Mark J Fernée
email: mjfernee@gmail.com.

Taras Plakhotnik

Yann Louyer

#,∇ Bradley N Littleton

∥,∇ Christian Potzner

Philippe Tamarat

∥,∇ Paul Mulvaney

Brahim Lounis

Mark J Ferneé

Spontaneous spectral diffusion in CdSe quantum dots

Keywords:

come

S ingle nanocrystal quantum dots (NCs) have been developed for a wide range of applications, from biological labeling to the light-absorbing materials in the next generation of photovoltaics. However, their adoption in certain applications has been hampered by a number of idiosyncrasies, photoluminescence blinking and spectral diffusion (i.e., environmentally induced fluctuation in the emission energy of individual quantum emitters). 1,2 In particular, photoluminescence blinking has received a considerable amount of attention 3 as it strongly impacts applications involving single NCs. The ultimate aim is to understand and, through NC engineering, 4,5 mitigate blinking. Until recently, spectral diffusion (SD) had received far less attention, and therefore, there is less understanding of the underlying photophysical properties of this phenomenon. At low temperatures, NCs have narrow spectral lines, [6][7][START_REF] Biadala | Band-Edge Exciton Fine Structure of Single CdSe/ZnS Nanocrystals in External Magnetic Fields[END_REF][START_REF] Louyer | Efficient Biexciton Emission in Elongated CdSe/ZnS Nanocrystals[END_REF] which could be exploited for quantum optical applications, [START_REF] Press | Complete Quantum Control of a Single Quantum Dot Spin using Ultrafast Optical Pulses[END_REF] but SD will be detrimental to such applications.

In colloidal NCs, spectral diffusion results from electric field fluctuations in the local environment of the NC that induces a shift of the band edge spectra due to the Stark effect. [START_REF] Empedocles | Quantum-Confined Stark Effect in Single CdSe Nanocrystallite Quantum Dots[END_REF] The fundamental cause of SD remains uncertain. Two distinct time scales for SD have been identified, a SD process that occurs on the microsecond time scale that results in broadening of the spectral line that is only observable using various highresolution spectroscopies [START_REF] Coolen | Emission Characterization of a Single CdSe-ZnS Nanocrystal with High Temporal and Spectral Resolution by Photon-Correlation Fourier Spectroscopy[END_REF] and a SD process that occurs at second time scales that results in discrete jumps of the spectral position. [START_REF] Ferneé | Charge Hopping Revealed by Jitter Correlations in the Photoluminescence Spectra of Single CdSe Nanocrystals[END_REF] This latter SD process is the most commonly studied and the most disruptive process due to the size of the spectral jumps and is the focus of this work. The spectral jumps have been attributed to individual charges moving around the surface, [START_REF] Empedocles | Influence of Spectral Diffusion on the Line Shapes of Single CdSe Nanocrystallite Quantum Dots[END_REF][START_REF] Muller | Monitoring Surface Charge Migration in the Spectral Dynamics of Single CdSe/CdS Nanodot/Nanorod Heterostructures[END_REF] although this attribution is merely speculative. For example, scanning electron force microscopy has shown that many, but not all, NCs have or develop surface charges, [START_REF] O 'brien | Charge and Photoionization Properties of Single Semiconductor Nanocrystals[END_REF] while SD is a ubiquitous phenomenon observed in all single NC studies. In addition, SD can be seen to persist through transitions between neutral and charged states, 7 suggesting that a free surface charge is not responsible. In general, the surface of a NC is a complex interface between crystal surface atoms and molecular ligands that bind to individual surface sites in order to passivate dangling bonds. It is known that the surface coverage is not complete; 17 thus, a NC has many possible surface configurations, which in turn have different associated charge distributions. Thus, a reasonable model of SD is one involving photoactivated rearrangement of surface ligands causing discrete jumps [START_REF] Ferneé | Charge Hopping Revealed by Jitter Correlations in the Photoluminescence Spectra of Single CdSe Nanocrystals[END_REF] between different surface configurations in a complex potential landscape. [START_REF] Plakhotnik | Anomalous Power-Law Statistics in Spectral Diffusion of Semiconductor Quantum Dots[END_REF] Such an interpretation is compatible with the SD model originally proposed by Empedocles and Bawendi. [START_REF] Empedocles | Influence of Spectral Diffusion on the Line Shapes of Single CdSe Nanocrystallite Quantum Dots[END_REF] Early reports of SD in NCs revealed a thermal activation energy below which the SD process is solely photoactivated. [START_REF] Empedocles | Influence of Spectral Diffusion on the Line Shapes of Single CdSe Nanocrystallite Quantum Dots[END_REF][START_REF] Muller | Monitoring Surface Charge Migration in the Spectral Dynamics of Single CdSe/CdS Nanodot/Nanorod Heterostructures[END_REF] This motivates the study of SD at cryogenic temperatures, where competing thermal effects are eliminated, effectively isolating the photophysical properties. A recent numerical study of the response of surface ligands to photoactivation has revealed that they can "jump" between different lattice sites. [START_REF] Voznyy | Mobile Surface Traps in CdSe Nanocrystals with Carboxylic Acid Ligands[END_REF] Such dynamics could also admit the possibility of spontaneous spectral diffusion for surface ligands promoted to an unstable higher-energy configuration can relax back to a more stable lower-energy configuration. Spontaneous SD is known in the field of single-molecule spectroscopy, where polymer matrixes are known to not be in a state of thermal equilibrium, allowing SD of a molecular transition to occur in the absence of photophysical pumping. [START_REF] Boiron | Are the Spectral Trails of Single Molecules Consistent with the Standard Two-Level System Model of Glasses at Low Temperatures?[END_REF] The ligands that passivate the surface of a NC have much in common with random polymer matrixes, and therefore, we may expect similar behavior with SD in NCs. Therefore, testing for spontaneous SD will give further insight into the physical origin of SD and ultimately aid the design of new materials that mitigate this effect.

In this Letter, we use cryogenic photoluminescence spectroscopy along with high-resolution resonant photoluminescence excitation spectroscopy (RPLE) to probe the origins of SD. An autocorrelation analysis of time series of spectral jumps is used to test the randomness of the SD phenomenon. We then test for spontaneous SD by comparing statistics of a spectral jump distribution obtained with continuous excitation with a spectral jump distribution comprising spectral shifts following brief periods with no pump laser. We also employ a complementary technique of RPLE of the band edge exciton state using a scanning single-frequency laser that can cause photoinduced jumps and detect any spontaneous return.

For statistical analysis, a time series of spectral peak positions is generated using a peak fitting routine to determine the positions of emission maxima in single NC spectra. A typical time series of spectral peak positions determined from a succession of single NC spectra obtained at cryogenic temperatures (here 5 K) is shown in Figure 1a. The time series appears to exhibit random walk-like behavior. This is borne out when we take the autocorrelation of the time series and see an apparent exponential decay, characteristic of a random walk (not shown). However, if we calculate the autocorrelation of the spectral jump time series (obtained as the series of successive difference between neighboring spectral positions), we see in Figure 1b that there is a pronounced anticorrelation at the first delay (lag 1). This indicates that, after a spectral jump, it is more likely for the next jump to return the emission to the previous position. Thus, the spectral jumps are not truly random but have some finite memory of previous configurations. The depth of this memory is tested using the partial autocorrelation of the spectral jump series, as shown in Figure 1c. The partial autocorrelation is a technique commonly used to determine the number of terms that need to be used in an autoregression model, AR(k), [START_REF] Brockwell | Time Series: Theory and Methods[END_REF] of the form,

x n = ∑ i=1 k a i x n-i + ε,
where k is the order of the model, ε is a normally distributed white noise term, and the a i are the model coefficients and represent a memory of previous values in the series. From the partial autocorrelation, we see that this depth of memory could be satisfied by an AR(5) model, indicating that the memory vanishes after approximately five spectral acquisitions.

We have recently reported correlations in SD time series that were related to Poissonian jump statistics. [START_REF] Ferneé | Charge Hopping Revealed by Jitter Correlations in the Photoluminescence Spectra of Single CdSe Nanocrystals[END_REF] In this case, the correlations were very sensitive to the experimental integration time and were easily lost. However, the anticorrelations reported in this Letter are relatively insensitive to the integration time and pump power. The robustness of the anticorrelations to variations in integration time and pump power is reflected in the partial autocorrelation result of Figure 1c, which gives an indication of the depth of the memory. We have checked this memory by increasing the integration time and find that the anticorrelations do indeed persist in accordance with the partial autocorrelation function. Furthermore, similar anticorrelations were found for most spectral time series (including data presented in refs 13 and 18; see Supporting Information Table S1), suggesting these anticorrelations to be a general characteristic of SD.

Monte Carlo simulations of the SD process were conducted in order to test the previously reported power-law-distributed two-level system (TLS) model of SD. [START_REF] Plakhotnik | Anomalous Power-Law Statistics in Spectral Diffusion of Semiconductor Quantum Dots[END_REF] In general, such models were only able to reproduce the anticorrelations by modifying the power law at high frequencies. Such a modification has no impact on the previously reported power law as the highfrequency region has little effect on those statistics (see Supporting Information Figure S1). Nevertheless, the observation of a robust anticorrelation in SD time series indicates a departure from the pure power law behavior previously reported. [START_REF] Plakhotnik | Anomalous Power-Law Statistics in Spectral Diffusion of Semiconductor Quantum Dots[END_REF] The anticorrelations revealed in Figure 1 indicate that spectral jumps are not purely random but retain a memory of past charge configurations. In particular, it is reasonable to assume a competition between the photophysical process driving the SD and a relaxation process toward a previous charge configuration. In order to test the possibility that the relaxation corresponds to spontaneous SD, we generate statistics for spectral jumps occurring across extended periods where the excitation laser is blocked, so-called dark jumps. This is conducted using a rotating beam block that introduces periods with no pump (ie. dark periods) interleaved with bright periods, as shown in Figure 2a. The statistics on the dark jumps are then compiled by calculating the difference between the last spectral peak position before a dark period and the f irst spectral position measured following a dark period. The periods of continuous laser excitation (ie. bright periods) were long enough to ensure a significant change in the local charge distribution (as indicated by significant SD in Figure 2a). Thus, there can be no memory of previous dark periods. Evidence for spontaneous spectral jumps will manifest in the dark jump distribution deviating from the bright jump distribution. In order to quantify this deviation, we compute two bright jump distributions [START_REF] Ferneé | Charge Hopping Revealed by Jitter Correlations in the Photoluminescence Spectra of Single CdSe Nanocrystals[END_REF][START_REF] Plakhotnik | Anomalous Power-Law Statistics in Spectral Diffusion of Semiconductor Quantum Dots[END_REF] corresponding to the difference between successive spectral peak positions (ie. j i = x ix i-1 ), called the 2I distribution and the 3I distribution, corresponding to the jumps calculated according to j i = x ix i-2 (indicating an additional period of illumination). These two SD distributions simply correspond to different illumination periods, where I represents the illumination period required to acquire a single spectrum. Resolving a spectral jump requires a minimum of two spectrum acquisitions, or 2I. Thus, the 3I jump distribution will be broader than the 2I jump distribution due to SD induced during the additional illumination period, and therefore, a dark jump distribution broader than the 2I distribution would indicate the presence of spontaneous SD.

In Figure 2b, we plot the cumulative distribution functions (CDFs) for 2I and 3I. The CDF is another form of the probability distribution that for some value, x′, gives the probability P(x ≤ x′). Thus, the CDF can be described in terms of the traditional probability distribution function as follows, P(x ≤ x′)=∫ -∞

x ′ P(x)dx. CDFs are used here as they allow all of the data to be represented as individual points without the need for binning into histograms, which can be misleading. This is especially useful for data sets with a small number of points, such as the dark jump data. The CDF is obtained from a concatenated time series of seven different NCs obtained under identical conditions. In order to highlight the dark jump behavior, we have also plotted the deviation away from the 2I CDF for both the 3I and dark jump CDFs in Figure 2c. Here, the dark jump CDF clearly matches the 3I CDF near the center of the distribution, but it more closely matches the 2I distribution at the extremities. This provides clear evidence for spectral jumps occurring during dark periods without laser illumination, although the extent to which this dark SD occurs is clearly limited. These data categorize the dark jumps into two energy regimes: Spectral jumps larger than ∼0.4 meV more closely follow the 2I distribution, indicating that these jumps are likely the result of irreversible charge redistributions (that may relax on longer time scales), while jumps smaller than ∼0.15 meV, which follow the 3I distribution, are characteristic of reversible charge redistributions that spontaneously relax to a prior charge distribution.

We also employed resonant excitation of a NC using a singlefrequency dye laser to scan through the spectral line while detecting the Stokes shifted emission in the LO phonon replica (ie. RPLE 6 ). This technique is complementary to the nonresonant excitation scheme used above in that the lack of signal indicates that the laser is not exciting the NC and hence represents a dark period. The technique can resolve single NC spectral lines with high resolution, as shown in Figure 3b. The stability of the NC emission is indicated in Figure 3c, representing an average of 18 scans (120 μeV range) with only a small increase in line width. However, if we slow the scan rate by a factor of 5, the NC is far less stable. In Figure 3a, we show a sequence of successive spectral scans obtained immediately after the faster scans used for Figure 3c. Under these conditions, each spectral position in successive scans is revisited every 6 s, which accounts for the scan and return times. The sequence of spectra in Figure 3a is remarkable as it shows a NC that undergoes large photoinduced spectral jumps to positions outside of the scan range but then returns. These returns occur spontaneously, without any pump laser influence. Although one or two such events may be possible with an unobserved pump influence (ie. near the edge of the line, where the count rate falls into the noise), the sequence of 10 successive scans showing this behavior rules out a statistical anomaly. Furthermore, there are clear periods where the spectral line has vanished only to reappear a number of scans hence. This behavior is consistent with the statistical behaviors shown in Figures 1 and2 and is an unambiguous example of spontaneous SD.

The observation of a spontaneous SD at temperatures as low as 2 K indicates that the process must either have a low activation energy or involve some alternative temperatureindependent mechanism. We speculate on the mechanism for spontaneous SD in the Supporting Information (section 3), invoking a model compatible with that originally proposed by Empedocles and Bawendi 14 based on highly asymmetric surface configuration potentials. However, we note here that spontaneous SD is observed from single molecules embedded in molecular glasses [START_REF] Boiron | Are the Spectral Trails of Single Molecules Consistent with the Standard Two-Level System Model of Glasses at Low Temperatures?[END_REF] at temperatures below 1 K. Given the complexity of the surface topology and NC/surface-ligand interface, it is reasonable to speculate that processes present in molecular glasses may also be relevant to NCs, such as tunneling between nearly energy degenerate surface configurations.

In conclusion, we have reported a hitherto unexpected memory effect in SD, which manifests as a robust anticorrelation in a time series of spectral jumps. This memory is attributed to relaxation to a low-energy surface charge configuration. The possibility of spontaneous relaxation was tested statistically by introducing periods where the pump laser was blocked. The dark jump distributions were consistent with limited spectral evolution due to spontaneous SD. Direct detection of this spontaneous SD was accomplished using resonant excitation scans of the NC spectral line, where following spectral jumps out of the scan range, the spectral line subsequently returned in successive scans. Thus, spontaneous SD is consistent with the relaxation to a surface electronic configuration of lower energy, which indicates that the phenomenon of SD is actually a progression though a series of stable and unstable surface charge distributions. This highlights new surface dynamics and hints at the effect that phonons have on surface ligands. Further insight into the SD process will require more advanced strategies beyond pure optical spectroscopic studies, such as correlating multiple nanoscale probes. [START_REF] Vanmaekelbergh | Single-Dot Microscopy and Spectroscopy for Comprehensive Study of Colloidal Nanocrystals[END_REF] Ultimately, a greater understanding of the NC surface is crucial for understanding effects such as luminescence blinking and for engineering improvements to both photophysical and photochemical stability in these materials.

■ EXPERIMENTAL METHODS

The CdSe/CdZnS nanocrystals used in this study were either sourced from a commercial supplier (Invitrogen ITK605 and ITK655 with core radii of approximately 2.4 and 4.1 nm, respectively [START_REF] Liptay | Anomalous Stokes Shift in CdSe Nanocrystals[END_REF] ) or were synthesized for this study as a CdSe core with a CdS/ZnS shell, as previously reported. [START_REF] Ferneé | Charge Hopping Revealed by Jitter Correlations in the Photoluminescence Spectra of Single CdSe Nanocrystals[END_REF][START_REF] Plakhotnik | Anomalous Power-Law Statistics in Spectral Diffusion of Semiconductor Quantum Dots[END_REF] Experiments using the conventional spectroscopy have been previously described, including sample preparation. [START_REF] Ferneé | Charge Hopping Revealed by Jitter Correlations in the Photoluminescence Spectra of Single CdSe Nanocrystals[END_REF] In order to collect the statistics for the dark jumps, a rotating shutter was employed, which was operated asynchronously with the acquisition. For the RPLE experiment, the samples (ITK655 NCs streptavidin conjugates) were spin-coated onto glass coverslips in a polyvinyl alcohol matrix, which helps stabilize the NCs for resonant excitation. The sample was mounted in a helium bath cryostat capable of achieving temperatures as low as 2 K. For the RPLE measurements, a narrow band dye laser (∼4 neV) was scanned across the NC resonance, and the optical phonon replica was detected with an avalanche photodiode, using a low-pass filter to remove the laser scatter. 6

Figure 1 .

 1 Figure 1. (a) Time series of spectral peak positions obtained with a pump irradiance of 52 W/cm 2 and a 1 s integration at 5 K. (b) Autocorrelation of the time series of spectral jumps calculated from the peak position time series in (a). (c) Partial autocorrelation calculated from the time series of spectral jumps, with 95% confidence limits indicated (red dashed lines). The shaded region indicates the extent of the anticorrelation.

Figure 2 .

 2 Figure 2. (a) Time series of spectra (vertical axis) obtained at 5 K with a pump irradiance of 340 W/cm 2 and an integration time of 1 s. During the acquisition, the pump laser was periodically blocked with ∼35 s of laser on and ∼18 s of laser off to introduce dark periods into the time series. (b) Cumulative distributions obtained from an ensemble of seven concatenated single NC data at 5 K, corresponding to 11746 spectral jumps and 292 dark jumps. The 2I (blue solid), 3I (green dashed), and dark jumps data are indicated (red dots). (c) Deviations of the 3I (green dashed) and dark jump (red dots) CDFs from the 2I CDF (solid blue line).

Figure 3 .

 3 Figure 3. Resonant PLE scans at 2 K with a pump irradiance of 100 W/cm 2 . (a) Series of successive slow scans (scan rate of 24 μeV/s). (b) Single fast scan (scan rate of 120 μeV/s) with a 16 μeV Gaussian fit (red dashed). (c) Average of 18 successive fast scans with a 20 μeV Gaussian fit (red dashed).

Author Contributions

∇ The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. These authors contributed equally.

Notes

The authors declare no competing financial interest.


Ibrahim Ahamada
email: ahamada@univ-paris1.fr.

Djamel Kirat
email: djamel.kirat@univ-paris1.fr.

Djamel Kirat Y Z May

Evidence of a nonlinear effect of the EU ETS on the electricity-generation sector

Keywords: Carbon Emission Trading, Energy prices, Nonlinear threshold model. JEL classi…cation: C13 C32 C51 Q49 Q58

 approach of sample splitting and threshold estimation. We …nd evidence of nonlinear threshold e¤ects in both countries. The estimated carbon price thresholds are 14:94e and 12:57e in France and Germany, respectively. In Germany, the carbon price does not a¤ect the electricity price below this threshold. In France, the price of emission allowances a¤ects the cost of electricity generation only below the carbon-price threshold, thus revealing speculative behavior by French electricity producers on the carbon-allowance market. This is not the case for German electricity producers.

Résumé

Cet article explore la présence d'e¤ets de seuil dans la relation liant le prix de l'électricité à celui des permis d'émission en France et en Allemagne pendant la seconde phase du SCEQE. Plus précisément, nous comparons un modèle linéaire de prix de l'électricité avec un modèle non linéaire à seuil en utilisant l'approche de Hansen (2000). Nous montrons que des e¤ets de seuil sont présents dans les deux pays. Les seuils de prix du carbone estimés sont de 14:94e et 12:57e en France et en Allemagne, respectivement. En Allemagne, le prix du carbone n'a¤ecte pas le prix de l'électricité en dessous de ce seuil. En France, le prix des quotas d'émission a¤ecte le prix de l'électricité seulement en-dessous du seuil de prix du carbone, révélant ainsi un comportement spéculatif des producteurs d'électricité français sur le marché du carbone. A l'inverse, les producteurs d'électricité allemands ne spéculent pas sur le marché des quotas d'émission.

Introduction

The European Union Emission Trading Scheme (EU ETS) is the world's largest emissions permit market to date. It concerns mainly energy and the major industrial emitters. It was set up in two phases: Phase 1, from January 1st, 2005 to December 31st, 2007; and Phase 2 from January 1st, 2008 to December 31st, 2012. Phase 1 is considered as a pilot phase before the introduction of Phase 2, which coincided with the …rst commitment period of the Kyoto protocol. A third phase was agreed in January 2008 which will start in 2013 and last up to 2020.

Electricity generation is the most polluting activity covered in the EU ETS. As such, we may expect the price of emission permits to impact on electricity prices. Most articles dealing with the relationship 2 Documents de Travail du Centre d'Economie de la Sorbonne -2012.47 between the price of emission allowances and electricity prices have appealed to linear models (OLS, VARs, VECM, etc.). [START_REF] Sijm | CO2 price dynamics: the implications of EU emissions trading for the price of electricity[END_REF][START_REF] Sijm | CO2 price dynamics: a follow-up analysis of the implications of EU emissions trading for the price of electricity[END_REF] use OLS to determine the fraction of the carbon price that is re ‡ected in electricity prices in Holland and Germany. [START_REF] Honkatukia | Impacts Of The European Emission Trade System On Finnish Wholesale Electricity Prices[END_REF] consider the long-and short-run dynamics of electricity, gas and coal prices and the price of carbon permits in the Finnish market via a VAR analysis. [START_REF] Bunn | A vector error correction model of the interactions among gas, electricity and carbon prices: an application to the cases of Germany and United Kingdom[END_REF] use a vector error correction model with allowances, electricity and gas prices in the United Kingdom (UK), with daily temperatures in London and seasonal dummies as the exogenous variables. More recent work has however suggested that the impact of the carbon price on electricity prices is nonlinear and depends signi…cantly on the country's energy mix. [START_REF] Kirat | The impact of the European Union emission trading scheme on the electricitygeneration sector[END_REF] With the linear modeling of an economic system or relationship, we impose the same model parameters across di¤erent groups and over time. If the parameters are truly those directly characterizing the economic relationship in question, a change in the underlying economic relationship from one state to another can be expressed as a change in the structural parameters of the empirical model. Econometrically, one way to take into account changes in the structure of parameters is to test for parameter instability in regression and timeseries models. There is a fundamental debate over the presence of nonlinearity due to structural change and that due to a threshold e¤ect. We can think of the structural change model (changepoint model) as a special case of the threshold model if we imagine time as the threshold variable. There is a substantial literature dealing with threshold models (see [START_REF] Hansen | Threshold autoregression in economics[END_REF]. Among these, [START_REF] Hansen | Sample splitting and threshold estimation[END_REF] develops a statistical theory for threshold estimation in the regression context and asymptotic distribution theory for the regression estimates.

This article compares a linear model of electricity prices, as in [START_REF] Kirat | The impact of the European Union emission trading scheme on the electricitygeneration sector[END_REF], to a nonlinear threshold model using [START_REF] Hansen | Sample splitting and threshold estimation[END_REF] approach of sample splitting and threshold estimation. Testing for threshold e¤ects depending on the price of carbon is of primary importance in the context of electricity-price models including the carbon price as a regressor. The model that we propose will allow us to see whether there exists a carbon price at which the behavior of electricity producers changes. We focus on the French and German electricity markets during the Kyoto commitment period of the EU ETS (2008)(2009)(2010)(2011)(2012). The results below reject the null hypothesis of linearity in favor of the alternative of a nonlinear threshold e¤ect in both countries. They also reveal speculative behavior by French electricity producers on the carbon market.

The paper is organized as follows. Section 2 sets out Hansen's method applied to the electricity-price 1 The …gure in Appendix A1 shows the quantities of electricity produced in both countries from various primary energy sources in 2004 and 2009. The share of each of these describes the energy-source mix in electricity generation. While in Germany more than 50% of electricity is generated using coal and lignite, France produces almost 80% of its electricity from nuclear energy, with fossil fuels accounting for just 9% to 10%. Moreover, producing electricity from fossil fuel plants is more costly and emits more CO2 compared to nuclear plants. Consequently, the electricity and carbon price relationships may be di¤erent in the two countries. 2 The threshold regression model

Threshold model

The linear model considered by [START_REF] Kirat | The impact of the European Union emission trading scheme on the electricitygeneration sector[END_REF] is very close to the following:

P elec t = 0 + 1 P elec t 1 + P gas t + P coal t + P carbon t + 1 T t + 2 T 2 t + 5 X j=2 j season j + " t (1)
where P y t is the logarithm of the price of commodity y in period t, and T is the temperature variable. The square of the temperature is included to capture the well-known nonlinear e¤ect of temperature on electricity prices. The seasonal dummies season j , j = 1; ::; 5, correspond to the …ve business days of the week (j = M onday; :::; F riday). This regression can also be written as follows:

P elec t = X t + " t (2)
where = ( 0 ; 1 ; ; ; ; 1 ; 2 ; 2 ; 3 ; 4 ; 5 ) and X t = (1; P elec t 1 ; P gas t ; P coal t ; P carbon t ; T t ; T 2 t ; season 2 ; season 3 ; season 4 ; season 5 ) 0 . We look for a possible nonlinear e¤ect of carbon price on electricity prices using the following threshold regression model:

P elec t = 8 < : 1 X t + " t if P carbon t p 2 X t + " t if P carbon t > p (3)
where (1) = ( 1 0 ; 1 1 ; 1 ; 1 ; 1 ; 1 1 ; 1 2 ; 1 2 ; 1 3 ; 1 4 ; 1 5 ) and (2) = ( 2 0 ; 2 1 ; 2 ; 2 ; 2 ; 2 1 ; 2 2 ; 2 2 ; 2 3 ; 2 4 ; 2 5 ). The threshold parameter p is considered to be unknown. It is convenient to rewrite (3) as follows:

P elec t = (2) X t + X t (p) + " t (4)
where = (1) (2) , X t (p) = X t I( P carbon t p) and I(.) is the indicator function. We want to estimate (1) , (2) and p if the null hypothesis of linearity is rejected, i.e. H 0 : = 0 in equation (4).

Nonlinearity Tests and Estimation

We …rst examine the null hypothesis of linearity in equation ( 4), H 0 : = 0. Without an a priori …xed value of p in regression (4), it is not easy to make any statistical inference regarding . In this case p is a nuisance parameter which is not identi…ed under the null hypothesis. To avoid this problem, [START_REF] Hansen | Inference when a nuisance parameter is not identi…ed under the null hypothesis[END_REF] developed a simulation technique producing a p-value statistic for the inference of . His approach does not require …xing an a priori value of p and allows for possible heteroskedasticity in [START_REF] Hansen | Sample splitting and threshold estimation[END_REF]. The computation method of the threshold estimate b p uses the concentrated sum of squared errors function from (4):

S(p) = T X t=1 P elec t d (2) (p)X t + b (p)X t (p) 2 (5)
and the threshold estimate b p is the value that minimizes S(p) :

b p = arg min p S(p) (6)
where is a bounded set of elements of fP carbon t ; t = 1; :::; T g and can be approximated by a grid (see [START_REF] Hansen | Sample splitting and threshold estimation[END_REF]. Finally, the slope estimates in the threshold model ( 3) can be computed via d Hansen (2000) also developed asymptotic distribution theory for the threshold estimate b p, and proposed asymptotic con…dence intervals by inverting the likelihood-ratio statistic. His approach again allows for possible heteroskedasticity in (4).

3 Application

Data

We use electricity prices in e/MWh from the day-ahead base-load 2 contracts covering the French and German markets which are traded on the EPEX spot exchange. 3 Day-ahead contracts are traded on a given day for the delivery of electricity one day ahead. The data we use here are of weekday frequency and run from March 3rd, 2008 to December 30th, 2010. The carbon spot price comes from the Bluenext environmental trading exchange expressed in e per ton. With respect to the primary energy markets, we appeal to the following price series expressed in e per MWh: i) the gas price of the month-ahead future contract traded on the Zeebrugge hub; and ii) the coal price of the month-ahead future contract Coal CIF ARA. The temperature information comes from the European Climate Assessment Dataset, 4 and is calculated as the average temperatures recorded at representative regional weather stations. Our …nal sample consists of 724 observations.

Results and comments

We …rst check that there is evidence of a threshold e¤ect associated with the emission permit price. We do so by employing both the F-test to consider a threshold under homoskedastic errors and the heteroskedasticityconsistent Lagrange multiplier (LM) test for a threshold of [START_REF] Hansen | Inference when a nuisance parameter is not identi…ed under the null hypothesis[END_REF][START_REF] Hansen | Sample splitting and threshold estimation[END_REF]. The p-values of teststatistics for the null H 0 : = 0 (conditional on p = b p) are computed using a bootstrap with 10000 replications. 2 The electricity base-load price is the price on the block for 24 hours. This is an arithmetic average price over the 24 hours of the day (from 0h to 23h). 3 EPEX Spot exchange is a holding company created by the collaboration between EEX Power Spot and Powernext SA, respectively the German and French electricity stock exchanges. 4 This table also contains the estimation results results from the corresponding linear models in columns [START_REF] Bunn | A vector error correction model of the interactions among gas, electricity and carbon prices: an application to the cases of Germany and United Kingdom[END_REF] and ( 5), and underlines the irrelevance of inference when nonlinearity is not taken into account. Row [START_REF] Hansen | Threshold autoregression in economics[END_REF] shows the estimated threshold b p and its 95% con…dence interval. The estimated carbon price thresholds are 12:57 and 14:94 e/ton in Germany and France, respectively. These thresholds are signi…cantly di¤erent 6 Documents de Travail du Centre d'Economie de la Sorbonne -2012.47 A detailed analysis of the results in Table (2) highlights the speculative behavior of French electricity producers on the emissions permit market. Speculation, as de…ned by David Newbery,5 is the purchase (or temporary sale) of goods for later resale (repurchase), rather than use, in the hope of pro…ting from the intervening price changes. When the price of emission permits is low (below the threshold of 14:94 e/ton),

French electricity producers expect it to rise. They buy permits in order to speculate and hedge the carbon risk-market by including the price of carbon allowance in their electricity-generation cost function: a rise of 1% in the emission-permit price results in 0:24% higher French day-ahead electricity prices. Hedging here is undertaken to reduce the risks arising from risky speculative activity. When the carbon spot price exceeds the threshold, it is no longer included in the cost function of electricity generation. At that time, French electricity producers sell their permits and take their pro…ts.

The behavior of German electricity producers is the opposite of their French counterparts and does not reveal any speculative conduct. When the emission-permit price is low (below the threshold of 12:57 e/ton), German electricity producers do not include the price of emission permits in their production cost function.

When the carbon price exceeds the threshold, it is a determinant of German electricity prices: 1% higher carbon prices result in 0:22% higher day-ahead electricity prices.

Conclusion

In this paper we have estimated the relationship between electricity prices and the prices of both the primary energies used in electricity generation and carbon dioxide emission permits in both France and Germany, using a nonlinear threshold model. The results reveal heterogeneity in the response of the electricity-generation sector to carbon constraints. French electricity producers behave speculatively, while their German counterparts do not. This behavior re ‡ects the composition of the French energy mix. The predominance of non-fossil energy sources in France means that there is less need to use emission permits, and so greater opportunity for speculation in the market for emission allowances.

  and Ahamada and Kirat (2012) consider the impact of carbon trading on electricity prices in France and Germany during both phases of the EU ETS: 2005-2007 and 2008-2012. 1 They …rst use a linear model before introducing nonlinearity via a structural change in the carbon spot price series, which break a¤ects the model parameters.

3

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.47 model in Kirat and Ahamada (2011). Section 3 then presents and discusses the results. Last, Section 4 concludes.

2 )

 2 (b p) and b (b p).

Figures ( 3 )

 3 Figures[START_REF] Hansen | Threshold autoregression in economics[END_REF] and[START_REF] Hansen | Sample splitting and threshold estimation[END_REF] show the graphs of the normalized likelihood-ratio statistic as a function of the threshold in the carbon-allowance price (in logs) in Germany and France, respectively. The estimates of the carbon thresholds (in logs) are the values that minimize these graphs, which occur at 2:531 (12:57 e/ton) and 2:704 (14:94 e/ton) in Germany and France, respectively. The dotted lines in the graphs represent the 95% critical values, so we can read o¤ the asymptotic 95% con…dence intervals from the graphs where the normalized likelihood-ratio sequence crosses the dotted lines. These con…dence intervals (in logs) are [2:5257; 2:5313] in Germany and [2:6925; 2:7555] in France. The corresponding 95% con…dence intervals in e/ton are [12:50; 12:58] and [14:77; 15:73],respectively. These results show that there is a reasonable evidence for a two-regime speci…cation in both countries. Figures[START_REF] Hansen | Threshold autoregression in economics[END_REF] and[START_REF] Hansen | Sample splitting and threshold estimation[END_REF] show that the con…dence intervals are fairly tight, so the uncertainty over the values of these thresholds is correspondingly small. Table[START_REF] Bunn | A vector error correction model of the interactions among gas, electricity and carbon prices: an application to the cases of Germany and United Kingdom[END_REF] present the estimation results of the threshold model of electricity prices in Germany and France.

Figure 1 :Figure 2 :

 12 Figure 1: Test for linearity against nonlinearity in Germany

Figure 3 :Figure 4 : 8

 348 Figure 3: Germany: Con…dence interval construction for the threshold (in logs)

Table 1 :

 1 Test results of no threshold against the alternative of a threshold

	Germany	France

Klein Tank et al., "Daily dataset of 20th-century surface air temperature and precipitation series for the European Climate Assessment", 2011, available at http://eca.knmi.nl. 5 Documents de Travail du Centre d'Economie de la Sorbonne -2012.47

Table ( 1

 ( ) and Figures[START_REF] Ahamada | The impact of phase II of the EU ETS on the electricity-generation sector[END_REF] and[START_REF] Bunn | A vector error correction model of the interactions among gas, electricity and carbon prices: an application to the cases of Germany and United Kingdom[END_REF] show the results of these tests of no threshold against the alternative of a threshold e¤ect in both Germany and France. These results strongly reject the null hypothesis of no threshold in favor of the alternative of a threshold at the 95% con…dence level in both countries. Figures[START_REF] Ahamada | The impact of phase II of the EU ETS on the electricity-generation sector[END_REF]

and (2) plot the F-test statistic as a function of the threshold in the carbon-allowance price in Germany and France, respectively. The dotted lines in the graphs represent the critical values at the standard signi…cance level of 95%. The null hypothesis of linearity is rejected in favor of the alternative of a threshold e¤ect in both countries. Linearity is rejected if the F-test statistic exceeds the critical value. Since the F-test is valid only with homoskedastic errors, it needs to be complemented by an LM test, as in Table

[START_REF] Ahamada | The impact of phase II of the EU ETS on the electricity-generation sector[END_REF]

. We thus consider the threshold-test results which are indicated by the results from the homoskedasticity tests which are shown in the last row of Table

[START_REF] Bunn | A vector error correction model of the interactions among gas, electricity and carbon prices: an application to the cases of Germany and United Kingdom[END_REF]

. Speci…cally, the relevant threshold tests are the F-test in Germany and the LM-test in France, since we do not reject homoskedasticity in the residuals of the threshold model in Germany but we do so for France.

Table 2 :

 2 Estimation Results Neither of theme appear in the 95% con…dence interval of the other price threshold.

	Country		Germany			France
			Linearity	Nonlinearity	Linearity	Nonlinearity
	Threshold (b p)		12.57e [12.50 ; 12.58]		14.94e [14.77 ; 15.73]
	Regime		Below threshold	Above threshold		Below threshold	Above threshold
	P elec t 1	0.575***	0.218	0.607***	0.730***	0.641***	0.794***
			(0.072)	(0.166)	(0.052)	(0.043)	(0.044)	(0.076)
	P gas t	0.215***	0.707***	0.174***	0.113***	0.205***	0.045
			(0.050)	(0.178)	(0.045)	(0.042)	(0.039)	(0.075)
	P coal t	-0.031	-0.780*	-0.031	-0.033	-0.197***	0.088
			(0.048)	(0.406)	(0.049)	(0.045)	(0.051)	(0.064)
	P carbon t	0.190***	-0.325	0.224***	0.182***	0.245***	0.048
			(0.046)	(0.250)	(0.055)	(0.042)	(0.053)	(0.063)
	T level	-0.006***	-0.025**	-0.006***	-0.016***	-0.025***	-0.005
			(0.001)	(0.0098)	(0.001)	(0.003)	(0.003)	(0.007)
	(T	level ) 2	0.0002***	0.0024*	0.0002***	0.0004***	0.0007***	0.000
			(0.000)	(0.0011)	(0.000)	(0.000)	(0.000)	(0.000)
	cons	0.767***	3.429***	0.671***	0.597***	0.889***	0.603***
			(0.120)	(0.920)	(0.080)	(0.073)	(0.151)	(0.144)
	season 2	-0.135***	-0.036	-0.144***	0.168***	-0.134***	-0.201***
			(0.019)	(0.040)	(0.019)	(0.016)	(0.017)	(0.028)
	season 3	-0.138***	-0.040	-0.147***	0.167***	-0.131***	-0.200***
			(0.020)	(0.043)	(0.020)	(0.017)	(0.018)	(0.029)
	season 4	-0.184***	-0.154	-0.184***	0.195***	-0.166***	-0.219***
			(0.025)	(0.124)	(0.020)	(0.016)	(0.020)	(0.027)
	season 5	-0.314***	-0.182***	-0.323***	0.314***	-0.277***	-0.353***
			(0.020)	(0.046)	(0.020)	(0.015)	(0.017)	(0.026)
		R 2	0.8128	0.4854	0.8475	0.8808	0.8316	0.8554
	Joint R 2		0.8225		0.8859
	Homoskedast-	0.014	0.069	0.000	0.000
	icity (p-value)

Note: Standard errors are in () ; values in [.] represent the 95% con…dence interval of the estimated threshold ; *, ** and *** refer respectively to the 10%, 5% and 1% signi…cance levels. The Joint R-squared is calculated from the residuals of model (4). 9 Documents de Travail du Centre d'Economie de la Sorbonne -2012.47 from each other.

Documents de Travail du Centre d'Economie de la Sorbonne -2012.47

See the article on "Futures markets, hedging and speculation" in "The New Palgrave Dictionary of Economics". 10 Documents de Travail du Centre d'Economie de la Sorbonne -2012.47

A APPENDICES A.1 Energy mixes in France and Germany

The French and German energy mix in 2004 & 2009 (source: IEA)


Tristan-Pierre Maury

Fabien Tripier
email: fabien.tripier@univ-nantes.fr

Giuseppe Bertola

Pierre Cahuc

Philipp William Hawkins

The quality e¤ect of intra…rm bargaining with endogenous worker ‡ows

Keywords: Matching, Intra…rm Bargaining, Worker Flows. JEL Classi…cation: J3, J6

This paper proposes a new explanation of the job quality issue in search and matching models, which is not based on market externalities but on strategic interactions within …rms through the intra…rm bargaining process. We develop a matching and intra…rm bargaining model in which large …rms hire workers on a frictional labour market and decide to destroy low productivity job-worker matches. The coexistence of entry and exit ‡ows of workers in a large …rm gives rise to a speci…c interaction between the …ring decision and the intra…rm bargaining process on wages, which causes ine¢ cient decisions to be made on hiring and …ring. The sources of ine¢ ciency in this economy are (i) the well-known quantitative e¤ect of intra…rm bargaining, namely the excessive size of the …rms concerned, and (ii) a new quality e¤ect, namely the poor quality of the job-worker matches selected by …rms.

Introduction

The quality of jobs is a highly debated topic in labor economics from both empirical and theoretical perspectives. The empirical debate started in the eighties, when some economists, as [START_REF] Bluestone | The Growth of Low-Wage Employment: 1963-1986[END_REF], have warned of the rising number of low-quality jobs; see [START_REF] Goos | Lousy and Lovely Jobs: the Rising Polarization of Work in Britain[END_REF] for an recent assessment of this view. The theoretical debate has focused on the ability of the competitive market to provide the e¢ cient quality of jobs. The seminal contributions of [START_REF] Stevens | Labour Contracts and E¢ ciency in on-the-Job Training[END_REF], [START_REF] Redding | The Low-skill, Low-quality Trap: Strategic Complementarities between Human Capital and R & D[END_REF] and [START_REF] Acemoglu | Good Jobs versus Bad Jobs[END_REF] showed that the quality of jobs is generally ine¢ cient with labor market search frictions because of market externalities. 1 In this paper, we provide a new explanation of the bias toward low-quality jobs, which is not based on market externalities but on the intra…rm bargaining mechanism described by Stole and Zwiebel (1996ab). 2 The bias toward low-quality jobs is the outcome of the strategic interactions within …rms between the bargaining process on wages with workers and the selection of the quality of worker-job matches by …rms. This result is obtained by extending the matching and intra…rm bargaining literature initiated by [START_REF] Smith | Search, Concave Production, and Optimal Firm Size[END_REF], Cahuc and Wasmer (2002) and [START_REF] Cahuc | A Theory of Wages and Labor Demand with Intra-…rm Bargaining and Matching Frictions[END_REF] to the case of endogenous workers ‡ows of entry and exit in …rms. 3 The empirical literature on labour market ‡ows has highlighted the importance of workers ‡ows and of its distinction with job ‡ows. Job ‡ows are associated with the net variation in the mass of jobs at the establishment level: job destruction only occurs in contracting establishments, in which the total mass of jobs falls, while job creation only occurs in expanding establishments, in which the total mass rises, see [START_REF] Davis | Gross Job Creation and Destruction: Microeconomic Evidence and Macroeconomic Implications[END_REF]. Job creation and job destruction cannot simultaneously occur within establishments, and only establishments heterogeneity can permit the reallocation of jobs. The picture is somewhat di¤erent for worker ‡ows because the entry and exit of workers may take place simultaneously in …rms. Among others, [START_REF] Burgess | Job Flows, Worker Flows, and Churning[END_REF] demonstrated that worker ‡ows largely exceed job ‡ows, and that regardless of its size or the growth in its number of employees, entry and exit worker ‡ows occur simultaneously in most establishments. 4 We develop a matching and bargaining model consistent with this speci…city of worker ‡ows (i.e., that entry and exit ‡ows occur simultaneously within …rms) and use this model to assess the potential ine¢ ciencies in the quantity and the quality of the job-worker matches in the labour market.

In order to study worker ‡ows within large …rms, we cannot use the traditional matching and bargaining model of the labour market developed by [START_REF] Diamond | Aggregate Demand Management in Search Equilibrium[END_REF], [START_REF] Pissarides | Equilibrium unemployment theory[END_REF], and [START_REF] Mortensen | Job Creation and Job Destruction in the Theory of Unemployment[END_REF], in which …rms are small and composed of one single job. 5 We must instead consider the decisions of large …rms composed of heterogeneous job-worker matches. Jobs and workers are homogeneous, but when they are matched, the e¤ective productivity of each job-worker pair is subject to idiosyncratic productivity shocks. It is widely-known that considering large …rms with a non trivial production function (i.e., a concave technology) has strong normative implications. While there exists a condition of e¢ cient bargaining when …rms are small 6 , the wage negotiation is generally ine¢ cient in large …rms. It proceeds from the strategic interactions between the …rm's decisions on the organisational design of production and the processes of wage bargaining with workers originally described by Stole and Zwiebel (1996a,b) and known as intra…rm bargaining. One of their key …ndings was that intra…rm bargaining causes …rms to overemploy workers in order to reduce their individual marginal productivities, thereby reducing the individual wage 4 These facts have been updated and con…rmed for the US economy by [START_REF] Davis | The Flow Approach to Labor Markets: New Data Sources and Micro-Macro Links[END_REF] who combine the most recent datasets including the Job Openings and Labor Turnover Survey. 5 Kiyotaki and Lagos (2007) and [START_REF] Burgess | Worker ‡ows, job ‡ows and unemployment in a matching model[END_REF] make the distinction between job ‡ows and worker ‡ows in matching models with small …rms by assuming that a job can survive after a separation with a worker and be …lled with another worker. 6 The e¢ cient labour contract of [START_REF] Hosios | On the E¢ ciency of Matching and Related Models of Search and Unemployment[END_REF] and [START_REF] Pissarides | Equilibrium unemployment theory[END_REF] has been de…ned in matching models with small …rms.

3 bargained with each worker. [START_REF] Smith | Search, Concave Production, and Optimal Firm Size[END_REF], Cahuc and[START_REF] Cahuc | Does Intra…rm Bargaining Matter in the Large Firm's Matching Model?[END_REF][START_REF] Cahuc | A Theory of Wages and Labor Demand with Intra-…rm Bargaining and Matching Frictions[END_REF] all studied the e¤ects of intra…rm bargaining on hiring in the matching model. 7 Our aim herein is to extend this literature to the case of endogenous worker reallocation driven by hiring and …ring. Because the …ring decision endogenises the average quality of the job-worker matches in …rms, we can identify a new qualitative e¤ect of intra…rm bargaining in addition to the traditional quantitative e¤ect associated with the size of …rms.

The novelty of our approach lies in the decision to address the issue of intra…rm bargaining in a setup where …rms simultaneously hire and …re workers. This is in contrast with the literature that has considered the issue of intra…rm bargaining in models of heterogeneous …rms that either create or destroy jobs in the tradition of [START_REF] Bertola | Cross-Sectional E¢ ciency and Labour Hoarding in a Matching Model of Unemployment[END_REF], such as for example, [START_REF] Bertola | Wages and the Size of Firms in Dynamic Matching Models[END_REF], [START_REF] Koeniger | Employment Protection, Product Market Regulation and Firm Selection[END_REF], [START_REF] Fujita | Worker ‡ows and job ‡ows: a quantitative investigation[END_REF]Nakajima (2009), Cosar et al. (2010), and Elsby and Michaels (2010). In such models, contracting …rms destroy jobs when hit by an adverse idiosyncratic shock that causes a drop in the demand of labour. An important property of this type of model is that workers in contracting …rms are unable to extract a positive rent from the Nash bargaining program on wages because the marginal value of a job is zero or negative for the …rm. The wage is consequently set at the worker's reservation level that makes its utility equal to that of an unemployed worker. Therefore, wages in contracting …rms are independent of the decisions of …rms and there is no room for strategic interactions between the …ring decision and wage bargaining.

For worker ‡ows, because …rms simultaneously hire and …re workers, our setup allows new interactions to take place between the …ring decision and wage bargaining. Existing matching models with large …rms that simultaneously hire and …re workers 8 are not suitable for our purposes because they generally assume a trivial production technology, with a constant marginal productivity of labour, which makes the individual 7 The strategic interaction between the …rm's decision on job creation and the wage bargaining process leads …rms to post an excessive number of vacancies or equivalently to overhire workers. 8 See in particular Merz (1999), [START_REF] Krause | Does intra-…rm bargaining matter for business cycle dynamics?[END_REF], [START_REF] Krause | Do search frictions matter for in ‡ation dynamics[END_REF]Faia et al. (2010). In Faberman and Nagypal (2008), the recruitment technology is non-linear (the marginal cost of creation of positions is increasing), but the production technology is linear with a constant marginal productivity of labor.

wage independent of the …rm's decisions and as a result removes the strategic interactions from the model.

The only exception is [START_REF] Helpman | Inequality and Unemployment in a Global Economy[END_REF] who studied the consequences of globalisation in a model where heterogeneous …rms screen workers. 9 We depart from several assumptions of [START_REF] Helpman | Inequality and Unemployment in a Global Economy[END_REF].

First, while we assume perfect competition on the goods market, they consider imperfect competition, hence intra…rm bargaining is the only source of ine¢ ciency in our setup. Second, we solve a dynamic rather than a static labour market search model. Third, we consider idiosyncratic productivity of matches at the origin of …ring instead of the screening technology introduced by them. Fourth, we model explicitly the utility of unemployed workers, which they normalize to zero. This last point matters because at equilibrium, the labour market tightness a¤ects wages through the endogenous value of unemployed worker utility.

In our model, we have brought together several di¤erent strands of the literature on matching and bargaining models. The process of hiring is modelled as in the standard matching model of the labour market; e.g., [START_REF] Pissarides | Equilibrium unemployment theory[END_REF]. Firms post costly vacancies on the labour market and unemployed workers search passively for a job. An aggregate matching function determines the ‡ow of new hirings according to the masses of vacancies and unemployed workers. The worker-job matches destruction process is taken from den [START_REF] Den Haan | Job Destruction and Propagation of Shocks[END_REF]. The productivity of matches is heterogeneous due to the presence of non persistent idiosyncratic shocks. 10 The distribution of productivity is the same for newly created matches and for existing ones. If the match productivity is too low, the worker is …red by the …rm, corresponding to the endogenous …ring carried out by …rms. Domestic production for unemployed workers ensures that temporary layo¤ are not preferred to permanent layo¤. In addition, there is an exogenous …ring rate in …rms that is independent of the match productivity. This structure of shocks was considered by den Haan et al.

( 2000) in a model applied to small …rms. We follow the approach of [START_REF] Krause | Does intra-…rm bargaining matter for business cycle dynamics?[END_REF], who applied the same structure of shocks for large …rms, but contrary to these authors, we consider a non-constant marginal productivity of labour. The concavity of the production technology gives rise to the intra…rm bargaining issue. The process of bargaining on wages is solved using the solution proposed by [START_REF] Cahuc | A Theory of Wages and Labor Demand with Intra-…rm Bargaining and Matching Frictions[END_REF] for matching and intra…rm bargaining models. As in [START_REF] Cahuc | A Theory of Wages and Labor Demand with Intra-…rm Bargaining and Matching Frictions[END_REF], there is no …rm entry. 11

We show that intra…rm bargaining induces ine¢ cient hiring and …ring rules in the economy. The individual wage solution of the intra…rm bargaining process depends on two variables that are decided by the …rm, namely its quantity of matches and its reservation productivity. In the intra…rm bargaining setup, the …rm tries to increase its pro…ts by reducing workers'productivity in order to push down the bargained wages. In our setup, the …rm has two means of reaching this aim: it can either increase the quantity of matches or decrease their average quality. The former is common in the intra…rm bargaining literature, whereas the second is new and speci…c to our setup of workers ‡ows. To lower the average wage, the representative …rm seeks to increase its production by posting a number of vacancies that is too high (as in other models of intra…rm bargaining and matching) and by choosing a level of reservation productivity that is too low (this e¤ect is speci…c to our setup). We use numerical simulations to quantify the e¤ects of intra…rm bargaining. For our benchmark calibration, intra…rm bargaining induces an excess supply of vacancies and an insu¢ cient quality of matches. This result is in the line with the classical overemployment result of Stole and Zwiebel (1996a,b). Our model shows that with endogenous …ring, overemployment is the outcome of an excessive supply of vacancies and a …ring rate that is too low. The source of ine¢ ciency in this economy is not only the excessive size of …rms, but also the poor quality of the jobs selected by them.

The remainder of our article is organised as follows. We describe our model in section 2 and in section 3, we show the resolution of the intra…rm bargaining process on wages, and give a formal de…nition of the equilibrium and its normative properties. In section 4, we provide some brief conclusions.

2 The Model

Hiring, Firing, and Production

In our model, …rms are composed of an endogenous number of job-worker matches, which are subject to idiosyncratic productivity shocks that are denoted a. At the beginning of each period, existing and newly formed matches draw a value for a from the cumulative distribution function G ( ), whose density function is denoted g (:). The …rm i chooses its reservation value for idiosyncratic productivity a it : all matches below this value are destroyed. The endogenous …ring rate is it = G (a it ) and the total separation rate is

it = + (1 ) G (a it ) (1)
where is the exogenous …ring rate. The survival rate for both exogenous and endogenous separation processes is (1 it ) :

The aggregate matching function is m (u

t ; v t ) = mu t v 1 t with v t = R 1 0 v it di,
which represents the mass of vacancies posted by all …rms with v it being the mass of vacancies posted by the …rm i, and u t being the mass of unemployed workers. t = v t =u t is the labour market tightness with

q ( t ) = m (u t ; v t ) =v t = m (1= t ; 1)
being the probability that a vacancy will be …lled in the next period.

The employment of …rm i is denoted n it , and is the sum of the employment masses over the range of admissible productivity levels a 2 [a it ; 1[

n it = Z 1 a it n it (a) da (2)
where n it (a) is the mass of worker-job with productivity a. The derivatives of the output and employment of …rm i with respect to n it (a) will be used to de…ne the …rm's contribution to the Nash bargaining process on wages. During the bargaining process, the …rm can decide to keep or not the marginal worker of productivity a during the bargaining process. For admissible values of a a it , all bargaining processes lead to wages that are accepted by both the …rm and the workers and the distribution of n it (a) depends on the endogenous variables fa it ; n it g and the exogenous density function g (a) :

Indeed, the …rm chooses the lower bound of productivity, a it , and the total number of matches, n it , through its supply of vacancies, v it , but it does not choose the allocation of workers within the range of admissible productivity levels [a it ; 1[, which is imposed exogenously. For a 0 > a 00 > a it , the …rm cannot control the relative sizes of the masses of matches, denoted n it (a 0 ) and n it (a 00 ), even if it would be pro…table to substitute n it (a 00 ) for n it (a 0 ) : The distribution of matches between [a it ; 1[ is non-persistent and is entirely determined by the exogenous power density function g ( ). The mass of matches of productivity a is given by

n it (a) = g (a) 1 G (a it ) n it ; for a 2 [a it ; 1] (3)
where n it is the aggregate employment and g (a) = [1 G (a it )] the power density function of a for the (endogenous) range of admissible values for the idiosyncratic productivity.

The evolution of …rm i employment level is

n it+1 = 1 it+1 (n it + v it q ( t )) (4)
in which we assume that each …rm gets a linear proportion v it =v t of the total matches m (u t ; v t ). Idiosyncratic productivity shocks are the same for both newly created (v it q ( t )) and existing jobs (n it ), and therefore the same proportion it+1 of jobs is destroyed at the beginning of the period t + 1.

Total output of …rm i at time t is

y it = f (h it ) = h it (5)
with 0 < 1 and h it is the amount of e¤ective labor input:

h it = h (n it ; a it ) = z Z 1 a it an it (a) da = zn it Z 1 a it a g (a) 1 G (a it ) da (6)
with z being the productivity of workers, which is common to all …rms.

The Firm' s Program

The discounted sum of the pro…ts of …rm i is

it = 1 X k=t k t z Z 1 a ik an ik (a) da Z 1 a ik e w ik n ik a 0 a 0 =1 a 0 =a ik ; a ik ; a n ik (a) da v ik (7) 1 X k=t 1 k+1 t ik+1 ( Z 1 a ik+1 n ik+1 (a) da (1 ) [1 G (a ik+1 )] Z 1 a ik n ik (a) da + v ik q ( k ) )
where k;t = (k t) is the discount factor at date k (the reference date is denoted t) and ik+1 is the multiplier of the employment evolution constraint. Two costs are taken into account. First, the recruitment of new workers is costly and the per-period cost of a vacancy is : Second, the wage for a match of productivity a is denoted e w ik ( ) and depends on the distribution of employment in the …rm fn ik (a 0 )g a 0 =1 a 0 =a ik , the current value for the reservation productivity fa ik g ; and the value for the idiosyncratic productivity a. According to the de…nition of n it (a) given in equation ( 3), the distribution of employment depends on the two endogenous variables fn it ; a it g and on the speci…cation of the exogenous power density function g ( ). We use this property and postulate a functional form of individual wage whose arguments are simply fn it ; a it ; ag rather than the entire distribution of employment fn it (a 0 )g a 0 =1 a 0 =a ik .

Claim 1 The wage for the value a of the idiosyncratic productivity shock in …rm i is a function of the two endogenous variables fn it ; a it g and writes as follows

e w it n it a 0 a 0 =1 a 0 =a ik ; a it ; a = e w it g (a 0 ) 1 G (a it ) n it a 0 =1 a 0 =a it ; a it ; a ! = e w it (n it ; a it ; a) (8)
This postulate for the individual wage function uses the property (3) and is consistent with the intra…rm bargaining approach; because the …rm does not decide on the whole distribution of matches, but only on the mass of matches and the lower bound of the idiosyncratic productivity level, the wage of a worker with a productivity of a does not depend on the distribution of jobs within the range of productivity [a it ; 1[ for a given …rm size. We will further prove that this speci…cation of the individual wage is consistent with the outcome of the Nash bargaining process. Therefore, the wage bill is

Z 1 a ik e w ik n ik a 0 a 0 =1 a 0 =a ik ; a ik ; a n ik (a) da = w it (n it ; a it ) n it (9)
where w it (n it ; a it ) is the average wage bill per employee de…ned by

w it (n it ; a it ) = Z 1 a it e w it (n it ; a it ; a) g (a) 1 G (a it ) da (10)
Once again, this expression only depends on the variables n it and a it due to the intra…rm bargaining assumption.

Finally, the …rm's problem is to choose its supply of vacancies v it ; its reservation value for idiosyncratic productivity a it , and its employment level n it to maximise the discounted sum of pro…ts de…ned by (7) under conditions (2), ( 3), ( 5), ( 6), and (10), that is

max v it ;n it ;a it+1 it = 1 X k=t k t zn ik Z 1 a ik a g (a) 1 G (a ik ) da w ik (n ik ; a ik ) n ik v ik (11) 1 X k=t 1 k+1 t ik+1 fn ik+1 (1 ) [1 G (a ik+1 )] (n ik + v ik q ( k ))g

The Asset Value of Jobs

To solve the Nash bargaining program, we de…ne the asset value of a job with productivity a for …rm i

J it (a) = za h (n it ; a it ) 1 e w it (n it ; a it ; a) (12) Z 1 a it @ e w it (n it ; a it ; a 0 ) @n it (a) n it a 0 da 0 | {z } Intra…rm + " 1 it+1 Z 1 a it+1 J it+1 (a) g (a) 1 G (a it+1 ) da #
where the …rst term is the marginal productivity of the match of productivity a, see ( 5)-( 6), the second term is the individual wage, the third term is associated with the "intra…rm e¤ect", and the last term is the discounted value for the match. With probability 1 it+1 , the match survives and the …rm gets the average value of future matches.

The third term, associated with the "intra…rm e¤ect", accounts for the impact of the marginal worker of productivity a on all the individual wages paid by the …rm i. Under conditions (2) and (3), and given the expressions (8) for the individual wage and (10) for the average wage bill, this term becomes

Z 1 a it @ e w it (n it ; a it ; a it+1 ; a 0 ) @n it (a) n it a 0 da 0 = w 1 it (n it ; a it ) n it , 8a 2 [a it ; +1[ (13)
that is the product of the marginal impact of employment on the average wage bill and the …rm's employment level -see Appendix A.1 for details. It is interesting to note that this expression does not depend on the idiosyncratic productivity level a. This crucial property of the model follows directly from the postulate on wage distribution (8). In the Nash bargaining program, we will use the following expression for the asset value of a job with productivity a for …rm i 12) and ( 13).

J it (a) = za h (n it ; a it ) 1 e w it (n it ; a it ; a) w 1 it (n it ; a it ) n it (14) + " 1 it+1 Z 1 a it+1 J it+1 (a) g (a) 1 G (a it+1 ) da # using (

The Nash Bargaining on Wages

To solve the Nash bargaining process on wages, we …rst de…ne the value of being employed or unemployed to a worker. The asset value of a match for a worker with productivity a is denoted W it (a) and de…ned by

W it (a) = e w it (n it ; a it ; a) + (1 ) [1 G (a it+1 )] Z 1 a it+1 W it+1 (a) g (a) 1 G (a it+1 ) da (15)
+ [ + (1 ) G (a it+1 )] U t+1
where U t is the expected return of being unemployed and is de…ned by

U t = b + t q ( t ) 1 t+1 Z 1 a t+1 W t+1 (a) g (a) 1 G (a t+1 ) da + 1 t q ( t ) 1 t+1 U t+1 ( 16
)
where b measures home production and the variables in bold show the average values of these variables in the economy. The wage solution of the bargaining program within …rm i is

W it (a) U t = 1 J it (a) (17)
where is the bargaining power for the worker and (1 ) is the bargaining power for the …rm.

Equilibrium

In this section, we …rst present the wage solution of the intra…rm bargaining process, then we show that it is consistent with our postulate for the individual wage speci…cation, and then we discuss our …ndings. We then de…ne the equilibrium and provide some conditions for its e¢ ciency.

3.1 The Wage Solution

It should be noted that the implicit functional form of e w it deduced from equation ( 18) is consistent with our postulate on the wage distribution (8). As is common in intra…rm bargaining models, there is a partial derivative of the wage function with respect to employment in the equation solution of the bargaining process. However, in our setup the partial derivative term is the derivative of the average wage w it ( ) with respect to employment, while the interest variable is the individual wage e w it ( ) : We must therefore …rst compute the average wage by aggregating the wages de…ned by the equation ( 18) using the de…nition (10)

of the average wage. The average wage solution of the Nash program (17) solves the partial di¤erential equation

w it (n it ; a it ) = (1 ) b + t + @f (h (n it ; a it )) @n it w 1 it (n it ; a it ) n it ( 19
)
whose solution is

w it (n it ; a it ) = (1 ) b + t + 1 (1 ) h (n it ; a it ) n it (20)
This expression for the average wage gives rise to the following equation for the individual wage solution of the equation ( 18)

e w it (n it ; a it ; a) = (1 ) b + t + @f (h (n it ; a it )) @n it (a) ( 1) 1 (1 ) @f (h (n it ; a it )) @n it (21)

The Impact of Firm' s Decisions on Wages

The average wage is a¤ected by the …rm's hiring and …ring decisions. The partial derivatives of the average wage function given by equation ( 20) with respect to a it and n it are

w 1 it (n it ; a it ) =
(1 ) 1 ( 1)

h (n it ; a it ) n 2 it 0 (22) w 2 it (n it ; a it ) = 2 1 (1 ) g (a it ) 1 G (a it ) h (n it ; a it ) n it 2 4 1 a it R 1 a it a g(a)
1 G(a it ) da

3 5 > 0 (23)
The negative e¤ect of employment on the average wage measured by w1 it ( ) corresponds to the classical "overemployment" result, originally described by Stole and Zwiebel (1996a,b) and afterwards restated in the context of matching frictions by [START_REF] Smith | Search, Concave Production, and Optimal Firm Size[END_REF] and [START_REF] Cahuc | A Theory of Wages and Labor Demand with Intra-…rm Bargaining and Matching Frictions[END_REF]. These authors showed that this e¤ect vanishes if the technology is linear ( = 1). The positive impact of the productivity reservation on the average wage measured by w 2 it ( ) is the sum of two e¤ects: The …rst e¤ect is associated with the "average quality" of matches and exists even if the production function is linear as in [START_REF] Krause | Does intra-…rm bargaining matter for business cycle dynamics?[END_REF]. 12 Increasing a it improves the average quality of the selected matches and therefore improves the average wage paid by the …rm: this e¤ect would hold without the intra…rm bargaining assumption. It is easy to see that this …rst term positively contributes to the impact of the productivity reservation on the average wage. In our speci…c setup, this …rst e¤ect is augmented by a second one linked to the impact of the productivity reservation a it on the individual wage e w it , which results directly from the intra…rm bargaining assumption. Using de…nition (21), we deduce that the contribution of this second e¤ect to the average wage is negative:

w 2 it (n it ; a it ) = g (a it ) 1 G (a it ) [w it (n it ; a it ) e w it (
Z 1 a it e w 2 it (n it ; a it ; a) g (a) 1 G (a it ) da = (1 ) (1 ) 1 (1 ) g (a it ) 1 G (a it ) h (n it ; a it ) n it 2 4 1 a it R 1 a it a g(a) 1 G(a it ) da 3 5 0 (25)
An increase in a it lowers the relative productivity of an a-type worker and consequently reduces the average wage. Therefore, the intra…rm bargaining assumption considered in an endogenous destruction setup contributes to the mitigation of the positive e¤ect of the productivity reservation on the average wage given by (24).

Discussion

The individual wage solution of the bargaining process given by equation ( 21) depends not only on the …rm's employment level (as is common in intra…rm bargaining models), but also on the …rm's decision on separation. This property makes our setup (based on workers'reallocation) di¤erent from other intra…rm bargaining models based on job reallocation such as that of [START_REF] Bertola | Cross-Sectional E¢ ciency and Labour Hoarding in a Matching Model of Unemployment[END_REF]. In these models of job reallocation, the average wage only depends on the …rm's level of employment, and not on the …rm's productivity threshold under which it destroys jobs or leaves the market. 13 Our wage equation is also di¤erent from that of [START_REF] Helpman | Inequality and Unemployment in a Global Economy[END_REF], who solved a static model in which the screening ability cuto¤ decided by the …rm simultaneously determines both the quality and the number of workers. Therefore, the wage equation has only one argument in [START_REF] Helpman | Inequality and Unemployment in a Global Economy[END_REF], namely the screening ability cuto¤. The novelty of our approach is that it provides a wage equation solution of the intra…rm bargaining process that depends both on the …rm's employment level and its …ring rate. This property will turn out to be decisive when we will assess the e¢ ciency of the competitive equilibrium, since we can detect two distortions in our 1 3 [START_REF] Bertola | Cross-Sectional E¢ ciency and Labour Hoarding in a Matching Model of Unemployment[END_REF] give the explicit expression for the wage in contracting …rms that does not depend on …rm's employment, see equation ( 19) in their paper, and the wage in expanding …rms that depends on the …rm's hiring e¤ort, see equation ( 21) in their paper. For similar wage solutions in models of job reallocation see equation ( 11) of [START_REF] Bertola | Wages and the Size of Firms in Dynamic Matching Models[END_REF], equation ( 10) of Elsby and Michaels (2010), equation ( 9) of [START_REF] Koeniger | Employment Protection, Product Market Regulation and Firm Selection[END_REF], and equation ( 22) of [START_REF] Fujita | Worker ‡ows and job ‡ows: a quantitative investigation[END_REF].

economy: the intra…rm bargaining assumption distorts both the hiring and …ring decisions.

The distortion of the hiring and …ring decisions could be interpreted in terms of both quantitative and qualitative e¤ects. The quantitative e¤ect proceeds from the impact of the quantity of matches on bargained wages and corresponds to the partial derivative ( 22). Because a greater number of matches contribute to a lowering of the average wage, …rms are inclined to post too many vacancies and consequently the equilibrium …rm size is above its optimal level. The interpretation of the quality e¤ect associated with the reservation productivity is rather less straightforward because it has two opposing consequences. First, the …rm's reservation productivity negatively a¤ects the individual bargained wages, as shown by partial derivative ( 25), and, this should therefore drive …rms to increase their reservation productivity. Second, the …rm's reservation productivity negatively impacts on the e¤ective labour input, de…ned by ( 6), as shown in the Appendix A.2, and should therefore drive …rms to decrease their reservation productivity in order to lower the average wage; see (20). In the following sections we provide a full analysis of these two opposite e¤ects of the reservation productivity and conclude that the second e¤ect generally dominates the …rst, and intra…rm bargaining pushes …rms to accept a quality of job-worker matches that is too low.

The E¢ ciency of the Competitive Equilibrium

Steady-state labour market allocations are determined by two variables: the productivity reservation and the labour market tightness (a; ). For given values of (a; ), the steady-state employment level n ( ; a) solution of equation ( 4) is 14

n ( ; a) = 1 (1 ) (1 G (a)) (1 ) (1 G (a)) 1 m + 1 1 (n)
with n 1 ( ; a) > 0 and n 2 ( ; a) < 0: The e¤ective labour input h (a; ) is given by equations ( 6) and (n)

h ( ; a) = z 1 (1 ) (1 G (a)) (1 ) (1 G (a)) 1 m + 1 1 Z 1 a a g (a) 1 G (a) da (h)
with h 1 ( ; a) > 0 and h 2 ( ; a) < 0: The steady-state employment level unambiguously increases with and decreases with a. Hence, h straightforwardly increases with :

The overall impact of a on the e¤ective labour input should be ambiguous because an increase in a lowers the employment rate, but raises the average quality of matches. We show in the Appendix A.2 that the overall impact is negative. The optimal and competitive values for (a; ) are now de…ned.

De…nition 1 The optimal values f ; a g, solution of the social planner program described in the Appendix A.4, solve

m ( ) 1 = (1 ) [1 G (a )] h ( ; a ) 1 z h ( ; a ) zn ( ; a ) a (H o ) m ( ) 1 + h ( ; a ) 1 za = b + (1 ) (F o )
where the functions h ( ; a) and n ( ; a) are given by Equations (h) and (n). The values f ; a g, solution of the competitive economy de…ned in Section 2, solve

m ( ) 1 = (1 ) (1 G (a )) (1 ) 1 (1 ) zh ( ; a ) 1 h ( ; a ) zn ( ; a ) a (H ) 1 4
It is computed as follows The steady-state of ( 4) is

n = (1 ) (n + vq ( )) (26)
The de…nition of the matching process implies vq ( ) = m (1 n) and the de…nition of given by ( 1) implies (1 ) =

(1 ) (1 G (a)). Therefore, (26) becomes

n = (1 ) m 1 (1 ) (1 m ) = m 1 (1 ) 1 + m = 1 1 m h 1 (1 )(1 G(a)) 1 i + 1 = 1 (1 ) (1 G (a)) (1 ) (1 G (a)) 1 m + 1 1 and …nally [manque qqchose ???] m ( ) 1 + (1 ) 1 (1 ) h ( ; a ) 1 za = (1 ) b + (F )
see the sections A.3 and A.4 of the Appendix for details of their solution.

The terms on the LHS of the Hiring equations (H ) and (H ) represent the average matching cost de…ned as the ratio of the per-period search cost to the job matching probability m 1 . Equilibrium hiring equalises this average matching cost to the discounted average value of a …lled job de…ned by the RHS term of (H ) for competitive …rms, and by the RHS term of (H ) for the social planner. The discount rate is the product of the subjective discount factor and the job survival rate to shocks, i.e.,

(1 ) (1 G (a)). The average marginal productivity of jobs has the same expression for the two equilibria, namely zh 1 (h= (zn) a), but the social planner considers the share of this productivity rather )] for competitive …rms. The term [1 ( 1)] 1 directly results from the intra…rm bargaining mechanism.

than the share (1 ) = [1 (1
The terms on the LHS of (F ) and (F ) represent the marginal value of the less productive job-worker match for competitive …rms and the social planner respectively. Keeping this job makes it possible to save the average matching cost -the …rst term ( =m) 1 . The second terms on the LHS of (F ) and (F ) are the shares of the marginal productivity of this match, given by h 1 za. As for the hiring equations, these shares are not identical for the social planner (i.e., ) and for competitive …rms:

(1 ) = [1 (1 )].
The RHS term of (F ) and (F ) account for opportunities outside the match, which consist in b the home production of unemployed workers and = p ( ) =q ( ), the product of the worker matching probability and the average matching cost. These two terms are weighted by for the social planner and for the competitive equilibrium.

Proposition 1 The competitive equilibrium is e¢ cient when the Hosios condition holds and the production technology is linear.

Proof. Equations (H )-(F ) are equivalent to (H )-(F ) if = 1 and the Hosios condition holds

(1 ) = .

With a linear production technology, the sole distortion in the economy (namely the trading externality associated with matching function) is e¢ ciently internalised in the labour contract if the bargaining power of agents correspond to their contribution to the trading activities; see [START_REF] Hosios | On the E¢ ciency of Matching and Related Models of Search and Unemployment[END_REF] and [START_REF] Pissarides | Equilibrium unemployment theory[END_REF]. Hence, the combination of intra…rm bargaining and endogenous …ring decisions generates no further distortion in the linear case. The competitive equilibrium with a concave production technology is ine¢ cient even if the Hosios conditions holds.

With a concave production technology, intra…rm bargaining prevents the labour contract from internalising trading externalities. Intra…rm bargaining creates a distortion in the economy because …rms take into account the impact of the marginal worker on the total mass of bargained wages with their employees.

Without intra…rm bargaining, the …rm internalises the fraction (1 ) of the marginal productivity of labour rather than (1 )] with intra…rm bargaining. The term [1 ( 1)] 1 corresponds to the ine¢ ciency induced by intra…rm bargaining and would be equal to 1 without intra…rm bargaining. It …rst appears in the hiring equation (H ) as in other models of matching and intra…rm bargaining models with exogenous separation; e.g. [START_REF] Smith | Search, Concave Production, and Optimal Firm Size[END_REF] and [START_REF] Cahuc | A Theory of Wages and Labor Demand with Intra-…rm Bargaining and Matching Frictions[END_REF]. Firms give a higher value to the marginal worker with intra…rm bargaining and are therefore more inclined to post vacancies. The originality of our model is that this term also appears in the …ring equation (F ). The RHS term of (F ) corresponds to the value of the match with the lowest productivity. Through the term [1 (1 )] 1 , intra…rm bargaining implies that …rms give a higher value to this match than they would do without intra…rm bargaining. Firms are therefore induced to accept matches with a productivity level below its optimal level. For given values of h and n; intra…rm bargaining unambiguously promotes an increase in labour market tightness and a fall in reservation productivity. The full impact is ambiguous, however, because the variables h and n are also a¤ected by variations of and a. The use of di¤erentiation may help to provide some further understanding of the e¤ect of the intra…rm bargaining assumption in our model.

) = [1 (1

A Graphical Representation of the Equilibrium

To characterise the equilibrium, we introduce a parameter ' that can take two values. If ' = (1 ), the solution is the outcome of the optimal equilibrium a; = a ;

. Otherwise ' = (1 ) = [1 (1 )] ;

and the solution is the outcome of the competitive equilibrium a; = n a ;

o : For a linear production technology and under the Hosios condition, the competitive equilibrium is optimal and is at the intersection of two decreasing curves in the plan (a; ). However, this may not be the case when the production technology is concave. Full details of the calculations are provided in the Appendix A.5.

The di¤erentiation of the Hiring Curves -either in the optimal (H ) or the competitive case (H ) - where the values of a; and ' vary for the optimal and competitive cases as explained above and assuming that the Hosios condition still holds. When = 1 the second bracketed term in the RHS of equation ( 27) is null and the slopes of the H-curves are negative. For < 1; the sign of d =da depends on the relative size of the two terms on the RHS, which notably depend on the chosen functional form of g (:), i.e., the density of productivity shocks. In the numerical section, we will show that under plausible parameterisations and assuming the log normality of idiosyncratic shocks, the overall sign of the RHS bracket term remains negative even for low values of . Consequently, the slopes of the H-curves appear to be negative in either case (optimal or competitive). Moreover, it is easy to see that the higher the value of '; the higher the value of for a given value of a (see Appendix A.5 for further details). Hence, the intra…rm bargaining assumption causes the H-curve (H ) to move upwards in the (a; ) plane, therefore contributing to overemployment for a given level of job destruction (this result con…rms the standard intra…rm e¤ect previously described in the creation locus represents the hiring one. This proves that for a standard value of the labour income share, the in ‡uence of the second term in the bracket of the RHS of equation ( 28) dominates that of the …rst term. This pattern of the locus then ensures the uniqueness of the equilibria for both the competitive and the optimal models. The labour market tightness is higher in the suboptimal intra…rm setup due to the analytically detailed upward position of the competitive hiring curve and leftward position of the competitive …ring curves. Hence, our exercise con…rms the higher labour market tightness e¤ect of intra…rm bargaining as already detailed in the "exogenous destruction" literature. At the same time, the reservation productivity appears to be lower in the intra…rm case because of the relative ‡atness of the H-curve (d =da is low in absolute values). This e¤ect on a is speci…c to our endogenous destruction setup and reinforces the standard over-tightness result: the overall impact of intra…rm bargaining on the employment rate (see Table 1) is positive due to the joint overposting of vacancies and the too low …ring rate. The usual "overemployment" result is worsened here due to the low quality of jobs selected by …rms.

Figure 2 (top panel) shows that, despite the instability of the signs of slope of the F-curve, the simultaneous e¤ect of intra…rm bargaining on and a holds a variety of values of . For all values of between 0.4 and 1, the competitive labour market allocations are characterised by excessive hiring, insu¢ cient …ring and consequently overemployment. 16 To put forward the implications of the …ring decision, we also consider the case of purely exogenous separations (i.e. a = 0). The middle panel of Figure 2 shows that the excessive size of …rms is quite similar between the two versions of the model (with or without endogenous destruction).

However, as shown by the bottom panel of Figure 2, the excess supply of vacancies is strikingly di¤erent between the two versions of the model. With endogenous destruction, the excess supply of vacancies is tiny and, for some values of , the supply of vacancies by …rms is even too small. The excess employment is mainly the outcome of a too low average separation rate in the economy rather than an excessive supply of jobs. This picture contrasts with the case of exogenous separation where the large excess of vacancies is clearly at the origin of the excessive size of …rms. Therefore, our setup of endogenous worker ‡ows provides a new view of the excessive …rms'size, which is more the outcome of an insu¢ cient amount of separations rather than an excessive supply of vacancies.

Conclusion

We have herein developed a matching and intra…rm bargaining model of the labour market to study the e¢ ciency of hiring and …ring by large …rms. The novelty of our approach lies in the consideration of …rms that simultaneously hire and …re workers as suggested by numerous empirical studies. This approach gives rise to speci…c interactions between the wage bargaining process and decisions on …ring. We …nd a negative impact on bargained wages of the productivity threshold under which workers are …red and show how this impact is included in the …rm's choice of this threshold. Because of these interactions, the equilibrium is ine¢ cient and both hiring and …ring decisions are distorted. Hence, our setup complements the standard "overemployment" result of the intra…rm bargaining literature with exogenous job destruction: we con…rm that the competitive employment rate is too high in a large …rm setting with endogenous …ring decisions, but also prove that the average quality of labour matches is too low in this case. Further researches should be undertaken on the usefulness of labour policy and employment legislation protection in this context. The quantitative e¤ect of a on h dominates its qualitative e¤ect.

A.3 The Competitive Equilibrium

This section de…nes the competitive equilibrium as the solution of the representative …rm's program de…ned by (11). The …rst order conditions for the …rm's program (11) are 1: Hiring and …ring curves for the competitive and optimal equilibria ( = 0:60).

v it : = q ( t ) (1 ) (1 G (a it+1 )) it+1 (A.

with n 1 (

 1 ; a) > 0 and n 2 ( ; a) < 0: The partial derivatives of h ( ; a) are h 1 ( ; a) = zn 1 (

  function $ α$Relative deviation with respect to the optimal case with endogenous or exogenous separation

Figure 2 :

 2 Figure 2: Relative deviation between the competitive and the optimal allocations on the labor market

  3.1.1 The Resolution of the Intra…rm Bargaining ProcessThe solution of the Nash program (17) is the individual wage function e w

it ( ) that satis…es e w it (n it ; a it ; a) + w 1 it (n it ; a it ) n it = (1 ) b + t + za h (n it ; a it ) 1

[START_REF] Fujita | Worker ‡ows and job ‡ows: a quantitative investigation[END_REF] develop a business cycle model with both worker ‡ows and job ‡ows, but since separations are exogenous there is no room for the strategic interactions between the …ring decision and wage bargaining considered in this paper.1 0 Considering persistent idiosyncratic shocks would clearly be a more realistic assumption, but that would complicate the analytics of the model making the identi…cation and the interpretation of the strategic interpretation between …ring and wage bargaining more di¢ cult .

1[START_REF] Smith | Search, Concave Production, and Optimal Firm Size[END_REF] proposes a model with an endogenous entry of …rms on the markets.

2 In[START_REF] Krause | Does intra-…rm bargaining matter for business cycle dynamics?[END_REF], the linearity of the production function ( = 1) makes the wage bill wit independent of nit, but not of ait.

6 Note that in some of the intermediate values of considered here (more precisely when is between 0:8 and 0:9), the uniqueness of the equilibrium cannot be guaranteed. The …gure only proposes a comparison based on a range of steady-states calibrated according to the rules described in the calibration subsection.

literature).

If we now turn to the impact of intra…rm bargaining on the F-curves, either in the optimal (F ) or the competitive case (F ), we get Once again, we …nd that d =da < 0 when = 1 because the second term on the RHS of equation ( 28) is null.

However, the role of this term if < 1 makes the slopes of the F-curves slope rather di¢ cult to determine.

As numerical experiments will show, the sign of d =da can turn out to be positive when is low, because the second term on the RHS dominates the …rst one. Moreover, the sign of d =da may remain ambiguous (positive for some parts of the curve and negative for others) for intermediate values of : the possibility of the existence of multiple equilibria can therefore not be excluded. Throughout the numerical exercise, we will mainly work on cases where the F-curves are monotonic and where an equilibrium exists and is unique. 15 Finally, the overall contribution of the intra…rm bargaining assumption can easily be identi…ed:

for a given value of , the higher the value of '; the lower the value of a. Hence, ' moves the F-curves to the left in the (a; ) plane. For a given level of market tightness, the intra…rm bargaining then contributes to a lowering of the average worker productivity. The total (i.e. simultaneously analysing the impact on the Hiring and Firing Curves) e¤ect of intra…rm bargaining will depend on the respective (and relative) slopes of the two curves and will be analysed in the numerical experiment below.

Numerical illustration

In order to illustrate the normative properties of our setup, we now proceed to our numerical simulations.

This will enable us to assess quantitatively the impact of the intra…rm bargaining assumption on the labour 1 5 We will leave the multiple equilibria analysis for further research.

20 market tightness and the productivity reservation.

Calibration

The model period is one quarter. We set the discount factor = 0:99, making the annual interest rate close to 4%. Labour income share is set to its standard value, = 0:6. Without any loss of generality, z is set to 1. For the other calibration constraints, we follow den [START_REF] Den Haan | Job Destruction and Propagation of Shocks[END_REF]. We choose an overall separation rate = 0:1 and set the exogenous …ring rate of = 0:068: Hence, the endogenous separation rate is G (a) = ( ) = (1 ) = 0:034. As in den [START_REF] Den Haan | Job Destruction and Propagation of Shocks[END_REF], the average matching rate q is set to 0:7. The unemployment rate u = 1 n is set to 0:12. This rate is higher than its o¢ cial US empirical counterpart, since it also includes jobseeking out-of-the-labour-force workers (see [START_REF] Cole | Can the Mortensen-Pissarides Matching Model Match the Business-Cycle Facts?[END_REF].

To ensure the optimality of the 'no intra…rm bargaining'setup, we suppose that workers'and …rms'shares of the bargaining surplus are similar and equal to 1/2 and impose that (1 ) = = 0:5. Finally, we assume that the idiosyncratic productivity a is iid lognormally distributed, with mean E [ln (a)] = 0 and standard error a = 0:1:

Table 1 gives the values of the parameters and steady-state interest variables of the intra…rm bargaining model deduced from the calibration procedure. The labour market tightness is derived from the dynamic equation for the employment rate. The endogenous separation rate gives a directly. Finally, the values of real cost per vacancy , and the home production utility ‡ow b are respectively deduced from the hiring and …ring equations. The optimal values of the interest variables (here denoted ; a and n ) are deduced from the no-intra…rm bargaining model through the use of equations (H ), (F ) and (n).

Comparison of the Intra…rm and Optimal Equilibria

Figure 1 provides a steady-state comparison of the hiring (H) and …ring (F ) curves in the intra…rm and optimal models. For both models, the hiring locus is decreasing and the …ring locus is increasing as in [START_REF] Mortensen | Job Creation and Job Destruction in the Theory of Unemployment[END_REF] where the job destruction locus represents the …ring one and the job

A Appendix

A.1 Calculus for the "intra…rm e¤ect"

The term associated with the "intra…rm e¤ect" in the value function ( 12) is

To get its expression given by ( 13), we proceed as follows. First, given the de…nition of aggregate employment

(2) we can rewrite this term as

where @n it =@n it (a) = 1. Second, introducing the expression of n it (a) given by ( 3) and the de…nition of the wage bill (8) lead to Z 1

because the functions g (:) and G (:) do not depend on n it , it is equivalent to

where the term in brackets is the average wage according to de…nition (10). Hence, the …nal expression given by ( 13): w 1 it (n it ; a it ; a it+1 ) n it :

A.2 Properties of the function h

The function h ( ; a) is

where the steady-state employment level n (a; ) is for the Cobb-Douglas matching function

The Hiring rule 17 is obtained by combining (A.11) and (A.12)

which can be interpreted as follows

The Firing rule 18 is obtained by combining (A.13) and ( 4)

and then introducing the expression of it given by (A.12)

and using (A.11), one obtains 0

The next equation is the counterpart of the Equation ( 13) of Krause and Lubick (2007).

1 8 The next equation is the counterpart of the Equation ( 14) of [START_REF] Krause | Does intra-…rm bargaining matter for business cycle dynamics?[END_REF] where @ it =@ait = (1 ) g (ait) :

29 which can be interpreted as follows

Marginal impacts on the average wage where the term on the LHS is the cost of increasing a and the term on the RHS is its bene…ts.

A.4 The Optimal Equilibrium

This section de…nes the optimal equilibrium as the solution of the social planner's program. Households are risk-neutral and discount future consumption at a rate . Therefore, the social planner maximises the intertemporal and discounted sum of consumption, which is equal to the output minus the cost of vacancies, plus the domestic production of unemployed workers.

The social planer's program is

The …rst order conditions are

The equilibrium de…nition provided in De…nition 1 is immediately deduced from these …rst order conditions.

A.5 Calculus for the Graphical Representation of the Equilibria

In order to ease the analysis of the model, we introduce a parameter ' that can take two values. If

da a ;

we can see on the LHS of this equation that for a given value of a, the higher the value of ' the higher the value of because m ( ) 1 h ( ; a) 1 is growing with . Hence, ' moves the H-curve upwards in the (a; ) plan.

A.5.2 The Firing Locus

Using the parameter ', the Firing curves (F ) and (F ) are particular cases of

The di¤erentiation of the F-curve (A.23) is

The sign of the bracketed term on the RHS cannot be deduced analytically because it depends on the speci…c functional form of G (). Nevertheless, we note that for = 1 and when the Hosios condition holds, we obtain we deduce that for a given value of , the higher the value of '; the lower the value of a. Hence, ' moves the F-curve to the left in the (a; ) plan.

B Table and Figures