Economic Growth and the Evolution of Preferences under Uncertainty. Stuart McDonald, Rodney Beard, and John Foster

Transcription

1 Economic Growh and he Evoluion of Preferences under Uncerainy Suar McDonald, Rodney Beard, and John Foser School of Economics Universiy of Queensland Qld 407 Ausralia January 003 Absrac: In his paper we show how i is possible o consruc a heory of preference evoluion from a model of economic growh under uncerainy. Our saring poin is he Solow-Meron model of growh under uncerainy. We use sochasic dynamic equaions ha we derive from his growh model o consruc a model of preference evoluion from a von Neumann-Morgensern expeced uiliy model of consumpion for a represenaive agen. We show ha he dynamics ha describe he change in he represenaive agen s preferences, lead o a canonical represenaion of preference, which is given by he Fokker-Planck equaion for consumpion. We hen provide a characerizaion of possible boundary condiions for he Fokker-Planck equaion. We hen show how i possible o use some of hese boundary condiions o derive a seady sae disribuion for consumpion from his evoluion equaion for preferences. This seady sae disribuion of consumpion can hen be used o solve he Ramsey problem of opimal savings under uncerainy, giving an equivalen resul o ha derived in Meron s paper. Hence, he Solow-Meron Neo-Classical growh model is a special case of our model. Keywords : Evoluionary models of economic growh, evoluion of preferences, economic growh under uncerainy, Solow-Meron growh model. JEL Classificaion: O4, E.

2 . Inroducion In an evolving economy new commodiies are creaed wih he passage of ime. Preferences for novel commodiies will no be defined, unless economic agens have previously imagined hese commodiies. These commodiies will no appear from nowhere. They mus be produced and hey mus be produced by labour, which will in urn evenually consume some of hese novel commodiies. Consequenly he creaion of new commodiies will depend upon and lead o he evoluion of preferences. This paper begins by re-examining he sochasic exension of he Solow s (956) exogenous growh model (Bourguigon 974, Meron 975, Chiang and Malliaris 987). The usual pah aken by he deerminisic exogenous growh lieraure is o characerise which values of he growh equaion s parameers lead o sable growh pahs. However, when uncerainy is incorporaed ino he exogenous growh model, he possible growh pahs are realizaions given by a sequence of random variables. Each realizaion in he growh pah is deermined by a disribuion ha is condiional on he prior hisory of he growh pah. This condiional disribuion is given by he Fokker-Planck equaion for he capial-labour raio. For example in he sochasic exension of he Solow s (956) exogenous growh model (Bourguigon 974, Meron 975, Chiang and Malliaris 987), he soluion of Fokker-Planck equaion is used o derive he seady sae condiion for economic growh, i.e. he disribuion on which he probabiliies converge asympoically. The sandard approach is o derive a sochasic differenial equaion describing he dynamics of capial accumulaion in erms of he capial-labour raio. The Fokker- Planck equaion ha provides equivalen represenaion of his sochasic differenial equaion can hen be used o derive asympoic disribuions for income and consumpion. In his paper, we begin wih a one-secor sochasic exogenous growh model as formulaed by Meron (975). From his model, we derive a sochasic differenial equaion describing he dynamics of capial accumulaion in erms of he capiallabour raio as a funcion of savings and sochasic changes in labour produciviy. This changing naure of labour is capured as a branching process. The inen is o capure wo effecs: a human capial effec and composiion of labour effec. I is argued ha labour is embodied wih human capial. In his paper cerain classes of labour are hardwired wih paricular skills, new commodiies can only be generaed if he composiion of labour changes in order o imagine novel producs. Innovaions o he ype of labour employed in producion hen lead o a sochasic represenaion of producion and consumpion and an inerpreaion of preferences as a probabiliy of consuming a paricular consumpion bundle.

3 From his sochasic differenial equaion describing per-capia change in capial, we hen derive a sochasic differenial equaion for consumpion. The equivalen represenaion is hen he Fokker-Planck equaion describing he probabilisic change in he consumpion pah. One inerpreaion of his Fokker-Planck equaion is ha in a dynamic, ever-changing world he probabiliy densiy of consumpion, capures he likelihood of consuming a paricular consumpion bundle in he nex ime period. The change in he likelihood of his consumpion is in urn driven by changes income via changes in he composiion of producion. However his explanaion does no explain he relaionship beween preference evoluion and he likelihood following a paricular consumpion pah. Our saring poin for his is he derivaion of a von Neumann-Morgensern expeced uiliy model of consumpion for he represenaive consume. Our main resul is a represenaion heorem showing ha he parial differenial equaion describing he change in uiliy wih respec o ime is equal o he Fokker-Planck equaion for consumpion. Hence he Fokker-Planck equaion for consumpion provides a canonical represenaion for he evoluion of preferences. In his represenaion, curren consumpion behaviour is condiioned on previous purchases. Our inerpreaion of preference evoluion in his represenaion ascribes he change in curren consumpion behaviour as being due o changes in likelihood o consume, wih respec o changes in key underlying variables. The change in preferences over ime will herefore describe he likely pah of consumpion. This pah of consumpion should be subjec o boundary condiions on he feasible consumpion se. These boundary condiions will be deermined by physical, behavioural, economic and insiuional consrains. In his paper we provide a characerizaion of he Fokker-Planck equaion subjec o differen boundary condiions on he feasible consumpion se. We explain how hese boundary condiions can be used o place consumpion consrains on he consumer. In doing his he possible impac on economic growh of behavioural phenomenon like fashion cycles or he impac of resource consrains on consumpion and growh. In his paper we use he boundary condiions ha are proposed by Meron (975) o secure a unique soluion for he saionary disribuion for he Fokker-Planck equaion of he capial-labour raio. As is shown by Meron (975) his asympoic disribuion on he capial-labour raio can be used o provide a soluion o he sochasic Ramsey problem. Moreover, because he soluion o he Fokker-Planck equaion is a seady sae soluion, i does no depend on ime. As a consequence, he dynamic opimal savings problem can be solved using saic opimisaion echniques. We show ha he seady sae soluion of he Fokker-Planck equaion, when subsiued ino he infinie horizon Ramsey model, gives an idenical soluion o ha provided by Meron (975). This leads o our conjecure ha he consumpion/savings ascribed by he Ramsey (98) model under he Solow-Meron exogenous growh

4 dynamics, is a special case of he consumpion behaviour ha can be described under our heory of growh wih preference evoluion. Solow-Meron Model of Growh Under Uncerainy We begin by deriving a one-secor sochasic Neo-Classical growh model. Producion Y = F K, N a ime [ 0, ) funcion of capial K( ) and labour produciviy defined by he producion funcion F(.,.) C R + [ 0, ) (975) and assume he following: i) F, F > 0, F, F < 0, F >0, in he one-secor economy is modelled as a K N KK NN KN L. Producion echnology is ii) F λk, λn = λf K, N, λ > 0.. We follow Meron Based on Meron (975), we assume ha he labour force L consiss of a finie number of individual workers, i=,..., N. A branching process is used o model labour produciviy of each worker L + h = nh+ ση h ; + νε h ;, i =,..., N, 0, where n, σ and ( ] ν are consans and η ( h ; ) and ( h ; ) i i i assume ha E ( η) E ( ν ) 0 and ( h ; ) i i = = and E ( η ) E ( νi ) h E ( ηνi) = E( νν i j) = 0, i ji, =,..., m. η is covariance saionary ( ) ε are random variables. We = = and ha E η h ; η + kh; h = 0, k =,,... i The sochasic differenial equaion for labour produciviy can be derived as follows N ( + ) = i( + ) = + ση( ; ) + νε i i ;, ( 0, ]. N h N L h N nh h h i= i= The condiional mean and variance for his equaion is given as follows: E N( + h) N = N nh and m var N + h N = N σ + N νi N h, ( 0, ]. We assume ha he N νi N O. i= i= ν i are bounded and approximaely he same size so ha N We can see ha under hese assumpions labour produciviy can be approximaed by he sochasic differenial equaion (SDE), = + σ dn N nd dw

5 where n is he deerminisic rend componen, σ governs he rae of diffusion and he dw N 0, d, 0,. noise erm W ( ) is a Wiener process [ ) We noe ha under his framework, a number of alernaive disribuions could be used o model he branching process governing individual worker produciviy. Two possible examples ha could be used are a Poisson birh-deah-migraion process and an Ehrenberg urn process. The Poisson birh-deah-migraion process would capure he churning effecs due o job creaion and desrucion and worker migraion. The urn process would model pah dependence in he hiring process of firms. Boh of hese processes could be used o provide a new ype of endogenous growh heory. Wih respec o he assumpions we have employed in his paper, our inerpreaion of he behaviour of he branching process is as follows. This branching process for individual labour produciviy may be inerpreed as follows. Labour is embodied wih human capial. If labour is hardwired o imagine paricular commodiies, new commodiies can only be generaed if labour somehow changes in order o imagine novel producs. Hence labour produciviy will vary over ime for each individual and we can associae each individual wih a paricular skill se and abiliy o imagine echnological producion possibiliies We model he dynamics of capial accumulaion as follows [ ) dk = I + δ K d, 0, where I d is he change in he sock of invesmen and depreciaion on capial sock. δ K d is he As he producion funcion F is linear homogeneous, per capia oupu can be defined as y = F( K, N ) = f ( k ), [ 0, ) N where k = K N( ) is he capial labour raio. The sochasic dynamics of k, [ 0, ) can be derived using Io s lemma o yield he following SDE ( ) ( δ ) σ σ where ( 0,] ( ), [ 0, ) dk = sf k n k d+ k dw s is he marginal propensiy o save. This is he Solow-Meron growh equaion. Here we le and ( ; ) ( ( ) ) µ k s = sf k δ n σ k σ ( k s) = σk s ( ] [ ) ;, 0,, 0,. Per capia consumpion is defined via he Keynesian demand ideniy as follows, ( ) [ ) c = ck ; s = s f k, 0,.

6 { } When aken ogeher he vecor process (, ) ; [ 0, ) growh pah for he economy. Combined wih he iniial condiions ( k0, c 0) process gives defines he admissible growh pah subjec o Pr{ k( 0) k 0 } 0 3 A Canonical Represenaion for he Evoluion of Preferences k c defines he feasible = >., he vecor In his secion we derive a canonical represenaion for he evoluion of preferences. This resul gives equivalence beween he ime derivaive of expeced uiliy and he Fokker-Planck equaion of he underlying sochasic process. Hence he Fokker- Planck equaion for consumpion lierally gives he evoluion of preferences in his sochasic economy. We now sae our resul in he form of a represenaion heorem for preference evoluion: Theorem 3..(Canonical Represenaion Theorem) The von Neumann-Morgensern expeced uiliy funcion for he represenaive agen is defined as follows: U( c ) = E uc ( ) = u( x) p( c = xdx ), where u C ( C ) { } ( ) Pr c = x = pc = x, x C wih C as he feasible consumpion se. If. + R, hen U ( c) = u( x) p( c = x dx is equal o he Fokker-Planck equaion of he sochasic process { ; [ 0, )} c : p( c = z c ) = α k ; s µ k ; s + β k ; s σ k ; s p c = z c ( ) ( ) ( ) s s ( α( k s) σ( k s) ) p ( c = z ; ;. z Proof: We begin by applying Io s lemma o he equaion for per capia consumpion. We express his SDE for per capia consumpion as follows: dc = α ( k ; s) µ ( k ; s) + β( k ; s) σ ( k ; s) d ( ) σ The coefficiens in his SDE are respecively defined as and ( ) [ ) + α k ; s k ; sdw, 0,. (( ) ) α k, θ = s k, θ f k f k sk ( ( )) ( ) β k, θ = s k, θ f k f k s + f k s. kk kk

7 We assume he exisence of a represenaive agen. We define an equaion for he represenaive agen s von Neumann-Morgensern expeced uiliy funcion as follows: U( c ) = E uc ( ) = u( x) p( c = xdx ), where { } ( ) Pr c = x = pc = x, x C wih C as he feasible consumpion se. The differenial equaion describing he evoluion of preferences can now be derived as follows: u( x) p ( c = x dx= lim u( x) { pc ( + = x pc ( = x } dx 0 = lim u( x) p ( c+ = x c = z) p( c = z dzdx 0 z C u( z) p( c = z dz, z C pc = x c = Pr c = x c, x C, is he condiion probabiliy of consuming where { } c s s = x a ime given ha c s was consumed a ime s, s. We now divide he inegral over x ino wo regions, x z ε and x z < ε We assume ha he uiliy funcion u (.) is wice coninuously differeniable wih respec o c. Hence for he region x z < ε his allows us o consruc a second-order Taylor expansion of u (.) wih respec o z u( x) = u( z) + u ( z) ( x z) + u ( z) ( x z) + x z Rxz (, ), R x z denoes he higher order remainder erm. We noe ha where (, ) x z 0. Rxz, 0 as Subsiuing his ino he preference evoluion equaion, we arrive a he following expanded version of his equaion: u( x) p ( c = x dx= lim ( u ( z)( x z) + u ( z)( x z) ) 0 x z < ε pc = x c = z p c = z c dzdx x z< ε x z< ε x z ε ( + ) ( s) (, ) ( + ) ( s) ( ) ( ) ( + ) ( s) + x z R x z p c = x c = z p c = z c dzdx + u z p c = x c = z p c = z c dzdx + s + u x p c = x c = z p c = z c dzdx u z p c = z cs dz z C

8 We noe ha by assuming uniform convergence we can ake he limi inside he inegral. As a consequence, we ge he following lim u ( z) ( x z) p( c+ = x c = z) p ( c = z dzdx 0 x z< ε = u ( z) α ( k ; s) µ ( k ; s) + β( k ; s) σ ( k ; s) p( c = z dz and lim u ( z) ( x z) pc ( + = x c = z) p ( c = z dzdx 0 x z< ε = u ( z) ( α( k ; s) σ( k ; s) ) pc ( = z dz. We also noe ha he remainder erm on he hird line of he consumpion evoluion equaion vanishes as ε 0 because lim x z R( x, z) p( c+ = x c = z) p( c = z dzdx 0 x z < ε lim x z pc ( + = x c = zdzdx ) max R( x, z) β( k, ) + O( ε) max Rxz, 0 x z< ε x z < ε x z < ε The remaining hree lines of he expanded preference evoluion equaion can now be given as u( z) dz w( x z) p( x w( x z) p( z dzdx. z C Upon inegraing by pars, he preference evoluion equaion hen reduces o u( x) p ( c ) ( ; ) ( ; ) ( ; ) ; x C = x cs dx= α k s µ k s + β k s σ k s u z ( α( k ; s) σ( k ; s) ) u ( z) pc ( = z dz ( s) ( s) + u z dz w x z p x c w x z p z c dzdx. z C We hen differeniae he preference evoluion equaion wih respec o z and cancel he uiliy funcions. Following he erminology of Gardiner (985), we hen arrive a a differeniable Chapman-Kolmogorov equaion

9 p( c = z c ) = α k ; s µ k ; s + β k ; s σ k ; s p c = z c x C ( ) ( ) ( ) s s ( α( k ; s) σ( k ; s) ) pc ( z + = ( ( s) ( s) ) + w z x p x c w x z p z c dx. The hird line of he differeniable Chapman-Kolmogorov equaion is a Maringale difference equaion and as he noise w is Brownian, by definiion his will be equal o zero ( ( s) ( s) ) w z x p x c w x z p z c dx = 0. Hence we arrive a he Fokker-Planck equaion for consumpion p( c = z c ) = α k ; s µ k ; s + β k ; s σ k ; s p c = z c? ( ) ( ) ( ) s s ( α( k s) σ( k s) ) p ( c = z ; ;. 4. Boundary and Barrier Condiions on he Feasible Consumpion Se We recall ha for he definiion of he von Neumann-Morgensern uiliy funcion we defined a feasible consumpion se x C R +. This se defines a region of sae space ha is deermined by culural, legal, economic physical and behavioural consrains. In oher words, You can only consume so much. Given ha his is he imporan quesion o ask is How much do we consume? The naure of he barriers on he consumpion se will deermine his. Hence he barriers on he consumpion se will deermine seady sae preferences. The Fokker-Planck equaion for consumpion is expressed as p( c = z c ) = α k ; s µ k ; s + β k ; s σ k ; s p c = z c ( ) ( ) ( ) s s ( α( k s) σ( k s) ) p ( c = z ; ;. We noe ha Fokker-Planck equaion can also be wrien as

10 p( c = z + V ( z, ) = 0 where V( z, ) = α ( k ; s) µ ( k ; s) + β( k ; s) σ k ; s pc = z c ( s) ( α( k s) σ( k s) ) pc ( z ; ; =. is a poenial difference equaion. The inegral form of he Fokker-Planck equaion is equivalen o a surface inegral on he feasible consumpion se C (see Gardiner 985, p.9) p( R cs ) = nv ( zds, ), x S where S is he boundary se of he se R C and n S. Hence anoher way of undersanding he Fokker-Planck equaion is as a ne flow of probabiliy across he consumpion se capuring he evoluion of preferences. Some examples of boundary condiions ha would be of ineres o economics are now given. Example 4.. (Reflecing Barrier) Consider he siuaion where z canno leave a region R C. Le S define he barrier. Thus we require ha nvz (, ) = 0, z Sn, S. This would he case where here was a budge consrain resricing iner-emporal consumpion. In his case, he orhogonal vecor n gives he vecor of prices. A similar condiion o his underlies he selecion of prices in saic consumer choice heory. Alernaively, a reflecing barrier could be used o find a minimum susainable consumpion level. Example 4.. (Absorbing Barrier) In his siuaion, when z reaches he barrier S, i is removed from he consumpion se, i.e. pz (, ) = 0, z S. This would be he when he bundle of goods is ouside he consumpion space. Example 4.3. (Periodic Barrier) This is he case of periodic consumpion cycles. A ypical example is seasonal flucuaions associaed wih seasonal goods. A subler example is fad goods wih long period cycles, such as he yo-yo. These boundary condiions are expressed as follows: i).lim px, = lim px, where [, ] C + x b x a = V ( x) ii).lim V x, lim,, + x b x a ab define he inerval on which he boundary is defined.

11 5. Evoluion of Preferences and he Ramsey Model of Opimal Savings As in he previous secion, we define he expeced uiliy derived from consumpion a ime by U( c ) = E u( c ) = u( x) p( c = xdx ), where u ( c) 0 and u ( c ) < 0 for all c C. The sochasic Ramsey problem (Ramsey 99, Meron 975) can be saed as follows: T J k,, T max E 0 U( c( k( τ) ; s) ) dτ = c C 0, subjec o a budge consrain where k( ) > 0 for all [ 0, T) k( 0) = k0 > 0 and 0 ( (( δ ) σ ) ) σ dk = f k n k c d+ k dw and he iniial and boundary condiions: k T wih probabiliy respecively., Following Meron (975) we solve his problem using he Bellman s equaion from dynamic programming. The derivaion of he Bellman funcion for his opimisaion problem is given as follows: + T JkT [,, ] = max E U( c( k( τ), s) ) dτ max E U( c( k( τ) ; s) ) dτ c C + + c C + = max E U ( c ( k, s) ) + Jk [, + T, ]. c C { } A his poin, we would like o focus he reader s aenion on he second erm in he las line of he Bellman equaion. We can Taylor expand his equaion wih respec o as follows: J[ k, + T, ] = JkT [,, ] + Jk k+ J + ( Jkk ( k) + J ( ) ) + Jk k + O( ), where = (, θ) ( δ ) σ + σ and W N( 0, ). Upon subsiuion of his erm ino [,, ] k sk f k n k k W JkT, we arrive a he following equaion δ JkT,, = max { E U ( c ( k, s) ) e + JkT [,, ] + Jk k [ ] c C ( ) + J + Jkk k + J Jk k + O. Upon rearranging his equaion, we arrive a he represenaion of he Bellman opimaliy condiion δ 0 = max E U ( c( k, s) ) e Jc c J ( Jcc ( c) J ) Jc c O. c C

12 We now ake he limi of he Bellman opimaliy condiion as 0 : δ 0= lim max E U ( c ( k, s) ) e + Jk k + J + ( Jkk ( k) + J( ) ) + Jk + k O( ) 0 c C δ ( ( )) = limmax E U c k, s e + J + Jk E k+ k 0 c C + Jkk E k ( ) k J JkE k ( ) k O δ max E U ( c ( k, s) ) e J Jk lim E k k c C = + Jkk lim E k ( ) k. + 0 Upon rearranging he las wo erms in his equaion HJB equaion, we arrive a he following opimaliy condiion J = max U ( c( k, s) ) + J kµ ( k, ) + σ ( k, ) Jkk, c C where µ ( k, ) = lim E k( ) k sk (, s) f ( k ) (( δ n) σ ) k 0 + = and σ ( k, ) = lim E ( ) k k σk 0 + =. To obain he dynamic programming soluion o he sochasic Ramsey problem, we he firs order necessary condiion is give by where k and T. ( ) = and s sk (, T ) U c du dc U ck, s J k = 0, = is he opimal savings rae as a funcion of In order o solve for s, we subsiue he firs order condiion back ino he HJB equaion, o obain he following PDE J = U ( c( k, s )) + µ ( k, U ) ( c( k, s )) + σ ( k, ) Jkk. We hen solve his PDE o obain a soluion for k which is hen subsiued back ino he firs order condiion o solve for s. Following Meron (975), we now examine he limiing case of he Ramsey problem where T. As he sochasic process k( ) is ime homogenous and U (.) is no ime dependen, we have ha ( ( )) J = U s kt, f k T.

13 Mirrlees (965, 973) has shown ha for an opimal policy o exiss under. δ n σ > 0. If his is he uncerainy, f mus saisfy he Inada condiions and case hen ( ) = lim s kt, s k T and, as Meron (975) as shown, here exiss an associaed seady sae disribuion for k. This implies ha where B is he bliss-poin. ( ) limj = U s k f k = B, T Subsiuing his ino he HJB equaion, we have ha J mus saisfy he following ODE as T 0 = U ( s ( k) f ( k) ) B+ J µ ( k, s ) + σ k J, where (, ) = ( ( ) ) µ ks s f k δ n σ k and he primes denoe derivaives wih respec o given by k. The firs order condiion is ( ) ( s J = U s f k s ) f ( k) + f ( k) k. Subsiuing for J and J in he above ODE, we now arrive a he following ODE ds 0 = ( ) ( ) σ k fu + fu σ kuf s + σ kuf U βk+ U B, dk where he coefficien β = n δ σ. We can solve his ODE o ge s. We noe ha under cerainy (i.e. when σ = 0 ) his ODE reduces o ( ) B s f n δ k = U U which is he Ramsey Rule for opimal savings. 6. Conclusion In his paper, we se ou o show how changes in consumer behaviour can impac on economic growh. Our quesion differs from he sandard approach ha has been used by growh heory is embedded in a radiion of supply-side change impacing on demand. In boh he Neo-Classical and endogenous growh lieraures, preferences over consumables are aken o be saionary over ime. In conras, we begin by providing a heorem for preference evoluion in erms of he Fokker-Planck equaion for consumpion. We also showed ha by changing he boundary condiions on he Fokker-Planck equaion we could accoun for differen demand behaviour like he impac of fashion cycles on preference formaion.

14 Furhermore, we showed ha by employing he boundary condiions used by Meron (975) we are able o derive he seady-sae disribuion for consumpion ha he used o predic he oucome he infinie horizon Ramsey savings model. Hence our model of preference evoluion and economic growh encompasses he Meron s model Neo- Classical growh under uncerainy. In addiion, by changing he ype of noise exhibied by he labour dynamics, we capure he ypes of growh paerns due o oher ypes of producion/facor relaionships. Hence, we can drive he growh dynamics for oher models aribuable o various paerns of job creaion and desrucion or pah dependence in hiring. This could lead o a new endogenous growh heory based on he microsrucure of he economy via he Theory of he Firm. References Chang, F.R. (988) The Inverse Opimaliy Problem: A Dynamic Programming Approach. Economerica 56, Chang, F.R. and Malliaris, A.G. (987) Asympoic Growh under Uncerainy: Exisence and Uniqueness. Review of Economic Sudies 54, Gardiner, C.W. (985) Handbook of Sochasic Mehods for Physics, Chemisry, and he Naural Sciences ( nd Ed.). Springer, Berlin. Malliaris, A.G. and Brock, W.A. (98) Sochasic Mehods in Economics and Finance. Norh-Holland, Amserdam Meron, R.C. (975) An Asympoic Theory of Growh under Uncerainy. Review of Economic Sudies 4, Ramsey, F.P. (98) A Mahemaical Theory of Savings. Economic Journal 37, Solow, R.M. (956) A Conribuion o he Theory of Economic Growh. Quarerly Journal of Economics 70,