The Agglomeration-Differentiation Tradeoff in ... - Yale University

CCPs would be used for firms’ beliefs about rivals’ actions in the next iteration. The n th iteration
of the standard NPL approach can be denoted by the following contraction mapping, M:
( Pn, n) ( Pn−1) where, n arg max L( Pn−1, ) ; Pn ( Pn−1,
n)
Θ =Μ Θ = Θ =Ψ Θ (30)
Θ
For a graphical illustration of the NPL iterations, suppose that the set ( P, Θ ) could be
‘collapsed’ onto one axis. In Figure 2(a) the X-axis corresponds to the vector Pn − 1,
the Y-axis
corresponds to the set ( Pn, Θ n)
, and the solid curve represents the contraction mapping ( P)
24
Μ .
The dotted lines represent the ‘track’ followed by the NPL iterations corresponding to a
particular starting value, P 0 . Note that a different starting value,
'
P 0 , would result in a different
track for the NPL iterations. With multiple iterations, if there is convergence, the contraction
* *
mapping would converge to an equilibrium solution or a NPL fixed point, ( , )
P Θ . In Figure
3(a), this is the point where Μ ( P)
intersects the 45 o line. Furthermore, if the fixed point is
* *
unique then it is, in fact, the global optimum, ( MLE , MLE )
2.2.2. Multiple Equilibria in the Data
P Θ .
The standard NPL approach, however, does not address the possibility of multiple
equilibria in the data which is when the contraction mapping in Equation (31) does not have a
unique NPL fixed point. The multiple eqilibria or the multiple NPL fixed points are essentially
the different ‘local optima’ of Equation (29). This is illustrated in Figure 3(b) where Μ ( P)
intersects the 45 o line at multiple points. Consequently, the NPL iterations may potentially
converge to a ‘local optima’ and not the global optimum. Further, as the track followed by the