are irrational expectations rational? David Demery Nigel W. Duck1 ...

Are Irrational Expectations Rational? 16was resetting its price. The first assumed RE - i.e. that the firm observesthetruevaluesofε t and u j,t separately and applies the correct α and γcoefficients to them to form its expectations of its future marginal costs. Thesecond assumes that the firm observes only the composite shock, η S j,t which(given our assumption that all previous shocks are zero) will equal ε t + u S j,t ,and then applies the ρ S s to this composite to form its expectations of futuremarginal costs. The third assumes a method of expectations formationclose to adaptive expectations in which E j,t ∆mc j,t+i = µ i ∆mc j,t and weimpose selected values of µ between 0 and 1. The fourth, naïve expectations,assumed that the a parameter µ =1.The four prices give rise to four streams of future profits for a particularstream of future realizations of ε and u j . We assumed that these futurerealizations were all zero and take the ratio of the streams of profits to thosegenerated by RE as our indicator of the firm’s expected loss of profits fromforming non-RE expectations. We express this ratio as DPF/DPF ∗ whereDPF ∗ denotes the discounted value of future profits from the optimal pricewhen both shocks are observed, and DPF denotes the discounted value offuture profits from the optimal price when some other method of formingexpectations is assumed. The closer is this ratio to 1 (it will always beless than 1), the smaller the loss of profits from not observing the shocksseparately. As our parameters we use estimates of the ρs andofβ and θpresented in Demery and Duck (2001, Table 1) and their estimates of theαs andσ 2 ε from an MA(q) empirical model for ∆mc t . The value of q wasdetermined by the number of significant ρs ineachcountry.FortheUKweestimated an MA(5) process for ∆mc t ;fortheUSweassumedanMA(3).Using these as our parameter values, and for different assumed values of σ 2 u,we then solved for the values of the γs whichareconsistentbothwiththeq-order invertible MA process driving u j,t and with our estimated values ofthe ρs. 12 We simulated 1000 values of DPF/DPF ∗ and report their meansin Table 5.12 We used numerical techniques to derive the required values of the γs. For the UK thefive αs were 0.325, 0.375, 0.3212,0.1007 and 0.1480; and the ρs were 0.807, 0.5204, 0.4013,0.2278, amd 0.1763. For the US the equivalent figures were 0.2696, 0.3199, and 0.0825;and 0.294, 0.1829 and 0.1499.