2. Mr. Kuchenfresser

This chapter introduces the concept of dynamic optimization through the engaging story of Mr. Kuchenfresser and his cake-eating dilemma. The text explores how to maximize utility over two periods, today and tomorrow, by making assumptions about Mr. Kuchenfresser's preferences and the nature of the cake. It delves into the mathematical modeling of the problem, including the construction of a Lagrangian function and the derivation of first-order necessary conditions for optimization. The chapter highlights the importance of the shape of the utility function, demonstrating how linear and non-linear utility functions lead to different solutions. It also introduces the concept of curvature in utility functions and its role in ensuring unique and optimal solutions. Additionally, the text discusses the equations of motion that connect different stages of the problem and the distinction between stationary and non-stationary problems. The chapter concludes with a discussion on the notation and nomenclature used in dynamic optimization problems, emphasizing the importance of clear specification and the use of functionals to represent value functions. Readers will gain insights into the practical application of dynamic optimization principles and the mathematical tools used to solve such problems.