INTRODUCTORY MATHEMATICS FOR ECONOMICS MSC S. LECTURE 5: DIFFERENCE EQUATIONS. HUW DAVID DIXON CARDIFF BUSINESS SCHOOL. SEPTEMEBER 2009.

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1 INTRODUCTORY MATHEMATICS FOR ECONOMICS MSC S. LECTURE 5: DIFFERENCE EQUATIONS. HUW DAVID DIXON CARDIFF BUSINESS SCHOOL. SEPTEMEBER

2 Difference Equaions. Much of economics, paricularly macroeconomics is dynamic. People save now o spend in he fuure: firms inves now o produce more in he fuure. Sudens learn now o earn more in he fuure. Difference equaions rea ime as discree =1,,3,..T. Firs Order Difference Equaion (FODE): y a by 1 An Equilibrium is defined as a siuaion where y remains unchanged over ime: y y 1 y*. a Hence: y* a by* y* 1 b Hence, a unique equilibrium exiss if and only if b 1. Quesions: will we ge o equilibrium if we sar from some/all iniial posiions? If we ge o equilibrium, will he pah be monoonic? Wha happens if no equilibrium exiss? 5.

3 Soluion of Difference Equaions (linear FODE). Suppose we sar from an iniial posiion y. Le us race ou wha happens: y1 a by y a by a b( a by ) ( 1) ( ) 1... y a by a b a by a ba b a by a ba b a b y y a ba b a b y This is called he definie soluion. Given an iniial posiion, we have a clear sequence, a pah of y raced ou over ime. This simplifies. If b=1 (and hence no equilibrium exiss), hen he ime pah is y a y If b 1? Well, le us firs look a he case of a homogeneous FODE where a=. In his case y*, and 5.3

4 y by So, wha happens as? Case 1: If <b<1 hen b monoonically. Case : If -1<b< hen b bu non-monoonically: i ges smaller in absolue value bu alernaes sign (negaive numbers raised o an even number are posiive, o an odd power are negaive). Case 3: b=, y jumps o he equilibrium in period 1. In all cases 1-3, we say ha he difference equaion is globally sable if he absolue value of b is less han 1, -1<b<1, or b 1. Wherever we sar from, we end up a he seady-sae equilibrium y*=. Les use an excel spreadshee o ake a look! 5.4

5 Sable FODE y*-y 1.5 b=.5 b= ime y by 1 b.5 y* 4 b.5 y* 4 / 3 5.5

6 Case 4: b>1. In his case y goes o infiniy (assuming iniial y is posiive). Case 5: b<-1. In his case y goes o plus/minus infiniy oscillaing beween and +. Now, i urns ou ha he sabiliy of an FODE is only deermined by he b consan. This is inuiive: he consan do no change, and so do no influence he dynamics. General soluion o he FODE y a by 1 If b 1, hen he general soluion is y y* Ab Where A is an arbirary consan. This is deermined by he iniial condiion (or in heory by a erminal condiion bu in economics i is usually an iniial condiion!). To work his ou we have a ime yo y* Ab y* A A ( y y*). 5.6

7 Example: Cobweb model. Supply in year depends on price in year -1: x x 1 dp sp 1 s Supply equals demand: 1 dp sp 1 P P 1 d 1 d Equilibrium: 1/ d 1 P* 1 ( s / d ) d s Is i sable? We have defined boh d,s posiive. Hence if d<b (he demand curve is sloped less seeply han he supply curve) hen he model is sable. Since he coefficien is negaive, he pah of oupu will oscillae Around he equilibrium. 5.7

8 Noe: Price on horizonal axis. A Q1, demand price is P, so Q is s(p), so demand price is P3 Oscillaes around equilibrium price and quaniy P,Q. This is sable, however no reason for he slope of demand o be more in absolue erms han slope of supply! In ha case you can ge explosive oscillaions. Try i wih a spreadshee.. s The soluion o he cobweb FODE is (assuming 1) d P ( P P*) b P* 1 s P* ; b d s d 5.8

9 Second Order Difference Equaions. These ake he form y by b y a 1 1 Now, he long-run equilibrium or paricular soluion is given by y* y y 1 y. Two possibiliies: b1 b 1 and b1 b 1 If b 1 b 1, (he general case), hen y * a 1 b b 1 If b 1 b 1, (he special case) hen we have wo soluions (see Dowling). We will assume ha he general case holds. The soluion is: where 11 y y* Ar A r 1 b 1 4b b ri 5.9

10 If b b 1 4b Then here are wo disinc real roos. If 1 4b here are no real roos, jus imaginary or complex roos. 1 4b hen here is a unique roo ( ri b b r 1 ). If And A i are arbirary consans o be deermined. Now, wih a second order difference equaion, you need iniial condiions: ( y, y 1), and hese will deermine he wo arbirary consans. Noe: if we compare he soluions o a firs order difference equaion: when r i, he wo soluions become idenical (check: if b, b 1 b ri b1,. Noe: he negaive sign on he b is here because of he way we wroe he equaions. 1 Sabiliy is derermined by he values of he roos 1 r i. Le assume ha here are wo disinc real roos. 1 The roos of he equaion are called eigenvalues: in general, if here is an nh order difference equaion, he soluion will be represened by a linear funcion of he n roos or eigenvalues. 5.1

11 11 y y* Ar A r If 1 r i 1(boh roos less han 1 in absolue size) we have global convergence: wherever you sar from, you ge o he equilibrium. If r 1 1, r 1 (one sable, one unsable) you have saddle-poin sabiliy. For iniial condiions where he unsable roo has a zero weigh ( A ), you proceed o equilibrium. Oherwise you explode. In economics, we ofen use saddle-pah soluions. Tha is because here is a unique pah o equilibrium, which is raced ou by he negaive roo. b 4b Wha if he roos are no real (imaginary): when 1. I will no go ino he deails here: however, basically you ge cycles in his case. These can be explosive or convergen. I is easies o see an example. y.5y 1.y y* ; y y 1. 1 Using excel spreadshee. In a column (row). Type in he iniial values 1,1 in he firs o rows (le us say y =C1=1, y1 C=1). Then ype in =-.5*C+.*C1: hen make char. o 5.11

12 Complex roos y().4 Series ime 5.1

13 Try: b.1; b.9: 1 Here we have explosive oscillaions! You can use a spreadshee o see wha happens o a second order difference equaion. Wih complex roos, you ge oscillaions, which can eiher end o he equilibrium, explode, or jus cycle and never converge

14 Conclusion. 1. Linear difference equaions: express he curren value as a linear funcion of pas (lagged) values.. The equilibrium soluion is derived by seing he curren and lagged values equal. 3. The sabiliy of he difference equaion is deermined by he coefficiens on curren and lagged values (no he consan). These deermine he roos or eigenvalues of he difference equaion. 4. A roo is sable if i is less han one in absolue value. If all roos are sable, hen he difference equaion is globally sable: wherever you sar from you will converge o he equilibrium. If he roo is posiive, you will converge monoonically, if negaive i will oscillae. 5. Wih a second (or higher) order difference equaion, you can ge complex roos which can give cycles. 5.14