Existence, uniqueness and stability of equilibria in an OLG model with persistent inherited habits

Transcription

1 Existence, uniqueness and stability of equilibria in an OLG model with persistent inherited habits Giorgia Marini y University of York February 7, 2007 Abstract This paper presents su cient conditions for existence and uniqueness of a steady-state equilibria in an OLG model with persistent inherited habits and analyses the implications of persistence of inherited habits for the local stability of the steady-state equilibria. The conditions for a stable solution are derived under the assumption that habits are transmitted both across and within generations. Under this assumption monotonic convergence to the steady state is not always assured. Both competitive and optimal equilibria may display explosive dynamics. keywords: Habits, Existence, Uniqueness, Stability, Overlapping generations jel classification: D50, D9, E3, E32, O4 Earlier versions were presented at the DeFiMS Wednesday Research Seminars in 2002, 2003, and 2004, at the University of Rome La Sapienza Seminar Series in 2005 and at the th Spring Meeting of Young Economists held in Seville, Spain, May I am indebt to Pasquale Scaramozzino for his invaluable supervision and to Giacomo Rodano for his constant encouragement and support. All errors are mine. y Centre for Health Economics, University of York, Heslington, YORK, YO0 5DD, UK. Phone: , Fax: , gm509@york.ac.uk

2 Introduction This paper derives su cient conditions for existence and uniqueness of a steadystate equilibria in an overlapping generations model with persistent inherited habits and analyses the implications of persistence of inherited habits for the local stability of the steady-state equilibria. The main assumption of this paper is that habits are transmitted both across and within generations: that is the situation in which habits are transmitted from one generation to the next one (inheritance) and individuals carry these habits into old age (persistence). This assumption is modelled in the paper with non-separable preferences both across and within generations. In the literature on stability of the equilibrium in OLG models with habits, Diamond s [2] OLG model represents the benchmark model with separable preferences both across and within generations. Galor and Ryder [5] develop suf- cient conditions that guarantee the existence of a unique and globally stable nontrivial steady-state equilibrium in Diamond s [2] model, and de la Croix and Michel [] prove that the optimal solution in Diamond s [2] model is always characterised by monotonic convergence to the steady state. Michel and Venditti [7] provide su cient conditions for stability of the equilibrium in an OLG model with separable preferences across generations only and prove that the optimal solution may be oscillating and optimal cycles may exist. de la Croix and Michel [] provide su cient conditions for existence and uniqueness of the equilibrium in an OLG model with separable preferences within generations only. They prove that the optimal solution may display damped oscillations even when the social planner does not discount the utility of future generations. The contribution of this paper is to provide su cient conditions for existence, uniqueness and stability of the equilibrium in an OLG model in which preferences are non-separable both across and within generations. Since the intergenerational transmission of habits generates an intergenerational externality, both the competitive equilibrium and the social planner problems are analysed. The paper derives the conditions under which the unique competitive equilibrium converges to or diverges from the steady state and shows that the competitive equilibrium may display either uctuations or explosive behaviour. Then it studies the conditions under which the unique optimal solution is stable. Under the assumption of persistence of inherited habits, convergence of the optimal solution to the steady state is not always assured. The optimal solution may display either locally explosive dynamics or damped oscillations. This paper shows that the assumption of persistence of inherited habits is not innocuous: introducing the assumption of persistence in the contest of an The assumption of persistent inherited habits in the contest of an OLG with production of a single good (consumption good such as food) is supported by plenty of empirical evidence on eating habits. This empirical evidence suggests that the risk of becoming obese is greater among children with obese parents (Dietz [3]) (inheritance) and that obesity beginning in childhood may persist through the life span (Law [6]) (persistence). 2

3 OLG model in which habits are inherited, dynamics of the model and stability of the equilibrium are considerably a ected. Therefore researchers should be more cautious in neglecting persistence of inherited habits. 2 The model The economy is populated by three generations, each one living three periods. The rate of growth of population is zero. The young generation passively inherits habits from their parents: e t = c a t ; 8t > 0. e t is the stock of habits inherited from the parents and c a t is the level of consumption of the parents. Since inherited habits are out of children s control, they generate an intergenerational externality. The adult generation derives utility from consumption of the quantity c a t, given the stock of inherited habits e t. The old generation derives utility not only from consumption of the quantity c o t+ but also from consumption in t + relative to consumption in t: h t+ = c a t ; 8t > 0. h t+ represents the stock of habits that individuals form during their adulthood and that persist into old age and c a t the level of consumption of the parents. Since habits persistence is under individual control, the stock h is internalised in the utility maximisation process and it does not generate any intragenerational externality. The intertemporal utility function of each individual is U c a t ; c o t+; e t ; h t+ = u (c a t e t ) + v c o t+ h t+ () in which 2 (0; ) is the parameter measuring the intensity of parental habits and 2 (0; ) is the parameter measuring the persistence of adult habits into old age. Note that the rate of depreciation of habits inherited from the parents is so high that parental habits do not directly a ect their children s consumption when their children are old: the function v is independent of the stock e. On the other hand, parental habits indirectly predispose children to a persistent behaviour from adulthood into old age: the function v depends on the stock h. Thus, intergenerational and intragenerational interactions work separately on the instantaneous utility functions u and v. Assumption (i) u : R 2 +! R is twice di erentiable over R 2 ++; (ii) u is strictly quasi concave over R 2 ++, so that u c a > 0; u c a c a < 0; (iii) u e < 0; u ee < 0 and u c a e > 0. Assumption 2 (i) v : R 2 +! R is twice di erentiable over R 2 ++; (ii) v is strictly quasi concave over R 2 ++, so that v c o > 0; v c o c o < 0; (iii) v h < 0; v hh < 0 and v c o h > 0. Conditions (iii) de ne reaction to the stocks e and h. The patterns inherited from the parents have a negative impact on the individual utility function (u e < 0) and an increase in the stock e of past experiences has a positive e ect on the desire for consumption (u ca e > 0). Similar comments hold for the assumptions v h < 0 and v co h > 0. 3

4 Assumption 3 (i) u e u c a; u ee 2 u ca c a; (ii) v h v c o; v hh 2 v co c o. Assumption 4 (Starvation is Avoided in Both Periods) lim c a t!0 u c a+v h = lim c o t+!0 v c o = ; 8 c a t ; c o t+ 2 R Assumption 5 (Second Period Consumption is a Normal Good) (u c a+ v h ) v c o h > v c o (u c a c a + v hh). Assumption 6 (E ective Consumption is Always Non-Negative) c a t e t > 0 and c o t+ h t+ > 0. Lemma Under Assumption 6, the indi erence curve associated to the intertemporal utility function () is downward sloping if < u c a=v c o. Proof. The slope of the indi erence curve is dc o t+ dc a t = u c a + t+ a t v c o = u c a v c o v c o (2) under Assumption 3 t+ =@c a t =. If = 0 (benchmark Diamond s [2] model with separable preferences), downward slope is always ensured by the standard set of assumptions on marginal utility, i.e. u c a; v c o > 0. If > 0, the slope is upward sloping if > u c a=v h, at if = u c a=v h and negative if < u c a=v h. Therefore, the indi erence curve is downward sloping only if 2 (0; u c a=v h ]. No restriction is necessary for the domain of. In the second period of their life (adulthood) each household supplies inelastic labour to the labour market for which they receive a wage. Individual rst and second period budget constraints are c a t + s t = w t (3) c o t+ = R t+ s t (4) in which s t are the savings, R t+ is the interest factor upon which consumers plan lifetime consumption and saving and w t is the real wage. The former constraint (3) states that individuals can decide how to allocate income between current consumption and saving; the latter constraint (4) states that during retirement agents spend their savings completely. Agents optimally choose the consumption plan c a t ; c o t+, solving the following problem max u (c a c a t e t ) + v c o t+ h t+ (5) t ;co t+ subject to the individual constraints (3) and (4), to the stock of persistent habits h t+ = c a t and given the external stock of inherited habits e t, wages w t and correctly foreseen interest rates R t+. 4

5 First-order conditions lead to the following Euler equation 2 u c a v c o = R t+ v c o (6) With respect to the standard Diamond s [2] model in which = 0, marginal utility of the young is lower, as v c o > 0: in order to achieve the same level of satisfaction when old, adults need to correct their satisfaction by the (negative) habit e ect. The saving function associated with the maximisation problem is de ned as s t = s (w t ; R t+ ; e t ): it is independent of h t+ since the constraint h t+ = c a t is under the agent s control, but it depends on e t since e t = c a t is an externality. Given (w t ; R t+ ), Assumptions and 2 imply that s t exists and is unique. Assumption 5 implies that it is an increasing function of the real wage w t s w = u c a c a + (R t+ + ) v co c o u ca c a + (R t+ + ) 2 v co c o > 0 (7) By Assumptions and 2 it may be an increasing or decreasing function of real return to capital h i v c o + v c o c oco t+ + R t+ s r = u ca c a + (R t+ + ) 2 (8) v co c o while by Assumption it is a decreasing function in the stock of externality e t s e = u c a c a u c a c a + (R t+ + ) 2 v c o c o < 0 (9) The e ect of the real wage is positive since an increase in real wage increases both current consumption and saving for retirement. The e ect of the interest rate is ambiguous since it depends on the intertemporal elasticity of substitution. The e ect of the consumption externality is negative: an increase in e implies an increase in the desire for consumption when an adult and therefore a reduction in saving, ceteris paribus. In each period a single consumption good is produced. Production of the single good is realised with a constant returns to scale technology. Per capita output y t is a function of capital intensity k t : y t = f (k t ) (0) Assumption 7 (i) f : R +! R + is twice continuously di erentiable; (ii) f k > 0 and f kk < 0 8k > 0; (iii) f (0) = 0; lim kt! f k = 0; lim kt!0 f k =. Firm maximises pro ts and inputs are paid their marginal products R t = f k (k t ) () 2 Second-order conditions are satis ed under Assumptions and 2. Strictly quasi concavity guarantees that the sequence of minors of the Hessian matrix is f ; +g and therefore that the matrix is de nite negative. 5

6 and w t = f (k t ) k t f k (k t ) (2) in which R t is the interest factor on capital and w t is the real wage paid to workers. Under the assumption that capital completely depreciates over time and that the economy is endowed with k 0 and e 0 = c a at time t = 0, the resource constraint of the economy is y t = c a t + c o t + k t+ (3) The consumption good can be either consumed by the adult and the old generations, or accumulated as capital for future production. 3 The competitive economy Considering the market clearing conditions (), (2) and the resource constraint (3), if follows that the capital accumulation is given by k t+ = s t (w t (k t ) ; R t+ (k t ) ; e t ) (4) De nition Under Assumptions, 2 and 7, the intertemporal equilibrium of the competitive economy is a sequence fk t ; e t g t= following from individual constraints (3) and (4), equilibrium of production factors () and (2), saving function and market clearing (4), in which k 0, e 0 and the condition e t = c a t are exogenously given: k t+ = s (f (k t ) k t f k (k t ) ; f k (k t ) ; e t ) (5) e t+ = f (k t ) k t f k (k t ) s (f (k t ) k t f k (k t ) ; f k (k t ) ; e t ) (6) De nition 2 Under Assumptions, 2 and 7, the steady-state equilibrium of the intertemporal equilibrium (5)-(6) is a combination of (k; e) 2 R 2 ++ such that k = s (f (k) kf k (k) ; f k (k) ; e) (7) e = f (k) kf k (k) s (f (k) kf k (k) ; f k (k) ; e) (8) 3. Existence, uniqueness and stability of the steady-state equilibrium (7)-(8) Proposition The steady-state equilibrium (7)-(8) exists and is unique if and only if det I J CE 6= 0, where J CE is the Jacobian matrix associated to the intertemporal equilibrium (5)-(6) and evaluated at steady state (k; e) (Galor [4]). Proof. See Appendix A. 6

7 Stability of the steady state (7)-(8) depends on parameters and. Since for some values of these parameters the hyperbolic condition may not be satis ed, it is necessary to look for the critical value of and, ^ and ^, at which the change in trajectory takes place (bifurcation) and the xed point becomes nonhyperbolic. Non-hyperbolicity may arise when mod ^; ^ = if the roots of the characteristic polynomial associated with the Jacobian matrix are real or when r = if these roots are complex. If one of these conditions is met, linear approximation cannot be used to determine the stability of the system. Otherwise, local stability properties of the linear approximation carry over to the non-linear system. Proposition 2 Suppose Proposition holds and suppose that s e = ( srf kk) kf kk, h s r > js f kk and ej s s rf kk < + wkjf kk j s rf kk i. Then the determinant of the Jacobian matrix J CE is positive and equal to, the trace T CE J is positive and smaller than 2, and the discriminant is negative. Proof. See Appendix B. Proposition 3 Suppose Proposition 2 holds. Then the eigenvalues of J CE are complex and conjugate and the steady state is stable if and only if mod ^; ^ < and unstable otherwise. Proof. If mod ^; ^ <, the system is characterised by a spiral convergence to the steady-state equilibrium; if mod ^; ^ =, the system exhibits a period orbit; and if mod ^; ^ >, the system exhibits a spiral divergence from the steady-state equilibrium (Galor [4]). The system is therefore characterised by a bifurcation identi ed at ^; ^ which are the roots of det J CE =. The competitive equilibrium is thus characterised by spillovers from one generation to the next and from adulthood to old age. The main components of the intergenerational spillovers are: the savings by old people nance the capital stock required for production in the next period directly, and wages of adults indirectly; past consumption of the adults feeds the inherited habits of the next generation. While the process transforming the savings by the old into income for the adult displays decreasing returns to scale due to the characteristics of the production function, the process transforming past consumption of the adults into consumption of the next generation displays constant returns to scale due to the characteristics of the utility function. The intragenerational spillovers is only given by the individual past consumption that feeds individual s habits from adulthood to old age. This process displays constant returns to scale, again due to the characteristics of the utility function. Thus, even though the intergenerational bequest in terms of higher wages will not be su cient to cover the intergenerational bequest in terms of higher inherited habits, the intragenerational spillover leaves a bequest in terms of higher persistence. The combination 7

8 of the positive bequests in terms of higher wages and higher persistence is suf- cient to o set the negative bequest in terms of the higher externality. This leads to an increase in saving to maintain future standards of consumption that induces an expansion. When the enrichment is strong enough, the externality has already reverted to higher levels, allowing a fall in savings and the start of a recession. As the e ect due to persistence is stronger than the e ect due to the externality, the model is characterised by converging cycles. Thus, the competitive equilibrium still displays uctuations, but the bifurcation corresponds to di erent critical values of and. Depending on the parameters and, the economy may converge to or diverge from the steady state. 4 The optimal solution As the inheritance from the parents introduces an externality in the model, the decentralised equilibrium is implicitly sub-optimal compared to the equilibrium that would maximise the planner s utility. Thus, this paper studies the social planner s optimisation problem. The social planner s goal is to choose the optimal allocation that maximises social welfare. Given the discount factor, the social planner chooses fc a t ; c o t g and fk t ; e t g such that X max u t (c a t e t ) + v (co t h t ) (9) c a t ;co t ;kt;et t=0 subject to the resources constraint (3), e t = c a t, h t = c a t and given e 0 and k 0. First order conditions are: u c a (c a t e t ) + u e c a t+ e t+ = (20) = v c o (co t h t ) v h c o t+ h t+ v c o (co t h t ) = v c o c o t+ h t+ fk (k t+ ) (2) Equation (20) represents the optimal intergenerational allocation of consumption between the middle aged and old, in which the social planner internalises the externality associated with e. Marginal utility of the adult, corrected for the externality through the additional term u e, is equal to the marginal utility of the old. Note that this social planner s rst-order condition does not respect equation (6). Moreover, with respect to the standard Diamond [2] model in which = = 0, marginal utility of the adult, u c a (c a t e t )+u e c a t+ e t+, is lower, as u e < 0, while marginal utility of the old, v c o (co t h t ) v h c o t+ h t+, is higher, as v h < 0. Equation (2) sets the optimal intertemporal allocation. De nition 3 Under Assumptions, 2 and 7, the intertemporal equilibrium of the optimal solution is a sequence fc a t ; c o t ; k t ; e t g t= following from rst order 8

9 conditions (20)-(2) and the resource constraint (3), under the condition e t = h t = c a t : u c a (c a t e t ) + u e c a t+ e t+ = (22) = v c o (co t h t ) v h c o t+ h t+ v c o (co t h t ) = v c o c o t+ h t+ fk (k t+ ) (23) e t = h t = c a t (24) k t+ = f (k t ) c a t c o t (25) De nition 4 Under Assumptions, 2 and 7, the steady-state equilibrium of the intertemporal equilibrium (22)-(25) is a combination (c a ; c o ; k; e) 2 R 2 ++ such that u c a (c a e) + u e (c a e) = v c o (co h) v h (c o h) (26) = f k (k t+ ) (27) e = h = c a (28) k = f (k) c a c o (29) Equation (26) shows that in an economy with persistent e ects of parental habits, the marginal utility of the adult is lower than the corresponding marginal utility in Diamond s [2] model: the inheritance represents a benchmark from which individuals want to depart. Even the marginal utility of the old is higher than the corresponding marginal utility in Diamond s [2] model: the same interpretation carries on. Once the externality associated with parents habits is internalised, persistence a ects marginal utility in the same way as the externality: they both induce consumers to save. Equation (27) is the modi ed Golden Rule. Equations (28) and (29) have been already discussed in the paper. 4. Existence, uniqueness and stability of the steady-state equilibrium (26)-(29) Proposition 4 The steady-state equilibrium (26)-(29) exists and is unique if and only if det I J SO 6= 0, where J SO is the Jacobian matrix associated to the intertemporal equilibrium (22)-(25) and evaluated at steady state (c a ; c o ; k; e) (Galor [4]). Proof. See Appendix C. Proposition 5 Assume that capital stock and the externality are state variables and that adult and old consumption are jump variables. Locally-explosive dynamics is possible, depending on the sign of the trace T SO J and of the element Z 9

10 of the Jacobian matrix J SO. If 0, the eigenvalues are real and local dynamics is either explosive or monotonic. If < 0, the eigenvalues are complex and conjugate and local dynamics displays either explosive or damped oscillation. Proof. See Appendix D. The above proposition states all the possible patterns of dynamics of the optimal steady-state equilibrium. Under the assumption that the trace T SO J and the element Z are both positive, locally explosive dynamics is identi ed by an unstable node if the eigenvalues are real and by an unstable focus if the eigenvalues are complex and conjugate. Under the assumption that T SO J and Z are both negative, the optimal solution is a stable saddle point, only if the constraints on the elements of the Jacobian matrix J SO respect the condition on negativity of the trace. Under the assumption that T SO J and Z have opposite sign, the optimal solution may be either stable or unstable. In the rst case, dynamics displays damping oscillation to the steady state. In the second, locally explosive dynamics occurs when the constraints on the elements of the matrix J SO do not respect the condition on negativity of the trace. The stability of the stationary point depends on the assumption that habits are transmitted both across and within generations, which a ects the sign of the trace T SO J and of element Z. Apart from the benchmark model (Diamond [2]), monotonic convergence to the steady state is ensured only under the assumption that the stock of habits that individuals form during their adulthood does not persist into their old age, i.e. only if = 0. Contrary to the competitive equilibrium, the optimal solution is characterised only by a positive intergenerational spillover: the savings by old people that directly nance the capital stock required for production in the next period, and that indirectly sustain wages of adults. The intergenerational and the intragenerational spillovers associated to consumption decisions are a priori internalised by the social planner in the maximisation problem (9). As in the competitive equilibrium, the process transforming the savings by the old into income for the adult displays decreasing returns to scale, due to the characteristics of the production function. However, the intergenerational bequest in terms of higher wages does not interact with any other spillover. The intergenerational bequest in terms of higher wages will lead to a constant increase in saving that induces a permanent expansion. The model is thus characterised by a diverging explosive dynamics. 5 Numerical example Assume that the utility function is logarithmic, ln (c a t e t )+ ln c o t+ h t+, and that the production function is Cobb-Douglas, y t = k t. The steady-state 0

11 = 0:25; = 0:43 = = 0 (Diamond s [2] model) = 0:25; = 0:43 = 0:65; = 0:3 y 0:609 0:628 w 0:457 0:47 R :09 :0 k 0:37 0:55 e 0:36 c a 0:39 0:36 c o 0:52 0:57 Table : Steady-state values in the competitive economy equilibrium (7)-(8) becomes (see Appendix E.) k = ( + ) " e = ( + ) 2 ( + ) + Once steady-state values of capital and stock of externality are found, use market clearing conditions () and (2), resource constraint (3) and individual constraints (3) and (4) to nd steady-state values of output, real wages, interest factor, consumption of the adult and old # y = ( + ) w = ( ) ( + ) R = ( + ) c a = ( ) ( + ) c o = ( + ) ( + ) Assigning values to the relevant parameters, it is possible to make a comparison with Diamond s [2] model (see Table ). Note that consumption of the adult is much higher than consumption of old people because individuals do not internalise the intergenerational externality e a ecting their own consumption (c a e) and therefore they tend to consume more than what is optimal.

12 Also note that the choice of the above parameters veri es Assumptions -7, Lemma, conditions (7)-(9), and Propositions and 2 as det I J CE = 0:449 6= 0 s r = 0:057 > 0:205 = js e j s r f kk = 0:270 < :57 = + T CE J = 0:57 < 2 f kk s w k jf kk j s r f kk = 0: = 3:733 < 0 if s e = kf kk s r f kk As < 0, the eigenvalues of J CE are complex and conjugate (Proposition 3), 0:379 0:250i, and the bifurcation is identi ed at ^ = 0:50; ^ = 0:00 which are the roots of ( ) det J CE ( ) ( ) ( + ) = ( + ) + [ ( + ) + ] 2 + = under = 0:25 and = 0:43. Therefore the system is characterised by a spiral convergence to the steady-state equilibrium if mod (0:50; 0:00) < ; the system exhibits a period orbit if mod (0:50; 0:00) = ; and the system exhibits a spiral divergence from the steady-state equilibrium if mod (0:50; 0:00) >. Analogously, when the utility function is logarithmic and that the production function is Cobb-Douglas, the steady-state equilibrium (26)-(29) becomes (see Appendix E.2) c o = c ( a c o = ( ) ( ) e = h = c a = ( ) ( ) k = ( ) ( ) ) ( ) Assigning values to the relevant parameters, it is possible to make comparisons between Golden Rule ( = ) and Modi ed Golden Rule ( = 0:5; 0:99) (see Table 2). Note that consumption of the adult is still higher than consumption of old people but intertemporal consumption plan is now smoother as the social planner internalises the intergenerational externality e. 2

13 = 0:25; = 0:43 = 0:65; = 0:3 = 0:5 = 0:25; = 0:43 = 0:65; = 0:3 = 0:99 = 0:25; = 0:43 = 0:65; = 0:3 = y 0:762 0:770 0:77 k 0:337 0:352 0:354 e = h = c a 0:234 0:226 0:224 c o 0:90 0:93 0:93 Table 2: Steady-state values in the optimal solution Modi ed Golden Rule ( = 0:5) 0:685 0:07i 2:85 0:446i Modi ed Golden Rule ( = 0:99) 0:536 0:79i :695 0:568i Golden Rule ( = ) 0:535 0:78i :682 0:56i Table 3: Eigenvalues in the optimal solution Also note that the choice of the above parameters veri es Propositions 4 and 5 as det I J SO = 0:200 6= 0 = 0:5 = 0:99 = T SO J 4:483 4:462 4:434 Z 7:228 7:48 7:06 As the eigenvalues of J SO are complex and conjugate (see Table 3) and as T SO J and Z are both positive, the dynamics of the social optimal is identi ed by a locally unstable focus. This result holds even when the social planner does not discount the utility of future generations, i.e. when = (the Golden Rule). 6 Concluding remarks This paper derives su cient conditions for existence and uniqueness of a steadystate equilibria in an overlapping generations model with persistent inherited habits and analyses the implications of persistence of inherited habits for the local stability of the steady-state equilibria. It derives the conditions for the steady-state stability of competitive equilibrium and shows that the competitive equilibrium may display either uctuations or explosive behaviour. Then it studies the conditions for the steady-state stability of optimal solution. It is proved that the optimal solution may display damped oscillations or locally explosive dynamics. This result depends on the assumption that habits are transmitted from one generation to the next one (inheritance) and individuals carry these habits into old age (persistence). This paper shows that the assumption of persistence of inherited habits is not innocuous: when we introduce persistence in the contest of an OLG model in which habits are inherited, dynamics of the model and stability of the 3

14 equilibrium are dramatically a ected. For this reason, researchers should be more cautious in neglecting this assumption. The results presented in this paper are therefore fundamental to understanding the mechanisms underneath models with habit formation and habit persistence, as habits seem to play a signi cant role in many aspects of macroeconomic theory. 4

15 A Proof of Proposition First, linearise the non-linear dynamic system (5)-(6) around the steady state (7)-(8) dk t+ = s w [ kf kk ] dk t + s r f kk dk t+ + s e de t (30) de t+ = kf kk dk t s w [ kf kk ] dk t s r f kk dk t+ s e de t (3) Then solve the rst equation by dk t+ dk t+ = s r f kk [ s w kf kk dk t + s e de t ] and substitute the solution into the second equation of the system (30) de t+ = s r f kk [(s w + s r f kk ) kf kk dk t s e de t ] The linearised system (5)-(6) around the steady state (7)-(8) is therefore dkt+ s = w kf kk s e dkt de t+ s r f kk (s w + s r f kk ) kf kk s e de t in which J CE = " s wkf kk s e s rf kk s rf kk s (s w +s rf kk )kf kk s rf kk e s rf kk # (32) is the Jacobian matrix evaluated at steady state (k; e). It is immediate to show that the matrix I J CE is equal to I J CE " # + swkf kk s e s = rf kk s rf kk (s w +s rf kk )kf kk s rf kk + se s rf kk and that its determinant is det I J CE = + s wkf kk s e + + s wkf kk (s w + s r f kk ) kf kk s r f kk s r f kk ( s r f kk ) 2 s e = = s rf kk + s e + (s w + s e ) kf kk s r f kk 6= 0 (33) under Assumptions -3 and 7, partial derivatives (7)-(9), equilibrium of production factors () and (2) and De nition 2. B Proof of Proposition 2 The determinant of the matrix J CE is equal to det J CE = s es w kf kk s e (s w + s r f kk ) kf kk ( s r f kk ) 2 ( s r f kk ) 2 5

16 the trace is equal to T CE J = and the discriminant is de ned as = T CE J Therefore, under the following conditions s w kf kk s r f kk s e s r f kk 2 4 det J CE s e = ( s rf kk ) s r > f kk js e j s r f kk < + kf kk s w k jf kk j s r f kk it is immediate to prove that the determinant is equal to det J CE = s ekf kk s r f kk = and that the trace becomes in absolute values smaller than 2 Finally, since T CE J = s wk jf kk j js e j + < 2 s r f kk s r f kk T CE J < 2 ) T CE 2 J < 4 the discriminant is negative 2 sw k jf kk j js e j = + 4 < 0 s r f kk s r f kk C Proof of Proposition 4 First, linearise the non-linear dynamic system (22)-(25) around the steady state (26)-(29): u c a c adca t + u c a ede t + u c a edc a t+ + u ee de t+ = v c o c odco t + v c o hdh t + v co hdc o t+ v hh dh t+ v c o c odco t + v c o hdh t v c o c of k (k t+ ) dc o t+ = v c o hf k (k t+ ) dh t+ + +v c of kk (k t+ ) dk t+ de t+ = dh t+ = dc a t dk t+ = f k (k t ) dk t dc a t dc o t 6

17 Then solve the rst equation by dc a t+ and the second by dc o t+, using the third and the fourth equations, under the assumption that de t =dh t : dc a vc t+ = o c o v c o h 2 u c a e u c a e f k (k t+ ) + v cof kk (k t+ ) v c o c of dc o t + k (k t+ ) + + vco h v 2 c o h 2 u c a e 2 u c a ev c o c of de t + k (k t+ ) uc + a c a u ee v hh v c o h vc of kk (k t+ ) u ca e u ca e u ca e u ca e v co c of k (k t+ ) + v co h dc a t + v c o h vc of kk (k t+ ) v c o c o u c a e v c o c of k (k t+ ) f k (k t ) dk t dc o vc of kk (k t+ ) t+ = v c o c of k (k t+ ) + dc o v c t + o h f k (k t+ ) v c o c of k (k t+ ) de t + vc of kk (k t+ ) v c + o h v co c of dc a v c of kk (k t+ ) t k (k t+ ) v co c o v co c of k (k t+ ) f k (k t ) dk t de t+ = dh t+ = dc a t dk t+ = f k (k t ) dk t dc a t dc o t The linearised system (22)-(25) around the steady state (26)-(29) is therefore de t de t 6 dc a [B( E) ] DB t+ 7 4 dk t+ 5 = A B (D E) C B ( + D) dc a t dc o 5 4 dk t 5 D t+ E D E + D dc o t in which A B v c o h > 0 u c a e u c a c a + u ee + v hh u ca e C v c o c o 2 < 0 u ca e D v c of kk (k t+ ) > 0 v co c o E v c o h v c o c o < 0 > 0 under the assumption that in steady state f k (k t+ ) = f k (k t ) =. As [B( E) ] DB J SO A B (D E) C B ( + D) = D E D E + D (34) 7

18 is the Jacobian matrix evaluated at steady state (c a ; c o ; k; e), it is immediate to show that the matrix I J SO is equal to I J SO [B( E) ] DB A + B (D E) C + B ( + D) = D E D + E D and that its determinant is det I J SO = A B (D E) + + ( + D) + B + CE 2 = = A + B (D E) D + B + CE 2 = = u c a c a + u ee + v hh + u ca e + v c o h vc of kk (k t+ ) v co h vc of kk (k t+ ) 6= 0 u ca e v co c o v co c o under Assumptions -3 and 7, conditions (28) and (29) and De nition 4. D Proof of Proposition 5 Study the characteristic polynomial P in the eigenvalues associated to the Jacobian matrix (34) P () = 4 T SO J 3 + Z 2 T SO J + det J SO = 0 in which det J SO = B + CE 2 = 2 > 0 T SO J = + + A B (D E) + D? 0 if + + A + D? B (D E) Z = 2 + (A + BE) + + (A C) D? 0 if 2 + (A C) D? (A + BE) + Then, factorise the polynomial P into P () = ( ) ( 2 ) = 0 2 which is equivalent to = 0 8

19 Now analyse all possible scenarios, due to the sign s ambiguity of the trace T SO J and of the element Z. Proof. First, assume that T SO J and Z are both positive and analyse the two possible cases:. 2 i 4 > 0; i = ; 2. The four eigenvalues are real and they can be: (a) four negative roots. This case implies that + 2 < 0 and 2 > 0. This case is excluded as it violates T SO J > 0. (b) two negative and two positive roots. This case implies that + 2? 0 and 2 < 0. This case is excluded as it violates Z > 0. (c) four positive roots. This case implies that + 2 > 0 and 2 > 0. This case is accepted as it respects both conditions on T SO J and Z i 4 < 0; i = ; 2. Look at the real parts only. Since the real part a = 2 i 6= 0; i = ; 2, the eigenvalues are complex and conjugate and they can be: (a) four negative roots. This case implies that + 2 < 0 and 2 > 0. This case is excluded as it violates T SO J > 0. (b) two negative and two positive roots. This case implies that + 2? 0 and 2 < 0. This case is excluded as it violates Z > 0. (c) four positive roots. This case implies that + 2 > 0 and 2 > 0. This case is accepted as it respects both conditions on T SO J and Z. Under the assumption that T SO J and Z are both positive, the only admissible case is c). It identi es an unstable node if the eigenvalues are real and an unstable focus if the eigenvalues are complex and conjugate. Locally explosive dynamics is highly likely. Then, assume that T SO J and Z are both negative and analyse the two possible cases:. 2 i 4 > 0; i = ; 2. The four eigenvalues are real and they can be: (a) four positive roots. This case implies that + 2 > 0 and 2 > 0. This case is excluded as it violates both conditions on T SO J and Z. (b) two negative and two positive roots. This case implies that + 2? 0 and 2 < 0. This case is admissible only if + 2 < 0 as T SO J < 0. (c) four negative roots. This case implies that + 2 < 0 and 2 > 0. This case is excluded as it violates condition on Z > i 4 < 0; i = ; 2. Look at the real parts only. Since the real part a = 2 i 6= 0; i = ; 2, the eigenvalues are complex and conjugate and they can be: 9

20 (a) four positive roots. This case implies that + 2 > 0 and 2 > 0. This case is excluded as it violates both conditions on T SO J and Z. (b) two negative and two positive roots. This case implies that + 2? 0 and 2 < 0. This case is admissible only if + 2 < 0 as T SO J < 0. (c) four negative roots. This case implies that + 2 < 0 and 2 > 0. This case is excluded as it violates condition on Z > 0. Under the assumption that T SO J and Z are both negative, the only admissible case is b), but only if + 2 < 0. It identi es a stable saddle point that ensures monotonic local convergence. Finally, assume that T SO J and Z have opposite sign and distinguish two possible cases:. 2 i 4 > 0; i = ; 2. The four eigenvalues are real and they can be: (a) four positive roots. This case implies that + 2 > 0 and 2 > 0. This case is excluded as it violates both conditions on T SO J and Z. (b) two negative and two positive roots. This case implies that + 2? 0 and 2 < 0. This case is admissible only if + 2 > 0 as is ensures T SO J > 0. (c) four negative roots. This case implies that + 2 < 0 and 2 > 0. This case is admissible only if T SO J < 0 and Z > i 4 < 0; i = ; 2. Look at the real parts only. Since the real part a = 2 i 6= 0; i = ; 2, the eigenvalues are complex and conjugate and they can be: (a) four positive roots. This case implies that + 2 > 0 and 2 > 0. This case is excluded as it violates both conditions on T SO J and Z. (b) two negative and two positive roots. This case implies that + 2? 0 and 2 < 0. This case is admissible only if + 2 > 0 as it ensures T SO J > 0. (c) four negative roots. This case implies that + 2 < 0 and 2 > 0. This case is admissible only if T SO J < 0 and Z > 0. Under the assumption that the trace and the element Z have opposite sign, case b) identi es an unstable solution as T SO J > 0. Locally-explosive dynamics is highly likely. Case c) identi es a stable node for real eigenvalues and a stable focus for complex and conjugate eigenvalues, and therefore it ensures damped convergence to the steady state. 20

21 E Closed-form solutions E. Intertemporal equilibrium (5)-(6) Assume that the utility function is logarithmic, ln (c a t e t )+ ln c o t+ h t+, and that the production function is Cobb-Douglas, y t = k t, and nd the steadystate values of k and e. The intertemporal equilibrium (5)-(6) becomes k t+ = e t+ = + [( ) ( ) k k t t e t ] + kt + ( + ) (35) ( ) kt 2 kt + ( + ) + + e t (36) and the associated system (7)-(8) becomes k = e = + [( ) k e] + ( ) k (k + ) ( + ) (37) ( ) k 2 (k + ) ( + ) + e (38) + Find e from equation (38) e = ( ) [ + ( )] k 2 k + = 0 k 2 k + (39) in which 0 ( ) = [ + ( )], and substitute e into the equation (37) and get the following second order equation in k [ 0 ( )] k 2( ) + ( + ) [ ( ) ] k + ( + ) = 0 Since 0 ( ) = [ + ( )], ( ) [ 0 ( )] = ( ) = + ( ) ( ) ( ) ( + ) = = + ( ) ( ) = [ + ] = + ( ) ( ) ( + ) ( ) = + ( ) and the second order equation in k becomes ( ) ( + ) ( ) k 2( ) + ( + ) [ ( ) ] k + ( + ) = 0 + ( ) ( ) ( ) [ + ( )] k2( ) [ ( ) ] + k + = 0 2

22 ( )( ) [ ( )] Setting a [+( )], b, c and k x, it is possible to rewrite the above second order equation in k as a quadratic equation ax 2 + bx + c = 0 (40) Equation (40) has the following set of possible solutions x ;2 = b p b 2 4ac 2a = [ ( )] r n [ ( )] 2 ( )( ) [+( )] o ( )( ) [+( )] Since capital cannot be an imaginary number, the discriminant 2 [ ( ) ] ( ) ( ) + 4 [ + ( )] must be non-negative. Since,,, ( ), ( ) and [ + ( )] are all positive, the discriminant is strictly positive. Now study the sign of the sequence fa; b; cg. By Descartes theorem, equation (40) has a positive and a negative solution, whatever the sign of the coe cient b. Since physical capital cannot be negative, the negative solution is excluded a priori. Therefore, the unique solution for equation (40) is [ ( )] + r n [ ( )] o ( )( ) [+( )] x = ( )( ) 2 [+( )] = 2 [ ( ) ] + ( ) ( ) ( ) + + ( ) 2 ( ) ( ) Using k k = + s [ ( ) ] 2 ( ) ( ) + 4 [ + ( )] x, the steady-state value of k is + ( ) [ ( ) ] + (4) 2 ( ) ( ) + s + ( ) [ ( ) ] ( ) ( ) = ) ( ) ( ) [ + ( )] Therefore, the solution of the steady-state system (7)-(8) is given by equations (4) and (39) k = ( + ) (42) " # ( + ) 2 e = (43) ( + ) + 22

23 in which 2 [ ( ) ] + ( ) ( ) ( ) s + ( ) [ ( ) ] 2 ( ) ( ) ( ) ( ) [ + ( )] ( ) + ( ) E.2 Intertemporal equilibrium (22)-(25) Assume that the utility function is logarithmic and that the production function is Cobb-Douglas, the intertemporal equilibrium (22)-(25) becomes c a t e t c a t+ e t+ = c o t h t = c o t + h t c o t+ h t+ c o kt+ t+ h t+ e t = h t = c a t k t+ = k t c a t c o t and the steady-state equilibrium (26)-(29) becomes c a e c a = e (c o h) + c o h (44) = k (45) e = h = c a (46) k = c a + c o + k (47) Then solve equation (44) by c o and get c o = c a (48) in which ( +)( ) +. Using the fact that in equilibrium c a = ( ) k k, substitute (48) into equation (47) and get k = ( + ) [( ) k k] + k which can be solved by k k = ( ) (49) 23

24 Now use (49) into (46) and (48) and nd the steady-state values of e and c o : e = h = c a = ( ) ( c o = ( ) ( ) ( ) ( ) ( ) ) 24

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