Cob-web Market Model

• Nicholas coined this term in 1934
• It is an economic model, that explains cyclical supply and demand, in a market where the amount produced must be decided before prices are observed. For example, it is common in certain types of markets, where there is a periodic fluctuation of certain commodities. Agricultural markets are a context where the Cob-web Market Model may apply.
• In Cob-web market situation,
  1. the output decision of a producer depends on the price of previous period.
  2. the demand for production depends on the price of current period
• $\therefore$ In Cob-web Market Model, $$\begin{array}{rlr}{Q}_d& =\alpha -\beta P_t& [\alpha ,\beta >0]\\ Q_s& =-\gamma +\delta P_{t-1}& [\gamma ,\delta >0]\end{array}\text{\hspace{1em}(1)(2)}$$

where, $Q_d$ is quantity demanded, $Q_s$ is Quantity supplied. $\beta =$ slope of demand function $\delta =$ slope of supply function

Equating $Q_d=Q_s$ we get

$$\begin{array}{rl}7\alpha -\beta P_t& =-\gamma +\delta P_{t-1}\\ -\beta P_t-\delta P_{t-1}& =-\gamma -\alpha \\ -\left(\beta P_t+\gamma P_{t-1}\right)& =-(\alpha +\gamma )\\ \beta P_t& =-\delta P_{t-1}+(\alpha +\gamma )\\ P_t& =-\frac{\delta }{\beta }{P}_{t-1}+\left(\frac{\alpha +\gamma }{\beta }\right)\end{array}$$

Let $b=-\frac{\delta }{\beta }$ and $a=\frac{\alpha +\gamma }{\beta }$

$$P_t=-b{P}_{t-1}+a\text{\hspace{1em}(3)}$$

So, (3) is a FOLDE and thus, the general solution for (3) is,

$$\begin{array}{rl}{P}_t& =\left(P_0-\frac{a}{1-b}\right)b^t+\frac{a}{1-b}\\ P_t& =\left(P_0-\frac{\frac{\alpha +\gamma }{\beta }}{1-\left(-\frac{\delta }{\beta }\right)}\right){\left(-\frac{\gamma }{\beta }\right)}^t+\frac{\frac{\alpha +\gamma }{\beta }}{1-\left(-\frac{\gamma }{\beta }\right)}\\ P_t& =\left(P_0-\frac{\frac{\alpha +\gamma }{\cancel{\beta }}}{\frac{\beta +\delta }{\cancel{\beta }}}\right){\left(-\frac{\gamma }{\beta }\right)}^t+\frac{\frac{\alpha +\gamma }{\cancel{\beta }}}{\frac{\beta +\delta }{\cancel{\beta }}}\\ P_t& =\left(P_0-\frac{\alpha +\gamma }{\beta +\delta }\right){\left(-\frac{\gamma }{\beta }\right)}^t+\frac{\alpha +\gamma }{\beta +\delta }\end{array}\text{\hspace{1em}(4)}$$

Since, $P_e=\frac{\alpha +\gamma }{\beta +\delta }$ is the inter-temporal equilibrium price. We ,write equation (4) is the time path of the Cob-web Market Model. Here, $P_0$ is the initial price and $P_e$ is the inter-temporal equilibrium price across time.

Now, the nature of the time path depends on $\frac{\delta }{\beta }$, i.e. ratio of slope of supply curve, $\delta$ and slope of demand curve, $\beta$.

There are three cases:

S.No.	Case	Graph
$I$	$\Vert \frac{\delta }{\beta }\Vert <1$ $;\delta <\beta$.	Transclude of Cob-web-Market-Model-2024-09-25-09.41.32.excalidraw
$II$	$\Vert \frac{\delta }{\beta }\Vert >1$ $;\delta >\beta$.	Transclude of Cob-web-Market-Model-2024-09-25-09.53.29.excalidraw
$III$	$\Vert \frac{\delta }{\beta }\Vert =1$ $;\delta =\beta$.	Transclude of Cob-web-Market-Model-2024-09-25-09.54.56.excalidraw

1. The time path is convergent and the market is dynamically stable.
2. The current path $P_t$ will diverge more from the equilibrium price. The time path is divergent and the market is not dynamically stable
3. In equation (4), the difference between $P_t$ and $P_e$ will remain same