Problem Set on Differential Equations

Transcription

1 Problem Se on Differenial Equaions 1. Solve he following differenial equaions (a) x () = e x (), x () = 3/ 4. (b) x () = e x (), x (1) =. (c) xe () = + (1 x ()) e, x () =.. (An asse marke model). Le p() denoe he price of equiy, le d() denoe he dividend paid a ime, and le r denoe he yield on a risk free bond. (a) Wha equaion of moion yields he following forward soluion for he price of equiy: rt ( ) rs ( ) p () lim pte ( ) dse () ds = +? T (b) Explain he economic inuiion behind his equaion of moion. (c) Wha assumpion abou he forward soluion implies ha he price of equiy is equal o he presen value of curren and fuure dividends? (d) Wha are he economic jusificaions for his assumpion? 3. (Solving general linear equaions). Consider equaions of he form, x () + px ()() = g (). (a) If g() is idenically zero, show ha he soluion is

2 PROBLEM SET ON DIFFERENTIAL EQUATIONS x () = Aexp psds (), where A is a consan deermined by boundary condiions. (b) If g() is no idenically zero, assume a soluion of he form x () = A ()exp psds (), where A is now a funcion of. Show ha A() mus saisfy he condiion A () = g ()exp psds (). (c) Find A() from his expression and hen use your answer o wrie an expression for x(). (d) Using he formulae jus obained, solve () x () + x = (Sabiliy of nonlinear equaions). For each of he following differenial equaions, analyze he global sabiliy of he seady sae: (a) ( ) ; (b) ( ) x () = b x () a x () = b x () a ; (c) ( ) 3 ; (d) ( ) 3 x () = b x () a x () = b x () a 5. (Log-linearizaion). Consider he nonlinear capial-sock equaion k () = sfk ( ()) δk (). A common analyical approach is log-linearize he equaion. To do so, one subsiues y()=lnk(), and hen linearizes around he seady sae value

3 PROBLEM SET ON DIFFERENTIAL EQUATIONS 3 of y. (a) Log-linearize his equaion. (b) Inerpre he resul. Why migh log-linearizaion be preferable o linearizaion? 6. (Populaion growh). a) Suppose x = x( α + βx). Derive an explici soluion for x and show ha i becomes infinie in finie ime. b) For he Gomperz growh equaion, x () = βx ()( α ln x ()), (i) Solve he equaion subjec o x()=x. (ii) Skech he graph and is associaed phase diagram. Derive he seady saes and esablish heir sabiliy or insabiliy. 7. (R&D driven growh). A well-known empirical regulariy in indusrial economics is ha firm R&D is more or less proporional o size. Making use of his regulariy wrie down a simple model of R&D-driven growh in a firm s marke share, s, ha incorporaes he following feaures: (i) marke share is bounded beween and 1; (ii) if a firm does no R&D i will lose marke share due o he R&D effors of oher firms; (iii) here are n firms; (iv) all firms have he same R&D abiliy. Solve (if possible) and characerize he soluion of he model. Wha is (are) he seady sae(s)? 8. (Sabiliy of a linear sysem). Solve and assess he sabiliy of he following differenial equaions: (a) x () = x () + y () and y () = 4 x () + y (); (b) x () = 3 x () + y () and y () = x () + y ();

4 PROBLEM SET ON DIFFERENTIAL EQUATIONS 4 (c) 1 x () = x () + y () and 1 y () = x () y (). 9. (Sabiliy of a nonlinear sysem). The following sysem has wo seady saes: x () x () + y () y () = x () y () + 1 a) Consruc he phase diagram for his sysem o fully characerize he sysem's behavior. b) Find he roos of he linearized sysem and verify your graphical characerizaion of he local properies of he sysem. 1. (Solow model wih human capial). Mankiw, Romer and Weil [ A Conribuion o he Empirics of Economic Growh. Quarerly Journal of Economics, 17():47-437] analyze he following version of he Solow model: α y () = k () h (), k () = sy () ( n+ g+δ)() k, k h () = sy () ( n+ g+δ) h (). h β where y is oupu per effecive uni of labor, h is human capial per effecive uni of labor, k is physical capial per effecive uni of labor, and s h and s k are he savings raes for physical and human capial. g is he rae of echnical change, n of populaion growh and δ he depreciaion rae.

5 PROBLEM SET ON DIFFERENTIAL EQUATIONS 5 a) How do he parameers of he model affec he seady sae income level, y*? b) Draw he phase diagram for his model and analyze he sabiliy of he seady sae(s). c) Mankiw, Romer and Weil poin ou ha he Solow model makes quaniaive predicions abou he speed of convergence o he seady sae. Specifically, a log linear approximaion around he seady sae yields dln( y( ) = ( n + g + δ)(1 α β) ( ln( y*) ln( y( )) ). d Derive his expression formally, and inerpre i.