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The economics of inaction. Stochastic control with fixed costs. (English) Zbl 1167.93002

Princeton, NJ: Princeton University Press (ISBN 978-0-691-13505-2/hbk). ix, 308 p. (2009).
The aim of the book is to provide an account of applications of optimal stochastic control techniques to economic problems where control is exercised only occasionally. This happens in situations where control actions entail a fixed cost and as a consequence large occasional changes are preferable to a sequence of smaller, more frequent changes. This behavior is typical in a number of important economic settings and the interest in this kind of models had a sensible increase in recent years. Examples of this type of problems are connected to price adjustment, investment behavior in manifacturing plants, job creation and destruction, individual portfolio behavior and attitude of consumers towards durable goods such as houses and automobiles.
The reading of the book requires rather advanced mathematical preliminaries, which are presented at a level that is acceptable for economists and rigorous enough to be useful in model building. All background material is given in the initial Chapters 2 to 5 while a few special topics are provided in two appendixes. Such material deals with basic results in stochastic analysis and stochastic control, namely Brownian motion and diffusions, stochastic calculus and martingales and Hamilton-Jacobi-Bellman equation. The applications to economic problems then follow.
Chapter 6 deals with the simplest example in the presence of a fixed cost, namely the problem of exercising a one time option of infinite duration. In this case action is taken only once and the action itself is fixed so that the only issue is timing. Two approaches are studied. The first is a direct approach similar to the one used to solve the analogous deterministic problem while the second approach is based on the use of the Hamilton-Jacobi-Bellman equation. A numerical example is also studied in detail.
Chapter 7 is concerned with more general models with fixed costs. Here, in particular, actions can be taken many times but there is an explicit fixed cost of adjustment. The size of the adjustment must also be chosen and decisions must be forward looking. A typical example is given by the so called menu cost model, where the profit of a firm depends on the price of its own product relative to a general price index, the latter being modeled as a geometric Brownian motion. Changing the price is assumed to entail a fixed cost but no variable costs. The problem consists in the maximization of the expected discounted returns net of adjustment costs and it is shown that its solution is characterized by an inaction interval (where the firm does nothing) and a return point inside the inaction interval. When the relative price leaves this interval the firm adjusts the nominal price so that the relative price is equal to the return price. Long run averages under the optimal policy can also be described, e.g., fraction of adjustments at each threshold, average time between adjustments, long-run density for relative price. The solution to the problem is obtained both using a direct approach, which characterizes the optimal policy and value function, and using the Hamilton-Jacobi-Bellman equation which characterizes also the long-run averages. Exogenous opportunities for costless adjustment are also discussed and finally a numerical example is developed in detail. Chapter 8 considers models with fixed as well as variable costs of adjustment and the problem is slightly more complicated than in the menu cost model, but with an optimal policy that still involves exercising control only occasionally. As a typical example a standard inventory model is examined in which a plant produces output, customers place order and the difference between supply and demand is the net inflow into a buffer stock (with negative stocks interpreted as back-orders). Control is exercised by a manager who can add to the stock by purchasing the good elsewhere or can reduce it by selling on a secondary market. The associated costs are holding costs depending on the size of the stock, fixed costs of adjustment and a variable cost proportional to the size of adjustment and representing the unit cost of purchasing goods from an outside source or the unit revenue from disposing of goods on a secondary market. The manager’s problem is to choose a policy for minimizing the expected discounted value of total costs: holding costs plus fixed and variable adjustment costs. It is shown that also in this case there exists an inaction interval where no control is applied. When the boundaries (lower and upper) of the inaction region are reached the stock is adjusted to a suitable (lower or upper) return point. As in the menu cost model the optimal policy and associated value function is derived both using a direct approach and using the Hamilton-Jacobi-Bellman equation. Chapter 9 deals mainly with the so called model of housing consumption and portfolio choice, that provides an example of models with continuous-time control variables. It is characterized by a consumer whose only consumption is the service flow from a house and whose only income is the return of a portfolio of two financial assets, a riskless asset paying a fixed interest rate and a risky asset with a stochastic return. Housing has interest and maintainance costs and the consumer is required to hold a minimum equity level in the house that is a fixed fraction of its value, but he can also hold additional wealth as housing equity, holding a mortgage on the remaining balance. Portfolio adjustment is costless and is carried out continuously, while the consumer can adjust the housing consumption flow only by selling the old house and buying a new one and this involves paying a cost that is proportional to the size of the old house. This problem is studied using the Hamilton-Jacobi-Bellman equation and it is shown that the optimal policy for housing transactions is characterized, as in the previous chapters, by upper and lower thresholds for the ratio of total wealth to housing wealth and a return point between the thresholds, while the optimal portfolio selection is derived as a function of the ratio in the inaction region. Several extensions and numerical examples are examined in detail.
Chapter 10 deals with a model similar to the inventory model studied in Chapter 8 but with no fixed costs of adjustment. Here the difference between inflows and outflows is described by a Brownian motion and it is first studied the control problem consisting in keeping it inside the inaction interval. On the basis of the results obtained in this case the optimal control problem consisting in the minimization of the expected discounted sum of various costs is studied. Three types of costs are considered: a unit cost for adding to the stock at the lower threshold, a unit cost for decreasing the stock at the upper threshold and a strictly convex carrying cost. As a result of the absence of fixed costs the size of the optimal upward and downward adjustments are zero and the regulation is just sufficient to avoid the exiting of the process from the inaction interval. A numerical example concludes the chapter.
In Chapter 11 several investment problems are studied using models similar to the inventory model of Chapter 10. They are characterized by a firm whose net revenue flow depends on its capital stock and a geometric Brownian motion describing demand. The problem is to choose the investment policy that maximizes the expected discounted value of the net revenue minus investment costs. Various assumptions about the investment costs are examined. The first considers a different purchase and sale price for capital and in this case the optimal policy has the same structure as in the inventory model in Chapter 10 with the thresholds being functions of the demand. If in addition to the capital price there is also a strictly convex cost of adjustment then the control consists in a continuous flow when the capital stock is outside the inaction interval. Special cases and further investment models are also considered in this chapter.
The final Chapter 12 considers aggregate versions of the menu cost model of Chapter 7. Since in this model the exogenous shocks experienced by individual firms are shocks to an economy wide price index, then in the aggregate version of that model shocks cannot be modeled as independent and identically distributed across agents, so that arguments based on a law of large numbers cannot be used for the derivation of a stationary cross sectional distribution and analytical results are much harder to obtain than in the case of agents subject to idiosyncratic shocks. The goal of the analysis of these models is the study of the hypothesis that monetary policy has real effects in the short run. Two economies are examined, identical except for the monetary policy adopted. In the first economy the money supply process is monotone and in this setting shocks to the money supply have no effect. In the second the log money supply follows a Brownian motion with zero drift and effects of shocks to the money supply follow and depend on the state of the economy when the shock arrives. A few numerical examples are provided and some variations of the above model are also studied.

MSC:

93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
91B02 Fundamental topics (basic mathematics, methodology; applicable to economics in general)
91B70 Stochastic models in economics
93E20 Optimal stochastic control
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
90B05 Inventory, storage, reservoirs
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