Question

1-a need Explanation of the variables of the Bellman equation

?(?,?)=??? ?(?)+??(?′,?′)

2-When to use it?

3-What are the main methods used for solving using the Bellman equation?

4-Give an economic example to which the Bellman equation can be applied and how

Answer 1

1.Bellman equation refers to a method for calculating value functions in dynamic environments. It gives a necessary condition for optimality associated with mathematical optimization method known as dynamic programming. It gives the value of a decision problem at a given point as a function of the payoff from initial options and value of remaining decision problem. The explanation of variables of Bellman equation may be stated as follows

V(x,y) – Gives the value of the state as a sum of rewards to a terminal state with reward of each successive state discounted. Ie; it gives the value for being in a certain state

u(c ) – The action which maximises the reward      

V(x’,y’) – Value of being in the next state after taking action in the state ‘s’

β-The discount factor

2. The following are the major application areas of Bellman equation in economics

· It was applied for the first time in consumption theory.

· It can be applied for the intertemporal capital asset pricing model, where the investors choses between the income today and the future income or capital gains.

· It is used in dynamic programming and used to model theoretical problems in economics using recursive methods.

· It can be used to solve many problems including optimal economic growth, public finance, business investment, asset pricing etc.

· It can be used for analysing capital budgeting.

· It can be applied for business valuation also

3. The following are the major methods used for solving using the Bellman equations

· Using first order equations associated with Bellman equation and then use the envelope theorem to eliminate the derivatives of value function which results in the development of ‘Euler equations’.

· Method of undetermined coefficients called as guess and verify method to solve infinite horizon equations.

· Using backward induction method to solve the equation and can be applied to artificial neural network approximations.

4. Solution using Bellman equation can be used in basic money in utility model. This can be explained as follows

· Here, the labour-leisure option is eliminated

· Family consumption and real money balances are used as utility functions.

· The uncertainty in currency impacts and technological changes can be avoided.

· First determine the total present utility value of family cycle.

· Determine the state and control variables [Here state variables represent the household’s initial level of resources.

· Compute the partial derivatives and use envelope theorem to solve it.

· The final expression indicated the marginal benefit of increased money holdings which will be equal to marginal utility of consumption over a period.