which is a fundamental tool of dynamic macroeconomics. "The term dynamic programming was originally used in the 1940s by Richard Bellman to describe the process of solving problems where one needs to nd the best decisions one after another. By 1953, he re ned this to the modern meaning, referring speci cally to nesting smaller decision problems inside larger decisions.
1Bellmans'(1957) and Bertsekas'(1976) contributions give us the mathematical theory behind it as a tool of solving dynamic optimization problems.
For economists, Sargent (1987), Stokey and Lucas (1989) contributed a valuable bridge between them.
2.1 Dynamic Programming Overview
Dynamic programming is used to solve complex problems by decomposing them into simpler sub-problems. The main idea behind it, is quite simple. In order to solve a given problem, we have to solve di erent parts of the problem
(sub-problems) and then to reach an overall solution we combine the solutions of these sub-problems. The dynamic programming approach aims to solve each sub-problem only once and therefore reduces the number of computations.
This is especially useful, as often the number of repeating sub-problems is exponentially large.
The basic idea of dynamic programming is to turn the sequence problem into a functional equation, i.e., one of nding a function rather than a sequence.
This often gives better economic insights, similar to the logic of com-
1From Wikipedia article on Dynamic Programming.
2. Stochastic Dynamic Programming 4 paring today to tomorrow. It is also often easier to characterize analytically or numerically. Some important concepts in dynamic programming are the time horizon, state variables, decision variables, transition functions, return functions, objective
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...he principle of optimality for dynamic programming. 6. The solution procedure begins by nding the optimal policy for the last stage. The optimal policy for the last stage prescribes the optimal policy decision for each of the possible states at that stage. The solution of this one-stage problem is usually trivial.
7. A recursive relationship that identi es the optimal policy for stage n, given the optimal policy for stage n + 1, is available.
In the context of mathematical optimization, dynamic programming often refers to the simpli cation of a decision by breaking it down into a sequence of decision steps over time. We de ne a sequence of value functions V1; V2; :::Vn, with an argument y which represent the state of the system at times i, i 2
1; :::; n. The de nition of Vn(y) is the value obtained at the last time n, in state
y. The values Vi at earlier times i = n
The Poole Model is a macroeconomic model where its main objective is to answer the discussion on whether monetary policy should be conducted using a money-supply rule or an interest-rate rule when managing the economy. In the Poole Model, the Central Bank’s objective is to minimize the loss function:
Class I CAs evolve4 to a uniform configuration of cell states, from nearly any initial configuration. This state can be thought of in dynamical systems terms as a ‘point attractor’, or ‘limit point’. As one would suspect, the rules for class I CAs map from most or all possible neighbour configurations to the same new state. Initial lattice configurations do exist for some class I CAs that lead to non-trivial cycles, but these are very rare.
C. once the action potential threshold is met an action potential of uniform and maximum intensity occurs.
i.e. K ̇(t)=sY(t)-δK(t), L ̇(t)=nL(t) and A ̇(t)=gA(t) it is important to consider the new assumptions that concern the newly added inputs.
Robert E. Lucas Jr.’s journal article, “Some Macroeconomics for the 21st Century” in the Journal of Economic Perspectives, uses both his own and other economist’s models to track and predict economic industrialization and growth by per capita income. Using models of growth on a country wide basis, Lucas is able to track the rate at which nations become industrialized, and the growth rate of the average income once industrialization has taken place. In doing so, he has come to the conclusion that the average rate of growth among industrialized nations is around 2% for the last 30 years, but is higher the closer the nation is to the point in time that it first industrialized. This conclusion is supported by his models, and is a generally accepted idea. Lucas goes on to say that the farther we get from the industrial revolution the average growth rate is more likely to hit 1.5% as a greater percentage of countries become industrialized.
Identify and sort out and summarize the problem(s). Decide which is the most important problem.
allow for each subsequent step to take place. And after each step it becomes increasingly
The second phase is the exponential phase also known as the log phase. This phase is known for its cell doubling. Everything is in place for the bacteria to start multiplying and doubling every few minutes. The doubling will continue at a consistent rate. This will ensure that both the number of cells and the rate of population increase. The actual rate of growth depends upon growth conditions. The frequency of cell division depends on the growth conditions as does the cells survival (Bacterial growth curve (2014)).
...derstand the behavior of a non-linear system you need in principle to study the system as a whole and not just its parts in isolation.
Over a short time interval, this variation ¡Ö C(wc-wo)t. Thus, the system continues to loop
Let be the size of a population at time and μ is the rate of growth of the population from one generation to another, the discrete logistic equation is the mathematical model in the form ( )
Problem-solving approaches presented by Takahashi, Adler et al. and Ruffolo et al. have six similar steps. They all include steps of identifying the problem, analyzing the problem, coming up with some solutions, evaluating the solutions, implementing the solution in action, and evaluating the outcome of the solution. Three approaches all give a useful procedure to solve a problem in group.
The economy tend to move from boom to recession, it is difficult for government to maintain and achieve macroeconomics objectives. At this time, there are “conflicts between government macroeconomic objectives”, which is this extended essay main theme. This essay will look at the government macroeconomic objectives, the conflicts between macroeconomics objectives, the best policy or mixture of policies to minimize the conflicts between macroeconomics objectives and recommendations, which are classified as main objectives and additional objectives.
For instance, the concentration of HCl produced after first time interval in data table 1:
remain in equilibrium condition, their history is rhythmic as a result of the mechanisms of