Learning and Convergence to a Full-Information Equilibrium are not Equivalent

Convergence to a full-information equilibrium (FIE) in the presence of persistent shocks and asymmetric information about an unknown payoff-relevant parameter θ is established in a classical infinite-horizon partial equilibrium linear model. It is found that, under the usual stability assumptions on the autoregressive process of shocks, convergence occurs at the rate n−1/2, where n is the number of rounds of trade, and that the asymptotic variance of the discrepancy of the full-information price and the market price is independent of the degree of autocorrelation of the shocks. This is so even though the speed of learning θ from prices becomes arbitrarily slow as autocorrelation approaches a unit root level. It follows then that learning the unknown parameter θ and convergence of the equilibrium process to the FIE are not equivalent. Moreover, allowing for non-stationary processes of shocks, the distinction takes a more stark form. Learning θ is neither necessary nor sufficient for convergence to the FIE. When the process of shocks has a unit root, convergence to the FIE occurs but θ can not be learned. When the process is sufficiently explosive and there is a positive mass of perfectly informed agents, θ is learned quickly but convergence to the FIE does not occur.