Cobweb Model: Concept, Theorem and Lagged Adjustments in Interrelated Markets

2 years ago

Atiqur Rahman

9 minutes

The three economists in Italy, Holland and the United States independently worked out the theoretical explanation, which has since come to be known as the “Cob-Web Theorem”.

All three originators of cobweb theory have followed the basic idea of carrying successive price, production and production readjustments back and forth between demand and supply curves.

Schultz has demonstrated cobweb by presenting a simple example of the convergent type, and he also plotted the resulting time series of price and quantities. Tinbergen’s analysis was considered more complete as he presented both convergent and divergent types. Ricci’s presented the diagrams of all three basic types- convergent, divergent and continuous.

No one out of three has considered the broader view of the cobweb theory. Schultz used it as an illustration of the difference between lagged readjustment and simultaneous readjustment of supply to demand. Tinbergen shows that when the production response lags behind the price change, ”instead of equilibrium being reached, a continuing movement of price and production is possible”. Ricci shows how important the precise values of the elasticities of demand and supply were since such greatly different economic consequences might follow from slight differences in their numerical values.

The cobweb model is also known as dynamic stability with lagged adjustment. It is the simplest model of economic dynamics when equilibrium reached over time between demand, supply and price is investigated.

Whenever demand or supply changes, equilibrium also changes as well. There is a time lag between a change in price and an appropriate adjustment in supply in response to it. Supply lag is the time gap between the decision to change the quantity supplied in response to a given price, and it is actually being supplied. The supply lag often results in oscillations in price and quantity over time.

Cobweb Model:

We assume that supply is a lagged function of price. It shows that supply responds to a change in price after a time lag.

𝑆_𝑡 = 𝑓(𝑃_𝑡−1)

On the other hand, there is no lag in the demand function; i.e. quantity demanded in this year depends on this year’s price only. The cobweb theorem can be explained in the form of three theorems:

Theorem I:

If the slope of the demand curve is less than the slope of the supply curve, then the equilibrium is stable: The system is convergent.

In Figure 2, price is measured on the y-axis and quantity on X-axis. DD and SS are the demand and supply curves. The initial equilibrium occurs at point E, where demand is equal to supply. This is the equilibrium in period t.

Suppose the price rises due to some reason to OP5; then equilibrium will be disturbed. At the new price OP5, demand is less than the expected supply by Q1Q6. Due to this, in period t+1, supply rises to OQ6 exceeding demand by Q1Q6. As a result, the price falls down to OP1 and causing a rise in demand for OQ6. But in response to the fall in price, supply in period t+2 decreases to OQ2. Now demand is more than supply by Q2Q6. As a result, the price rises to OP4, causing an increase in supply in period t+2 by Q2Q5. It is the price now that has to adjust itself to existing demand and supply conditions.

This whole process is repeated period after period. Each time the process of adjustment is repeated, the magnitude of change in supply, price and demand is decreasing. In period t+1, supply increases by Q1Q5. In period t+2, it decreases by Q2Q6 and in period t+3, it increases by Q2Q5 such that Q1Q6>Q2Q5>Q3Q. The same is true for demand and price. The decreasing magnitude of changes in demand, price and supply converges the equilibrium point at E. The equilibrium position is stable.

Theorem II:

If the supply curve has a smaller slope than the slope of the demand curve, the equilibrium is unstable. The adjustment process is divergent or oscillatory.

If the supply curve has a smaller slope than the slope of the demand curve, the equilibrium is unstable. The adjustment process is divergent or oscillatory. When the slope of the supply curve is less than the slope of the demand curve, then the process of adjustment makes the price and quantity diverge away and away from the equilibrium position. In this case, the magnitude of changes in price and quantity around the equilibrium point goes on to increase. Thus, the new equilibrium position is unstable. This is shown in the figure.

In Figure 3, price is measured on the y-axis and quantity on X-axis. DD and SS are the demand and supply curves. The initial equilibrium occurs at point E, where demand is equal to supply. This is the equilibrium in period t; suppose there is an increase in demand because of some reason. This increase in the demand curve shifts the demand curve to the right to D1D1.

An increase in demand results in price in period t from OP to OP3. At this price, supply is more than demand. This excess supply forces the price to fall to OP1. In the next period t+2, supply decrease by Q1Q3, i.e. reduction of supply by CD amount. Now demand is more than supply, and therefore price rises to OP4.

We can observe from the fluctuation in demand and supply that the amplitude of changes in price and quantity goes on increasing. This causes the movement of price quantity combinations away and away from the equilibrium point. Therefore, the equilibrium position is unstable.

Theorem III:

If the slope of the demand curve is equal to the slope of the supply curve: equilibrium is non-damped oscillating.

The undamped oscillating equilibrium is the equilibrium which, when displaced, keeps shifting in a circular way around the original equilibrium point with a constant change in demand, quantity and price. It is shown in Figure 4.

In Figure 4, the initial equilibrium is at point E, where the demand is equal to the supply. The equilibrium price is OP2, and the equilibrium quantity is OQ2. Let us suppose that the equilibrium is displaced either by a change in price or by a change in quantity. In both cases, equilibrium will keep circulating around its original point E.

Suppose the price rises from OP2 to OP3, then demand decreases from OQ2 to OQ1. This also results in an increase in supply from OQ2 to OQ3. This would result in an excess supply equal to AB. This excess supply exerts pressure on the prices to fall by BC. This fall in price resulted in a rise in demand and a decrease in supply, i.e. excess demand of Q1Q3. This excess demand in Q1Q3 causes the price to go up by P1P3. This process of change in price and quantity continues indefinitely.

Cobweb Model: Concept, Theorem and Lagged Adjustments in Interrelated Markets