Uncertainty, Multi-dimensional Capacity and Profits: A Model-output Oriented Approach 1

Transcription

1 Uncertainty, Multi-dimensional Capacity and Profits: A Model-output Oriented Approach 1 Achim I. Czerny 2,ErikT.Verhoef 3 and Anming Zhang 4 February 4, We thank Alex Anas, Vincent van den Berg, Jan Brueckner, Mogens Fosgerau, Marc Ivaldi, Sergio Jara-Diaz, Paul Koster, Robin Lindsey, Yu Morimoto, Se-il Mun, Jos van Ommeren, Ken Small, and Jaques Thisse for helpful comments and suggestions. We are also grateful to the participants of the EUREKA seminar at the VU University Amsterdam, research seminar of the Department of Logistics and Maritime Studies at the Hong Kong Polytechnic University, Urban Economics Workshop at Kyoto University, the OPTION conference, and the ATARD Conference. Partial financial support from the European Research Council (ERC, AdG Grant OPTION) is gratefully acknowledged. 2 Department of Logistics and Maritime Studies, Hong Kong Polytechnic University, Hong Kong, achim.czerny@polyu.edu.hk. 3 Department of Spatial Economics, VU University Amsterdam, and Tinbergen Institute, The Netherlands, e.t.verhoef@vu.nl. 4 Sauder School of Business, University of British Columbia, Canada, anming.zhang@sauder.ubc.ca.

2 1 Introduction Many economic decisions have to be taken under imperfect information. The modelling of such decision making is crucial for understanding the effects of uncertainty on economic behaviour. Our paper presents a new model-output oriented approach for dealing with uncertainty in economic analysis. This approach itself poses no restrictions on the types of uncertainty and underlying mechanisms and processes that can be considered, but allows an efficient way of assessing the impacts of uncertainty for wide classes of models and functional forms of underlying behavioural relations. An application of our method is concerned with the assessment of uncertainties from the firms perspectives, and the capacity or inventory choice of a risk-neutral firm. The present study captures that capacity decisions are typically long term relative to pricing decisions: Firms can relatively easily adjust prices to respond to volatile market fluctuations. But, capacity often limits output in periods of high demand, whilst excess capacity exists in periods of low demand, which usually cannot be easily fixed in the short run. 1 On top of this, periods of high and low demand may be difficult to predict, meaning that capacity decisions have to be made under uncertainty. Airlines, rail operating companies, cruise and passenger lines, container lines, freight forwarders and logistics companies can serve as examples. These firmsneedtoplantheirfleets of vehicles long term in both the number of vehicles and their sizes. Furthermore, they typically face significant fluctuations in demand and cost conditions, some of which are caused by predictable seasonal patterns but others are hard to foresee at the time when capacity decisions are made. Also in infrastructure supply (airports, seaports, highways, rail tracks, etc.) and in other businesses and institutions such as hotels, cinemas, schools, hospitals and fire fighters, where capacity may again be determined by both the size and number of facilities, similar issues emerge. This paper is most closely related to two strands of the literature. The first strand 1 Meyer (1975) shows that it can indeed be optimal for a monopoly firm to invest in production capacity that exceeds production in the case of low demand. 1

3 evaluates uncertainties from the firms perspectives by analyzing how an increase in uncertainty changes expected profits. Oi (1961) considers firms under perfect competition and shows geometrically that price instabilities can be a vice rather than a virtue because they can increase the firms expected profits. A feature of his analysis is that a positive shock will increase both prices and quantities relative to their expected values, while a negative shock reduces both prices and quantities relative to their expected values. Alexandrov (2015) considers a monopoly firm and shows that the same uncertainty increases expected profit is true when demand is linear, marginal costs are constant and stochastic, and prices are determined ex-post, that is, under perfect information. A feature of this analysis is that an upward cost shock is associated with a small price markup on marginal costs and a small quantity relative to the situation with a downward cost shock. Alexandrov also develops a model based on general functional forms and uses an ad hoc example to show that an increase in uncertainty need not always increase expected profits. The second strand of the literature refers to the large body of literature on the socalled newsvendor problem. More specifically, the newsvendor problem describes a periodic-review inventory model, where a firm decides its inventory facing uncertain demand for perishable products (for example, Massé, 1946; Arrow et al., 1951; Dvoretzky et al., 1952a and 1952b; Within, 1955). In this strand of the literature (and elsewhere), it is common to distinguish between two model-input oriented types of uncertainties called additive and multiplicative uncertainties. For example, let the model input be a demand function. Furthermore, let denote the deterministic part of the demand function and be a noise term, then additive uncertainty means that the noise term is added to the deterministic part of the demand function, that is, demand is +, while multiplicative uncertainty means that the demand function is determined by the product of the deterministic and the stochastic parts of the demand, that is, the demand function is. In this example, the additive uncertainty can be interpreted as a situation where the effect of uncertainty (usually measured by the standard deviation of the noise term) on the willingness to pay is the same for all customers and, 2

4 thus, independent of the level of willingness to pay. With multiplicative uncertainty the willingness to pay of customers with a high willingness to pay may be more affected by the noise term than the willingness to pay of customers with a low willingness to pay. We consider this approach as a model-input oriented approach because (implicitly) the utilities and cost functions are considered as stochastic, which then leads to uncertainties in the model outputs, for example, equilibrium prices and demands. Mills (1959) considered a monopolistic firm and additive demand uncertainty, and found that uncertainty can reduce the ex ante optimal price (that is, the price is optimized before uncertainty has resolved), because in this way the firm reduces the probability of unsold production. Karlin and Carr (1962) found that the opposite is true with multiplicative uncertainty, where uncertainty increases the ex ante optimal price. Also Zabel (1972) found that monopoly behavior depends on the structure of stochastic processes: output increases in uncertainty with additive uncertainty, whilst output decreases in uncertainty with multiplicative uncertainty. For public utility pricing, Brown and Johnson (1969) found that ex ante optimized prices can be decreasing in uncertainty and ex ante optimized capacity increasing in uncertainty independent of whether demand uncertainty is additive or multiplicative. 2 The present study develops a model-output oriented approach (Section 2). It is output oriented in the sense that it explicitly models uncertainties in the ex post optimized values (that is, values are optimized after uncertainty has resolved). We will show that both ways, model-input and model-output orientation, can characterize the same stochastic process and the same set of optima/equilibria. However, by taking the model-output oriented approach, analytical or numerical results can be obtained 2 For passenger management in the airline industry, the newsvendor model has been used for singleleg revenue management problems (McGill and van Ryzin (1999) provide a survey of this literature). For example, Wong et al. (2009) consider a newsvendor model of multiple items with capacity constraints to analyze airline passengers and cargo supply. They address multiple fare classes and pricedependent demand and find, empirically, that airlines could increase profit by a reduction of passenger baggage limits for large aircraft. 3

5 and characterized that could be consistent with wide sets of underlying input-oriented model descriptions (type of stochasticity, functional forms, etc.), which qualifies it as an efficient way of obtaining and representing results for wide classes of underlying cases. For example, the model-output oriented approach makes use of an uncertainty-type parameter that provides a convenient way to capture that a high optimized monopoly price markup may be associated with a high or a low optimized monopoly quantity (all relative to the expected value of their optimized values). The model-output oriented approach can then be used to show that an increase in uncertainty increases expected monopoly profits if and only if a price markup above its expected value is associated with a quantity above its expected value. Section 3 analyzes the model inputs that are required to produce a specific type of model-output oriented uncertainty. More specifically, this section identifies how, conditional on given functional forms for benefits and costs, stochasticity should enter parameter values in order to produce a specific value of the uncertainty-type parameter. Here, we show that the same value of the uncertainty-type parameter can emerge from different commonly used functional forms for benefits and costs. In this sense, specific assumptions on functional forms for benefits and costs, as they are common in the literature, can sometimes be made without loss of generality. To further illustrate the role of uncertainty types for economic outcomes, Section 4 analyzes the role of the uncertainty-type parameter for ex ante optimized levels of capacities (or inventories). This part analyzes one- and two-dimensional capacities. The latter case refers back to airlines with a single fare class where capacities depend on the number and the size of aircraft. This case is used to show how uncertainty types can affect capacity structures. While the theory is developed for a (risk neutral) monopoly firm, 3 the model-output oriented approach can be applied to a wide range of other economic problems. 4 This 3 Baron (1970) and Sandmo (1971) showed that production output can be reduced by risk aversion. 4 To show that our theory of uncertainty types is not tied to the monopoly problem and thus can be considered as a more general theory, it is applied to the policy problem of externality internalization in Appendix C. 4

6 References Aguirre, I., Cowan, S. and Vickers, J. (2010), Monopoly price discrimination and demand curvature, American Economic Review 100: Alexandrov, A. (2015), When should firmsexposethemselvestorisk?, Management Science 61: Arnott, R., de Palma, A. and Lindsey, R. (1996), Information and usage of free-access congestible facilities with stochastic capacity and demand, International Economic Review 37: Arrow, K.J., Harris, T. and Marschak, J. (1951), Optimal inventory policy, Econometrica 19: Bagnoli, M. and Bergstrom, T. (2005), Log-concave probability and its applications, Economic Theory 26: Baron, D.P. (1970), Price uncertainty, utility, and industry equilibrium in pure competition, International Economic Review 11: Brown, G. Jr. and Johnson, M.B. (1969), Public utility pricing and output under risk, American Economic Review 59: Brueckner, J.K. (2004), Network structure and airline scheduling, Journal of Industrial Economics 52: Czerny, A.I. (2010), Airport congestion management under uncertainty, Transportation Research Part B 44: Czerny, A.I., Forsyth, P., Gillen, D.W. and Niemeier, H.-M. (eds.) (2008), Airport Slots: International Experiences and Options for Reform, Ashgate Studies in Aviation Economics and Management, Ashgate. 29

7 Douglas, G.W. and Miller, J.C. (1974), Quality competition, industry equilibrium, and efficiency in the price-constrained airline market, American Economic Review 64: D Ouville, E.L. and McDonald, J.F. (1990), Effects of demand uncertainty on optimal capacity and congestion tolls for urban highways, Journal of Urban Economics 28: Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1952a), The inventory problem: I. Case of known distributions of demand, Econometrica 20: Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1952b), The inventory problem: II. Case of unknown distributions of demand, Econometrica 20: Karlin, S. and Carr, C. (1962), Prices and Optimal inventory policy, in Arrow, K.J. et al. (eds.), Studies in Applied Probability and Management Science, Stanford University Press. Kraus, M. (1982), Highway pricing and capacity choice under uncertain demand, Journal of Urban Economics 12: Massell, B.F. (1969), Price stabilization and welfare, Quarterly Journal of Economics 83: Massé, P. (1946), Les réserves et la rétulation de l avenir dans la vie économique, Paris: Hermann, 1946, Vol. 2, Avenir Aléatoire. McGill, J.I. and van Ryzin, G.J. (1999), Revenue management: Research overview and prospects, Transportation Science 33: Meyer, R.A. (1975), Monopoly pricing and capacity choice under uncertainty, American Economic Review 65: Mills, E.A. (1959), Uncertainty and price theory, Quarterly Journal of Economics 73:

8 Oi, W.Y. (1961), The desirability of price instability under perfect competition, Econometrica 29: Proost, S. and Van der Loo, S. (2010), Transport infrastructure investment and demand uncertainty, Journal of Intelligent Transport Planning and Operation 14: Rietveld, P. and Roson, R. (2002), Direction dependent prices in public transport: A good idea? The back haul pricing problem for a monopolistic public transport customer, Transportation 29: Sandmo, A. (1971), The theory of the competitive firm under price uncertainty, American Economic Review 61: Waugh, F.V. (1944), Does the consumer benefit from price instability?, Quarterly Journal of Economics 58: Weitzman, M.I. (1974), Prices vs. quantities, Review of Economic Studies 41: Within, T.M. (1955), Inventory control and price theory, Management Science 2: Wong, W., Zhang, A., Hui, YV and Leung, L. (2009), Optimal baggage limit policy: Airline passenger and cargo allocation, Transportation Science 43: Xiao, Y., Fu, X. and Zhang, A. (2013), Demand uncertainty and airport capacity choice, Transportation Research Part B 57: Zabel, E. (1972), Multiperiod monopoly under uncertainty, Journal of Economic Theory 5: Zhang, A. and Czerny, A.I. (2012), Airports and airlines economics and policy: An interpretive review of recent research, Economics of Transportation 1: