and the elasticity of demand as ${\displaystyle {\frac {dQ^{D}/Q^{D}}{dP^{D}/P^{D}}}}$. If we evaluate these two elasticities at the equilibrium point, that is ${\displaystyle P^{S}=P^{D}=P>0}$ and ${\displaystyle Q^{S}=Q^{D}=Q>0}$, then we see that the convergent case requires

${\displaystyle {\frac {dQ^{S}/Q}{dP^{S}/P}}<\left|{\frac {dQ^{D}/Q}{dP^{D}/P}}\right|,}$

whereas the divergent case requires

${\displaystyle {\frac {dQ^{S}/Q}{dP^{S}/P}}>\left|{\frac {dQ^{D}/Q}{dP^{D}/P}}\right|.}$

In words, the convergent case occurs when the demand curve is more elastic than the supply curve, at the equilibrium point. The divergent case occurs when the supply curve is more elastic than the demand curve, at the equilibrium point (see Kaldor, 1934, page 135, propositions (i) and (ii).)

One reason to be skeptical of this model's predictions is that it assumes producers are extremely shortsighted. Assuming that farmers look back at the most recent prices in order to forecast future prices might seem very reasonable, but this backward-looking forecasting (which is called adaptive expectations) turns out to be crucial for the model's fluctuations. When farmers expect high prices to continue, they produce too much and therefore end up with low prices, and vice versa.

In the stable case, this may not be an unbelievable outcome, since the farmers' prediction errors (the difference between the price they expect and the price that actually occurs) become smaller every period. In this case, after several periods prices and quantities will come close to the point where supply and demand cross, and predicted prices will be very close to actual prices. But in the unstable case, the farmers' errors get larger every period. This seems to indicate that adaptive expectations is a misleading assumption—how could farmers fail to notice that last period's price is not a good predictor of this period's price?

The fact that agents with adaptive expectations may make ever-increasing errors over time has led many economists to conclude that it is better to assume rational expectations, that is, expectations consistent with the actual structure of the economy. However, the rational expectations assumption is controversial since it may exaggerate agents' understanding of the economy. The cobweb model serves as one of the best examples to illustrate why understanding expectation formation is so important for understanding economic dynamics, and also why expectations are so controversial in recent economic theory.

• Adaptive expectations
• Cobweb plot
• Lotka–Volterra equation
• Pork cycle
• Tatonnement

• W. Nicholson, Microeconomic Theory, 7th ed., Ch. 17, pp. 524–538. Dryden Press: ISBN 0-03-024474-9.
• Jasmina Arifovic, "Genetic Algorithm Learning and the Cobweb Model", Journal of Economic Dynamics and Control, vol. 18, Issue 1, (January 1994), 3-28.
• Arthur Hanau (1928), "Die Prognose der Schweinepreise". In: Vierteljahreshefte zur Konjunkturforschung, Verlag Reimar Hobbing, Berlin.
• Mordecai Ezekiel (February 1938). "The Cobweb Theorem" (PDF). Quarterly Journal of Economics. 52 (2): 255–280. doi:10.2307/1881734. JSTOR 1881734. Archived from the original (PDF) on 2015-06-16. Retrieved 2015-03-05.
• Nicholas Kaldor, "A Classificatory Note on the Determination of Equilibrium", Review of Economic Studies, vol I (February 1934), 122-36. (See especially pages 133–135.)
• Marc Nerlove (1958). "Adaptive Expectations and Cobweb Phenomena". Quarterly Journal of Economics. 72 (2): 227–240. doi:10.2307/1880597. JSTOR 1880597.
• C.P. Wellford, 'A Laboratory Analysis of Price Dynamics and Expectations in the Cobweb Model', Discussion Paper 89-15 (University of Arizona, Tucson, AZ).
• Muth, J. F. (1961). "Rational Expectations and the Theory of Price Movements". Econometrica. 29 (3): 315–35. doi:10.2307/1909635. JSTOR 1909635.
• Pashigian, B. P. (2008). "Cobweb Theorem". The New Palgrave Dictionary of Economics (2nd ed.).
• Rosen, S.; Murphy, K.; Scheinkman, J. (1994). "Cattle Cycles". Journal of Political Economy. 102 (3): 468–92. doi:10.1086/261942. S2CID 219377989.
• Ricci, Umberto (1930). "Die 'synthetische Ökonomie' von Henry Ludwell Moore". Zeitschrift für Nationalökonomie. 1 (5): 649–668. doi:10.1007/BF01318499. S2CID 154514568.
• Schultz, H. Der Sinn der statistischen Nachfragekurven. p. 34.
• Tamari, Ben (1981). "Cycles, Prices And Quantities In Israel Housing Market - Cobweb model" (PDF). {{cite journal}}: Cite journal requires |journal= (help) update March 2011.