Neoclassical Growth Model

Transcription

1 Neolaial Growh Model I. Inroduion As disued in he las haper, here are wo sandard ways o analyze he onsumpion-savings deision. They are. The long bu finie-lived people who leave heir hildren no beque. 2. The infiniely lived families wih parens aring abou heir hildren s uiliy. Having inrodued he firs onsru in he previous haper, we now presen he infiniely-lived family onsru. The model ha is sudied in his haper is he Neolaial Growh Model. Effeively, he Neolaial Growh Model is jus he Solow Model bu where savings is deermined by he uiliy maximizing hoies of households. Alhough he Neolaial Growh Model is he key measuring devie used in he sudy of busine yles, we pospone his appliaion o a laer haper. In his haper, we sudy is seady sae properies wih he purpose of onsidering alernaive ax poliies. More speifially, we will use he model o evaluae alernaive ways of generaing ax revenues o pay for a neeary level of governmen expendiures. The quesion we will be answering is

2 wheher eliminaing or lowering of he apial gain ax will make he average person in he US beer off. II. Model Reall, ha he Neolaial growh model is jus he Solow model bu where we drop he aumpion ha people save a fixed fraion of heir inome. Insead, hey hoose he opimal level of savings based on maximizing heir uiliy. Alhough he mos general form of he Neolaial Growh model allows for exogenous populaion hange and ehnologial hange, we will shu down boh fores in his haper. This grealy simplifies he noaion and analysis. This simplifiaion, however, does no ome a he expense of meaningful answers. For busine yle analysis and fisal poliy relaed quesions, he answer ha he alibraed simple Neolaial Growh model gives o quesions is no muh differen han he alibraed more omplex version wih ehnologial hange and populaion growh. In wha follows we will develop a se of neeary ondiions for a ompeiive equilibrium by onsidering he maximizaion problems faing families and firms. We also add a finanial seor of he model and onsider he maximizing problems of ha seor. This se of ondiions will be used o deermine he unique balaned growh ompeiive equilibrium for a parameri se of model eonomies. We onsider he problem faing eah seor in urn. Households We will aume ha he size of families is onsan and equal o one in order o simplify

3 he noaion and analysis. The household s preferenes are defined over infinie sreams of onsumpions { } 0 and leisure { l } 0. Households preferenes are ordered by 2 3 u( 0, l0 u(, l u( 2, l2 u( 3, l3... For shorhand, his infinie series is wrien as 0 u(, l The parameer 0 < β < is he subjeive ime disoun faor. The smaller i is he more people prefer onsumpion now relaive o onsumpion in he fuure. The funional form for wihin period uiliy is he same as in Chaper 9, namely, u, l ln( ln( l ( In our se-up, we will have he household having one uni of ime in eah period o spli beween work and leisure. Thus, in any period l h The household will earn wage inome in eah period equal o w h. In addiion, he household will earn ineres inome on is deposis held a banks. We le d denoe he deposi of he household has in his bank aoun a he sar of period. In his period, he will earn wage inome, w h, and he will onsume. The amoun by whih he will add deposis or subra deposis o his bank aoun will be he differenes beween wage inome and onsumpion. Thus, a he end of period, he household will have he amoun d + w h in he bank. Beween his period and he sar of nex period, he household will earn ineres on his deposi. We le i denoe he real ineres rae on he deposi of he household. Hene, he deposi he household begins wih in period + is d ( i [ d w h ]

4 This is he household s budge onsrains in eah period. Two ses of neeary firs order ondiions for maximizaion are (H u / u / dl w (H2 u u / d / /( i i (H desribes onsumpion-leisure subsiuion in equilibrium: he marginal rae of subsiuion beween onsumpion and leisure mus equal heir prie raio. Sine one uni of leisure oday o w onsumpion oday (ha is how muh onsumpion you forego by deiding o enjoy his one uni of leisure insead of working, we have (H. Similarly, (H2 desribes h e o p i m a l ineremporal subsiuion of onsumpion. The marginal rae of subsiuion beween onsumpion oday and onsumpion omorrow mus equal heir prie raio. Sine one uni of onsumpion oday o + i unis of onsumpion omorrow (ha is how muh onsumpion omorrow you forego if you deide o enjoy one uni of onsumpion oday insead of saving i ill omorrow, we have he raio of pries on he righ hand side of (H2. Firms The firms fae a sequene of sai maximizaion problems, one for eah dae. The dae problem is max{ F( k, h w h r k } where F (k, h is a neolaial aggregae produion funion a dae. W e w i l l o n i n u e o u s e h e s a m e f u n ional form for produion ha was used in he

5 ( Solow Model in Chaper 3, namely, F k, h Ak h, The neeary and suffiien firs order ondiions for a maximum are (F w F k, h h( (F2 r F k, h k ( Given ha F is a neolaial produion funion, and herefore displays onsan reurns o sale, paymens o faors exhaus he produ and here are no dividends. Banks To esablish a key relaionship beween he renal rae of apial and he ineres rae, we add a banking seor o he model. The addiion of his seor is no eenial o he resuls; we would ge he same resuls if we had he household own he apial and ren i direly o firms. Insead, we shall have he banks own he apial and ren i o he firms. In rening a uni of apial o firms, a fraion δ of he apial wears ou, i.e. depreiaes, in he period. In addiion o buying apial and rening i o firms, banks also aep deposis from households and pay ineres i. A he end of period he households deposi whaever inome hey did no onsume, namely, w h d The banks ake hese deposis and buy apial o be rened o firms in he nex period, +. Thus, he apial rened o firms in period + is exaly how muh he household deposis in he bank a he end of he period, namely, k w h d In period + banks ren k + unis of apial o firms and earn renal inome equal o r+

6 k +. The undepreiaed apial (-δk + is hen sold by he banks. In effe, he undepreiaed par of apial ( - δ k is reversed engineered a no os bak ino he final good and beomes par of he supply. This will beome apparen in he marke learing ondiion desribed laer on in his seion. Banks mus pay households ineres on heir deposis. As he deposis of households made a he end of period are equal o w h d, and his is exaly he amoun apial bough by banks, k +, i follows ha he ineres and prinipal paymens o households in period + are ( i [ wh d ] ( i k Addiionally, sine he balanes in he household s banking aoun a he sar of period + are d ( i [ w h d ], i follows ha (B d ( i k We have only one more imporan resul o derive from he banking indusry. We aume ha he banking indusry is perfely ompeiive and he inermediaion ehnology has onsan reurns o sale. An impliaion of hese aumpions is ha in equilibrium banks neiher make a profi nor suffer a lo. They break even in equilibrium. Hene, paymens mus equal he reeips. Reall paymens in period + are (+i k + and reeips are r + k + +(-δk +. The zero profi ondiion hus requires ha 6

7 (B2 r i This jus says ha he renal rae on apial mus over boh ineres o and depreiaion o if he bank is o break even. A final aumpion is ha iniially balanes are d 0 = (r 0 +- δ k 0. The reason for his aumpion is ha i resuls in he ne worh of banks being zero and, herefore, zero dividends paid by banks. Goods Marke Clearing A ime he supply of goods is given by oupu in period and he undepreiaed apial (-δk sold by banks. The demand for goods is given b y household onsumpion and bank purhases of apial for omorrow s use. Hene, (M k k F( k, h, ( Definiion of Compeiive Equilibrium A Compeiive Equilibrium is a sequene of household hoies {, h, l, d 0 }, a sequene of bank variables { k, d 0 }, a sequene of firm hoies { k, h } 0, and pries { w, r, i} 0 ha saisfy (i (ii The family maximizes uiliy subje o is budge onsrains Banks maximize profis subje o heir ehnology onsrains 7

8 (iii (iv Firms maximize profis subje o heir ehnology onsrains Markes lear Marke learing means he following: ( deposis of households equal deposis held by banks; (2 labor sold by households equals labor purhased by firms; (3 goods marke learing (M and (4 apial servies sold by he banks equal apial servies bough by he firms a every dae. Solving for he Seady Sae Equilibrium In he previous seion we developed seven neeary and suffiien ondiions for he ompeiive equilibrium. Using he speifi funional forms for uiliy and he produion funion, hese 7 ondiions are (H (H2 h w i (B d ( i k (B2 r i (F k w ( A h (F2 r h A k (M Ak h ( k k 8

9 Reall ha in he Solow model, here was no savings/onsumpion deision by he households; households were aumed o always save a fraion s of heir inome. This made he model rivial o solve, no only he balaned growh pah equilibrium bu he nonbalaned growh pah equilibrium, i.e., he ransiional dynamis. In fa, we ould mehanially solve he model using exel: given K and N in period, K + was easily deermined by he apial sok law of moion equaion K + =(-δk +sy and N + was given by he aumpion ha populaion grew exogenously a rae n. Wih he family hosen he opimal amoun of savings, solving his model beomes a hallenge. Even hough he law of moion of he apial sok is he same, he fa ha he household is maximizing over an infinie horizon means ha here is no lear link beween K and K +. I is no generally poible o solve he equilibrium pah for any iniial K 0 and N 0 wih pen and penil. The exepion is he seady sae where if we sar wih he righ amoun of apial, he equilibrium pries and quaniies eah period never hange. The following seps allow us o solve for he seady sae equilibrium of he model. We sar wih (H2. In he seady sae, onsumpion is onsan and so i follows from (H2 ha (i i / Now ha we have solved for he seady sae real ineres rae use (B2 o solve for r (r r i / ( 9

10 Nex use (F2 and he soluion for r o solve for k /h. As r h ( A k i follows ha (k /h k h A ( /(. This allows us o solve for h as a funion of k, namely, h /( ( k. A Nex use (F2 wih our soluion for (k /h o solve for w, (w w A( A ( /( Nex use (H o solve for. ( w ( h. Nex, rewrie (M o be a funion of h using our expreion for ( and (k /h. This is w ( h h k h Ah k h ( h k h, whih simplifies o w ( h h k h Ah k h. We wan o solve for h. This is h A w ( k / h ( k / h w We an now subsiue he soluions for w and (k /h ino (h and have he soluion for 0

11 h in erms of he parameers. Insead of doing his, we shall go hrough a numerial soluion using he following parameer values A=, β=.96, δ=.0, α=.0 and θ =.30. Sep : Use (i and he value of β=.96 o arrive a i ( / Sep 2: Use i =.04 and δ=.0 ogeher wih (r o arrive a r Sep 3: Use r =.4, θ =.30, and A =.0 ogeher wih (k /h o arrive a k / h (.3/.4 / Sep 4: Use k /h =2.9 wih (w.3 o arrive a w.7( Sep 5. Use w =.96 and α=.0 wih ( o arrive a.96( h.96 Sep 6: h (2.9 ( Calibraion The numerial example above sared wih parameer values and hen solved for he equilibrium of he model. In Sep 4 of he alibraion exerise, we effeively do he reverse. We have he equilibrium values for he model eonomy; hese are jus hose from Sep 3 of he alibraion exerise obained from adjusing he NIPA aouns. Sep 4 hen finds he parameri values ha guaranee he equilibrium ouomes. Alhough we do no have a quesion in mind here, i is neverhele o go hrough he model alibraion. Convenienly, he adjusmens o he NIPA for he Neolaial growh model are he same as he ones we made in Chaper 3 for he Solow model. The NIPA

12 readjusmens we made in Chaper 3 implies x/y=.25, k/y=2.75, and rk/y=/3. Addiionally, given ha we have leisure in he model we need o add an observaion for hours work per period. For he US, he average hours worked per week by he working age populaion is roughly 25, auming ha people have 00 hours of non-sleep and non-personal are available. As a model period is one year, and our ime endowmen has been normalize o, he orresponding observaion for hours worked is h =.25. There are five parameers of he model: A, β, δ, α and θ. As we did in he Solow model, we are free o normalize he TFP parameer, A=. The aignmen of he ehnology parameer, θ, proeeds along he same logi as in he parameer aignmens in he Solow model: beause in he model θ=rk/y and rk/y in he daa is /3, he alibraed value of θ=/3. Nex, we find he value for δ. Here we use he apial sok law of moion ogeher wih observaions k/y=2.75 and x/y=.25. The apial sok law of moion in he seady sae redues o k y x y. Using he observaions for k/y and x/y, we an solve for δ=.25/2.75=.09. Nex we find he subjeive ime disoun faor, β. This we do by firs impuing he renal rae for apial. Sine /3=rk/y and k/y=2.75, r=.2. Nex we use (B2 o solve for he real ineres rae, i=r- δ=.2-.09=.03. From here we use (H2 o solve for he value of β using he resul ha in he seady sae is onsan. (H2 implies β=/(+i=/(+.03=.97. This leaves he leisure preferene parameer, α. Here we use ondiion (H ogeher wih he observaions ha /y=.75 and h=.25 and he model resul ha labor share, wh/y=2/3. In pariular, divide boh sides of (H by y, whih implies 2

13 h w y y. Nex we muliply he lef hand side by h/h so ha h h w h y y. Now h w h.75 2 we an solve for α, This omplees he aignmen h y / y of parameer values. III. Adding Governmen Poliy Here we use he growh model presened above bu appropriaely modified o inlude a governmen o sudy quesions relaed o publi finane. We inrodue hree differen ax raes ino he model: a ax on labor inome, τ h ; a ax rae on apial inome, τ k, and a ax rae on onsumpion, τ. Addiionally, we shall aume ha he governmen purhases some of he eonomy s goods, g, and makes lump-sum ransfers bak o he household, T. Reall ha he empirial ounerpar of governmen ransfers inludes any purhase by he governmen ha is a subsiue for privae onsumpion whereas governmen expendiures provide no uiliy. This means ha governmen expendiures on eduaion, healh, polie and he judiiary are inluded in governmen ransfers. Miliary expendiures, in onras, are inluded in g beause hey do no subsiue for privae onsumpion. The ax on apial is paid by he bank. Following he ax ode, we aume ha he ax 3

14 on apial inome is ne of depreiaion. Wih hese axes, he zero profi ondiion of he banking seor is (B2 rk k k( rk k ( k ( i k This simplifies o (B2G ( k ( rk i The household pays axes on labor inome and is onsumpion. This leads o he following modifiaion of he household budge onsrain. d ( i [ d ( w h ( T ] h The oher key hange o he model is he inlusion of he governmen budge onsrain. As we are going o deal wih seady sae omparisons, we impose ha he governmen run a balaned budge every period so ha ax reeips equal oulays. The governmen budge onsrain is hus (GBC g T ( r k w h k k h The inroduion of governmen ino he model hanges several of he equilibrium ondiions. Firs i hanges he household opimizaion ondiions (H and (H2. The new ondiions are (H h ( w h (H2 i 4

15 Beause banks pay he axes, he equaion (B does no need o be modified. This is (B d ( i k Addiionally, he goods marke learing ondiion (M needs o be modified o inlude he purhase by he governmen. This is (M Ak h ( g k k Definiion of a Compeiive Equilibrium. Given parameer values for (A,β,δ,α,θ and he sequene of governmen poliies {g,t,τ k,τ h,τ }, he ompeiive equilibrium onsi of household variables {, h, l, d 0 }, a sequene of bank variables { k, d 0 }, a sequene of firm hoies { k, h } 0, and pries { w, r, i} 0 ha saisfy (i (ii (iii (iv (v The family maximizes uiliy subje o is budge onsrains Banks maximize profis subje o heir ehnology onsrains Firms maximize profis subje o heir ehnology onsrains The Governmen budge onsrain is saisfied every period. Markes lear To illusrae, we solve for he ompeiive equilibrium wih a pariular se or parameer values and poliy parameers. The parameers are A=, β=.96, δ=.0, α=.0 and θ =.30. The governmen poliy is given by g =.0F(k,h, τ k =/3, τ h =/3 and τ =/5. We do no speify he ransfers beause given values for g, τ k, τ h and τ, is value is deermined by he governmen budge onsrain (GBC. 5

16 We again sar wih (H2-. In he seady sae, onsumpion is onsan and so i follows from(h2 ha (i i /. 04 Now ha we have solved for he seady sae real ineres rae use (B2G o solve for r (r i.04 r /3 k Nex use (F2 and he soluion for r o solve for k /h. As h r A k i follows ha /( /.7 (k /h k A h r.6 Nex use (F2 wih our soluion for (k /h o solve for w, k h (w w A( Nex use (H o solve for. ( w ( h (. ( h.3 Nex, rewrie (M o be a funion of h using our expreion for ( and (k /h. This is w ( h( h ( h k h.9ah k h ( h k h, whih simplifies o w ( h ( h ( h k h.9ah k h. We wan o solve for h. 6

17 This is h ( [ ( k / h ( h w.9a( k / h ] w (2/3.92 ( (.2[.(2.45.9(2.45 h ] (2/ IV. Calibraion In his seion we use he growh model o evaluae ax poliy. We shall go hrough he five seps of he alibraion in deail and in urn. Sep : Pose a Quesion: The quesion we shall addre involves a hange in he urren he ax sysem, wih a high reliane on apial and labor inome axes wih one ha axes only onsumpion. The ax hange we envision mus be revenue neural. I mus ensure ha he governmen is able o generae he same amoun of ax revenues in order o oninue o buy he same amoun of goods, g, and provide he same amoun of lump sum ransfers. Speifially, he quesion we ask is: Wha is he effe of replaing he urren ax sysem based on inome wih one ha axes onsumpion only? Sep 2: Choie of measuring devie: This is jus he model of Seion 4. Sep 3. Define Consisen Measures: Inroduing he governmen poliy requires ha we make several adjusmens o he NIPA in order for he daa o onform o our model eonomy. The firs adjusmen is ha onsumpion in he model is no onsumpion in he NIPA. Le C be NIPA onsumpion expendiures and model onsumpion. Beause of 7

18 he onsumpion ax, heir relaion is C (. In effe, he model ounerpar is NIPA C le onsumpion axes. Wha are onsumpion axes in aualiy? Consumpion axes are sales axes, exise axes and value added axes, even hough in he laer ase, par of he value added ax is olleed on he busine ha generaed he value added. In he NIPA, hese onsumpion axes are inluded in he inome aegory ha goes under he heading of Taxes on produion and impors. The Taxes on Produion and Impor aegory also inludes propery axes. Propery axes will affe he prie he onsumer pays, and so suh axes are likewise onsidered as onsumpion axes. Of ourse, hese axes apply o a busine when hey purhase an invesmen good, so no all he axes in he Taxes on Produion (Tax P aegory fall on onsumpion. However, sine we do no have a ax on invesmen goods, we will simply aume ha 75% of he axes on produion on ax aegory apply o onsumpion. Wih his rule, τ =.75 x Tax P. In he NIPA (Table B ERP, Taxes on Produion are 7.0% of GDP over he period. Using 75% rule, onsumpion axes average 5% of GDP over he period. There are wo imporan addiional adjusmens ha we mus make on aoun of reduing onsumpion by axes paid on onsumpion. Sine expendiures equal oupu, and we have redued onsumpion by he onsumpion ax, hen we mus redue NIPA GDP by he onsumpion ax o be omparable o model oupu. This means ha GDP-.75 x Tax P equals model y. 8

19 Addiionally sine inome equals oupu we mus subra his omponen from he inome side. Hene, we mus subra ou his amoun from he inome side. Reall, ha in alibraing he Solow model and he Neolaial growh model, he Taxes on Produion aegory is an ambiguous aegory in ha i is no lear wheher o aribue i o labor inome or apial inome. Given ha we are aribuing 75% of his aegory o he onsumpion ax, i means we only have o aign 25% of his aegory o labor inome and apial inome. We will mainain he aumpion ha he ambiguous aegory is spli equally. Wih his aignmen rule, we sill mainain he raios of his 25% being spli beween labor inome and apial inome. Making hese adjusmens, we sill arrive a apial share of inome equal o /3. Anoher key adjusmen involves governmen expendiures. In he model, governmen expendiures provide no value o households or firms. In he NIPA, hese are bes approximaed by he defense expendiures, G mil. Those governmen onsumpion expendiures ha are of value o onsumers suh as eduaion, healh, and judiial orrespond o he model lump-sum ransfers, T. In he 202 Eonomi Repor of he Presiden, G mil /GDP averaged 6% over he period (Table 2. Governmen onsumpion, no inluding defense expendiures 2% of GDP and Governmen Invesmen is 3%. This governmen onsumpion is par of lump-sum ransfers in he model of GDP over his period. 9

20 Wih hese adjusmens, he empirial ounerpar of model,, is =C+G -.75 Tax P and y=gdp -.75Tax P =GDP-.05GDP=.95GDP sine.75tax P =. 05GDP Using he daa in he 202 ERP and averaging over he period, we find ha /y=.75, g/y=.05 and x/y=.20. Also, sine K/GDP in he US is 2.75 and y=.95gdp, we arrive a k/y=2.90 Sep 4: Aign Parameers. The preferene and ehnology parameers of he model are β, θ, α, δ and A. In addiion, here are he poliy parameers, g, τ, τ h, τ k and Tr. For he poliy parameers, we use he observaion ha g/y in he NIPA (afer adjusmens equals.05. For he axes on produions, we use he observaions ha /y=.75 as well as ha τ /GDP=.05 as well as y=.95gdp. This yields τ =.05/(.95x.75=.07. For he labor inome ax and he apial inome ax, we appeal o esimaes in he lieraure. From he OECD Taxing Wages 204, he marginal ax rae on labor inome for he Unied Saes is τ h =.3 and from he Tax Foundaion he ax rae on apial inome τ k =.29. The final poliy parameer is he lump sum ransfers o housheholds. We do no speify a value for his poliy parameer beause we require ha he governmen s budge be balaned eah period. Given he oher poliy parameers, and he implied ax revenues hey generae, he lump-sum axes are equal o he value ha guaranees ha (GBC is saisfied. Aually, he adjusmens imply ha g/y=.063, bu sine we like round numbers we use

21 Turning o he ehnology and preferene parameers: Saring wih he TFP parameer, we oninue o follow he onvenion of seing A=. For he apial share parameer, i is se o mah apial share of inome in he NIPA (afer our adjusmens. Thus, θ=/3. The apial sok law of moion in he seady sae redues o k y x y. Using he observaions for k/y and x/y, we an solve for δ=.20/2.75=.07. Nex we find he subjeive ime disoun faor, β. This we do by firs impuing he renal rae for apial. Sine /3=rk/y and k/y=2.90 i implies r=.. Nex we use (B2G using τ k =.29 o solve for he real ineres rae, i=(-τ k (r- δ=.7(.-.07=.3. From here we use (H2 o solve for he value of β using he resul ha in he seady sae is onsan. (H2 implies β=/(+i=/(+.03=.97. This leaves he leisure preferene parameer, α. Here we use ondiion (H ogeher wih he observaions ha /y=.75 and h=.25 and he model resul ha labor share, wh/y=2/3. In pariular, divide boh sides of (H by y, whih implies h w y ( ( y h. Nex we muliply he lef hand side by h/h so ha h h wh y ( ( y h. Now we an solve for α, h h wh y ( h ( / y This omplees he aignmen of parameer values. 2

22 Sep 5: Tes Theory/Poliy Evaluaion: To evaluae alernaive poliies, we need o solve for he seady sae equilibrium using he alibraed preferene and ehnology parameers deermined in Sep 4. We jus use he new poliy parameers and repea he seps ha are oulined on pages 5-6. One iue in evaluaing poliy is ha we will wan he new poliy o be revenue neural, namely, o generae he same amoun of ax revenue as he urren US ax poliy. Addiionally, we will require ha he governmen spending be he same absolue amoun as under he urren poliy. The idea here is ha under he alernaive ax poliy, he governmen sill needs o provide he same level of publi onsumpion. Alhough here is nohing diffiul wih numerially solving for he seady sae given he ax raes, and governmen spending, i is no rivial o find he new ax raes ha are revenue neural. This is beause for any se of ax raes, here is a differen seady sae apial sok, labor hours, and pries. For his reason, i is more onvenien o auomae he soluion and le he ompuer find he ax raes ha are revenue neural. The ourse webpage onains an ineraive pyhon ode where you an do jus his. The ode is ineraive in wo pars. Firs, i asks you o inpu he observaions o be used in Sep 4, inluding he ax raes. Then i asks you o inpu he new ax raes on apial inome and labor inome. The program finds he ax on onsumpion ha leaves ax revenues unhanged for he alibraed poliy in Sep 4. Imporanly, i akes he spending from Sep 4 and aumes ha level in ompuing he new 22

23 seady sae equilibrium. IV Conlusion In his haper, we have presened he Neolaial Growh model. This is an example of an eonomy where he onsumer lives forever. No one lives forever, a leas no ye. The idea of an infiniely lived onsumer is really ha of a dynasy- where he urren generaion ares abou is hildrens uiliy and akes ha ino aoun in is deisions. In doing so, i will leave beque o is hildren. We have used he model o perform a number of omparaive sai exerises where we ompare seady saes under alernaive governmen poliies. This is an appliaion of he model o he area of publi finane. In he nex hapers we will apply he model o he sudy of he busine-yle. 23