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## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
The research described in this book contributes to the scientific field of optimal control theory applied to dynamic models of the firm. In 1963, Jorgenson first wrote about the use of optimal control theory in order to analyze the dynamic investment behaviour of a hypothetical firm. A decade later, reports appeared of work on more realistic models of the firm carried out by, amongst others, Lesourne [1973] and Bensoussan et al. [1974].
Mark W. J. Blok, A. T. Kearney

### Chapter 2. Mathematical Background to Dynamic Optimization

Abstract
This chapter starts with a brief description of solving time-continuous optimal control problems with pure state constraints analytically (Section 2.2). For a more detailed treatment of this type of problem, refer to Feichtinger & Hartl [1986]. Subsequently, discretization of time-continuous problems is explained so that numerically solving with the aid of a computer and specific programs for optimizing non-linear functions of a finite number of variables under nonlinear subsidiary conditions will be possible (Section 2.3). Then Section 2.4 gives a general economic interpretation of adjoint variables as shadow prices, both for continuous and discrete time problems. The last section deals with the procedure followed with the various models of the firm in this book.
Mark W. J. Blok, A. T. Kearney

### Chapter 3. The Basic Model

Abstract
In this chapter, the model of the firm that serves as the starting point of the research work reported in this book will be treated briefly. Essentially, this basic model can be traced back to Lesourne [1973]; later, it was also used as the foundation of most models by, amongst others, Van Loon [1983], Van Schijndel [1988], Kort [1989] and Van Hilten [1991]. Contrary to the research described in literature, here, attention will be paid to relatively short planning periods.
Mark W. J. Blok, A. T. Kearney

### Chapter 4. A Model with Start-up Costs

Abstract
In the model of the firm from the previous chapter, a static relationship between the production rate Q(t) and the stock of capital assets K(t) is suggested:
$$Q\left( t \right) = {k^{ - 1}}K\left( t \right)$$
(4.1)
This means that the utilization rate defined as:
$$\frac{{kQ\left( t \right)}} {{K\left( t \right)}}$$
(4.2)
is always 100%.
Mark W. J. Blok, A. T. Kearney

### Chapter 5. Models with a Business Cycle

Abstract
In this chapter, four models of the firm are examined in which an exogenous business cycle is presented by means of an explicitly time-dependent price function. The model described in the next section is actually the basic model from Chapter 3. The other three models have been derived from it, but each contains a different development. The developments are: a variable utilization rate (Section 5.3), a cash balance (Section 5.4) and an inventory of finished goods (Section 5.5), respectively. An interesting feature of these models is that jumps in in the course of the costate variables occur in certain situations. Great emphasis is put on the economic interpretation of those jumps. Finally, in Section 5.6, the most important conclusions are summarized.
Mark W. J. Blok, A. T. Kearney

### Chapter 6. A Model with Increasing Returns to Scale, an Experience Curve and a Production Life Cycle

Abstract
In this chapter, a model of the firm is studied which is an elaborated version of the basic model in Chapter 3. Essentially, it is refined in two places, the first concerns the production function of the model which describes the relationship between the production rate and the minimum quantity of production factors required. The second refinement concerns the price function which deals with the returns per product unit as a function of the sales rate (equal to the production rate) and time.
Mark W. J. Blok, A. T. Kearney

### Backmatter

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