ing
Consider the scenario in which the manufacturer has all information that the sup- plier has at each period; i.e., Fs
n ⊆ Fnm for every n. Both decision makers obtain new demand information from a third-party research firm (such as Gartner; see http://www.gartner.com) over time, but the manufacturer has additional informa- tion because of her proximity to the market and the insider information about her
own product. In other words, the manufacturer has privatedemand information that is not available to the other decision maker. We refer to this forecast evolution model as the Martingale Model of Asymmetric Forecast Evolution (MMAFE). We denote the difference between the two decision makers’ forecasts by An ≡Xnm−Xns. Then, we have the following properties of the MMAFE:
Theorem 3.3. If (Xs
n, Xnm) constructed by (XN+1,Fns,Fnm) is an MMAFE, then we
have the following properties for every n: (a) E[Xm
n+l|Fns] =E[Xnm+l|Xns] =Xns for every l ≥0.
(b) E[XN+1|Xns, An] =E[XN+1|Fnm] =Xnm.
(c) E[An] = 0 and An is uncorrelated with Fns.
Part (a) implies that the supplier’s estimate of the manufacturer’s forecast is the same as his own forecast. Part (b) implies that by knowing the value ofAn, the supplier can make the best forecast. Part (c) implies that An is uncorrelated with the supplier’s information set,Fs
n.
For the case of multiplicative MMFE, we can model the asymmetric information scenario by settingδns,nm = 1 for everynm > ns. In other words, the supplier obtains
no information earlier than the manufacturer. Hence, the information obtained by the supplier at period n consists of δn,0, δn,1, . . . , δn,n, where each δn,nm has already
been obtained or is being obtained at the same time by the manufacturer. We refer this case by the multiplicative Martingale Model of Asymmetric Forecast Evolution
(m-MMAFE) and we will use this model in the second part of the chapter. The information structure of the m-MMAFE is provided in Figure 3.2(b). We refer the reader to this figure for better understanding of the following discussion.
The manufacturer’s private demand information represents the information asym- metry between the two decision makers. The manufacturer’s demand uncertainty is also the demand uncertainty faced by the system. Recall that the multiplication of
δm
n, δnm+1, . . . , δmN represents the demand uncertainty faced by the manufacturer at the beginning of period n, and we denote it by n ≡
QN
k=nδmk. From the manufacturer’s perspective, demand is XN+1 = Xnmn, where Xnm is her current forecast, which is
deterministically known to her. The remaining market uncertaintyn is resolved over periods n toN as the manufacturer obtains information, i.e. the forecast updates.
In contrast, the demand uncertainty faced by the supplier at the beginning of periodnisQN
k=nδks. The manufacturer has already obtained part of this information. To distinguish the known part, we rewrite
N Y k=n δks = N Y k=n n−1 Y nm=0 δk,nm N Y nm=n δk,nm ! = N Y k=n n−1 Y nm=0 δk,nm ! | {z } ξn N Y k=n N Y nm=n δk,nm ! | {z } n . (3.1)
The first part of Equation (3.1) represents the demand information that is already obtained by the manufacturer. The second part represents the demand information that is not yet obtained by the manufacturer. Because δns,nm = 1 for nm > ns, the
second part of (3.1) is N Y k=n N Y nm=n δk,nm = N Y nm=n N Y k=n δk,nm = N Y nm=n N Y k=0 δk,nm = N Y nm=n δmnm,
which is equal ton. The first part of (3.1) is the manufacturer’s private information, and we denote it by ξn. Then, demand can be represented as XN+1 = Xnsξnn. From the supplier’s perspective, Xs
n is deterministic, ξn and n are uncertain. By construction, Xns, ξn and n are independent. Note also that Xnm = Xnsξn. The supplier obtains only part of the information of n and ξn during period n and he obtains the full information of n and ξn over periods n toN.
For notational simplicity, we denote the standard deviation of log(Z) of a log- normal random variableZ asσZ throughout this chapter. The value ofσn represents
the degree of demand uncertainty of the system at period n, and the value of σξn
represents the degree of information asymmetry between the supplier and the man- ufacturer at period n. By construction, σn always decreases in n. In contrast, σξn
can either increase or decrease in n depending on the values of σδs
the supplier obtains more information than the manufacturer during period n, i.e.,
σδs
n > σδnm, we have σξn+1 < σξn, and vice versa.