Optimal strategies for general price-quality decision models of new products with learning production costs

The question of product quality permeates every level of business and is becoming crucial for the survival of modern manufacturing firms in automotive and high-tech industries. In this paper, we deal with the optimal price and quality policies for the introduction of a new product. On the supply side, the firm wants to determine the unit price and quality level over time given that unit cost declines along a learning curve, and increases if quality is made greater. On the demand side, dynamic demand is related to price and quality, as well as to cumulative sales (which represent diffusion and saturation effects). We will model this problem in a general framework that includes several previous results as special cases.By applying the maximum principle, we will derive the optimal price and quality policies and discover the interactions between these two major strategic marketing instruments, and the diffusion process. Several fundamental theoretical results will be established for the model. Under certain specified conditions higher prices do imply higher quality, and under other conditions the optimal price declines over time while the product quality improves. To illustrate these results, the theoretical results are applied to two specific-cases: the first one is a simple nonseparable demand growth function in price and quality, the other is a separable demand function of price and quality.     

Optimal Dynamic Investment Policy for Different Tax Depreciation Rates and Economic Depreciation Rates

This paper analyzes the investment policy consequences of incorporating a tax depreciation rate different from the economic depreciation rate. Most often, firms choose their tax depreciation rate in a strategic way. Therefore, it would be a coincidence, should the optimization process lead to a tax depreciation rate that equals the economic depreciation rate. The implications of a difference between tax depreciation rate and economic depreciation rate are investigated in an optimal control model for the determination of the firm investment policy over time.     

Proportional economic growth under conditions of limited natural resources

The paper is devoted to economic growth models in which the dynamics of production factors satisfy proportionality conditions. One of the main production factors in the problem of optimizing the productivity of natural resources is the current level of resource consumption, which is characterized by a sharp increase in the prices of resources compared with the price of capital. Investments in production factors play the role of control parameters in the model and are used to maintain proportional economic development. To solve the problem, we propose a two-level optimization structure. At the lower level, proportions are adapted to the changing economic environment according to the optimization mechanism of the production level under fixed cost constraints. At the upper level, the problem of optimal control of investments for an aggregate economic growth model is solved by means of the Pontryagin maximum principle. The application of optimal proportional constructions leads to a system of nonlinear differential equations, whose steady states can be considered as equilibrium states of the economy. We prove that the steady state is not stable, and the system tends to collapse (the production level declines to zero) if the initial point does not coincide with the steady state. We study qualitative properties of the trajectories generated by the proportional development dynamics and indicate the regions of production growth and decay. The parameters of the model are identified by econometric methods on the basis of China’s economic data.     

The effect of tax depreciation on the stochastic replacement policy

The optimal replacement policy for an asset subject to a stochastic deteriorating operating cost is determined for three different tax depreciation schedules and a known re-investment cost, as the solution to a two-factor model using a quasi-analytical method. We find that tax depreciation exerts a critical influence over the replacement policy by lowering the operating cost thresholds. Although typically a decline in the corporate tax rate, increase in any initial capital allowance, or decrease in the depreciation lifetime (increase in depreciation rate) results in a lower operating cost threshold which justifies replacing older equipment, these results are not universal, and indeed for younger age assets the result may be the opposite. An accelerating depreciation schedule may incentivize early replacement in a deterministic context, but not necessarily for an environment of uncertainty.     

Cost efficiency in data envelopment analysis under the law of one price

To impose the law of one price (LoOP) restrictions, which state that all firms face the same input prices, Kuosmanen, Cherchye, and Sipiläinen (2006) developed the top-down and bottom-up approaches to maximizing the industry-level cost efficiency. However, the optimal input shadow prices generated by the above approaches need not be unique, which influences the distribution of the efficiency indices at the individual firm level. To solve this problem, in this paper, we developed a pair of two-level mathematical programming models to calculate the upper and lower bounds of cost efficiency for each firm in the case of non-unique LoOP prices while keeping the industry cost efficiency optimal. Furthermore, a base-enumerating algorithm is proposed to solve the lower bound models of the cost efficiency measure, which are bi-level linear programs and NP-hard problems. Lastly, a numerical example is used to demonstrate the proposed approach.     

Leader–Follower Dynamic Game of New Product Diffusion

The objective of this paper is to study optimal pricing strategies in a duopoly, under an asymmetric information structure, where the appropriate solution concept is the feedback Stackelberg equilibrium. In order to take into account effects such as imitation (e.g., word of mouth) and saturation, the demand (state equation) is assumed to depend on past cumulative sales, market potential, and both players' prices. We assume also that the unit production cost decreases with cumulative production (learning effects). Each player maximizes his total discounted profit over the planning horizon.The problem is formulated as a two-player discrete-time finite-horizon game. Existence results are first obtained under rather mild conditions. Since the solution of this problem is intractable by analytical methods, we use a numerical approach. Thus, we design a numerical algorithm for the computation of feedback Stackelberg equilibria and use it to obtain strategies in various representative cases. The numerical results presented are intented to give some insights into the optimal pricing strategies in the context of an asymmetrical feedback information structure.     

Dynamic pricing with uncertain production cost: An alternating-move approach

This article studies a two-firm dynamic pricing model with random production costs. The firms produce the same perishable products over an infinite time horizon when production (or operation) costs are random. In each period, each firm determines its price and production levels based on its current production cost and its opponent’s previous price level. We use an alternating-move game to model this problem and show that there exists a unique subgame perfect Nash equilibrium in production and pricing decisions. We provide a closed-form solution for the firm’s pricing policy. Finally, we study the game in the case of incomplete information, when both or one of the firms do not have access to the current prices charged by their opponents.     

Joint determination of optimal safety stocks and production policy for an imperfect production system

In this article, we develop an imperfect economic manufacturing quantity (EMQ) model for an unreliable production system subject to process deterioration, machine breakdown and repair and buffer stock. The basic model is developed under general process shift, machine breakdown and repair time distributions. We suggest a computational algorithm for determination of the optimal safety stock and production run time which minimize the expected cost per unit time in the steady state. For a numerical example, we illustrate the outcome of the proposed model and perform a sensitivity analysis with respect to the model-parameters which have direct influence on the optimal decisions.     

A model of spatial flows and growth of capital and labor stocks

This article deals with optimal spatio-temporal development of capital and labour stocks in a production economy with spatial extension. Current stocks of capital and labour are used to produce a commodity, partly invested to replace worn capital, partly consumed. These stocks can be relocated in space, but relocation uses up some of the inputs themselves. Under these constraints the objective is to maximize a utility measure derived from per capita consumption and aggregated over individuals, space and time.The necessary conditions for optimum are derived as Euler equations of a continuous variational problem. They concern choice of production scale and technology, rate of reinvestment, and optimal flows of labour and produced commodities through space. The Lagrangian multipliers of the constraints are interpreted as imputed wages and commodity prices.The whole structure of optimum depends on these imputed wages and prices, and their solution can be derived from a pair of dependent non-linear partial differential equations. The spatial flow portrait at each moment depends on the time parameter and on the parameters of the model (net reproduction and capital depreciation rates). It can undergo sudden changes described by the elliptic and hyperbolic umbilic catastrophes.     

规模效应下产品具有可替代性的竞争企业的外包决策

针对产品具有一定替代性的两个竞争企业(分别为企业1和企业2)和存在规模效应的上游供应商的外包决策问题, 构建了企业1 外包前后各方的利润模型,求解了下游企业的外包和自产的最优策略以及供应商的最佳批发价格,分析了企业1 的外包策略对企业2 和供应商的外包决策的影响,比较了产品替代性对外包前后各决策变量的影响。研究发现:当企业的单位生产成本高于外包成本时,企业也可能选择自产;而当企业的单位生产成本低于外包成本时,企业也可能选择产品外包。并对模型进行进一步的拓展,比较了下游企业作顺序和同时外包决策两种情景的异同。     

An EMQ model with inspections and random machine failures

In the classical Economic Manufacturing Quantity (EMQ) model, it is assumed that all items produced are of perfect quality and the production facility never breaks down. However, in real production, the product quality is usually a function of the state of the production process which may deteriorate over time and the production facility may fail randomly. In this paper, we study the effect of machine failures on the optimal lot size and on the optimal number of inspections in a production cycle. The formula for the long-run expected average cost per unit time is obtained for a generally distributed time to failure. An optimal production/inspection policy is found by minimising the expected average cost.     

Controlled predatory pricing in a multiperiod Stackelberg game: an MPEC approach

We analyze a multiperiod oligopolistic market where each period is a Stackelberg game between a leader firm and multiple follower firms. The leader chooses his production level first, taking into account the reaction of the followers. Then, the follower firms decide their production levels after observing the leader’s decision. The difference between the proposed model and other models discussed in literature is that the leader firm has the power to force the follower firms out of business by preventing them from achieving a target sales level in a given time period. The leader firm has an incentive to lower the market prices possibly lower than the Stackelberg equilibrium in order to push the followers to sell less and eventually go out of business. Intentionally lowering the market prices to force competitors to fail is known as predatory pricing, and is illegal under antitrust laws since it negatively affects consumer welfare. In this work, we show that there exists a predatory pricing strategy where the market price is above the average cost and consumer welfare is preserved. We develop a mixed integer nonlinear problem (MINLP) that models the multiperiod Stackelberg game. The MINLP problem is transformed to a mixed integer linear problem (MILP) by using binary variables and piecewise linearization. A cutting plane algorithm is used to solve the resulting MILP. The results show that firms can engage in predatory pricing even if the average market price is forced to remain higher than the average cost. Furthermore, we show that in order to protect the consumers, antitrust laws can control predatory pricing by setting rules on consumer welfare.     

Optimal production control of a dynamic two-product manufacturing system with setup costs and setup times

This paper deals with the optimal control of a one-machine two-product manufacturing system with setup changes, operating in a continuous time dynamic environment. The system is deterministic. When production is switched from one product to the other, a known constant setup time and a setup cost are incurred. Each product has specified constant processing time and constant demand rate, as well as an infinite supply of raw material. The problem is formulated as a feedback control problem. The objective is to minimize the total backlog, inventory and setup costs incurred over a finite horizon. The optimal solution provides the optimal production rate and setup switching epochs as a function of the state of the system (backlog and inventory levels). For the steady state, the optimal cyclic schedule is determined. To solve the transient case, the system's state space is partitioned into mutually exclusive regions such that with each region, the optimal control policy is determined analytically.     

On automorphisms of a distance-regular graph with intersection array {204, 175, 48, 1; 1, 12, 175, 204}

The research is focused on the question of proportional development in economic growth modeling. A multilevel dynamic optimization model is developed for the construction of balanced proportions for production factors and investments in a situation of changing prices. At the first level, models with production functions of different types are examined within the classical static optimization approach. It is shown that all these models possess the property of proportionality: in the solution of product maximization and cost minimization problems, production factor levels are directly proportional to each other with coefficients of proportionality depending on prices and elasticities of production functions. At the second level, proportional solutions of the first level are transferred to an economic growth model to solve the problem of dynamic optimization for the investments in production factors. Due to proportionality conditions and the homogeneity condition of degree 1 for the macroeconomic production functions, the original nonlinear dynamics is converted to a linear system of differential equations that describe the dynamics of production factors. In the conversion, all peculiarities of the nonlinear model are hidden in a time-dependent scale factor (total factor productivity) of the linear model, which is determined by proportions between prices and elasticities of the production functions. For a control problem with linear dynamics, analytic formulas are obtained for optimal development trajectories within the Pontryagin maximum principle for statements with finite and infinite horizons. It is shown that solutions of these two problems differ crucially from each other: in finite horizon problems the optimal investment strategy inevitably has the zero regime at the final stage, whereas the infinite horizon problem always has a strictly positive solution. A remarkable result of the proposed model consists in constructive analytical solutions for optimal investments in production factors, which depend on the price dynamics and other economic parameters such as elasticities of production functions, total factor productivity, and depreciation factors. This feature serves as a background for the productive fusion of optimization models for investments in production factors in the framework of a multilevel structure and provides a solid basis for constructing optimal trajectories of economic development.     

Performance modeling of state dependent system with mixed standbys and two modes of failure

The present investigation deals with a multicomponent repairable system with state dependent rates. For smooth functioning of the system, mixed standbys (warm and cold) are provided so that the failed units are immediately replaced by standbys if available. To prevent congestion in the system due to failure of units, permanent along with additional repairmen are provided to restore the failed units. It is assumed that the units may fail in two modes. The units have exponential life time and repair time distributions. The failed unit may balk in case of heavy load of failed units. The failed units may also wait in the queue and renege on finding the repairmen busy according to a pre-specified rule. The Chapman–Kolmogorov equations, governing the model in the form of matrix are constructed using transition flow rates of different states. The steady state solution of queue size distribution is derived using product formula. A cost function is suggested to determine the optimal number of warm and cold standbys units required for the desired level of quality of service. The numerical illustrations are carried out to explore the effect of different parameters on performance measures.     

Technology selection and capacity investment under uncertainty

We analyze the problem of technology selection and capacity investment for electricity generation in a competitive environment under uncertainty. Adopting a Nash-Cournot competition model, we consider the marginal cost as the uncertain parameter, although the results can be easily generalized to other sources of uncertainty such as a load curve. In the model, firms make three different decisions: (i) the portfolio of technologies, (ii) each technology’s capacity and (iii) the technology’s production level for every scenario. The decisions related to the portfolio and capacity are ex-ante and the production level is ex-post to the realization of uncertainty. We discuss open and closed-loop models, with the aim to understand the relationship between different technologies’ cost structures and the portfolio of generation technologies adopted by firms in equilibrium. For a competitive setting, to the best of our knowledge, this paper is the first not only to explicitly discuss the relation between costs and generation portfolio but also to allow firms to choose a portfolio of technologies. We show that portfolio diversification arises even with risk-neutral firms and technologies with different cost expectations. We also investigate conditions on the probability and cost under which different equilibria of the game arise.     

The dynamics of price, quality and productivity improvement decisions

Although quality has received significant attention during the last decades and its economic benefits are beyond any doubt, lots of questions have remained unanswered as to how much, when, and in what to invest to maintain sustainable competitive advantage. A model is introduced here to guide a firm in addressing these questions. The firm produces a single product and operates in a market where monopolistic competition is effective. Demand for the product in the industry depends on both price and performance quality. Increasing productivity knowledge decreases unit production cost, but demand for the company’s product decreases over time, as competitors will be able to offer products with similar performance. Productivity and quality knowledge can be developed through induced and autonomous learning in order to strengthen company position. The paper provides an optimal control formulation of the problem and develops necessary conditions for optimality and characterizes the dynamics of optimal price, quality and investment decisions.     

Layoff costs and underutilization of labour in fisheries

The cost of reducing the labour force during a transition from an overexploited fishery to a bionomic fishery is taken into account. This affects both the long run steady state and the optimal approach to steady state. These effects are illustrated using the case of the NortheastArctic cod stock as a stylized example. The method outlined represents an operational way to assess harvest quotas as well as effort quotas both in the steady state and not least on the path to steady state. In the steady state analysis completely general functional forms are used, whereas in the optimal path analysis the objective function is required to be quadratic in the control variable. This requirement, however, incorporates the most important sources of nonlinearities such as downward sloping demand and increasing marginal costs.     

The ability to compensate for suboptimal capacity decisions by optimal pricing decisions

Certain companies have high capacity cost and rather moderate production cost. These companies usually assume that deciding about their capacity is quite critical. Frequently, however, they are able to adjust the demand for their products to the available capacity by setting appropriate prices, that is higher (lower) than current prices in the presence of under-capacity (over-capacity). We argue that appropriate prices can reduce the adverse effects of non-optimal capacities. We analyze the sensitivity of profit in such a situation for a company in a monopolistic market, selling a non-storable product and facing fluctuating but interdependent demand across two time periods which allows to profitably differentiate prices. Therefore, we state optimality conditions for prices in situations of variable and given capacities and describe a procedure to determine them. The main suggestion of this analysis is that, within the bounds of the normative models and specific parameters examined, optimal prices can substantially reduce the adverse effects of capacity deviating from its optimum. In this way, profit is rather insensitive to deviations of capacity from its optimum. The implications of this finding are discussed for a number of situations.     

EXHAUSTIBLE RESOURCES,LEARNING, AND THE THEORY OF THE SECOND BEST

We formulate and characterize the solutions to optimal control models of social welfare maximization in a resource industry composed of two parts — conventional, with unit cost that increases with cumulative production, and backstop, whose unit cost decreases with cumulative production, due to learning. Since marginal cost pricing leads to losses in backstop operations, we consider three “second best” constraints that ensure non-negative profits. These constraints introduce distortions away from marginal cost pricing. Two constraints cause a discontinuous upward jump in price at the time of transition from conventional to backstop, and a supply side distortion-violation of cost minimization to meet demand.