MACROECONOMIC ANALYSIS IN THERMODYNAMIC MODEL REVIEW BY USING FUZZY STATISTICS

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1 I J A B E R, Vol. 13, No. 2, (2015: MACROECONOMIC ANALYSIS IN THERMODYNAMIC MODEL REVIEW BY USING FUZZY STATISTICS Sutanto *, Purnai Widyaningsih * and Ririn Stiyowati Abstract: Application of thory and physics law in conoics is known as conophysics. Econophysics applis if phnona in physics and conoics ar analogous. On xapl of conophysics is application of thory and law on throdynaic in conoics. Throdynaics odl applid in conoics is Grand Canonical Partition Function (GCPF. GCPF that is xtnsiv and unitary ts fuzzy statistics critria. GCPF in fuzzy statistics is usd for dtrining xtnsiv variabl valu in conoics basd on acroconoic indicator valus. Macroconoic indicator usd as intnsiv variabl in conoic is Gross Dostic Product (GDP pr capita, pric lvl, vlocity of ony and intrst rat issud by BI. Dtrination of xtnsiv variabl, such as th aount of ony supply, utility, ntropy, is don by doing fuzzy statistics siulation.kywords : Econoic, Macroconoic, Throdynaic, Grand Canonical Partition Function, Fuzzy statistics 1. INTRODUCTION Th dynaics of a country s conoic growth, ithr acroconoic or icroconoic, shows that applid athatics play a rol in conoic ara. Macroconoic dvlopnt of a country is shown by four ain indicators, such as Gross Dostic Product pr capita, pric lvl, vlocity of ony, and intrst rat issud by BI. Th four indicators ar rlatd to total asst lvl (utility, goods quantity, and ony supply. Sukirno [10] xplains th corrlation aong pric lvl, goods quantity, vlocity of ony and ony supply xprssd in quantity thory of ony dvlopd by Irving Fishr. Manwhil, quantity thory of ony dvlopd by Marshall is rviwd fro inco aspct that is xplaining th corrlations aong vlocity of ony, ony supply, conoic growth rat and GDP. Th cobination of th two thoris according to Bryant[1] dscribs idal gas thory in throdynaics. Idal gas thory dscribs corrlations aong tpratur, prssur, volu and particl quantity in a syst whil throdynaic is a study in physics about all activitis occur in a syst rlatd to nrgy chang du to hat transfr and th work don. In throdynaics, thr ar two variabls, thos ar xtnsiv * Dpartnt of Mathatics, Faculty of Mathatics and Natural Scincs, Sblas Mart Univrsity, Surakarta, Indonsia, E-ail : sutanto@uns.ac.id; por@uns.ac.id; ririns64@gail.co

2 640 Sutanto, Purnai Widyaningsih and Ririn Stiyowati variabl and intnsiv variabl. In throdynaics statistics rviw, th corrlation btwn th two variabls is dscribd in Grand Canonical Partition Function (GCPF. GCPF ts xtnsiv and unitary fuzzy statistics critria [8]. Yakovnko [11] proposs that th study in throdynaics is analogous with th study in conoic that conoic is a study that larns all activitis occur in an conoic syst rlatd to production and consuption procsss. That statnt is supportd by th rsarch don by Saslow [7] rlatd to throdynaic analogy in conoic. Thrfor, GCPF in throdynaic can b usd for dscribing corrlation aong variabls in conoic. Furthror, fuzzy statistics is usd for dtrining valu fro xtnsiv variabl in conoic bcaus GCPF is xtnsiv and unitary and siulation toward xtnsiv variabl with varid paratr valu is don and th rsults of siulation ar intrprtd. 2. THERMODYNAMICS ANALOGY IN ECONOMICS According to Sunarto and Babang Stiono[2], conoics is a scinc that studis huans fforts to fulfill thir unliitd nds. Econoics studis conoic activitis ntirly, ithr production or consuption larnd in acroconoic. Th study cannot b sparatd fro conoic actor s Illustrativ. That study is larnd in icroconoic. Four indicators in acroconoic ar Gross Dostic Product (GDP, pric lvl, SBI intrst rat, and vlocity of ony [3]. Thos indicators dtrin th ony supply, th quantity of goods and th total assts of a country xprssd in utility (U valu. Th corrlation btwn goods quantity (G with ony supply (M is xplaind in quantity thory of ony dvlopd by Irving Fishr in 1990 [10] that is xprssd in quation pg=vm with p for pric lvl and v for vlocity of ony. Fro inco aspct, quantity thory of ony dvlopd by Marshall xplains corrlation btwn th ony supply and ral inco shown by GDP (Y xprssd in quation vm=cy with c for conoic growth rat. GDP valu is not othr than ultiplication of GDP pr capita T and th total population (n. Fro th cobination of th two thoris, th corrlation can b writtn PG = c(nt (3.1 According to Bryant [1], quantity thory of ony quation in quation (3.1 is analogous with idal gas thory in throdynaics which is xprssd in quation PV = nkt with T for tpratur, V for volu, P for prssur and n for nubr of particls. Th statnt is supportd by th rsarch don by Saslow [7], Miks [5], and Yakovnko [11] about analogy of throdynaics in conoics. Th rsarch xplains that throdynaics and conoics ar analogous.

3 Macroconoic Analysis in Throdynaic Modl Rviw by using Fuzzy Statistics 641 According to Moran t al. [6], throdynaics is priary knowldg of physics rlatd to nrgy chang phnonon du to hat transfr and th work don. In throdynaics, thr ar two variabls; thos ar xtnsiv and intnsiv variabls. Extnsiv variabl is stat variabl that is influncd by ass or volu. Thos variabls ar intra nrgy (E, ntropy (S, and particl nubr (N. Manwhil, intnsiv variabl is stat variabl that is not influncd by volu, such as T, P and chical potntial µ. Mljanac [4] stats that corrlation btwn xtnsiv variabl and intnsiv variabl in throdynaic statistics, both Bos- Enstin statistics and Fri-Dirac statistics ar dscribd in GCPF xprssd in quation Z i i ( i µ (1, 1,2,... (3.2 with i is nrgy in i-tr stat and assuption that thr is no intraction aong particls. Th particl nubr N in Bos-Einstin statistics and Fri-Dirac statistics conscutivly ar dfind as N Bos dan N ( i µ Fri ( i µ i 1 i 1 (3.3 Rfrring to Satriawan [8], GCPF in quation (3.2 is xtnsiv and unitary so that it t th critria of fuzzy statistics. Whras GCPF in quation (3.2 in fuzzy statistics is prsntd p q 1 ajx x i i Z( x1, x2,, x 1 b x (3.4 i 1 k 1 j 1 k i p q Unitary condition is t if, a, b 0 and 1a 1 b 1. P and j k j j k k q valu stat fuzzy statistics that is latr known as fuzzy statistics (p, q. Bos-Enstin statistics only rgs on paratr b and a paratr for Fri- Dirac statistics whn anothr paratr is zro that is fuzzy statstics (p, 0 for Bos-Einstin and (0, q for Fri-Dirac. Statistika fuzzy (p, q is Bos-Fri statistics. Du to th analogous charactristic btwn throdynaics and conoics, GCPF can b applid in conoics with assuption that thr is no intraction aong conoic actors. Manwhil, throdynaics variabls analogy rlatd to GCPF in Econoics basd on th studis don by Stiyowati [9] is prsntd in Tabl 1. Thr by, i in quation (3.2 in conoics xprsss utility quantity in i tr stat that is furthr rotatd with u i.

4 642 Sutanto, Purnai Widyaningsih and Ririn Stiyowati Throdynaic Tabl 1 Throdynaics variabls analogy in conoics Econoic Enrgy (E Utility (U Tpratur (T GDP pr capita (T Entropy (S Entropy in conoic (S Prssur (P Pric lvl/ihk (p Volu (V Goods quantity (G Chical potntial (µ Vlocity of ony (v Particl nubr (N Mony Supply (M 3. EXTENSIVE VARIABLE CALCULATION IN ECONOMICS In conoic quantity calculation that is xtnsiv variabl, such as utility (U function, ntropy (S and ony supply (M that is within a country s conoy us GCPF which ts fuzzy statistics (p, q. Fuzzy statistics (p, q has uniqu charactristics in trs of xtnsiv GCPF. Th statistics can giv dscription of conoic condition basd on a and b paratrs. In this discussion, conoic quantity calculation is dtrind only by using fuzzy statistics (1,1. Th dtrination of conoic quantity for fuzzy statistics (1,1 bgins fro GCPF logarith in quation (3.4 with = 0. GCPF logarith notatd with q(t, G, z that is q( T, G, z ln Z ln(1 ax ln(1 bx (3.5 i i i 1 i 1 ( i with u v x and i 1 T with T is th valu of GDP pr capita as wll as a+b=1. Furthror, it is dfind z = and z 1 = a z as wll as z 2 = b z so th quation (3.5 bcos ui 1 2 i 1 i 1 ui q( T, G, z ln(1 z ln(1 z. (3.6 By using quation (3.3, th avrag of ony supply M is M( T, G, z. 1 ui 1 ui i 1 z1 i 1 z2 (3.7 For th syst with larg goods quantity and th total quantity in quations (3.6 and (3.7 is a for of intgral. Th for of intgral is 0 f ( u du

5 Macroconoic Analysis in Throdynaic Modl Rviw by using Fuzzy Statistics 643 with f(u is assud having valu of 1 gg 2 3 u rfrring to throdynaic thory and 0. g valu in conoic probl is considrd having 1 valu. Thrby, quations (3.6 and (3.7 bco G u u q( T, G, z du du, z 1 z 1 ( u i 0 1 u i and 2 2 G u u M( T, G, z du du. z 1 z 1 ( u i 0 1 u i For xapl, u = x. Intgral tr in quation (3.8 and (3.9 is writtn s 1 1 x fs( z dx x and ( z s 1 g ( z s s 1 1 x dx x with s = 5/2 for intgral ( z s 2 tr in quation (3.8. Thrfor, quation (3.8 and (3.9 can b writtn G q( T, G, z f ( z g ( z 3 5/2 1 5/2 2 and G M( T, G, z f ( z g ( z. ( /2 1 3/2 2 Aftr q(t, G, z and M(T, G, z ar obtaind, utility and ntropy valus can b dtrind. With U( T, G, z ln q( T, G, z z, G dfinition, is obtaind 3M f5/2( z1 g5/2( z2 U( T, G, z. 2 f ( z g ( z 3/2 1 3/2 2 ( Manwhil ntropy valu S is obtaind with th dfinition that S ( U W T with W = pg + vm. Basd on quation (3.10 and (3.11, is obtaind

6 644 Sutanto, Purnai Widyaningsih and Ririn Stiyowati 5M f5/2( z1 g5/2( z2 S( T, G, z. 2 f ( z g ( z 3/2 1 3/2 2 (3.12 So far, xtnsiv variabl calculation in conoics uss fuzzy statistics (1,1 has finishd. Nxt, to find out M(T, G, z, U(T, G, z, and S (T, G, z, siulation in cas application is don. 4. SIMULATION 4.1. Siulation Scnarios In th application of this cas, a, v and T valus ar takn fro bi.go.id and bps.go.id sits yar Fro thos sits, a, v and T found conscutivly ar 0.065, and 27 illion pr prson with a valu is takn fro BI rat. BI rat is chosn to rprsnt a paratr valu bcaus all conoic quantitis cannot b sparatd fro th influnc of rat valu dtrind by BI. Thrfor, M(T, G, z, U(T, G, z, and S (T, G, z valus in fuzzy statistics (1,1 ar dtrind by 3 paratrs. This siulation is don with two paratr valus which ar varid and on fixd paratr valu. In th first siulation, fixd a and T valus ar dtrind, whil v is fluctuat Coputational Rsults By using quation (3.10 and (3.11, siulation toward M(T, G, z and U(T, G, z valus ar prsntd in Figur 1. Figur 1: Illustrativ M/G (lft and U/M (right in fuzzy statistics (1,1 vrsus with a and T fixd

7 Macroconoic Analysis in Throdynaic Modl Rviw by using Fuzzy Statistics 645 Figur 1 (lft xplains that for fuzzy statistics (1,1, whn vlocity of ony is qual to zro, th ony supply pr unit is qual to illion. It shows that condition of Indonsia in 2010 is dficit that causs conoic instability of ony supply pr unit dcrass as vlocity of ony incrass. Figur 1 lft also shows ony supply pr unit dcrass as vlocity of ony incrass. Mony supply valu pr unit is dfind whn vlocity of ony is lss than 1.8. Quantity thory of ony dvlopd by Irving Fishr xplains that in pric lvl and total fixd goods, th ony supply dcrass as vlocity of ony incrass. Basd on that thory, th ony supply for fuzzy statistics (1,1 is according to quantity thory of ony. Manwhil, Figur 1 right xplains that for fixd T and a valus, th incras of vlocity of ony causs th dcras of utility valu pr unit of ony supply. Bsids, utility valu pr unit of ony supply for fuzzy statistics (1,1 is dfind for vlocity of ony lss than 1.8. Furthror, by using quation (3.12, ntropy Illustrativs in conoics with fixd T and a paratr valus ar prsntd in Figur 2. Figur 2 Entropy pr ony supply in fuzzy statistics (1,1 vrsus v with a and T fixd Figur 2 xplains that th dcras of vlocity of ony for fuzzy statistics (1,1 is followd by th dcras of ntropy valu in conoics. Basd on Figur 1 and Figur 2, it is found that th incras of vlocity of ony dcrass utility valu, ony supply and ntropy. Nxt, th scond siulation is don with fixd paratr valus of a and v, whil T valu is fluctuat. Th siulation is prsntd in Figur 3 and Figur 4.

8 646 Sutanto, Purnai Widyaningsih and Ririn Stiyowati Figur 3: Illustrativ M/G (lft and U/M (right in fuzzy statistics (1,1 vrsus with a and v fixd Figur 3 lft xplains that ony supply pr unit for fuzzy statistics (1,1 incrass as GDP pr capita incrass. Figur 3 lft xplains that th ony supply pr unit will hav fixd valu of illion for GDP pr capita or than 80 illion pr prson. Manwhil, Figur 3 right xplain that th incras of GDP pr capita dcrass utility valu pr unit of ony supply for fuzzy statistics (1,1. That ans th quantity of GDP valu pr capita in Indonsia dos not rflct total assts of Indonsia. Basd on Figur 4, it is shown that th incras of GDP valu pr capita incrass ntropy valu pr unit of ony supply total. Figur 4: Entropy pr ony supply in fuzzy statistics (1,1 vrsus T with a and fixd Th third siulation uss a valu which is fluctuat and fixd v, T paratr valus. Th chang on ony supply pr unit and utility pr unit of ony supply toward th chang of intrst rat is prsntd in Figur 5.

9 Macroconoic Analysis in Throdynaic Modl Rviw by using Fuzzy Statistics 647 Figur 5: Illustrativ M/G (lft and U/M (right in fuzzy statistics (1,1 vrsus a with T and v fixd It can b sn in Figur 5 lft that th incras of ony supply pr unit is proportional with th incras of intrst rat. Bsids, Figur 5 lft also dscribs that intrst rat which is or than 0.6, ony supply pr unit has positiv valu. Manwhil, Figur 5 right xplains that th incras of intrst rat in fuzzy statistics (1,1 is followd by th incras of utility valu pr unit of ony supply. Th utility valu pr unit of ony supply has positiv valu for intrst rat or than Nxt, th chang of ntropy valu pr ony supply is prsntd in Figur 6. Figur 6: Entropy pr ony supply in fuzzy statistics (1,1 vrsus a with T and v Fixd

10 648 Sutanto, Purnai Widyaningsih and Ririn Stiyowati Th chang of ntropy valu pr ony supply in Figur 6 shows that th incras of intrst rat incrass ntropy valu that ans th incras of intrst rat incrass conoic instability of th country. 5. CONCLUSION In this papr, w hav introducd a nw approach for conophysics. Basd on th discussion abov, it can b concludd that throdynaics and conoics ar two analogous things so that Grand Canonical Partition Function in throdynaics can b usd for dscribing condition in conoy. Manwhil, basd on th rsults of siulation, it can b concludd that for fuzzy statistics (1,1, th valu of ony vlocity or than 1.8 causs th valus of ony supply, utility and ntropy cannot dfind that ans in that condition Indonsia s conoy is unstabl. Th intrst rat or than 0.6 causs th valu of ony supply unstabl, whil utility and ntropy ar also in that condition whn th intrst rat or than Mony supply and ntropy ar constant if GDP pr capita is or than 80 illion pr prson. Rfrncs [1] Bryant, J., Throconoics, VOCAT Intrnational Ltd, USA, [2] Sunarto dan Babang Stiono, Ekonoi akro, 3 d., Pusdiklatwas BPKP, Bogor, [3] Bank Indonsia, Laporan prkonoian indonsia [4] Mljanac, S., M. Stojic, and D. Svrtan, Partition functions for gnral ulti-lvl systs, (1996, RBI-TH [5] Miks, J., Econophysics and sociophysic: Trnds and prspctivs, Willy VCH Vrlag GbH, Winhi, [6] Moran, M. J. and Howard N. Shapiro, Fundantal of Enginring Throdynaics, John Wily & Sons, Inc, [7] Saslow, W. M., An Econoic Analogy to Throdynaics, A. J. Phys 67 (1999, no. 12, [8] Satriawan, M., Bos-lik condnsation in half-bos half-fri statistic and in fuzzy bosfri statistic, Prsntd at th workshop on Bos Enstin Condnsation (12-16 Novbr 2007, Institut of Mathatical Scincs, National of Singapor, [9] Stiyowati, R., Analisis Ekonoi Mnggunakan Modl Trodinaika, Skripsi, Jurusan Matatika FMIPA UNS, [10] Sukirno, S., Makrokonoi tori pngantar, 3 d., PT Raja Graûndo Prsada, Jakarta, [11] Yakvnko, V. S., Econophysics, Statistical Mchanic Approach to, Dpartnt of Physics, Univrsity oy Maryland, Maryland, USA, 2008.