# Cobweb theorem and rational expectation

In the cobweb theorem it is assumed that the producers follow an adaptive expectation, however if the price they look to determine their production reflects all the available information at the moment, so this price is in the context of the efficient market hypothesis and therefore it is formed under the rational expectation. What is wrong here?

• EMH stems from rational expectations, so without them you don't have that result. Cobweb stems from adaptive, so you don't get EMH. – VCG Aug 30 '16 at 19:17
• Definitely, however what is wrong with my reasoning? – unmark1 Aug 30 '16 at 19:59
• I still don't know what "price in context" means. – VCG Aug 30 '16 at 20:14
• – VCG Aug 30 '16 at 20:44
• I've found this which offers a good explanation: developing-trade.com/wp-content/uploads/2014/11/… – unmark1 Sep 2 '16 at 14:40

In the basic "Cobweb Theorem", the assumption is that suppliers must decide now next period production, and that they assume that next period market price will be equal to current period market price.

$$p^e_{t} = p_{t-1} \tag{1}$$

Now, we know that if price follows a random walk without drift, then the above a priori assumed expectations formation hypothesis coincides with the Rational Expectations Hypothesis:

$$\text {If}\;\;\; p_{t} = p_{t-1} + u_{t} ,\;\;\; u_{t} \sim WN \implies E_{t-1}p_{t} = p_{t-1} \tag{2}$$

But in the cobweb model, the actual price process is determined endogenously, given the a priori assumption $(1)$: we assume $(1)$ and we let the equilibrium of the model give us the model-consistent price process.

Assuming linear demand (say, with a white noise preference shock), and supply function we have

$$D_t = a-bp_t +u_t,\;\;\; S_t = \gamma + \delta p^e_t$$

Imposing market clearing (an additional assumption no less) we have

$$D_t = S_t \implies p_t = \frac {a-\gamma}{b} - \frac {\delta}{b} p^e_t +u_t$$

and applying the a priori expectations formation assumption $(1)$ we obtain

$$p_t = \frac {a-\gamma}{b} - \frac {\delta}{b} p_{t-1} + u_t\tag{3}$$

$(3)$ is the model-consistent price process here (i.e. given $(1)$), not $(2)$. Only in the case that $a=\gamma, b=-\delta$ do we get that the price process emerges as a random walk without drift and so that the expectations formations hypothesis used in the model coincides with the Rational Expectations hypothesis.

The price that suppliers use is the current market clearing price indeed. But how they use it (hypothesis $(1)$), does not "reflect all available information", which is embodied in $(3)$ which suppliers ignore.