ECON 5068: Investment, Finance and Asset Prices
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
ECON 5068: Investment, Finance and Asset Prices
1. Introduction to Difference Equations
Given a function y = f(t), its first difference is defined as the difference between the value of the function when the argument assumes the value t + h, h 持 0, and the value of the function corresponding to the value t of the argument. In symbols, Ay = f(t + h) 一 f(t). It should be noted that it is not important if whether the values run forwards or backwards, namely we could also define the first difference of the function as Ay = f(t) 一 f(t 一 h).
Without any loss of generality we can assume unit increments of the dependent variable, i.e. Ay = f(t + 1) 一 f(t) or equivalently Ay = f(t) 一 f(t 一 1).
If we consider successive equally – spaced values of the independent variable (t + 1, t + 2, t + 3 … ) we can obtain successive first differences:
Ayt = f(t + 1) 一 f(t) = yt+1 一 yt
Ayt+1 = f(t + 2) 一 f(t + 1) = yt+2 一 yt+1
Ayt+2 = f(t + 3) 一 f(t + 2) = yt+3 一 yt+2
and so on. We can also compute the second differences, namely the sequence of differences between two successive first differences:
A2yt = Ayt+1 一 Ayt = (yt+2 一 yt+1) 一 (yt+1 一 yt) = yt+2 一 2yt+1 + yt
and so on . The superscript 2 means that the operation of the computing the difference has been repeated twice, i .e. that the difference operator A has been applied twice .
Proceeding on a similar way we could compute the differences between two successive second differences and obtain the third differences and so on .
We can now define an ordinary difference equation as a functional equation involving one or more of the differences Ayt,A2yt etc., of an unknown function of time. Since the argument t varies in a discontinuous way, taking on equally spaced values, it follows that our unknown function will be defined only corresponding to these values of t (i.e. the graph of the function will be succession of separate points).
We call that equation ordinary because the unknown function is only a function of one argument. When the partial differences of a function having more than one argument are involved, the equation becomes a partial difference equation .
The order of a difference equation is that of the highest difference appearing in the equation. If, for example, the highest difference contained is the second, then the equation is the second order; note that the equation is of the second order independently of the fact that the lower – order differences are or are not contained in the equation.
Usually we find the difference equation, as we have already mentioned above, expressed in terms of values of the function at different points of time . In that form, the order of an equation is given by the highest difference between time subscripts .
For instance the ordinary difference equation yt+2 一 2yt = b is a difference equation of a second order, as the highest difference between the time subscripts is 2.
Theorem A
The general solution of a difference equation of order n is a function of t involving exactly n arbitrary constants.
1.1.The Homogeneous Equation
Definition
A difference equation is called homogeneous when the following Theorem B is satisfied for any A.
Theorem B
If y1(t) is a solution of the homogeneous equation, then Ay1(t), where A is an arbitrary constant, is also a solution.
Theorem C
If y1(t),y2(t) are two distinct (i.e. linearly independent) solutions of the homogeneous equation (n 持 1), then A1y1(t) + A2y2(t) is also a solution for any two constants A1,A2.
1.2.The Non – Homogeneous Equation
Theorem D
If t is any particular solution of the non – homogeneous equation, the general solution of the same equation is obtained adding t to the general solution of the corresponding homogeneous equation, namely
y(t) = t + f(t;A1,A2, … ,An)
is the general solution of the non – homogeneous equation.
Thus the general solution of the homogeneous equation is only a part of the general solution of the non – homogeneous equation, and so it is not “general” with respect to the latter. This means that the expression “general solution” must always be qualified. As a matter of terminology note that the expression “particular solution” is used: in the sense of a solution obtained from the general solution by giving specific values to the arbitrary constants and in the sense of any single non – general solution of the homogeneous equation. The expression “complementary function” is used to indicate the general solution of the homogeneous equation when considered as a part of the general solution of the non – homogeneous equation.
1.3.Method for solving non – homogeneous equations
I . Find a particular solution t of the non – homogenous equation.
II . Put g(t) 三 0 and solve the resulting homogeneous equation.
III . Add the two results.
The particular solution of the non – homogeneous equation will depend on the form of the known function g(t).
2. The Method of Undetermined Coefficients
The method of undetermined coefficients is extremely useful for solving linear models in economics. That method suggests the following general approach: to find a particular solution of the non – homogeneous equation, try a function having the same form of 6)?( but with undetermined constants (i.e. if 6)?( is a constant, try an undetermined constant; if it is an exponential function, try the same exponential function with an undetermined multiplicative constant, and so on). Substitute this function, in the non –homogeneous equation and determine the coefficients so that the equation to be satisfied.
The above described method is called the method of undetermined coefficients. In general to apply that method first guess the form of the solution and then verify the guess and solve the undetermined coefficients.
2.1.Example using the Method of Undetermined Coefficients
Polynomial Guess
Solve 人才+t 一 S人才 = t? + Z subject to 人0 = 0.
First we solve the Homogenous Equation (H.E .) which is as follows:
Let 人才 = Y才 丰 0 which is a non – trivial solution and then substitute into the Homogeneous Equation.
人才+t 一 S人才 = 0 一
Y才+t 一 SY才 = 0 一
Y才 )Y 一 S( = 0 一
Y = S
Hence the solution of the H.E. i.e. the complementary function equals: 人才(o) = VY才 = VS才
In the second step we will solve the Non – Homogeneous Equation (N.H.E) by guessing a solution in the form of a first order polynomial guess function in ?. (Why we do so?? Because the non – homogenous part of our equation is of this form) .
人才 = D? + q Dup 人才+t = D)? + t( + q
Substitute now the above relations into our first order difference equation and it follows that:
yt+1 一 5yt = 3t + 2 一
[a(t + 1) + b] 一 5(at + b) = 3t + 2 一
at + a + b 一 5at 一 5b = 3t + 2 一
一4at 一 4b + a = 3t + 2 一
(一4a)t + (a 一 4b) = 3t + 2.
Now we have to equate the coefficients from both sides which yield a linear system in two unknown parameters, which are the so called undetermined coefficients.
(a一4一) 一 (a = 一)
Therefore, the solution of the N.H.E., i.e. the particular solution (P .S.), equals:
t = 一 t 一
The General Solution (G.S.) equals to the sum of the C.F . and the P.S. and it is as follows:
yt = yt(c) + t 一
yt = A5t 一 t 一
We can also use the initial condition in order to determine the arbitrary constant A in the G.S., and that will be as follows:
y0 = 0 = A50 一 0 一 一 0 = A 一 一 A =
Therefore the G.S. equals
yt = 5t 一 t 一 Vt
Homework
You could try to solve yt+1 一 3yt = 4t subject to y0 = 0. Hint: Use an
exponential guess function and follow the method.
There also other methods that you could apply to solve Difference Equations of first or higher order. Such methods are the Method of Lag Operators and the Method of Forward or Backward Substitution (depending on the nature of the variable, if they are forward or backward looking).
3. Discrete Time Dynamic Programming
In mathematics, computer science, economics and other sciences, Dynamic Programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems and optimal substructure. Dynamic programming algorithms are used for optimization.
The term Dynamic Programming was originally used in the 1940s by Richard Bellman to describe the process of solving problems where one needs to find the best decisions sequentially . The word dynamic was chosen by Bellman to capture the time – varying aspect of the problems. The word programming referred to the use of the method to find an optimal program.
Dynamic Programming is both a mathematical optimization method and a computer programming method. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler subproblems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively; Bellman called this the “Principle of Optimality” . If subproblems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the subproblems. In the optimization literature this relationship is called the Bellman equation .
In terms of mathematical optimization , dynamic programming refers to simplifying a decision by breaking it down into a sequence of decisions steps over time. That is done by defining a sequence of value functions V1, V2, … Vn, with an argument y representing the state of the system at times i e [1,n]. The definition of Vn(y) is the value obtained in state y at the last time n. The values at earlier times can be found by working backwards, using a recursive relationship called the Bellman equation .
The method of Dynamic Programming is best understood by studying finite – horizon problems first. These problems are often encountered in making life – cycle planning decisions on optimal consumption, savings, portfolio choice etc. Below we are going to present a motivation example.
3.1.Application in Dynamic Programming
Problem
Suppose that there is a consumer who lives over the periods t = 0, 1, 2, … T and must decide how much to consume and how much to save in each period.
Let ct denote the consumption in period t and assume that consumer derives utility from consuming which is given by the following utility function u(ct) = ln(ct) as long as the consumer lives.
Assume also that the consumer discounts the future utility by a factor b (shows the degree of impatience of the consumer) each period, where 0 想 b 想 1.
Let wt be an asset in period t. Assume that the initial value of the asset w0 > 0 and suppose that this period asset and consumption determine next period’s asset as follows:
wt+1 = wt 一 ct
Assume also that this asset cannot be negative and that wT 三 0.
Answer the following questions .
1. Formulate and write explicitly the dynamic consumer’s decision problem.
2. Apply the Method of Dynamic Programming and formulate the Bellman Equation.
3. Solve the problem applying the Method of Undetermined Coefficients .
Hint: Try the guess function of the form Vt (wt) = A + Bln(wt) and determine coefficients A, B.
2022-11-17