Monotone comparative statics

Roughly speaking, a sufficient condition for monotone comparative statics is supermodularity of players payoffs in strategies and a parameter. Note that, if there are multiple equilibria, we cannot claim that every equilibrium is monotone in a rather, a set of all equilibria is monotone in the sense of Theorem 10. A convenient way to think about the last Theorem is through the... 

In GT models, just as in the non-competitive SCM models, many of the managerial insights and results are obtained through comparative statics such as monotonicity of the optimal decisions w.r.t. some parameter of the game. 

Figure 27 shows the base shear coefficient (WHO - roof drift (0 ) behavior of the conventional MRF and the SC-MRF from nonlinear monotonic (pushover) static analysis. V is the base shear and W is the seismic weight. An inverted triangular force distribution along with roof displacement control was used in these analyses. The MRF and the SC-MRF have comparable base shear strengths and comparable... 

Non-monotonic density profiles are unstable. Since, however, the influence of the wall decays exponentially with the distance, the dynamics is practically frozen whenever the interphase boundary is separated from the wall by a layer thick compared to the characteristic width of the diffuse interface. A static solution with a fixed h exists only at a certain fixed value of i, which can be determined using a solvability condition of the first-order equation as in Section 1.3. In a wider context, an appropriate solvability condition serves to obtain an evolution equation for the nominal position h of the interphase boundary. The technique of derivation of solvability conditions for a problem involving a semi-infinite region and exponentially decaying interactions is non-standard and therefore deserves some attention. 

Nonlinear static monotonic (pushover) analysis shows that the conventional MRF and the SC-MRF have comparable base shear strength and initial stiffness. The conventional MRF experiences significant damage in beams at the DBE drift. On the other hand, the SC-MRF has damage-free beams for drifts even higher than the MCE drift. 

As the discussion in Sect. 2.1.1, and Eq. (2.A.12) show, in order for a rotating fluid element to maintain its equilibrium (static position in the r-direction), the pressure on its surface at higher r must exceed that on its surface a lower r. Thus the static pressure must increase monotonically with increasing radius. This, in fact, is borne out by experiment—a classic example of which is the data of Ter Linden (1953), a sample of which is presented in Fig. 3.1.2. Here the lower curves contained within each set of curves represents the variation in static pressure, p, with radial position the upper curves, the total pressure, p+(l/2)y0 (static plus dynamic). Comparing with Eq. (2.1.3) and realizing, as before, that the second term in Bernoulli s trinomial is small, we see from the profiles of total pressure in Fig. 3.1.2 that Bernoulli s trinomial is almost constant in the outer, nearly loss-free part of the vortex, while it decreases significantly in the center. This is as we would have expected. 

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