Potential entrants in markets that opened up in the years after 2000 tend to have higher levels of experience at vertically integrated entry, which is likely to lower their costs of entering as an integrated firm. If the effect of past vertical entry experience is not fully captured by the Downstream Experience and Upstream Experience variables – which would be the case if vertically integrated entry generates cost-lowering effects that are not gained from separately entering the downstream and upstream segments of different markets – it is possible for the Post-2000 dummy variable to pick up the unexplained portion of the effect. The coefficients on the two firm-level experience variables all have the same expected sign. Past downstream entry experience has a significantly positive impact on unintegrated downstream and vertically integrated payoffs. Past upstream entry experience similarly affects the payoffs of unintegrated upstream entrants and vertically integrated entrants in a significantly positive manner. The preceding results indicate that firms’ entry actions are consistent with the existence of efficiency effects and the absence of foreclosure effects. This implies that vertical integration in this industry is likely to be procompetitive from a static point of view. Nevertheless, vertical integration can still have an anti-competitive market structure effect.
For instance, independent upstream firms may be deterred from entering when they anticipate tough competition from vertically integrated rivals. This, in turn, vertical grow rack may reduce the expected profits of independent downstream entrants and lead to fewer entrants in the downstream segment. Given the stylized fact that the prices of generic drugs fall monotonically with the number of downstream entrants , competition authorities are likely to be concerned if vertical integration tends to reduce the equilibrium number of entrants.36 On the other hand, it is possible for vertical integration to increase the equilibrium number of downstream entrants, because its efficiency effects may benefit unintegrated downstream entrants through positive spillovers. If vertical integration is found to promote entry into the downstream segment, we can conclude that vertical integration has a procompetitive overall effect. This is because the combination of greater downstream entry and significant efficiency effects unambiguously implies lower prices and/or higher quality for the final product. To examine how vertical integration affects market structure formation, I conduct a policy simulation. Specifically, I simulate the effect of a hypothetical policy that bans any firm from entering both vertical segments of the same market.
To my knowledge, no such policy has yet been contemplated for the generic pharmaceutical industry. However, it is similar in spirit to vertical separation regulations found, for example, in the electric utility industry. Recent antitrust cases such as FTC v. Mylan et al. , involving exclusive dealing contracts between API manufacturers and finished formulation firms, have shown that vertical practices can have highly anti-competitive effects in the generic drug industry. This suggests that vertically integrated entry might come under stronger antitrust scrutiny in the future. If vertical integration has an entry-reducing effect, the ban on vertically integrated entry should increase the equilibrium number of downstream entrants relative to the status quo where vertically integrated entry is allowed. Conversely, the ban would reduce the number of downstream entrants in equilibrium if vertical integration has an entry-promoting effect. To simulate the effect of the policy, I run two sets of predictions on equilibrium market structures. In the first set, firms are allowed to enter as a vertically integrated entity. In the second set, I simulate the ban by removing “vertically integrated entry” from the choice set of every potential entrant. For both sets, I make predictions for 50 draws of the error term vector , and for each draw I compute all pure strategy Nash equilibria of the entry game. The parameter values that I use are the modes of the marginal posterior distributions.37 Figures 3.7 and 3.8 present the results of policy simulation.
To compare the predicted equilibrium market structures with and without the ban on vertically integrated entry, I count the number of entrants that are predicted to be present in each vertical segment. A vertically integrated entrant is counted as both an upstream entrant and a downstream entrant. For the 85 markets in the dataset, I calculate the mean number of entrants in each segment, averaging over draws as well as over multiple equilibria. Each dot in Figures 3.7 and 3.8 represents a sample market, with the horizontal axis measuring the market’s user population and the vertical axis measuring the change in number of entrants caused by the vertical entry ban. In Figure 3.7, we see that the number of upstream entrants tends to be greater when vertically integrated entry is banned. Of the 85 sample markets, 24 experience an increase in the predicted number of entrants, only four experience a decrease, and 52 experience no change.38 The magnitude of impact tends to be greater for positive changes: among those markets where the expected number of upstream entrants increases, the average change is +0.4472, whereas among the markets experiencing a decrease the average change is -0.3167. The efficiency effect of vertical integration appears to be sufficiently strong to deter some firms from entering as an unintegrated upstream supplier. By removing this entry deterrent effect, the vertical entry ban succeeds in promoting upstream entry. On the other hand, Figure 3.8 shows that the number of downstream entrants tends to decrease in response to the vertical entry ban. The number of downstream entrants decreases in nineteen markets , while it increases in seven and remains unchanged in 54 . The absolute value of the change is larger in markets experiencing a decrease: while the average positive change is +0.0919, the average negative change is -0.2368. It thus appears that the efficiency spillovers from vertical integration are so large that, despite decreased upstream entry, more downstream entry occurs when vertically integrated entry is allowed. It is also possible that the problem of double marginalization is more severe under the vertical entry ban, leading to fewer downstream entrants despite greater entry in the upstream segment. The policy of banning vertically integrated entry is therefore counterproductive. By decreasing the number of downstream entrants and depriving the opportunities for efficiency enhancement through vertical integration, the ban is likely to have a negative impact on market outcomes.Accurate detection of physical interference between two or more bodies is crucial in the design of many engineering systems. Modeling interference between physical bodies is, therefore, an important problem in computational design. Non-interference constraints appear in numerical optimization problems that manipulate an object within an environment containing other objects such that there is no collision. Efficient and accurate modeling of the non-interference constraints is critical for fast and reliable solutions in the overarching optimization problem. Prior literature on these problems describe these constraints using inconsistent terminology, e.g., anatomical constraints, spatial integration constraints, hydroponic shelf system boundary constraints, and interference checks. We observe that these terms represent the same underlying constraint. We propose to call these constraints geometric non-interference constraints since they are employed in design optimization to ensure a design where there exists no interference between two or more geometric shapes or paths of motion. In our study, a geometric shape is associated with the design configuration of an engineering system at a particular instance of time. The geometric shapes of interest in this paper are curves in two dimensions, or orientable surfaces in three dimensions. We assume that the geometric shapes are non-self-intersecting but make no assumptions on whether they are open or closed. A path of motion or trajectory is the set of points that traces the motion of a point on the engineering system as the system changes configuration over time. The paths considered in this paper are simply curves in two or three dimensions.
We use the term layout to refer to a set of geometric shapes. Based on the definitions above, we identify three major classes of optimization problems with geometric non-interference constraints: layout optimization, shape optimization, and optimal path planning. All three classes are parts of the scope of problems we address in this paper. Layout optimization optimizes the positions of geometric shapes via translation subject to geometric non-interference, with or without additional boundary constraints. For example, the wind farm layout optimization problem consists in positioning the wind turbines within a wind farm in an optimal way such that interference between turbines and the boundary of the farm is avoided. Another example of a layout optimization problem is the packing problem. Packing problems consist of positioning objects within a space to ensure the minimum space is occupied or the maximum number of objects are placed without geometric interference. Shape optimization seeks to optimize geometric shapes subject to geometric noninterference, with or without additional boundary constraints. For example, shape optimization of an aircraft fuselage optimizes the shape of a fuselage with constraints ensuring that the passengers, crew, payload, and all the subsystems fit inside the fuselage. Optimal path planning optimizes the trajectory of a point or a set of points subject to geometric non-interference, with or without additional boundary constraints. Robot motion planning is a class of problems that falls under optimal path planning and is widely researched. The design optimization of surgical robots is an example of a problem involving robot motion planning that has had recent attention. In the design optimization of surgical robots, non-interference constraints are imposed such that the robot does not collide with the anatomy of a patient during operation. Additionally, it is desirable for the robot to maintain a safe distance from the anatomy, motivating the use of a distance-based non-interference constraint formulation in such problems. The problems just mentioned are solved using numerical optimization algorithms. Historically, gradient-free algorithms have been more commonly used to solve such problems, e.g., in layout optimization and in robot motion planning. As models become more complex, that is, with more disciplines and design variables, solutions become impracticable with gradient-free algorithms since these algorithms scale poorly with the number of design variables. However, the recent emergence of modeling frameworks such as Open MDAO has enabled efficient design of large-scale and multidisciplinary systems using gradient-based optimization, including some of the aforementioned problems with geometric non-interference constraints. Geometric non-interference constraint functions for gradient-based optimization require special consideration. These functions must be continuously differentiable or smooth in order to be used with a gradient-based optimization algorithm. They should also be efficient to compute because optimization algorithms evaluate constraint functions and their derivatives repeatedly over many optimization iterations. During some iterations, the optimizer may violate an interference constraint, and useful gradient information on such iterations is still required despite it being infeasible. Consequently, any noninterference constraint function must be defined in the event of an overlap between objects and provide necessary gradient information. Figure 1.1 shows a diagram with two iterations of a design body in an optimization problem. One of the designs shown is feasible while the other is not. The feasible design is the one where the design body is completely inside the feasible space whereas the infeasible design has at least one point on the design body lying outside the feasible space.The constant ϵ can be any small positive value appropriate for a given problem. Existing non-interference constraint formulations suffer from various limitations. The formulation of quasi-phi-functions by Stoyan et al. provides an analytical form to represent an interference for simple geometric shapes. Quasi-phi-functions are continuous but only piece wise continuously differentiable. These functions are also not generalized to represent any arbitrary shape. The formulation by Brelje et al. is generalized to any triangulated 3D geometric shape, but has computational limitations. The computational complexity of their method is O, where NΓ is the number of elements in the triangulation. They are able to overcome this scaling issue by making use of graphics processing units but demonstrate their formulation on a geometric shape with only 626 elements in the triangulation. In their recent work on the WFLOP, Risco et al. formulate a generic explicit method for geometric shapes in 2D, but the method suffers from the same scaling issues as in [3] and contains discontinuous derivatives. The formulation by Bergeles et al. employs a distance potential function that is calculated with the k-nearest neighbors.