A Dynamic Network Oligopoly Model with Transportation Costs, Product Differentiation, and Quality Competition Anna Nagurney John F. Smith Memorial Professor and Dong (Michelle) Li Doctoral Student Department of Operations & Information Management Isenberg School of Management University of Massachusetts Amherst, Massachusetts 01003 Raytheon MTN Symposium University Track, Andover, MA October 9, 2013 University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
Acknowledgments This research was supported, in part, by the National Science Foundation (NSF) grant CISE #1111276, for the NeTS: Large: Collaborative Research: Network Innovation Through Choice project awarded to the University of Massachusetts Amherst. This support is gratefully acknowledged. University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
This presentation is based on the paper: Nagurney, A. and Li, D., 2012. A Dynamic Network Oligopoly Model with Transportation Costs, Product Differentiation, and Quality Competition, Computational Economics , in press, where a full list of references can be found. University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
Outline Motivation The Dynamic Network Oligopoly Model Stability Analysis The Algorithm Numerical Examples Summary and Conclusions University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
Motivation Oligopolies constitute fundamental industrial organization market structures of numerous industries world-wide. In classical oligopoly problems, the product is assumed to be homogeneous. However, in many cases, consumers may consider the products to be differentiated according to the producer. University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
Motivation Quality is emerging as an important feature in numerous products, and it is implicit in product differentiation . University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
Literature Review Banker, R. D., Khosla, I., and Sinha, K. J. (1998). Quality and competition. Management Science, 44(9), 1179-1192. Hotelling, H. (1929). Stability in competition. The Economic Journal, 39, 41-57. Nagurney, A., Dupuis, P., and Zhang, D. (1994). A dynamical systems approach for network oligopolies and variational inequalities. Annals of Regional Science, 28, 263-283. Dafermos, S. and Nagurney, A. (1987). Oligopolistic and competitive behavior of spatially separated markets. Regional Science and Urban Economics, 17, 245-254. University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
Literature Review Nagurney, A. and Yu, M. (2012). Sustainable fashion supply chain management under oligopolistic competition and brand differentiation. International Journal of Production Economics, 135, 532-540. Masoumi, A.H., Yu, M., and Nagurney, A. (2012). A supply chain generalized network oligopoly model for pharmaceuticals under brand differentiation and perishability. Transportation Research E, 48, 762-780. University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
Motivation Cabral (2012) recently articulated the need for new dynamic oligopoly models, combined with network features, as well as quality. In this research, we develop a network oligopoly model with differentiated products and quality levels. We present both the static version, in an equilibrium context, which we formulate as a finite-dimensional variational inequality problem, and then we develop its dynamic counterpart, using projected dynamical systems theory. University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
The Quantification of Quality Quality level is quantified and incorporated in the model. Quality level is defined and quantified as the “quality conformance level”, the degree to which a specific product conforms to a design or specification (Juran and Gryna (1988)), and it should be within 0 and 100 percent of defects levels. University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
The Network Structure of the Dynamic Network Oligopoly Problem with Product Differentiation ♠ ♠ ♠ . . . · · · m Firms 1 i ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ ✟ ❩❩❩❩❩❩❩❩❩❩❩ ✚ ❏ ✡ ✟ ✟ ✚ ❏ ✡ ✟ ✚ ✟ ❏ ✡ ✟ ✚ ✟ ✚ ❏ ✡ ✟ ✟ ✚ ✡ ❏ ✟ ✚ ✟ ✡ ❏ ✟ ✚ ✟ ✚ ✡ ✟ ❏ ✟ ✚ ✡ ✟ ❏ ✚ ✙ ✢ ❄ ✡ ✟ ✟ ❏ ❫ ✚ ❂ ❄ ❍ ⑦❄ ❥ ❩ ♠ ♠ ♠ . . . j · · · n Demand Markets 1 University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
The Dynamic Network Oligopoly Model Conservation of flow equations n � s i = Q ij , i = 1 , . . . , m , (1) j =1 d ij = Q ij , i = 1 , . . . , m ; j = 1 , . . . , n , (2) Q ij ≥ 0 , i = 1 , . . . , m ; j = 1 , . . . , n . (3) We group the production outputs into the vector s ∈ R m + , the demands into the vector d ∈ R mn , and the product shipments into + the vector Q ∈ R mn + . University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
The Dynamic Network Oligopoly Model Production cost function for firm i ˆ f i = ˆ f i ( s , q i ) , i = 1 , . . . , m . (4) We assume, hence, that the functions in (5) also capture the total quality cost, since, as a special case, the above functions can take on the form ˆ f i ( s , q i ) = f i ( s , q i ) + g i ( q i ) , i = 1 , . . . , m . (5) The production cost functions (4) (and (5)) are assumed to be convex and continuously differentiable. University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
The Dynamic Network Oligopoly Model Interestingly, the second term in (5) can also be interpreted as the R&D cost (cf. Matsubara 2010), which is the cost that occurs in the processes of the development and introduction of new products to market as well as the improvement of existing products. Evidence indicates that the R&D cost depends on the quality level of its products (see, Klette and Griliches 2000; Hoppe and Lehmann-Grube 2001; Symeonidis 2003). University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
The Dynamic Network Oligopoly Model Nonnegative quality level for firm i ’s product q i ≥ 0 , i = 1 , . . . , m . (6) We group the quality levels of all firms into the vector q ∈ R m + . Demand price function for firm i ’s product at demand market j p ij = p ij ( d , q ) , i = 1 , . . . , m ; j = 1 , . . . , n . (7) We allow the demand price for a product at a demand market to depend, in general, upon the entire consumption pattern, as well as on all the levels of quality of all the products. The generality of the expression in (6) allows for modeling and application flexibility. The demand price functions are, typically, assumed to be monotonically decreasing in product quantity but increasing in terms of product quality. University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
The Dynamic Network Oligopoly Model Transportation cost function ˆ c ij = ˆ c ij ( Q ij ) , i = 1 , . . . , m ; j = 1 , . . . , n . (8) The demand price functions (7) and the total transportation cost functions (8) are assumed to be continuous and continuously differentiable. University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
The Dynamic Network Oligopoly Model The strategic variables of firm i are its product shipments { Q i } where Q i = ( Q i 1 , . . . , Q in ) and its quality level q i . Utility function n n � � U i = p ij d ij − f i − g i − ˆ c ij . (9) j =1 j =1 In view of (1) - (9), one may write the profit as a function solely of the shipment pattern and quality levels, that is, U = U ( Q , q ) , (10) where U is the m -dimensional vector with components: { U 1 , . . . , U m } . University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
Definition: A Network Cournot-Nash Equilibrium Let K i denote the feasible set corresponding to firm i , where K i ≡ { ( Q i , q i ) | Q i ≥ 0 , and q i ≥ 0 } and define K ≡ � m i =1 K i . Definition 1 A product shipment and quality level pattern ( Q ∗ , q ∗ ) ∈ K is said to constitute a Cournot-Nash equilibrium if for each firm i ; i = 1 , . . . , m , i , ˆ i ) ≥ U i ( Q i , q i , ˆ U i ( Q ∗ i , q ∗ q ∗ q ∗ ∀ ( Q i , q i ) ∈ K i , Q ∗ Q ∗ i , ˆ i , ˆ i ) , (11) where ˆ i ≡ ( Q ∗ 1 , . . . , Q ∗ i − 1 , Q ∗ i +1 , . . . , Q ∗ Q ∗ m ); and ˆ i ≡ ( q ∗ 1 , . . . , q ∗ i − 1 , q ∗ i +1 , . . . , q ∗ q ∗ m ) . (12) University of Massachusetts Amherst A Dynamic Network Oligopoly Model with Quality Competition
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