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 temporal summation is defined as two or more excitatory potentials occurring in succession. When this occurs, an action potential can be observed. The initial time interval used was 1000 µs, but the first elicited compound action potential occurred at a time interval of 400 µs (Figure 2). As the interval was decreased by 100 µs, the amplitude increased along an upward trend.
3.11 Repeatability: The flexibility of a system to travel to an equivalent location in recurrent tries.
for n = 3 and the switch voltage is equal to five times the input voltage for all cases.
Macroeconomic Forecasting Abstract Annual data was gathered on the United States' Gross Domestic Product and the economic indicators of unemployment, employment growth, inflation, and interest rates. Using 2004 as the base year, forecasts for the next two years were taken from three different forecasting organizations and compared to historical figures. Differences in projected data were addressed, as well as relationships between forecasts and among the targeted indicators. The results of the economic forecasts were applied to current Motorola operations and plans.
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.
Only what to produce and how to produce, since distribution is not the task of economics.
Identify and sort out and summarize the problem(s). Decide which is the most important problem.
To understand the world we live in today, we need to understand what economics is and where it came from. Economics is the social science concerned with the production and consumption of goods, services, and the analysis of the commercial activities of a society. Economics also deals with the choices we make to fulfill our wants and needs and how we spend and invest our money. It is split into two main parts known as macroeconomics and microeconomics. Macroeconomics is the study of national or international economies while microeconomics studies individuals or firms within the economy. Adam Smith is widely known as the founding father of modern day economics, but it is actually an Irish banker Richard Cantillon. Richard Cantillon wrote his book “Essai sur la Nature du Commerce en General" which translates to “An Essay on Economic Theory” in the 1730’s
Macroeconomic Equilibrium Introduction Macroeconomic equilibrium for an economy in the short run is established when aggregate demand intersects with short-run aggregate supply. At the price level Pe, the aggregate demand for goods and services is equal to the aggregate supply of output. The output and the general price level in the economy will tend to adjust towards this equilibrium position. If the price level is too high, there will be an excess supply of output.
Difficulties in Formulating Macroeconomic Policy Policy makers try to influence the behaviour of broad economic aggregates in order to improve the performance of the economy. The main macroeconomic objectives of policy are: a high and relatively stable level of employment; a stable general price level; a growing level of real income (economic growth); balance of payments equilibrium, and certain distributional aims. This essay will go through what these difficulties are and examine how these difficulties affect the policy maker when they attempt to formulate macroeconomic policy. It is difficult to provide a single decisive factor for policy evaluation as a change in political and/or economic circumstances may result in declared objectives being changed or reversed. Economists can give advice on the feasibility and desirability of policies designed to attain the ultimate targets, however, the ultimate responsibility lies with the policy maker.
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 ( )
...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.
For instance, the concentration of HCl produced after first time interval in data table 1:
Solving problems is a particular art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice…if you wish to learn swimming you have to go in the water, and if you wish to become a problem solver you have to solve problems. -Mathematical Discovery
Problem-solving help the students to create their own representation or illustration (De Corte, Vrerschaffel, De Win 1985; Hegarty, Mayer, Monk, 1995; Pape, 2003) based on how they interpret or understand the given problem (Pape, 2003; Van der Schoot, Bakker Arkema, Horsley, Van Lieshout, 2009). Problem-solving also tests their critical thinking skills on how they look for another strategy or ways to solve the problem easier. Problem-solving helps the problem solver to develop characteristics of a good problem solver which includes open-mindedness, optimistic, persistent, not afraid to commit mistakes and systematic person since he is following a certain step in solving the