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Lecture 8

Lecture 8.pdf

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Leslie Shiell

ECO 3145 Mathematical Economics I Lecture 8 FirstOrder Difference EquationsChiang ch 171173 176SummaryI Dynamic analysis II Solution method repeated iteration III Stability IV General solution method V Qualitative analysis phase diagrams VI ApplicationsI Dynamic Analysis1 static analysis no consideration of timevariables of interest can take only one value eg equilibrium price quantity2 dynamic analysis passage of time is accounted for in the analysisinstead of singlevalued variables focus on paths of each variable over timeeg time path of consumptionthe value of a variable in one period may be related to the value in an adjacent perioddifference equations and differential equations are tools for analysing how the value of a variable in one period is related to its value in other periods as well as the values of other variables3 time measurement continuous timetime represented by a variable t defined over the real number lineunits of time are infinitesimally small no durationthe path of a variable y over time is represented by y or the function yt t because the units of t are infinitesimally small y can change from moment to moment change in y is measured by a derivative or differential 31458doc Shiell 1 discrete time time a series of distinct periods identified by integer values eg t123each period has durationor ytthe path of a variable y over time is represented by yt the value of y remains constant for the duration of each period of time it can only change when the period changeschange in y measured by difference ie y y yor y y y 1 t tt 1 t example how old are you 4 sequence definition set of values showing the path of a variable in discrete time notationorwhere each term xK x x x XK x x x X indicates the value of x n3 2 13 2 1for the duration of time period n K63971217 51 X egjx xx x xKj 1 j 1 i iti t 5 difference equation DE definition an equation which specifies the change in a variable from one period to the next examplewhereis a sequence of exogenous variablesis a y x xyxt 1 t tttsequence of endogenous variables because its value depends upon y and previous values of itself andrepresents a constant order the difference between the earliest and latest time period in the equation y x x exampleis a firstorder DE t 1 t t example y x x x is a secondorder DE t 2 t 1 t tlinear vs nonlinear DEegis linear y x xt 1 t t2 egis nonlinear y x xt1 t t 6 solution definitiona representation of each element of xas a function of t and the sequence of exogenous tvariables y t ie what sequencewould satisfy the DE xt 31458doc Shiell 2
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