Gilles Saint

Gilles Saint-Paul

How Darwinian should an economy be?

Keywords: E02, E14, E32 Adaptive learning, Cobweb model, selection, aggregate dynamics

This paper studies aggregate dynamics in a cobweb model where learning takes place through a selection mecanism, by which more successful …rms are replicated at a higher rate. The structure of the model allows to characterize analytically the aggregate dynamics, and to compute the e¤ect on welfare of alternative levels of selectivity. A central aspect is that greater selectivity, while bringing the distribution of …rm types closer to the optimal one at a given date, tends to make the economy less stable at the aggregate level.

, …rms di¤er in their labor/capital ratio. They do not choose it optimally, rather it is a characteristic of a …rm. The distribution of …rms evolves over time in a way that favors the most pro…table …rm types. Selection may be inadequate because …rms are being selected on the basis of incorrect market signals. Selection itself may reinforce such mispricing, thus generating instability.

I compare economies that di¤er in the volatility and persistence of their productivity shocks, as well as the elasticity of labor supply. The key …ndings are as follows.

First, a trade-o¤ arises since greater selection allows to better track shocks and limits mutational drift in …rm types; on the other hand, selection may strengthen cobweb oscillatory dynamics.

Second, there seems to be a value in maintaining a diverse "ecology of …rms", in order to cope with future shocks.

These observations explain the key results. Optimal selectivity is larger, the less "cobweb unstable" the economy, i.e. the more elastic the labor supply. Second, optimal selectivity is larger, the more persistent the aggregate productivity shocks. Finally, optimal selectivity is larger, the lower the variance of productivity innovations.

The model can be extended to allow for …rm entry and trend productivity growth, and a selection process with memory. Empirical evidence suggests

Introduction

In most of macroeconomics, agents are considered as su ciently intelligent to carry all required calculations and compute the rational expectations equilibrium (REE). An important literature, however, questions that assumption and tries to examine the extent to which the economy can "learn" such an equilibrium 1 . In many cases, for example, a reduced form law of motion for the variables of interest is postulated and the agents learn its parameters, typically by using least squares or Bayesian techniques. This paper asks the following question how does an economy behave, when learning takes place through a Darwinian selection mechanism by which less pro…table …rms are eliminated and more pro…table ones replicate themselves? Does greater selection systematically bring the economy closer to the rational expectations equilibrium? When it does not, what are the dynamic properties of aggregate uctuations, and how does welfare depend on the parameters that govern the selection process? A naive "as if" argument would predict that the more selective the economy, the closer it is to the REE. Yet such an argument overlooks the fact that the market signals that are driving selection need not be the correct ones, because the environment in which selection takes place, as determined by the current value of the shocks and the current distribution of the individual …rms characteristics, is not the same that will apply to the …rms that have been selected. Furthermore, selection of a given type of …rm perturbs the market signals in such a way that mistakes in the selection process can be reinforced 2 .

I study these issues in the context of a simple partial equilibrium economy with potential cobweb cycles. As in [START_REF] Nelson | An Evolutionary Theory of Economic Change[END_REF], …rms di¤er in their labor/capital ratio. They do not choose it optimally, rather it is a characteristic of a …rm (that is, this behavior is hard wired in their DNA).

Firms whose labor/capital ratio is further away from the pro…t-maximizing one are selected out. As a result the distribution of …rms evolves over time in a way that favors the most pro…table …rm types. A single parameter, called selectivity, captures how stringly the most pro…table types are favored.

Which …rm type is most pro…table depends on current wages and on the current realization of an aggregate productivity shocks. Therefore, selection may be inadequate because di¤erent wages and productivity levels will prevail in the future. Furthermore, excess selection may be destabilizing because it may induce a cobweb cycle when wages are low the most labor-intensive …rms are selected, which leads to too high wages and too high labor demand in the subsequent period, where the least labor-intensive …rms will be selected, thus intensifying the cycle. Such cycles illustrate that …rms are being selected on the basis of incorrect market signals and that selection itself also contributes to this mispricing.

I compare economies that di¤er in the volatility and persistence of their productivity shocks, as well as the elasticity of labor supply. For each of those economies I can characterize their aggregate dynamics as a function of their degree of selectivity. I can also compute aggregate welfare and the selectivity level which maximizes that welfare.

The key …ndings are as follows.

First, there is a trade-o¤ due to the fact that, on the one hand, greater selection allows the economy to better track aggregate shocks (as long as they have some persistence) and limits the mutational drift in the cross-sectional variance of …rm types; on the other hand, as ust pointed out, selection may strengthen cobweb oscillatory dynamics, which leads to increased volatility and potentially unstable dynamics.

Second, there seems to be a value in maintaining a diverse "ecology of …rms", because the …rm types that will be more adequate in future (uncertain) environments have to be drawn from the pool of existing …rms. If selection is too extreme in the current environment, the …rms that are best adapted to a given future environmental change, yet performing poorly in present circumstances, will be very scarce, and it will take longer for the economy to produce a large number of such …rms in the new environment.

These observations help us to understand the results. We …nd that Optimal selectivity is larger, the less "cobweb unstable" the economy, i.e. the more elastic the labor supply. This is because the more elastic the labor supply, the less distorted the wage signals may be and the more dampened the oscillations of the economy in response to an initial misalignment in wages. Therefore, the less destabilizing a given degree of selectivity will be. Indeed, if labor supply were in…nitely elastic, that would pin down wages at their correct social value; they could not be distorted by wrong decisions on the demand side of the labor market.

Optimal selectivity is larger, the more persistent the aggregate productivity shocks. This is because selection that takes place now a¤ects the distribution of …rms in the future. If shocks are more persistent …rms that do better today are also more likely to do better in the future, hence selectivity is more valuable.

Optimal selectivity is larger, the lower the variance of productivity innovations. This is the "biodiversity" e¤ect. When productivity shocks are more volatile, the future is more uncertain and this makes it more valuable to keep a su cient mass of …rms of various types, because it is more likely that one of them will be the optimal one.

The model can also be extended to account for an endogenous capital stock, economic growth, and a selection process "with memory", i.e. which rewards past performance in addition to ust current performance. It is shown that the optimal selectivity level should go up (resp. down) with economic growth, if capital accumulation is unresponsive (resp. responsive) to growth. Also, my numerical simulations indicate that if selectivity is chosen optimally, faster growing economies will tend to have more volatile uctuations. Finally, memory tends to raise the optimal level of selectivity, because it introduces another mechanism for raising inertia.

While selection is governed by a mechanical process, its parameters can be intuitively related to economic institutions. For example, we may think that greater selectivity is the outcome of more competitive markets or more stringent credit conditions. The results imply that which institutions work best at delivering a sound macroeconomic performance depends on the structure of the shocks and of the productivity growth process faced by the economy. I provide some suggestive evidence that the nexus between institutions, selectivity, and aggregate inertia may be at work in a cross-section of countries I show that there exists a positive correlation between the ranking of a country in a number of indicators of product market regulation, on the one hand, and its inertia in the aggregate labor/capital ratio, on the other hand. This is consistent with the model provided one assumes that product market regulations reduce selectivity.

Related literature

Selection naturally intervenes in all models where some relevant dimension of economic activity is sub ect to an extensive margin (see for example ovanovic (1982), Caballero andHammour (1994), M litz (2003)). However, as long as agents optimize, such selection is ust a by-product of the existence of nonconvexities and …xed costs. The assumption that …rms are rational optimizing agents is not adequate to analyze the central role of selection as a mechanism for error correction in a capitalist economy3 . By contrast in models of bounded rationality and adaptive learning, selection is an essential ingredient of the process by which the economy evolves. This is not the …rst paper which studies those issues in the context of the cobweb model. Following the standard results of [START_REF] Ezekiel | The cobweb theorem[END_REF] and Muth (19 1), the literature has analyzed whether the cobweb cycle converges depending on learning processes (Carlson (19 9), Bray and Savin (198 )).

More recently, [START_REF] Arifovic | Genetic algorithm learning and the cobweb model[END_REF] has addressed the same issue using genetic algorithms, that is, applying Darwinian selection mechanisms to the learning strategies being used4 . Her simulations indicate that, in the absence of shocks, the economy generally converges to the rational expectations equilibrium, even though the parameters may be such that it is cobweb-unstable. [START_REF] Franke | Coevolution and stable ad ustments in the cobweb model[END_REF], building on this work, provides a number of interesting simulations that typically (but not systematically) imply that the economy does not deviate much from the Walrasian equilibrium.

Our simple framework allows us to parametrize selectivity by a single number, derive linear dynamics at the aggregate level, and compute the selectivity parameters that deliver the highest welfare. The price to be paid is that the strategies that the …rms follow are …xed (as in [START_REF] Arifovic | Genetic algorithm learning and the cobweb model[END_REF]) so that the richer adaptive learning strategies of [START_REF] Franke | Coevolution and stable ad ustments in the cobweb model[END_REF] are not consid-ered S . Besides showing that greater selectivity reduces inertia and raises the likelihood of unstable aggregate oscillations, this paper s key contribution is to show how the optimal degree of selectivity depends on the economic environment, as de…ned by (i) the elasticities of supply and demand, (ii) the volatility and persistence of aggregate shocks, (iii) the variance of mutations, and (iv) the trend rate of productivity growth.

A simple selection process

There is a continuum of …rms of total mass equal to one. Each …rm i has one unit of capital. Its production function at date t is

y it = A t l i ;
where A t is an aggregate productivity shock. Firms do not optimize, instead their behavior is pinned down by their "type". Here this means that each …rm pursues a …xed employment policy l i : More generally this could stand for various aspects of the "DNA" of a …rm, such as managerial practices, etc. Optimization then only takes place indirectly, through the way markets select …rms. At date t; total labor supply is such that

w t = !L t ;
where L t is total employment and w t is the wage. Given w t ; the pro…t of a …rm of type l at date t is given by t (l) = A t l w t l:

Note however that Franks results themselves, somewhat fascinatingly, show that a large proportion of …rms survive following …xed output strategies, despite the other …rms pursuing more exible behavioural rules.

The most pro…table …rm type, at t, is the one such that l = l t = A t w t 1 1 : I assume that the distribution of …rm type at t is given by f t (x); where x = ln l: f t () evolves over time for two reasons. First, …rm types are sub ect to random mutation. Second, selection by markets raises the frequency of the most pro…table …rms. This selection process is formalized as follows the greater the distance between a …rm type and the most pro…table type, the more its frequency is reduced in the following period. More speci…cally, let x t = ln l t and let 2 t = V ar t (x) be the cross-sectional variance of the distribution of log …rm size: At the end of period t; random mutation takes place, so that x is replaced by x + "; where " is a random noise with density h("): That is, the distribution of …rm types at the end of period t is given by

g t (x) = Z + f t (y)h(x y)dy:
After this mutation takes place, selection operates so as to favor the …rm types that were closer to the most pro…table one T . Hence the distribution at the beginning of t + 1 is given by

f t+1 (x) = g t (x) exp( (x x t ) 2 2 ) D t ; (1)
where

D t = R g t (x) exp( (x x t ) 2 2
)dx: The parameter captures the intensity of selection. At = 0 no selection takes place, and the distribution of …rms keeps spreading under the in uence of random mutations. At = +1 only …rms that have the optimal employment level survive, and next period all …rms will have that type.

Strictly speaking, it would be more rigorous to assume that f t (x) is altered by a multiplicative factor which is increasing in t (e x ); but replacing pro…ts by the distance to the optimal employment level is a handy approximation.

The model is silent about how this selection process operates. There are three potential margins imitation (by either new entrants or existing …rms), exit, and growth of the most successful …rms. The relative importance of these three margins is irrelevant here.

It is easy to see that if f t () is normal, i.e. f t (x) = 1 p 2 t exp( (x xt) 2 2 2 t ); and if h is normal, that is h(") = 1 p 2 m exp( " 2 2 2 m
); then so is f t+1 : Furthermore the mean log employment level then evolves according to

x t+1 = x t + ( 2 t + 2 m ) x t 1 + ( 2 t + 2 m ) ; ( 2
)
while the variance of the distribution evolves according to

2 t+1 = 2 t + 2 m 1 + ( 2 t + 2 m ) : (3)
We are now in a position to solve for the equilibrium of this economy. It will be useful to use the following parameters We have that

x t = 1 1 (ln + a t ln w t ) ; ( 4
)
where a t = ln A t ;

ln L t = ln E t l = E t ln l + 2 t 2 = x t + 2 t 2 : ( )
This determines the wage at t :

ln w t = x t + 2 t 2 + ln !: ( )
Finally, substituting ( ) into (4) and then into (2) we get

x t+1 = b t + ( 2 t + 2 m ) (1 + ( 2 t + 2 m ))(1 ) a t + 1 ( 2 t + 2 m ) (1 + ( 2 t + 2 m ))(1 ) x t ; (7)
where

b t = ( 2 t + 2 m ) (1 + ( 2 t + 2 m ))(1 ) ln ! 2 t 2 :
Equations ( 7) and ( 3) characterize the dynamics of the system. While they are non linear, the system is asymptotically univariate and linear. This is because (3) implies that 2 t evolves deterministically and converges monotonically to

2 = 2 m 2 + p 4 m + 4 2 m = 2 : (8)
This asymptotic cross-sectional dispersion of …rms is larger, the greater the "mutation rate" 2 m and the smaller the selectivity parameter : It becomes in…nite as ! 0 and nil as ! 1:

Also, the deterministic (log) employment component b t converges to b = 2 1 ln ! 2 =2 (9)
Asymptotically, then, the evolution equation of x becomes

x t+1 = b + 2 (1 ) a t + x t ; (10)
where

= 2 2 + 2 m 2 1 = p 1 + 4=d 1 2 1 + p 1 + 4=d : (11)
This formula shows the …rst result of this paper PROPOSITION 1 Assume the stochatic process for a t is stationary.

The dynamics of x t are stable if and only if > 1; or equivalently

1 d > 2 1 4 : (12)
Furthermore, if

1 d < + 2 ; ( 13
)
then the dynamics are oscillatory, i.e. < 0.

PROOF The AR1 term,

2 2 + 2 m 2 1 ;is always < 1: Dynamics are stable i¤ it is > 1: This is equivalent to < 1 1 2 + 2 m + 1 2
; which by ( 8) is equivalent to (12). Dynamics are oscillatory i¤

2 2 + 2 m 2 1
< 0;i.e.

< 1 1 2 + 2 m
; which is equivalent to (13). ED

The case where = +1 and where there are no shocks delivers the standard cobweb cycle [START_REF] Ezekiel | The cobweb theorem[END_REF] by selecting only the most pro…table …rm at t; markets set labor demand a t + 1 at the level that corresponds to wages at t; if wages are high at t; employment is low at t + 1; and wages are low at t + 1: It is well known that this cycle converges if and only if < 1 ; i.e. < 1: Indeed, if < 1; (12) always holds. In the sequel I will label an eonomy such that the cobweb cycle converges "cobweb-stable".

Furthermore, the greater ; the greater the absolute value of the root of such a cycle. Hence is an index of cobweb instability.

As shown by Proposition 1, if the economy is cobweb-stable, then it is even more stable under …nite selectivity. Otherwise, it will be stable provided selectivity remains below a given threshold. The economy is more stable, the lower its selectivity and the lower the mutation rate. Therefore, the selectivity threshold below which the economy is stable is lower, the greater the mutation rate. In generating instablility, mutation plays a somewhat similar role as selectivity. A greater mutation rate means that the pool of …rms pursuing today s optimal policy rather than yesterday s will be larger, which increases the number of …rms that will pursue this policy tomorrow, and therefore the likelihood of instability.

If we interpret, plausibly, economies with a larger or a larger mutation rate as being more of the "capitalist" kind, then Proposition 1 provides some foundations for the often heard claim that "capitalist economies are inherently instable". This instability comes from the fact that selection takes place on the basis of incorrect prices that is, on the basis on wages at t; instead of the REE wages at t + 1 and by inducing too many …rms to adopt the incorrect type, prices next period are made even more incorrect.

Suppose now a central planner wants to choose how capitalist the economy is, i.e. what the optimal value of should be. This central planner can be viewed as a metaphor for the outcome of "systems competition" la Sinn (2003). That is, it is reasonable to assume that economies that do poorly will eventually adopt the institutions of economies that do well. By looking at the optimum, I am silent about the way this macro-selection process operates, as the model is equally silent about how the selectivity parameter relates to actual institutions.

Our central planner clearly faces a trade-o¤ the economy is more stable when is lower, on the other hand it takes longer for it to learn the correct price, and it is less reactive to shocks. Finally, a lower value of also raises the asymptotic dispersion of …rm size because less selective pressure is exerted against mutations. The central planner would want all …rms to choose the Walrasian employment level (which also maximizes total surplus) given by

x t = 1 1
(ln + a t ln w t ) and ln w t = x t ; that is

xt = 1 1 + (ln + a t ): (14)
I assume that the central planner s welfare is given by

E + X t=0 t ln(S t + t );
where S t is the workers surplus and t total pro…ts. It is shown (Appendix 1) that maximizing this expression can be approximated by the following

problem min E + X t=0 t ( 2 t + (1 + )( x t xt ) 2 ): (1 )
I assume that a t follows an AR1 process

a t = a t 1 + " t ; (1 )
with " t N (0; 2 " ): Furthermore, the central planner acts "asymptotically" by minimizing the long-term welfare ow on the RHS of ( 1). That is, the government minimizes lim t3

2 t + (1 + )E( x t xt ) 2 ;which is equal to L = 2 + (1 + )((Eu) 2 + V ar(u)); ( 17
)
where u t = x t xt is the "average log size gap" (ALSG), that is, the di¤erence between the average log size of a …rm in our economy and the (common to all …rms) log size in the Walrasian economy. The preceding derivations allow us to characterize the law of motion for the ALSG

u t+1 = u 1 1 + " t+1 + 1 1 + a t + u t ; (18)
where

u = 2 4 : (19)
From the RHS of ( 17), we see that the asymptotic losses to the social planner come from three sources 1. The cross-sectional dispersion of …rm size, given by

2 = 2 m 2 ( 1 + p 1 + 4=d): (20)
This is larger, the larger the mutation rate and the smaller the selectivity level.

The average output gap, given by

Eu = u 1 : (21)
This formula tells us that the average mistake made by a …rm setting employment in this economy compared to the Walrasian one is larger, the more cobweb-unstable (the larger ) and the less selective (the lower ) the economy. Why is that so? The average output gap is negative (meaning …rms on average are too small compared to the Walrasian allocation) because log …rm size dispersion per se tends to raise aggregate employment due to ensen s inequality7 . This tends to raise wages above the Walrasian level, which reduces average log …rm employment. This e¤ect is stronger, the greater the reaction of wages to employment, i.e. the greater the instability

; and the greater the asymptotic cross sectional variance of log employment, i.e. the lower the selectivity parameter d:

3. The volatility of the ALSG, given by 1) :

V ar(u) = 2 " (1 + ) 2 2 (1 + )(1 + )(
(22)

The e¤ect of and

Table 1 reports the optimal value of d as a function of and for a typical set of simulations. The other parameters have been set to = 0:5; 2 " = 0:02 and 2 m = 0:05: The optimal value of d is driven by the trade-o¤ between long-term cross-sectional dispersion in …rms type and aggregate stability of employment dynamics. Two key patterns emerge.

First, optimal selectivity is larger, the more persistent the productivity shocks. This is because the more persistent those shocks, the more the most pro…table …rms at date t are likely to be the optimal …rm type in the subsequent periods. Note that even if the shocks are not persistent some selectivity is optimal, since otherwise mutations would accumulate and the dispersion of …rm types would become in…nite, which is ine cient. Furthermore, as long as it is not strong enough to generate instability, some selectivity brings the …rm types in line with the correct ones on average. Second, the greater ; i.e. the more cobweb-unstable the economy, the lower the selectivity level. The greater ; the more wages react to a deviation in aggregate employment, and the greater the welfare loss in subsequent periods from picking the wrong …rm type at a given date. Since selection never operates on the basis of the correct market signals, one is more willing to mitigate it when is larger. We can also note that when is large, selectivity is very inelastic to the degree of persistence in productivity shocks. which is equal to + 1+ : 8 We see that in some cases it is optimal to pick d so as 8 This is due to the fact that if

xt = x t E x; then xt+1 = d 1 ât + xt ; so that V ar(x t ) = d 2 (1+ ) 2 " ( 1
) 2 (1 )(1 2 ) and E xt xt+1 = V ar(x t ) + d 2 2

"

(1 ) 2 (1 )( 1) :

to allow for negative autocorrelation. This is the case for not too large and large. In other cases the optimal choice of d will increase the persistence of employment relative to its Walrasian counterpart (where it is simply equal to ). Note also that empirically observing a positive autocorrelation, i.e.

+ 1+

> 0; does not preclude the endogenous dynamics of employment to be oscillatory, i.e. < 0:

n 0 0.2 0.
1 2 0.01 0.28 0.12 -0.07 -0.28 -0. 4 -0.77 0.2 0.18 0.02 -0.1 -0.3 -0. 9 -0.77 0.4 0.07 -0.07 -0.24 -0.42 -0. 9 -0.77 0.

0.01 -0.17 -0.32 -0.47 -0.

-0.77 0.8 0.01 -0.24 -0.38 -0. 1 -0.71 -0.77 0.99 0.01 -0.30 -0.41 -0.

-0.71 -0.77 Table 2a Persistence parameter at the optimal choice of d: Same parameters as Table 1. 0.000 0.009 0.02 0.0 2 0.10 0.2 0.8 0.000 0.007 0.023 0.049 0.097 0.2 0.99 0.000 0.004 0.021 0.04 0.097 0.2 4 Average ALSG ( ) at the optimal choice of d: Same parameters as Table 1 3.0.2 The e¤ect of 2 m and 2 " :

How do the variances of mutations and aggregate shocks a¤ect the optimal selectivity level? We can again answer this question by running numerical simulations. Also, in some special cases we can get an analytical solution.

For = = 0 we have that

Eu = 0 and V ar(u) = 2 "
(1 ) 2 1 :

Therefore, one replace the choice variable by and the asymptotic ob ective is

min 2 m 2(1 ) + 2 2 " ( 1
2 )(1 + ) :
The optimal value of is 1) m

= 2 " (1 ) m 2 " + (
and the corresponding value of d is 9

d = 2 "
(1

) 2 2 m 1 4 1 if 2 " > (1 ) m = +1 if 2 " (1 ) m :
This expression suggests that optimal selectivity is a decreasing function of the " = m ratio. The greater the variance of productivity shocks, the greater the "biodiversity" value of maintaining a large enough pool of …rms at various employment level, in order to better react to future "changes in the environment", i.e. productivity shocks that call for a change in the optimal …rm size. One can also show, based on the preceding formula, that d =d 2 m > 0: Thus selectivity clearly goes up with the variance of mutational shocks. Bigger mutations raise the long-term cross sectional dispersion of …rm size, which induces an increase in selectivity so as to limit the associated welfare losses.

Are these results con…rmed in more general parameter con…gurations? To answer that question we again have to use numerical simulations. Figure 1 shows the evolution of d as a function of 2 " for = 0:5 and twelve di¤erent 9 As = p 2 How can we explain this pattern? For a given ; an increase in the size of mutations makes the cross-sectional distribution of …rm size less persistent, which in turn reduces the persistence parameter : For < 0 this makes aggregate dynamics more unstable, with the associated welfare losses captured by ( 22). This e¤ect is stronger, the more negative ; i.e. the greater : To o¤set it, one must pick a lower value of : This e¤ect dominates for large enough, explaining the negative dependence of on 2 m : On the other hand, for low, the dynamics are unlikely to be oscillatory, and the contribution of ( dominates selectivity goes up when mutations are larger, to o¤set the increase in the long run cross-sectional dispersion of …rm size.

Extensions 4.1 Endogenous number of …rms and the e¤ect of growth

In the above model, the number of …rms is …xed. In this section I make it more general by endogenizing the number of …rms. This extension will then prove convenient to analyze the consequences of the introduction of an exogenous growth trend in total factor productivity.

I assume that a fraction s of total output is saved (as in the Solow (19 ) model), and that these savings are used to accumulate capital, which means more …rms here. But instead of a linear relationship between savings and capital accumulation, I assume that it is non linear and sub ect to decreasing returns to scale, so that the following relationship holds

K t+1 = Y t ;
(23) with 2 [0; 1]: In general should depend on s; but this dependency is immaterial, and we ignore it here. The above model is a special case for s = 0; = 1 and = 0: The parameter captures how sensitive the number of …rms is to the output level. The evolution of the distribution of …rm size is unchanged from the previous analysis, meaning that the RHS of ( 1) drives the distribution of …rm size among new entrants as well as survivors.

We clearly have that Y t = A t K t E t l i : Taking logs and denoting by y t = ln Y t , this clearly implies that

y t = a t + k t + x t + 2 2 t 2 : (24)
Similarly, L t = K t E t l i and therefore

ln L t = k t + x t + 2 t 2 :
This equation allows us to derive the equilibrium wage and pro…t-maximizing log …rm size, that are respectively equal to

ln w t = ln ! + (k t + x t + 2 t
2 )

and

x t = 1 1 (ln ! + a t (k t + x t + 2 t 2
)):

Using the same derivations as in Section 3 we get the aymptotic law of motion for x t :

x t+1 = ( 12

) (1 + )) x t 2 ) k t + 2 ) 1 a t + b ) ; (2 )
where b ) is de…ned by (9).

Using ( 23) we also get an evolution equation for k t = ln K t :

k t+1 = a t + k t + x t + c ) ; (2 )
where We note that ( 27) is more likely to hold, the greater : the more entry is sensitive to economic activity, the more stable the economy. Capital accumulation raises inertia, making it less likely that the economy oscillates.

c ) = ln + 2
The preceding extension can be applied to an economy where TFP a t has a deterministic growth trend. More speci…cally, I assume that a t = a Ct + gt;

where g is the trend growth rate and a Ct the cyclical component. I assume that the law of motion for a Ct is given by the same AR1 process as above, i.e. eq. ( 1).

We want to know how the optimal selectivity depends on growth; we also want to know how this dependence a¤ects the economy s cyclical properties.

For this we …rst need to extend the welfare criterion derived in (17) to the case with an endogenous capital stock. This is done in the Appendix. 8, one should …rst note that the deterministic growth component of the average log …rm size is equal to (see Appendix) 1)) ( 1)(1 + (1 + ( 1))) :

g x = g 1 (1 + (
This component is the same in our equilibrium and in the Walrasian one our economy does not diverge from its Walrasian counterpart over time.

If < 1 1+ ( 1) ; g x is positive and grows with g : the capital stock is not responsive enough to trend growth for the size distribution of …rms to remain constant. Firm size grows over time. We have seen in Table 4 that …rm size is typically smaller, on average, than the Walrasian benchmark. Since the distribution of …rm size that produces at t + 1 is selected from the pool of …rms at date t; an increase in g tends to widen the gap between the average …rm size at t and its Walrasian counterpart. This raises the gain to the social planner of being more selective, because greater selectivity reduces the ALSG Eu in absolute value; as illustrated by ( 19) and ( 21).

For > 1 1+ ( 1) ; however, entry "overshoots" the level that would deliver a constant distribution of x through time. As a result, the average …rm size shrinks over time, more so, the greater g: The lag in …rm selection now tends to o¤set the fact that the ALSG is smaller in our equilibrium than in the Walrasian one. This o¤setting factor is stronger, the greater g: Therefore, as g goes up, the ALSG shrinks in absolute value, which reduces the bene…ts of selectivity.

Finally, for = 1 1+ ( 1) ; both our economy and the Walrasian one are in a balanced growth path where the cross-sectional distribution of …rm size (which is degenerate in the Walrasian case) has no deterministic trend. As a result the growth rate has no impact on the ALSG and consequently no impact on the optimal selectivity level. For our parameter values this corresponds to = 2=3 and is reported in the last column of Table 8. gn 0 0.2 0. 0.7 0.9 1 2/3 0 0.00 0.010 0.023 0.039 0.072 0.103 0.03 0.02 0.00 0.011 0.023 0.039 0.07 0.1 9 0.03 0.0 0.00 0.011 0.024 0.039 0.07 0.1 9 0.03 0.1 0.007 0.012 0.02 0.038 0.07 0.1 9 0.03 0.1 0.008 0.013 0.02 0.037 0.07 0.1 9 0.03 0.2 0.009 0.014 0.02 0.03 0.07 0.1 9 0.03 Table 9 Output volatility as a function of g and Table 9 reports the corresponding volatility of output. For < 1 1+ ( 1) = 2=3; selectivity grows with g; and we observe that this is associated with an increase in volatility. For = 0 this is due to the increase in the absolute value of the negative root, i.e. an ampli…cation of the cobweb cycle, as discussed above. For > 0 the roots are complex and the results are more di cult to interpret. Overall, though, the results suggest that economies that grow faster should be more volatile, although this is not systematic (see column ).

Selection with memory

I now study an extension of the model where selection takes place on the basis of current and past pro…tability, instead of current pro…tability alone.

Intuitively, this means that investors have a "long memory" and that the …rms that are selected for are those that have a greater pro…t over some period of time10 . A natural way to formalize this is to assume that instead of ( 1) we have

f t+1 (x) = g t (x) exp( (x xt) 2 2 ) D t ; (28)
,where the most selected type is a geometric weighted average of current and past most pro…table types

xt = xt 1 + (1 )x t = xt 1 + 1 1 (ln + a t ln w t ) :
is a parameter which captures the memory, or horizon, of the selection process. It is obvious that ( 3) is unchanged and that in [START_REF] George | Learning and Expectations in Macroeconomics[END_REF] x t has to be replaced by xt : Consequently, asymptotically we end up with a two dimensional system which evolves as

x t+1 = 1 + p 1 + 4=d 1 + p 1 + 4=d ! x t + d 2 ( 1 + p 1 + 4=d)x t xt+1 = (1 ) d 2 ( 1 + p 1 + 4=d) xt (1 ) 1 + p 1 + 4=d 1 + p 1 + 4=d ! x t (1 ) 2 m 4 ( 1 + p 1 + 4=d) + 1 1 (ln + a t+1 ) ;
We can already analyze how the stability of the system is a¤ected by the selection process parameters ( ; ): The following proposition can be proved.

Proposition 3 A su cient condition for the system to be stable is

> p 1 + 4=d + p 1 + 4=d : (29)
PROOF See Appendix 2.

We note that this condition is satis…ed for any provided < p 1 + 4=d;

which is equivalent to condition (12). Condition ( 12) is associated with the special case = 0: Intuitively, the longer the memory of the selection process, the less likely it is that the economy will oscillate and the more likely it is to be stable 11 . A longer memory thus enhances the range of selectivity values compatible with stability the trade-o¤ between selectivity and inertia is eased. As (29) makes clear, the maximum level of selectivity consistent with stability becomes in…nite as converges to 1.

11 From the Proof of Proposition 3, we can partition the ( ; ) plane in various regions. Negative roots prevail for high values of and low values of : If the economy is cobweb unstable, there is a region where is too large and too low for these oscillations to be stable. But there is also a region where they are dampened. For larger and smaller, the roots are positive and the economy is always stable. Finally there is a region where the roots are complex, and in this case the economy is again stable, with dampened oscillations at a frequency lower than a 2-cycle.

It is then natural to expect that the optimal selectivity level will depend positively on the length of memory : The following While the preceding discussion obviously rests on a very speci…c model, which ignores many features of an economy as well as many dimensions of a …rm s characteristics upon which selection may take place, it is interesting to illustrate empirically one key proposition emerging from the above analysis, namely that there is a negative relationship between the intensity of selection and the degree of inertia at the macroeconomic level.

While how to measure selection in a real economy is open to discussion, I am more particularly interested in documenting the view that some institutions such as barriers to entry, lack of investor protection, etc, are likely to reduce the selection level. If the above analysis is correct, we should expect countries with more regulated markets to exhibit a higher level of macroeconomic inertia.

My strategy is to use the Penn World Table to estimate a simple version of Equation ( 2) for a number of countries13 , and correlate the implied inertia in x (measured as the labor/capital ratio from the PWT) with some indices of product market regulation from the OECD product market regulation database. I remain relatively agnostic regarding the determinants of the preceding period s optimal value of x; and I proxy it by a set of three variables TFP, the real wage, and the real exchange rate. The coe cient of interest is the degree of inertia, i.e. the coe cient on the lagged value of x: According to the model, it should be equal to

H = 1 1 + ( 2 2 + 2 m ) = p 1 + 4=d 1 p 1 + 4=d + 1 :
This is clearly decreasing from 1 to 0 as d goes up from zero to in…nity.

I then rank the countries by this estimated persistence coe cient (hence more highly ranked countries are interpreted as being more selective) and correlate this with their rank in a number of OECD indices of product market regulation. The results are reported in Table 11.

Indicator

Rank correlation with H State control 0.22 (0.1) Barriers to entrepreneurship 0.24 (0.11) Barriers to international trade -0.1 (0.11) Regulation of legal professions 0.14 (0.11) Regulation of retail trade 0.3 (0.1) Regulation of energy, transport and communication 0.28 (0.11) Table 11 Mean rank correlation between estimated inertia and OECD indicators of product market correlation. Standard deviations in parentheses.

Both means and standard deviations estimated with bootstrap simulations to account for the sampling error in the estimation of H . We see that the rank correlation, although not very large is generally positive, reaching 0.3 for retail trade. The only indicator that exhibits a negative correlation with inertia is that of barriers to international trade.

However while this type of regulation forces domestic prices to deviate from international prices, there is no reason to believe that it would reduce the severity of selection among domestic …rms, contrary to the other indicators.

That this indicator, unlike the others, is not positively correlated with the degree of inertia, therefore comes as no surprise.

While these results hardly constitute a proof of the model (we can think of mechanisms other than selection through which regulation would a¤ect inertia), they are consistent with its general message that there exists a link between selectivity at the micro level and inertia at the macro level.

Conclusion

In this paper I have developed a model that allows us to study the interactions between the intensity of selection at the microeconomic level and aggregate dynamics, as well as to discuss what level of selectivity is most desirable depending on the economic environment. This exercise has its limits because it assumes a mechanical rule for the evolution of the cross-sectional distribution of …rm characteristics. This is the price to be paid for analytical transparency. An important direction for future research consists in providing foundations for the selection mechanism based on real world institutions such as the rules governing bankruptcies and corporate governance, as well as labor and product market regulations.

Appendix

7.1 Derivation of (15).

The workers surplus is equal to

S t = w t L t ! t L 1+ t 1 + : Therefore t + S t = Y t ! t L 1+ t 1 + : (30)
Furthermore

Y t = A t Z +4 54 f t (x)e x dx:
Therefore

ln Y t = a t + ln El = a t + x t + 2 2 t 2 :
Similarly ln L t = x t + 2 t =2:

For convenience we rewrite ( 14)

xt = 1 1 + (ln + a t z t ):
Using this and the preceding derivations, (30) can be rewritten as

t + S t = A 1+ 1 + t ! 1 + t 1 + exp( ( x t xt ) + 2 2 t
2 )

A 1+ 1 + t ! 1 + t 1+ 1 + 1 + exp((1 + )( x t xt + 2 t 2
)):

We can check that 2 < 1: As for 1 ; it must be greater than 1 for the system to be stable. This is equivalent to

3 + x(1 + ) > p (x):
This inequality is violated if x 3+ 1+ : Suppose that x < 3+ 1+ : Then the preceding inequality is equivalent to

(3 + x(1 + )) 2 > (x):
Rearranging, this latter condition is equivalent to

x < x : (32)
Since, on the one hand, x < 3+ 1+ ; and, on the other hand, x > x 2 and the system is stable for x 2 (x 1 ; x 2 ); it follows that the system is stable i¤ (32) holds. Then, we note that x = d+ @ d 2 +4d 2

; substituting into (??) and rearranging, we get that the system is stable i¤

1 d > + 1+ 2 1 4 :
As > ; this is more likely to hold, the greater :

7.3 Computing social welfare in the model with endogenous capital

Relative to Appendix 1, consumption now di¤ers from output. We have to subtract savings from it. Therefore, the ow of total welfare is given by

ln ( t + S t sY t ) = ln((1 s)Y t ! L 1+ t 1 + ):
We follow the approach of Appendix (ln + a t ln wt )

and ln wt = ln ! + (k t + x t ): Therefore,

xt = 1 1 + ln ! + a t kt : (33)
The law of motion for capital in the Walrasian equilibrium is

kt+1 = ln + kt + a t + xt : (34)
For convenience we reproduce our dynamical system

x t+1 = (1 2 A (1 + )) x t 2 A k t + 2 A 1 a t + b A ;
(3 )

k t+1 = a t + k t + x t + c A : (3 )
It is easy to compute the trend growth rates for x and k from those equations. One gets 1)) ( 1)(1 + (1 + ( 1)))

g x = g 1 (1 + (
and

g k = g
(1 + ( 1)) ( 1)(1 + (1 + ( 1))) > 0:

One can also check from ( 33) and ( 34) that the trend growth rates of xt and kt+1 are the same.

Let x = x x and k = k k: Then we have that

Y t = exp(k t + a t + x t + 2 2 B =2) = exp( kt + a t + xt ) exp( kt + xt + 2 2 B =2):
Similarly,

L 1+ t = exp((1 + )( kt + xt )) exp((1 + )( kt + xt + 2 B =2)):
Now using (33) we get that

kt + a t + xt = 1 + ln ! + M t and (1 + )( kt + xt ) = 1 + 1 + ln ! + M t ;
where

M t = 1 + 1 + a t + (1 + ) (1 ) 1 + kt :
Clearly, then,

(1 s)Y t ! L 1+ t 1 + = ! 1 + exp M t : (1 s) exp(k t + a t + x t + 2 2 B =2) 1+ exp((1 + )( kt + xt + 2 B =2))
:

Only the expression in brackets depends on : Therefore, maximizing ln ( t + S t sY t ) is equivalent to maximizing that term. Denoting this term by exp( ); computing its second-order Taylor expansion and keeping only terms of order 0,1, and 2 in ( k; x; C ) we get

exp 1 s 1 + 1 a 1 k + a 2 x + a 3 2 C + a 4 k2 + a 5 x2 + a 6 xk ; (37)
where

a 1 = 1 s 1 s =(1 + ) a 2 = s 1 s =(1 + ) a 3 = 2 (1 s) 2 (1 s =(1 + )) a 4 = 1 s (1 + ) 2(1 s =(1 + )) a 5 = 2 (1 s) (1 + ) 2(1 s =(1 + )) a 6 = (s + ) 1 s =(1 + ) :
Applying again a second-order Taylor expansion for ln(1 + x) to (37) and taking expectations we get, neglecting again terms of order greater than 2

E ln 1 s 1 + + a 1 E k + a 2 E x + a 3 2 C (38) + a 5 a 2 2 2 E x2 + a 4 a 2 1 2 E k2 + (a 6 a 1 a 2 ) E kx:
This is the quantity being maximized in the numerical simulations.

To compute the moments that appear in the RHS of (38) we use the following 4-dimensional system

kt+1 = kt + xt + 2 2 C 2 (39) xt+1 = 4 D 2 + 1 1 + g + 1 + a Ct 1 1 + a Ct+1 2 D kt + (1 2 D (1 + ))x t + 1 + kt + 1 + xt (40) kt+1 = kt + xt + (g + a Ct ) (41) xt+1 = 1 1 + g + 1 1 + a Ct+1 1 + a Ct (42) 1 + kt 1 + kt
To get (39), subtract (34) from (3 ). To get (41), subtract (34) lagged once from itself. To get (42), subtract (33) at t from itself at t + 1; then replace kt+1 in the resulting expression with the RHS of (41). To get (40), subtract (33) at t+1 from (3 ), then replace xt+1 in the resulting expression with the RHS of (42).

This system can be rewritten in matrix form as

V t+1 = AV t + B + t+1 ; where V = (x; k; x; k) E ; = 0 B B @ 1 + a Ct 1 1 + a Ct+1 0 1 1 + a Ct+1 1 + a Ct a Ct 1 C C A E ;
and the coe cients of matrices A and B are obtained straightforwardly from the above expressions.

It can be checked that, denoting by L the lag operator,

= (1 L) 1 (L)";
where (L) = 0 + 1 L+ 2 L 2 ; 0 = (

1 1 + ; 0; 1 1 + ; 0) F ; 1 = ( 1+ 1 + ; 0; 1+ 1 + ; ) F ; 2 = ( 1 + ; 0; 1 + ; ) F : Therefore, V t EV = (I A) 1 (1 L) 1 (L)" t = +G X j=0 A j 0 " t j + +G X j=1 A j 1 ( 1 + 0 )" t j + +G X j=0 A j +G X i=2 i ( 1 )" t i j ! = Q(L)" t :
The last term can be computed as 1): Altogether, this expansion allows us to get all the coe cients Q i of Q and then to compute Let us now turn to the regime where the eigenvalues are real, i.e. where (43) is violated. We have to distiguinsh between two cases.

P +G k=2 ( I A) 1 ( k 1 I A k 1 ) 2 (
V ar(V ) = +G X j=0 Q i Q F i ! 2
Case 1

( Q + ) > 0:
This inequality holds i¤

> (1 ) (1 + )(1 ) + = pos ( ):
This de…nes a threshold for which is increasing in and converges to 1 + as ! 1: Then both eigenvalues are positive, and the largest one is

x 1 = Q + + p 2 ;
where = ( Q + ) 2 4 1+ : For the system to be stable we need that x 1 < 1; i.e. p < 2 Q :

A necessary condition is that 2 Q 0: This is equivalent to (1+2 )( 1)+ (1+ )( 1)+

; which always holds since the RHS is 1: Given that, the preceding inequality holds i¤

= ( Q + ) 2 4 1 + < (2 Q ) 2 ;
or equivalently

1 + < 1 ( Q + ); but 1 ( Q + ) = 1 + (1 ) (1 + )(1 ) 1 1 + > 1 + :
This proves that dynamics are stable in the zone where the eigenvalues are positive.

Next, note that at the frontier of this zone, we have Q + = 0; implying that (43) holds. Therefore, c ( ) < pos ( ) < + c ( ) for pos ( ) > 0:

This means that for > (1 )= ; the eigen values are complex, and stable, for c ( ) < < + c ( ); and positive, and stable for > + c ( ); while they are negative for < c ( ): On the other hand, for < (1 )= ; the eigenvalues are complex and stable for c ( ) < < + c ( ); and positive and stable for < c ( ) and > + c ( ):

Case 2

( Q + ) < 0;
i.e.

<

(1 ) (1 + )( 1) + :

Then both roots are negative, and the largest one in absolute value is

x 2 = Q + p 2 :
We have x 2 > 1 i¤ p < 2 + Q + :

7.5 Computing welfare in the case with memory

We rewrite the dynamical system as = N:

We then get that the variance-covariance matrix of ŷ; V; is the solution to the linear equation

V = AV A Q + N + AM + M Q A Q ;
which can be solved by vectorization.

The asymptotic social welfare is then given by ( 17), i.e. 0,01 0,04 0,07 0,1 0,13 0,16 0,19 0,22 0,25 0,28 0,31 0,34 0,37 0,4 0,43 0,46 0,49 0,52 0,55 0,58 0,61 0,64 0,67 0,7 0,73 0,76 0,79 0,82 0,85 0,88 0,91 0,94 0,97 1

  value of d as a function of and ; = 0:5; 2 " = 0:02, 2 m = 0:05:

  8 0.70 0. 1 0.48 0.22 0.07 0.99 0.99 0.98 0.98 0.97 0.94 0.92 Table2bEmployment autocorrelation at the optimal choice of d: Same parameters as Table1

  e¤ects of …rm size dispersion which is discussed above. As selectivity reduces dispersion, it also reduces the size of the ALSG.

;

  if the optimal is negative, it cannot be reached by picking d: I such cases, one cannot to better than setting d = +1: set of parameter values for 2 m ; and : These values are reported in Table .

1 As

 1 Figure 1 reports an inverse measure of d (1=(1 + d)), it con…rms our intuition that selectivity falls with the variance of productivity shocks.

Figure 2

 2 Figure 2 reports the evolution of as a function of 2 m : The corresponding parameter values are reported on Table7. We see that selectivity goes up

2 ) 2 :

 22 Proposition 2 characterizes the stability properties of the dynamical sys-

  Then, in order to compute social welfare, one needs to compute the stochastic steady state moments of the relevant vector (k t ; x t ; a t ): The relevant formulas are also derived in the Appendix. One can then use those formulas to compute the welfare-maximizing level of d: The next table shows how it depends on g and : The other parameters were = 0:5; = 1; s = 0:5; = 0:5; ! = 1; = 0:01; 2 " = 0:02 and 2 m = 0:05: d as a function of g and To interpret Table

  1 and express welfare as a function of the moments of the deviation between the endogenous variables x t ; k t and their Walrasian counterparts, denoted by xt and kt : In the Walrasian equilibrium with wages equal to wt ; we clearly have that xt = 1 1

  I A) 1 B: This allows us to get all the moments in (37). Next, to compute to compute the moments of the equilibrium, we rewrite (3 )-(3 ) as well as the law of motion for a as v t+1 = M v t + C + N " t + Zt; where v = ( x; k; a) F : The variance-covariance matrix of the cyclical component in v; ; is solution to = M M F + N N F 2 " : By rewriting (24) as y t = Dv t + E; we then compute detrended output volatility as 2 y = D D F : there exists an interior interval of values of included in [0; 1] ; [ c ( ); + c ( )]; over which (43) holds and therefore the eigenvalues of A are complex. Furthermore, we can check that for = 0; Q = 1 and = 1=(1 + ); and we have a double root equal to 1. On the other hand for ! 1; we have Q ! 1 + 1 and ! 1 ; and we get the double root =Q = 1 + : Hence the banana-shaped lens on Figure 1.

:

  Let u t = x t xt and v t = xt xt : Let y t = (u t v t ) P : Theny t+1 = Ay t + w t+1 ;Let ŵ = a Ew and ŷ = y Ey: Then ŷt+1 = Aŷ t + ŵt : Furthermore,

Figure 1 -

 1 Figure 1 --Effect of the variance of aggregate shocks on the optimal selectivity level (1/(1+d) on the vertical axis

Figure 2 -

 2 Figure 2 --Effect of the mutational variance on optimal selectivity ((1/ 1+δ on the vertical axis)

Table 2a

 2a

reports the corresponding value of ; the AR1 coe cient in aggregate employment dynamics. Note that this is not the autocorrelation in x t ; but instead the part of it that is induced by selectivity (the other part comes from the autocorrelation in the shocks a). Under = +1 it would be equal to : As Table

2a

makes clear, this contribution is most of the time negative. Table

2b

reports the autocorrelation of average log employment x t ;

Table 3

 3 reports the corresponding asymptotic cross-sectional dispersion of …rm size, & ; while Table 4 reports the average ALSG. Given (20), the interpretation ofTable 3 is straightforward. Table 4 con…rms the negative

Table 3

 3 Cross-sectional log employment variance at the optimal choice of d: Same parameters as Table1

Table 7 .

 7 We see that selectivity goes up

	with 2 m for low values of but falls with 2 m for large values of :

Table 7

 7 Parameter values for Figure

  Table reports simulations of the optimal d for di¤erent values of and ; and it con…rms this intuition: 12

	0 0.0 1.8 1.08 0. 1 0.32 0.14 0.03 0.2 0. 1 2 n
	0.2	.2 2.33 1.13 0.	0.2 0.0
	0.4 99	13.3 3.17 1.28 0. 2 0.14
	0.	99	99	99	. 9 1.	0.3
	0.8 99	99	99	99	99	2.12
	0.99 99	99	99	99	99	99
	Table 10 Optimal selectivity d as a function of	and ;	= 0:5;

See for example[START_REF] Marcet | Convergence of Least-Squares Learning in Environments with Hidden State ariables and Private Information[END_REF],Evans and Honkahpoja (2001),[START_REF] Marcet | Recurrent Hyperin ations and Learning[END_REF].

This issue arises in[START_REF] Saint-Paul | How does the allocation of credit select between boundedly rational …rms?[END_REF], in a model where competing …rms set their prices subj ¡t to mistakes.

As pointed out by Caballero and Hammour (I¢¢£¤¥ in general entry and exit will not be e¦ cient (as compared to some optimal benchmark) if there are market failures; this is an instance of inadequate selection, but only as a consequence of the market failure.

The Darwinian view according to which …rms, rather than optimizing, are characterized by an array of …xed business strategies that one may interpret as their "DNA" is pervasive in the business literature. See for example[START_REF] Marks | Business Darwinism: Evolve or Dissolve[END_REF].To be sure, the strategies described in such a book are far more complex and qualitative than §¨st deciding o©e s capital/labor ratio. Nevertheless they exemplify how capitalism is a trial-and-error process through which unpro…table behavioral rules are eventually abandoned, while pro…table ones are replicated. This paper is a …rst step at analyzing the consequences of this trial-and-error process for aggregate dynamics.

By the same token, log …rm size dispersion would tend to push aggregate employment above its Walrasian level. But we have proved that the social plann!"#s ob$!%tive can be expressed as a function of the cross-sectional deviation of log …rm size and the absolute value of the output gap. Thus taking those into account we can ignore employment.

m

u001ing track of memory can be implemented in the economy through institutions such as money, or more generally, wealth. See uocherlakota (1998).

How to compute the welfare criterion is described in the Appendix.

See the Appendix for the actual speci…cation and the estimated values for inertia.

where

We note that for v 0; (u; v) (0; 0) = ln(1 1+ ); that 6 u (0; 0) = 0; 6 v (0; 0) = : Hence for u; v << 1

The other terms of the Taylor expansion are all negligible relative to either u or v: Maximization of E is therefore equivalent to minimizing E(v+

Proof of Proposition 2

The relevant matrix to be studied is, from ( 2)-( 2)

:

The eigenvalues of M are solution to

The corresponding discriminant is

This quantity is positive i¤

It is easy to see that (1+ ) ( 2 ) = b 2 < 0 for all 2 [0; 1] and

2 ): Therefore we have that (x) < 0 for x 2 (x 1 ; x 2 ); with

: Computations show that this quantity is smaller than

In the zone where x 2 (x 1 ; x 2 ); the two roots of (31) are complex con ugate since (x) < 0: Their module is equal to (1 x(1 + )) < 1: Therefore the system is stable. In the zone where x = 2 (x 1 ; x 2 ); the roots of ( 31) are real and given by

Proof of Proposition 3

The characteristic equation for the eigenvalues of matrix A is

where 1) :

Note that if the roots are complex their common module is 1+ 1=2 < 1:

Therefore the dynamics are stable. Let us characterize this regime …rst.

Roots are complex i¤

This will be the case provided lies between the roots of

We note that if Q 2 1+ < 0 and if these roots are real, they are both positive, and one of them is lower than one since their product is 2 < 1:

) 2 > 0: Therefore both roots, if they exist are between 0 and 1:

: Thus we always have that Q 2 1+ < 0: Next, the roots are real if and only if

This clearly always holds for 0: Furthermore, for > 0; it is equivalent to Q < 1 1+ ; which we ust proved above. This proves that for any ; 1) + :

If this condition is violated, the dynamics are necessarily unstable. If it holds, then they are stable if

This is equivalent to

or equivalently

We note that 1) + < stab ( ) < pos ( ):

Furthemore, at = stab ( ) > 0; we have that 1+ = 1 + Q + ;

implying that ( Q + ) 2 = 1 + 1+ 2 > 4 1+ : Therefore (43) holds, implying (since stab ( ) < pos ( ) < + c ( )), that we must have stab ( ) < c ( ):

2( 1) ; the eigenvalues are negative and unstable for < stab ( ); negative and stable for stab ( ) < < c ( ); complex and stable for c ( ) < < + c ( ); and positive and stable for > + c ( ): On the other hand, if 1 < < 3(1 ) 2( 1) ; the eigenvalues are negative and stable for < c ( ); complex and stable for c ( ) < < + c ( ); and positive and stable for > + c ( ): This completes the proof of Proposition 3.


Brillouin Étrodiffusion

Une Dans

Optique Fibre

Énure Chalcog

As

Microstructur Ée

J C Tchahame

J-C Beugnot

L Brilland

D Méchin

J Troles

H Maillotte

T Sylvestre

CARACT ÉRISATION DE LA R

Keywords: Effet Brillouin, verres chalcogénures, fibres optiques microstructurées

R ÉSUM É

Dans ce document, nous présentons les résultats de caractérisation de la rétrodiffusion Brillouin dans une fibre optique microstructurée (FOM) en verre de chalcogénure As 42 S 58 . Les résultats obtenus montrent que cette fibre optique serait appropriée pour des applications en optique non-linéaire basées sur l'effet Brillouin.

INTRODUCTION

Les verres chalcogénures ont été récemment identifiés comme des matériaux optiques très attractifs pour de nombreuses applications aux domaines des capteurs, des télécommunications optiques et de la spectroscopie dans le moyen-infrarouge [START_REF] Eggleton | Chalcogenide photonics[END_REF]. Leurs avantages par rapport aux verres à base de silice sont d'une part une grande fenêtre de transparence allant de 1 µm jusqu'à 12 µm et, d'autre part, de très fortes non-linéarités optiques [START_REF] Slusher | Large Raman gain and nonlinear phase shifts in high-purity As2Se3 chalcogenide fibers[END_REF]. A titre d'exemple, L'indice non-linéaire Kerr n 2 et le gain Raman g R peuvent atteindre des valeurs 100 fois supérieures à celles de la silice [START_REF] Slusher | Large Raman gain and nonlinear phase shifts in high-purity As2Se3 chalcogenide fibers[END_REF]. Par conséquent, de nombreuses études sont actuellement consacrées à l'étude ces verres et en particulier aux développement de fibres optiques pour notamment réaliser des sources de lumière supercontinuum [START_REF] Mouawad | Multioctave midinfrared supercontinuum generation in suspended-core chalcogenide fibers[END_REF]. Parmi les non-linéarités optiques du troisième ordre, l'effet Brillouin est une interaction acousto-optique fondamentale qui survient également dans les fibres optiques et qui se traduit par une diffusion inélastique de la lumière par des ondes acoustiques. Malgré ses applications potentielles aux domaines des lasers et des capteurs [START_REF] Tow | Relative intensity noise and frequency noise of a compact Brillouin laser made of As38Se62 suspendedcore chalcogenide fiber[END_REF], l'effet Brillouin a été relativement peu étudié dans les fibres de verres chalcogénures [START_REF] Ogusu | Brillouin-gain coefficients of chalcogenide glasses[END_REF][START_REF] Abedin | Observation of strong stimulated Brillouin scattering in single-mode As 2 Se 3 chalcogenide fiber[END_REF][START_REF] Beugnot | Reduction and control of stimulated brillouin scattering in polymer-coated chalcogenide optical microwires[END_REF].

Dans cette communication, nous présentons les résultats de l'étude expérimentale réalisée dans une nouvelle fibre optique microstructurée avec une composition originale de type As 42 S 58 . Un des principaux avantages de ce verre par rapport aux autres verres chalcogénures, par exemple à base de Selenium, est sa bien meilleure tenue au flux de puissance. Nos mesures de spectres Brillouin révèlent des coefficients de gain 20 fois supérieurs à des fibres monomodes de silice à 1550 nm, et des largeur spectrale de gain d'environ 80 MHz.

R ÉSULTATS EXP ÉRIMENTAUX

Afin d'observer et caractériser la rétrodiffusion Brillouin dans la fibre As 42 S 58 , nous avons réalisé le montage expérimental schématisé sur la figure 1, et dont le principe de mesure repose sur la détection hétérodyne [START_REF] Beugnot | Complete experimental characterization of SBS in photonic crystal fiber[END_REF]. Pour ce faire, on utilise un laser à fibre émettant en régime continu à la longueur d'onde de 1550 nm avec une largeur de raie relativement étroite ( 10 kHz). Le faisceau laser est ensuite injecté dans la fibre As 42 S 58 à l'aide d'un circulateur optique et d'une fibre à grande ouverture numérique (HNA) afin d'assurer un bon taux de couplage. Les pertes relatives entre les deux fibres sont d'environ 17%. Une caméra infrarouge située en sortie de montage nous permet également de vérifier le caractère unimodal de la fibre. Le signal rétrodiffusé et décalé en fréquence du coté Stokes par effet Doppler est redirigé par le circulateur vers un coupleur 90/10 afin d'effectuer un battement optique avec la source laser initiale. La détection hétérodyne s'effectue ainsi dans le domaine radio-fréquence (RF) au moyen d'une gain Brillouin en fonction de la puissance incidente. Plus la puissance incidente se rapproche du seuil de puissance critique (théoriquement estimé à 28dBm), plus le signal rétrodiffusé est prononcé. Le décalage en fréquence Brillouin ν B induit par l'onde acoustique longitudinale est de 7, 605GHz. Cette mesure est en bon accord avec l'équation de la fréquence Brillouin donnée par ν B = 2n eff v A /λ , où v A est la vitesse acoustique longitudinale (v A = 2600 m.s -1 ) et λ la longueur d'onde du laser dans le vide [START_REF] Beugnot | Reduction and control of stimulated brillouin scattering in polymer-coated chalcogenide optical microwires[END_REF]. Un traitement numérique incluant moyennage et débruitage nous permet d'extraire la largeur à mi-hauteur ∆ν B des spectres et d'estimer théoriquement le gain Brillouin g B à partir de la relation suivante 1.

g B = 2πn 7 eff p 2 12 λ 2 P ρ 0 v A c∆ν B (1)
Dans le tableau suivant nous avons regroupé les résultats de mesures. Les coefficients P 12 = 0,299, ρ 0 = 3198kg/m 3 , v A = 2600 m.s

CONCLUSION

Nous avons observé et caractérisé la rétrodiffusion Brillouin dans une fibre optique microstructurée en verre chalcogénure As 42 S 58 , révélant ainsi un facteur de gain Brillouin de 4, 9 × 10 -10 m/W , 20 fois supérieur à celui d'une fibre monomode de silice. Ces résultats montrent un potentiel applicatif intéressant de ces fibres chalcogénures pour la réalisation de laser Brillouin à fibre dans le proche et moyen-infrarouge ainsi que des capteurs répartis à fibre pour mesurer la température ou la contrainte.

R ÉF ÉRENCES

FIGURE 1 :

 1 Photodiode

FIGURE 2 :

 2 FIGURE 2 : Spectres Brillouin expérimentaux de la fibre microstructurée As 42 S 58 de longueur 8 m. (a) Mesures du spectre pour plusieurs puissances incidentes, les réflexions de Fresnel sont prises en compte. (b) Approximation lorentzienne du spectre à 15,47 dBm pour la détermination de la largeur à mi-hauteur.

  -1 [10] sont ceux de la composition As 2 S 3 , qui est stoechiométriquement proche de As 42 S 58 . Dans le tableau 1, on peut constater que ∆ν B croit avec la puissance incidente,

	Puissances incidentes (dBm) 15,47 16,66 17,41
	∆ν B (MHz)	74	77	94
	g B (×10 -10 m/W )	5,3	5,1	4,2

TABLE 1 :

 1 Largeur à mi-hauteur des spectres mesurés à 1550 nm pour plusieurs puissances incidentes et estimations du gain Brillouin. n eff 2, 36 à 1550 nm, c = 3 × 10 8 m/s. contrairement à ce qui est généralement observé dans les fibres optiques de silice. Nous trouvons également un gain Brillouin moyen de 4, 9 × 10 -10 m/W qui est presque 20 fois supérieur à celui d'une fibre monomode de silice (g silice = 2, 71 × 10 -11 m/W[START_REF] Beugnot | Reduction and control of stimulated brillouin scattering in polymer-coated chalcogenide optical microwires[END_REF]), malgré une largeur spectrale de gain relativement importante.