Xavier Pichon

Marie-Cécile Robert

Edouard Bertrand

Robert H Singer

Evelina Tutucci
email: evelina.tutucci@einstein.yu.edu

New generations of MS2 variants and MCP fusions to detect single mRNAs in living eukaryotic cells

Keywords: MS2-MCP system, mRNA labelling, single molecule, single cell, S. cerevisiae, mammalian cells, gene expression, mRNA localization, transcription

Live imaging of single RNA from birth to death brought important advances in our understanding of the spatiotemporal regulation of gene expression. These studies have provided a comprehensive understanding of RNA metabolism by describing the process step-by-step. Most of these studies used for live imaging a genetically encoded RNA-tagging system fused to fluorescent proteins. One of the bestcharacterized RNA-tagging systems is derived from the bacteriophage MS2 and it allows single RNA imaging in real-time and live cells. This system has been successfully used to track the different steps of mRNA processing in many living organisms. The recent development of optimized MS2 and MCP variants now allows the labeling of endogenous RNAs and their imaging without modifying their behavior.

In this chapter, the improvements observed in the detection of single mRNAs with different variants of MCP and fluorescent proteins that were tested in yeast and mammalian cells are discussed. Moreover, we describe protocols using MS2/MCP systems improved for real-time imaging of single mRNAs and transcription dynamics in S. cerevisiae and mammalian cells, respectively.

Introduction

Cells are the basic unit of life. Within a single cell, networks of molecules control how the environment is sensed through signaling and how cells adapt via metabolic changes and modulation of gene expression.

Quantitative methods are required to model these fundamental processes, which often involve only tens or less molecules [START_REF] Tutucci | Imaging mRNA in vivo, from Birth to Death[END_REF][2][3]. To detect variations in gene expression, one approach is to measure mRNA levels. Even though these are not always a proxy for protein expression, i.e. when a delay exists between mRNA accumulation and protein production [4], mRNA measurements provide information about the rate of transcription, accumulation and decay, revealing modes of gene expression regulation. Bulk mRNA measurements, i.e. northern blots, quantitative PCR and RNA sequencing are informative to measure multiple mRNA species from a single RNA preparation and to perform relative comparisons of mRNA levels in different conditions. However, these approaches, which average millions of cells, have fundamental limitations when it comes to precisely measure RNAs at the level of single cells, or within cellular compartments i.e. nucleus vs cytoplasm, cellular protrusions such as the bud of S. cerevisiae, the leading edge of fibroblast or neuronal dendrites and axons.

To achieve quantitative subcellular mRNA measurements, several approaches based on fluorescence microscopy have been developed over the past decades. In fixed cells, a standard method to visualize and count individual mRNAs is single molecule fluorescent in situ hybridization (smFISH) [START_REF] Femino | Visualization of single RNA transcripts in situ[END_REF]6]. Briefly, this approach allows detecting single endogenous mRNAs by hybridizing tens of fluorescently labelled DNA oligos onto the target molecule. By using sensitive digital cameras and wide-field microscopy, it is possible to detect single mRNAs as diffraction limited spots, allowing their subcellular localization and quantification in thousands of cells. This method can be applied to single isolated cells (i.e. [START_REF] Gandhi | Transcription of functionally related constitutive genes is not coordinated[END_REF][8][START_REF] Senecal | Transcription factors modulate c-Fos transcriptional bursts[END_REF][10][START_REF] Yang | Single molecule fluorescence in situ hybridisation for quantitating post-transcriptional regulation in Drosophila brains[END_REF][12][START_REF] Duncan | A method for detecting single mRNA molecules in Arabidopsis thaliana[END_REF]) as well as to tissues [14][START_REF] Moffitt | High-performance multiplexed fluorescence in situ hybridization in culture and tissue with matrix imprinting and clearing[END_REF][START_REF] Long | Quantitative mRNA imaging throughout the entire Drosophila brain[END_REF]. Several modified smFISH protocols exist, which use DNA probes of different length and complexity (i.e. 20 mer, 50 mer, branched DNA, RNA scope [17,10,8]) or fluorescence amplification systems to detect weak signals (i.e. hybridization chain reaction, HCR [18]). An important difference between these techniques is whether the probes are fluorescently labelled or not, in which case they need to be detected with a secondary fluorescent oligo. A simple, reliable and affordable protocol that uses indirect labeling has recently been published [19]. Furthermore, smFISH can be multiplexed to simultaneously visualize different mRNA species within single cells (up to 10,000; [START_REF] Moffitt | High-performance multiplexed fluorescence in situ hybridization in culture and tissue with matrix imprinting and clearing[END_REF][20][START_REF] Lubeck | Single-cell in situ RNA profiling by sequential hybridization[END_REF][START_REF] Shah | seqFISH Accurately Detects Transcripts in Single Cells and Reveals Robust Spatial Organization in the Hippocampus[END_REF][23][24]), or it can be also combined to protein detection by immuno-fluorescence (see [START_REF] Bayer | Single-Molecule RNA In Situ Hybridization (smFISH) and Immunofluorescence (IF) in the Drosophila Egg Chamber[END_REF][START_REF] Eliscovich | Imaging mRNA and protein interactions within neurons[END_REF] and this issue). These approaches revealed asymmetric RNA distribution within single cells, as well as significant cell-to-cell variability existing in tissue or even isogenic populations [START_REF] Buxbaum | In the right place at the right time: visualizing and understanding mRNA localization[END_REF][START_REF] Weil | Making the message clear: visualizing mRNA localization[END_REF][START_REF] Martin | mRNA localization: gene expression in the spatial dimension[END_REF][START_REF] Lecuyer | Global analysis of mRNA localization reveals a prominent role in organizing cellular architecture and function[END_REF][START_REF] Samacoits | A computational framework to study sub-cellular RNA localization[END_REF][START_REF] Das | The travels of mRNAs in neurons: do they know where they are going?[END_REF]. Furthermore, by using smFISH, other aspects of gene expression have been characterized, such as the 'bursty' nature of transcription [START_REF] Halpern | Bursty gene expression in the intact mammalian liver[END_REF][START_REF] Raj | Stochastic mRNA synthesis in mammalian cells[END_REF][START_REF] Senecal | Transcription factors modulate c-Fos transcriptional bursts[END_REF]8], the mechanisms controlling mRNA export form the nucleus to the cytoplasm [START_REF] Paul | Altered RNA processing and export lead to retention of mRNAs near transcription sites and nuclear pore complexes or within the nucleolus[END_REF][START_REF] Halpern | Nuclear Retention of mRNA in Mammalian Tissues[END_REF] as well as the control of mRNA degradation [START_REF] Trcek | Single-Molecule mRNA Decay Measurements Reveal Promoter-Regulated mRNA Stability in Yeast[END_REF][START_REF] Trcek | Temporal and spatial characterization of nonsensemediated mRNA decay[END_REF]. For more in-depth reviews see [START_REF] Pichon | A Growing Toolbox to Image Gene Expression in Single Cells: Sensitive Approaches for Demanding Challenges[END_REF][START_REF] Tutucci | Imaging mRNA in vivo, from Birth to Death[END_REF]2]; see also Chapter 3 by Bleckmann et al. and Chapter 4 by Tutucci and Singer. However, fixed cells provide limited information about highly dynamic and rare events controlling mRNA metabolism. To follow mRNAs in living cells, several labeling strategies have been developed over the past decades. The best characterized system is a genetically encoded reporter based on the multimerization of RNA stem-loops derived from the bacteriophage MS2 [START_REF] Bertrand | Localization of ASH1 mRNA particles in living yeast[END_REF][START_REF] Peabody | The RNA binding site of bacteriophage MS2 coat protein[END_REF][START_REF] Fusco | Imaging of single mRNAs in the cytoplasm of living cells[END_REF]. To visualize single mRNAs in living cells, 24 MS2 stem-loops are used to tag an RNA of interest, which is then detected by co-expression of a specific RNA-binding protein, the MS2 coat protein (MCP), fused to fluorescent proteins (FP, i.e. eGFP, mCherry, tdTomato, HALO, photoactivatable proteins) [START_REF] Pichon | A Growing Toolbox to Image Gene Expression in Single Cells: Sensitive Approaches for Demanding Challenges[END_REF][START_REF] Tutucci | Imaging mRNA in vivo, from Birth to Death[END_REF]. Several MS2 array variants are available, and the recommendation for using a particular one depends on the model organism, the mRNA, and the step of the mRNA life cycle under investigation [START_REF] Tutucci | Imaging mRNA in vivo, from Birth to Death[END_REF][START_REF] Tutucci | Single-mRNA detection in living S. cerevisiae using a reengineered MS2 system[END_REF]. High-affinity MS2-MCP variants have been successfully used to measure dynamics of mRNA transcription and splicing [START_REF] Brody | The in vivo kinetics of RNA polymerase II elongation during co-transcriptional splicing[END_REF][START_REF] Darzacq | In vivo dynamics of RNA polymerase II transcription[END_REF][START_REF] Schmidt | Real-time imaging of cotranscriptional splicing reveals a kinetic model that reduces noise: implications for alternative splicing regulation[END_REF], export [START_REF] Grunwald | In vivo imaging of labelled endogenous beta-actin mRNA during nucleocytoplasmic transport[END_REF][START_REF] Smith | In vivo single-particle imaging of nuclear mRNA export in budding yeast demonstrates an essential role for Mex67p[END_REF], localization [START_REF] Lionnet | A transgenic mouse for in vivo detection of endogenous labeled mRNA[END_REF][START_REF] Park | Visualization of dynamics of single endogenous mRNA labeled in live mouse[END_REF], and translation in mammalian cells [START_REF] Halstead | Translation. An RNA biosensor for imaging the first round of translation from single cells to living animals[END_REF][START_REF] Katz | Mapping translation 'hot-spots' in live cells by tracking single molecules of mRNA and ribosomes[END_REF][START_REF] Morisaki | Real-time quantification of single RNA translation dynamics in living cells[END_REF][START_REF] Pichon | Visualization of single endogenous polysomes reveals the dynamics of translation in live human cells[END_REF][START_REF] Wu | Translation dynamics of single mRNAs in live cells and neurons[END_REF]. They are also recommended if the experimental set-up involves FRAP. However, several groups reported that the high affinity MS2-MCP variants are not optimal to visualize mRNAs in rapidly dividing organisms such as S. cerevisiae [START_REF] Garcia | MS2 coat proteins bound to yeast mRNAs block 5' to 3' degradation and trap mRNA decay products: implications for the localization of mRNAs by MS2-MCP system[END_REF][START_REF] Haimovich | Use of the MS2 aptamer and coat protein for RNA localization in yeast: A response to "MS2 coat proteins bound to yeast mRNAs block 5' to 3' degradation and trap mRNA decay products: implications for the localization of mRNAs by MS2-MCP system[END_REF][START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF][START_REF] Heinrich | Stem-loop RNA labeling can affect nuclear and cytoplasmic mRNA processing[END_REF]. For this reason, we recently generated an improved MS2 array (MS2 binding sites V6, MBSV6) with decreased affinity for MCP. This allowed measuring the half-life of rapidly decaying mRNAs while preserving single molecule mRNA detection in living cells [START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF][START_REF] Tutucci | Single-mRNA detection in living S. cerevisiae using a reengineered MS2 system[END_REF]. Low affinity MS2-MCP variants have also been used to tag mRNAs in mammalian cells, specifically to generate arrays containing up to 128 MS2 stem-loops in a single transcript, with the aim of monitoring transcription with high temporal resolution for long periods of time and with minimal photo-bleaching [START_REF] Tantale | A single-molecule view of transcription reveals convoys of RNA polymerases and multi-scale bursting[END_REF].

To visualize more than one mRNA species at the time, in single living cells, several orthogonal systems are available. Another system was generated by multimerizing RNA stem-loops derived from the bacteriophage PP7, detected by the cognate protein PP7 coat protein (PCP) fused to fluorescent proteins [START_REF] Chao | Structural basis for the coevolution of a viral RNA-protein complex[END_REF]. This reporter has been used to study transcription dynamics [START_REF] Larson | Real-time observation of transcription initiation and elongation on an endogenous yeast gene[END_REF], mRNA export [START_REF] Saroufim | The nuclear basket mediates perinuclear mRNA scanning in budding yeast[END_REF] and, translation [START_REF] Wang | Real-Time Imaging of Translation on Single mRNA Transcripts in Live Cells[END_REF][START_REF] Yan | Dynamics of Translation of Single mRNA Molecules In Vivo[END_REF]. It has also been used to create an homozygous mice where the immediate early gene Arc was endogenously tagged with 24 PP7 loops, allowing to visualize its response to synaptic activity [START_REF] Das | A transgenic mouse for imaging activity-dependent dynamics of endogenous Arc mRNA in live neurons[END_REF].

Alternative genetically encoded RNA labelling strategies use arrays generated from other RNA sequences, such as the BglG stem-loop [START_REF] Chen | High efficiency of HIV-1 genomic RNA packaging and heterozygote formation revealed by single virion analysis[END_REF], the  BoxB RNA [START_REF] Lange | Simultaneous transport of different localized mRNA species revealed by live-cell imaging[END_REF] and the U1A loop [START_REF] Brodsky | Identifying proteins that affect mRNA localization in living cells[END_REF][START_REF] Takizawa | The myosin motor, Myo4p, binds Ash1 mRNA via the adapter protein, She3p[END_REF]. Other RNA labelling strategies are reviewed elsewhere [START_REF] Pichon | Visualization of single endogenous polysomes reveals the dynamics of translation in live human cells[END_REF][START_REF] Tutucci | Imaging mRNA in vivo, from Birth to Death[END_REF][START_REF] Tutucci | Single-mRNA detection in living S. cerevisiae using a reengineered MS2 system[END_REF].

The optimization of single mRNA visualization in living cells relies also on the expansion of the FP and fluorophores palette [START_REF] Rodriguez | The Growing and Glowing Toolbox of Fluorescent and Photoactive Proteins[END_REF][START_REF] Specht | A Critical and Comparative Review of Fluorescent Tools for Live-Cell Imaging[END_REF]. In a recent publication, the brightness, photostability, pH resistance and monomeric properties of more than 40 FP have been systematically quantified [START_REF] Cranfill | Quantitative assessment of fluorescent proteins[END_REF]. These measurements are valuable to design mRNA imaging reporters with precise photochromatic properties. Furthermore, the development of fluorescent dyes partly bypasses some of the common problems encountered using fluorescent proteins, i.e. wide excitation and emission spectrum, sensitivity to photobleaching, tendency to multimerization, since these molecules generally have small size, high brightness and photostability and narrow spectrum.

In the following paragraphs we report our tests aimed at optimizing the visualization of single mRNAs in living eukaryotic cells. We tested several MCP variants in the model organism S. cerevisiae as well as multiple green FP fused to MCP both in S. cerevisiae and mammalian cell lines. These comparisons revealed that for efficient mRNA detection, MCP variants with high affinity for the RNA reporter remain the best option. In addition, we found that the green FP Envy shows improved brightness compared to other GFP variants, both in yeast and mammalian cells.

mRNA detection using different MCP variants

We previously demonstrated that it is possible to efficiently detect single mRNAs in S. cerevisiae by using the latest MS2 variant, MBSV6, in combination with the expression of MCP-NLS-2xyeGFP [START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF][START_REF] Tutucci | Single-mRNA detection in living S. cerevisiae using a reengineered MS2 system[END_REF] (Figure 1 a-d). To improve the long-term detection of single mRNAs in living cells (i.e. brightness and photostability) we generated other MCP constructs that we compared to MCP-NLS-2xyeGFP for their brightness and propensity to form aggregates. Even though we find that the MCP-NLS-2xyeGFP performs better than other constructs tested thus far, here we report the advantages and disadvantages of other tested reporters.

To improve the brightness of single mRNAs, we generated an MCP variant fused to 3xyeGFP. This plasmid was transformed in the yeast strain expressing MDN1 tagged with 24xMBSV6 and Nup49-tdTomato (Figure 1e-f). We compared the brightness of MDN1 mRNAs detected either with MCP-2xyeGFP or MCP-3xyeGFP (Figure 1g). As expected, mRNAs labelled with MCP-3xyeGFP shows a 25% increase in brightness compared to mRNAs labelled with MCP-2xyeGFP (310950± 155349 a.u. and 263373±109890 a.u., respectively). In addition, the number of MDN1 mRNAs per cell counted with the MCP-3xyeGFP reporter is similar to the mRNAs counted with MCP-2xyeGFP or by smFISH (mean ± S.D. 9.2±6.1 mRNAs/cell, Figure 1h). However, we found that the MCP-3xyeGFP reporter has the tendency to induce cytoplasmic aggregates, likely due to the propensity of GFP to multimerize (Figure 1f, orange arrowheads). These aggregates are similar to the ones that we previously described [START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF] and that can lead to artifactual conclusions about mRNA localization in yeast. It may still be possible to use the MCP-3xyeGFP reporter for mRNAs that are less abundant than MDN1, but we recommend always comparing the live imaging results to smFISH to avoid false conclusions.

Comparison of FP variants in yeast

Many FP variants now exist, and we tested a few promising ones to see whether this would improve the signal: sfGFP, mNG2, eGFP, muGFP and Envy. sfGFP is the superfolder GFP and muGFP is one of its ultrastable monomeric variants [START_REF] Scott | A Novel Ultra-Stable, Monomeric Green Fluorescent Protein For Direct Volumetric Imaging of Whole Organs Using CLARITY[END_REF]. mNG is a bright and stable variant derived from a Branchiostoma lanceolatum fluorescent protein [START_REF] Feng | Improved split fluorescent proteins for endogenous protein labeling[END_REF], and Envy is an FP that performs particularly well in S. cerevisiae [START_REF] Slubowski | Plasmids for C-terminal tagging in Saccharomyces cerevisiae that contain improved GFP proteins, Envy and Ivy[END_REF]. The cDNA coding for these fluorescent proteins was cloned into pET296, replacing the yeGFP tag, such that they are expressed as monomeric variants lacking NLS (MCP-1xFP). The plasmids were then transformed into yeast cells expressing the DOA1 gene tagged with 24xMBSV6 as described in [START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF].

Cells were then grown, fixed and observed under the microscope (Figure 2 a-c). We found that mRNAs were undetectable with MCP-1xeGFP, MCP-1xsfGFP and MCP-1xmuGFP, while they were nicely visible with the MCP-1xEnvy and MCP-1xmNG2 (with best results for MCP-1xEnvy).

Comparison of low and high affinity MCP variants in yeast

The original MS2 system was designed with high-affinity variants of both the MBS and the MCP [START_REF] Bertrand | Localization of ASH1 mRNA particles in living yeast[END_REF].

Indeed, this MCP variant carried the V29I mutation that enhances binding stability by 5-10 fold [START_REF] Lim | Mutations that increase the affinity of a translational repressor for RNA[END_REF]. Since a too high affinity of the MCP-MBS interaction was found to cause artefacts in yeast [START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF], we tested the effect of reverting the V29I mutation. A plasmid expressing MCP-I29V-2xyeGFP was generated and transformed in yeast cells expressing the DOA1 gene tagged with MBV6. No signals corresponding to single mRNAs could be detected, suggesting that the affinity of the MS2V6/MCP-I29V is probably too low for an efficient mRNA detection. Overall, MBSV6/MCP-V29I appears to have the best affinity compromise to allow for an artefact-free mRNA detection in yeast.

Comparison of FP variants in mammalian cells

MCP-FP variants displayed distinct performance in yeast, and we thus tested how these variants [START_REF] Cormack | FACS-optimized mutants of the green fluorescent protein (GFP)[END_REF][START_REF] Zacharias | Partitioning of lipid-modified monomeric GFPs into membrane microdomains of live cells[END_REF][START_REF] Pedelacq | Engineering and characterization of a superfolder green fluorescent protein[END_REF][START_REF] Slubowski | Plasmids for C-terminal tagging in Saccharomyces cerevisiae that contain improved GFP proteins, Envy and Ivy[END_REF][START_REF] Scott | A Novel Ultra-Stable, Monomeric Green Fluorescent Protein For Direct Volumetric Imaging of Whole Organs Using CLARITY[END_REF][START_REF] Shaner | A bright monomeric green fluorescent protein derived from Branchiostoma lanceolatum[END_REF] performed in mammalian cells (Figure 3 In light of these recent developments, we describe here protocols to visualize and quantify mRNAs labelled with the low affinity MS2 systems that we recently developed both for S. cerevisiae and mammalian cells. By describing the visualization of mRNA in these model organisms, we highlight the general rules and recommendations that can improve live imaging of single mRNAs. While elsewhere we described in depth the steps required to endogenously tag an mRNA with the MS2 reporter in yeast [START_REF] Tutucci | Single-mRNA detection in living S. cerevisiae using a reengineered MS2 system[END_REF][START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF] or in mammalian cells [START_REF] Lionnet | A transgenic mouse for in vivo detection of endogenous labeled mRNA[END_REF], here we focused on the protocols used to perform live imaging and to quantify the number and the brightness of single mRNAs and transcription sites.

For S. cerevisiae we outline a protocol for the visualization of mRNAs labelled with the low-affinity 24xMBSV6 (see paragraphs 2.1 and 3.1). The constitutive and well-characterized mRNA MDN1 is used as a model gene. MDN1 was endogenously tagged with 24xMBSV6 as previously described in [START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF]. Here For mammalian cells, we outline a protocol for the visualization of mRNAs labelled with the low-affinity MS2x64 reporter, which is particularly useful to analyze transcription dynamics (see paragraphs 2.2 and 3.2). To this end, a stable cell lines expressing the HIV-1 MS2x64 reporter gene was created in a HeLa H9 AAVS1-Tat cell line (3.2.1). Next, we describe how to produce lentiviruses encoding the MCP-FP (3.2.2), how to generate a stable cell line expressing the MCP-FP construct (3.2.3), how to prepare the cells (3.2.4) and to perform live image acquisition (3.2.5).

Materials

Materials for visualizing MBSV6-labelled mRNAs in yeast

1. Yeast cells: all strains used in this protocol are derived from the S. cerevisiae background BY4741 MATa; his3Δ1; leu2Δ0; met15Δ0; ura3Δ0.

2. YPD medium: 50 g/L of the YPD mix (i.e. Clonetech). Sterilize by autoclaving.

3. LEU-medium (drop-out media lacking leucine): 6.7 g/L Yeast Nitrogen Base (YNB) with ammonium sulfate, drop-out mix lacking leucine, 20 g/L glucose. Sterilize by autoclaving. 17. Image analysis software: FISH-quant [START_REF] Mueller | FISH-quant: automatic counting of transcripts in 3D FISH images[END_REF], free software developed in the MATLAB programming language (MathWorks). Download the FISH-quant package (http://code.google.com/p/fish-quant/) together with the MCRInstaller, which allows one to run a MATLAB algorithm without separately installing MATLAB onto the computer. 18. Image analysis software: Fiji (Java software for image-processing analysis; freely available at https://fiji.sc/)

Methods

Visualizing MBSV6-labelled mRNAs in yeast

Yeast transformation

1. Grow yeast expressing the mRNA of interest tagged with 24xMBSV6 in 5 ml of YPD at 26 °C until an OD 600 of 0.6-0.8 is reached. Do not use cells grown over OD 600 >1. The transformation efficiency will be significantly reduced. The method to tag the mRNA with the MS2 system and to verify that the expression of the mRNA is not affected is described in more details in [START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF][START_REF] Tutucci | Single-mRNA detection in living S. cerevisiae using a reengineered MS2 system[END_REF] (see

Note 7).

2. Centrifuge the cells for 3 minutes at 7000g. Discard the supernatant and resuspend the cells in 5 ml of Lithium-TE. 12. Incubate at 26 °C for 3-4 days. The transformed cells can be used to start cultures for live imaging (see Note 8).

Growing yeast cells for live imaging

1. Grow a low-density culture of the yeast strain expressing the endogenously tagged mRNA and MCP-GFP in selective LEU-medium overnight at 26 °C. Apply constant shaking at 210 rpm. It is important to keep the cells growing in exponential phase (OD 600 <1) at all times.

2. In the morning, dilute the cells with fresh medium to an OD 600 ~0.1 and allow to grow until OD 600 0.2-0.3. At this concentration, the auto-fluorescence of the cells is minimal and the expression of the MCP is homogenous (see Note 9).

Coating of Delta-T dishes and plating of cells

1. Incubate the Delta-T dishes with 400 µl of ConA at a final concentration 1 mg/ml for 10 minutes at room temperature.

2. Aspirate the excess and let the dish air dry completely.

3. Activate the ConA coating, by incubating the dish with 400 µl of ConA activating solution, for 10 minutes at room temperature (see Note 10).

4. Aspirate the excess and let the dish air dry completely.

5. Wash the dish twice with sterile DDW and let air dry completely.

6. Plate 500 µl of cells at OD 600 0.2-0.3 (see Note 11).

7. Place the dish on the microscope stage and let the cells attach for at least 15-30 minutes. Using the Delta-T temperature control system, allow the temperature to stabilize at 26 °C. It is important to wait until cells attach and restart the cell cycle.

Live imaging acquisition

The live imaging conditions need to be adapted based on the expression of the mRNA under investigation. We recommend testing the mRNA of interest first by smFISH. Here, we outline the imaging conditions for the constitutive MDN1 mRNA. We previously characterized the expression of this mRNA by smFISH [START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF]8] and this protocol is detailed in Chapter 4 of this book as well as in [START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF][START_REF] Tutucci | Single-mRNA detection in living S. cerevisiae using a reengineered MS2 system[END_REF]. Figure 1 a-b shows an example of smFISH for the MDN1 mRNA in wild-type cells, for side by side comparison with the live imaging results shown in Figure 1c-f. For live imaging, the MDN1 mRNA was tagged at the 3' UTR with 24xMBSV6 (Figure 1c) [START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF]. To distinguish the nucleus from the cytoplasm, the nuclear pore protein Nup49 is endogenously fused to the red fluorescent protein tdTomato (YET443 MATa; his3Δ1; leu2Δ0; met15Δ0; ura3Δ0 NUP49::NUP49-tdTomato KAN-CRE recombined MDN1:: MDN1 3UTR 24MS2V6 KAN-CRE recombined; Ycp-Lac111 CYC1p MCP-NLS-2xyeGFP).

1. For short-term imaging and fast acquisition to follow mRNAs with high temporal resolution, stream one single Z-plane, 50 ms exposure. Detection of single mRNAs tagged with MCP-GFP is achieved by using 10% of 100 mW 491 laser (~1-2 mW/cm 2 measured at the objective). To detect Nup49-tdTomato, use 1% of 50 mW 561 laser (~0.5 mW/cm 2 measured at the objective). Under these conditions the mRNAs in the cytoplasm and the transcription sites in the nucleus can be visualized for 2-3 minutes before significant photobleaching occurs (see Video 1).

2. For long-term imaging i.e. over the course of a complete cell cycle, and to cover the whole cell width, take 15 Z-stacks every 0.5 m every two minutes (~90 Z-stacks total). An exposure of 50 ms for each Z-plane was used under our conditions. To visualize MBSV6-MCP-NLS-2xyeGFP labelled mRNAs use a 491 nm wavelength laser. For visualization of single mRNA molecules, set the laser to 10% power (~1-2 mW/cm 2 measured at the objective). To detect Nup49-tdTomato, use 1% of 50 mW 561 laser (~0.5 mW/cm 2 measured at the objective). Acquire Z-planes at different stage positions and use them to detect the number, the position and the brightness of mRNAs in living cells (Figure 1d and see Note 12).

Imaging analysis

1. To improve signal-to-noise ratio, restore the images using a deconvolution software such as the Huygens software package. Automatically compute the theoretical point spread function based on your microscope settings. Restore the images using the Classic Maximum Likelihood Estimation algorithm (i.e. number of iterations = 99; signal/Noise ratio = 15).

2. To measure the number, position and the brightness of the mRNAs in single cells, use the freely available software FISHquant running on Matlab [START_REF] Mueller | FISH-quant: automatic counting of transcripts in 3D FISH images[END_REF]. Deconvolved images can be analyzed with FISHquant without further filtering. Cell outlines can be created using FISHquant or using the freely available software CellProfiler [83]. Recent versions of FISHquant, have a built-in plugin converting CellProfiler outlines for FISHquant analysis. 3D gaussian fitting of the single spots generates a text file containing the x,y position, brightness and number of spots identified in each cell. In our experiments, the counting of MDN1 mRNA molecules per cell detected with MCP-2xyeGFP revealed a mean ± S.D. = 10.8±6.1 mRNAs/cell. Counting the MDN1 mRNAs per cell by smFISH gave similar results, i.e. a mean ± S.D. = 9.5±4.4 mRNAs/cell (Figure 1h). These results show that live imaging faithfully reports on MDN1 mRNA expression.

Visualizing MS2-labelled RNAs in mammalian cells, with a focus on analyzing transcription dynamics

Lentiviral-production of MCP-GFP

1. Seed HEK-293T cells at 4x10 6 cells in 100 mm tissue culture plate and incubate cells for 24 hours.

2. Change HEK-293T cells into 5 ml of fresh medium without antibiotics.

XL1-blue competent cells are bacteria of choice for transformation of plasmid containing MS2

repeats. Note that the bacteria can be grown at 30°C if the plasmid is unstable.

3.

Other Flp-in mammalian cell lines can be used.

4.

A stable Flag-Tat expressing cell line was created by CRISPR genome editing using an AAVS1

repair vector [START_REF] Tantale | A single-molecule view of transcription reveals convoys of RNA polymerases and multi-scale bursting[END_REF] in HeLa Flp-in H9 cell line (available upon request). Individual clones were picked and analyzed by immuno-fluorescence with an anti-Flag antibody. One clone was further characterized and used for following experiments. Isogenic stable cell lines expressing the HIV-1

MS2x64 reporter gene were created using the Flp-in system in a HeLa H9 AAVS1-Tat cell line [START_REF] Tantale | A single-molecule view of transcription reveals convoys of RNA polymerases and multi-scale bursting[END_REF]. Flp-in integrants were selected on hygromycin (150 µg ml -1

). The MS2 tag is located in the intron of the reporter and thus label only the pre-mRNA. Note that splicing of this reporter occurs post-transcriptionally and is not disturbed by the MS2 repeat [START_REF] Tantale | A single-molecule view of transcription reveals convoys of RNA polymerases and multi-scale bursting[END_REF]. We found that this set-up was the most appropriate to visualize transcription because it allows the use of large tags without compromising the mRNA fate. Individual clones were picked and analyzed by in situ hybridization. One clone was further characterized and used for following experiments.

5.

A ZEISS Axioimager Z1 wide-field microscope equipped with a Plan Apochromat 63× objective, NA 1.4 oil-immersion objective (ZEISS) was used with a ZEISS VSG HBO 100/001.26E

illuminating System, and Zyla 4.2 sCMOS Camera (2048 x 2048 pixels; 6.5 µm-pixel size, from Andor). We acquire data using 21 optical sections with a z-step size of 0.3 μm. MetaMorph (Molecular Devices) software is used for instrument control as well as image acquisition.

6. An inverted OMX Deltavision microscope in time-lapse mode with temperature-controlled chamber with CO2, together with a x100, NA 1.4 objective and EMCCD cameras Evolve 512 x 512 was used for live-cell imaging. Spinning disk confocal or HiLo microscopes equipped with 60x or 100 x objectives with a NA >1.3 are also suitable.

7.

To maximize the brightness of the tagged mRNAs, we recommend tagging the mRNA of interest with 24xMBSV6. However, as we discussed in previous publications [START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF][START_REF] Tutucci | Single-mRNA detection in living S. cerevisiae using a reengineered MS2 system[END_REF], for mRNAs that are strongly expressed and that have a short half-life (i.e. GAL1 mRNA), tagging with 24 stem loops can be suboptimal. This is because when many mRNAs have to be rapidly degraded, the presence of 24 MS2 loops, even if they are low-affinity variant, can cause a delay in the degradation of the MS2 array. In this case the mRNAs can be tagged with 12xMBSV6. We recommend always testing whether the insertion of the MS2 loops affects the stability, the localization and the expression of the mRNA of interest by comparing the tagged mRNA (with or without the expression of MCP) to the untagged mRNA by smFISH [START_REF] Tutucci | An improved MS2 system for accurate reporting of the mRNA life cycle[END_REF][START_REF] Tutucci | Single-mRNA detection in living S. cerevisiae using a reengineered MS2 system[END_REF].

8.

Transformed cells can be frozen at this stage. Glycerol stocks can be prepared by mixing 1 ml of exponentially growing culture with 1 ml of 60% glycerol in YPD. Mix thoroughly and freeze in cryo-tubes at -80°C. We did not notice a reduction in live imaging quality if the cells are thawed instead of using fresh transformations.

9.

Background strains that are ADE2+, thus not producing the red pigment accumulating in mutant cells, show reduced cellular background during live imaging.

10.

For short term imaging, not requiring strict temperature control, other types of glass bottom dishes can be used i.e. MatTek.

11.

High quality and reproducible imaging is achieved only if the cells are imaged while growing exponentially.

12. The parameters for imaging single mRNAs in living cells (laser power, exposure time) can be modified to increase the fluorescence intensity of single mRNAs. Choose parameters that will keep the fluorescence intensity signal in the dynamic range of the camera while minimizing photo-bleaching of the sample.

13.

At this point active lentivirus is being produced so strict adherence to biosafety class II is necessary. All materials in contact with virus containing liquid must be bleach treated prior to disposal.

14.

Virus may be stored at 4°C for short periods (hours to days). For long-term storage, it should be frozen at -80°C. 17. We recommend a stable integration of the MCP-GFP with viral infection followed by sorting a pool of low expressing MCP-FP cells. This gives much better results than a crude transient transfection. Optimization of the MCP-GFP expression further improves the signal-to-noise ratio of single molecule detection and enhances the quality of the data recorded.

18.

We recommend stacks of 11 planes with a Z-spacing of 0.6 µm. This size allows accurate quantification of single mRNA molecules when using 100x, NA 1.4 objectives.

19. These short and long movies are important if one wants to capture the entire promoter dynamics (see [START_REF] Tantale | A single-molecule view of transcription reveals convoys of RNA polymerases and multi-scale bursting[END_REF]), and they are required since at high temporal resolution, bleaching limits the acquisitions to 30 minutes.

20.

After being processed, each movie should be checked to ensure that no drift or loss of the focal plan appeared during the microscope acquisition (especially for long time acquisition movies). To measure the intensity of single RNA molecules, a straight line with a width of 2 pixels was used, and the maximal value of the intensity plot was recorded. This value was divided over the mean grey value measured in the nucleus of the cell following the application of a Gaussian blur of 3 pixels (maximum intensity projected images). The background value (outside the cell nucleus) was subtracted from each of these values (total of 133 single molecules quantified per construct).

Legends

Table 1

List of the FP variants

NLS

3 ,

 3 , Table1). Transient transfections of the different NLS-MCP-FP variants construct were evaluated in a HeLa H9 cell line without RNA reporter gene (Figure 3, top panels). A diffuse fluorescent signal was seen in the nucleus, indicating a good solubility of the all NLS-MCP-FP variants constructs. When the various NLS-MCP-FP were expressed in a derivative of this cell line expressing the HIV-1 MS2x64 RNA reporter (Figure 4; Figure 3, middle panels), the NLS-MCP-GFPEnvy showed the highest contrast (see signal quantifications in Figure 5), followed by the NLS-MCP-eGFP and NLS-MCP-sfGFP. Single RNA molecules were only barely detectable with the NLS-MCP-muGFP and NLS-MCP-mNG construct, while single RNA molecules were observed by smFISH (Figure bottom panels).

  we detail how to transform yeast cells with the constructs expressing MCP-FP (3.1.1), grow cells for live imaging (3.1.2), prepare the coated dishes (3.1.3), perform live imaging (3.1.4) and count single mRNAs in living cells (3.1.5).

4 .

 4 LEU-plates (drop-out agar plates): 6.7 g/L Yeast Nitrogen Base (YNB) with ammonium sulfate, drop-out mix lacking leucine, 20 g/L glucose, 20 g/L of bacteriological agar. Sterilize by autoclaving.5. 100% glycerol stock: sterilize by autoclaving. Store at room temperature, protected from light. 6. Plasmids: pET264-pUC 24xMS2V6 Loxp KANr Loxp (Addgene ID:104393), pET251-pUC 12xMS2V6 Loxp KANr Loxp (Addgene ID:104392); pET296-YcpLac111 CYC1p-MCP-NLS-2xyeGFP (yeast optimized eGFP) (Addgene ID:104394); pET511-YcpLac111 CYC1p-MCP-NLS-3xyeGFP; pET518-YcpLac11-CYC1p-MCP-1x-eGFP; pET519-YcpLac11-CYC1p-MCP-1xEnvy; pET521-YcpLac11-CYC1p-MCP-1xmuGFP; pET522-YcpLac11-CYC1p-MCP-1xmNG; pET523-YcpLac11-CYC1p-MCP-1xsfGFP (available upon request). 7. Lithium-TE: 100 mM LiAc, 10 mM Tris-HCl, pH 7.5, 1 mM EDTA. Sterilize by autoclaving. 8. Lithium-TE-PEG: 100 mM LiAc, 10 mM Tris, pH 7.5, 1 mM EDTA, 50% PEG 3350-4000. Sterilize by autoclaving. 9. Salmon sperm DNA (ssDNA): 10 mg/ml lyophilized, sheared, organically extracted and denatured ssDNA is resuspended in double distilled water (DDW). Store 100 µl aliquots at -20°C. 10. Centrifuges (table-top): up to 20,000g for samples 1.5 ml. 11. Heat blocks at 42°C and 95°C. 12. Temperature controlled shaker for yeast cultures. 13. Temperature controlled, Delta-T dishes (i.e. Bioptech Cat# 04200417C). 14. Concanavalin A (ConA) stock: 10 mg/ml in sterile DDW (10x stock). Store 500 µl aliquots at -20°C. 15. ConA-coated plates activation solution: 50 mM CaCl 2 , 50 mM MnCl 2 in DDW. Filter sterilize, store at room temperature. 16. Fluorescent widefield microscope of choice for live-cell image acquisition (see Note 1).

3 . 7 .

 37 Centrifuge for 3 minutes at 1000g and resuspend in 150 µl of Lithium-TE. 4. Put 450 µl of Lithium-TE-PEG in an Eppendorf tube. 5. Add 5 µl of 10 mg/ml ssDNA denatured at 95 °C for 10 minutes. 6. Add 500 ng-1 µg of plasmid expressing MCP-GFP (i.e. YcpLac111 MCP-NLS-2xyeGFP) for each transformation. Add 150 µl of cells to the tube and mix by gentle vortexing (speed 5 out of 10). 8. Incubate at room temperature for 30 minutes. 9. Heat shock the cells at 42 °C for 15 minutes. 10. Centrifuge the cells for 3 minutes at 7000g. 11. Discard the supernatant and resuspend the pellet in 100 µl of DDW and plate the entire transformation on selective LEU-plates.

3 .

 3 Transfect HEK-293T cells with JetPRIME (Polyplus transfection; according to the supplier's recommendations) by preparing a mix of 10 µg DNA total with 8 µg of pHAGE-Ubc-NLS-MCP-GFP, 0,4 µg of pHDM-Hgpm2, 0,4 µg of pHDM-tat1b, 0,4 µg of pRC-CMV-rev1b, 0,8 µg of pHDM-G, and 500 µl jetPRIME buffer.

4 . Notes 1 .

 41 Yeast microscopy experiments were performed on a home-built microscope built around an IX71 stand (Olympus). For excitation, a 491nm laser (CalypsoTM, Cobolt) and a 561nm laser (JiveTM, Cobolt) were combined and controlled by an acoustic-optic tunable filter (AOTF, AOTFnC-400.650-TN, AA Opto-electronic) before being coupled into a single mode optical fiber (Qioptiq). The output of the fiber was collimated and delivered through the back port of the microscope and reflected into an Olympus 150x 1.45 N.A. Oil immersion objective lens with a dichroic mirror (zt405/488/561rpc, 2mm substrate, Chroma). The tube lens (180mm focal length) was removed from the microscope and placed outside of the right port. A triple band notch emission filter (zet405/488/561m) was used to filter the scattered laser light. A dichroic mirror (T560LPXR, 3mm substrate, Chroma) was used to split the fluorescence onto two precisely aligned EMCCDs (Andor iXon, Model DU-897U-CS0, pixel size 16 µm) mounted on alignment stages (x, y, z, θand φangle). Emission filters FF03-525/50-25 and FF01-607/70-25 (Semrock) were placed in front of green and red channel cameras respectively. The two cameras were triggered for exposure with a TTL pulse generated on a DAQ board (Measurement Computing). The microscope was equipped with a piezo stage (ASI) for fast z-stack and a Delta-T incubation system (Bioptech) for live cell imaging. The microscope (AOTF, DAQ, Stage and Cameras) was automated with the software Metamorph (Molecular Devices).

15 . 16 .

 1516 Polybrene increases the efficiency of retrovirus-mediated gene transfer; this should be optimized for each cell type. Expression levels of the MCP-FP transgene should show a strong spot for the transcriptional start site (TSS), smaller dots throughout the nucleus (single-molecule mRNA) and a low nuclear background of free unbound MCP-FP. If needed, the best pool can be re-sorted by FACS to obtain a more homogenous population. Positive cells can be FACS-sorted as one cell per well in a 96-well plate to create pure clones. Select the most suitable sub-cloned cells display the optimal signal-to-noise ratio for the live-cell imaging acquisition.

Video 1 MDN1

 1 mRNAs tagged with 24xMBSV6 in the 3'UTR and labelled with MCP-NLS-2xyeGFP are shown in gray, Nup49-tdTomato is shown in red. One single Z-plane was streamed at 100 ms intervals for a total of 100 frames. The white arrowhead indicates a transcription site in the nucleus, brighter than single spots in the cytoplasm. The blue arrowheads indicate single mRNAs in the cytoplasm. Scale bar 3 µm.

Figure 1 MDN1

 1 Figure 1 MDN1 mRNA detection using the MS2-MCP system in S. cerevisiae. (a) Schematic representation of the MDN1 locus. smFISH probes labeled with Q760 anneal all along the CDS. Probes list is provided in [58]. (b) Overlap of the DAPI signal in the nucleus (blue), smFISH for the MDN1 CDS (green) with the

Figure 2 Comparison

 2 Figure 2 Comparison of MCP-FP variants in yeast. Images are from yeast cells expressing the DOA1 gene tagged with 24xMBSV6 transformed with (a) MCP-NLS-2xyeGFP plasmid, (b) MCP-NLS-GFPEnvy plasmid (c) MCP-NLS-eGFP plasmid. Panels are maximum intensity projections of a Z-stacks. Scale Bar 5 µm.

Figure 3 Comparison

 3 Figure 3 Comparison MCP-FP variants in mammalian cells. Images are maximum intensity projections from HeLa Flp-in cells transfected with the indicated NLS-MCP-FP variants constructs. Top panels: images are from cells expressing the indicated NLS-MCP-FP variants without RNA reporter (parental HeLa H9 AAVS1-Tat cell line). Middle panels: images are from cells expressing the indicated NLS-MCP-FP variant in HeLa H9 AAVS1-Tat cell line expressing the HIV-1 MS2x64 reporter. Bottom panels: images are smFISH signals obtained with probes against the MS2 tag. Scale bar 5 µm. Contrast adjustment is identical for all images of the top and middle panels.

Figure 4 Schematic

 4 Figure 4 Schematic representation of pIntro-MS2x64 RNA reporter construct. (a) Generation of the MS2x64 RNA tag. The MS2x32 stem-loop sequence with low affinity binding to MCP is multimerized to generate repeats of 64 and 128 stem-loops. (b) Schematic of the pIntro-MS2x64 reporter construct. The striped box represents the MS2 repeat, the grey and green spot, MCP-GFP; the

Figure 5 Signal

 5 Figure 5 Signal-to-noise ratio (SNR) of single RNA molecules detection with MCP-FP variants. The bar plots show the mean values of the SNR of single RNA molecules (error bar: standard deviation). NS: nonsignificant detection of single RNAs occurred with the NLS-MCP-muGFP and NLS-MCP-mNG constructs.

  Figure 1

  Figure 4

  For each cell, determine the correct focal plane, as well as the boundaries of the Z-stacks. (Note 18)For transcriptional studies, two types of movies can be recorded: short movies (Fast acquisition) where a Zstack is recorded every 3 seconds for 30 minutes, long movies where a Z-stack is recorded every 3 minutes for >8 hours (see Note19). Analyze the time-lapse movies of transcription sites with dedicated software tools that are available upon request (MS2-quant, RampFinder, RampFitter, ON-quant, see[START_REF] Tantale | A single-molecule view of transcription reveals convoys of RNA polymerases and multi-scale bursting[END_REF])

	6. Harvest medium containing lentivirus and filter it through a 0.45 µm filter into a sterile falcon tube
	and keep it at 4 °C.
	7. Add 5 ml of fresh medium without antibiotics to the packaging cells and incubate at 37 °C for 24
	hours for a new round of lentivirus production.
	8. Repeat steps 6 and 7 once. After the harvest of virus, discard the HEK-293T cells with 10%
	bleach.
	9. Pool the viral harvests of this and the previous days.
	10. To concentrate lentiviral stocks, add Lenti-X Concentrator (Clontech) to viral harvest and incubate
	at 4 °C for 30 minutes to overnight.

4. Mix by vortexing, add 20 µl of JetPRIME, vortex for 5 seconds, spin down briefly, and incubate for 10 minutes at room temperature. 5. Add the transfection mix dropwise onto the HEK-293T cells. Gently rock the plate back and forth and incubate at 37 °C for 24 hours. (see Note 13). 11. Centrifuge the mixture at low speed (1,500x g for 45 minutes at 4 °C) and discard the supernatant. (Note 20).

Acknowledgments

This work was supported by NIH Grant GM57071 to R.H.S, and by an ANRS Grant from to E.B. E.T was supported by Swiss National Science Foundation Fellowships P2GEP3_155692 and P300PA_164717. X. P. was supported by a fellowship from the Labex EpiGenMed Montpellier/Université de Montpellier, E.B had a travel grant from the Philippe Foundation.


Andrea Attar
email: andrea.attar@tse-fr.eu.

Thomas Mariotti
email: thomas.mariotti@tse-fr.eu.

François Salanié
email: francois.salanie@tse-fr.eu.

The Social Costs of Side Trading *

Keywords: Adverse Selection, Side Trading, Second-Best Allocations. JEL Classification: D43, D82, D86

We study resource allocation under private information when the planner cannot prevent bilateral side trading between consumers and firms. Adverse selection and side trading severely restrict feasible trades: each marginal quantity must be fairly priced given the consumer types who purchase it. The resulting social costs are twofold. First, second-best efficiency and robustness to side trading are in general irreconcilable requirements. Second, there actually exists only one budget-feasible allocation robust to side trading, which deprives the planner from any capacity to redistribute resources between different types of consumers. We discuss the relevance of our results for insurance and financial markets.

Introduction

The theory of incentives identifies the holding of private information by economic agents as a fundamental constraint on the allocation of resources [START_REF] Hurwicz | The Design of Mechanisms for Resource Allocation[END_REF]). Standard aggregate resource constraints must accordingly be supplemented by incentive-compatibility constraints that reflect the agents' ability to conceal their private information [START_REF] Myerson | Incentive Compatibility and the Bargaining Problem[END_REF][START_REF] Myerson | Optimal Coordination Mechanisms in Generalized Principal-Agent Problems[END_REF]). The problem of the optimal allocation of resources then reduces to that of characterizing informationally constrained efficient, or second-best, allocations [START_REF] Harris | Resource Allocation Under Asymmetric Information[END_REF]). The key finding is that a tradeoff arises between redistribution and incentives [START_REF] Mirrlees | An Exploration in the Theory of Optimum Income Taxation[END_REF]).

A crucial assumption of theories of the second-best is that, although individual types are unobservable, individual trades can be perfectly monitored by the planning authority.

Because few, if any, economic institutions have the required ability to monitor all individual trades, this calls for an explicit consideration of the role of side trading in the theory of resource allocation under private information, as first pointed out by [START_REF] Hammond | Straightforward Individual Incentive Compatibility in Large Economies[END_REF].

To this end, we consider a general environment in which firms can provide a divisible good to privately informed consumers who may be of two types. Consumers' preferences satisfy a single-crossing condition, and there is adverse selection in that consumers who are more willing to trade are also more costly to serve; private values arise as a limiting case when consumers' types are not payoff-relevant for the firms selling to them. This framework encompasses many applications, including the standard [START_REF] Rothschild | Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information[END_REF] insurance economy as a prominent example.

In this setting, we characterize the allocations that can be achieved by a planner who observes neither consumers' types nor the trades they may conduct with firms. To do so, we refine the standard notion of incentive-feasibility by focusing on allocations that are robust to side trading. This reflects two additional constraints on resource allocation. First, the planner cannot force consumers to trade with him; this is the case, for instance, when consumers can opt out of a publicly provided health-insurance plan, as in the current German system. Second, he cannot prevent them from engaging in mutually advantageous additional trades with a firm. We formalize these constraints by requiring the planner to offer a tariff such that no firm, acting as an entrant, can guarantee itself a positive profit by offering complementary side trades. This approach provides us with a modified criterion of incentive feasibility which is useful for evaluating the social costs of side trading. Focusing on twotype environments affords us a straightforward comparison with standard characterizations of the second-best efficiency frontier in both private-value [START_REF] Bierbrauer | The Pareto-Frontier in a Simple Mirrleesian Model of Income Taxation[END_REF]) and common-value [START_REF] Prescott | Pareto Optima and Competitive Equilibria with Adverse Selection and Moral Hazard[END_REF], [START_REF] Crocker | The Efficiency of Competitive Equilibria in Insurance Markets with Asymmetric Information[END_REF], Bisin and are typically not robust to side trading, so that the planner's inability to monitor consumers' trades has significant welfare implications. Second, only one budget-feasible allocation is robust to side trading, so that the third-best efficiency frontier reduces to a point: the threat of side trading effectively deprives the planner from any capacity to redistribute resources between different types of consumers. The allocation we characterize is thus the natural candidate for a competitive equilibrium, but, being the only feasible one under side trading, little, if anything, can be argued about its desirability.

A distinctive feature of our approach is that we model side trade as bilateral contracts between a consumer and a firm. This reflects our dissatisfaction with the standard way of representing unobservable side trades as transactions on Walrasian markets, which would call for a centralized market institution to ensure that all these trades take place at the same price.

Bilateral trading plays a key role in our analysis. Our key Lemma 1, in particular, shows that budget-feasible allocations robust to side trading have a very peculiar price structure: each marginal quantity, or layer, is priced at the cost of serving the types who purchase it. This form of competitive pricing, reminiscent of [START_REF] Akerlof | The Market for "Lemons": Quality Uncertainty and the Market Mechanism[END_REF], implies that there are no crosssubsidies between these layers, though there may be cross-subsidies between types. When the allocation is interior and separating, linear pricing can emerge only in the private-value limiting case.

Lemma 1 has a simple but important implication that we state in Theorem 1: no secondbest allocation in which only one incentive compatibility-constraint binds is robust to side trading. The reason is that, by the standard efficiency-at-the-top property, consumers for which this constraint binds must trade at the margin at the cost of serving them. As a result, the layer that connects the trades of the two types cannot be priced at the cost of serving the type who is the most willing to trade. But then, by Lemma 1, there always exists some side trade that a firm finds it profitable to conduct with at least one consumer type. For instance, in the [START_REF] Rothschild | Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information[END_REF] insurance economy, any second-best allocation in which the high-risk type's incentive-compatibility constraint binds can be exploited by an entrant offering complementary coverage at a premium rate slightly higher than the high-risk fair premium rate, which this type is willing to trade along with the coverage provided by the allocation for the low-risk type.

Our second main result, Theorem 2, states that, among the allocations that feature no cross-subsidies between layers, only one is robust to side trading, namely, the Pareto-efficient one that maximizes the utility of the consumer type who is the less willing to trade. This uniqueness result stands in stark contrast with the nondegenerate second-best efficiency frontier that emerges from the standard tradeoff between redistribution and incentives. To complete our characterization, we evaluate whether this allocation can be second-best, hence considering the situations not covered by Theorem 1. Theorem 3 shows that a second-best allocation robust to side trading must either feature pooling of the two consumers' types, or each type purchasing her first-best quantity. We argue that these situations can only occur under very special assumptions on preferences and costs. Thus the unique third-best allocation typically does not belong to the second-best efficiency frontier.

Related Literature

While the constraints induced by private information on resource allocation are by now well understood, less is known about the impact of side trading on feasibility and redistribution.

Starting with the early contributions of [START_REF] Hammond | Straightforward Individual Incentive Compatibility in Large Economies[END_REF][START_REF] Hammond | Markets as Constraints: Multilateral Incentive Compatibility in Continuum Economies[END_REF], [START_REF] Allen | Repeated Principal-Agent Relationships with Lending and Borrowing[END_REF], and Jacklin (1987), several authors have attempted to identify the limits to risk sharing generated by consumers' side trading in financial markets. [START_REF] Cole | Efficient Allocations with Hidden Income and Hidden Storage[END_REF], [START_REF] Golosov | Optimal Taxation with Endogenous Insurance Markets[END_REF], [START_REF] Farhi | A Theory of Liquidity and Regulation of Financial Intermediation[END_REF] have analyzed different private-value environments in which the planner is constrained by the existence of Walrasian markets on which privately informed consumers can complement their trades with the planner by trading linearly priced commodities. This raises two sets of questions, which we address in this paper. First, the existence of unobservable transactions is at odds with the idea that all trades take place in a centralized Walrasian market where a single market-clearing price is quoted-the issue is not so much that the quantities traded are unobservable, which is consistent with the functioning of a Walrasian market, but rather that side trades may take place at different prices and that bilateral contracting prevents the planner from monitoring the terms of these trades. Second, if one sticks to the Walrasian paradigm, extending the approach to incorporate common values and adverse selection presents severe conceptual difficulties [START_REF] Prescott | Pareto Optima and Competitive Equilibria with Adverse Selection and Moral Hazard[END_REF], [START_REF] Rustichini | General Equilibrium in Economies with Adverse Selection[END_REF]); a possible way out is to rely on an Akerlof-like (1970) equilibrium, which however only obtains under linear utilities and/or indivisibilities (Attar, Mariotti, and Salanié (2011), Philippon and Skreta (2012), Tirole (2012)). We hence depart from the above literature in two ways. First, we offer an alternative representation of side trading, which we essentially regard as a bilateral rather than a centralized process. Second, we focus on trade under common values; hence, in a bilateral relationship, a firm's profit directly depends on the types of the consumers it trades with, which is arguably a prominent feature of insurance and financial markets.

The constraints induced by private contracting on redistribution between workers who are privately informed of their productivities have also been investigated by [START_REF] Stantcheva | Optimal Income Taxation with Adverse Selection in the Labour Market[END_REF] in the context of optimal taxation. In her framework, however, the State retains full observability of workers' total incomes, as in [START_REF] Mirrlees | An Exploration in the Theory of Optimum Income Taxation[END_REF], which severely limits their ability to engage in side trading.

Our requirement that an allocation must be implementable by an entry-proof tariff to be robust to side trading is in line with the definition by Kahn and Mookherjee (1998) or [START_REF] Bisin | Moral Hazard and Nonexclusive Contracts[END_REF] of third-best allocations in moral-hazard environments. In any such allocation, the planner's tariff must prevent consumers from complementing it with an additional profit-making contract provided by a firm. We extend this notion to privateinformation environments, and, in addition, we put no restriction on the side trades a firm can make available.

The unique third-best allocation that we characterize has been formerly derived by [START_REF] Glosten | Is the Electronic Open Limit Order Book Inevitable?[END_REF] and Attar, Mariotti, and Salanié (2019a) as the outcome of competitive financial markets where market makers are restricted to post collections of limit orders and insiders thus face a convex aggregate tariff. In contrast with these authors, we adopt a fully normative perspective. This requires imposing no restriction on sides trades, which we assume to be fully bilateral. In line with this assumption, we allow firms to react to the planner's tariff by posting arbitrary tariffs. [START_REF] Jaynes | Equilibria in Monopolistically Competitive Insurance Markets[END_REF], [START_REF] Hellwig | A Note on the Specification of Interfirm Communication in Insurance Markets with Adverse Selection[END_REF], and Stiglitz, Yun, and Kosenko (2018) show that this allocation can also be obtained in an equilibrium of a modified [START_REF] Rothschild | Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information[END_REF] insurance economy in which insurance companies can exchange information about their customers. Although the allocation is the same, the logic of our approach is entirely different. First, these authors allow firms to react to the information disclosed by their competitors by possibly enforcing exclusivity clauses, which is at odds with the very notion of side trading that we emphasize. Second, as noticed above, we are interested in the normative implications of side trading and not in characterizing the equilibrium of a given extensive-form game. We return to the issue of the decentralization of the third-best allocation in Section 4.

Our analysis shares with the common-agency literature [START_REF] Martimort | Multi-Contracting Mechanism Design[END_REF]) the idea that what can be implemented by a principal-here, the planner-crucially depends on the trades made available by other principals-here, an entrant. Our normative approach, however, differs from the fully strategic approach adopted, in a similar environment, by Attar, Mariotti, and Salanié (2014). The idea is that when a planner contemplates implementing an allocation by a tariff, he always anticipates that an entrant, acting as a follower, may complement this tariff by providing consumers with further trading opportunities. The requirement that the planner's tariff be entry-proof then captures in a natural way the additional constraints that the possibility of side trades imposes on the planner.

The paper is organized as follows. Section 2 describes the model. Section 3 defines our concept of robustness to side trading, fully characterizes the only feasible allocation satisfying this requirement, and shows that it is typically not second-best. Section 4 discusses the relevance of our results for insurance and financial markets. Proofs not given in the text can be found in the Appendix.

The Economy

Consumers There is a continuum of consumers who can purchase a divisible good in exchange for monetary transfers. Each consumer is privately informed of her type i = 1, 2 and the proportion of type i among consumers is m i > 0. Type i's preferences over quantitytransfer bundles (q, t) ∈ R + × R are represented by a strictly quasiconcave and continuously differentiable utility function u i , with ∂ t u i < 0. Hence her marginal rate of substitution

τ i ≡ - ∂ q u i ∂ t u i
is well defined and strictly decreasing along her indifference curves. We impose the Inada condition that τ i (q, t) vanishes as q grows large along any such curve. Hence, whatever her endowment point, type i's demand at any price p > 0 is finite. The following strict single-crossing condition is the key determinant of consumer demand:

For all q and t, τ 2 (q, t) > τ 1 (q, t).

Thus type 2 is more willing to increase her purchases than type 1.

Firms The supply side of the economy is described by a constant-return-to-scale technology, with unit cost c i > 0 of serving type i. Type 2 is weakly more costly to serve than type 1:

c 2 ≥ c 1 . (2)
Together with (1), (2) typically generates adverse selection, whereas values are private in the limiting case c 1 = c 2 . We let c ≡ m 1 c 1 + m 2 c 2 be the average cost of serving a consumer.

These assumptions hold in the Rothschild and Stiglitz (1976) insurance economy: c i is type i's riskiness, with c 2 > c 1 , q is the amount of coverage she purchases, and t is the premium she pays in return. Our model encompasses many other specifications and is relevant for a broad spectrum of insurance, financial, and labor markets.

Incentive Feasibility and Efficiency A contract is a pair (q, t) for some q ≥ 0, and with unit price t/q if q > 0. An allocation is a pair of contracts, one for each type. An allocation

(q i , t i ) i=1,2 is budget-feasible if m 1 (t 1 -c 1 q 1 ) + m 2 (t 2 -c 2 q 2 ) ≥ 0.
To this aggregate resource constraint, we must add, following [START_REF] Myerson | Incentive Compatibility and the Bargaining Problem[END_REF][START_REF] Myerson | Optimal Coordination Mechanisms in Generalized Principal-Agent Problems[END_REF] and [START_REF] Harris | Resource Allocation Under Asymmetric Information[END_REF], constraints reflecting that the allocation of resources takes place under asymmetric information. An allocation (q i , t i ) i=1,2 is incentive-compatible if u 1 (q 1 , t 1 ) ≥ u 1 (q 2 , t 2 ) and u 2 (q 2 , t 2 ) ≥ u 2 (q 1 , t 1 ).

We denote these constraints by IC 1→2 and IC 2→1 , respectively. An allocation is incentivefeasible if it is budget-feasible and incentive-compatible. A second-best allocation is Paretoefficient among incentive-feasible allocations. This is the relevant notion of efficiency for a planner who perfectly monitors trades, but does not observe consumer types [START_REF] Prescott | Pareto Optima and Competitive Equilibria with Adverse Selection and Moral Hazard[END_REF]). In the [START_REF] Rothschild | Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information[END_REF] insurance economy, the second-best efficiency frontier consists of a continuum of allocations [START_REF] Crocker | The Efficiency of Competitive Equilibria in Insurance Markets with Asymmetric Information[END_REF], [START_REF] Bisin | Efficient Competitive Equilibria with Adverse Selection[END_REF]).

Tariffs A tariff T is a schedule specifying a transfer T (q) to be paid in return for a quantity q, with T (0) = 0 in case a consumer chooses not to trade along the tariff and T (q) = ∞ in case the tariff does not allow consumers to purchase the quantity q. A tariff T implements the allocation (q i , t i ) i=1,2 if For each i, q i ∈ arg max {u i (q, T (q)) : q ≥ 0} and t i = T (q i ).

To ensure that the various maximization problems we will encounter have solutions, we impose the mild requirement that a tariff be lower semicontinuous, with T (q)/q bounded away from 0 as q grows large; this notably holds true if T has a compact domain.

Second-Best Allocations and Side Trading

When side trading is feasible, the planner can no longer monitor trades between consumers and firms. This imposes two additional constraints on resource allocation. First, the planner cannot force consumers to trade with him, reflecting that they can opt out of any mechanism he could propose. To model this constraint, we require that the planner offer a tariff T P , the key restriction being T P (0) = 0. Second, the planner cannot prevent consumers from engaging in mutually advantageous additional trades with a firm. To model this constraint, we require that T P be such that no firm, acting as an entrant, can guarantee itself a positive profit by offering complementary side trades.

Side trades are usually assumed to take place on Walrasian markets [START_REF] Hammond | Straightforward Individual Incentive Compatibility in Large Economies[END_REF][START_REF] Hammond | Markets as Constraints: Multilateral Incentive Compatibility in Continuum Economies[END_REF], [START_REF] Allen | Repeated Principal-Agent Relationships with Lending and Borrowing[END_REF], [START_REF] Jacklin | Demand Deposits, Trading Restrictions, and Risk Sharing[END_REF], [START_REF] Cole | Efficient Allocations with Hidden Income and Hidden Storage[END_REF], [START_REF] Golosov | Optimal Taxation with Endogenous Insurance Markets[END_REF], Farhi, Golosov, and Tsyvinski (2009)); in our context, this would amount to impose that the entrant must post a linear tariff. We find this at odds with the idea that side trades cannot be monitored and instead allow the entrant to post an arbitrary tariff T E ; the taxation principle [START_REF] Hammond | Straightforward Individual Incentive Compatibility in Large Economies[END_REF], [START_REF] Guesnerie | On Taxation and Incentives: Further Reflections on the Limits to Redistribution[END_REF], [START_REF] Rochet | The Taxation Principle and Multi-Time Hamilton-Jacobi Equations[END_REF]) ensures that this involves no loss of generality. Notice that we do not thereby prevent the entrant from offering a linear tariff; rather, the key idea is that the entrant can choose the terms of side trades and restrict the set of quantities consumers can choose from. This motivates the following definition.

Definition 1 The planner's tariff T P is entry-proof if, for any entrant's tariff T E , there exists a solution (q P i , q E i ) to each type i's problem max {U i (q P + q E , T P (q P ) + T E (q E )) : q P ≥ 0 and q E ≥ 0}

such that entry is not profitable:

m 1 [T E (q E 1 ) -c 1 q E 1 ] + m 2 [T E (q E 2 ) -c 2 q E 2 ] ≤ 0. ( 4
)
An allocation is robust to side trading if it can be implemented by an entry-proof tariff.

It may at first be objected that we are tilting the odds in the planner's favor by de facto assuming that he is able, in case consumers are indifferent, to coordinate their behavior on an allocation in which the entrant does poorly. However, this assumption plays no role in the characterization results we offer in Theorems 1-3: indeed, Theorem 2 shows that this arguably weak entry-proofness concept already singles out a unique budget-feasible allocation robust to side trading; moreover, the necessary conditions we derive are obtained using contracts that are profitable for the entrant no matter the consumers' best responses.

By contrast, strengthening our entry-proofness concept-for instance, by requiring that the planner's tariff be robust to entry no matter the consumers' best responses-would threaten the very existence of a budget-balanced allocation robust to side trading. This situation, however, is hardly specific to our model: indeed, when sustaining equilibria of competitive adverse-selection models, it is often necessary to assume that firms cannot sort out the least costly types at the deviation stage in case consumers are indifferent. 1Any allocation robust to side trading is incentive-compatible. The question we ask is whether such an allocation can also be second-best. Our argument is twofold.

On the one hand, budget-feasible allocations robust to side trading have the following price structure.

Lemma 1 In any budget-feasible allocation (q i , t i ) i=1,2 robust to side trading,

t 1 = cq 1 and t 2 -t 1 = c 2 (q 2 -q 1 ). (5)
Proof. Because an allocation (q i , t i ) i=1,2 robust to side trading is incentive-compatible, it satisfies q 2 ≥ q 1 by single crossing. Moreover,

t 1 ≤ cq 1 . (6)
Otherwise, an entrant can supply q 1 at a price slightly above c: this profitably attracts type 1 as T P (0) = 0, and remains profitable even if type 2 is attracted. Similarly,

t 2 -t 1 ≤ c 2 (q 2 -q 1 ). (7)
Otherwise, an entrant can supply q 2 -q 1 at a price slightly above c 2 : this profitably attracts type 2 along with the contract (q 1 , t 1 ), and is even more profitable if type 1 is also attracted.

Rewriting the resource constraint as

t 1 -cq 1 + m 2 [t 2 -t 1 -c 2 (q 2 -q 1 )] ≥ 0
and taking advantage of ( 6)-( 7) yields [START_REF] Attar | Entry-Proofness and Market Breakdown under Adverse Selection[END_REF]. The result follows.

Hence pricing is competitive, in the sense that the prices of the layers q 1 and q 2 -q 1 reflect the costs of serving the types who purchases them. However, if c 2 > c 1 and q 1 > 0, then the quantities q 1 and q 2 are not priced competitively: as q 1 is sold at the average cost c > c 1 , type 1 subsidizes type 2.

On the other hand, second-best allocations satisfy the following efficiency-at-the-top property.

Lemma 2 In any second-best allocation (q i , t i ) i=1,2 ,

(i) If IC 2→1 is slack, then τ 1 (q 1 , t 1 ) ≤ c 1 , with equality if q 1 > 0. (ii) If IC 1→2 is slack, then τ 2 (q 2 , t 2 ) = c 2 .
Proof. If IC 2→1 or IC 1→2 is slack, then q 2 > q 1 by incentive compatibility and single crossing. If IC 2→1 is slack and τ 1 (q 1 , t 1 ) > c 1 , then ((q 1 + ε, t 1 + c 1 ε), (q 2 , t 2 )) is incentivefeasible for ε > 0 small enough and Pareto-dominates (q i , t i ) i=1,2 , a contradiction. Thus

τ 1 (q 1 , t 1 ) ≤ c 1 . Moreover, if q 1 > 0 and τ 1 (q 1 , t 1 ) < c 1 , then ((q 1 -ε, t 1 -c 1 ε), (q 2 , t 2 )) is
incentive-feasible for ε > 0 small enough and Pareto-dominates (q i , t i ) i=1,2 , once again a contradiction. This proves (i). The proof of (ii) is similar, using q 2 > 0, and is therefore omitted. The result follows.

Combining Lemmas 1 and 2 yields our first main theorem.

Theorem 1 A second-best allocation in which only one incentive-compatibility constraint binds is not robust to side trading.

Proof. Suppose first that only IC 1→2 binds. Then q 2 > q 1 by incentive compatibility and single crossing, and τ 1 (q 1 , t 1 ) ≤ c 1 by Lemma 2(i). Moreover, because type 1's preferences are strictly convex and IC 1→2 binds, we have t 2 -t 1 < c 1 (q 2 -q 1 ) ≤ c 2 (q 2 -q 1 ). By Lemma 1, (q i , t i ) i=1,2 is not robust to side trading.

Suppose next that only IC 2→1 binds. Then q 2 > q 1 by incentive compatibility and single crossing, and τ 2 (q 2 , t 2 ) = c 2 by Lemma 2(ii). Moreover, because type 2's preferences are strictly convex and IC 2→1 binds, we have t 2 -t 1 > c 2 (q 2 -q 1 ). By Lemma 1, (q i , t i ) i=1,2 is not robust to side trading. Hence the result. To study these cases, we strengthen Lemma 1 by establishing that a unique allocation is budget-feasible and robust to side trading. In this allocation, the first layer is optimal for type 1 at price c, while the second layer is optimal for type 2 at price c 2 , conditional on her purchasing the first layer. This allocation is thus Pareto-efficient-maximizing type 1's utility-among those satisfying [START_REF] Attar | Entry-Proofness and Market Breakdown under Adverse Selection[END_REF].

Theorem 2 The third-best allocation defined by

q * 1 ≡ arg max {u 1 (q, cq) : q ≥ 0}, ( 8
)
t * 1 ≡ cq * 1 , (9)
q * 2 ≡ q * 1 + arg max {u 2 (q * 1 + q, t * 1 + c 2 q) : q ≥ 0}, ( 10
)
t * 2 ≡ t * 1 + c 2 (q * 2 -q * 1 ), ( 11
)
is the only budget-feasible allocation robust to side trading.

Proof. (Uniqueness) Because an allocation (q i , t i ) i=1,2 robust to side trading is incentivecompatible, it satisfies q 2 ≥ q 1 by single crossing. Moreover,

u 1 (q 1 , t 1 ) ≥ max {u 1 (q, cq) : q ≥ 0}. ( 12
)
his inability to monitor trades, which effectively prevents any kind of redistribution between different types of consumers.

The existence of such an allocation for any binary distribution of types is also noteworthy.

Nonexclusivity, or consumers' ability to combine the contracts offered by an entrant with those offered by the planner, is key to this result. While this enlarges the set of contracts an entrant can use to attract consumers, this also gives the planner more instruments to deter entry. These take the form of latent contracts, which are not meant to be traded but only to make entry unprofitable. Of course, the planner must make sure that, by offering latent contracts, he does not create new profitable entry opportunities. The third-best tariff [START_REF] Emanuel | Health Care Vouchers-A Proposal for Universal Coverage[END_REF] strikes a balance between these two requirements.

In the adverse-selection case c 2 > c 1 , type 1's and type 2's marginal rates of substitution at the third-best allocation are strictly ordered, τ 1 (q * 1 , t * 1 ) < τ 2 (q * 2 , t * 2 ). In particular, we have

τ 1 (q * 1 , t * 1 ) = c < c 2 = τ 2 (q * 2 , t * 2 )
if the third-best allocation is interior and separating. This contrasts with private-value models where side trades take place on Walrasian markets, which calls for an equalization of marginal rates of substitution [START_REF] Hammond | Straightforward Individual Incentive Compatibility in Large Economies[END_REF][START_REF] Hammond | Markets as Constraints: Multilateral Incentive Compatibility in Continuum Economies[END_REF]).

Yet incentive-compatible gains from trade between types 1 and 2 are exhausted at the thirdbest allocation, subject to the side-trading constraint. Indeed, supposing that consumers have access to the same constant-return-to-scale technology as firms, the minimum price at which type 1 would be willing to sell a small additional quantity to type 2 is c 2 , and at this price type 2 is not willing to buy. In that sense, the third-best allocation is the only candidate for a competitive equilibrium.

Regarding the proof of Theorem 2, an interesting duality is that the third-best allocation is the only candidate for a budget-feasible allocation robust to side trading even if the entrant can only offer a single contract, while the third-best tariff is entry-proof even if the entrant can post an arbitrary tariff. This differs from [START_REF] Glosten | Is the Electronic Open Limit Order Book Inevitable?[END_REF], who in his analysis of limitorder markets requires the entrant's tariff to satisfy a property he dubs single crossing and that generalizes convexity. Another important difference is that Theorem 2 does not require consumers' preferences to be quasilinear, which makes it relevant for standard insurance economies.

We are now ready to address the remaining cases not covered by Theorem 1.

Theorem 3 If a second-best allocation is robust to side trading, then it coincides with the third-best allocation and one of the following conditions holds:

(i) The third-best allocation is pooling, that is, τ 2 (q * 1 , t * 1 ) ≤ c 2 .

(ii) The third-best allocation is separating and first-best, that is, c 1 = c 2 or τ 1 (0, 0) ≤ c 1 .

the case of fully observable trades: for any second-best allocation, there exists a system of transfers ensuring that this allocation obtains in an equilibrium of a decentralized economy in which firms compete by offering exclusive contracts. However, in the polar case where firms can only observe their own trades with consumers, decentralizing a third-best allocation is a more delicate task. To perform it, one needs to explicitly model firms' behavior when no information on consumers' aggregate trades is available to them.

In perfectly competitive markets, it is standard to assume, following [START_REF] Bisin | Competitive Equilibria with Asymmetric Information[END_REF], that nonexclusive competition forces prices to be linear. Yet, as Theorem 3 shows, this is consistent with third-best efficiency only in the special private-value case. A natural alternative is to consider strategic models in which firms compete through menus of bilateral contracts. Side trading is then captured by letting consumers free to trade with any subset of firms, in contrast with the exclusivity assumption made in standard competitive screening models following [START_REF] Rothschild | Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information[END_REF].

Attar, Mariotti, and Salanié (2014) provide a general analysis of this nonexclusivecompetition scenario. Their main result is that a positive level of trade for type 2 can be supported in a pure-strategy equilibrium only if type 1 is left out of the market. Indeed, equilibrium requires that type 1 not be willing to consume at price c, and type 2 purchasing her optimal quantity at price c 2 . Given ( 8)-( 11), the resulting equilibrium allocation is a degenerate third-best one in which (q * 1 , t * 1 ) = (0, 0). A necessary and sufficient condition for decentralization is τ 1 (0, 0) ≤ c, that is, [START_REF] Akerlof | The Market for "Lemons": Quality Uncertainty and the Market Mechanism[END_REF] condition for a market breakdown in which only the worse-quality goods are traded. In all other cases, a pure-strategy equilibrium fails to exist altogether.

An alternative strategy that has been pursued in the literature consists in designing extensive-form games that incorporate inter-firm communication. This approach, initiated by [START_REF] Jaynes | Equilibria in Monopolistically Competitive Insurance Markets[END_REF] and [START_REF] Hellwig | A Note on the Specification of Interfirm Communication in Insurance Markets with Adverse Selection[END_REF], and further developed by [START_REF] Stiglitz | Characterization, Existence, and Pareto Optimality in Insurance Markets with Asymmetric Information with Endogenous and Asymmetric Disclosures: Revisiting Rothschild-Stiglitz[END_REF], allows each firm to dynamically react to the information about its customers' trades disclosed by its competitors. This may induce a firm to withdraw its initial offers or to enforce exclusivity clauses, which suggests that these models are better interpreted as models of endogenous exclusivity than of side trading. In addition, enforcing any such mode of communication would require a very sophisticated institutional setting to properly take into account firms' incentives to reveal information over time.

By contrast, we suggest below some simple market interventions that decentralize the third-best allocation. The novelty of our proposal consists in the explicit consideration of public programs that the agents can complement by resorting to the private sector, therefore endogenously determining the aggregate level of trade. Their simplicity lies in the fact that little information is required for public authorities. We develop our discussion in the context of insurance and financial markets.

Insurance Markets In modern health-insurance systems, public insurance schemes for the provision of basic coverage do not prevent an active role for the private sector. In Germany, consumers can opt out from the public-insurance scheme to buy basic coverage designed and priced by private insurance companies. Consumers also often have the option to complement basic coverage with additional privately provided coverage, such as mutuelles in France. Finally, different forms of mandatory health insurance, whereby consumers are not allowed to remain uninsured, are in place in several systems, as in France, Germany, Japan, Netherlands, and Switzerland. 2Our analysis suggests a simple intervention that achieves a mix of public and private insurance, with no need for observability requirements. It consists in letting the State offer any amount of basic coverage up to q * 1 at the average premium rate c. As private insurance companies are willing to provide any amount of complementary coverage at the high premium rate c 2 , the State together with any insurance company make the third-best tariff available.

Because this tariff is entry-proof, no insurance company has an incentive to deviate and entry is impossible. Implementing the third-best allocation is therefore compatible with letting consumers free to choose their preferred level of coverage. This is reminiscent of the universal health-care vouchers advocated by Emanuel andFuchs (2005, 2007), whereby universal coverage is provided while letting consumers free to purchase additional services or amenities on private insurance markets.

Financial Markets

In the aftermath of the recent crisis, the opportunity for agents to opt out of a public program and trade in private markets has been acknowledged as a key constraint for the design of financial institutions in the presence of adverse selection. In this respect, recent works have suggested a rationale for liquidity-injection programs that provide a credible signal to uninformed lenders by rejuvenating the relevant markets. An optimal intervention then typically consists in attracting only the least profitable borrowers, either through direct lending (Philippon and Skreta (2012)), or by repurchasing low-quality assets (Tirole (2012)). By participating in a bailout program, a borrower may however end up signalling her financial weakness to the market, creating a stigma effect with potentially perverse implications [START_REF] Gorton | Stress for Success: A Review of Timothy Geithner's Financial Crisis Memoir[END_REF]).

While bailout policies are derived under the assumption that public and private liquidity are mutually exclusive, our approach offers a general theoretical framework for evaluating public interventions in situations where privately informed borrowers may complement a public program with additional funds raised on private markets. A possible intervention would require public liquidity provision to involve a price sufficiently low, c, so as to attract all borrowers, and a borrowing limit q * 1 such that no overborrowing by the least profitable ones is possible. Further borrowing may then take place on private markets at price c 2 .

Overall, such an intervention would implement the third-best allocation, thereby achieving budget balance, unlike those proposed by [START_REF] Philippon | Optimal Interventions in Markets with Adverse Selection[END_REF] and [START_REF] Tirole | Overcoming Adverse Selection: How Public Intervention Can Restore Market Functioning[END_REF], and inducing all types of borrowers to participate. This in turn would make it harder to infer their individual financial conditions, mitigating the impact of the stigma effect. Finally, the corresponding allocation of funds is the only one that can be reached under a budgetbalanced program under the threat of side trading.

Concluding Remarks

We would like to conclude by mentioning two avenues for future research. First, we have followed [START_REF] Rothschild | Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information[END_REF] and many authors since in assuming that there are only two types of consumers. This, in particular, enabled us to directly compare our results to the characterization of the second-best efficiency frontier in [START_REF] Crocker | The Efficiency of Competitive Equilibria in Insurance Markets with Asymmetric Information[END_REF]. Yet the question naturally arises to which extent our results are robust to this assumption. Attar, Mariotti, and Salanié (2019a) show that, in general, there exists a unique convex tariff for the planner that is entry-proof and implements a budget-feasible allocation. However, it is not straightforward to extend this uniqueness result to more than two types when the planner's tariff is nonconvex. 3 Preliminary work suggests that imposing more structure on the consumers' preferences, as arises for instance in a Rothschild and Stiglitz (1976) economy, might be instrumental for such an extension. Second, we have throughout the analysis postulated that, given the planner's tariff, an entrant can make a take-it-or-leave-it tariff offer to consumers. In that sense, bilateral trading is frictionless in our model. An alternative to this assumption, and to the standard Walrasian approach to side trading, would be to assume that bilateral trades are the outcome of a dynamic matching and bargaining game between firms and consumers searching for trade partners, in the spirit of [START_REF] Gale | Strategic Foundations of General Equilibrium: Dynamic Matching and Bargaining Games[END_REF]. The planner may then be able to manipulate the outcome of this game by making search more costly, possibly enabling him to enlarge the set of implementable allocations and to achieve redistribution. More generally, the optimal manipulation of the market for side trades, which we have by assumption ruled out from our analysis, is an important topic for future investigations.

This in particular the case when the equilibrium allocation involves some amount of pooling, as in[START_REF] Akerlof | The Market for "Lemons": Quality Uncertainty and the Market Mechanism[END_REF],[START_REF] Miyazaki | The Rat Race and Internal Labor Markets[END_REF],[START_REF] Wilson | A Model of Insurance Markets with Incomplete Information[END_REF],[START_REF] Spence | Product Differentiation and Performance in Insurance Markets[END_REF], and Attar, Mariotti, and Salanié (2011).

We refer to the surveys of[START_REF] Thomson | Private Health Insurance in the European Union[END_REF] and Thomson, Osborne, Squires, and Jun (2013) for institutional details and cross-country evidence.

Technically, this is because such a tariff may cause the consumers' indirect utility functions, as defined in the proof of Lemma 3, to fail to satisfy a single-crossing condition. Notice in any case that the existence of a budget-balanced allocation robust to side trading is always guaranteed.

authors. We thank the Editor, Gilat Levy, and three anonymous referees for very thoughtful and detailed comments. Financial support from the ANR (Programme d'Investissement d'Avenir ANR-17-EURE-0010) and the Chaire SCOR-TSE is gratefully acknowledged.

Otherwise, an entrant can offer a contract with unit price slightly above c that profitably attracts type 1 as T P (0) = 0, and remains profitable even if type 2 is attracted. Similarly, u 2 (q 2 , t 2 ) ≥ max {u 2 (q 1 + q, t 1 + c 2 q) : q ≥ 0}. [START_REF] Crocker | The Efficiency of Competitive Equilibria in Insurance Markets with Asymmetric Information[END_REF] Otherwise, an entrant can offer a contract with unit price slightly above c 2 that profitably attracts type 2 along with the contract (q 1 , t 1 ), and is even more profitable if type 1 is also attracted. Finally, if (q i , t i ) i=1,2 is budget-feasible, then [START_REF] Attar | Entry-Proofness and Market Breakdown under Adverse Selection[END_REF] holds, so that ( 12)-( 13) are equalities. Thus (q i , t i ) i=1,2 is the third-best allocation defined by ( 8)- [START_REF] Caillaud | Government Intervention in Production and Incentives Theory: A Review of Recent Contributions[END_REF].

(Existence) By ( 8)-( 11), the piecewise-linear convex tariff

implements the third-best allocation. Now, suppose that an entrant posts a tariff T E . The following monotonicity property is established in the Appendix.

Lemma 3 There exists a solution ((

Let us fix such a solution in what follows. As T P allows type 1 to purchase her optimal quantity q * 1 at price c, we must have

Moreover, because q E 2 ≥ q E 1 , type 2 could alternatively obtain the same aggregate quantity q P 2 + q E 2 as in her best response by purchasing q E 1 from the entrant and q P 2 + q E 2 -q E 1 from the planner, paying overall T P (q P 2 + q E 2 -q E 1 ) + T E (q E 1 ). As she chooses to pay T P (q P 2 ) + T E (q E 2 ) instead, we must have

Because T P is convex with slope at most c 2 and q E 2 ≥ q E 1 ,

Collecting ( 15) and ( 16)- [START_REF] Gale | Strategic Foundations of General Equilibrium: Dynamic Matching and Bargaining Games[END_REF] yields

which is (4). This shows that T P is entry-proof. Hence the result.

The uniqueness of the third-best allocation contrasts with the multiplicity of second-best allocations, which form a nondegenerate frontier. The planner is thus severely constrained by Proof. By Theorem 2, if a second-best allocation is robust to side trading, then it coincides with the third-best allocation. By Theorem 1, we only need to consider two cases.

(i) If IC 1→2 and IC 2→1 bind, then q * 2 = q * 1 by incentive compatibility and single crossing. Hence the third-best allocation is pooling, which amounts to τ 2 (q * 1 , t * 1 ) ≤ c 2 by ( 10)-( 11). (ii) If IC 1→2 and IC 2→1 are slack, then q * 2 > q * 1 by incentive compatibility and single crossing. Hence the third-best allocation is separating. Two cases can arise. If q * 1 > 0, then τ 1 (q * 1 , t * 1 ) = c 1 by Lemma 2(i) and τ 1 (q * 1 , t * 1 ) = c by ( 8)-( 9), so that c 1 = c 2 . If q * 1 = 0, then τ 1 (0, 0) ≤ c 1 by Lemma 2(i). In either case, each type i trades efficiently at cost c i , so that the third-best allocation is first-best. Hence the result.

Condition (i) is clearly extreme. It cannot hold in a Rothschild and [START_REF] Rothschild | Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information[END_REF] insurance economy, because the optimal coverage of type 1 at the average premium rate c is only partial, while type 2 is willing to purchase additional coverage at the fair premium rate c 2 until she reaches full insurance. In the case of quasilinear preferences, condition (i) together with the condition τ 1 (q * 1 , t * 1 ) ≤ c implied by ( 8)-( 9) entails that type 1's first-best quantity is at least as large as type 2's, and strictly larger if q * 1 > 0, a case of nonresponsiveness (Caillaud, Guesnerie, Rey, and Tirole (1988)).

Condition (ii) is also extreme. Indeed, the third-best allocation is then first-best, and the third-best tariff is linear with slope c 2 . In the private-value case c 1 = c 2 , each type trades efficiently at marginal cost. In the adverse-selection case c 2 > c 1 , a separating second-best allocation is robust to side trading only if type 1 is not willing to trade at cost c 1 and hence is in some sense irrelevant.

This answers the question we raised in this section: second-best efficiency and robustness to side trading are irreconcilable requirements, except in very special cases. Overall, our results suggest that the threat of side trading constitutes a serious obstacle to efficiency and redistribution in private-information economies. In the limiting case of private values, side trading poses no threat to efficiency, as it leads to a first-best allocation; yet the requirement that there be no cross-subsidies between layers precludes the planner from redistributing resources between different consumer types. By contrast, under adverse selection, the social costs of side trading are twofold: first, the threat of side trading moves the economy away from the second-best efficiency frontier; second, it precludes redistribution.

Decentralization and Market Intervention

The Decentralization Problem Determining whether constrained-efficient allocations can be supported in an equilibrium of an adverse-selection economy is a central issue for welfare economics. [START_REF] Bisin | Efficient Competitive Equilibria with Adverse Selection[END_REF] provide a positive answer to this question in Appendix Proof of Lemma 3. Each type i evaluates any contract (q E , t E ) she can trade with the entrant through the indirect utility function z -P i (q E , t E ) ≡ max {u i (q P + q E , T P (q P ) + t E ) : q P ≥ 0}. (A.1)

Because u i is strictly quasiconcave and T P is convex, the maximum in (A.1) is attained at a unique q P i (q E ), and ( q P i (q E ), q E i ) is a solution to (3) if and only if

According to Attar, Mariotti, and Salanié (2019b, Lemma 1), the convexity of the tariff T P and the strict single-crossing condition for the functions u i imply the following single-crossing condition for the functions z -P i :

For all q E < q E , t E , and t E , z -P 1 (q E , t E ) < z -P 1 (q E , t E ) implies z -P 2 (q E , t E ) < z -P 2 (q E , t E ).

To conclude, suppose that q E 2 < q E 1 at some solution ((q P i , q E i )) i=1,2 to (3). By (A.2), z -P 2 (q E 2 , T E (q E 2 )) ≥ z -P 2 (q E 1 , T E (q E 1 )).

Because q E 2 < q E 1 , the above single-crossing condition then implies z -P 1 (q E 2 , T E (q E 2 )) ≥ z -P 1 (q E 1 , T E (q E 1 )).

Thus ( q P 1 (q E 2 ), q E 2 ) is also a solution to (3) for type 1. The result follows.