Author: Mohan Gopalakrishnan
Publisher:
ISBN:
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Languages : en
Pages : 0
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Motivated by direct interactions with practitioners and real-world data, we study a monopoly firm selling multiple substitute products to customers characterized by their different social network degrees. Under the multinomial logit model framework, we assume that the utility a customer with a larger network degree derives from the seller's products is subject to more impact from her neighbors and describe the customers' choice behavior by a Bayesian Nash game. We show that a unique equilibrium exists as long as these network effects are not too large. Furthermore, we study how the seller should optimally set the prices of the products in this setting. Under the homogeneous product-related parameter assumption, we show that if the seller optimally price-discriminates all customers based on their network degrees, the products' markups are the same for each customer type. Building on this, we characterize the sufficient and necessary condition for the concavity of the pricing problem, and show that when the problem is not concave, we can convert it to a single-dimensional search and solve it efficiently. We provide several further insights about the structure of optimal prices, both theoretically and numerically. Furthermore, we show that we can simultaneously relax the multinomial logit model and homogeneous product-related parameter assumptions and allow customer in- and out-degrees to be arbitrarily distributed whilemaintaining most of our conclusions robust.
Author: Chenhao Du
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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We consider a seller's problem of determining revenue-maximizing prices for an assortment of products that exhibit network effects. Customers make purchase decisions according to a multinomial logit (MNL) customer choice model, modified -- to incorporate network effects -- so that the utility each individual customer gains from purchasing a particular product depends on the market's total consumption of that product. In the setting of homogeneous products, we show that if the network effect is comparatively weak, then the optimal pricing decision of the seller is to set identical prices for all products. However, if the network effect is strong, then the optimal pricing decision is to set the price of one product low, and to set the prices of all other products to a single high value. This boosts the sales of the single low-price product in comparison to the sales of all other products. These results can be compared to the optimal pricing policy for the classical MNL model (without network effects) in which it is optimal to set identical prices and obtain identical sales quantities for homogeneous products. We obtain comparative statics results that describe how optimal prices and sales levels vary with a parameter that determines the strength of the network effects. We extend our analysis of settings with heterogeneous products as well as to settings with inter-product network effects, and we describe efficient computational methods.
Author: Mengzi Amy Guo
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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We consider an infinite-horizon multi-product dynamic pricing problem with reference effects in a monopolistic setting, where the reference price is an exponentially weighted average of historical prices. In each period, the demand follows the multinomial logit (MNL) model, where the utility depends on both the current price and the reference price, and consumers can have product-differentiated sensitivities to the price and the reference price. We conduct thorough analyses of the myopic pricing policy, including its solution, long-run convergence behavior, and performance guarantee compared to the long-term discounted revenue of the optimal pricing policy. Furthermore, we establish the structural properties of the optimal pricing policy. When consumers are loss-neutral towards all products, we explicitly characterize the optimal pricing policy if it converges to a steady state, and based on our characterization we show that the steady state price can be computed efficiently by a binary search. Interestingly, we find that such a convergence behavior of the optimal pricing policy heavily relies on the upper bound of the admissible price range, and a low price upper bound facilitates the policy to converge. In contrast, when consumers are gain-seeking towards all products, we prove that the optimal pricing policy admits no steady state regardless of the price range. Nevertheless, if consumers are only gain-seeking towards certain but not all products, the optimal pricing policy can potentially be convergent. In addition, our numerical experiments show that loss-aversion over all products does not rule out price fluctuations. This finding is at odds with the classic belief on loss-averse consumers and hence, highlights the significance of accounting for cross-product effects through the MNL demand.
Author: Zhenyu Cui
Publisher:
ISBN:
Category :
Languages : en
Pages : 25
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Book Description
We obtain exact semi-explicit solutions to the optimal pricing problem for the multinomial logit (MNL) consumer choice model with network effects, through a novel conditioning argument and the use of the Lambert W function. Then we manage to characterize the exact optimal solution in the presence of negative, positive and weak or positive and strong network effects, and also in the setting of heterogeneous market parameters. Furthermore, we provide a semi-explicit expression of equilibrium sales quantities in terms of equilibrium prices, based on which we establish sufficient conditions for the uniqueness of market equilibria. In addition, we provide an exact semi-explicit expression for the steady-state choice probability, based on which we discuss the uniqueness of the steady-state choice probability.
Author: Guillermo Gallego
Publisher:
ISBN:
Category :
Languages : en
Pages : 26
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Book Description
We study firms that sell multiple substitutable products and customers whose purchase behavior follows a Nested Logit model, of which the Multinomial Logit model is a special case. Customers make purchasing decision sequentially under the Nested Logit model: they first select a nest of products and subsequently purchase one within the selected nest. We consider the multi-product pricing problem under the general Nested Logit model with product-differentiated price sensitivities and arbitrary nest coefficients. We show that the adjusted markup, defined as price minus cost minus the reciprocal of price sensitivity, is constant for all products within a nest at optimality. This reduces the problem's dimension to a single variable per nest. We also show that the adjusted nest-level markup is nest-invariant for all the nests, which further reduces the problem to maximizing a single-variable unimodal function under mild conditions. We also use this result to simplify the oligopolistic multi-product price competition and characterize the Nash equilibrium. We also consider more general attraction functions that include the linear utility and the multiplicative competitive interaction models as special cases, and show that similar techniques can be used to significantly simplify the corresponding pricing problems.
Author: Philipp Wilhelm Keller
Publisher:
ISBN:
Category :
Languages : en
Pages : 204
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Book Description
We consider a retailer offering an assortment of differentiated substitutable products to price-sensitive customers. Prices are chosen to maximize profit, subject to inventory/ capacity constraints, as well as more general constraints. The profit is not even a quasi-concave function of the prices under the basic multinomial logit (MNL) demand model. Linear constraints can induce a non-convex feasible region. Nevertheless, we show how to efficiently solve the pricing problem under three important, more general families of demand models. Generalized attraction (GA) models broaden the range of nonlinear responses to changes in price. We propose a reformulation of the pricing problem over demands (instead of prices) which is convex. We show that the constrained problem under MNL models can be solved in a polynomial number of Newton iterations. In experiments, our reformulation is solved in seconds rather than days by commercial software. For nested-logit (NL) demand models, we show that the profit is concave in the demands (market shares) when all the price-sensitivity parameters are sufficiently close. The closed-form expressions for the Hessian of the profit that we derive can be used with general-purpose nonlinear solvers. For the special (unconstrained) case already considered in the literature, we devise an algorithm that requires no assumptions on the problem parameters. The class of generalized extreme value (GEV) models includes the NL as well as the cross-nested logit (CNL) model. There is generally no closed form expression for the profit in terms of the demands. We nevertheless how the gradient and Hessian can be computed for use with general-purpose solvers. We show that the objective of a transformed problem is nearly concave when all the price sensitivities are close. For the unconstrained case, we develop a simple and surprisingly efficient first-order method. Our experiments suggest that it always finds a global optimum, for any model parameters. We apply the method to mixed logit (MMNL) models, by showing that they can be approximated with CNL models. With an appropriate sequence of parameter scalings, we conjecture that the solution found is also globally optimal.
Author: Hai Jiang
Publisher:
ISBN:
Category :
Languages : en
Pages : 34
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Book Description
We study the multiproduct price optimization problem under the multilevel nested logit model, which includes the multinomial logit and the two-level nested logit models as special cases. When the price sensitivities are identical within each primary nest, that is, within each nest at level 1, we prove that the profit function is concave with respect to the market share variables. We proceed to show that the markup, defined as price minus cost, is constant across products within each primary nest, and that the adjusted markup, defined as price minus cost minus the reciprocal of the product between the scale parameter of the root nest and the price-sensitivity parameter of the primary nest, is constant across primary nests at optimality. This allows us to reduce this multidimensional pricing problem to an equivalent single-variable maximization problem involving a unimodal function. Based on these findings, we investigate the oligopolistic game and characterize the Nash equilibrium. We also develop a dimension reduction technique which can simplify price optimization problems with flexible price-sensitivity structures.
Author: Natalie Mizik
Publisher: Edward Elgar Publishing
ISBN: 1784716758
Category : Business & Economics
Languages : en
Pages : 713
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Book Description
Marketing Science contributes significantly to the development and validation of analytical tools with a wide range of applications in business, public policy and litigation support. The Handbook of Marketing Analytics showcases the analytical methods used in marketing and their high-impact real-life applications. Fourteen chapters provide an overview of specific marketing analytic methods in some technical detail and 22 case studies present thorough examples of the use of each method in marketing management, public policy, and litigation support. All contributing authors are recognized authorities in their area of specialty.
Author: Gad Allon
Publisher:
ISBN:
Category :
Languages : en
Pages : 27
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Book Description
In this paper, we postulate a general class of price competition models with Mixed Multinomial Logit demand functions under affine cost functions. We first characterize the equilibrium behavior of this class of models in the case where each product in the market is sold by a separate, independent firm and customers share a common income level. We identify a simple and very broadly satisfied condition under which a Nash equilibrium exists while the set of Nash equilibria coincides with the solutions of the system of First Order Condition equations, a property of essential importance to empirical studies. This condition specifies that in every market segment, each firm captures less than 50% of the potential customer population when pricing at a level which, under the condition, can be shown to be an upper bound for a rational price choice for the firm irrespective of the prices chosen by its competitors. We show that under a somewhat stronger, but still broadly satisfied version of the above condition, a unique equilibrium exists. We complete the picture, establishing the existence of a Nash equilibrium, indeed a unique Nash equilibrium, for markets with an arbitrary degree of concentration; under sufficiently tight price bounds. We then discuss two extensions of our model: unequal customer income and a continuum of customer types. A discussion of the multi product case is included in the appendix. The paper concludes with a discussion of implications for structural estimation methods.
Author: Yuhang Ma
Publisher:
ISBN:
Category :
Languages : en
Pages : 163
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Book Description
In most E-commerce scenarios such as hotel booking and online shopping, products are not offered to customers simultaneously. Instead, they are divided into different webpages and presented to customers sequentially. In this thesis, we focus on solving a common problem faced by online retailers: when products are revealed to customers sequentially, which products should the retailers display at each stage and what prices should the retailers charge for each product so that the expected revenue can be maximized? To solve those problems, we generalize the classical multinomial logit model to capture the customer's choice behavior under the sequential setting and present efficient algorithms for different generalized choice models and different operational constraints.
