Overlapping Generations Models An Introductory Course

Transcription

1 Overlapping Generations Models An Introductory Course Alfredo Medio LUISS University, Rome GREDEG-CNRS, University of Nice

2 Outline 1 Introduction 2 A Pure exchange OLG Model 3 A leisure consumption OLG model 4 Intertemporal Equilibria as Dynamical Systems 5 An OLG Model with Production by Capital and Labor

3 Introduction Overview of the Course I In this course, we will discuss some basic ideas and methods belonging to the field of economic theory known as overlapping generations models (OLG). Influential ideas have often many precursors. In this case, Allais (1947) and Samuelson (1958) are generally quoted as the founding fathers of OLG. OLG models belong to the class of intertemporal general equilibrium models and, within the neoclassical tradition of equilibrium theory, the OLG approach has become the strongest competitor of the canonical Arrow Debreu paradigm.

4 Introduction Overview of the Course II However, OLG models retain most fundamental neoclassical assumptions, namely: (i) rationality of choices (agents take their decisions maximizing certain object functions, such as utility or profit functions, subject to certain budget or technological constraints); (ii) price taking (in their decisions agents take prices as given parameters); (iii) perfect foresight (or rational expectations in the presence of uncertainty); markets clearing (demand is equal to supply); The crucial distinguishing feature of the OLG paradigm concerns the demographic structure: instead of a fixed set of agents, in OLG models we have an infinite succession of generations partially overlapping in time.

5 Introduction Overview of the Course III This innovation is the at the root of some dramatic differences between the properties of the canonical Arrow-Debreu models and those of the OLG models. In particular, certain welfare optimality properties of the former may not exist for the latter The study of OLG models is a very vast and growing area of research, including innumerable applications to virtually all field of theoretical and applied economics This course will not have an encyclopedic character but will rather discuss in detail a small number of basic models, assuming that the students attending the course have little or no knowledge of the subject matter.

6 Introduction Overview of the Course IV In particular, we will present an introductory analysis of three versions of two-generations, two-periods OLG models, namely: pure-exchange, leisure consumption and productions with labor and capital. Time and progress permitting, we will briefly treat also a three generations model. Like all intertemporal equilibrium models, the OLG models have an intrinsic dynamical nature in the sense that an equilibrium is an infinite time sequence of values of the state variables, satisfying the equilibrium conditions. Thus, equilibria can be seen as solutions of dynamical systems, usually taking the form of difference equations.

7 Introduction Overview of the Course V Economists attention has been mainly focused on the properties of stationary equilibria, whereas non stationary equilibria have been rather neglected We will try to compensate this bias in the economic literature by devoting special attention to the investigation of the dynamical properties of non stationary equilibria and to an intriguing problem arising in some basic class of OLG models known as backward dynamics.

8 Introduction Overview of the Course VI For this purpose, we will be doing numerical graphical exercises based on special cases of the OLG models discussed in the course, making use of a dedicated, open source software called idmc The literature on OLG models is extremely vast. The references at the end of these lectures notes are far from being exhaustive or representative of that literature. They are just, more or less random examples of useful reading for the students wishing to further pursue the study of the many interesting questions arising from our introductory discussion.

9 Introduction Overview of the Course VII Last but not least: there are points discussed in the quoted articles that will not be dealt with in the course (and vice versa). Nevertheless, a good understanding of the teaching material provided during the course should be enough to answer the exam questions satisfactorily.

10 A Pure exchange OLG Model Pure exchange: Assumptions I H1, Demography: a constant population of individuals (identical except for their age), living two periods of time and divided at each period into two equally numerous classes, respectively labeled young and old H2, Consumption: at each period t, the young agent receives a fixed endowment w 0 0 of a unique perishable good, and consumes a quantity c t 0 of it. The corresponding quantities for the old agent are κ t 0, w 1 0. Young agent s and old agent s excess demand at each time t are denoted by z t = c t w 0 and ζ t = κ t w 1, respectively. Within limits, z t and ζ t can be positive, zero or negative. A negative/positive excess demand corresponds to savings/dissavings.

11 A Pure exchange OLG Model Pure exchange: Assumptions II H3, Exchange: at times t and t + 1 the consumption good is traded at a price per unit p t, and p t+1, respectively. The ratio ρ t = p t /p t+1 denotes the interest factor, i.e. the rate of exchange between a unit of good today and the same quantity tomorrow. If r t = (p t p t+1 )/p t+1 ) = ρ t 1 denotes the interest rate over the period (t, t + 1). H4, Preferences: for each generation living through the periods (t, t + 1), preferences are defined by smooth, strictly increasing and concave utility functions V (c t ), U(κ t ). Unless otherwise noted, we assume that the functions V and U satisfy the conditions: V (c t ) > 0, U (κ t ) > 0; V (c t ) < 0, U (κ t ) < 0; lim ct 0 V (c t ) = ; lim κt 0 U (κ t ) =

12 A Pure exchange OLG Model Pure exchange: Assumptions III H5, Maximization: At each t, the young agent chooses the present and future levels of consumption (or, equivalently, the present and future levels of excess demand) as functions of the observed current price p t and the anticipated future price p t+1, subject to a two period budget constraint requiring that the value of savings/dissavings today be equal to that of dissavings/savings tomorrow. H6, Perfect foresight: for each t, Perfect foresight: for each t, the young agent s expectations concerning the values of the variables at t + 1 are perfectly fulfilled

13 A Pure exchange OLG Model Pure exchange: Assumptions IV H7, Market clearing: at each t, total supply (w 0 + w 1 ) is equal to total demand (c t + κ t ). H1 and H7 imply that, at each t, 0 c t w 0 + w 1, 0 κ t w 0 + w 1 or, equivalently, w 0 z t w 1 and w 1 ζ t w 0

14 A Pure exchange OLG Model Financial Institutions Given the demographic structure of OLG models, the equilibrium conditions require that borrowers (agents with a positive excess demand) will settle their obligations, paying back to agents having a negative excess demand, who are different from those from whom, directly or indirectly, they borrowed. To complete the model, therefore, we require some negotiable asset performing the function of a store of value, such as debt or money, and a social institution holding the assets issued by borrowers or issuing them to savers, such as a central clearing house. Aggregate savings may be negative, zero or positive.

15 A Pure exchange OLG Model Notation c t 0 : consumption of the young agent at time t κ t 0 : consumption of the old agent agent at time t w 0 0 : endowment for the young agent, constant w 1 0 : endowment for the old agent, constant p t > 0 : price of the consumption good at time t p t+1 > 0 : price of the consumption good at time t + 1 ρ t 0 : interest factor r t = ρ t 1 : interest rate V (c t ), U(κ t+1 ) : utility functions z t = c t w 0 : young agent s excess demand (dissaving) at time t ζ t = κ t+1 w 1 : old agent s excess demand at time t

16 A Pure exchange OLG Model Intertemporal optimization max {V (c t ) + U(κ t+1 )} c t,κ t+1 s.t. p t (w 0 c t ) + p t+1 (w 1 κ t+1 ) = 0, budget constraint (1) Market clearing condition (demand=supply) requires that t c t + κ t = w 0 + w 1 or, equivalently z t = ζ t (2) N.B. Notice that equation (1) refers to one young agent s decision over the two periods of his/her life, whereas equation (2) defines a macroeconomic condition and refers to the compatibility of the decisions of two different agents, one young and the other old, at the same period.

17 A Pure exchange OLG Model From (1), we can write the Lagrangian function L(c t, κ t+1, λ) = {V (c t ) + U(κ t+1 ) + λ[p t (c t w 0 ) +p t+1 (κ t+1 w1)]} (3) The first order conditions (F.O.C.) for this problem are: L c t = V (c t ) + λp t = 0 L κ t+1 = U (κ t+1 ) + λp t+1 = 0 (4) L λ = [p t (c t w 0 ) + p t+1 (κ t+1 w 1 )] = 0

18 A Pure exchange OLG Model From equations (4), it follows that: and ρ t = p t p t+1 = V (c t U (κ t+1 ) = ζ t+1 z t = z t+1 z t = ζ t+1 ζ t (5) H(c t, κ t+1 ) = V (c t )(c t w 0 ) + U (κ t+1 )(κ t+1 w 1 ) = 0 (6)

19 A Pure exchange OLG Model In view of the fact that z t = c t w 0 ; ζ t = κ t w 1, t, the intertemporal equilibrium condition (6) can be conveniently re-written in terms of the excess demand of young and old agents at two successive periods of time. Thus we have: H(z t, ζ t+1 ) = V (z t + w 0 )z t + U (ζ t+1 + w 1 )ζ t+1 = 0 (7) Also, since z t = ζ t, t, equation (7) can be conceived as an implicit difference equation in a single variable (z t or, equivalently, ζ t ). Finally, notice that the condition c t, κ t 0 (no negative consumption admitted) implies the following restrictions on excess demand: z t [ w 0, w 1 ], ζ [ w 1, w 0 ].

20 A Pure exchange OLG Model Intertemporal, perfect foresight equilibrium (IPFE) Definition An intertemporal, perfect foresight equilibrium (IPFE) is an infinite sequence of the state variable z t or ζ t that satisfies the condition (7) and the restrictions on the relevant domains

21 A Pure exchange OLG Model The problem of initial condition In the logic of OLG models, fixing an initial condition gives rise to some conceptual difficulties that are not present in other type of economic dynamical models. To see this, consider that the value of the relevant variable at each time t, say ζ t, is justified only by the existence of a sequence of perfectly anticipated future values (ζ t+1, ζ t+2,...) such that each successive pair satisfies the equilibrium conditions of the model, and this rule, strictly speaking, should apply to the initial conditions as well. Accordingly, ζ 0 should be interpreted as the excess demand of the old agent at time t = 0 that, correctly anticipated, justified the choice of saving z 1 made in the same agent s youth, and so on, backward in time.

22 A Pure exchange OLG Model Strictly speaking, the story told by the OLG model should have no beginning and no end. There exist two basic approaches to this conceptual difficulty that are relevant for our discussion. The first of them nicknamed here unlimited horizon hypothesis (UHH) is to consider sequences of intertemporal equilibria stretching indefinitely both in the future and in the past. This would pose additional restrictions to sequences that are forward and backward admissible The UHH is logically impeccable but its realism and fruitfulness can be questioned.

23 A Pure exchange OLG Model The second approach nicknamed single shock hypothesis (SSH) introduces the ad hoc assumption that, at some instant of time t = 0, after the young agent s decision has been made, the economy is perturbed by an unexpected random shock, whose precise nature depends on the details of the model, and that no further disturbances occur. The SSH allows one to treat initial conditions as random (within certain constraints), at the price of violating, for the initial period only, the hypothesis of agents perfect foresight. In what follows, most of time we adopt the SSH, but we also briefly discuss the implications of the UHH for our argument.

24 A Pure exchange OLG Model Notice the following: from the combination of budget constraint and market clearing condition, it follows that the signs of the excess demand of young and old agents are constant all along the infinite time sequence of each IPFE in the sense that, if z t > 0 at a any time t (and therefore ζ t < 0) it must be positive (and ζ t negative) at any other time and vice versa if z t < 0 (and ζ t > 0.

25 A Pure exchange OLG Model Before investigating the properties of IPFE in general, we consider a special class of equilibria that, for good or bad reasons, attracts most of economists attention, namely: stationary IPFE, i.e. infinite sequences of values of the state variables (with the possible exception of prices) constant w.r.t. time and satisfying (7). An obvious candidate for a stationary IPFE is the so-called no-trade (or, non monetary, or autarkic ) equilibrium, i.e., z 1 = ζ 1 = 0. This solution exists for (7) on condition that lim V (z + w 0 )z = 0 z 0 lim ζ 0 U (ζ + w 1 )ζ = 0 which of course depends on the properties of the utility functions and is generally verified under our assumptions.

26 A Pure exchange OLG Model A second, hypothetical solution, z 2 = ζ 2 0, must satisfy the equation V ( z 2 + w 0 ) z 2 + U ( ζ 2 + w 1 ) ζ 2 = 0 (8) or, dividing throughout by z 2 (or by ζ 2 ), we get or, equivalently, V ( z 2 + w 0 ) = U ( z 2 + w 1 ) (9) V ( ζ 2 + w 0 ) = U ( ζ 2 + w 1 ) (10) A sufficient condition for existence and uniqueness of a non stationary solution is that lim ct 0 V (c t ) = and lim κt 0 U (κ t ) =.

27 A Pure exchange OLG Model Trade and No Trade Intertemporal Equilibria The no-trade stationary equilibrium z 1, ζ 1 describe a situation in which, given the interest factor (price ratio), young and old agents are satisfied with their endowments and therefore excess demand is zero at all time In the trade stationary equilibrium, young and old agents trade part of their endowments, saving and dissaving within their budget constraints The trade stationary equilibrium is also called optimal because, as we shall see, it satisfies the so called golden rule requiring that the interest rate is equal to the rate of growth of the economy. In our case, ρ = 1 + r and there is no growth so the golden rule is satisfied if ρ = 1.

28 A Pure exchange OLG Model Classical and Samuelson IPFE Classical case: the maximizing young agent is impatient, dissaves and borrows in his/her youth so as to consume more than his/her endowment (i.e. z t > 0 )and, under the budget constraint, must save in the old age (i.e. ζ t+1 < 0). Samuelson case: the young agent is thrifty, saves and lends in the first period (z t < 0) so as to be able to consume more than his/her endowment in the old age (ζ t+1 > 0). Example Examples of trade stationary equilibria, and lack of them, are shown in Figs. 1 3 below

29 A Pure exchange OLG Model Figure 1: Trade stationary equilibrium, Samuelson case

30 A Pure exchange OLG Model Figure 2: Trade stationary equilibrium: Classical case

31 A Pure exchange OLG Model Figure 3. No trade stationary equilibrium: Classical case

32 A Pure exchange OLG Model The differences between the classical and Samuelson cases are important and are not limited to the stationary equilibria. To avoid repetitions, however, we will discuss them only partially here, deferring a more detailed investigation after describing the second basic type of overlapping generation model, that we nickname leisure-consumption.

33 A Pure exchange OLG Model Non stationary IPFE From equilibrium conditions to difference equations Start again from equation (7) defining the equilibrium condition. H(z t, ζ t+1 ) = V (z t + w 0 )z t + U (ζ t+1 + w 1 )ζ t+1 = 0 and ask the question: is it possible to derive from (7) a difference equation of the following form? z t+1 = F(z t ) (11)

34 A Pure exchange OLG Model The answer is yes if the function U(ζ) = U (ζ t+1 + w 1 )ζ t+1 is invertible. If so, putting V = V (z t + w 0 )z t we can write and, using the fact that ζ t+1 = z t+1, ζ t+1 = U 1 [ V(z t )] (12) z t+1 = F(z t ) = U 1 [ V(z t )] (13)

35 A Pure exchange OLG Model Considering that, by assumption, U (ζ) > 0, U (ζ) < 0, we can conclude that in the classical case (where z t > 0 and therefore ζ t < 0), U(ζ) is monotonically increasing over the relevant domain and therefore invertible. To verify this, consider that du(ζ) dζ = U (ζ) + ζu (ζ) (14) which is positive if U (ζ) > 0, U (ζ) < 0 and ζ < 0 as assumed here.

36 A Pure exchange OLG Model Forward in time moving IPFE Iterations of F describe the sequences of values of z (and implicitly those of ζ) that move forward in time, starting from any given initial value [remember our caveat!] and satisfy the intertemporal equilibrium requirements. As we shall see later, those orbits can have a very complicated structure if F is non invertible.

37 A Pure exchange OLG Model Following the same line of reasoning, it is easy to show that in the Samuelson case, where ζ > 0, the function V(z t ) is invertible. Using the fact that z t = ζ t, t, we can derive the difference equation ζ t = G(ζ t+1 ) (15) where G(ζ t ) = V 1 [ U(ζ t+1 )]. In this case, the iterations of G describe the sequences of values of ζ t (and implicitly those of z t ) backward in time.

38 A Pure exchange OLG Model Only if G is invertible can we define a function T (ζ t ) = G 1 (ζ t ) such that ζ t+1 = T (ζ t ) (16) generating equilibrium sequences moving forward in time. But this is not guaranteed because the sign of du(ζ) dζ = U (ζ) + ζu (ζ) is uncertain when ζ > 0 as in the Samuelson case, and it can vary when ζ changes. The problem of backward dynamics and its economic significance is complex. Because it also appears in the leisure-consumption OLG models, we postpone its discussion.

39 A leisure consumption OLG model Leisure-consumption versus pure exchange OLG models The pure exchange OLG model can be transformed into the leisure consumption model by replacing the hypothesis of given endowments of the consumption good without production with the alternative hypothesis of a consumption good produced by variable amounts of labor. As we shall see, the resulting intertemporal equilibrium conditions (and the dynamical equations derived from them) are essentially the same in the two models

40 A leisure consumption OLG model New Hypotheses I More specifically, the assumptions H1 and H6 remain invariant whereas H2-H5 and H7 and the associate notation are changed as follows. H2, Consumption: At each period t, the young agent consumes a quantity c t 0 of the unique perishable good, and a quantity of leisure ( l l t ) 0, where l 0 denotes the constant labor endowment and l l t 0 is labor supply. The corresponding quantities for the old agent are κ t 0, ῑ 0 and ι t 0.

41 A leisure consumption OLG model New Hypotheses II H3, Production: Production takes place by means of current labor only; at each time t, output is traded at a price p t, and the wage rate is w t ; physical units of measure of output and labor are normalized so that one unit of labor yields one unit of output. In this case, profit maximization implies that w t = p t, t.

42 A leisure consumption OLG model New Hypotheses III H4, Preferences: For each generation living through the periods (t, t + 1) preferences are defined by the following utility functions: V [c t, ( l l t )]; U[κ t+1, (ῑ ι t+1 )] (4) where V and U are smooth, concave functions strictly increasing in each argument H5, Maximization: At each t, the young agent chooses the present and future levels of consumption and labor supply as functions of the observed current price (= wage rate) and the perfectly anticipated future price p t+1 (= perfectly anticipated wage rate), subject to a two period budget constraint.

43 A leisure consumption OLG model New Hypotheses IV H7, Market clearing condition: At each t, supply (l t + ι t ) and demand (c t + κ t ) for the consumption good are equal.

44 A leisure consumption OLG model Differences between the maximizing problem in the leisure consumption model (LCM) and that of the pure exchange model (PEM): In the PEM, each of the utility functions have a single argument (consumption) and, because supply of the consumption good is given, fixing consumption in each period is equivalent to fixing excess demand. In the LCM, each utility functions have two arguments consumption and leisure and therefore, within the given constraints, the same level of excess demand in each period can be obtained by different combinations of consumption and leisure (and therefore different amount of labor supply and output)

45 A leisure consumption OLG model Thus, the young agent maximization problem can be more conveniently be solved in two stages: first, maximizing utility (as a function of consumption and leisure (or labor supply) in the two periods, subject to the constraint of given levels of the excess demand in the two periods, say, z and ζ. Under the standard assumptions on the utility functions, this uniquely determines functions c t (z ), ( l l t )(z ) and κ t+1 (ζ ), (ῑ ι t+1 )(ζ ), respectively for the first and the second period The second stage of the problem consists in maximizing utility (as a function of excess demand in the two periods) subject to the intertemporal budget constraint. The derived utility functions Ṽ (z t) = V [c t (z t ); ( l l t )(z t )]; Ũ(ζ t+1) = U[κ t+1 (ζ t+1 ); (ῑ ι t+1 )(ζ t+1 )] will inherit the fundamental properties of the original functions.

46 A leisure consumption OLG model As we shall see, at this point the solution of the maximizing problem and the resulting difference equations describing intertemporal equilibria will be formally identical to those obtained for the PEM. Thus, unless otherwise indicated, our considerations concerning one model can also be applied to the other.

47 A leisure consumption OLG model The Maximizing Problem in the LCM: First Step I Formally, we have: max V [c t, ( l l t )] + U[κ t+1, (ῑ ι t+1 )] c t, ( l l t ) κ t+1, (ῑ ι t ) s.t. c t + ( l l t ) = z + l κ t+1 + (ῑ ι t+1 ) = ζ + ῑ with a Lagrangian function (17) L = { V [c t ; ( l l t )] + U[κ t+1 ; (ῑ ι t+1 )] + λ 1 [c t + ( l l t ) (z + l)] +λ 2 [κ t+1 + (ῑ ι t+1 (ζ + ῑ)]} (18)

48 A leisure consumption OLG model the F.O.C. of this problem are: L/ c t = V / c t + λ 1 = 0 L/ ( l l t ) = V / ( l l t ) + λ 1 = 0 L/ λ 1 = c t + ( l l t ) (z + l) = c t l t z = 0 L/ κ t+1 = U/ κ t+1 + λ 2 = 0 L/ (ῑ ι t+1 ) = U/ (ῑ ι t+1 ) + λ 2 = 0 L/ λ 2 = κ t+1 + (ῑ ι t+1 ) (z + ῑ) = κ t+1 ι t+1 ζ = 0 (19)

49 A leisure consumption OLG model From the F.O.C. we derive that, for each period the marginal utility of the two goods, consumption and leisure must be equal (a standard result of microeconomics). Moreover, from the assumed properties of the utility functions V and U, we can derive uniquely four functions c t (z t ), l t (z t ) and κ t+1 (ζ t+1 ), ι t+1 (ζ t+1 ) and thereby we can define a maximization problem in term of the two excess demand functions z t, ζ t+1, as follows:

50 A leisure consumption OLG model The Maximizing Problem in the LCM: Second Step max z t,ζ t+1 Ṽ (z t ) + Ũ(ζ t+1) s.t. p t (z t ) + p t+1 ζ t+1 = 0 l z t ῑ ῑ ζ t l c t, κ t 0; l t l; κ t ῑ (20) where Ṽ, Ũ have been defined above.

51 A leisure consumption OLG model The derived utility functions Ṽ, Ũ have the same fundamental properties as those of V, U and, formally, the problem above is essentially the same as that investigated for the pure exchange OLG model. In particular, the intertemporal equilibrium condition for the leisure consumption model can be written as H(z t, ζ t+1 ) = Ṽ (z t )z t + Ũ (ζ t+1 )ζ t+1 = 0 (21) which corresponds to equation (7) above.

52 Intertemporal Equilibria as Dynamical Systems The problems discussed in what follows can be referred to either of the two models and, for simplicity s sake, whenever possible we will use the same notation for both. As we know, from intertemporal equilibrium conditions we can derive certain difference equations F(z t ), G(ζ t+1 ) whose iterations describe the dynamics of excess demand (saving/dissaving) of successive generations of (young or old) different agents. Before discussing the properties of those dynamics, however, we will use the functions F, G to study the relation between the interest factor ρ and the saving/dissaving decisions over a two period life t, t + 1 of an individual agent.

53 Intertemporal Equilibria as Dynamical Systems Excess demand and interest factor Recall that the young agent s decisions concerning the excess demand in the two periods depend on the present and future prices and thereby on the interest factor ρ. From the F.O.C. of the maximizing problem, the relation between excess demand and interest rate are implicit in the following equation: ρ = p t = V (z t ) p t+1 U (ζ t+1 ) = ζ t+1 = z t+1 = ζ t+1 (22) z t z t ζ t We will discuss it for the Classical and the Samuelson case.

54 Intertemporal Equilibria as Dynamical Systems Excess demand and interest factor: the Classical case In the Classical case z t 0 ζ t 0. We assume that both the trade and the no trade stationary equilibria exist From (7) (or (21), we have derived a difference equation z t+1 = F(z t ) whose curve is shown in Figs. 4-5 From the market clearing condition, we have z t+1 = ζ t+1 and that equation can also be written as ζ t+1 = F (x t ) Thus, the curve of the function F in the plane (z t, z t+1 ) describes the relation between the excess demand of the same agent over the two periods of his/her life Making use of (22), to each point on the curve of F there corresponds a unique value of ρ t

55 Intertemporal Equilibria as Dynamical Systems Figure 4. Classical case: the function F is monotone

56 Intertemporal Equilibria as Dynamical Systems Figure 5. Classical case: the function F is non monotone

57 Intertemporal Equilibria as Dynamical Systems Excess demand and interest factor We have the following main situations: at the no trade equilibrium, in the Classical case we have ρ = V (0) U (0) > 1 [proof to follow] for any other point on the curve, the interest factor is measured by the tangent of the angle formed by the abscissa and the segment joining that point and the origin at the trade equilibrium that lies on the bisector, clearly ρ = 1 (and the interest rate is zero)

58 Intertemporal Equilibria as Dynamical Systems Excess demand and interest factor: the Samuelson case I In the Samuelson case, z t < 0, ζ t > 0. From the equations (7) or (21) we can derive a function ζ t = G(ζ t+1 ) or, using the market clearing condition, z t = G(ζ t+1 ). From (22) we can see that, for any other point on the curve of G(ζ t+1 ), the interest factor ρ is measured by the tangent of the angle formed by the ordinate axis and the segment joining the origin and that point. In the present case, at the no trade equilibrium (z t = ζ t+1 = 0), the interest factor ρ < 1 [proof to follow].

59 Intertemporal Equilibria as Dynamical Systems Excess demand and interest factor: the Samuelson case II Also from the hypothesis of strict concavity of the utility functions we can deduce that ρ t always increases as ζ t+1 increases and vice versa. Thus, dζ t+1 /dρ t > 0, i.e., the higher is the interest factor, the higher the old agent s excess demand. As concerns the effects of changes in ρ t and the young agent s excess demand, z t, two situations are possible:

60 Intertemporal Equilibria as Dynamical Systems Excess demand and interest factor: the Samuelson case III the function G(ζ t+1 ) is monotonically increasing (see Fig. 6). Then, dζ t /dρ t < 0, i.e., the old agent s excess demand is inversely related to the interest factor over the entire relevant domain G(ζ t+1 ) is not monotone: it is increasing for low values of ζ t+1 up to a certain critical value of ζ t+1 = ζ and decreases thereafter (see Fig. 7). Then, we have dζ t /dρ t > 0 and therefore dz t /dρ t < 0, for ζ < z t+1 < 0 but dz t /dρ t > 0, for z t+1 < ζ.

61 Intertemporal Equilibria as Dynamical Systems Excess demand and interest factor: the Samuelson case IV As we have already said, which of these two situations prevails depends of the relative strength of two effects of changes of ρ, the substitution effect and the income effect. We return to this question later

62 Intertemporal Equilibria as Dynamical Systems Figure 6. Samuelson case: the function G is monotone

63 Intertemporal Equilibria as Dynamical Systems Figure 7. Samuelson case: the function G is non monotone

64 Intertemporal Equilibria as Dynamical Systems Monotonicity of F (z t ) and G(ζ t+1 ) and risk aversion The mathematical properties of the functions F(z t ) (classical case) and G(ζ t+1 ) (Samuelson case) and in particular whether they are or not monotonic can be related to an attribute of the utility functions V, U known as Arrow-Pratt-DeFinetti relative risk aversion coefficient (though in the OLG models we are studying here there is no risk!) Denoting that coefficient by R V and R U for the utility functions V (z) and U(ζ), respectively, we have R V (z) R U (ζ) = V (z)z V (z) 0 = U (ζ)ζ U (ζ) 0 (23)

65 Intertemporal Equilibria as Dynamical Systems The numerators of the two fractions above measure (the absolute value) of the income effect and the denominators the substitution effect. The former effect prevails on the latter or the other way around according to whether R V (or R U ) is greater or smaller that one. R V = 1 (or R U = 1) indicates that the two effects are equal. Consider now the intertemporal equilibrium condition H(z t, ζ t+1 ) = Ṽ (z t )z t + Ũ (ζ t+1 )ζ t+1 = 0 (24) Making use of the implicit function theorem, for the Classical case for which z t 0, ζ t 0 t we can write dζ t+1 dz t = = { } H/ zt H/ ζ t+1 { V (z t )+z t V (z t ) U (ζ t+1 )+ζ t+1 U (ζ t+1 ) } (25)

66 Intertemporal Equilibria as Dynamical Systems Derivation of the result: Classical case Dividing throughout by V (z t ), we have { } dζ t+1 1 R V (z t ) = dz t [U (ζ t+1 ) + ζ t+1 U (ζ t+1 ]/V (z t ) Considering that dζ t+1 /dz t = dz t+1 /dz t, we have { } dz t+1 1 R V (z t ) = dz t [U (ζ t+1 ) + ζ t+1 U (ζ t+1 ]/V (z t ) (26) (27) Since, under our assumptions about the utility functions, in the Classical case, where z t > 0, ζ t < 0, t, the denominator of the fraction above is always positive, we conclude that dz t+1 dz t 0 iff R V (z t ) 1 (28)

67 Intertemporal Equilibria as Dynamical Systems Essentially the same procedure can be followed in the Samuelson case. Starting again from the intertemporal equilibrium condition H(z t, ζ t+1 ) = Ṽ (z t )z t + Ũ (ζ t+1 )ζ t+1 = 0 (29) and applying the inverse function theorem, in the Samuelson case (where z t < 0, ζ > 0 and therefore V (z t ) + z t V (z t ) > 0), we can write dz t dζ t+1 { H/ ζt+1 = = } H/ z t { U (ζ t+1 )+ζ t+1 U (ζ t+1 ) [V (z t )+z t V (z t )]/U (ζ t+1 } (30)

68 Intertemporal Equilibria as Dynamical Systems Derivation of the result: Samuelson case Dividing throughout by U (ζ t+1 ), and using the fact that dz t /dζ t+1 = dζ t /dζ t+1, we have { } dζ t 1 R U (ζ t+1 ) = dζ t+1 [U (ζ t+1 ) + ζ t+1 U (ζ t+1 ]/V (z t ) Thus, we conclude that (31) dζ t dζ t+1 0 iff R U (ζ t+1 ) 1 (32)

69 Intertemporal Equilibria as Dynamical Systems In the Classical case, substitution and income effects on the young agent s excess demand work in the same direction and dz t /dρ t < 0, independently of the level of z t, whereas the effects on the old agent s excess demand are conflicting and the sign of dζ t+1 /dρ t may depend on the level of z t. For certain utility functions, the substitution effect prevail throughout, dz t+1 /dρ < 0 and therefore dζ t+1 /dρ t > 0 for all levels z t in the relevant domain. For other functions, however, there may exists a critical level of z t beyond which the income effect prevails and dz t+1 /dρ > 0 and therefore dζ t+1 /dρ t < 0.

70 Intertemporal Equilibria as Dynamical Systems In the Samuelson case, we have a somewhat symmetrical situation: substitution and income effects on the old agent s excess demand work in the same direction and dζ t+1 /dρ t > 0, independently of the level of ζ t+1, whereas those effects on the young agent s excess demand are conflicting and we may have dz t /dρ t < 0 for small levels of ζ t+1, but, after a certain critical value of ζ t+1, we may have dz t /dρ t > 0. As we shall see, the cases in which there are inversion of the sign of dz t /dρ t or dζ t+1 /dρ t are particularly interesting because they are related to the complexity of the dynamics of excess demand throughout the successive generations of agents.

71 Intertemporal Equilibria as Dynamical Systems Proof that ρ NT 1 z 0 ( ζ 0) I Classical case I: ρ NT > 1 z > 0 Let: ρ NT be the interest factor at the no trade stationary equilibrium at which z = ζ = 0; be z = ζ the excess demand of the young and the old agent, respectively, at the trade stationary equilibrium. Remember that, in both the Classical and the Samuelson cases, at the trade stationary equilibrium, at which V ( z) = U ( ζ), ρ T = 1 Proof of the if part ( z > 0 ρ NT > 1). To move from the trade to the non trade equilibrium, we have to reduce the value of z and increase that of ζ. But in view of the assumptions V (z) < 0; U (ζ) < 0, it follows that V (0) > V ( z) and U (0) < U ( ζ). Therefore, ρ NT > 1.

72 Intertemporal Equilibria as Dynamical Systems Proof that ρ NT 1 z 0 ( ζ 0) II Classical case II: ρ NT > 1 z > 0 Proof of the only if part (ρ NT > 1 z > 0). We use an argumentum ad absurdum and suppose that ρ NT > 1 and z < 0. But the latter hypothesis implies that V ( z) > V (0) and U ( ζ) < U (0) and therefore by definition ρ NT = V (0) U (0) < V ( z) U ( ζ) = 1 This implies ρ NT < 1 : a contradiction

73 Intertemporal Equilibria as Dynamical Systems Proof that ρ NT 1 z 0 ( ζ 0) III Samuelson case: ρ NT < 1 ζ > 0 The proof of our proposition for the Samuelson case where z < 0, ζ > 0) is already implicit in the argument above. The if part of this case (z < 0 ρ NT < 1) is established by the only if part of the Classical case (where it is argued that ( z < 0) ρ NT < 1). The only if part (ρ NT < 1 z < 0)is implied by the if part of the Classical case from which we deduce that, assuming ρ NT < 1 and z > 0, (or, equivalently, ζ < 0),we run into a contradiction. Q.E.D.

74 An OLG Model with Production by Capital and Labor OLG Model With Production: Assumptions I Assumptions H1 (demography) and H6 (perfect foresight) remain unchanged. The others are modified as follows: H2, consumption and labor: Only young people work, earning a unit wage w, and save all of it; only old people consume

75 An OLG Model with Production by Capital and Labor Assumptions II H3, production: in each period t, a certain amount of a single homogeneous output x t, which can be either consumed or saved and invested. Output is produced by means of inputs of current labor l t, and capital k t 1 saved and invested in the previous period. The stock of capital depreciates entirely during one period of time. To fix ideas, think of output as corn, whose harvested seeds can be eaten (consumed), or sown (invested). To simplify the analysis, we assume a linear, fixed coefficient technology.

76 An OLG Model with Production by Capital and Labor Assumptions III Formally, we adopt the production function: x t = min{a l t, b k t 1 }, (33) where the constants a and b denote the output/labor and the output/capital coefficients, respectively. To ensure that the economy is viable we assume that b > 1, (e.g., to produce a certain amount of output, we have to invest a smaller quantity of it). To economize in the use of parameters, and without loss of generality, we choose the arbitrary unit of measure of labor so that a = 1.

77 An OLG Model with Production by Capital and Labor Assumptions IV Output is traded each time at a unit price p t. Physical units of measure of output and labor are normalized so that one unit of labor yields one unit of output. In this case, profit maximization implies that w t = p t, t (real wage, w/p must be equal to the marginal productivity of labor which is equal to 1). Thus, l t = x t = bk t 1, t. H4, preferences: For each generation living through the periods (t, t + 1) preferences are defined by the utility function [U(c t+1 ) V (l t )] where V and U are smooth functions, increasing in each argument; U is concave and V convex. [Notice that V (l) measures the disutility of labor.].

78 An OLG Model with Production by Capital and Labor Assumptions V H5, maximization: At each time t, the young agent, given the the actual and expected prices p t, p t+1, must choose the amount of current labor supplied in view of the expected consumption in his/her old age, c t+1. The problem can be formalized as follows: max{u(c t+1 ) V (l t )} c t+1,l t s.t. p t+1 c t+1 = p t l t, c t, l t 0, t (P2)

79 An OLG Model with Production by Capital and Labor Assumptions VI From the F.O.C. of this problem we obtain the intertemporal equilibrium condition: U(c t+1 ) = V(l t ) (34) where U(c t+1 ) = U (c t+1 )c t+1 ) and V(l t ) = V (l t )l t. H7 Market clearing: At each time t, the amount of output saved is entirely invested, i.e.: k t = x t c t (35)

80 An OLG Model with Production by Capital and Labor Dynamic Equations I From the results above, we can now try to obtain a system of two difference equations describing the evolution forward in time of labor and consumption (or, equivalently, of any pair of variables out of the quartet l, c, k, x (for each t, the values of any two of them determine automatically the values of the remaining two). From the production function and the market clearing condition we can obtain l t+1 = b(l t c t ) (36) determining the value of labor at time t + 1 as a function of the values of labor and consumption at time t.

81 An OLG Model with Production by Capital and Labor Dynamic Equations II We now need a second equation determining c t+1 as a function of c t and l t (or just one of them). Whether this is possible or not, depends on whether the function U(c t+1 ) is invertible For example, if we adopt a utility function of the constant relative risk aversion (CRRA) type, such as U(c) = (1/α)c α, 0 < α < 1), and we put V (l t ) = (1/β)l β t ) β > 1, simple calculations lead to the equation:

82 An OLG Model with Production by Capital and Labor Dynamic Equations III A Forward in Time Dynamical System c t+1 = f (l t ) = U 1 V(l t ) = [V(l t )] 1/α (37) and therefore to the complete system: c t+1 =l γ t l t+1 =b(l t c t ) (38) with γ = β/α > 1.

83 An OLG Model with Production by Capital and Labor Dynamic Equations IV Stationary Equilibria For this system only two stationary equilibria exist, namely: E 1 : l 1 = 0; c 1 = 0, i.e. the origin of the coordinates E 1 : l 2 = ( ) b 1 1/(γ 1) b ; c2 = ( ) b 1 γ/(γ 1) b Under our assumptions, both l 2 and c 2 are positive

84 An OLG Model with Production by Capital and Labor Dynamic Equations V Stability of Equilibrium Local stability can be established by considering the Jacobian matrix J evaluated at each of the two stationary equilibria and verifying the following conditions: (1) 1 TrJ + DetJ > 0 (2) 1 + TrJ + DetJ > 0 (3) DetJ < 1

85 An OLG Model with Production by Capital and Labor Dynamic Equations VI Equilibrium E 1 For E 1 we have: ( ) 0 0 J 1 = b b We can immediately see that for b > 1 condition 1 is always violated, hence E1 is unstable.

86 An OLG Model with Production by Capital and Labor Dynamic Equations VII Equilibrium E 2 For E 2, the Jacobian matrix is ( 0 γ( b 1 J 2 = b b b ) ) It can be easily shown that, for γ > 1; b > 1, conditions 1 and 2 are always verified, thus stability depends on condition 3. E 2 is more likely to be unstable the larger the values of the output/capital ratio b and the coefficient γ (hence the larger β and the smaller α).

87 An OLG Model with Production by Capital and Labor Although for this case there is a rigorous proof of the existence of periodic and aperiodic non chaotic solutions, the most exciting dynamics are found when U(c) is not invertible. This occurs, for example, if the function U belongs to the class of constant absolute risk aversion (CARA) utility functions, such as: U(c) = r e c, (39) where r is a parameter > 0, whence U(c) = U (c)c = r c e c (40)

88 An OLG Model with Production by Capital and Labor A Backward in Time Dynamical System The function U(c) has a one-hump form and does not admit a global inverse. In this case, we can use the fact that the function V(l t ) is always invertible (this can be gathered by considering that V(l t ) = V (l t )l t and therefore dv(l t )/dl t = V (l t ) + l t V (l t ) which, under the postulated assumptions, is always positive).

89 An OLG Model with Production by Capital and Labor Therefore, we can now write the system of two difference equations: l t =g(c t+1 ) where g(c t+1 ) = [rc t+1 e c t+1] 1/β. c t =g(c t+1 ) (1/b)l t+1 (41)

90 An OLG Model with Production by Capital and Labor Stationary Equilibria I For this systeme there exist two stationary equilibria. It is easy to see that the first of them is the origin of coordinates, i.e. E 1 : c 1 = l 1 = 0. On the contrary, it may be difficult or even impossible to define exactly the second stationary equilibrium in terms of two functions of the system parameters. Nevertherless, we can find out all we are interested in here, namely the existence of a unique non zero stationary equilibrium E 2 : ( c 2, l 2 ) with c 2 and l 2 both positive for r > 0, b > 1, β > 1.

91 An OLG Model with Production by Capital and Labor Stationary Equilibria II This result can be established by considering that at the equilibrium E 2, we must have f 1 (c) = f 2 (c) where f 1 (c) = c β 1 (b/(b 1)) β and f 2 (c) = re c. It easy to show that there exists one and only one intersection of the curves of these two functions in the positive quadrant of the space (f (c), c). (See Fig. 8). (The first of the two dynamic equations of the system defines l 2 as a function of c 2.)

92 An OLG Model with Production by Capital and Labor Stationary Equilibria III From the equilibrium condition f 1 (c) = f 2 (c) we can also derive a function ( β r : [0, ) [0, ); r(c) = e c β 1 2 c b 2 b 1) which is monotonically increasing and can be used to study stability and bifurcations of equilibrium E 2 as functions of the parameter r. (See Figure 9.)

93 An OLG Model with Production by Capital and Labor

94 An OLG Model with Production by Capital and Labor Figure 9. The function r( c 2

95 An OLG Model with Production by Capital and Labor Stability of Stationary Equilibria The Jacobian matrix of this system can be written as: ( 0 g J 2 = ) (c) 1/b g (c) Considering that lim c 0 g (c) =, we conclude that E 1 is always unstable.

96 An OLG Model with Production by Capital and Labor To establish the stability conditions for E 2, we can write the Jacobian matrix as follows: ( 0 b β(b 1) J 2 = (1 c ) 2) b 1/b β(b 1) (1 c 2) Thus, we have TrJ 2 = ( b DetJ 2 = ( 1 β(b 1) β(b 1) ) (1 c 2 ) = TrJ 2 /b. ) (1 c 2 ) and From this we can canclude that: Stability condition (1) is always satisfied under our assumptions concerning the parameters r, b, β and c 2 > 0. Condition (2) is verified if c 2 > 1 + β(b 1)/(1 + b) and condition (3) if c 2 > 1 β(b 1)

97 An OLG Model with Production by Capital and Labor Given the initial conditions for l and c, this system will determine the dynamics of the variables (c, l) in the past, a situation we have already encountered in the two preceding OLG models when, in the Samuelson case, the function U was not invertible. In what follows, we will devote some special sessions to the discussion of the various types of interesting dynamics occurring in OLG models and to the special problem of backward dynamics.

98 An OLG Model with Production by Capital and Labor Selected References I Allais, M., Economie et Interet, Paris, Imprimerie National Benhabib J. and R. Day, A characterization of erratic dynamics in the overlapping generations model, Journal of Economic Dynamics and Control, 4, Blanchard, O.J. and S. Fischer, Lectures on Macroeconomics, MIT Pres, Ch. 3, , Ch. 4, and Ch. 5, Cass, D., On Capital Overaccumulation in the Aggragative, Neoclassical Model of Economic Growth: A Complete Characterization, Journal of Economic Theory, 4,

99 An OLG Model with Production by Capital and Labor Selected References II Diamond, P., National debt in a neoclassical growth model, American Economic Review, 55 (5), Gale, D., Pure exchange equilibrium of dynamic economic models, Journal of Economic Theory, 6, Geanakoplos, J., 2008, Overlapping Generations Models of General Equilibrium, Cowles Foundation Discussion Papers N Gazzola, G. and A. Medio, Global Sunspots in OLG Models. Theory and Computational Analysis, Journal of Macroeconomics, 28 (1), Grandmont, J.M., On endogenous competitive business cycles, Econometrica, 53,

100 An OLG Model with Production by Capital and Labor Selected References III Grandmont, J.M., Local bifurcations and stationary sunspots, in: W.A. Barnett, J. Geweke and K. Shell (eds.), Economic complexity: chaos, sunspots, bubbles and nonlinearity, Cambridge University Press Medio, A. and B. Raines, Backward Dynamics in Economics. The inverse Limit Approach, Journal of Economic Dynamics and Control, 31, Reichlin, P., Equilibrium cycles in an overlapping generations economy with production, Journal of Economic Theory, 40, Samuelson, P. A An Exact Consumption-Loan Model of Interest With or With out the Social Contrivance of Money, Journal of Political Economy, 66(6),

101 An OLG Model with Production by Capital and Labor Selected References IV Weil, P. 2008, Overlapping Generations: The First Jubilee, Journal of Economic Perspectives, 22, Woodford, M., Indeterminacy of Equilibrium in the Overlapping Generations Model: A Survey, Columbia University, mimeo.