The consumpion-based deerminans of he erm srucure of discoun raes: Corrigendum Chrisian Gollier Toulouse School of Economics March 0 In Gollier (007), I examine he effec of serially correlaed growh raes on he efficien long discoun rae. I develop he idea ha posiive serial correlaion ends o magnify he long erm risk. This ends o induce he pruden represenaive agen o accumulae more precauionary savings. This raises he efficien discoun rae for long horizons. Various illusraions of his inuiion were explored in he paper, wih dynamic growh processes such as mean-reversion, Markov, and random walks wih parameric uncerainy. The discoun rae r associaed o ime horizon is wrien as r Eu'( c ) ln, () u'( c ) 0 where is he rae of impaience, c is consumpion a dae, and u is he increasing and concave uiliy funcion of he represenaive agen. Le x i denoe he coninuous-ime growh rae of consumpion beween dae i- and i. This implies ha we can rewrie he above equaion for = as follows: wih r ln Eh ( x, x ), ( 0 (,. u'( c0 ) hx x u' c exp( x x ) (3) Obviously, any change in he disribuion of ( x, x ) ha raises he expecaion of hx (, x reduces he long discoun rae r. In he general expeced uiliy model, he coefficien of correlaion beween wo random variables as x and x is usually insufficien o characerize he I hank Louis Eeckhoud and Chrisoph Heinzel for helpful commens.
role of he saisical relaionship on an expecaion as Eh( x, x. The full join disribuion funcion is generally required o deermine he forward discoun rae. In Gollier (007), I defined he noion of posiive firs-degree sochasic dependence o solve his problem. To do his, consider a disribuion funcion F for he pair of random variables ( x, x, wih F (, ) P[ xx ]. Le also Fi () denoe he marginal disribuion of x i, i=,. Definiion (Gollier (007): There is posiive firs-degree sochasic dependence beween x and x if Px x is decreasing in for all. In Lemma of Gollier (007), I misakenly claimed ha his condiion was necessary and sufficien o raise Eh( x, x for all supermodular funcions h. I is in fac sufficien bu no necessary, as I show now. In fac, he necessary and sufficien condiion for a change in he join disribuion of ( x, x ) wih fixed marginals o raise he expecaion of hx (, x for all supermodular funcions h already exiss in he lieraure. This is he concep of concordance inroduced by Tchen (980) and Epsein and Tanny (980). Definiion (Tchen (980)): Consider wo pairs of random variables ( x, x ) and ( x, x ) respecively wih disribuion funcions F and F wih he same marginal. We say ha ( x, x is more concordan han ( x, x if (, ), F(, ) F (, ). (4) ( x, x is concordan if i is more concordan han is corresponding pair of independen random variables wih he same marginals han ( x, x, i.e., if F (, F ( ) F( for all (,. Observe ha posiive firs-degree sochasic dependence implies concordance, as shown in he following Proposiion. This is equivalen o he noion of posiive quadran dependence proposed by Lehmann (966).
Proposiion : If ( x, x exhibis posiive firs-degree sochasic dependence, hen ( x, x ) is concordan. Proof: Suppose ha ( x, x exhibis posiive firs-degree sochasic dependence, so ha Px x is decreasing in for all. I have o prove ha his implies ha F (, ) F ( ) F( ) for all (, ). Afer dividing by F ( ), his condiion can be rewrien as follows: Px x df( ) Px x df( ) (5) G ( ) G( ). F( ) F( ) I would be done if G would be decreasing. Observe ha G is negaive if and only if Px x df( ) Px x. F ( ) (6) Observe ha he lef-hand side of his inequaliy is a weighed mean of Px x for,. Because his funcion is decreasing in, his mean is larger han is value a he upper bound of his semi-inerval. Because his is rue for all, his concludes he proof. Posiive firs-degree sochasic dependence is sufficien for concordance, bu i is clearly no necessary. Here is a counerexample: The suppor of x is, 0, and he suppor of x is 0,. Suppose ha P(,0) P(,) 3/, P(,) P(,0) /, P(0,0) 4 / and P(0,) 0. I is easy o check ha his pair of random variables exhibis concordance bu no posiive firs-degree sochasic dependence. I now show ha concordance is necessary and sufficien for a change in saisical relaion in ( x, x o yield an increase in Eh( x, x for all supermodular funcions h. The formal proof of he following lemma is in Tchen (980), Epsein and Tanny (980) or Meyer and Srulovici (0). Proposiion : Consider a bivariae funcion h. The following condiions are equivalen:
For any wo pairs of random variables ( x, x ) and ( x, x ) such ha ( x, x ) is more concordan han ( x, x, Eh( x, x Ehx (, x. h is supermodular. The proof of his resul is based on he observaion ha Eh( x, x Ehx (, x h(, F (, F (, dd. (7) This can be obained by a double inegraion by pars. I implies ha our resuls in Gollier (007) are valid modulo he swich of all occurrences of he erms posiive firs-degree sochasic dependence by he erm concordance. For example, here is he correc version of our Proposiion 3 in Gollier (007): Proposiion 3: The presence of any concordance in changes in log consumpion reduces he longerm risk-free rae if and only if relaive prudence is larger han uniy. Posiive firs-degree sochasic dependence is sufficien for he resul, bu no necessary. In Gollier (007), I also developed he noion of posiive second-degree sochasic dependence. The associaed resuls suffer from he same deficiency. Denui, Eeckhoud, Teslin and Winkler (00) and Heinzel (0 provide a correc and complee characerizaion of his case. Bibliography Denui, M., L. Eeckhoud, I. Tselin, and R.L. Winkler (00), Mulivariae Concave and Convex Sochasic Dominance, Working paper, INSEAD. Epsein, L.G. and S.M. Tanny, (980), Increasing Generalized Correlaion: A Definiion and Some Economic Consequences, Canadian Journal of Economics, 3, 6-34. Gollier, C., (007), The consumpion-based deerminans of he erm srucure of discoun raes, Mahemaics and Financial Economics, (, 8-0. Heinzel, C., (0), Term srucure of ineres raes under mulivariae s -ordered consumpion growh, mimeo, Toulouse School of Economics. Joe, H., (997), Mulivariae models and dependence conceps, Chapman and Hall/CRC.
Lehmann, E.L., (966), Some conceps of dependence, Annals of Mahemaical Saisics, 37, 53-73. Meyer, M., and B. Srulovici, (0), The supermodular sochasic ordering, Nuffield College, Oxford. Tchen, A.H., (980), Inequaliies for disribuions wih given marginals, The Annals of Probabiliy, 8, 84-87.