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Universidad de La Habana

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## Universidad de La Habana

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**Universidad**de La Habana Lectures 5 & 6 : Difference Equations Kurt Helmes 22nd September - 2nd October,2008**CONTENT**Part 1: Introduction Part 2: First-Order Difference Equations Part 3: First-Order Linear Difference Equations**1**Introduction Difference Equations (Prof. Dr. K. Helmes)**Part 1.1**An Example Difference Equations (Prof. Dr. K. Helmes)**Example 1 (Part 1)**cf. compound interest Dagobert- Example**interest factor**Starting Point: Given: K0 initial capital ( in Euro ) p interest rate ( in % ) r**Objective:**Find .... • 1. The amount of capital after 1 year. • 2. The amount of capital after 2 years. • n. The amount of capital after n years.**Solution:**After oneyear the amount of capital is: How much capital do we have after 2 years?**Solution:**After one year the amount of capital is: After twoyears the amount of capital is:**Solution:**After one year the amount of capital is: After twoyears the amount of capital is:**Solution:**After nyears the amount of capital is:**The solution formula can be**rewritten in the following way: is given, is given, Observation: special difference equation recursion formula**Part 1.2**Difference Equations Difference Equations (Prof. Dr. K. Helmes)**Illustration:**A difference equation is a special system of equations, with • (countably) infinite many equations, • (countably) infinite many unknowns.**Hint:**The solution of a difference equation is a sequence (countably infinite many numbers).**An equation, that relates for any the**nthterm of a sequence to the (up to k)preceding terms, is called a (nonlinear) difference equation of order k. Definition: Difference Equation Explicit form: Implicit form:**2**First-Order Difference Equations Difference Equations (Prof. Dr. K. Helmes)**Part 2.1**A Model for the„Hog Cycle“ Difference Equations (Prof. Dr. K. Helmes)**Example 2**cf. Microeconomic Theory „Hog Cycle“ (Example)**ratio**16 12 Avg 8 year Starting Point: Given: Hog-corn price ratio in Chicago in the period 1901-1935:**price ratio**time Starting Point: Stylized:**Starting Point:**• Find: • A (first) model, which „explains“ / describes the cyclical fluctuations of the prices (ratio of prices).**in units at time**in units at time in units at time in units at time Model (Part 1): Supply and Demand The suppply of hogs: The demand of hogs:**The supply at time depends on the hog price**at time . Model (Part 2): Supply and Price Assumption:**i.e. it is determined by and , and p(t).**is given Model (Part 2): Nature of the dependance Assumption: The supply function is linear:**parameter**Model (Part 3): Demand and Price Assumption: For the demand we assume: If the hog price increases, the demand will decrease, thus:**for all**Model (Part 4): Equilibrium Postulate: Supply equals demand at any time:**Model (Part 4): Equilibrium**The equilibrium relation yields a defining equation for the price function:**is given**Solution (Part 4): Equilibrium Thus we obtain the following difference equation:**Model (Part 4): Equilibrium**This difference equation is: • first-order • linear • inhomogeneous**Model**(Part 5): Analysis solution formula:**a**= : 1 e.g. b = 3 g = 2 d = 5 Iteration rule Figure 2:**„stable“: The values converge to the**equilibrium state when . Model (Part 5): Analysis Results: The equation / solution is stable. The equation / solution is unstable.**1**0,8 0,6 0,4 0,2 0 -0,2 -0,4 0 10 20 30 40 5 15 25 35 Figure 3:Price development for:**1**0,8 0,6 0,4 0,2 0 -0,2 -0,4 16 0 4 10 14 20 2 6 8 12 18 Figure 4:Price development for:**40**30 20 10 0 -10 -20 -30 0 10 20 30 40 5 15 25 35 Figure 5:Price development for:**The term has an alternating**sign, . Summary: The given difference equation has a unique solution; it can be solvedexplicitly. The price is the sum of a constant and a power function.**CONCLUSION:**We can model and analyze dynamic processeswith difference equations.**Part 2.2**Definitions und Concepts for First-Order Difference Equations Multivariable Calculus: The Implicit Function Theorem (Prof. Dr. K. Helmes)**Definition:**A (general) first-order nonlinear difference equation has the form : (F is defined for all values of the variables.)**Important Questions:**• Does at least one solution exist? • Is there a unique solution? • How many solutions do exist? • How does the solution change, if „parameters“ of the system of equations are changed (sensitivity analysis)?**Important Questions:**• Do explicit formulae for the solution exist? • How do we calculate the solution? • Does the system of equations has a special structure ? e.g.: a) linear or nonlinear, b) one- or multidimensional ?**”fixed number”,**Remark: If the initial value of the solution (sequence) of a difference equation is given, i.e. then we call our problem an ” initial value problem ” related to a first-orderdifference equation.**If is an arbitrary fixed number, then there exists a**uniquely determined function/sequence , that is a solution of the equation and has the given value for . Remark: The initial value problem of a first-order difference equation has a unique solution.**In general there exists for each choice of a different**(corresponding) unique solution sequence. Remark:**For time homogeneous nonlinear difference equations**we call points which satisfy the equation Definition: Invariant Points invariant points. F ”right-hand side”.