Peter M. Kort

• Tilburg University

We examine the optimal behavior of carbon-emitting companies operating under the European Union Emission Trading System (EU ETS), under which firms are obliged to purchase emission permits on the secondary market if their emissions exceed their allowance. Specifically, we consider the scenario where firms are endowed with the (real) option to under...

We propose a theoretical model to study individual lifestyle choices related to calorie intake and physical activity, depending on personal fitness level and body weight. The model builds on the rational eating literature and can generate a variety of behaviors that are consistent with the empirical evidence. In particular, we show that engaging in...

The objective of this paper is to analyze an incumbent–entrant model under uncertainty. The entrant observes the realization of the random variable(s) prior to making decisions regarding market entry and capacity selection, leaving the uncertainty entirely on the incumbent’s side. The sources of this uncertainty pertain to the characteristics of th...

Modeling Cooperative and Noncooperative Incentives in Investment Projects: A Novel Approach Using Cooperative Game Theory and Real Options Theory In “Dynamic Stability of Cooperative Investment under Uncertainty,” Ketelaars, Borm, and Kort study an investment project involving multiple stakeholders. The research was prompted by the fact that many i...

In response to increasingly stringent green regulations, ocean carriers are transitioning their emissions reduction strategy from reducing ship speeds to investing in green equipment. However , because carriers face uncertain demand and a competitive freight market, it is challenging for them to determine the timing and size of green equipment inve...

This paper considers a firm’s investment decision determining the timing and capacity level in a dynamic setting with demand uncertainty. Its investment is financed by borrowing from a lender that has market power, generating a capital market inefficiency. We show that the firm’s investment is subject to double marginalization in the sense that the...

We present and analyze a life cycle model of a scientific career to investigate how the budget available for research related activities affects a scientist’s long run success in academia. Recognizing that reputation has a positive impact on obtaining research funds as well as on producing scientific output, we determine the optimal efforts of a sc...

International environmental agreements can lead to the design of environmental policy targets for individual countries. One of the most timely examples are targets aiming at increasing the share of renewable energy production in the overall energy mix. To reach these targets, corresponding govern- ments must stimulate the private sector to underta...

The COVID-19 pandemic has devastated lives and economies around the world. Initially a primary response was locking down parts of the economy to reduce social interactions and, hence, the virus' spread. After vaccines have been developed and produced in sufficient quantity, they can largely replace broad lock downs. This paper explores how lockdown...

This paper analyzes investment decisions under uncertainty in a triopoly market. We determine the investment timing and the investment size of the firms under the condition that firms hold an asymmetric cost structure. We build on Shibata (European Journal of Operational Research, 2016) who solves this problem with exogenous investment size. He fin...

This paper considers a capital accumulation game where the installation costs of investments are lowered by the firm’s own capital stock because of learning and by the competitor’s capital stock because of spillover effects. To properly understand the impact of the two capital stocks, we consider six information structures which differ in whether a...

An important development in the real options theory is the notion that an investment decision is not only about timing but also about size. This plays for instance a crucial role when the firm has market power, implying that quantity affects the output price. Analyzing the resulting problem requires the inclusion of a demand function in the real op...

Today’s economy is innovation driven. Besides the launch of new innovative products, this implies that established products have a finite lifetime. This paper investigates its implications for optimal firm investments while explicitly taking into account that an investment decision is not only about timing, but also involves choosing the investment...

It is often argued that compared to a carbon tax, a volatile carbon price under an emissions trading system poses a problem in the transition towards a low carbon economy. However, this paper shows that, when sufficiently positively correlated with the electricity price, carbon price uncertainty diminishes overall volatility because of a diversific...

We present closed-form solutions to some double optimal stopping problems with payoffs representing linear functions of the running maxima and minima of a geometric Brownian motion. It is shown that the optimal stopping times are th first times at which the underlying process reaches some lower or upper stochastic boundaries depending on the curren...

We study the inter-temporally optimal innovation strategies of incumbent manufacturing firms that compete in an established market and can extend their product line through product innovation. Firms invest in production capacity and R&D knowledge stock, where the R&D knowledge stock and the current R&D investment determine the hazard rate of innova...

In determining the durability of its product a firm faces a trade off. Performing a policy of planned obsolescence by making their products less durable implies that the consumer needs to replace them earlier, which thus enhances demand. However, a lower quality of the product will result in a lower reputation, which in turn will affect demand nega...

Most nations have responded to the COVID-19 pandemic by locking down parts of their economies starting in early 2020 to reduce the infectious spread. The optimal timing of the beginning and end of the lockdown, together with its intensity, is determined by the tradeoff between economic losses and improved health outcomes. These choices can be model...

This paper considers investment problems in real options with non-homogeneous two-factor uncertainty. We derive some analytical properties of the resulting optimal stopping problem and present a finite difference algorithm to approximate the firm’s value function and optimal exercise boundary. An important message in our paper is that the frequentl...

The use of recycled plastics is critical in the transition to a circular economy. However, for certain types of plastics, the recycling process is economically unviable. Government-driven incentives, such as a policy imposing a minimum fraction of recycled plastics to be used in production processes of plastic goods, offer an exit from this impasse...

The mothballing option has been studied in the literature, but mainly in decision theoretic frameworks. This paper looks at it from a strategic point of view and applies it to an incumbent-entrant framework. In particular, based on the recent strategic interactions between OPEC and the shale oil industry, we conduct a case study where the incumbent...

The paper considers a firm that has the option to invest in a project with an unknown profitability, which is affected by general market uncertainty. The project has the adverse effect that it can cause environmental damage. In case the firm has the option to undertake preventive investment at the time of market entry, we get that preventive invest...

This paper studies a discrete-time dynamic duopoly game with homogenous goods. Both firms have to decide on investment where investment increases production capacity so that they are able to put a larger quantity on the market. The downside, however, is that a larger quantity raises pollution. The firms have multiple objectives in the sense that ea...

In abnormal optimal control problems it is necessary to basically ignore the objective for certain state values in order to be able to determine the optimal control. In the past, abnormal problems were considered to be degenerated problems that did not fit to any real application. In the present paper we discuss reasons for the occurrence of abnorm...

Subsidies initially installed to stimulate green capacity investments tend to be withdrawn after some time. This paper analyzes the effect on investment of this phenomenon in a dynamic framework with demand uncertainty. We find that increasing the probability of subsidy withdrawal incentivizes the firm to accelerate investment at the expense of a s...

The paper concerns the study of equilibrium points, or steady states, of economic systems arising in modeling optimal investment with vintage capital, namely, systems where all key variables (capitals, investments, prices) are indexed not only by time but also by age. Capital accumulation is hence described as a partial differential equation (brief...

One of the principal ways nations are responding to the COVID-19 pandemic is by locking down portions of their economies to reduce infectious spread. This is expensive in terms of lost jobs, lost economic productivity, and lost freedoms. So it is of interest to ask: What is the optimal intensity with which to lockdown, and how should that intensity...

This article considers an incumbent’s product innovation decision within an uncertain framework, where the firm decides whether to continue selling the established product. The model being dynamic allows to analyze the trade-off between an early innovation where the new product only slightly improves the existing one, or innovating late with a much...

Nations struggled to decide when and how to end COVID-19 inspired lockdowns, with sharply divergent views between those arguing for a resumption of economic activity and those arguing for continuing the lockdown in some form. We examine the choice between continuing or ending a full lockdown within a simple optimal control model that encompasses bo...

The observation that a socioeconomic agent with a high reputation gets a disproportionately higher recognition for the same work than an agent with lower reputation is typical in career development and wealth. This phenomenon, which is known as Matthew effect in the literature, leads to an increasing inequality over time. The present paper employs...

We study capacity investment decisions among oligopoly firms under conditions of cost heterogeneity and output flexibility within capacity constraints. Output flexibility causes the value of the firm to be convex in the state of demand, which implies that the firm invests in larger capacity when the economic environment is more uncertain. Under cos...

We present closed-form solutions to some discounted optimal stopping problems for the running maximum of a geometric Brownian motion with payoffs switching according to the dynamics of a continuous-time Markov chain with two states. The proof is based on the reduction of the original problems to the equivalent free-boundary problems and the solutio...

The aim of the present paper is to analyze how firms that sell durable goods should optimally combine continuous-time operational level planning with discrete decision making. In particular, a firm has to continuously adapt its capacity investments and sales strategy, but only at certain times it will introduce a new version of the durable good to...

This paper analyzes how the transferability of production capacities from an established to a new product influences the incentives of a firm to invest in R&D. A dynamic duopoly model is considered, where initially both firms offer a homogeneous product. The firms invest in production capacities and simultaneously in R&D which determines their inno...

The technological developments observed in the last two decades contributed to the digitalization of products and the introduction of various mobile devices designed for the consumption of this digital content. Many online retailers launched their own mobile devices, which had a direct effect on their multi-product pricing strategies, but also an e...

Rapid technological developments are inducing the shift in consumer demand from existing products towards new alternatives. When operating in a declining market, the profitability of incumbent firms is largely dependent on the ability to correctly time the introduction of product innovations. This paper contributes to the existing literature on tec...

The marketing-mix of price-quality and advertising-quality relationship is well studied. Less understood is the price-advertising-quality relationship. This article fills the gap, investigating the interplay between price, advertising, and quality in an optimal control model. Our results generalize the condition of Dorfman-Steiner in a dynamic cont...

Flooding events can affect businesses close to rivers, lakes or coasts. This paper provides an economic partial equilibrium model, which helps to understand the optimal location choice for a firm in flood risk areas and its investment strategies. How often, when and how much are firms willing to invest in flood risk protection measures? We apply Im...

Well-planned port capacity investments are required to accommodate growing maritime trade, especially in developing countries. Such investments prove to be expensive, irreversible and subject to uncertainty. Under these conditions, the application of real options is better suited than the traditionally used NPV approach, given that specific port an...

The paper develops an optimization model of the career of a scientist. Recognizing that research efforts and networking get more efficient if the scientist is more knowledgeable, history dependent solutions are developed. We give a theoretical underpinning of the four different research patterns detected in Way et al. (2017, Proceedings of the Nati...

We analyze the problem of a firm that sells durable goods. In particular, we investigate how this firm optimally combines continuous-time operational-level planning (continuously deciding on capacity investment) with discrete decision making (when to launch a new generation of the product, how to price a particular generation of the product). We fi...

This paper analyzes a dynamic model of the firm. We focus on the effect of investment in green energy. We explicitly take into account that green energy has a positive side effect, namely that it contributes to the goodwill of the firm and thus increases demand. Different models are proposed and the solutions range from monotonic saddle point conve...

We investigate the optimal investment decision in renewables under market demand uncertainty, in the context of the Italian strategy for renewable deployment under the EU policy. The firm has to decide about the time and size of the investment. We find that a higher subsidy level induces the firm to in- vest earlier with a smaller investment capaci...

The paper concerns the study of equilibrium points, or steady states, of economic systems arising in modeling optimal investment with \textit{vintage capital}, namely, systems where all key variables (capitals, investments, prices) are indexed not only by time $\tau$ but also by age $s$. Capital accumulation is hence described as a partial differen...

Ports worldwide operate in an uncertain environment and compete with nearby ports to attract cargo. The extent of competition is influenced by the geographical location and differentiated services offered at ports. In this paper, we study the flexible investment decision of two ports with an option to delay investment in a capacity level that is no...

The paper considers the problem of an investor that has the option to acquire a firm. Initially this firm is run as to maximize shareholder value, where the shareholders are risk averse. To do so it has to decide each time on investment and dividend levels. The firm’s capital stock can be financed by equity and debt, where less solvable firms pay a...

Since port capacity investments involve expensive large projects with high uncertainty and irreversibility, we use real options calculations to study the optimal size and timing of the investment decision in new port capacity. This paper focuses on cases where the managing port authority (PA) is responsible for the investment in infrastructure on t...

Subsidies initially installed to stimulate green capacity investments tend to be withdrawn after some time. This paper analyzes the effect on investment of this phenomenon in a dynamic framework with demand uncertainty. We find that increasing the probability of subsidy withdrawal incentivizes the firm to accelerate investment at the expense of a s...

This paper examines a dynamic incumbent-entrant framework with stochastic evolution of the (inverse) demand, in which both the optimal timing of the investments and the capacity choices are explicitly considered. We find that the incumbent invests earlier than the entrant and that entry deterrence is achieved through timing rather than through over...

This paper considers the effect of investment in solar panels on optimal dynamic firm behavior. To do so, an optimal control model is analyzed that has as state variables goodwill and green capital stock. Following current practice in companies like Tesla and Google, we take into account that the use of green energy has positive goodwill effects. A...

We investigate the optimal investment decision in renewables under market demand uncertainty, in the context of the Italian strategy for renewable deployment under the EU policy. Upon investing, the firm has to decide about the time and size of the investment. We find that a higher subsidy level induces the firm to invest earlier with a smaller inv...

This paper presents a numerical method for the characterization of Markov-perfect equilibria of symmetric differential games exhibiting coexisting stable steady states. The method relying on the calculation of ‘local value functions’ through collocation in overlapping parts of the state space, is applicable for games with multiple state variables....

This paper addresses the problem of a video game producer who starts out with a subscription-based business model but considers when, if ever, to switch to a free-to-play model, which price discriminates between typical users, who play for free, and heavy users who pay for acquiring extra features. The videogame producer has the possibility to adve...

One of the principle characteristics of additive manufacturing is that customers get access to their own design, i.e., they need not choose from a small number of standard products. The paper considers a framework where an incumbent with a standard technology produces a limited number of standard products, and faces a potential entrant with an addi...

The paper considers a firm that operates in a market with fluctuating prices (for instance energy). If the price is sufficiently high, the firm is an active producer. When the price is low the firm has the choice between temporary abandoning activities or choose for irreversible exit. The former case is a fruitful opportunity especially when prices...

The paper provides a framework that enables us to analyze the important topic of capital accumulation under technological progress.We describe an algorithm to solve Impulse Control problems, based on a (multipoint) boundary value problem approach. Investment takes place in lumps and we determine the optimal timing of technology adoptions as well as...

This paper considers the investment decision of a firm where it has to decide about the timing and capacity. We obtain that in a fast-growing market, right after investment the firm produces below capacity, where the utilization rate (the proportion of capacity that is used for production right after the investment) increases with market uncertaint...

This paper examines the existence of Markov-Perfect-Equilibria that give rise to coexisting locally stable steady states in asymmetric differential games. The strategic interactions between an incumbent in a market and a potential competitor, which tries to enter the market through product innovation, are considered. Whereas the potential entrant i...

In the literature on marketing models, the assumption of mixed word-of-mouth has been limited to the Bass diffusion model. Yet explicit leveraging of the originating factors of such assumption is lacking. Apart from that example, mixed word-of-mouth has been disregarded in contagion sales models. This paper bridges the gap by suggesting a sales mod...

For certain goods or services, the quality of the product can be assessed by customers only after consumption. We determine the optimal time paths for pricing, advertising and quality for a profit-maximizing firm facing demand that is influenced both by this experience quality as well as by advertising. In particular, there may exist two optimal tr...

This article studies strategic investment behavior of firms facing an uncertain demand in a duopoly setting. Firms choose both investment timing and the capacity level while facing additional uncertainty about market participants, which is introduced via the concept of hidden competition. We focus on the analysis of possible strategies of the marke...

The paper takes a real option approach to consider optimal capacity investment decisions under uncertainty. Besides the timing of the investment, the firm also has to decide on the capacity level. Concerning the production decision, we study a flexible and an inflexible scenario. The flexible firm can costlessly adjust production over time with the...

This paper considers a firm that faces a declining profit stream for its established product. The firm has the option to invest in a new technology with which it can produce an innovative product while having the option to exit at any point in time. In the presence of an exit option, earlier work determined the optimal timing to invest, where it wa...

In the Introduction to their seminal 1986 book on Optimal Control Feichtinger and Hartl paraphrase Joseph Schumpeter’s famous quote that dealing with capitalist economies without taking into account dynamics is like Hamlet without the prince of Denmark. A similar statement certainly applies also to the analysis of firm behavior. Dynamic aspects are...

This volume collects research papers addressing topical issues in economics and management with a particular focus on dynamic models which allow to analyze and foster the decision making of firms in dynamic complex environments. The scope of the contributions ranges from daily operational challenges firms face to strategic choices in dynamic indust...