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 theta 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 theta from prices becomes arbitrarily slow as autocorrelation approaches a unit root level. It follows then that learning the unknown parameter theta 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 theta 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 theta can not be learned. When the process is sufficiently explosive and there is a positive mass of perfectly informed agents, theta is learned quickly but convergence to the FIE does not occur.