Capacity expansion and dynamic monopoly pricing
Introduction
The so-called plant-size problem introduced in the 1950s by Chenery (1952) studies the optimal capacity to install in order to serve an exogenous demand increasing over time, with economies of scale in plant construction. This problem arose in the context of developing countries: explosive demographics forced the governments to regularly increase productive capacity over time, in order to face increase in demand resulting from such population booms. Increasing returns to scale in plant construction opened the door to the possibility of anticipating the increments of capacity required to meet future demand levels, and exploiting thereby the resulting economies of scale. In this context, it was reasonable to assume that demand expansion was pushed exogenously by the population growth.
Today, in developed countries, similar demand booms are still observed in several industries, but for other reasons than an exogenous population growth. These booms now often follow from the globalization of trade: while markets in developed countries expanded initially around local or national demands, they are now progressively concerned by larger and larger geographical areas. In some sense, firms operating in these industries are faced today in developed countries with a problem akin to the problem met by governments in developing countries in the past: they also expect progressive expansion of demand consecutive on trade globalization, while benefiting as well from economies of scale in plant construction. However, in this new context, it is not reasonable to view demand increases addressed to firms operating in such industries as exogenous. These firms are no longer public institutions, as in the original situation, but rather profit maximizing institutions. As such, they benefit from specific instruments which were not at the disposal of the governments. In particular, they can now accompany their investment policy by a price policy through which they can manipulate the levels of demand through time.
In order to capture this new strategic possibility, we study in the present paper a similar plant size problem as in Chenery (1952) and Manne (1961), but now allowing the profit-maximizing firm producing the good to combine its investment policy with a product price policy adjusting demand upwards, or downwards, over time. This extension has to cope with the difficulty that, even if the volume demanded is exogenously increasing (or decreasing) at each price over time, the firm can now decrease or increase it by instantaneous increases or decreases in the price pattern. We characterize the optimal price and investment policies of the monopolist under the assumption that the firm while controlling the size of the investments to be undertaken, does not control the dates, which are assumed to be equally spaced through time, at which these investments have to be consented. Our findings are as follows. The optimal price policy, through which the monopolist can possibly dampen or enhance the expansion of demand through time, leads to an investment pattern with increments of capacity which are either constant over time at each investment date, or start to become constant after a finite set of investment dates. Furthermore, the optimal price pattern between two dates at which new capacity is installed falls into two categories: either the price pattern is such that total existing capacity is fully used at each instant between these dates; or the instantaneous monopoly price is used for some period within the cycle and, thereafter, until the end of the cycle, the price which dampens instantaneous demand at the level of installed capacity is adopted. Finally, we show that the optimal constant increment of capacity is smaller than the one which would have been selected by a planner facing a demand growing exogenously at a rate corresponding to the instantaneous monopoly price, as in Manne (1961). These conclusions are a priori far from being evident. Even constrained to invest at equally spaced dates, the monopolist could have as well preferred to manipulate the price, according to the value of the interest rate, in order to decrease or increase the size of the investments through time rather than using a constant-cycle investment size policy.1
Research on the plant-size problem has been pursued without discontinuity since the sixties. Manne (1961, 1967) faced the capacity expansion problem which was met by several manufacturing facilities in India. Assuming an exogenous demand growing linearly over time and economies of scale in investment costs, he finds that the optimal investment policy is constant-cycle: successive investments are all of the same size and undertaken at equally spaced points of time. These findings have been extended afterwards in several directions. Srinivasan (1967) proves that the constant-cycle property of the optimal plan extends to the case where demand is growing geometrically over time. Gabszewicz and Vial (1972) extend the analysis to exogenous technological progress which can arise either at a deterministic date or at a random time in the future. In both cases, the constant-cycle property of the optimal policy still holds. Around the same period a considerable amount of research work has been devoted to the problems of lead times and uncertain demand (see in particular Nickell (1977), Freidenfelds (1981), Bean et al. (1992), Chaouch and Buzacott (1994) and Ryan (2002)).
Although these research contributions represent important advances in the understanding of the plant-size problem, they all assume that demand for capacity increases is exogenously given, and cannot be adjusted through some price policy. Of course, this assumption was a natural entry point since past contributions were mainly concerned with finding the solution of a planning problem met in the framework of a country development programme, and not with identifying the solution of a profit-maximizer monopolist. Nevertheless, as shown in the present paper, the plant-size problem can be reformulated in order to take into account this market alternative interpretation, in which the firm is allowed to manipulate instantaneous demand through price. So, departing from the above research lines, we propose to relax the assumption of exogenous demand increase. We replace it by an assumption about the time expansion of a price-quantity relation specifying how instantaneous demand varies with price at each point of time. More precisely, we suppose that, at each point of time, market demand is given by a linear function of instantaneous price, the intercept of which increases linearly with time. Moreover, we make the assumption alluded above concerning the size of the investments, namely, we fix a sequence of equally spaced points of time at which the investments for adding capacity may be undertaken and characterize the optimal policy of the monopolist in terms of instruments: price regime and investment size.
The model is described in Section 2. In the same section, we fully characterize both the optimal price and investment policies. We summarize our findings in the conclusion and propose some paths for further research.
Section snippets
The model
We consider a monopolist facing a demand function defined by with denoting continuous time and instantaneous price: demand at each price is accordingly expanding through time because the intercept of the demand function increases at a rate proportional to . At each instant of time , the productive capacity of the firm is bounded by the existing amount of equipment, which we denote by . While the existing capacity may exceed the current demand level , no
Conclusion
The plant size problem introduced in the fifties was designed in order to solve planning problems raised by an exogenous expansion of demand due to the demographic trend in developing countries. In this paper we have considered an alternative version of the plant size problem, now formulated in the context of a market environment. A monopolist, facing demand expansion and increasing returns, has the opportunity to combine his investment policy with a price policy aiming at manipulating the
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