# Agent-based cobweb model

Long time ago theoretical background of the cobweb model has somewhat troubled me. So I wanted to explore my doubts. At the time I didn't have any idea how to do this properly, but recently I think I have figured it out.

In this post I will construct quite simple agent-based model of the price formation in the free market. This time the approach works, at least in part, and from these we can uncover hidden assumption made in the classical cobweb model.

## Agent-based cobweb model

Note that this is not a true agent-based model as buyers and suppliers are not heterogeneous and are effectively represented by a single agent. Nevertheless let us continue with the introduction of the model.

From the cobweb model we have the supply and the demand laws. Our suppliers will produce in accordance to the current market price and observe how well their product is received. They will observe for a single unit of time.

During the observation period our buyers will buy goods at rate predicted by the demand law. Based on how many goods were sold and how fast they were sold, the suppliers will estimate the true demand of the buyers at the current price:

\begin{equation} \hat{D}_i = \frac{G_i}{T_i} . \end{equation}

In the above \( G \) stands for the number of goods sold and \( T \) stands for the time it took to sell those goods. The index simply enumerates the observation periods (time). This estimate is used by the suppliers to calculate the excess demand:

\begin{equation} \mathrm{ED}_i = \hat{D}_i - S_i . \end{equation}

Suppliers are aware that they could increase their sales, if the could make use of this "uncovered" (excess) demand. But they are also aware the demand will shrink if they increase production and price as per supply law. Let us assume that they increase the price at rate \( \beta \):

\begin{equation} P_{i+1} = P_i + \beta \cdot \mathrm{ED}_i . \end{equation}

Note that the excess demand can be negative, in which case the price would decrease.

## Dynamics of the model

Now, the dynamics of this model are a lot nicer than the ones we had in the previous approach. While at the first glance they somewhat contradict the intuitions of the original cobweb model, but we can quickly fix that.

First of all let us define inverse supply and inverse demand laws. Original supply and demand laws are linear functions of quantity:

\begin{equation} S(Q) = \alpha_s [ Q - Q_{eq} ] + P_{eq} , \end{equation}

\begin{equation} D(Q) = \alpha_d [ Q_{eq} - Q ] + P_{eq} . \end{equation}

Respective inverse laws are functions of price:

\begin{equation} S^{-1}(P) = \frac{P - P_{eq}}{\alpha_s} + Q_{eq}, \end{equation}

\begin{equation} D^{-1}(P) = \frac{P_{eq} - P}{\alpha_d} + Q_{eq}. \end{equation}

Price in the original cobweb model is updated as follows:

\begin{equation} P_{i+1} = D(S^{-1}(P_i)) = \frac{\alpha_d (P_{eq} - P_i)}{\alpha_s} + P_{eq} . \end{equation}

As we can see price deviation from the equilibrium is multiplied by a coefficient \( K = \frac{\alpha_d}{\alpha_s} \). It should be trivial to see that if \( | K | > 1 \) then the model is unstable. Obviously the same conclusion follows from the graphical analysis covered in the original post.

For the agent-based model we have introduced in this post, ignoring the inherent randomness, we find a somewhat different the update rule:

\begin{equation} P_{i+1} = P_i + \beta \cdot [ D^{-1}(P_i) - S^{-1}(P_i) ] , \end{equation}

which expands into:

\begin{equation} P_{i+1} = \frac{[ \alpha_d + \alpha_s ] \cdot \beta}{\alpha_d \alpha_s} [ P_{eq} - P_i ] . \end{equation}

Qualitatively the rule is similar, so after few algebraic operations we can easily find the coefficient. We see that:

\begin{equation} K = \frac{ (\alpha_d + \alpha_s) \beta - \alpha_d \alpha_s}{\alpha_d \alpha_s}. \end{equation}

Now we just have to require that \( | K | < 1 \) for the model to be stable. This condition is used by the app below to estimate when the market should be stable. Note the "should": if \( | K | \) is close to \( 1 \), the market can still collapse due to the randomness and discreteness effects.

The next step is to compare to the coefficients for the two models. After brief examination it is easy to see that if \( \beta = \alpha_d \), the coefficient for the agent-based cobweb model becomes identical to the one arising from the original model. This actually lends an interpretation for the rate \( \beta \): it is the guess of a supplier about the flexibility of the buyer. Note that, the model is usually stable when the guess underestimates \( \alpha_d \), though in some cases small overestimation also can yield stability. While gross overestimation usually will result in the market collapse.

## Conclusion

Cobweb model seems to have a hidden assumption that suppliers know the demand law (or at least its slope) of the buyers. But if they know the law fully, they should be able to immediately know where the equilibrium point lies. This is likely the thing that originally bothered me about the cobweb model.

Another catch is that after few initial observation periods the suppliers could be even smarter. They could used linear fit to make a rather good guess of the demand law. Thus these smarter suppliers could find equilibrium point even quicker.

In some cases, near \( | K | = 1 \), price seems to exhibit bursty behavior. I wonder if they price changes exhibit stylized facts common to financial markets. This we will explore in the next post.

Finally recall that behavior of the agents is governed by the supply and the demand laws, but they themselves might be also put through a more detailed inquiry as it is not totally evident why suppliers can't (or don't want to) charge higher prices. We have discussed this point from game theory perspective in an earlier post. Both of these approaches could be combined and a more complete (and complicated) model could be analyzed, but at this point I believe that we have explored the problem enough to get basic intuitions.

## Interactive app

At this time we invite you to explore the agent-based cobweb model using an interactive app below. Feel free to verify our intuitions as well as to develop your own.

## Disclaimer

Note that this model is a product of my inquiries to the nature of the cobweb model. This exploration might not make a lot of sense to a person more familiar with Economics than myself. Also it is quite possible that someone has already made a similar inquiry.