Mathematical Economics ECN 505
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Date Created: 10/21/15
ECN 505 HANDOUT SPRING 2009 ynamic Systems AN INTRODUCTION TO DYNAMC SYSTEMS IN ECONOMICS EXAMPLE ADJUSTMENT IN THE ISLM FRAMEWORK Consider the ISALM Framework presented as an application when we looked at comparative statics Recall the goods and money markets are in equilibrium if Y CY239 IY239 G where we assume no taxes for simplicity and LY 2 respectively We assume the usual signs on the partials of the functions If the goods market is not in equilibrium we assume that income Y changes if the money market is not in equilibrium we assume that income 239 changes Speci cally we assume that YaCY239IY2 G7Y 1 a dY where Y E and a gt 0 1s the speed of adjustment 1n the goods market To s1mphfy notation we do not i explicitly denote the fact that Y and 239 depend on t Also assume that a M 39 L Y 39 7 2 z 5 lt 2 P lt a d39 where 239 E 1 and gt 0 is the speed of adjustment in the money market We rewrite 1 and 2 in matrix form a aCY239 IY239 G7Y Y M l 2 l 5 Lama ln Y239 space we rst plot the Y points such that the goods market is in equilibrium ie the IS curve or the Y points such that Y 0 recall that under the usual assumptions on partials the slope of the Y 0 IS curve is negative Now we plot the Y points such that the money market is in equilibrium ie the LM curve or the Y239 points such that 239 0 recall that under the usual assumptions on partials the slope of the 239 0 LM curve is positive We sketch both curves on the graph which follows i B 1 20 LM 39A C 13 If the economy is not on the Y 0 curve the goods market is not in equilibrium and Y will change If the economy is not on the 0 LM curve the money market is not in equilibrium and 239 will change We determine the direction in which Y and 239 will change these directions are indicated in the preceding diagram First notice that in moving from point A to point B in the diagram 239 increases while Y is constant Then consumption and investment must decrease leaving aggregate demand less than Y in the goods market therefore Y will decrease It follows that above to the right of the Y 0 curve Y decreases Similarly below to the left of the Y 0 curve Y increases Finally on the Y 0 curve Y doesn t change Now notice that in moving from point C to point D in the diagram Y increases while 239 is constant Then real demand for money must increase leaving it greater than the real supply of money therefore 239 will increase It follows that below to the right of the 0 curve 239 increases similarly above to the left of t e 0 curve 239 decreases Finally on the 0 curve 239 doesn t change Olech s Theorem Given the nonlinear system Jsvltwem its equilibrium point 35 yquot is asymptotically globally stable if fag gy lt 0 for all 121 figy 7 fygi gt 0 for all 121 and figy 7E 0 for all 3021 or fygi 7E 0 for all We apply Olech s Theorem to see that the lS7LM dynamic system is globally stable Let fY239 E aCY239 IY239 G 7 Y M gY2 E LY2 7 Then fy 9 aCy y 7 l Li lt 0 for all Y239 fygi 7 figy a Cy Iy 7 l Li 7 02Ci 1 Ly gt 0 for all Y239 and fygi aCy y 7 l Li 7E 0 for all Y239 ANOTHER EXAMPLE In contrast to the last example consider the following example y DYNAMIC OPTIMIZATION THE PONTRYAGIN MAXIlVIUM PRINCIPLE We give a brief exposition of a dynamic optimization problem For more comprehensive treatments see the references at the end of the handout The Pontryagin Maximum Principle refers to the solution method presented in this handout It is illustrated through con ideiiu time 39 A 39 39 model Consider the following formulation of an optimal control problem see Takayama pp 456458 for this formulation notice how it is a speci c formulation of a more general problem given on pp 4537456 In it ut is the control variable the ut function de ned on some time interval t0T is to be chosen from a set of admissible functions U unless otherwise speci ed assumed to be piecewise continuous functions 1t is the state variable it can only be controlled through the choice of ut which affects the rate of change of 1t with respect to time This problem is called a nite horizon problem because the nal time T is known and it is nite T mgag fto f 1t ut t dt target functlon st out g actut t constraint 10 constraint or initial condition The triple ut xtpt is called the optimal triplet ut is the solution pair Theorem 1 The Maximum Principle Given the speci cation of the problem in order that ut be a control solution of the problem with the corresponding state solution xt it is necessary that there exists a continuous function pt gt 0 for all t lt T auxiliary or costate variable such that a pt together with 1t and ut solve the following Hamiltonian system 8Hquot 8Hquot f 7D 7 8p p 81 where H the Hamiltonian is de ned by Hi7ui7i7pi E f 0714370 1409 0714070 81 6Hltxltt7ultttpltz and g 6Hltxltt7ultztpltz 8p 7 8 I P 81 81 where b The Hamiltonian is maximized with respect to ut that is HWWUWWUJWUD 2 HWWUWULLPUD for all ut E U c pT 0 the transversality condition d 1t0 do Please see Takayama for sets of suf cient conditions in addition see his text for many related slightly modi ed problems Example As a representation of most of these ideas consider the following example in which we ll see slightly different conditions Consider an optimal growth problem speci ed in the following way i Y Ct It ii Y FN Kt exhibits constant returns to scale ie it is homogeneous of degree 1 iii 1 I iv ampn Nt It can be shown see Takayama pp 355356 that in this model M fUW where y is per capita worker output at time t and k is the capital to labor ratio at time t We assume that for every k 2 0 1 gt 0 f k lt 0 3 also assume that f0 0 f 0 007 lim f k 0 k7gtoo Under the speci cation of the model it can be shown that see Takayama pp 355356 12tfkt nkt Ctv 4 where c is per capita consumption at time t We assume that the representative consumer has the utility function uc where c is per capita consump7 tion u gt 0 u lt 0 and u 0 00 in addition assume that there exists 12 such that uc lt 12 for every 0 2 0 We ask that given 4 and an initial level of capital kg and the restrictions 0 2 0 k 2 0 the present value of utility be maximized in a world without end given control over the consumption time path 0 that is given a strictly positive personal discount factor p the problem is the following maax 10 uce ptdt st lac fk 7 nk 7 c k0 k0 kt Z 07 C 2 0 In this case 0 is the control variable k is the state variable Note that unlike the set up in the preceding section this is set up as an in nite horizon problem therefore the necessary conditions will be somewhat different from the ones in the theorem of the preceding section Also notice the presence of the nonnegativity constraints on per capita consumption and capital The Hamiltonian in this case is H E ucte pt Pt fk 7 nk 7 0 The following conditions are necessary for an optimum in this example Notice that all but the last two are given by the theorem of this section the difference in this example is due to the in nite rather than nite time horizon and the constraints on c and kt a kZC solve the following Hamiltonian system that is they are such that 8H a 12 kquot k7nk7c 5 apt t f t t t the constraint and 9 8H 0 p77 7p7pltf ltk7n7amp7n7f w 6 8k Pt b H is maximized WRT c 2 0 that is Hkfyc 7t7pn 2 Hk 70c7t7pn for all c 2 0 for all t 3 BHquot 7 BHquot a qu cte pt7p 0 act ct0 for all t slnce lt can be argued that lt ls never the case that c 0 thls condltlon lrnplles that 8Q u cE t 7 0 p ucna m h 7 notlce that thls relatlonshlp tells us that the Pontryagln multlpher ls the present value of the margmal utlllty of consumptlon at tlme t lt ls the opportunlty cost of accumulatlon of capltal c Ic0 Icu d The followlng condltlons hold 111m n 2 0 111m we 0 8 ek201020 It can be shown that these condltlons are sumclent for an optlrnal solutlon glven the restrlctlons on u and We use these equatlons to set up a phase dlagram wlth c on the horlzontal axls and k on the vertlcal 8x15 whlch wlll glve us some ldea of the nature of the solutlon path We drop the s from the solutlons for notatlonal slmpllclty leferentlate 7 WRT t to nd that New 7 WWW 75 9 Dlvlde 9 by 7 to nd that u cr ra quot 7 WWW W005 7 7 War 7 War 71 10 Use 6 and 10 to nd that 7 folcacti 77u 0 7 n 7 n HEP Wot Piaf unqfkz r ll settlng thls equatlon equal to 0 allows us to sketch the 039 0 curve where 1 rs such that f E n p or ere ct Recall that equatlon 5 ls k M2 nkz or 12 settlng thls equatlon equal to 0 allows us to sketch the in 0 curve In thls framework by 7 8 and the assumptlons on the utlllty functlons the fourth condltlon becomes 111m u cca quotIn 0 13 Please convmce yourselves that under the assumptlons the 039 0 and 1 0 curves have the shapes lndlcated In the followmg dlagram It can be shown that wlth the glven dlrectlon of change posslbllltles on the dlagram and the resultlng tune paths for c and 1c the condltlons cannot beisatls ed lf the path ever enters quadrants lor III In the followmg phase dlagram It follows that lflcn lt gtIc the path P1 P2 should chosen Note that lf the functlons were glven expllcltly then you could use 11 and 12 to so ve expllcltly for c and ks You ll have the opportunlty to do thls type of exerclse REFERENCES Your text Chiang7 Dynamic Optimization Takayama7 On reserve