# dynamic programming optimal consumption and saving

Example 4.1. As we shall see, the theory of dynamic programming uses this insight in a dynamic context. This paper provides a number of tools to speed up the solution of such models. Optimal consumption and saving A mathematical optimization problem that is often used in teaching dynamic programming to economists (because it can be solved by hand ) concerns a consumer who lives over the periods and must decide how much to consume and how much to save in each period. households and ﬁrms. Firstly, I use that many consumption models have a nesting structure implying that the continuation value can be efficiently pre-computed and the consumption … Consumption-saving models with adjustment costs or discrete choices are typically hard to solve numerically due to the presence of non-convexities. Each period he receives uncertain labor income. Examples include consumption-saving problems with many assets, business cycle models with numerous sectors or countries, multiproduct menu-cost models, corporate nance models with various types of capital goods and bonds of It does not matter in which period the extra cake is eaten since, due to optimality, the return (in terms of the value function) of eating extra cake is equalised across periods. borrow or save in period tby buying/selling bonds, B t.These bonds cost q t units of consumption (which serves as the numeraire); B t units of bonds brought into period t+ 1 pays out B t units of income in period t+1. When b is higher, the agents save more. A consumer is initially endowed with some savings. Hence, a greedy algorithm CANNOT be used to solve all the dynamic programming problems. 2.1 Consumers Consumer choice theory focuses on households who solve: V(I,p)=max c u(c) subject to: pc = I where c is a vector of consumption goods, p is a vector of prices and I is income.1 The ﬁrst order condition is given by Dynamic Programming – Analytic Solution Assume the following problem for the social planner: {1} 0 0, 0 1 1 0 ... solve for the optimal policy rules for consumption and capital. B tcan be positive or negative; a positive value means that the agent saves, a negative value means that the agent borrows. Both Atsumi (1965) and McKenzie (1968) recognized that this ... dynamic programming (often referred to as BeIlman's optimality of savings of a nation is generally regarded as the paper which ... accumulation oriented models to consumption oriented optimal growth models of the Ramsey-type, this key concept remained. An optimal consumption and investment problem with partial information. Optimal consumption and savings with ... a tractable consumption rule via continuous-time dynamic programming, which sharpens the underlying economic mechanism and develops new economic intuition, and (3) generating new quantitative implications and empirical predictions consistent with data. Part of: Hamilton-Jacobi theories, including dynamic programming; Stochastic systems and control; Mathematical finance; Stochastic analysis; Hiroaki Hata (a1) and Shuenn-Jyi Sheu (a2) dimensional dynamic programming problems. Below we give an example to illustrate the use of dynamic programming method to solve the optimal control problem. 1 allows consumption in any period to increase, therefore, 0 (1)= − 1 0( ). ... our savings rate is ab. However, we prove that dynamic constraints are binding. Extra Space: O(n) if we consider the function call stack size, otherwise O(1). 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