Zhixiang Liu

Laurent Ciarletta
email: lau-rent.ciarletta@loria.fr

Chi Yuan
email: chiyua@encs.concordia.ca

Youmin Zhang
email: ymzhang@encs.concordia.ca

Didier Theilliol
email: didier.theilliol@univ-lorraine.fr

Path Following Control of Unmanned Quadrotor Helicopter with Obstacle Avoidance Capability

This paper proposes a new path following methodology combining with a obstacle avoidance scheme for unmanned quadrotor helicopter (UQH) capable of working in the cluttered and hazardous environments. A new cross-track error prediction based mechanism, where the cross-track error is estimated by utilizing the extend Kalman filter (EKF), is first developed for the path following scheme. Then, the UQH is equipped with obstacle avoidance capability employing a lightcomputational approach, the visibility graph algorithm. The priority of UQH is to switch to obstacles avoidance maneuvering in the presence of obstacles, and continue to execute the assigned mission after avoiding all hazardous objects blocking the desired path. The control system developed for attitude and position control of UQH is also introduced. Finally, extensive simulation studies on a nonlinear model of UQH with a series of dangerous scenarios are conducted to demonstrate the effectiveness of the proposed methodology.

I. INTRODUCTION

The last decades have seen a tremendous progress in the development of unmanned aerial vehicles (UAVs). A growing number of research institutes, universities, governments, and commercial entities across the world are developing and employing UAVs for a diverse range of applications, such as scientific research [START_REF] Gomez-Balderas | Tracking a ground moving target with a quadrotor using switching control[END_REF], environmental monitoring and surveillance [START_REF] Yuan | A survey on technologies for automatic forest fire monitoring, detection and fighting using UAVs and remote sensing techniques[END_REF], [START_REF] Yuan | Aerial images-based forest fire detection for firefighting using optical remote sensing techniques and unmanned aerial vehicles[END_REF], natural resources exploration [START_REF] Rango | Unmanned aerial vehicle-based remote sensing for rangeland assessment, monitoring, and management[END_REF], post-disaster search and rescue [START_REF] Shakernia | Landing an unmanned air vehicle: Vision based motion estimation and nonlinear control[END_REF], and military missions [START_REF] Escareño | Trajectory control of a quadrotor subject to 2D wind disturbances[END_REF]. Unmanned quadrotor helicopters (UQHs), as an important group of UAV, possess tremendous advantages comparing with other manned/unmanned aerial vehicles including affordable cost of development [START_REF] Liu | A learning-based fault tolerant tracking control of an unmanned quadrotor helicopter[END_REF], decreased mechanical structure complicacy [START_REF] Alexis | Switching model predictive attitude control for a quadrotor helicopter subject to atmospheric disturbances[END_REF], easy-to-fly [START_REF] Zhang | Development of advanced FDD and FTC techniques with application to an unmanned quadrotor helicopter testbed[END_REF], enhanced maneuverability and outstretched deployability [START_REF] Castillo | Real-time stabilization and tracking of a four-rotor mini rotorcraft[END_REF], [START_REF] Dierks | Output feedback control of a quadrotor UAV using neural networks[END_REF], as well as experimental platform for newly developed techniques [START_REF] Hoffmann | Precision flight control for a multi-vehicle quadrotor helicopter testbed[END_REF].

As the complexities of application grow, UQHs are accordingly required to deploy in more cluttered, sophisticated, and hazardous environments, this situation strongly demands UQHs to be equipped with enhanced reliable and safe instruments as well as more robust guidance and control algorithms for guaranteeing satisfactory performance of tasks execution and preventing collision with buildings, mountains, trees, other manned/unmanned vehicles, etc.. However, semiautonomy up to date is still favored over full autonomy due to the diverse nature of missions and limited obstacle avoidance capabilities of UQHs. Therefore, this work is intended to establish the study of methodology with satisfactory and safe performance to improve the autonomy of current UQHs including increasing application diversity and minimizing the amount of human supervision.

The existing mission execution can be generally partitioned into three predominant sorts: set-point stabilization, trajectory tracking, and path following [START_REF] Liu | Unmanned surface vehicles: An overview of developments and challenges[END_REF]. This study falls into the path following problem of UQH. From the literature, there exist two categories of design methods: 1) individual design of guidance and control systems, which separates the system into an inner control loop and an outer guidance loop; 2) integrated design of guidance and control systems. In order to simplify the design procedure and separately manage the guidance-related disturbances and uncertainties problems in the guidance system without introducing them to control system and complicating the design and debugging procedures, this study chooses to separately design guidance and control systems. Numerous existing research works have been dedicated to the field with individual design philosophy. In [START_REF] Consolini | Path following for the PVTOL aircraft[END_REF], the robust nonlinear control theory is used for maneuvering a UQH to follow a class of smooth Jordan curves. [START_REF] Sanghyuk | Performance and Lyapunov stability of a nonlinear path following guidance method[END_REF] presents a modified purepursuit path following method which uses inertial velocity in the calculation of commanded lateral acceleration. With this method, both straight and curved line paths can be well followed. A nonlinear output-feedback control method, which employs the global exponential observer, Lyapunov's direct method and backstepping technique, is developed in [START_REF] Duc | On global tracking control of a VTOL aircraft without velocity measurements[END_REF] for path following application. In [START_REF] Nelson | Vector field path following for miniature air vehicles[END_REF], a vector-field based path following scheme is devised to generate desired course references to attitude control in the inner-loop system. [START_REF] Kaminer | Path following for small unmanned aerial vehicles using L1 adaptive augmentation of commercial autopilots[END_REF] introduces a path following control law which relies on a nonlinear control strategy derived at the kinematic level.

Regarding obstacle avoidance methods, a large number of heuristic obstacle avoidance methods have been developed for UAVs, such as particle swarm optimization (PSO) [START_REF] Foo | Path planning of unmanned aerial vehicles using B-splines and particle swarm optimization[END_REF], genetic algorithm (GA) [START_REF] Lai | Time-optimal control of a hovering quad-rotor helicopter[END_REF], artificial potential field (APF) [START_REF] Franchi | Shared control: Balancing autonomy and human assistance with a group of quadrotor UAVs[END_REF], [START_REF] Liu | Fault-tolerant formation control of unmanned aerial vehicles in the presence of actuator faults and obstacles[END_REF], and probabilistic roadmap-based method [START_REF] Hrabar | 3D path planning and stereo-based obstacle avoidance for rotorcraft UAVs[END_REF]. Considering the low computational capability of the onboard processor of UQH as well as high maneuverability of UQH, this study selects a light-computational obstacle avoidance method, the visibility graph [START_REF] Tomás | An algorithm for planning collisionfree paths among polyhedral obstacles[END_REF], as the obstacle avoidance design.

Following the concept of hierarchical structure design, a hybrid system, which consists of three functionality modules including the path following, obstacle avoidance, and motion control, is developed in this study. The first module of the proposed strategy is devised for operating the UQH to follow the desired path with satisfactory performance. To improve the performance of path following, a path following method with the capability of predicting cross-track error is developed; the cross-track error is estimated by an extended Kalman filter (EKF). The second module is designed for obstacle avoidance by modifying the reference heading angle to control system when there are any obstacles blocking the desired path. In order to significantly reduce the excessive computation burden of onboard processing devices, a relatively simple but effective method, the visibility graph, is utilized for the design of obstacle avoidance mechanism. The last module is the motion control system which maneuvers UQH to follow the reference path assigned from the first and second modules. The traditional proportional-integralderivative (PID) control approach is used for the height maintenance, while a fuzzy proportional-derivative (FPD) control method is chosen for operating the attitude of UQH, expecting that the UQH is capable of adapting with different working conditions. Finally, numerical simulations on a nonlinear UQH model is carried out to verify the efficacy of the proposed methodology. The remainder of this paper is arranged as follows: Section 2 provides some preliminaries for the design of proposed methods. Section 3 introduces the design details of proposed approaches. Section 4 addresses the conducted numerical simulations and their corresponding performance analyses. The last section summarizes the conclusions and potential future works.

II. PRELIMINARIES A. Nonlinear Model of Unmanned Quadrotor Helicopter

The UQH, as concisely depicted in Fig. 1, is normally actuated by four motor-driven propellers which are located at its four corners (front, rear, left, and right, respectively). These propellers generate corresponding thrusts u 1 , u 2 , u 3 , and u 4 . The motion of UQH can be briefly illustrated as follows: 1) distributing identical amount of control signal to each motor to obtain the vertical translation; 2) assigning a different amount of control signals to the opposite motors to achieve the horizontal translation [START_REF] Liu | A learning-based fault tolerant tracking control of an unmanned quadrotor helicopter[END_REF].

As described in [START_REF] Liu | A learning-based fault tolerant tracking control of an unmanned quadrotor helicopter[END_REF], a common UQH dynamical model with respect to the earth-fixed coordinate system can be The applied torque in θ direction u φ (t)

The applied torque in φ direction u ψ (t)

The applied torque in ψ direction Kn

ẍ = (sinψsinφ + cosψsinθcosφ)u z (t) -K 1 ẋ m ÿ = (sinψsinθcosφ -cosψsinφ)u z (t) -K 2 ẏ m z = (cosθcosφ)u z (t) -K 3 ż m -g φ = u φ (t) -K 4 φ I x θ = u θ (t) -K 5 θ I y ψ = u ψ (t) -K 6 ψ I z . (1)
Furthermore, from the relationship between accelerations and lift/torques, one can obtain that:

    u z (t) u θ (t) u φ (t) u ψ (t)     =     1 1 1 1 L -L 0 0 0 0 L -L C m C m -C m -C m         u 1 (t) u 2 (t) u 3 (t) u 4 (t)     . (2)
The propeller force and its corresponding pulse width modulation (PWM) signal have the following relationships:

u i (t) = K m ω m s + ω m u ci (t). (3)
The above-mentioned symbols are all defined in Table I.

III. PATH FOLLOWING SCHEME DESIGN

As outlined in Fig. 2, the path following and control schemes are separately designed, for the purpose of avoiding any errors defecting the reference heading angle before introducing it to the control system.

A. Cross-Track Error Prediction Based Path Follow Rule

In order to successfully follow the assigned path, UQV is required to move through a set of predefined waypoints along the path with a constant forward speed. As illustrated in Fig. 2: Illustration of the proposed path following scheme. Fig. 3: Geometry of the proposed path following scheme. Fig. 3, a two-dimensional (2D) continuous C 1 parametrized path, which is chosen in this study, goes through a set of successive waypoints P (x k , y k ) for k = 1, ..., i.

For an arbitrary position of UAV locates at P (x, y), the cross-track error, which is defined as the orthogonal distance to the path tangential reference frame, is calculated by:

ε = (y k+1 -y k )x + (x k -x k+1 )y + y k x k+1 -x k y k+1 (y k+1 -y k ) 2 + (x k -x k+1 ) 2 ,
(4) where P (x k , y k ) and P (x k+1 , y k+1 ) denote the last and next waypoints, respectively.

In order to follow the desired path, a reference heading angle is required. Before calculating this reference angle, in this study, a prediction of cross-track error is first defined as follows:

ε = ε + U f × T s × sin(∆ψ), (5)
where ε is the prediction of cross-track error, U f represents the forward velocity of UAV, T s , which denotes the updating rate, is selected to calculate ε for the next period. ∆ψ = ψ ref -ψ is the error between the path-tangential angle ψ p and real heading angle ψ of UAV. The path-tangential angle ψ p is constant when the reference path is straight, while ψ p varies in each sampling time when the desired path is a curve. Based on the prediction of cross-track error, the reference heading angle ψ d can then be obtained:

ψ d = ψ p + K c ε, (6)
where K c is a coefficient used for tuning the conversion from cross-track error to reference heading angle.

In practice, however, the fixed gains T s and K c can be well tuned for rapidly reducing the cross-track error, while an undesirable fluctuation around the reference path may be caused, vice versa. Thereby the prediction of cross-track error is intended to be redesigned for the purpose of fast decreasing the cross-track error, while avoiding the unexpected oscillation around the path as well. The improvement to Eq. ( 6) is then made by designing the following saturation function:

f (ε) = T 1 ε |ε|(1 + e -T2(|ε|-T3) ) , (7)
where T 1 , T 2 , and T 3 are selected according to the practical situation including forward speed.

It is worth-mentioning that the introduction of saturation function to the prediction of cross-track error can improve the performance of path following from two aspects: 1) guarantee a more rapid operation of UAV approaching to the desired path when it is far away from the path; 2) smoothly manoeuvre the UAV to follow the desired path when it is close to the path.

Consequently, Eq. ( 6) with combination of Eq. ( 7) becomes:

ψ d = ψ p + K c f (ε). (8)
B. Improving Path Following Rule with EKF Although Eq. ( 5) provides a solution of predicting the cross-track error, an improvement is still desirable due to the fact that this kinematics based method tends to be sensitive to environmental disturbances, noises, and various of uncertainties. In this study, EKF is further employed to modify the design of prediction. The reason behind this selection is EKF's advantage in estimation based on its statistical characteristic, and it has also been popularly and successfully applied in a variety of practical applications [START_REF] Ahn | Online tuning fuzzy PID controller using robust extended Kalman filter[END_REF].

First, the following system model and measurement model are established [START_REF] Grewal | Kalman filtering: Theory and practice using MATLAB[END_REF]:

x i = f (x i-1 ) + ω i-1 , d i = h(x i ) + ν i , (9)
where the stochastic variable x i is the system state at time t i . ω i-1 and ν i denote the process and measurement noises, respectively. f (•) and h(•) represent the nonlinear vector functions of the states.

Then, the problem introduced by EKF is to find an estimate xi of x i given d j (0, ..., i).

If the nonlinearities of system (9) are sufficiently smooth, this system can be expanded around the state estimate xi using Taylor series as:

f (x i ) = f (x i ) + F i × (x i -xi ) + higher order terms, h(x i ) = h(x i ) + H T i × (x i -xi ) + higher order terms, ( 10
) where F i = ∂f (x) ∂x | x=xi , H T i = ∂h(x) ∂x | x=xi .
Neglecting the higher order terms in Eq. ( 10), Eq. ( 9) can then be approximated by:

x i = F i-1 x i-1 + ω i-1 + φ i-1 , d i = H T i x i + ν i + ϕ i-1 , (11)
where

φ i-1 = f (x i-1 ) -F i-1 xi-1 , ϕ i-1 = h(x i-1
) -H T i-1 xi-1 . Therefore, the desired estimate xi can be obtained by the following EKF recursive equations [START_REF] Grewal | Kalman filtering: Theory and practice using MATLAB[END_REF]:

                 F i-1 = ∂f (x) ∂x x=x i-1|i-1 , H i = ∂h(x) ∂x x=x i|i-1 , K i = P i|i-1 H T i (R i + H i P i|i-1 H T i ) -1 , xi|i = f (x i-1|i-1 ) + K i [d i -h(x i|i-1 )], P i|i = F i-1 (P i-1|i-1 -K i H i P i-1|i-1 )F T i-1 + Q i-1 , (12
) where d i represents the observation vector, K i is the Kalman gain, P i denotes the covariance matrix of state estimation error, and the estimated state xi|i is the optimal solution which approaches the conditional mean value

E[x i |(d 0 , d 1 , . . . , d i )].
Ultimately, choosing the state of EKF as x = [U f , ∆ψ] T , and applying the Kalman recursion [START_REF] Hoffmann | Precision flight control for a multi-vehicle quadrotor helicopter testbed[END_REF], the prediction of cross-track error ε can be obtained for the computation of reference heading angle. Fig. 4: Geometry of the adopted obstacle avoidance method.

IV. OBSTACLE AVOIDANCE SCHEME DESIGN

Maintaining the safety of UQHs is generally a priority over mission execution including trajectory tracking and path following. Therefore, in the event of encountering obstacles, the current objective is first to avoid any potential collisions, then return to the assigned mission. Considering the computational capability of the processor onboard UQH and high maneuverability requirement in practice, this study utilizes a relatively simple but effective roadmap-based obstacle avoidance method, the visibility graph, to avoid any obstacles approaching to the UAV.

The general concept of visibility graph method is to connect UAV and the next objective waypoint, two possibility can occur: 1) the next objective waypoint is visible when there is no obstacles between UAV and the waypoint, and the path is approachable; 2) otherwise, the path to objective waypoint is blocked by obstacles, then an optimal path (the Fig. 5: Composition of controllers to follow the desired path.

shortest path) of avoiding the obstacle and moving towards to objective waypoint is achievable by comparing all of the available connections from UAV, through one of the detected vertex of obstacle, to objective waypoint.

As illustrated in Fig. 4, in the absence of obstacles, UAV follows the desired path by tracking the reference heading angle ψ d generated according to the desired path; but in the presence of obstacles, UAV stops following the desired path and changes to follow the obstacle avoidance heading angle to avoid the approaching obstacles.

In order to calculate the corresponding heading angle to keep the UAV away from colliding with obstacles, the following rule is established:

ψ d = ψ l if δ l ≤ δ r , ψ r if δ l > δ r , (13)
where δ l = |ψ l -ψ| and δ r = |ψ r -ψ|. ψ l and ψ r denote the leftmost and rightmost vertex of obstacle combining with a safety distance with respect to UAV, respectively. This safety distance is chosen by inflating the obstacle with a specific radius according to the forward velocity of UAV, minimum turning radius of UAV, and the practical safety requirement.

V. CONTROL SYSTEM DESIGN

As shown in Fig. 5, the tasks of control system design in this study include:

1) To maintain a desired height and follow the reference yaw angle. 2) To keep a constant forward motion by achieving an acceptable performance of pitch angle control. 3) To guarantee the stability of UAV with small deviations from the hovering flight.

A PID controller is chosen for height control, controller gains can be determined by:

u z = K h P (z d -z) + K h I (z d -z)dt + K h D (-ż), (14)
where z d is the desired height, and K h P , K h I and K h D are the gains of PID controller.

A PD controller is designed for the attitude operation (yaw and pitch angle control), controller gains is calculated as:

u θ = K θ P (θ d -θ) + K θ D (p d -p), u ψ = K ψ P (ψ d -ψ) + K ψ D (r d -r), (15)
where K ψ P and K ψ D are the gains of PD controller for yaw angle, while K θ P and K θ D are the gains for pitch angle. p and r denote the angular velocities of pitch and yaw.

In order to improve the robustness of attitude controller against structural changes and uncertainties in system parameters as well as environmental disturbances, the attitude controller is further tuned by the fuzzy logic control (FLC) rule. The following simple linear transformation is used for calculating controller gains:

K P = (K max P -K min P )∆K P + K min P , K D = (K max D -K min D )∆K D + K min D , (16)
where

[K min P , K max P ], [K min D , K max D
] represent the predefined ranges of K P and K D , respectively. ∆K P and ∆K D are determined by the following linguistic rules of FLC:

If e k is A i and ∆e k is B i , then ∆K P is C i and ∆K D is D i , (17)
where A i , B i , C i , and D i represent the fuzzy sets corresponding to e k , ∆e k , ∆K P , and ∆K D .

Due to the space limit, further details of the design procedure is omitted here. Readers can refer to other literature for the similar design method [START_REF] Amoozgar | Fault-tolerant fuzzy gain-scheduled PID for a quadrotor helicopter testbed in the presence of actuator faults[END_REF].

VI. SIMULATIONS

To demonstrate the effectiveness of the proposed method, numerical simulations on a nonlinear UQH model are conducted. Two scenarios are selected in the simulation: 1) Scenario 1: In the first scenario, as shown in Fig. 6, the UAV starts at the left bottom corner (5m, 20m), while the end point is located at the right top corner (250m, 150m). One static obstacle (the red rectangle) situates at point (160m, 80m) of the map.. 2) Scenario 2: In this scenario, there are three static obstacles (see Fig. 8) situating at points (50m, 40m), (95m, 60m), and (170m, 70m), respectively.

UQH is set at a constant forward speed (10m/s) and a constant height. UQH is assumed to be deployed in 2dimensional (2D) space. A straight line starting from (0, 0) and ending at (250m, 150m) is chosen as the desired path for both scenarios.

A. Results of Scenario 1

From Fig. 6, it shows that the desired path is followed by the UQH, while the obstacle blocking the path is also successfully avoided, finally, the UQH can return back to the desired path. Fig. 7 displays the cross-track error is effectively and promptly estimated with an acceptable residual.

B. Results of Scenario 2

Fig. 8 shows UQH safely passes the three obstacles, and then continues to path following execution.

As observed from Fig. 9, the cross-track error can still be well estimated even UQH has experienced a series of sophisticated manipulation.

VII. CONCLUSIONS

In this paper, a new cross-track error prediction based path following method is presented. Before distributing to the control system, the reference heading angle is generated by the path following module, and to be modified by the obstacle avoidance module in the presence of obstacles. A fuzzy proportional-derivative control approach is then employed for tracking the desired heading angle. Simulations have validated the effectiveness of the proposed approaches.

Potential future work can be extended to the validation of the developed algorithm on a real unmanned quadrotor helicopter in the field test environment. Another possible research direction is suggested to improve the current reference heading angle calculation rule making the path following strategy capable of adapting to various of working and environmental conditions with superior performance.

Fig. 1 :

 1 Fig. 1: Schematic diagram of the UQH.

  (n = 1, 2, ..., 6) Drag coefficients u i (t) (i = 1, 2, 3, 4) Thrust of each rotor L Distance of UQH's gravity and propeller Cm Thrust-to-moment scaling factor g Acceleration of gravity m UQH mass Ix Moment of inertia along x direction Iy Moment of inertia along y direction Iz Moment of inertia along z direction ωm Actuator bandwidth Km A positive gain u ci (t) (i = 1, 2, 3, 4) PWM signals distributed to each rotor written as follows:

Fig. 6 :Fig. 7 :Fig. 8 :

 678 Fig. 6: Performance of path following and obstacle avoidance schemes.

Fig. 9 :

 9 Fig. 9: Performance of cross-track error prediction.

TABLE I :

 I Nomenclature (earth-fixed coordinate system)

	Symbols	Explanation
	x, y, z	Coordinates of UQH at center of mass
	θ	Pitch angle
	φ	Roll angle
	ψ	Yaw angle
	uz(t)	Total lift force
	u

θ (t)

Zhixiang Liu and Laurent Ciarletta are with the Lorraine Research Laboratory in Computer Science and its Applications (LORIA),


Gilles Saint-Paul

Saint- Paul Gilles

Satiation Secular

Gilles Saint

Secular Satiation

Keywords: E13, E14, E21, E22, E23 Growth, satiation, innovation, new products, consumer society, leisure society, labor supply, multiple equilibra, strategic complementarities JEL -E13, E14, E21, E22, E23 Keywords -Growth, satiation, innovation, new products, consumer society, leisure society, labor supply, multiple equilibra, strategic complementarities. 1

Introduction

Limits to growth are generally considered to lie on the supply side: limited resources, running out of good ideas, etc. Yet as societies become more a-uent, the question arises as to whether consumers will be able to absorb an ever increasing output level. Indeed we observe that over time growth becomes increasingly qualitative as opposed to quantitative.

Productivity improvements are matched by greater product quality and variety as well as a secular decline in working time.

This paper analyzes growth dynamics when needs may be subject to satiation. I assume that utility derived from any individual good reaches a maximum at a …nite consumption level. An immediate implication is that there may be two kinds of long-term steady states: a non satiated one where the capital stock is determined by the standard Keynes-Ramsey condition, and a satiated one where the marginal utility of consumption is zero, and there may be excess capital (in that its net marginal product is negative) in the long run.

I then allow for new products to be introduced. While each product is subject to satiation, there is no satiation in the taste for variety: utility can potentially be raised without bounds if enough new goods are introduced. I show that a social planner will never elect a trajectory which converges to a satiated steady state. In such a steady state, the marginal utility of consumption is zero, and so is the opportunity cost of introducing an additional variety, which in turn has a …rst order positive e¤ect on utility.

I then consider a decentralized economy where, in a standard fashion, new varieties are introduced by pro…t-seeking innovators, who derive subsequent monopoly rents. It is shown that the economy may converge to satiated steady states. At the margin of those equilibria, the market does not deliver the required innovation level to lift the economy out of its satiation trap, despite that the social opportunity cost of innovation is zero.

The associated market failure goes beyond the usual appropriability and business stealing issues that the literature has identi…ed since [START_REF] Dixit | Monopolistic competition and optimum product diversity[END_REF]. Relative prices are pinned down by the production side, and equal to marginal rates of transformation. But marginal rates of substitution are not de…ned since the marginal utility of consuming any variety is zero. The price system does not convey the information that the marginal utility of consumption is zero. A reduction in the consumption of existing goods entails secondorder welfare losses, while a¤ecting these resources to innovation generates …rst-order welfare gains. By pricing innovation using prevailing prices, however, markets evaluate costs and bene…ts as being of the same order of magnitude; they do not internalize the facts that new products would eliminate satiation and restore a strictly positive marginal utility of consumption, while absent innovation opportunity costs remain second order.

The model also highlights an important strategic complementary, which generates multiple equilibrium trajectories. Unlike vertical innovation, horizontal innovation raises the marginal utility of aggregate consumption. Consequently, expecting more varieties to be introduced in the future raises the incentives to save. Higher savings in turn reduces interest rates, which favors innovation. I show that equilibrium trajectories that converge to a non satiated equilibrium where the modi…ed golden rule holds coexist with trajectories that converge to satiated equilibria. In the latter, the capital stock and the number of varieties are lower than in the former, while interest rates are higher and wages are lower.

The model can be extended in several ways. I discuss the consequences of consumer heterogeneity in rates of time preference, characterize balanced growth paths when TFP follows a deterministic trend, and discuss the implications of endogenous labor supply.

In particular, if TFP grows, endogenous labor supply does not rule out balanced growth paths such that consumers are asymptotically satiated and the economy grows at a positive rate. It may be that labor supply does not fall to zero because wages grow faster than the decline in the marginal utility of consumption. It may also be that while labor supply falls to zero, output nevertheless grows due to technical progress and capital accumulation.

Under endogenous labor supply, a "consumer society" balanced growth trajectory such that working time remains constant and consumers are not satiated may coexist with a "leisure society" trajectory such that working time trends downwards, converging to zero, while consumers are asymptotically satiated. This feature is reminiscent of debates about the relative merits of the "American model" vs. the "European model", and suggests that people may have coordinated on di¤erent equilibrium paths on di¤erent sides of the Atlantic. It is shown that the leisure society is associated with lower growth in output, physical capital and the number of varieties than the consumer society, while being more intensive in innovation relative to physical capital.

Satiation in the consumption of many category of goods is consistent with the study of Engel curves. For example, [START_REF] Moneta | The evolutions of Engel curves and its implications for structural change[END_REF] estimate cross-sectional Engel curves for various range of products in various years and …nd that they reach an interior maximum in many instances. Of course, a mode may indicate that the good becomes inferior beyond some income level. However, this is implausible for some goods (such as travel services, which has a mode in 1968 and 1988), while for some other goods the Engel curve ‡attens out rather than having a strict mode, which is more consistent with satiation (this is the case for leisure goods or household goods in 1968)1 . Interestingly, these curves shift over time, in such a way that a category of goods which exhibits satiation may no longer do so, or vice-versa. This may be due to factors that are unrelated to the theme of this paper, such as shifts in relative prices. For example, energy has a mode in the 1968 Engel curve, but the post oil shock 1978 one is strictly monotonic. But it is plausible that the mode disappears because innovation lifts the economy out of satiation, as is probably the case for "leisure services": The range of goods included in this category, as argued by [START_REF] Gallouj | Innovation in Services[END_REF], has presumably widened over time, explaining why the mode that appears in its 1968 Engel curve has disappeared. Of course, this evidence does not tell us how this wave of innovation relates to low real interest rates nor whether another trajectory with less innovation, higher interest rates and converging to satiation might have existed. Nevertheless it is consistent with the mechanisms highlighted in this paper.

The present paper is related to several strands of literature. Non homothetic preferences have been introduced into growth models in order to explain structural change [START_REF] Echevarria | Changes in Sectoral Composition Associated with Economic Growth[END_REF], [START_REF] Laitner | Structural Change and Economic Growth[END_REF], [START_REF] Ngai | Structural Change in a Multisector Model of Growth[END_REF], [START_REF] Foellmi | Structural Change, Engel's Consumption Cycles and Kaldor's Facts of Economic Growth[END_REF]) and to study the interplay between income distribution and growth through demand channels (Matsuyama (2002), [START_REF] Foellmi | Income Distribution and Demand-Induced Innovation[END_REF], [START_REF] Saint-Paul | Distribution and Growth in an Economy with Limited Needs: Variable Markups and the End of Work[END_REF]).

Yet to my knowledge the role of satiation as a limit to growth and its relationship to innovation has been under-researched by the mainstream of economics 2 . Rather, research has focused on supply-side limits to growth such as complementarities and bottlenecks [START_REF] Baumol | Unbalanced growth revisited: Asymptotic stagnancy and new evidence[END_REF] and whether innovation may endogenously lift those limits [START_REF] Acemoglu | Directed Technical Change[END_REF]). 3Alternative approaches have paid more attention to the possibility of saturation, perhaps, since at least [START_REF] Veblen | The Theory of the Leisure Class: An Economic Study of Institutions[END_REF], as a by-product of a more general critique of "consumer society". Hence, saturation of demand for existing goods, and the implied need to reallocate R and D e¤ort towards new goods, plays an important role in [START_REF] Pasinetti | Structural change and economic growth[END_REF] theory of structural change4 . An important theme of this literature is how capitalism may manipulate preferences and institutions so as to avoid satiation and maintain a large consumer basis [START_REF] Galbraith | The A-uent Society[END_REF], [START_REF] Scitovsky | The Joyless Economy: An Inquiry into Human Satisfaction and Consumer Dissatisfaction[END_REF], [START_REF] Schor | The Overspent American: Why We Want What We Don't Need[END_REF], Lee (2000)) 5 . In contrast, this paper pursues a standard neo-classical approach: preferences are exogenous and di¤er from usual ones only in that they include a bliss point6 .

The paper is organized as follows. Section 2 lays out the basic model without innovation and analyzes whether the economy's growth trajectory converges to a satiated steady state. Section 3 extends the model by allowing for new goods to be introduced. It derives the paper's two key results: …rst, that the decentralized equilibrium may fail to lift the economy out of satiation through innovation even though a central planner would always choose to do so; second, that the strategic complementarity between innovation and savings may lead to multiple equilibrium trajectories associated with multiple steady states.

Section 4 studies how the economy reacts to an increase in TFP and to a fall in the cost of R and D. In equilibria such that the economy is not satiated in the long run, these shocks unambiguously increase output, capital and product variety. Things are more mixed, however, for equilibria with satiation: In trajectories such that innovation fails to pick up, an increase in TFP is met by a fall in investment; in a trajectory which converges to a satiated steady state with innovation, a reduction in the cost of R & D eventually reduces the number of varieties. In Section 5, I consider what happens if agents di¤er in their rate of time preference. If the most patient type is patient enough, the long-term equilibrium exhibits segmentation: agents below a critical discount rate are all satiated, while those above it end up with negligible wealth and consumption. While no satiation equilibria disappear, the strategic complementarity that we have highlighted remains in another form: a higher interest rate reduces the net present value of innovation, but, as it boosts incentives to save, raises the long-run proportion of satiated consumers and therefore the market size for innovation. As a result, multiple equilibrium interest rates may coexist, if the latter e¤ect dominates over some range. Section 6 shows that if there is a trend of TFP growth, the steady states that have been constructed can be generalized as balanced growth paths. Section 7 discusses how the features of those trajectories change when one allows for endogenous labor supply. I highlight the possibility of a "leisure society" trajectory with positive growth in output, capital, and product variety, negative growth in labor supply, and asymptotic satiation. Section 8 concludes.

The model without innovation

How does the possibility of satiation a¤ect the analysis of the standard neoclassical growth model? My starting point is the Ramsey model. A representative consumer, endowed with one unit of labor, maximizes max

Z +1 0 u(C t )e t dt;
(1) where C t is consumption at date t: Output Y t is produced using labor and capital, and the representative consumer is endowed with one unit of labor:

Y t = AK t L 1 t = AK t : (2)
Each unit of output can be converted into one unit of consumption or one unit of capital, which depreciates at rate : Hence the consumer maximizes (1) subject to the law of motion of capital given by

_ K t = AK t C t K t : (3)
Let us assume that needs may be satiated, by picking a quadratic speci…cation for u() :

u(C) = C C 2 2 :
Note that labor supply is assumed exogenous. Consequently, the economy may not escape satiation through working time reduction. This may occur if there exists a minimum level of working hour for which there is no disutility, or if, as suggested by [START_REF] Schor | A New Analytic Basis For: An Economic Critique of Consumer Society[END_REF], institutional barriers prevent working time from being reduced. In Section 7 I relax the assumption of exogenous labor supply, and I show that as long as there is productivity growth, it does not rule out asymptotic satiation.

Clearly the solution to this problem is

C t = max( 1 t ; 0);
where t , the marginal value of capital, follows the law of motion

_ t = t ( r(K t )); (4)
where r(K) = AK 1 denotes the net marginal product of capital.

Two types of steady state are possible. In a non satiated steady state, > 0; C < 1= and the capital stock satis…es the standard modi…ed golden rule:

K = r 1 ( ) = K M GR :
Aggregate consumption then is equal to C M GR = AK M GR K M GR : In a satiated steady state, = 0; C = 1= and the equilibrium capital stock must solve AK K = 1= :

Whenever a solution exists, I will denote by K S1 and K S2 the smaller and larger solutions.

Clearly K M GR < K S2 : The solution is then described by the following proposition:

Proposition 1 -Let K = arg max AK t K t = r 1 (0) be the Golden Rule capital stock and C = AK K the corresponding consumption level. Let K 0 be the initial capital stock.

A. If 1= > C then the economy converges to a unique, non satiated steady state

K = K M GR : B. If 1= < C and K S1 > K M GR then (i) If K S1 < K 0 < K S2
the economy converges to a satiated steady state such that K = K S2 : Throughout the convergence path, the consumer is satiated: C t = 1= :

(ii) If K 0 < K S1 the economy converges to a non satiated steady state such that

K = K M GR : The consumer is never satiated. C. If 1= < C and K M GR > K S1 then (i) If K 0 > K S1
the economy converges to a satiated steady state such that K = K S2 :

Throughout the convergence path, the consumer is satiated: C t = 1= :

(ii) If K 0 < K S1 the economy converges to a satiated steady state such that K = K S1 :Throughout the convergence path, C t < 1= :

The Proof is in the Appendix. Figure 1 illustrates Proposition 1. The convergence path is always unique. In case A the satiation level is too high to be feasible in the long run: the only steady state is a non satiation one. In cases B and C satiation is sustainable or not depending on whether the initial capital stock is high enough. In case B people are impatient enough so that they would never elect to remain satiated in the long run unless they can do so throughout the entire accumulation path. Consequently there are two long-run stable steady states. If initial capital is large, setting consumption at C = 1= throughout does not deplete the capital stock. The economy ends at K = K S2 ;

an apparently dynamically ine¢ cient steady state (but because of satiation there is no point in picking a path with higher consumption). Otherwise, the economy converges to the modi…ed golden rule steady state. In case C people are patient enough so that the marginal utility of consumption falls to zero asymptotically. For high enough capital they can a¤ord to be satiated throughout. Otherwise they gradually accumulate capital so as to reach satiation asymptotically.

Innovation

I now introduce the possibility of product innovation. I assume that there is a bliss point for each good, however one may in principle escape from satiation by consuming a greater variety of goods. I …rst compute the …rst-best allocation and show that the economy cannot settle in a satiated steady state. If that were to happen, it would be optimal to devote some resources to innovation and broaden the range of goods being consumed.

Starting from a satiated situation, it is always optimal to deviate by introducing more goods and reducing the consumption of each good. The losses from doing so are second order while the gains are …rst order.

Next, I consider the case of market-determined innovation. I show that the economy may converge to a satiated steady state without any innovation despite that in such a situation the marginal social cost of innovation is zero. Depending on initial condition, such an equilibrium may be at the margin of innovation (type I), i.e. innovation takes place along the convergence paths but eventually stops despite satiation, or it may be such that innovation never occurs (type II).

The reason for this discrepancy between optimal and equilibrium outcomes is that the set of relative prices that drive production and innovation no longer re ‡ect the consumers' marginal willingnesses to pay. Producers and innovators fail to internalize the fact that the marginal productivities of the activities they undertake are multiplied by that of consumption, which is zero. That is, while relative prices are equal to marginal rates of transformation, marginal rates of substitution are not de…ned. Economic choices are driven by equilibrium prices, that are themselves determined from the production side;

because both marginal costs and marginal bene…ts, in terms of utility, are equal to zero, economic decisions by producers and innovators do not internalize the e¤ect of those decisions on welfare. In particular, they do not internalize the fact that more innovation has a …rst-order e¤ect on welfare by just lifting consumers out of the satiation trap.

Finally, I show that the economy may converge to a (type III) more conventional steady state with positive innovation, no satiation, and a capital stock equal to its modi…ed golden rule level, and discuss how the same economy with the same initial conditions may, for some parameter range, follow either a trajectory converging to such a steady state or, alternatively, a trajectory with long-run satiation.

The …rst best

In this subsection, I introduce the relevant assumptions and notations and I compute the …rst best allocation.

At any date t there are N t goods and the consumer's total utility is

U (fc it g) = Z Nt 0 u(c it )di;
where c it denotes consumption of good i and again u(c) c c 2 =2:

The individual goods are produced using a single good, "output", which is again produced according to (2) and can also be transformed into capital or new varieties.

Therefore, the capital accumulation equation is:

_ K t = AK t C t K t R t ; (5)
where R t denotes the resources devoted to innovation ("R and D") and C t is equal to the amount of output devoted to the production of varieties.

The production function for any variety i is

c it = z it ;
where z it is the amount of output used in the production of good i: Therefore, by symmetry, it is optimal to consume the same amount of each good, de…ned as c t = C t =N t : We can then rewrite utility as

U (C t ; N t ) = C t C 2 t 2N t : (6)
Clearly, U 2 > 0; re ‡ecting the consumers'taste for diversity.

The production process for new goods is as follows: One unit of output in the R and D sector produces new goods per unit of time. Consequently, the law of motion for N t is given by

_ N t = R t : (7)
The social planner maximizes max

Z +1 0 U (C t ; N t )e t dt; (8)
subject to the law of motions ( 5) and ( 7) and the constraint R t 0:

Proposition 2 characterizes the optimal solution in a steady state:

Proposition 2 -Let c be the unique solution in the (0; 1= ) range to the following equation

2 c 2 = (1 c): (9)
Then in any steady state c c and

N N = C M GR =c : Furthermore K = K M GR and C = C M GR : Proof -See Appendix
Intuitively, c is the minimum consumption level per variety beyond which, in a steady state, it is socially pro…table to innovate. Equation ( 9) equates the marginal bene…t per unit of time of introducing a new variety, 2 c 2 ; to its annuity cost in terms of utility, equal to the product of the discount rate and the marginal utility of consumption 1 c: Clearly, the solution is always strictly smaller than 1= : In the long run, the representative consumer will never elect a consumption level per item c greater than c ;

because in such a situation it is preferable to sacri…ce some consumption units to raise N;

which eventually brings down c below c : Consequently, a satiated steady state does not exist.

Equilibrium satiation

I now compute the equilibrium level of innovation and I show that if innovation is determined in a decentralized fashion, a satiated steady state is no longer ruled out. This, despite that, as illustrated by ( 9), the social marginal cost of innovating in such a situation is zero.

Following the literature, I assume that innovators get a permanent patent on the variety they have invented. However, competitive imitators can produce the same good without authorization, at a marginal cost in terms of output equal to > 1: If is not too large, the patent holder will charge at a markup equal to : Using output as the numéraire, is also equal to the price of the good. Consequently, the pro…ts generated by any variety at date t are given by t = ( 1)c t :

The market value of a patent at t; V t , follows the usual law of motion

r t V t = _ V t + t ; (10)
where r t is the real interest rate which, in equilibrium, equates the net marginal product of capital:

r t = r(K t ):
At date t; the cost of producing a new variety is 1= : Therefore the economy can be in one of two regimes:

A. The no innovation regime: V t < 1= and R t = 0:

B. The innovation regime: V t = 1= and R t > 0:

The consumer maximizes (8) with respect to fC t g; or equivalently fc t g; subject to his intertemporal budget constraint:

Z +1 0 c t N t e R t 0 rudu dt Z +1 0 w t e R t 0 rudu dt + N 0 V 0 + K 0 ; (11)
where w t is the wage, also equal to labor income. 7

From this problem, denoting by the Lagrange multiplier of (11), we get the optimality conditions for consumption:

c t = max 1 e t R t 0 rudu ; 0 ! : (12)
In particular, as long as c > 0 it satis…es the following Euler equation:

_ c t = 1 c t (r t ) : (13)
In partial equilibrium, if r > ; the consumer converges to satiation regardless of his income. Satiation only depends on people's degree of patience: if they are patient 7 As will be clear below, because of satiation, the economy may converge to a steady state such that r 0: In this case the RHS of ( 11) is not de…ned. But N t is bounded and the intratemporal budget constraint _ W t = r t W t + w t N t c t is stable and the consumer can clearly pick c = 1= throughout. Therefore, this special situation does not invalidate (13).

enough relative to the market interest rates, they will eventually have accumulated enough resources to reach satiation.

The following Proposition establishes the existence of long-run steady states that are satiated 8 . Furthermore, they are stable in the sense that there exist trajectories, starting from the initial values of the capital stock K 0 and of the number of varieties N 0 ; that converge to these steady states, for sets of initial conditions of positive measure. Proof -See Appendix.

Proposition 3 -Let K S = r 1 ( 1) : Let N S = (AK S K S ): Assume N 0 < min( C M GR ; N S ): Let K < K S such that N 0 = (A K K): A. If K < K 0 < K S
In the trajectories described in Proposition 3, agents are patient enough to accumulate enough capital so as to be satiated in the long run. However, there is no point in accumulating wealth to consume beyond the satiation level. In case B, no innovation takes place and people target an equilibrium level of the capital stock so as to be asymptotically satiated. At this level, interest rates are too high for innovation to be pro…table. In case A, initial capital is high enough for people to choose a path such that they are satiated throughout while accumulating capital at a positive rate. At some point the interest rate falls to a level where innovation becomes pro…table. It is then optimal to devote 8 In Propositions 3 and 4 I only focus on the regimes that are most relevant. For example, if C M GR < N 0 < N S ; implying in particular K S > K > K M GR , and > ( 1) for K 0 < K the economy follows a standard Ramsey path with no innovation and convergence of the capital stock to K M GR :

Also, I do not analyze paths with falling capital stocks, which may arise if the initial capital stock is large, because one would then have to either allow for instantaneous conversion of a mass of capital into new goods, or for a corner regime where no gross investment takes place, and the marginal product of capital is lower than the interest rate which would be pinned down by the innovation equilibrium condition V = 1= : all investment to innovation (any further accumulation of physical capital would make innovation strictly superior to capital accumulation) while raising aggregate consumption so as to maintain consumption of each variety equal to its satiation level. As this process continues, less and less resources remain available for innovation, which eventually dies out as the economy approaches steady state.9

Equilibrium non-satiation

It is also possible to construct equilibrium trajectories which converge to a modi…ed golden rule steady state at the margin of the innovation regime. These trajectories are characterized in Proposition 4:

Proposition 4 -Assume < ( 1) : Let N N = C M GR ( 1) : If N 0 N N and K 0 K M GR
then there exists an equilibrium trajectory such that.

(i) Until some date T; _ K > 0; _ N = 0; and _ c = 0

(ii) For any t T; K t = K M GR ; c t = ( 1) = c N < 1 , _ N t > 0 (iii)
The economy converges to a (type III) steady state such that

K = K M GR > K S ; c = c N < 1 ; C = C M GR > C S ; N = N N > N S :
Proof -See Appendix

Figure 3 illustrates the trajectory elicited in Proposition 4. This trajectory converges in …nite time to the modi…ed Golden rule capital stock, with a consumption level which is lower. The surplus of savings at K = K M GR after date T is devoted to innovation. As innovation proceeds, aggregate consumption gradually goes up as people consume a greater variety of goods, each in the same quantity. This process continues until, asymptotically, no resources are left for further innovations.

As long as < ( 1) and N 0 < N S ; these trajectories coexist with those converging to satiated steady states and characterized in Proposition 311 . Therefore, there exist multiple self-ful…lling trajectories that converge to di¤erent steady states. A trajectory that converges to a type I satiated steady state is associated with lower innovation and lower capital accumulation than a trajectory converging to a non satiated steady state.

In a satiated steady state, the market size for innovation is higher than in a non satiated one, but that is compensated by higher interest rates since such a steady state is less capitalistic.

This strategic complementarity is based on the following mechanism: Suppose people expect more goods to be introduced in the future. This raises the marginal utility of future aggregate consumption C and therefore the incentives to save12 . These incremental savings in turn reduce real interest rates, which makes innovation more pro…table13 .

Conversely, expecting a lower level of innovation or no innovation at all would lead to lower savings and higher interest rates, which would deter innovation.

The possibility of satiation is important for this strategic complementarity to lead to multiple steady states. Otherwise, there would be a unique long-run equilibrium capital stock given by the modi…ed golden rule level K M GR : It is easy to check that this is consistent with a unique value of c and a unique value of N ; which may or may not be in the innovation regime depending on initial conditions.

As in [START_REF] Dixit | Monopolistic competition and optimum product diversity[END_REF], the equilibrium number of varieties in Proposition 4 may be either higher or lower than the social optimum. An appropriability e¤ect which tends to make innovation too low has to be balanced against a business stealing e¤ect which tends to make it too high.

Some comparative statics

Whenever the economy converges to a satiated equilibrium, the conventional results regarding the e¤ects of greater productivity are partially overturned.

In the standard Ramsey model, a permanent increase in TFP raises capital accumulation. In a type II steady state, however, a strong income e¤ect dominates: In the long run, absent innovation, one can …nance the same satiated level of consumption with a lower capital stock.

In standard models of horizontal innovation, the number of varieties goes up when the cost of R and D falls. In a type I steady state, however, the number of varieties goes down, because the economy accumulates less physical capital.

E¤ect of productivity shocks

I …st study the e¤ect of productivity shocks. I show that they have di¤erent qualitative e¤ects on the equilibrium depending on whether or not the economy lies at the margin of the innovation regime in the long-run. This is summarized in Proposition 5:

Proposition 5 -Consider a small increase in A: Then Proof -This follows immediately from Propositions 3 and 4.

The e¤ect of TFP growth is quite di¤erent depending on the type of steady state.

If there is no innovation and people are eventually satiated (type II); the capital stock falls so as to leave aggregate consumption C unchanged in the long run. The extra output allowed by productivity growth is essentially consumed, allowing to consume more over the convergence trajectory. Accumulating more capital would have zero value since people are satiated in the long run and there is no innovation. In a type III steady state, however, higher productivity makes it worthwhile to accumulate more capital, because this eventually allows to invent more varieties. Indeed, starting from steady state, consumers start saving more at the time of the shock, which further speeds up capital accumulation.

The economy transitions from the capital accumulation regime to the innovation regime with a higher capital stock, which allows to devote more output to innovation14 . The economy ends up with both more capital and more varieties, while consumption of each variety remains unchanged, since there is only one market size which delivers zero net pro…ts from innovating at the equilibrium interest rate r = :

To summarize: a positive productivity shock generates an investment boom if innovation is expected to occur, and an investment slump if that is not the case.

E¤ects of shocks to the cost of R and D

I now study an increase in the productivity of R and D, i.e. an increase in : A shock to also has very di¤erent e¤ects on capital and innovation depending on whether or not the economy converges to a steady state with satiation:

Proposition 6 -Consider a small increase in : Then Proof -Immediate from Propositions 3 and 4.

To understand these paradoxical results, consider …rst the most standard case, i.e.

a type III steady state (…gure 4a). As the cost of innovation falls, people anticipate that more varieties will be introduced in the future, which raises their marginal utility of aggregate consumption in the future compared to the present. For this reason they save more, reducing their consumption of each individual good c: As capital is accumulated more quickly, the critical date when innovation kicks in is reached earlier. Furthermore, agents coordinate their consumption plans so that r and K are still equal to their modi…ed Golden Rule levels, otherwise the subsequent trajectory would not converge to a steady state. Thus the equilibrium value of c must be lower: as the cost of innovation is lower and the interest rate is the same, market size must fall for the free entry condition in innovation to be satis…ed. This in turn frees more resources available for innovation: in the long run, more varieties have been introduced.

Consider now (Figure 4b) an economy that converges to a type I steady state. A higher shifts the II locus down and to the left, meaning that since innovation is less costly, the economy will innovate at higher interest rates, i.e. lower capital stocks. Hence the trajectory hits point E earlier than if had remained smaller; people bene…t from an increase in the number of varieties earlier. However that is a mixed blessing because the economy stops raising its capital stock after this date, investing all savings in innovation.

In the long run, as the capital stock is lower, there are also fewer resources left to innovate, and this also reduces the equilibrium number of varieties -As the economy follows a self-ful…lling trajectory with satiation in all varieties, the mechanism allowing for variety to go up thanks to a reduction in c does not operate.

Heterogeneous agents

I now discuss how the analysis changes if agents are heterogeneous. I assume that people di¤er in their rate of time preference : I assume that it is distributed with c.d.f. F and density F 0 = f: To simplify matters I make two key assumptions:

ASSUMPTION 1 -F has full support over (0; +1) and no mass point:

ASSUMPTION 2f is hump-shaped, implying F is S shaped.

Let us now consider steady states. For any equilibrium interest rate r > 0; there exists a positive measure of agents such that < r: According to (12), they are necessarily satiated. By the same token, agents such that > r have a zero consumption level.

While in the Ramsey model with heterogeneous agents the most patient dynasty own all the wealth in the long run, while other dynasties have zero consumption, here consumers for an entire range of values of have positive consumption in the long run, and they are all satiated.

Given that a fraction F (r) of consumers consume 1= of each good and the others zero, the pro…t level for any variety is equal to

= 1 F (r):
Let us focus on a steady state where there is innovation at the margin. The free entry

condition V = 1= reads 1 F (r) = r : (14)
This condition determines the equilibrium interest rate. Interestingly, both sides are increasing functions of r; so that the equilibrium interest rate may not be unique. This is due to the following: While a higher interest rate reduces the net present value of innovating through the usual discount e¤ect, it also induces a greater fraction of the population to save in order to eventually be satiated, which in turn raises the market size for innovations and hence the ‡ow of pro…ts it generates15 .

Figure 5 depicts the determination of r: Under Assumption A2 there are at most three equilibrium values for r : 0; r 1 and r 2 : It is easy to see that r = 0 cannot be an equilibrium 16 . Intuitively, r 1 can be ruled out on economic stability grounds. Consider a small perturbation, say an increase in r: Then in the neighborhood of equilibrium r 1 the LHS of ( 14) goes up by more than its RHS, meaning that the market size e¤ect on innovation dominates the e¤ect of the cost of capital. We expect innovators to innovate more, which raises the demand for loanable funds and should push r further up. That is, let V (r)

1 F (r)
r : If we de…ne a stable equilibrium as an equilibrium r which satis…es V 0 (r) < 0 in addition to V (r) = 1= ; then r 1 may be ruled out and, since F is S shaped, the only stable equilibrium is such that r = r 2 : From there it is easy to compute the equilibrium capital stocks and number of varieties 17 :

K 2 = r 1 (r 2 );
(15)

N 2 = F (r 2 ) (AK 2 K 2 ) : (16)
As in the analysis of Section 4.2, an increase in raises the equilibrium r: Again from ( 15)-( 16) fewer varieties exist in equilibrium and the capital stock is lower. On the other hand, the proportion of agents that are satiated, F (r); goes up.

Balanced growth paths

This section extends the results of Section 3 by allowing for growth in the TFP parameter A: Clearly, since the critical capital stock K goes to zero as A goes up, a type II balanced growth path (BGP) is precluded: as consumption of each good is bounded, capital accumulation would push interest rates down to zero; however innovation would pick up before such a point is reached. On the other hand, it is easy to construct BGPs of types I and III. These are characterized by the following propositions.

Proposition 7 -Assume A t = A 0 exp(g A t): Assume g A < 1

( 1) + ( 1) :

Then there exists an equilibrium trajectory such that 16 At r = 0 (14) does not in fact apply. The RHS of any consumer's budget constraint becomes in…nite. All consumers therefore choose c = 1= : Then = 1 > 0; implying V = +1; which contradicts the requirement that V = 1= : 17 An equilibrium with no innovation can also be computed. Let N 0 be the initial number of varieties. Then the equilibrium capital stock K(N 0 ) is the unique solution to 0 = AK N 0 F (r(K))= K: For no innovation to indeed take place in equilibrium, K(N 0 ) must be such that r(K) > ( 1)F (r(K))= :

(i) K t = kA 1 1 t ; where k = + ( 1) 1 1 ; (ii) c t = 1= (iii) r t = ( 1) (iv) Y; K and N grow at common rate g = g A 1 = g CS ; (v) N t = nA 1 1 t ; where n = k ( ( 1) + )= g 1= +g= > 0: Proof-See Appendix Proposition 8 -Assume A t = A 0 exp(g A t): Assume g A < (1
) and < ( 1) :

Then there exists an equilibrium trajectory such that

(i) K t = k 0 A 1 1 t ; where k 0 = + 1 1 ; (ii) c t = ( 1) < 1 (iii) r t = (iv) Y; K and N grow at common rate g CS ; (v) N t = n 0 A 1 1 t ; where n 0 = k 0 ( + )= g ( 1) +g= > 0: Proof -See Appendix.
The BGPs constructed in Propositions 7 and 8 generalize the type I and type III steady states, respectively. Again, as the conditions for Proposition 7 and Proposition 8 to hold are not mutually exclusive, the two trajectories may coexist in the same economy 18 . Furthermore, it is easy to check that the non satiated BGPs have more innovation and capital than the satiated ones. 19 18 The condition g A < ( 1) is the standard one that is needed for a Ramsey MGR BGP to deliver summable utility. However if 1

( 1) + ( 1) < (1 ) and > ( 1) ; then a BGP cannot be constructed for 1

( 1) + (1 ) < g A < (1 ) :

19 Since < ( 1) ; k 0 > k: Furthermore, in both BGPs, N = Y (g+ )K g= +r=( ( 1)) : Since r is lower in the non satiated BGP, the numerator is lower. Since K is higher in the non satiated BGP but, since r > g; lower than the Golden Rule level, the numerator is higher in the non satiated BGP. Therefore, N is higher in the non satiated BGP than in the satiated one.

7 Long-term growth, near satiation and the leisure society

In this section I make labor supply endogenous, while still assuming the existence of a productivity growth trend. I assume all units of labor have a disutility. Absent growth, then, satiation would not arise because if it did, labor supply would be equal to zero, and the economy could not produce enough output for people to be satiated. Under growth, however, the story is di¤erent.

I assume that the consumer's ‡ow of utility is now given, instead of (6), by

U (C t ; N t ; L t ) = C t C 2 t 2N t v(L t );
where L t denotes labor supply and v(L t ); the marginal disutility of labor, is given by

v(L t ) = L 2 t 2 if L t 1 = +1 if L t > 1:
At any date t; the real consumption wage us w t = and the marginal utility of consumption is 1 c t : Therefore, labor supply is given by

L t = 1 if w t (1 c t )= > ; = w t (1 c t ) if w t (1 c t )=
Since, along a non satiated BGP, w grows over time while c is constant and lower than 1= ; the equilibria described in Proposition 8 still exist, provided the initial level of wages satis…es w 0 (1 c)= > : In those equilibria, labor supply is at its maximum level. In contrast, equilibria with satiation no longer exist. Interestingly, however, we can construct trajectories that converge to satiation while output does not go to zero, either because labor supply is at its maximum, or because TFP growth more than o¤sets the e¤ect of shrinking labor supply in the long run. These trajectories are characterized in the following two propositions, again proved in the Appendix:

Proposition 9 -Assume that

g A > (1 ) ( 1) : (17)
Then there exists ; " > 0 and a decreasing function A(); such that for all k 0 2 (k; k+ )

and n 0 2 (n "; n + "); and for A 0 > A(k 0 ) and A t = A 0 exp(g A t); there exists an equilibrium trajectory (K t ; c t ; N t ; L t ) such that

(i) K 0 = k 0 A 1 1 0 (ii) N 0 = n 0 A 1 1 0 (iii) _ c t > 0 and lim t!1 c t = 1= (iv) L t = 1, 8t (v) _ N t > 0 and lim t!1 Nt A 1 1 0 = n (vi) lim t!1 Kt A 1 1 0 = k (vii)
The economy is in the innovation regime throughout. In particular, r t = (

1)c t = A t K 1 t :
Proposition 9 has constructed a trajectory which is near the BGP of Proposition 7 and therefore near satiation. Furthermore this trajectory converges to that BGP and consumers are asymptotically satiated. Nevertheless, it is rational for them to supply one unit of labor forever, because wage growth overtakes the decline in the marginal utility of consumption.

If ( 17) is violated, but growth is not too low, we can construct a trajectory such that labor supply shrinks over time, and converges to zero, while consumers are asymptotically satiated. Growth is strictly positive because technical progress and capital accumulation more than o¤set the attrition in labor supply.

Proposition 10 (Leisure society) -Assume that < ( 1) ; that A t = A 0 exp(g A t); and that 1 2

( 1) < g A 1 < ( 1) : (18) Let g = 2 1 g A ( ( 1) ) = g LS > 0 and z = 1 ( ( 1) + ) g 1= + g= > 0:
Then there exists ; ; " > 0 such that for any initial conditions (K 0 ; N 0 ) such that

K 0 < min( A 2 1 0 ; A 1 1 0
) and N 0 2 (K 0 z(1 "); K 0 z(1 + ")); there exists an equilibrium trajectory (K t ; c t ; N t ; L t ) from those initial conditions such that (i) _ c t > 0 and lim t!1 c t = 1=

(ii) L t < 1; and lim t!1

_ Lt Lt = 1 1 g A ( ( 1) ) = g L < 0: Therefore, lim t!1 L t = 0 (iii) lim t!1 _ Kt Kt = lim t!1 _ Yt Yt = lim t!1 _ Nt Nt = g: (iv) lim t!1 Nt Kt = z (v)
The economy is in the innovation regime throughout. In particular, r t = (

1)c t = A t K 1 t :
Proposition 10 constructs a leisure society where, despite innovation, capital accumulation proceeds at a faster pace than the introduction of new goods, eventually leading to satiation. Over time, labor supply falls to zero, but at a lower rate than the contribution of TFP growth and capital accumulation to GDP, thus making room for long-run satiation.

As long as

< ( 1) < 3 ; there exists an interval of values of g A for which a leisure society trajectory characterized by Proposition 10 coexists with a consumer society characterized by Proposition 8. Since for the leisure society to exist it must be that

g A < (1 ) ( 1)
; one clearly has g LS < g CS : the leisure society grows less fast than the consumer society. By comparing the expression for z to its counterpart for the consumer society, which may be obtained from (v) in Proposition 2, we …nd that the leisure society has a greater range of goods, relative to K; than the consumer society. This is due to several factors: …rst, the market size for new goods is larger in the leisure society, because people are satiated in those goods. Second, as the economy grows less fast, it invests less in physical capital, which leaves more resources for innovation. Nevertheless, over time, the consumer society will widen its gap relative to the leisure society both in terms of capital and number of varieties.

The above analysis also suggests that, contrary to the claims by [START_REF] Schor | A New Analytic Basis For: An Economic Critique of Consumer Society[END_REF], the persistence of high labor supply despite satiation may well be an equilibrium phenomenon rather than the outcome of institutional constraints. Not only the economy may never approach satiation because it will endogenously introduce new goods, but in the con…guration of Proposition 9 near satiation does not preclude labor supply from being in its maximum. It is also interesting to note that di¤erences between European-style "leisure societies" and American-style "consumer societies" may be explained by multiple equilibria instead of di¤erent exogenous institutions.

Conclusion

The possibility of satiation of needs substantially changes our analysis of growth models.

The market may fail to introduce new goods so as to lift the economy out of a satiation trap. As expectations of introducing new goods generate a strategic complementarity between savings and innovation, there are multiple equilibria. When one allows for endogenous labor supply, satiated ("Leisure Society") equilibria may involve a lower growth rate, and an ever decreasing labor supply, compared with non-satiated ("Consumer Society") equilibria.

There is no overproduction in the economy analyzed here, since it is always at a full employment equilibrium. Nevertheless, I have shown that because of satiation, an increase in total factor productivity may lead to a reduction in the long-term capital stock, thus triggering an investment slump. Furthermore, for this slump to occur it must be that the economy remains in the (type II) no innovation regime. This is reminiscent of both [START_REF] Schumpeter | Business Cycles. A Theoretical, Historical and Statistical Analysis of the Capitalist Process[END_REF] Assume > : Since 0, > 0: Therefore, the steady state is non satiated. Then

clearly = r(K); implying K = K M GR : Since _ K = _ N = 0; it must be that C = C M GR :
In particular, c = C=N > 0; so that = 1 c: Since = c 2 2 ; = 1 c and > ;

it must be that (1 c) > c 2 2 ; or equivalently c < c : A. We …rst construct a trajectory such that R = 0; N = N 0 ; and c = 1= : Along this trajectory, the law of motion for K is

Therefore N = C=c = C M GR =c > C M GR =c = N :
_ K t = AK t N 0 K t :
Since K 0 > K; one has _ K > 0 throughout and the trajectory converges to K; which is the larger solution to N 0 = (AK K): In particular, r( K) < 0; implying K > K S :

Let T be the date at which, along this trajectory, K t = K S :

Assume that the trajectory followed by the economy is such that (c; K) are along this saddle-path and N = N 0 : To prove that this is an equilibrium trajectory, we only need to show that V 1= throughout. From (10) we have that

V t = Z +1 t ( 1)c t e R s
t rudu ds:

Since N 0 < N S ; K < K S : Therefore r t = r(K t ) > r( K) > ( 1) and c t < 1= : Clearly, then, V t < 1= : QED.

Proof of Proposition 4

First, we can construct a unique saddle path (K t ; c t ) such that (i) (K t ; c t ) satis…es the system (19-20), (ii) K = K 0 initially and (iii) c = ( 1) at K = K M GR : Given our assumption that K 0 < K M GR and N 0 < N N ; along this trajectory _ c > 0 and _ K > 0: If the mathematical solution is such that c < 0; we replace it by c = 0 as in the proof of Proposition 3, B. Furthermore, at c = ( 1) ; _ K = (N N N 0 ) ( 1) > 0; implying that the point (K M GR ; c) is reached at a …nite date T:

For t T we assume the economy follows this trajectory.

For t T we assume that c and K remain constant at c = ( 1) ; K = K M GR ; while

_ N = C M GR N ( 1

) :

Since r = and c is constant, the consumer's optimality conditions are clearly satis…ed for t T: Since c is C1 as a function t; they are also satis…ed locally around T:

Since the point c = ( 1) ; K = K M GR lies on II; i.e. r(K M GR ) = = ( 1)c N ;

we clearly have, for t T; V = ( 1)c N r = 1 ; consistent with the free entry condition for innovation.

Last, we need to prove that V < 1= for t < T: As in the proof of Proposition 3, A, we have that for t < T

V t = Z T t ( 1)c t e R s t rudu ds + 1 e R T t rudu : Since K t < K T = K M GR ; r t > : Furthermore, c t < ( 1) : Therefore V t < R T t e (s t) ds+ 1 e (T t) = 1= : Finally, the property that N N > N S is straightforward, since N N = C M GR ( 1) > C M GR > C S = N S : QED 9.5 Proof of Proposition 7
Consider a trajectory such that (i), (ii) and (v) hold. It follows from (i) that the net marginal product of capital satis…es (iii). It is also true from (ii) that the consumption Euler equation ( 13) holds. Property (iv) holds by construction and from the production

function Y t = A t K t : In particular Y is proportional to K; Y = r+ K = yA: From (iii)
and (ii) we get that the equilibrium condition for innovation V = =r = 1= holds.

Given the equilibrium values of r and g; the property n > 0 follows from the assumption that g A < 1

( 1) + ( 1) : To complete the proof, just substitute the values of N = nA; K = kA; Y = yA into the capital accumulation equation ( 5), which can be rewritten as

_ K t = A t K t C t K t _ N t = ; (21)
and check that it is satis…ed. QED.

Proof of Proposition 8

The proof is similar to that of Proposition 8, with the following di¤erences. The Euler equation now holds because r = : The equilibrium consumption level is the one which guarantees equilibrium innovation at r = : The assumption < ( 1) guarantees that this level is below satiation. In addition, the consumer's transversality condition has to hold (it always holds under satiation). For this to be the case we need that r > g;

which is implied by the assumption that g A < (1 ) : This in turn guarantees that ( + )= g > 0:

QED.

Proof of Proposition 9

We construct a trajectory based on a path fc t g: From the Euler equation, provided its initial value is between =( ( 1)) and 1= ; c t ! 1= : Furthermore, let us impose that the economy is in the innovation regime throughout. Then r t = ( 1)c t ; implying that

K t = A t ( 1)c t + 1 1 : ( 22
)
Since K is a state variable, this pins down the initial c; c 0 : Furthermore, for k 0 = K 0 =A 1 1 0 above and close enough to k; c 0 will be below and close to 1= ; and therefore above =( ( 1)); implying indeed that c t ! 1= :

To have L t = 1; throughout, we need that w t (1 c t ) > : Note that close enough to k: Along our trajectory c goes up and K=A 1 1 falls. Therefore, we can always pick initial conditions so that these two quantities are arbitrarily close to 1= and k; respectively. In such a case, the inequality w t (1 c t ) >

w t = (1 )A t K t = ( 1)c t + 1 (1 )
will hold for all t; provided it holds initially, that is

( 1)c 0 + 1 (1 )A 1 1 0 (1 c 0 ) > 0:
Clearly, given the initial value of k 0 and hence of c 0 ; the preceding inequality holds for A 0 large enough.

To summarize, there exists such that we can choose k 0 2 (k; k + ) and A 0 > A(k 0 )

in such a way that for K 0 = k 0 A 1 1 0 ; the trajectory for K t de…ned by ( 22), with fc t g the unique solution to (13) such that (22) holds at t = 0; is such that the representative consumer would pick L = 1 throughout. Furthermore, by construction the economy is in the innovation regime throughout, and since c t ! 1= ; k t = K t =A 1 1 t ! k:

To complete our construction, we have to check the feasibility of the implied trajectory for N t : Let g = g A =( 1) and n t = N t =A If n t is su¢ ciently close to n; the two terms in brackets are arbitrarily small. Since the …rst term is > 0; this can be made larger than any negative number. In particular, larger than g= : Thus we can pick k 0 and n 0 simultaneously close enough to k and n to make sure that _ N =N = _ n=n + g > 0:

QED

Proof of Proposition 10

We construct our trajectory in a similar fashion as for Proposition 9. Given a path for c t ;

we make sure that we are in the innovation regime and that there is an interior solution for labor supply. That is

r t = A t K 1 t L 1 t = (

  there exists an equilibrium trajectory such that (i) c t = 1= throughout, (ii) there exists a critical date T such that _ K > 0; _ R = 0 for t < T and _ K = 0; _ R > 0 for t > T; (iii) the economy converges to a satiated (type I) steady state such that N = N S and K = K S B. If K 0 < K there exists an equilibrium trajectory such that (i) c t < 1= throughout, (ii) _ R = 0 throughout (iii) the economy converges to a satiated (type II) steady state such that c = 1= ; K = K; and N = N 0 :

Figure 2 illustrates

 2 Figure 2 illustrates Proposition 3 in the (K; c) plane 10 . The II schedule is the steady free-entry condition in the R and D sector, i.e. r(K) = ( 1)c. In steady state, the economy cannot be above this locus, otherwise one would have V > 1= : The horizontal line SS is the satiation level c = 1= : The economy's trajectory cannot be above this schedule. The KK(N ) schedule is the _ K = 0 locus associated with a stationary mass of varieties equal to N: Whenever the economy is innovating, this schedule shifts to the right. In case A, the trajectory is below KK(N 0 ): At date T the economy hits point E which is the intersection of II and SS: Thereafter both K and c remain constant while N goes up. The _ K = 0 schedule gradually shifts right and asymptotically converges to KK(N S ) which intersects II and SS at point E: In case B, the equilibrium trajectory is an asymptotically satiated saddle path without innovation. Throughout this trajectory V t < 1= and innovation does not take place. Dynamics are given by the Euler equation (13) and the KK(N 0 ) schedule.

( i )

 i If the economy converges to a type I steady state, in the long run K; Y; C and N go up. c is unchanged at c = 1= : (ii) If the economy converges to a type II steady state, in the long run K and Y fall. C; N; and c = 1= are unchanged. (iii) If the economy converges to a type III steady state, in the long run K; Y; C and N goes up, c is unchanged at c = ( 1) : Furthermore, if the economy is initially in steady state and A goes up, then c falls upon impact.

  (i) If the economy converges to a type I steady state, in the long run K; Y; C and N fall. c is unchanged at c = 1= : (ii) If the economy converges to a type II steady state, this steady state is unchanged. (iii) If the economy converges to a type III steady state, in the long run K is unchanged as is Y: C is unchanged, c falls and N goes up.

  Assume = : Since = c 2 =2; one cannot have c = 0: Otherwise one would have = = 0; implying c = 1= ; a contradiction. Therefore c > 0; implying > 0 and therefore > 0 and satiation cannot hold. Again = r(K); K = K N and C = C N : Since = 1 c = = c 2 =(2 ); c = c and N = C=c = C N =c = N : QED 9.3 Proof of Proposition 3

45°Figure 5 -

 5 Figure1-A

  ), the consumer is either always satiated or never satiated. Since K would become equal to zero in …nite time if he were always satiated, the consumer is never satiated. Assume the economy converges to a satiated steady state, implying lim t*1 K =

	9 Appendix The representative consumer maximizes H subject to R t	0 and C t	0: Let c t =
	9.1 Proof of Proposition 1 C t =N t ; r(K) = AK 1 : The …rst order conditions are
	A. Remember that C = max K AK c t = max(0; K: Therefore if 1= > C the equation AK 1 t )
	K = 1= has no solution and the only steady states are non satiated. Consequently, _ t + t = t r(K t );
	K = K M GR in steady state. B. (i). Assume C t = 1= throughout. Then _ t t and ( t K t = AK t ) R t = 0; K 1= : Over (K S1 ; K S2 ]; K clearly converges to K S2 : Furthermore, this trajectory is optimal as it delivers the _ t + t = c 2 t 2 :
	maximum possible utility level. Consider any steady state. Then either >	or = :
	analysis of long waves and recent New Keynesian analyses of contractionary technical progress 20 . Indeed, a natural direction for further research would be to analyze the short and medium term implications of satiation, in models where sticky prices may generate overproduction and underemployment. (ii) By (4which therefore cannot be optimal. Consequently, the economy necessarily converges to
	a satiated steady state.
	QED
	9.2 Proof of Proposition 2
	The Hamiltonian is given by
	H = (C t	C 2 t 2N t	)e t + t e t (AK t	C t	K t R t ) + t e t R t

K S;i : Since K S2 > K S1 > K M GR ; for some T and t > T; K > K M GR : Then _ C < 0 for t > T; which contradicts the fact that lim t!1 C = 1= : Therefore the economy converges to a non satiated steady state, implying lim K = K M GR :

C. (i) Same proof as B (i)

(ii) Again the consumer is never satiated. Assume the economy converges to a non satiated steady state. Then lim t*1 K = K M GR : Since K M GR > K S1 ; for some T and t > T; K S2 > K > K S1 : Let t 0 > T: Consider the following deviation Ct from the actual trajectory: Ct = C t ; t < t 0 ; Ct = 1= ; t t 0 : Then the corresponding capital stock Kt is such that Kt = K t ; t < t 0 : Therefore K S2 > K t 0 > K S1 : Clearly, then lim K = K S2 : Hence the deviation is feasible. Furthermore, as C t < 1= throughout, Ct > C t for t t 0 : It follows that the deviation delivers a strictly higher utility level than the initial trajectory,

  A

							1
							1 t	:
				1
	A su¢ cient condition for the quantity A t 1	( (		1)c t + ) 1 (1	c t ) to grow over
	time is
	1	1	g A 1	1 c t ( (	1)		=c t ) ( (	1)c t	) > 0:
	Clearly, since we have assumed that g A > (1	)	h	( 1)

i

; this holds if c close enough to 1= ; i.e. again K=A 1 1

  Using (21) we get that_ n t = = k t k t _ k t gk t c t n t gn t = :Note that n = k (g+ )k 1= +g= Clearly, then, as c t ! 1= and k t ! k; n t ! n: For the trajectory for N to be feasible, we need that _ N t 0 throughout. Note that we can always choose small enough such that the quantity k t (g + )k t is arbitrarily close to (1= + g= )n and c t is arbitrarily close to 1= : Then

						1
						1
	_ n t n t	=	_ k t n t	+	k t (g + )k t n t	(1= + g= )n n t	+	(1= + g= )n n t	(c t +	g ) :

t :

If utility is smooth, one cannot be satiated in one good unless one is satiated in all other goods. This is what occurs in this paper's model. However, if utility has a kink at the satiation point, this no longer holds: one may be at the satiation point for some goods but not others. This rationalization is consistentn with the structuralist literature a la Pasinetti.

As an example, Gordon (2012) lists six "headwind" factors that will slow down US economic growth, none of them coming from consumer demand.

[START_REF] Aoki | Demand saturation-creation and economic growth[END_REF] study a growth model with "saturation of demand" in each good and horizontal innovation. However, what they refer to is not satiation, but an S-shaped demand pattern following the introduction of each good. In their model, which is in the fashion of[START_REF] Young | Learning by doing and the dynamic e¤ects of international trade[END_REF] models of product life cycle, demand for each good is bounded in equilibrium because there is no vertical innovation, and the S-shape comes from the assumption that the utility derived by consumers from any individual good, which is logarithmic and therefore has no bliss point, is itself multiplied by a logistic factor. Furthermore, horizontal innovation is assumed to be entirely exogenous. Accordingly, none of the results and welfare analysis studied here hold.

See Andersen (1998) for an evolutionary learning model, based on Pasinetti's ideas.

See Benhabib and Bisin (2002) for a formalization of those approaches based on endogenous preferences.

6 At some level of abstraction, however, one can always interpret the introduction of new goods as implying a change in the consumers'preferences. For horizontal innovation, the formalism of new growth theory does not allow to distinguish between the preferences for new products being genuine versus arti…cially created.

For simplicity, nonnegativity constraints on investment have been ignored. For K 0 > K S ; we conjecture that in equilibrium all capital in excess of K S would be immediately converted into new varieties.

10 It has been assumed that K S < K M GR ; i.e. <( 1)

If < ( 1) then K S < K M GR and N S < C M GR ; implying that if N 0 < N S then N 0 < C M GR ;so that Proposition 3 applies.

In that respect, horizontal innovation is very di¤erent from vertical innovation. The latter raises the future physical amount of goods available to the economy, and interest rates must be higher for consumers to absorb those goods. In contrast, horizontal innovation raises the marginal utility of future consumption, which makes it more valuable to save at any given interest rate.

This accumulation process in anticipation of a future wave of innovation is reminiscent ofGreenwood and Yorukoglu's (1994) analysis of the IT revolution. The convergence path highlighted here has two phases: a capital accumulation phase and an innovation phase. If the contribution of innovation is mismeasured, the second phase may be wrongly interpreted as a 'slump'.

Thus, in Regime III, a "vertical" innovation, i.e. an increase in TFP, triggers a phase of capital accumulation boosted by higher savings, followed by a halt to capital accumulation and a phase of vertical innovation. This bears some similarities to Schumpeterian long waves such as those analyzed by[START_REF] Aghion | What Do We Learn From Schumpeterian Growth Theory?[END_REF].

In some sense, type I and III equilibria in the preceding section are extreme versions of the ones highlighted here.

Gali (1999),[START_REF] Basu | Are Technology Improvements Contractionary?[END_REF]..

1 This research has bene…tted from …nancial support from CEPREMAP, Paris Sciences Lettres and the OSE Labex under grant ANR-10-LABX-93. I am grateful to Florin Bilbiie for helpful discussions.

Our equilibrium trajectory coincides with that trajectory for t T: Thereafter, we assume that c t = 1= ; K t = K S ; and therefore

Clearly, N converges monotonically to N S > N 0 :

We now show that this trajectory satis…es all the equilibrium conditions.

First, the Euler condition ( 13) is always satis…ed since _ c = 0 and c = 1= : The law of motion ( 5) is also satis…ed. For t T; we have that r t = r(K S ) = ( 1) and t = ( 1)= : Therefore, from (10), V t = 1= : The economy is in the innovation regime, consistent with our assumption that _ N > 0: For t < T; integrating (10) yields

Consequently, the economy is in the no innovation regime for t < T; consistent with _ N = 0:

Thus, the constructed trajectory satis…es all the equilibrium conditions.

B. Assume now that K 0 < K and consider the following system

This is a standard system and there exists a unique saddle-path converging to its steady state such that c = 1= and K = K: Note that since

We can express this trajectory as c = c SP (K); where c 0 SP > 0:

by a zero value. It is then straightforward to check that the trajectory de…ned by

where the marginal productivity condition w t = (1 )A t K t L t has been used to derive this last expression. In turn, we can solve for K and L as a function of c :

The composite parameters ' and are de…ned as follows:

We note that (23) de…nes K as a decreasing function of c which maps (0; 1= ) onto (0; +1): Therefore, there exists a unique c 0 associated with the initial capital stock K 0 :

for 0 c 0 is arbitrarily close to 1= :

Since (13) here is equivalent to

we clearly have that _ c > 0 throughout and c t ! 1= : Then, using ( 25) and (23, we have that

Therefore

where the inequality comes from the assumption that g A > 1 2 ( ( 1)= ): A corollary is that for 0 < g 0 < g; if c t is close enough to 1= ; the RHS of ( 26) is > g 0 :

Therefore, we can always pick 0 small enough so that _ Kt Kt > g 0 > 0 throughout our entire constructed trajectory.

It is also possible to compute Y

; and we get

; where = ' 1 : Again lim t!1 _ Yt Yt = g: Turning now to labor supply, we have to make sure that L t as de…ned by (24) remains < 1 throughout. We have that

Consequently,

where the inequality comes from the assumption that g A < (1 )( ( 1)= ):

Clearly, then, if c t is close enough to 1= ; the RHS of ( 27) is negative. Since c t grows over time, all we need for this to be the case for all t is that c 0 is close enough to 1= ;

or equivalently, again, that 0 is not too large. Then, for L t to be lower than 1, we just need it to be lower than 1 initially. Substituting (23), into (24), we get that

To conclude the proof, we need to check that the implied trajectory for N t is feasible.

Let z t = N t =K t : The law of motion (21) can be rewritten as

The expression on the RHS is always larger than (1 ) ( 1) ; which is > g A by (18).

Recall that by picking 0 small enough, we can have c t arbitrarily close to 1= and _ Kt Kt arbitrarily close to g throughout the entire trajectory. If in addition to that, we pick z 0 close enough to z; _ z t =z t will be arbitrarily close to zero. Consequently, N t = z t K t will grow at a strictly positive rate throughout. This proves that the trajectory for N is feasible.

QED