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The author introduces the “recursive” model that has emerged within micro-economics over the past few decades. This modern recursive equilibrium model is contrasted with the neoclassical model, in terms of the optimization and time periods involved. The modern, multi-period consumer savings problem is introduced, as well as the “cake eating” problem and basic pricing equation. The author argues these form the basis of a modern microeconomic theory, and that the stochastic discount factor that emerges from the basic pricing equation provides a valuable insight that is lacking in the neoclassical and classical worlds. As with other valuation principles, the author tests the principle as a practical valuation tool for three actual businesses, demonstrating that is provides an incomplete basis for valuation of private firms.

Dynamic optimization is widely used in many fields of economics, finance, and business management. Typically one searches for the optimal time path of one or several variables that maximizes the value of a specific objective function given certain constraints. While there exist some analytical solutions to deterministic dynamic optimization problems, things become much more complicated as soon as the environment in which we are searching for optimal decisions becomes uncertain. In such cases researchers typically rely on the technique of dynamic programming. This chapter introduces the principles of dynamic programming and provides a couple of solution algorithms that differ in accuracy, speed, and applicability. Chapters 8 to 11 show how to apply these dynamic programming techniques to various problems in macroeconomics and finance. To get things started we want to lay out the basic idea of dynamic programming and introduce the language that is typically used to describe it. The easiest way to do this is with a very simple example that we can solve both ‘by hand’ and with the dynamic programming technique. Let’s assume an agent owns a certain resource (say a cake or a mine) which has the size a0. In every period t = 0, 1, 2, . . . ,∞ the agent can decide how much to extract from this resource and consume, i.e. how much of the cake to eat or how many resources to extract from the mine.We denote his consumption in period t as ct. At each point in time the agent derives some utility from consumption which we express by the so-called instantaneous utility function u(ct). We furthermore assume that the agent’s utility is additively separable over time and that the agent is impatient, meaning that he derives more utility from consuming in period t than in any later period.We describe the extent of his impatience with the time discount factor 0 < β < 1.