Franc ¸ois Rivet Antoine Lhomel
email: antoine.lhomel@ims-bordeaux.fr

Nathalie Deltimple

D-Band CMOS Power Amplifiers: Challenges, State-of-the-Art, Technological Limitations and Trends

Keywords: D-Band, CMOS Power Amplifier (PA), subterahertz (sub-THz), unilateral power gain (U), gain-boosting

To achieve ultra-high-speed wireless communications, millimeter-wave bands have emerged as bands of interest for these new applications. However, a technological breakthrough linked to power amplifiers has appeared. Indeed, working near the maximum oscillation frequency of the transistor leads to low intrinsic gain and poor efficiency for these devices. This paper presents an overview of sub-terahertz CMOS power amplifiers with a focus on gain enhancement techniques. The paper will also feature state-of-the-art D-Band CMOS power amplifier architectures. The goal is to present how the gain enhancement techniques are used in a complete architecture. Based on the three techniques and the state-ofthe-art presented in the paper, a focus on three architectures is proposed. The aim is to highlight the benefits of each method in the design. The final part summarizes the paper and opens up future perspectives and trends identified through the paper.

I. INTRODUCTION

The upper millimeter-wave spectrum [150GHz -300GHz] and terahertz [0.3THz -3THz] bands have recently emerged as promising frequencies for various applications, including high-speed communications, high-resolution sensing, and spectroscopy. Focusing on high-speed communications, the D-Band [110GHz -170GHz] is particularly suitable for such applications thanks to its high carrier frequency and wide available spectrum. Fig. 1, extracted from [START_REF] Fujishima | Key Technologies for THz Wireless Link by Silicon CMOS Integrated Circuits[END_REF], shows the evolution of the data rate over the years. The data rate increase through the years needs higher bandwidth and spectrum to achieve the required performances. This highlights the interest in D-Band communication in continuous improvement of wireless data rate.

To enable the implementation of such systems in large volumes, CMOS technology has been used in numerous works [START_REF] Seo | A 150 GHz Amplifier With 8 dB Gain and +6 dBm Psat in Digital 65 nm CMOS Using Dummy-Prefilled Microstrip Lines[END_REF][START_REF] Xu | A Three Stage, Fully Differential 128-157 GHz CMOS Amplifier with Wide Band Matching[END_REF][START_REF] Van Thienen | A 160-GHz three-stage fullydifferential amplifier in 40-nm CMOS[END_REF][START_REF] Moghadami | A 0.2-0.3 THz CMOS Amplifier With Tunable Neutralization Technique[END_REF] due to the low cost of manufacture. However, due to the operating frequency being close to the maximum oscillation frequency (f max ) of the transistor, there is a low power gain available from each transistor. The consequences are low output power for the transistor and high power consumption, leading to low power efficiency of the overall architecture. To tackle the mentioned problem, most of the work is focused on increasing the gain of each transistor when it operates near f max . Based on S. Mason's work [START_REF] Mason | Power Gain in Feedback Amplifier[END_REF], the unilateral gain (U) is defined for all 2-port networks and so, this metric can be used for transistors used in common source configuration. Multiple techniques are used to enhance the gain of transistors [START_REF] Sadlo | One stage gain boosted power driver at 184 GHz in 28 nm FD-SOI CMOS[END_REF][START_REF] Rashtian | Gain Boosting in Distributed Amplifiers for Close-to-fmax Operation in Silicon[END_REF][START_REF] Kim | Analysis and Design of Dual-Peak Gmax-Core CMOS Amplifier in D-Band Embedding a T-Shaped Network[END_REF][START_REF] Simic | Analysis and Design of Lossy Capacitive Over-Neutralization Technique for Amplifiers Operating Near fMAX[END_REF][11], such as gain-boosting networks or neutralization. The gain-boosting method aims to reach the maximum power gain of the 2-port active network (2-PAN) based on an optimized linear, lossless, and reciprocal (LLR) network. Neutralization is based on compensating the C gd capacitance of the transistor, which becomes a limiting factor at high frequencies and causes a drop in gain. However, the implementation of these methods requires calculating the sizes of the transistors to achieve targeted performances, determining the convenient network and taking into account the losses brought by interconnections and passive components. But the higher the frequency, the higher the losses in passive components due to the electromagnetic (EM) effect, coupling, and resistivity of metal level. Moreover, targeting high gain, output power, and efficiency requires a more complex architecture and meticulous work to optimize the passive components for matching, combining, and dividing the power.

The paper is organized as follows. Section II overviews the transistor's behavior at sub-THz frequencies and the basic gain definitions for a 2-port active network. It describes the methodology to design single-peak gain-boosting and dual-peak gain-boosting. It also highlights gain enhancement based on the neutralization technique. Section III presents a State-of-the-Art of D-Band PAs with different technologies. Based on the State-of-the-Art, D-Band PA architectures are presented using the proposed gain-boosting method. Finally, the paper is concluded in Section V.

Digital Object Identifier 10.29292/jics.vXXiX.X

II. GAIN ENHANCEMENT METHOD AT SUB-THZ FREQUENCIES

A. Definition of gain parameters U, G ma , and G max A transistor can be seen as a 2-port active network (2-PAN) characterized by its Y-matrix as shown in Fig. 2.

2-port Active Network

Fig. 2: Block Scheme of a 2-Port Active Network.

Mason's gain or unilateral power gain (U) can be written in terms of Y parameters based on the following equation:

U = |Y 21 -Y 12 | 2 4(Re(Y 11 ) * Re(Y 22 ) -Re(Y 21 ) * Re(Y 12 )) (1)
Based on [START_REF] Amakawa | Theory of gain and stability of small-signal amplifiers with lossless reciprocal feedback[END_REF], the formula proposed to calculate U can be written in the same way using Z and S parameters. This gain can be seen as a Figure of Merit (FoM) for the transistor to compare their performances and moreover evaluate the f max .

The f max can be evaluated as the frequency where U(f max ) is equal to 1 or 0dB. Two other gains can be derived from U. The first one is the maximum available gain G ma defined, using the stability factor K, as:

G ma = |A| K + √ K 2 -1 with A = Y 21 Y 12 (2)
The last gain defined here is the maximum achievable gain G max considered as the maximum power gain that can be reached from a transistor. This gain is defined using Mason's gain as follows:

G max = (2U -1) + 2 U (U -1) (3)
These 3 gains can be plotted simultaneously, resulting in the curve in Fig. 3. The transistor's size used is W = 20µm and L = 30nm in the 28nm FDSOI process from STMicroelectronics. The G max is the maximum power gain that can be pulled out from a single transistor. The purpose of the enhancement gain technique is to reach this gain based on the enhancement of G ma at a specified frequency.

The next three parts will focus on the enhancement gain technique mentioned in Section I.

B. Single-peak gain-boosting method

Based on the 2-PAN described in the previous part, a gainboosting network can be added between the input and the output of the 2-PAN to increase the G ma and assure stability. Fig. 4 highlights the block schematic of the method. The purpose of the single-peak gain-boosting method is to determine a value of the inductance L boost in order to increase G ma at the desired frequency. The schematic of the transistor with a gain-boosting network is shown in Fig. 5. A capacitance is added for decoupling gate and drain supplies. Design methodology are presented in [START_REF] Sadlo | One stage gain boosted power driver at 184 GHz in 28 nm FD-SOI CMOS[END_REF], [START_REF] Amakawa | Theory of gain and stability of small-signal amplifiers with lossless reciprocal feedback[END_REF] and [START_REF] Amakawa | Graphical approach to analysis and design of gain-boosted near-fmax feedback amplifiers[END_REF]. Two ways can be used to determine the optimum L boost at a targeted frequency. To illustrate the methodology, we took an example at a specified frequency of 152GHz is used. Considering the same transistor (W = 20µm) and the same process (28nm FDSOI), U, G ma , and G max are plotted in Fig. 6.

2-port Active Network

As mentioned previously, the purpose of this method is to enhance the G ma to get closer to G max at one frequency. Without gain enhancement, the G ma is equal to 6.8dB, and the increase at f c = 152GHz is 3.7dB reaching 10.5dB with the gain-boosting network. It can be noticed that the gainboosting network does not impact the unilateral gain U and the maximum achievable gain G max confirming the theory developed in [START_REF] Mason | Power Gain in Feedback Amplifier[END_REF]. However, the increase of G ma with this kind of network induces a low bandwidth. The resonance made with the single-peak gain-boosting network is targeted

C. Dual-peak gain-boosting method

The previous gain-boosting network is calculated at a single targeted frequency and intrinsically leads to a small bandwidth. In order to increase the bandwidth of the PA architecture, a dual-peak gain-boosting network is proposed. This network, inspired by [START_REF] Kim | Analysis and Design of Dual-Peak Gmax-Core CMOS Amplifier in D-Band Embedding a T-Shaped Network[END_REF], is not targeted at a single frequency, but two located at the band edges. Based on a T-shaped network with two capacitances and an inductor, the network is presented in Fig. The two couples L/C 1 and L/C 2 are resonating at the two targeted frequencies. Based on the 28nm FDSOI process from STMicroelectronics and a transistor's size of W = 20µm, Fig. 8 presents U, G ma , and G max for the new gainboosting method.

An enhancement of 1.1dB on G ma is seen at f c = 152GHz, this enhancement is located around the band of interest . Two advantages can be found for this structure from the previous one: the first one is the wideband behavior of the proposed stages and the second one, there are multiple possibilities to achieve the wanted G ma behavior. However, the obtained G ma is lower than the singlepeak network dropping from 10.5dB to 7.7dB at the same frequency, but this network leads to a higher bandwidth on the G ma going from 21GHz to 44GHz considering the 3dB around 152GHz.

D. Neutralization method

The last method addressed here is the most commonly used in D-Band architecture: the neutralization method. The purpose is to neutralize the intrinsic capacitance C gd of the MOS transistor by adding a capacitance with the same value as represented in Fig. 9. For lower-frequency applications, these two neutralization capacitances (C n ) are used to stabilize the differential pair. However, it can also help to increase the G ma of the stage [START_REF] Zhang | A 120-150 GHz Power Amplifier in 28-nm CMOS Achieving 21.9-dB Gain and 11.8-dBm Psat for Sub-THz Imaging System[END_REF][START_REF] Tang | A 140 GHz T/R Front-End Module in 22 nm FD-SOI CMOS[END_REF][START_REF] Zhang | A 124-152 GHz ¿ 15-dBm Psat 28-nm CMOS PA Using Chebyshev Artificial-Transmission-Line-Based Matching for Wideband Power Splitting and Combining[END_REF]. Fig. 10 shows the increase of G ma at a fixed frequency, depending on the value of C n . It also highlights the value of k to determine if the stage is stable or not. The stability factor k and G ma are dependent on C n but the maximum of each is not achieved for the same value. Based on this, a compromise needs to be found to ensure a higher gain and stability. Considering the proposed case, a capacitance of 8fF is chosen to increase the G ma of the differential pair and ensure stability. U, G ma , and G max are plotted in Fig. 11 to highlight the improvement. Without gain enhancement, the G ma is equal to 6.8dB, and the increase at f c = 152GHz is 1.5dB reaching 8.3dB with the neutralization method.

Cn

Three methods are presented here to improve the intrinsic gain of a single transistor or a differential pair. The achieved performances of each method are summarized in Table 1. Even though U and G ma are characterized by small-signal parameters, their values are close to the expected power gain value in large signal simulations. Considering D-Band power amplifier, these metrics are the first approach to evaluate the total gain available for an architecture.

The next Section proposed to overview the performances of D-Band PA through the literature and described architectures of D-Band PA using the proposed gain enhancement methods, which have performances at the forefront of Stateof-the-Art.

III. STATE-OF-THE-ART D-BAND POWER AMPLIFIERS

To highlight the expected performances of D-Band PA, a State-of-the-Art is done targeting three main performances: output Gain, saturation power (P sat ) and Power-Added Efficiency (PAE). Different technologies are taken into account, extracted from [START_REF] Seo | A 150 GHz Amplifier With 8 dB Gain and +6 dBm Psat in Digital 65 nm CMOS Using Dummy-Prefilled Microstrip Lines[END_REF][START_REF] Xu | A Three Stage, Fully Differential 128-157 GHz CMOS Amplifier with Wide Band Matching[END_REF][START_REF] Van Thienen | A 160-GHz three-stage fullydifferential amplifier in 40-nm CMOS[END_REF][START_REF] Moghadami | A 0.2-0.3 THz CMOS Amplifier With Tunable Neutralization Technique[END_REF] and .

Fig. 12 depicts the gain achieved by the various studies collected. In the D-band, the gain varies between 10 dB and 25 dB, with a slight decrease when the proposed works are in the higher part of the band. It is also important not to overlook studies conducted at over 200 GHz, which may exhibit significant gain values. However, these studies still struggle to attain adequate output power and efficiency. About the P sat , Fig. 13 illustrates the results obtained for various architectures. A clear trend line is evident, indicating that P sat decreases as the operating frequency increases. This phenomenon is thus correlated with the need to optimize f max and, consequently, reduce the margin for current density passing through the transistor. This current density limitation is inherently linked to the maximum output power of the PA. The same observation can be made for the PAE (Power-Added Efficiency), presented in Fig. 14, with a decreasing trend line as a function of frequency. For D-band works, it does not exceed 15% in CMOS technology. These values can be explained by the transistor's low intrinsic gain and the need for high biasing of the stages used.

The next section will present architectures that are on the cutting edge of State-of-the-Art. The purpose is to demonstrate the usefulness of each method presented in Section II.

IV. ARCHITECTURE OF D-BAND POWER AMPLIFIERS

A. D-Band Power Amplifier using single-peak gain boosting

Based on the work in [START_REF] Bameri | A High-Gain mm-Wave Amplifier Design: An Analytical Approach to Power Gain Boosting[END_REF] and to illustrate the single-peak gain-boosting method, a 4-stage D-Band PA is presented slightly above the D band. It highlights a G ma increase of 3.7dB for each stage before implementing it in the complete architecture. The first step to achieve this type of architecture is to evaluate the correct network to increase the G ma . Fig. 15 shows the evolution of the intrinsic gain of the transistor depending on the value of inductors around the transistor. The dual purpose of the network is to increase the gain and also ensure the stability of the transistor. Once the network is evaluated and implemented, a 4-stage single architecture is proposed using input and output stub matching as shown in Fig. 16.

The PA is manufactured in 65nm CMOS. Fig. 17 shows the setup used for measurements. The obtained S-parameters are presented in Fig. 18. The amplifier has a maximum measured small-signal gain of 9.2 dB at 257 GHz and the input and output reflection coefficients of -5.5 and -8.5 dB, respectively. The 3dB bandwidth achieved by the architecture is 12.2GHz. As shown in Fig. 19, at 255 GHz, the maximum PAE is 1.35% and the maximum P sat is -3.9 dBm with V DD = 1 V. The amplifier consumes 27.6 mW at VDD = 1 V.

The purpose is to show the feasibility of the method and implement it in a complete PA architecture. Despite the low bandwidth displayed, the architecture proposes small and large signal measurements that validate the functionality of the circuit.

Another interest of this single-peak embedding is the possibility of using them in addition to other gain enhancement techniques [START_REF] Balasubramanian | A 160-GHz Power Amplifier with 32-dB Gain and 9.8% Peak PAE in 28-nm FD-SOI[END_REF]. In this work, the proposed architecture is a All stages are based on neutralized differential pairs; however, the distinctive feature of this architecture lies in stages 2 and 3. To enhance the G ma of this section, a single-peak gain-boosting network is employed around these two stages. The resulting layout is presented in Fig. 22. The four stages are visible, with special attention given to the A2-A3 couple, where the gain-boosting network is incorporated to enhance the power gain. This implementation results in a 3.1dB enhancement at 160GHz, in addition to the initial improvement achieved by the neutralization capacitance. As mentioned by the author, the network was designed to ensure a stability factor equal to 1. This low k-factor can lead to instability in measurements and is sensitive to process variations. The paper proposes Post-Layout Simulations (PLS) at 160GHz, considering large signal parameters of the PA: Gain, P sat , and PAE. The results obtained are shown in Fig. 22. The PA achieves an OP 1dB of 5 dBm and P sat of 11 dBm with a gain of 32dB. The peak PAE is 9.8%. The PA consumes power of 112 mW from a 1.1 V supply. A. ,

B. D-Band Power Amplifier using dual-peak gain boost-ing

Another interesting work done in [START_REF] Kim | Analysis and Design of Dual-Peak Gmax-Core CMOS Amplifier in D-Band Embedding a T-Shaped Network[END_REF] gives an example of a dual-peak gain boosting network to overcome the performance limitations of CMOS technology at frequencies exceeding 100 GHz. The concept of a T-embedding network has been explored to enhance the G ma . In this study, an innovative Gmax-core incorporating a T-shaped gain-boosting network that provides two G ma peaks is analyzed and demonstrated in the D-band using a 28nm FD-SOI CMOS process. The proposed methodology for the multistage design is the following: each stage has its own G ma behavior, enabling achieving a wider bandwidth at the end. Fig. 23 shows the modification of G ma behavior, starting from a single transistor to a transistor with the proposed Tembedding network. This type of network provides another useful feature: the input and output impedances can be adjusted based on the Tnetwork combinations. Indeed, a good choice of network can provide a lower frequency dependency of the input and output impedance of the stage. It results in a wideband matching thanks to impedance transformers commonly used in the literature.

Based on these considerations, the schematic view of the PA is presented in Fig. 24 and the layout view in Fig. 25. The proposed D-Band PA is a 3-stage broadband amplifier using the method presented above. The proposed T-shaped network is implemented in 28nm FD-SOI CMOS. Small signal S-parameters were measured using a WNA and a WR-6.5 frequency extender. The obtained results are presented in Fig. 26.

The amplifiers exhibit a power gain and 3dB bandwidth of 14.5 dB and 26 GHz [117GHz -143 GHz], respectively, with a power dissipation of 21.6 mW. Despite the great performances achieved by the PA in small signal parameters, no information is available for large signal behavior and the P sat , compression point, and PAE can not be evaluated. To conclude on single and dual-peak gain-boosting networks, the trend indicates that these types of networks can be used for D-Band architecture. However, most architectures utilizing these networks exclusively apply them across all stages. As mentioned earlier, considering the active part with gain-boosting modifies the k-factor, bringing it closer to 1. This low stability factor results in lower power levels due to the circuit's sensitivity when a high power level is applied. The major risk is an oscillating power amplifier (PA). Alternatively, gain-boosting can also be employed in addition to neutralized stages, leading to high-gain stages. It is important to highlight that this is primarily used for gain stages. If power levels are too high, the risks are similar to those of other architectures.

C. D-Band Power Amplifier using neutralization capacitances

This part focuses on a 4-stage D-Band PA achieving the highest performances in Gain, P sat and PAE [START_REF] Li | A 130-151 GHz 8-Way Power Amplifier with 16[END_REF]. Based on neutralization capacitances (C n ), the PA first proposes a 4stage differential cell which is combined in a 4-way PA to achieve the obtained performances. This work highlights the two points targeted in this paper. The first one is the gain enhancement using the neutralization method. This method is commonly used in numerous works in D-Band, but also at lower frequency.

Based on the method presented in Section II, the first goal is to evaluate the optimal value of C n . Fig. 27 shows the behavior of G ma and k depending on the value of C n . Fig. 27: k and Gma Depending on Cn [START_REF] Li | A 130-151 GHz 8-Way Power Amplifier with 16[END_REF].

The value of C n needs to be chosen to increase G ma and ensure the stability of the stage. In this work, due to the increase of transistor's sizes through the stage, the capacitances are chosen equal to 7fF for the two gain stages, 9fF for the driver and 10fF for the power stage. The schematic of the differential PA single cell is presented in Fig. 28. As for the previous work and due to the low intrinsic gain of each stage, the architecture needs multiple stages to achieve enough power gain. This leads to the design of passive components such as impedance transformers and power combiners. The first one helps to achieve impedance matching between the stages and the second one helps to achieve higher P sat . In both cases, losses in these components must be minimized to not compromise the work performed on the active part and also try to keep power efficiency as high as possible. This point will be developed from now on. Based on the previous differential PA cell, a 4-way power combination is proposed to achieve a P sat of 17.5dBm at 140GHz. To do so, the schematic of the complete architecture is presented in Fig. 29 and the fabricated PA, in 45nm RFSOI, is shown in Fig. 30.

Three cases need to be distinguished. Input and output baluns are used for impedance matching and single-todifferential conversion. Interstage transformers are used for interstage matching. Power combiners are used to combine the four output ways to increase the maximum output power.

Starting with input and output baluns, the target is to have highly coupled baluns to achieve low loss and power matching. High coupling coefficients are achieved using symmetrical baluns. However, the bandwidth of the baluns is intrinsically linked to the coupling coefficient between the two inductors. A high coupling coefficient results in narrowband matching, but methods are proposed to increase the bandwidth of such baluns [START_REF] Eissa | A 13.5-dBm 200-255-GHz 4-Way Power Amplifier and Frequency Source in 130-nm BiC-MOS[END_REF]. About the interstage transformer, the preferred topology is asymmetric transformer. It enables both impedances simultaneously without adding other components. Fig. 31 shows an example of an asymmetric transformer used in the design.

However, it requires being careful of the coupling coefficient and quality factor of the inductors used for the design. Losses of the transformers are linked to these two parameters. Thick levels of metals are to be preferred for this design to reduce the skin effect, increase the quality factor of the inductors, and manage properly the coupling coefficient. The last part highlights the design of power combiners used in D-Band architectures. Most of the combiners are 0°c ombiners based on Wilkinson topologies. Working at high frequencies (above 100GHz) enables the integration of such combiners based on λ 4 lines which became small enough to be integrated. The main default of these combiners is the non-possibility of matching. However, [START_REF] Eissa | A 13.5-dBm 200-255-GHz 4-Way Power Amplifier and Frequency Source in 130-nm BiC-MOS[END_REF] and [START_REF] Yu | A 211-to-263-GHz Dual-LC-Tank-Based Broadband Power Amplifier With 14.7-dBm PSAT and 16.4-dB Peak Gain in 130-nm SiGe BiCMOS[END_REF] proposed solutions to co-design power combinations and matching networks. An example of these combiners is proposed in Fig. 32. This structure combines 4 differential ways with insertion losses of 1.4dB which is acceptable for D-band design considering the number of combinations.

Based on these considerations, the PA proposed in [START_REF] Li | A 130-151 GHz 8-Way Power Amplifier with 16[END_REF] achieves the performances in small-signal parameters shown in Fig. 33 and continuous wave (CW) measurements are shown in Fig. 34. The PA achieves a 3dB bandwidth of 19GHz from 131GHz to 150GHz with a gain ripple of 1dB. S 11 and S 22 are below -10dB in-band.

Considering the CW, gain and PAE are plotted according to the input power P in . The characterization is made at f = 140GHz. The power gain achieved is 24dB at 140GHz with P sat of 17.5dBm and maximum PAE of 13.4%. The proposed D-Band PA achieves high output power, linearity and efficiency using neutralization enhancement gain technique as well as low loss 8-way power combining. This work highlights the best FoM seen in State-of-the-Art.

As a conclusion to this Section, all the performances achieved by the presented architectures are summarized in Table II. For comparison, a Figure of Merit (FoM) is introduced. The architecture employing a neutralization method emerges as the optimal solution for applications requiring higher power, while the boosting method proves particularly well-suited for gain stages operating at lower power. This dual approach provides optimal flexibility to meet diverse performance requirements.

V. ASSESSMENT OF STATE-OF-THE-ART AND PROSPECTIVE

This paper presents an overview of D-Band PAs with the challenges and technological limitations appearing through the design. Multiple complete PA architectures are presented after the characterization of various enhancement gain techniques such as single and dual-peak gain boosting or neutralization capacitances by simulations.

Trends followed by State-of-the-Art can then be highlighted. Considering the gain-boosting method, most of the work do not highlight large-signal measurements and limits the study to S-parameters. However, it brings multiple possibilities for the designer like bandwidth increase and input and output impedance control of the power cell. Best-in-class PA processed in 45nm RF-SOI uses neutralization capacitances. It highlights the best performances around 140GHz, with the idea to simplify the active part with a well-known method and focus the major part of the work on passive components.

Considering the various points addressed in this paper, it is clear that the intrinsic gain of the transistor alone is not sufficient. Gain-boosting networks must be added to complete architectures with efficiency exceeding 10%. In the context of future work, it is essential to validate the operation and understanding of these networks, particularly those relying on conditions close to transistor oscillation. The designer's task will be to ensure the model's validity and proper use of the networks, especially during high-output power tests. Finally, the role of passive components should not be underestimated. Given the state of the art, the combination of mul-tiple paths appears to be the best approach to obtaining highperformance architectures. However, this combination must be carefully managed to minimize losses and avoid further degradation of the limited available gain from active stages.

Fig. 1 :

 1 Fig. 1: Evolution of Data Rate of Wired Communication and Wireless Communication Through the Year.

Fig. 3 :

 3 Fig. 3: U, Gma, and Gmax Plotted for a Transistor With W = 20µm and L = 30nm.

Fig. 4 :

 4 Fig. 4: Block Scheme of a 2-Port Active Network With a Gain-Boosting Network.

Fig. 5 :

 5 Fig. 5: Schematic of a Transistor With a Single-Peak Gain-Boosting Network.

Fig. 6 :

 6 Fig. 6: U, Gma, and Gmax Plotted for a Transistor with W = 20µm and L = 30nm with Single-Peak Gain-Boosting Network.

7

 7

Fig. 7 :

 7 Fig. 7: Schematic of a Transistor with a Dual-Peak Gain-Boosting Network.

Fig. 8 :

 8 Fig. 8: U, Gma, and Gmax Plotted for a Transistor with W = 20µm And L = 30nm Using Dual-Peak Gain-Boosting Network.

Fig. 9 :

 9 Fig. 9: Schematic of a Differential Pair Using Neutralization Capacitance.

Fig. 10

 10 Fig. 10: k, Gma and Gmax Plotted For a Differential Pair With W = 20µm and L = 30nm Depending on the Neutralization Capacitance Value.

Fig. 11

 11 Fig. 11: U, Gma, And Gmax Plotted for a Differential Pair with W = 20µm And L = 30nm Using Neutralization Capacitances.

Fig. 12 :Fig. 13 :Fig. 14 :

 121314 Fig. 12: Gain Achieved for Power Amplifier in Different Technologies.

Fig. 15 :

 15 Fig. 15: Evolution of Gma for Different Inductors Values [27].

Fig. 16 :

 16 Fig. 16: Schematic of the 4-Stage Single Peak Gain-Boosting PA [27].

Fig. 17 :

 17 Fig. 17: Test Setup for Large-Signal Measurements [27].

Fig. 18 :

 18 Fig. 18: Measured and Simulated S-Parameters for the 4-Stage Power Amplifier [27].

Fig. 19 :

 19 Fig. 19: Power Added Efficiency, Output Power, and Large-Signal Gain of the Amplifier for V DD = 1V [27].

4 -

 4 stage PA working at 160GHz. The block view of the PA is presented in Fig.20.

Fig. 20 :

 20 Fig. 20: Block view of the PA [32].

Fig. 21 :

 21 Fig. 21: Layout view of the PA [32].

Fig. 22 :

 22 Fig. 22: Large signal gain, Power Added Efficiency and output power of the PA [32].

Fig. 23 :

 23 Fig. 23: Strategy of Broadband Amplifier Design [9].

Fig. 24 :

 24 Fig. 24: Schematic of the Proposed D-Band Amplifier With T-Shaped Embedding Network [9].

Fig. 25 :

 25 Fig. 25: (a) Core Layout (b) Chip Microphotograph [9].

Fig. 26 :

 26 Fig. 26: Measured S-Parameters [9].

Fig. 28 :

 28 Fig. 28: Schematic of the Differential PA Single Cell [18].

Fig. 29 :

 29 Fig. 29: Schematic of the Complete 4-Way PA [18].

Fig. 30 :

 30 Fig. 30: Layout of the Complete 4-Way PA [18].

Fig. 31 :

 31 Fig. 31: Interstage Matching Asymmetrical Transformer [18].

Fig. 32 :

 32 Fig. 32: Zero-Degree Combiner with Matching Network [28].

Fig. 33 :

 33 Fig. 33: S-Parameters of the 4-Way Differential PA [18].

Fig. 34 :

 34 Fig. 34: Continuous Wave Characterization of the 4-Way Differential PA at 140GHz [18].

Table I .

 I : Summary of Achieve Gma with Gain Boosting Networks.

		Gma at fc = 152GHz	3dB BW @ fc
	No gain-boosting network	6.8dB	35GHz
	Single-peak gain-boosting	10.5dB	46GHz
	Dual-peak gain-boosting	7.7dB	72GHz
	Neutralization	8.5dB	49GHz

ACKNOWLEDGEMENTS

The project leading to this application has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No 964246. This document reflects only the author's view, the European Commission is not responsible for any use that may be made of the information it contains.


Chiara Canta

Helmuth Cremer

Firouz Gahvari

Welfare improving tax evasion *

Keywords: JEL classication: H20, H21, H26 Optimal taxation, tax evasion, audits, welfare-improving

We study optimal income taxation in a two-group framework where the private cost of misreporting income is positively correlated with productivity. We show that, if highwage types always reveal their income truthfully, letting low-wage types cheat would lead to Pareto-superior outcomes regardless of the audit costs (as compared to deterring them).

When there is no cheating, redistribution takes place on rst-or second-best frontiers with the low-wage types always ending up worse o than the high-wage types. Letting lowwage types conceal their income reduces the need to recourse to second-best mechanisms for redistribution. Additionally, it increases the reach of rst-best redistribution to outcomes at which low-wage types are better o than high-wage types.

Introduction

It is nearly ve decades since [START_REF] Mirrlees | An exploration in the theory of optimum income taxation[END_REF] and [START_REF] Allingham | Income tax evasion: a theoretical analysis[END_REF] launched the literatures on the optimal general income tax and on tax evasion. The two, which have now grown into two of the most fertile subdisciplines in the area of economics of taxation, are both concerned with ecient and fair ways to raise tax revenues for the government. Yet, rather curiously, they have gone their own separate ways. The focus of the optimal tax literature has been on the formulation of income tax schedules, and that of the tax evasion literature on the design of enforcement policies taking the tax schedule as given. One fundamental reason for this parting of ways is no doubt their diametrically dierent foundational assumptions. Whereas the Mirrleesian optimal taxation literature assumes incomes are publicly observable, the tax evasion literature has unobservability of incomes as its raison d'être.

Over this period of time, the attempts to bring these two literatures together have been few and far between. One, published some twenty ve ago, is [START_REF] Cremer | Tax evasion and the optimum general income tax[END_REF] who, using the [START_REF] Stiglitz | Self-selection and Pareto ecient taxation[END_REF] two-group reformulation of [START_REF] Mirrlees | An exploration in the theory of optimum income taxation[END_REF], allow for incomes to be misreported and observed only through costly audits. They investigate the properties of the resulting optimal policy with a general income tax schedule, an audit policy conditioned on reported incomes, and punishment for misreporters as its instruments. Chander and Wilde (1998) consider nonlinear taxation with a continuum of individuals and establish some properties of the optimal tax schedule for the case where individuals are risk neutral. [START_REF] Schroyen | Pareto ecient income taxation under costly monitoring[END_REF] also allows for non-linear taxation but restricts the penalty to be proportional to the tax evaded.

The aim of this paper is to cast doubt on a widely-accepted view about tax evasion in the literature and the public. This is the view that tax evasion is a bad thing; that is, it lowers social welfare and must be deterred when not too costly. We ask if this is always the case: are there indeed no circumstances under which tax evasion can be a good thing which should be glossed over even if it can be deterred at a low cost (indeed costlessly)? The answer lies in a hitherto ignored role that tax evasion might play in the design of an optimal general income tax schedule. This role is due to the existence of private evasion costs and comes into play when high-wage individuals nd it more costly to engage in evasion activities than the low-wage individuals. Allowing for it, as we show below, opens up an avenue for tax evasion to become a determining factor for designing optimal tax/deterrence policies. Tolerating some tax evasion can be at times socially optimal.

The key insight comes from the realization that optimal tax systems are inherently distortive 1 [START_REF] Casamatta | Optimal income taxation with tax avoidance[END_REF] considers costly but legal avoidance so that the issue of auditing does not arise. In his setting, concealment costs are the same for everyone. [START_REF] Gahvari | Wage endogeneity, tax evasion, and optimal nonlinear income taxation[END_REF] rule out audits by following a riskless approach to evasion and assume identical concealment technologies.

as they have to ensure certain incentive compatibility constraints are satised (unless, of course, the constraints do not bite). Tax evasion can ease these constraints and alleviate the distortions.

In this way, tax evasion can improve social welfare. Needless to say, tax evasion is also, by nature, a costly activity that reduces social welfare. In designing an optimal tax system, when incomes are publicly unobservable, both of these positive and negative eects need to be taken into account. An important point to consider though is that it is not just the magnitude of misreporting costs per se that matters in the trade-o between the gains and losses. How the misreporting costs aect individual of dierent types also plays a crucial role in this calculus.

It is this aspect that helps determine which incentive compatibility constraints will loosen and thus the possibility and the extent of redistribution through the tax system.

We consider this problem within the two-group reformulation of [START_REF] Mirrlees | An exploration in the theory of optimum income taxation[END_REF] optimal income tax problem by [START_REF] Stiglitz | Self-selection and Pareto ecient taxation[END_REF]a setting we shall refer to as MS (for Mirrlees/Stiglitz).

We adopt its informational structure about the public unobservability of ability types and labor supplies, but drop its observability of incomes assumption thus allowing for the possibility of tax evasion. The key feature of our setup is that the two ability groups face dierent evasion costs. At the most general level, the two sources of heterogeneity (ability and cost of income misreporting) can be uncorrelated or correlated (positively or negatively). The popular perception is that the well-to-do have more opportunities to engage in tax evasion. Others reject this. In a recent study, for example, [START_REF] Gsottbauer | Social class and (un)ethical behavior: Causal versus correlational evidence[END_REF] nd no support for a negative correlation between social status and ethical behavior. Some believe the relationship goes the other way. Andreoni et al.

(2017) have argued that the rich are more likely to behave socially than the poor. They attribute this behavior to the diminishing marginal utility of income. Another argument that supports a correlation of this nature is the stylized fact that low-skill workers are more likely to work in the underground economy, while high-skill workers may nd it dicult or even impossible. A number of studies point to a negative relationship between education and informal labor provision; see for instance [START_REF] Pedersen | The Shadow Economy in Germany, Great Britain and Scandinavia: A Measurement Based on Questionnaire Service[END_REF], [START_REF] Oviedo | Economic informality: causes, costs, and policiesa literature survey[END_REF], [START_REF] Heigner | Driving forces of informal labour supply and demand in Germany[END_REF], and [START_REF] Kolm | Informal unemployment and education[END_REF]. To put it dierently, the educated rich nd it more costly to evade as compared to the uneducated poor.

The aim of this paper is not to take side on this ndingonly to explore its implication for devising optimal general income tax policies in the presence of tax evasion. Specically, we focus on the surprising implications of income misreporting costs for the design of optimal income tax/deterrence policies in a setting where these costs induce low-productivity types to evade but not high-productivity types. A stylized setting in which (i) the misreporting costs for the high-productivity types are large enough to prevent them from income misreporting under any circumstance and (ii) low-skill individuals can misreport their income at no cost, is the simplest vehicle for this purpose.

2 In what follows, we shall refer to high-productivity individuals who face a prohibitively large evasion cost and thus reveal their income truthfully as honest, and to our setting as EL (denoting evasion by the low-wage type).

To see how tax evasion may be welfare-enhancing in our model, consider a redistributive tax system wherein individuals who report a low income face lower taxes (or higher subsidies).

Low-skill individuals are in a position to report a low income without incurring any evasion cost which entitles them to pay a low tax or receive a high transfer. They can do this while earning more than they report (by working, for instance, in the informal market). High-skill individuals, by contrast, cannot do this. Evasion imposes a prohibitively high cost on them. To report a low income level, they must also earn a low income level. They may thus prefer to earn/report a high income and to pay a high tax, rather than earn/report a low income in order to pay a low tax. This type of evasion enables the tax system to redistribute a higher amount of resources to low-skill individuals than would otherwise be possible. One should then tolerate it rather than eradicate it through audits (even if the audits entail no costs).

That the simultaneous existence of honest in addition to dishonest taxpayers matter for equilibrium outcomes and policy design have long been recognized in the literature, but not within the Mirrleesian optimal tax paradigm. Gordon (1987) introduces a psychic cost of evasion into [START_REF] Allingham | Income tax evasion: a theoretical analysis[END_REF] and study how that changes the latter paper's results. [START_REF] Erard | Honesty and Evasion in the Tax Compliance Game[END_REF] posit a game theoretic framework to study and compare the equilibrium solutions for a model with dishonest taxpayers and a model with a mix of honest and dishonest taxpayers. Honesty is dened in terms of truthful reporting of incomes which are exogenously determined. The tax system is given with constant proportional tax and penalty rates. The policy design is limited to audits based on reported incomes. [START_REF] Alger | Screening ethics when honest agents care about fairness[END_REF] study the importance of honesty in a wider context. They consider a principal and agent framework wherein the agent has certain private information. Agents may or may not feel compelled to reveal their private information truthfully. If they do, they are referred to as honest; otherwise as dishonest. The authors introduce another layer of complication to this setup by assuming that honest agents may or may not feel compelled to reveal that they are honest. They show that the distinction matters signicantly and that this latter conditional honesty drastically aects the set of implementable allocations.

In our setup, honesty refers only to truthful reporting of incomes; the taxpayer's type always remains hidden. Incomes are endogenous and not publicly observable. The policy design includes 2 The intuition we develop in the paper for our results makes it clear that these extreme assumption are not necessary for tax evasion to be welfare-enhancing. Indeed, we present an example in Appendix D where both types face identical misreporting costs and show that there continue to be circumstances under which tax evasion improves welfare.

the tax system, whose sole restriction is incentive compatibility (with respect to the ability type).

The tax administration is able to uncover true incomes through an audit policy conditioned on reported incomes. All individuals, regardless of their type, choose their labor supply and the amount of income they want to report. Low-wage individuals have no cost associated with misreporting their income. This is not the case for high-wage individuals, whose cost of misreporting is prohibitive. 3 Consequently, their reported income is the same as their true income. If they want to choose the same consumption/reported-income bundle as low-wage individuals (mimic them as the terminology goes), high-wage individuals will have to work less hours than low-wage individuals in order to actually earn what the latter types report.

We show that allocations that can be implemented in the EL setting include the set of implementable allocations under MS. The inclusion is strict; there are rst-best allocations that cannot be implemented in the MS setting with full observability of incomes, but are implementable under EL when low-wage individuals' incomes are not observable. While these results rely on our specic assumptions about misreporting costs, we show in Appendix D that the mechanism for achieving them remains optimal for a wider range of misreporting costs. Interestingly too, auditing is never optimal in this setting. This is surprising a priori because one would expect less information to yield a worse outcome. The intuition for the result is that, if the income of low-wage individuals is not observable, the tax schedule does not aect their labor supply choice, which can then be set at its rst-best level. Hence, their reported income can be distorted down, to relax the incentive constraints of high-wage individuals, at no welfare loss. Put dierently, one can decrease the utility of the mimicker without hurting the mimicked individual. We derive conditions under which the no-audit solution implements the rst best.

These results have an interesting methodological implication whose scope goes beyond our specic setting. The existing auditing models, which consider instruments that are restricted only by the information structure (and a cap on penalties), ensure that the revelation principle applies and show that there is truthful reporting of incomes. The fact that the revelation principle and truthful reporting go hand in hand is of course trivial when there is only adverse selection (with individuals' types being their true incomes). [START_REF] Border | A theory of auditing and plunder[END_REF] Nevertheless, models that allow for moral hazard as well, as in [START_REF] Mookherjee | Optimal auditing insurance and redistribution[END_REF] or [START_REF] Cremer | Tax evasion and the optimum general income tax[END_REF], have similar properties in that the revelation principle applies and there is truthful reporting of incomes in equilibrium. In our model too, policies are restricted only by the information structure and a 3 In developing countries high income individuals can and do send their income away from their country. This is not due to their low-cost concealment technology, but to the fact that authorities turn a blind eye on their activities.

A necessary requirement for ghting evasion is to have incorruptible tax administrators. Almost all models of tax evasion, including ours, assume a well-meaning and welfare-maximizing tax authority (government). [START_REF] Slemrod | Optimal taxation and optimal tax systems[END_REF] points out, quite correctly, that a well-functioning tax system requires one to specify the enforcement mechanism. We need a completely dierent theoretical guide for addressing corruption at the top.

4 See for instance [START_REF] Border | A theory of auditing and plunder[END_REF]. maximum penalty, and the revelation principle continues to apply. Yet we do not have truthful reporting of incomes. [START_REF] Casamatta | Optimal income taxation with tax avoidance[END_REF] The crucial point here is that, because of misreporting costs, incomes are no longer just a message which can be changed at no cost. This result is not quite the same as the one of [START_REF] Green | Partially Veriable Information and Mechanism Design[END_REF], who show that, when the message space is type specic, the revelation principle may not apply. In our setting, it is only the cost of income misreporting which is type specic. Perfectly mimicking another type, by earning their assigned income, bears no cost in itself; its cost arises only when this income is misreported.

Related literature

The optimal tax evasion/deterrence literature is vast and even a concise survey of it is beyond the scope of this paper; but a few papers must be mentioned to put our ndings in proper perspective.

6 Our focus in this regard is on the existence of tax evasion in the equilibrium of their models. Most of the early literature on tax evasion, starting with [START_REF] Allingham | Income tax evasion: a theoretical analysis[END_REF], where income is exogenously given, and subsequently [START_REF] Sandmo | Income tax evasion, labour supply, and the equity-eciency tradeo[END_REF] and [START_REF] Cremer | Tax evasion, concealment and the optimal linear income tax[END_REF], where income is determined endogenously, restrict the income tax schedule to be linear. They also concentrate on purely random audits where each tax return is audited with a certain probability regardless of the amount that is reported.

7 These papers do not address the optimality of audits and, by their nature, have tax evasion in equilibrium.

Reinganum and Wilde (1986) took a crucial step forward in the study of optimal audits. They cast the problem within a principal agent setting with exogenous incomes and restrict audits to follow a simple cuto rule wherein all, and only, the reports below a certain level are audited. They show that this policy dominates purely random audits and that there is tax evasion in equilibrium. 8 Later, Chander and Wilde (1998) and many subsequent papers, took a more general approach and restricted policies only by the available information. 9 While the possibility of misreporting in these models restricts the set of feasible policies, the revelation principle applies and there is eectively no tax evasion in equilibrium. 10

Among the more recent models is [START_REF] Lehmann | Tax me if you can! Optimal nonlinear income tax between competing governments[END_REF], who consider a two-country randomutility multi-principal model wherein there is no tax evasion per se, but individuals can avoid 5 Recall that to prove the revelation principle, one shows that every mechanism that involves misreporting can be replicated by a mechanism that induces truth telling. This is achieved by giving the individuals the same payo in the incentive compatible mechanism as they would have obtained with their (possibly false) report in the initial mechanism. [START_REF] Chander | A General Characterization of Optimal Income Tax Enforcement[END_REF] We thank a reviewer who suggested this and provided helpful guidance.

7 An interesting and critical overview of this literature is provided by [START_REF] Cowell | The Economic Analysis of Tax Evasion[END_REF].

8 It is of course never optimal to report incomes above the cuto level. 9 A maximum penalty is, however, always imposed and this constraint is binding (the so called principle of maximum deterrence). 10 Marhuenda and Ortuno-Ortin (1997) show that this depends on the way the penalty function is restricted.

For some penalty functions, there will be tax evasion in equilibrium. taxes through migration. The migration cost in their setting is the counterpart to the concealment cost in ours. They show that the results crucially depend on the income elasticity of this evasion cost. The case where this elasticity decreases with the skill level is similar to our setting where evasion costs increase with wages.

11 A mention must also be made of Dhami andAl-Nowaihi (2007, 2010) who point out that the expected-utility approach the literature has followed fails to account for the fact that most individuals do not cheat even with simple audit strategies that imply a low probability of detection (so that cheating would increase expected utility).

They show that this apparent paradox can be resolved by considering prospect theory to model individuals' choices under uncertainty; see also [START_REF] Piolatto | Optimal income tax enforcement under prospect theory[END_REF].

12 These studies share a common message; namely, that the possibility of tax evasion can only lower social welfare. This is the message we challenge in the current paper. [START_REF] Davidson | Ecient Black Markets?[END_REF] have previously challenged this message but in a dierent context. They show that black markets may have a positive impact on social welfare and that they should not necessarily be done away with even if audit costs are zero. Their approach diers from ours in that they concentrate on commodity taxes so that the driving factor in their model is the quality of the products that can be sold in the black market. By the rst theorem of welfare economics, the competitive equilibrium is on PF ; it is shown by point a which is above the 45 degree line because w h > w ℓ . In what follows, we concentrate on the part of the frontier to the right of the competitive equilibrium. This implicitly assumes that the weights are such that the solution involves redistribution from the high-wage to the low-wage individuals so that T * h > T * ℓ . We know from Stiglitz (1982) that this includes the utilitarian FB obtained when α i = n i . We also assume I * ℓ > 0.

max T h ,I h ,T ℓ ,I ℓ W = α h u I h -T h , I h w h + α ℓ u I ℓ -T ℓ , I ℓ w ℓ , s.t. n h T h + n ℓ T ℓ = 0.
The Mirrlees-Stiglitz problem assumes that incomes I i = w i L i are publicly observable at no cost, but w i and L i are not (for i = h, ℓ). To determine the MS allocations, one has to add an incentive compatibility constraint (IC) to problem P F , which yields problem P M S dened as max

T h ,I h ,T ℓ ,I ℓ W = α h u I h -T h , I h w h + α ℓ u I ℓ -T ℓ , I ℓ w ℓ s.t. u I h -T h , I h w h -u I ℓ -T ℓ , I ℓ w h ≥ 0, (1)
n h T h + n ℓ T ℓ = 0.
It is common practice to refer to the utility of the h-type evaluated at the allocation intended for the ℓ-type in the incentive constraint as the utility of a ctitious mimickera terminology that we also follow. [START_REF] Stiglitz | Self-selection and Pareto ecient taxation[END_REF]

[(x * ℓ , I * ℓ ), (x * h , I * h )] = [(I * ℓ -T * ℓ , I * ℓ ), (I * h -T * h , I * h )].
frontier (PF). This is depicted by that part of PF in Figure 1 which ends at point b. In the other, the solution is given by the FOC of problem P M S along with the binding incentive compatibility constraint (1) and the resource constraint. This is depicted by the MS curve in Figure 1 that starts from point b and lies everywhere above the 45 degree line. An allocation with u l > u h would obviously violate the incentive constraint so that it cannot be achieved under MS; see footnote 18 for a formal proof. [START_REF] Border | A theory of auditing and plunder[END_REF] The model with low-wage type evaders (EL)

Consider a setting with public unobservability of ability types and labor supplies, as in MS, but with the possibility of tax evasion. Assume that the two groups dier in their cost of concealing their true incomes. High-wage individuals, who cannot be identied by the policy designer, always reveal their true income I h = w h L h , because their cost of misreporting is prohibitive. On the other hand, low-wage individuals, who are not identiable either, can misreport their income

I ℓ = w ℓ L ℓ at no cost.
Denoting reported income by I, we thus have I h = I h but I ℓ may dier from I ℓ . Assume 0 ≤ I ℓ ≤ I ℓ to rule out negative-and over-reporting of income. 14 Concentrate again on the case where the binding incentive constraint (if any), is from the h-type to the ℓ-type.

Assuming no audits are performed (we show below that this is in fact the optimal policy), the policy problem is max

T h ,I h ,T ℓ , I ℓ W = α h u I h -T h , I h w h + α ℓ u I ℓ -T ℓ , I ℓ w ℓ s.t. u I h -T h , I h w h -u I ℓ -T ℓ , I ℓ w h ≥ 0, (2)
I ℓ ≥ 0, n ℓ T ℓ + n h T h = 0,
and referred to as P ℓ . The incentive compatibility constraint ensures that high-wage individuals do not mimic low-wage individuals. Since high-wage individuals have a prohibitively high cost of misreporting, if they mimic low-wage individuals they have no choice but to earn exactly the reported income of low-wage individuals, I ℓ . 15 14 The no-overreporting constraint simplies the expressions; one can easily show that it will not be binding.

15 Without loss of generality, we concentrate on self-selecting policies. Formally, proceeding as in Laont and Martimort (2002, pp. 4850) or Salanie (1998; pp. 1718), one can show that, for any policy that does not entail truthful reporting, an incentive compatible policy can be constructed that yields the same outcome. Note that, unlike in much of the literature on optimal auditing, the revelation principle in our setting does not imply that incomes are reported truthfully. It merely implies that both types select the contract designed for them.

While the policy designer does not set I ℓ directly, it is eectively set; albeit indirectly. Lowwage individuals are induced to choose and report I ℓ and T ℓ . Then, given these values, they choose I ℓ to maximize u (I ℓ -T ℓ , I ℓ /w ℓ ). Importantly, u (I ℓ -T ℓ , I ℓ /w ℓ ) is the only term in problem (2) that depends on I ℓ . 16 Consequently, the optimal choice of I ℓ by low-wage individuals is tantamount to maximization of W with respect to I ℓ . This allow us to reformulate problem P ℓ by including I ℓ in the list of decision variables. This is represented by problem P ′ ℓ , which is the same as P ℓ except that there is an extra decision variable.

The Kuhn-Tucker expression for problem

P ′ ℓ is £ = α h u I h -T h , I h w h + α ℓ u I ℓ -T ℓ , I ℓ w ℓ + λ u I h -T h , I h w h -u I ℓ -T ℓ , I ℓ w h + γ I ℓ + µ (n h T h + n ℓ T ℓ ) . (3)
Observe that P ′ ℓ is similar to P M S except that I ℓ replaces I ℓ in the utility of the mimicker; see while it has no impact on the utility of the mimicked.

It is clear that, in the above expressions, µ > 0 because the resource constraint must be binding. However, it is possible that the other multipliers λ and γ are equal to zero. Consequently, Problem P ′ ℓ may yield dierent solution regimes depending on the pattern of the binding and non-binding constraints. To study this issue in the most ecient way, we organize our analyses around the results that are already known for the MS problem and examine if or how they may change.

Solution regimes

As with the MS setting, we distinguish between rst-and second-best regimes.

16 Unobservability of I ℓ disconnects it from the incentive compatibility constraint.

First best

The solutions under EL will be rst best if λ = γ = 0 in problem P ′ ℓ . The notable point about them is that they include all rst-best allocations that are implementable under MS. First, we know from [START_REF] Stiglitz | Self-selection and Pareto ecient taxation[END_REF] that, because the competitive equilibrium satises the incentive compatibility constraint with strict inequality, the Pareto ecient allocations in the neighborhood of this equilibrium can be implemented under the information structure in MS. Intuitively, the IC constraint is not violated when the amount of redistribution is small. The FB allocations satisfying the IC constraint are represented by the segment ab on the Pareto frontier in Figure 1; they are such that

u I * h -T * h , I * h w h ≥ u I * ℓ -T * ℓ , I * ℓ w h , (4)
where a star denotes the rst-best value of a variable.

To see that these allocation are also implementable under EL, one can simply duplicate them by setting

[(T h , I h ), (T ℓ , I ℓ )] = [(T * h , I * h ), (T * ℓ , I * ℓ )].
Although low-wage individuals can cheat at no cost, they will not do so because I ℓ = I * ℓ maximizes their utility regardless of their report (as long as

T ℓ = T * ℓ ).
The more interesting question is whether there are rst-best allocations that can be implemented under EL but not under MS. Consider a Pareto ecient allocation [(T * h , I * h ), (T * ℓ , I * ℓ )] for which (4) does not hold and thus is not implementable under MS. Then formulate an EL policy that consists of the tax function

T ( I) = T * ℓ if I = 0, = T * h if I > 0,
and no audits. In words, an individual who reports a zero income pays T * ℓ (which may be negative), while any other reported income is associated with a tax equal to T * h . The ℓ-types' best option under this scheme is to report I = 0 and pay T * ℓ which then leads them to earn I * ℓ . To report I > 0 and pay T * h > T * ℓ will only reduce their utility. As to the h-types, recall that they cannot misreport their income. Consequently, if they were to report I = 0 in order to receive -T * ℓ , they must also earn I h = 0. Their options are thus either (i) pay T * h and earn I * h or (ii) receive and consume -T * ℓ and earn I h = 0.

The rst-best solution will be implementable under EL if and only if

u I * h -T * h , I * h w h ≥ u (-T * ℓ , 0) . (5)
Now, as long as I * ℓ > 0, the right-hand side of the incentive compatibility constraint (4) is larger than that of (5):

u I * ℓ -T * ℓ , I * ℓ w h > u I * ℓ -T * ℓ , I * ℓ w ℓ > u 0 -T * ℓ , 0 w ℓ .
Condition ( 5) is thus strictly weaker than (4). Consequently, there must exist FB allocations that can be implemented under EL even though they are not under MS. These are the allocations that satisfy (5) but not (4). They are represented by segment bc on the Pareto frontier in Figure 1.

While ( 5) must hold for some Pareto ecient allocations, it will not hold for all. As long as u (-T * ℓ , 0) > 0, there will be a non-empty segment below and to the right of point b which is implementable under EL. However, when the rst-best utility of high-wage individuals gets suciently close to zero, the direction of inequality (5) will unavoidably be reversed. In words, when the utility level of high-wage individuals is suciently small, they would be better o by exiting the labor market altogether and consume the transfer -T * ℓ which is intended for the low type. The point where ( 5) is satised as equality is point c on the Pareto frontier. Any point to its right cannot be implemented through EL. This is illustrated in the numerical example given in Section 6 below.

Finally, we have thus far assumed that no audits are performed. This is in fact the optimal policy because audits can only do harm. With a positive probability of audits, low-wage individuals might nd it optimal to report I ℓ > 0. Which will then mean that they have to pay T * h rather than T * ℓ , making the implementation of [(T * h , I * h ), (T * ℓ , I * ℓ )] no longer feasible. Moreover, when I ℓ > 0, the incentive compatibility constraint (5) would have to be amended, thus making the mimicking option more attractive for the h-type. This in turn will reduce the set of FB allocations that can be implemented.

The results derived thus far are summarized in the following proposition.

Proposition 1 Consider the EL and MS settings as dened above. (i) The set of rst-best allocations that can be implemented under EL includes the set of allocations that is implementable in the MS setting. (ii) As long as I * ℓ > 0, the inclusion is strict so that there exist FB allocations that can be implemented under EL but not under MS. (iii) Not all FB allocations are implementable under EL. (iv) Auditing is never desirable.

That the unobservability of I ℓ leads to the implementability of rst-best allocations, unattainable when I ℓ is observable at no cost, is a rather striking result. Having less information is expected to bring about a worse outcome, not a better one. To garner intuition for this, remember that, in the MS setting, distorting the low-wage individuals' labor supply downwards is needed to make mimicking less attractive to high-wage individuals. The distortion is no longer needed for this purpose, nor is it possible to induce it, under EL. The incentive constraint is manipulated through I ℓ in a way that makes mimicking even more costly than under MS. Low-wage individuals report I ℓ and choose their most desired level of labor supply which would be FB (regardless of the marginal tax rate on I ℓ ).

Second best

We now turn to the case where FB allocations violate condition (5) so that they cannot be implemented. These are the allocations that lie to the right of and below point c on the Pareto frontier in Figure 1. The Kuhn-Tucker conditions of problem P ′ ℓ stated in Appendix A yield second-best solutions at which condition ( 5) is binding and λ > 0. Of course, with the resource constraint always binding, µ must also be positive. Lemma 1, stated and proved in Appendix B, proves that γ > 0 as well. This is because the optimal policy continues to imply I ℓ = 0, though this time as a corner solution. Given I ℓ = 0, it follows from the Kuhn-Tucker conditions that 17

u c I h -T h , I h w h + 1 w h u L I h -T h , I h w h = 0, (6)
u c I ℓ -T ℓ , I ℓ w ℓ + 1 w ℓ u L I ℓ -T ℓ , I ℓ w ℓ = 0, (7)
u I h -T h , I h w h -u (-T ℓ , 0) = 0. (8)
Equations ( 6)(8), along with the resource constraint n ℓ T ℓ + n h T h = 0, determine a unique set of values for I h , T h , I ℓ , T ℓ . In particular, these values are independent of the weights assigned to u h and u ℓ . Moreover, these equations are precisely the same equations that determine the rst-best allocation [(T * h , I * h ), (T * ℓ , I * ℓ )] at point c in Figure 1. Recall that c is the boundary point on the Pareto frontier satisfying the IC constraint as an equality (beyond it the IC constraint will be binding).

Intuitively, equation ( 6), along with the resource constraint, implies that I h is set at its rstbest level. Consequently, equation ( 8 any larger value of α ℓ continues to yield the solution given by point c. When redistribution is limited by a positive value of I ℓ , it will no longer be possible to make the ℓ-types any better-o by decreasing I ℓ notwithstanding the fact that higher values of α ℓ call for it. The numerical example of Section 6 below illustrates this point. The oor to redistribution is thus dictated by 17 In Appendix A, set I ℓ = 0 in the Kuhn-Tucker conditions (A1a)(A1g). This simplies equations (A1b), (A1e)(A1f) into equations ( 6) [START_REF] Cremer | Tax evasion and the optimum general income tax[END_REF].

I ℓ = 0 with I ℓ ≤ I ℓ .
Observe that the EL policy can increase the utility of the ℓ-types beyond what is feasible under MS. There, the ℓ-types always remain less well-o than the h-types. 18 In contrast, point c is necessarily below the 45 degree line. At point c, where the incentive constraint starts to bind, we have

u h = u I * h -T * h , I * h w h = u (-T * ℓ , 0) < u ℓ .
The inequality follows because I ℓ = 0 makes the consumption bundle (-T * ℓ , 0) available to the ℓ-types under EL. The inequality is strict as long as I * ℓ > 0. Compared to the competitive equilibrium, the ranking of utilities is reversed. This property also implies that EL can implement the Rawlsian FB solution (where the 45 degree line intersects the Pareto frontier). It follows that the FB solutions not implementable under EL are rather extreme and go beyond the usual notion of income redistribution in the sense that they reverse the order of inequality between the two types.

Finally, observe that auditing has only the eect of decreasing the utility of the mimicked individual without aecting the incentive constraint. Consequently, the optimal policy entails no audits. The results derived above are summarized in the following proposition. (ii) Point c is below and to the right of the point where the 45 degree line intersects the Pareto frontier, so that u ℓ > u h ; consequently the MS frontier, which lies above the 45 degree line, must lie everywhere to the left of point c in Figure 1. 18 To see this, observe that with w h > w ℓ , and the fact that utility changes negatively with labor supply, we have

u I ℓ -T ℓ , I ℓ w h > u I ℓ -T ℓ , I ℓ w ℓ .
Moreover, the IC constraint requires

u I h -T h , I h w h ≥ u I ℓ -T ℓ , I ℓ w h .
It follows from these two inequalities that

u I h -T h , I h w h ≥ u I ℓ -T ℓ , I ℓ w h > u I ℓ -T ℓ , I ℓ w ℓ .
In this regard, Figure 2.1 in Stiglitz (1987, p. 2.1) which shows the FB and SB frontiers is misleading. Under MS, the IC constraint ( 4) is satised as a strict inequality if and only if α ℓ < 0.42.

Under EL, the IC constraint (5) holds as a strict inequality if and only if α ℓ < 0.54. Consequently, for all α ℓ ∈ [1/3, 0.42], the rst-best is implementable under both MS and EL settings. These allocations correspond to the points on ab segment of the Pareto frontier. When α ℓ exceeds 0.42, the rst-best allocations can no longer be implemented under MS. However, as long as α ℓ ∈ [0.42, 0.54], the rst-best is implementable under EL. These allocations are shown in Figure 2 as segment bc on the PF. When α ℓ exceeds 0.54, the EL setting too cannot implement the corresponding FB allocations. These are shown by segment cf on the PF in Figure 2.

Turning to allocations that are second-best, they will be attained under MS for α ℓ > 0.42 and shown in Figure 2 by the bb ′ curve that lies everywhere below the PF. It approaches the 45 degree line as α ℓ → 1. The limiting ℓ-types' allocation is I ℓ = 0, c ℓ = -T ℓ = 1.53 resulting in u ℓ = 2.47. The corresponding values for the h-types are I h = 10, c h = 8.48, and u h = 2.49.

Under EL, on the other hand, the second-best allocation is unique and given by point c. This is the case for all α ℓ above 0.54. The reason for it is that at α ℓ = 0.54, I ℓ = 0. Interestingly though, whereas u ℓ = 2.48 for α ℓ = 1 under MS, u ℓ = 2.95 > 2.48 for α ℓ ≥ 0.54 under EL.

Table 1 illustrates the laissez-faire allocations as well as the rst best, MS, and EL allocations for dierent welfare weights. When low-wage individuals have a welfare weight equal to α ℓ = 0.35 < 0.42, the rst best is implementable under both MS and EL. When α ℓ = 0.5 > 0.42, the rst-best allocation continues to be implementable under EL but not MS. This is the Rawlsian FB solution with u ℓ = u h = 2.72. Observe that the second-best MS solution for α ℓ = 0.5 entails a lower utility level for the ℓ-types as compared to what they can attain under EL (2.34 versus 2.72), but the h-types enjoy a higher utility level (3.05 versus 2.72.) At α ℓ = 0.54, the EL solution remains rst-best with u ℓ = 2.95 and u h = 2.48. This corresponds to point c on PF and caps the utility level the ℓ-types can attain under EL. Observe also that u ℓ > u h at this point.

Raising α ℓ further does not change the optimal allocations under ELnot even as a secondbest solution. Table 1 illustrates this point by nding the solution for α ℓ = 0.65 for which we continue to have u ℓ = 2.95 and u h = 2.48 under EL as we had with α ℓ = 0.54. However, for α ℓ = 0.65, this allocation is no longer FB. At the FB allocation for this value of α ℓ , the ℓ-types attain a higher utility level equal to u ℓ = 3.39 (and the h-types a lower utility level equal to u h = 1.82.) The second-best MS allocation at α ℓ = 0.65 results in u ℓ = 2.42 (and u h = 2.93),

which is worse for the ℓ-types as compared to EL.

7 Discussion: positive and non-prohibitive misreporting costs

The results we have derived on the implementability of FB allocations under EL, rest on the fact that in equilibrium no resources are spent on misreporting (the high-wage does not misreport and the low-wage misreports at zero cost). It is plainly obvious that when solution entails costly misreporting, one cannot attain the FB utility levels. Nevertheless, it may still be possible to retrieve the FB allocations of consumptions and labor supplies. [START_REF] Kolm | Informal unemployment and education[END_REF] The crucial comparison, 19 This depends on whether evasion costs are specied in terms of utility or real resources.

u ℓ = 2.34, u h = 3.05 u ℓ = 2.37, u h = 3.02 u ℓ = 2.42, u h = 2.93 LF I ℓ = 4, I h = 10, u ℓ = 1.59, u h = 3.50
Table 1: Incomes, taxes, and utility levels in rst-best, EL, MS, and laissez-faire allocations.

Utilities have the functional form u = 2(c + γ ln(1 -L)) 0.5 , with w ℓ = 14, w h = 20, and γ = 10.

however, is with the MS solutions and whether or not EL dominates MS. We briey discuss this question below, relegating the formal presentation to Appendix D. As we shall see, our mechanism remains optimal for a wide range of misreporting costs. 20

The rst extension in Appendix D is to replace the ℓ-type's zero cost of misreporting with costly misreporting. As long as the misreporting cost is reletively small, our mechanism continues to be implementable. The reason is that, when low-wage individuals face a small cost of misreporting under our mechanism (for earning a positive income while reporting zero), the cost can fall short of the gain they make by attaining the rst-best level of labor supply. Interestingly too, the low-cost of misreporting allows the mechanism to dominate MS. The intuition for this is that the labor supply decision of the low-wage individuals is distorted under MS (due to incentive compatibility constraint). Our mechanism enhances social welfare by eliminating this distortion.

At the same time, it lowers social welfare because of the cost of misreporting. As long as the latter costs are relatively small, the gain outweighs the cost.

We also consider the case where both types face positive misreporting costs. We show that our mechanism continues to be feasible and dominates MS if the misreporting cost is neither too large (to ensure that low-wage types misreport), nor too small (to ensure that high-wage types do not misreport), and if the weight of the low-wage types in the social welfare function is not too large (to ensure that the misreporting costs incurred by the ℓ-types do not lower social welfare by an exceedingly large amount).

20 We thank a reviewer for suggesting this extension.

Concluding remarks

Three decades ago Slemrod (1990, p. 157) wrote ...in its current state, optimal tax theory is incomplete as a guide to action concerning the questions that began this paper and for other issues in tax policy. It is incomplete because it has not yet come to terms with taxation as a system of coercively collecting revenues from individuals who will tend to resist. The few attempts that have been made to address this problem, since it was identied by Slemrod, remain far and between. In this paper, we have tried to take a step forward in this direction by integrating one particular factor that enters individuals' tax evasion decision making into the design of optimal general income tax schedules.

The factor in question is the private cost of evasion; particularly the way it aects dierent ability types dierently (whether in terms of real resource cost to conceal the evasion or in terms of an internal utility cost). These dierentials can help in making a redistributive tax system less distortionary. What makes a tax system distortionary are the incentive compatibility constrains embedded into its structure. To the extent that private evasion costs aect dierent ability types dierently, these costs can be used to devise policies that ease the incentive compatibility constraints into a more desired direction that allow for more redistribution.

To bring out the factors that come into play, in the simplest possible way, we have considered a setup in which high-wage types face so high an evasion cost as to make them always report their incomes truthfully. The low-wage types, on the other hand, face no evasion costs. Interestingly, and rather surprisingly, we have found that for every allocation that is feasible when both types are truthful (MS solutions), there exists a corresponding weakly Pareto-superior allocation if the low-wage types cheat but the high-wage types do not (EL solutions). Moreover, auditing is never desirable even if it can be done at no cost. Specically, we have shown that: (i) Every rst-best allocation that can be implemented under MS can also be implemented under EL. Neither of these type of solutions are available under MS.

Note that we do not undo tax evasion in our analysis; quite the opposite. In most of the audit literature, there is eectively no tax evasion in equilibrium. In these models, the revelation principle implies truthful reporting of incomes but the possibility to evade aects the incentive constraints. We follow this tradition in that we do not impose any ad hoc restrictions on the policy and derive the best allocation that can be achieved given the available information. The interesting feature is that, while the revelation principle continues to apply when it comes to the reporting of types, it no longer implies truthful reporting of income. Consequently, the optimal policy involves tolerating some evasion rather than getting rid of it by auditing individuals that report a low income.

There are many directions in which this work can be extended to integrate tax evasion into the Mirrleesian optimal income tax framework. The rst obvious direction is to relax the assumption of perfect correlation between ability type and unwillingness to misreport income; Appendix D is a tiny step in this direction. Either one of the two individual types, or both, may include honest and dishonest income reporters. Another avenue is to explore the implications of introducing some kind of conditional honesty along the lines of [START_REF] Alger | Screening ethics when honest agents care about fairness[END_REF]. The message of [START_REF] Slemrod | Optimal taxation and optimal tax systems[END_REF] remains as relevant today as it was three decades ago.

Solving equations (A5)(A6) along with the resource constraint for c h , c ℓ , and T ℓ yields

T * ℓ = α 2 h [w ℓ -β + β ln (β/w ℓ )] -α 2 ℓ [w h -β + β ln (β/w h )] α 2 ℓ + α 2 h , c * h = w h -β + T * ℓ , c * ℓ = w ℓ -β -T * ℓ .
Given the specication for the utility function [START_REF] Cremer | Tax evasion, concealment and the optimal linear income tax[END_REF], and using the rst-best values of c h , c ℓ , L h , and L ℓ from above equations, we have

u 2 h + u 2 ℓ = 4 [w h + w ℓ -2β + β ln(β/w h ) + β ln(β/w ℓ )] .
The Pareto frontier is found from this equation to be

u ℓ = 2 w h + w ℓ -2β + β [ln(β/w h ) + ln(β/w ℓ )] - u 2 h 4 . 5
Using the above values for c * h , L * h , c * ℓ , L * ℓ in the utility function ( 9), we have:

u c * h , I * h w h = 2α h (w h + w ℓ -2β) + β [ln (β/w ℓ ) + ln(β/w h )] α 2 ℓ + α 2 h .5 , (A7) u c * ℓ , I * ℓ w h = 2 β ln w h -w ℓ + β w h + α 2 ℓ [w h + w ℓ -2β + β ln (β/w h )] -βα 2 h ln (β/w ℓ ) α 2 ℓ + α 2 h 0.5 , (A8) u (-T * ℓ , 0) = 2 α 2 ℓ [w h -β + β ln (β/w h )] -α 2 h [w ℓ -β + β ln (β/w ℓ )] α 2 h + α 2 ℓ 0.5 . (A9)
Comparing (A8) with (A9), one nds that u (-T * ℓ , 0) < u c * ℓ ,

I * ℓ w h if and only if w ℓ -β + β ln 1 - w ℓ -β w h > 0, (A10)
which is true for all w h > w ℓ > β > 0 (as required for I * ℓ > 0). 21 21 With w h > w ℓ and w ℓ -β,

w ℓ -β + β ln 1 - w ℓ -β w h > w ℓ -β + β ln β w ℓ > 0.
Second best under MS Under MS, the rst-best allocation is implementable as long as (A10) is satised. If this is not the case, the problem of the social planner is max

I ℓ ,I h ,T 2α h I h + T + β ln 1 - I h w h 0.5 + 2α ℓ I ℓ -T + β ln 1 - I ℓ w ℓ 0.5 (A11) s.t. I h + T + β ln 1 - I h w h ≥ I ℓ -T + β ln 1 - I ℓ w h .
It is straightforward to show that the second-best allocation is characterized by no distortion at the top, i.e., I M S h = w h -β. Using this condition, the incentive compatibility constraint can be rewritten as

w h -β + T + β ln β w h ≥ I ℓ -T + β ln 1 - I ℓ w h ,
which, when binding, implies that

T = I ℓ + β -w h -β ln(β) + β ln(w h -I ℓ ) 2 .
Substituting this expression for T and I M S h = w h -β in (A11), the problem of the social planner can be rewritten as max

I ℓ 2α h w h -β + β ln(β) 2 -β ln (w h ) + I ℓ + β ln(w h -I ℓ ) 2 0.5 + 2α ℓ w h -β + β ln(β) 2 + I ℓ -β ln(w h -I ℓ ) 2 + β ln 1 - I ℓ w ℓ 0.5
The rst-order condition, for an interior solution, is

1 2 + β 2(w h -I M S ℓ ) α h u h + 1 2 + β 2(w h -I M S ℓ ) + β w ℓ -I M S ℓ α ℓ u ℓ = 0.
Second best under EL The second-best outcome occurs when I ℓ = 0 and we have a unique solutions with I EL h and I EL ℓ set at their rst-best levels with T being determined from (A1f) as

T EL = - I * h + β ln(1 -I * h /w h ) 2 = - w h -β + β ln(β/w h ) 2 .

D Positive misreporting costs D.1

Positive misreporting cost for the ℓ-type

Let δ i > 0 denote the misreporting cost of type i and assume this is a xed utility cost which is incurred whenever the individual misreports. In this case, the utility of individual i reporting type i is

u i =        u(I i -T i , I i w i ) -δ i if I i ̸ = I i u(I i -T i , I i w i ) if I i = I i .
Recall that the EL mechanism consists of the tax function

T ( I) = T * ℓ if I = 0, = T * h if I > 0.
With a prohibitively high δ h , the high-wage individuals will never misreport. Our mechanism thus achieves to decentralize the rst-best consumption and labor supply levels if low-wage individuals nd it optimal to misreport when I ℓ = 0. This requires

u(I * ℓ -T * ℓ , I * ℓ w ℓ ) -δ ℓ ≥ u(-T * ℓ , 0) ⇐⇒ δ ℓ ≤ δ ℓ ≡ u(I * ℓ -T * ℓ , I * ℓ w ℓ ) -u(-T * ℓ , 0).
Now, because δ ℓ > 0, there must exist positive levels of misreporting costs such that the mechanism we have characterized decentralizes the rst-best consumption and labor levels. However, the rst-best utility level cannot be achieved. This is because our mechanism implies misreporting by low-wage individuals and that is costly. Social welfare under this mechanism is then

W F B -α ℓ δ ℓ .
Given that social welfare now falls short of W F B , the EL solution no longer necessarily dominates the MS solution under which incomes are observable. Denoting the welfare achievable under MS by W M S , EL dominates only if

W F B -α ℓ δ ℓ ≥ W M S ⇐⇒ δ ℓ ≤ δ ℓ ≡ W F B -W M S α ℓ .
This condition can never be satised for values of α ℓ such that the FB allocation on the Paretofrontier lies to the left of point b. Such allocation are implementable under MS so that W M S = W F B , which implies that δ ℓ = 0 and there exists no δ ℓ > 0 that satises the above condition.

To the right of point b, the rst-best allocations are not implementable under MS so that δ ℓ ≡ (W F B -W M S )/α ℓ > 0 and the above condition can be satised.

Considering all these restrictions, EL dominates MS only if δ ℓ ≤ min[δ ℓ , δ ℓ ] and α ℓ large enough, so that the FB allocation lies on the Pareto-frontier to the left of point b. Start rst by considering point a in Figure 1 which corresponds to the competitive equilibrium. At this point, T * ℓ = T * h = 0 and U hℓ -U h = 0, so that the rst condition is always satised.

It is thus sucient that the second condition holds at point a for our mechanism to decentralize the rst-best consumption and labor supply levels. This requires δ to be suciently small.

Next, let α ℓ increase from its value at point a, so that the rst-best allocations move to the right of point a along the Pareto frontier. It is easy to show that as α ℓ increases, U hℓ increases while U h decreases. Hence U hℓ -U h increases in α ℓ .

To determine how an increase in α ℓ aects U ℓ -u(-T * ℓ , 0), partially dierentiate it with respect to α ℓ : With U hℓ -U h increasing and U ℓ -u(-T * ℓ , 0) decreasing in α ℓ , there must exist a threshold α ℓ such that U hℓ -U h ≤ U ℓ -u(-T * ℓ , 0) if and only if α ℓ ≤ α ℓ . As a consequence, our mechanism cannot decentralize the rst-best consumption and labor supply levels if α ℓ > α ℓ . When α ℓ ≤ α ℓ , the mechanism is successful in decentralizing the rst-best consumption and labor supply levels if and only if

U hℓ -U h ≤ δ ≤ U ℓ -u(-T * ℓ , 0).
Finally, for the EL mechanism to dominate MS when audits are costless, one has to compare the welfare it achieves, W F B -δ, with the one achievable under MS. Denote the welfare attainable under MS by W M S , misreporting is optimal only if

W F B -δ ≥ W M S ⇐⇒ δ ≤ δ ≡ W F B -W M S .

3

  The benchmark model (MS) Consider an economy with two types of individuals, denoted by i = h, ℓ, who dier in their productivity w h and w ℓ with w h > w ℓ . There are n h individuals of type h and n ℓ individuals of type ℓ. Preferences over consumption x and labor supply L are represented by the utility function u(x, L), satisfying the standard properties. Denote pre-tax incomes by I i = w i L i , tax payments by T i and assume purely redistributive taxes. The full information Pareto-frontier is obtained by maximizing a weighted sum of utilities with weights α h and α ℓ such that α h + α ℓ = 1 subject to the resource constraint. It is determined by solving problem P F dened as

Figure 1 :

 1 Figure 1: Pareto frontier and implementable allocations under MS and EL.

( 1 )

 1 and (2). Importantly, though, problem P ′ ℓ contains an extra choice variable in comparison with P M S . This variable is I ℓ and can always be set equal to I ℓ to obtain the MS allocation. The idea we develop below is the possibility of choosing I ℓ such that u I ℓ -T ℓ , I ℓ /w h < u (I ℓ -T ℓ , I ℓ /w h ). If this is possible, reducing u I ℓ -T ℓ , I ℓ /w h relaxes the otherwise binding incentive constraint and allows for increased redistribution to enhance welfare. Intuitively, reducing the reported income of the low wage type, I ℓ , hurts the mimicker (who cannot misreport)

  ) is simply condition[START_REF] Casamatta | Optimal income taxation with tax avoidance[END_REF] holding as an equality which is precisely the denition of point c. At this point, the high-wage individuals are indierent between paying the high tax that goes with working and receiving a transfer without working. In other words, increasing α ℓ above its value at point c does not change the values of [(T * c ℓ , I * c ℓ ), (T * c h , I * c h )];

Proposition 2

 2 In the second best: (i) The solution under EL, [(T * c h , I * c h ), (T * c ℓ , I * c ℓ )], is unique. It is represented by point c in Figure 1 regardless of h-and ℓ-types' weights in the social welfare function.

6

  A numerical example This section illustrates our results through a numerical example. Details of the derivations are presented in Appendix C. Assume preferences are quasilinear and represented by the following utility function: u = 2 [c + 10 ln(1 -L)] 0.5 , (9) where 0 < β < w ℓ . The two ability-types are of equal size with the population size being normalized at one. Hence n ℓ = n h = 1/2. Additionally, set the parameter values w ℓ = 14 and w h = 20. Given these values, the Pareto frontier is represented by u 2 h + u 2 ℓ = 14.82 as depicted in Figure 2. The lowest attainable utility level is 0 and the highest 3.85.

Figure 2 :

 2 Figure 2: Pareto frontier and implementable allocations under MS and EL. Utilities have the functional form u = 2(c + β ln(1 -L)) 0.5 , with w h = 20, w ℓ = 14, and β = 10.

FBI ℓ = 4 , I h = 10 T = - 2 . 24 I ℓ = 4 ,

 4102244 I h = 10 I ℓ = 4, I h = 10 u ℓ = 3.39, u h = 1.82 EL I ℓ = 4, I h = 10 T = -1.22 T = -1.53 I ℓ = 4, I h = 10 T = -0.20 u ℓ = 2.72, u h = 2.72 u ℓ = 2.95, u h = 2.48 T = -1.53 u ℓ = 1.82, u h = 3.39 u ℓ = 2.95, u h = 2.48 MS I ℓ = 3.52, I h = 10 I ℓ = 3.27, I h = 10 I ℓ = 2.67, I h = 10 T = -0.74 T = -0.80 T = -0.92

  (ii) Every utility level that low-wage individuals can have under a second-best MS solution is available to them as a rst-best EL solution. (iii) The rst-best EL solutions include the Rawlsian outcome as well as outcomes wherein low-wages individuals are better o than high-wage individuals.

D. 2

 2 Positive misreporting cost for both typesAssume that both types face a utility cost equal to δ when misreporting. As previously, low-wage individuals who report I = 0 will choose to evade ifu(I * ℓ -T * ℓ , I * ℓ w ℓ ) -δ ≥ u(-T * ℓ , 0).

  to high-wage individuals, they have two options if they were to report I = 0 and pay T * ℓ .They can choose not to earn any income or earn an extra income and hide it. The rst option gives them a utility level equal to u(-T * ℓ , 0), and the second option u(I * hℓ -T * ℓ , I * hℓ /w h ) -δ where I * hℓ denotes the income they earn in this case. Consequently, high-wage individuals will choose to report I > 0 and pay T * h if they prefer the (I * h , T * h ) allocation to either of the two they canhave if they report I = 0. That is, -δ ≤ u(-T * ℓ , 0). (A14)But since the mimicker has a higher utility than the mimicked, u(I * hℓ -T * ℓ , I * hℓ /w h ) > u(I * ℓ -T * ℓ , I * ℓ /w ℓ ), there cannot exist any positive δ for which (A12) and (A14) hold simultaneously.Consequently, if our mechanism is implementable, it must be the case that if high-wage individuals report I = 0, they will earn a positive income and evade. That is, -δ ≥ u(-T * ℓ , 0).The conditions under which our mechanism is feasible are thus:u(I * h -T * h , I * h w h ) ≥ u(I * hℓ -T * ℓ , I * hℓ w h ) -δ, (A15) u(I * ℓ -T * ℓ , I * ℓ w ℓ ) -δ ≥ u(-T * ℓ , 0). (A16) Let U hℓ ≡ u(I * hℓ -T * ℓ , I * hℓ /w h ), U h ≡ u(I * h -T * h , I * h /w h ), and U ℓ ≡ u(I * ℓ -T * ℓ , I * ℓ /w ℓ ). Using these notations, conditions (A15)(A16) can be rewritten as δ ≥ U hℓ -U h , δ ≤ U ℓ -u(-T * ℓ , 0).

∂

  (U ℓ -u(-T * ℓ , 0)) ∂α ℓ = -[u c (I * ℓ -T * ℓ , I * ℓ /w ℓ ) -u c (-T * ℓ , 0)] ∂T * ℓ ∂α ℓ ,where u c (•, •) denotes the partial derivative of u(•, •) with respect to its rst argument (consumption). The bracketed expression on the right-hand side of the above equation shows the change in the marginal utility of consumption as income increases. It is negative as long as the marginal utility of consumption is increasing in leisurean assumption we maintain. Now, because ∂T * ℓ /∂α ℓ ≤ 0, it follows that U ℓ -u(-T * ℓ , 0) decreases as α ℓ increases.

However, since they consider a continuum of types and, most signicantly, introduce strategic interaction between the two principals, their results cannot be directly compared with ours.

They use the same mechanism design problem as in Chandler and Wilde (1998) but with agents who behave under prospect theory.

We thank the editor and the reviewers for their helpful comments and suggestions. Helmuth Cremer gratefully acknowledges the nancial support from Chaire Marché des risques et creation de valeur of the FdR/SCOR. as well as the funding received by TSE from ANR under grant ANR-17-EURE-0010 (Investissements d'Avenir program).

Appendix

A First-order conditions of Problem [START_REF] Andreoni | Are the rich more selsh than the poor, or do they just have more money? A natural eld experiment[END_REF] The rst-order (Kuhn-Tucker) conditions are

B Lemma 1: statement and proof Lemma 1 If the Lagrange multiplier λ in Problem P ′ ℓ is positive, the Lagrange multiplier γ is also positive.

Proof. Observe rst that I ℓ has to be set to minimize the utility of the mimicker u hℓ I ℓ -T ℓ , I ℓ /w h which is the only term in problem P ′ ℓ that it aects. Specically, given our assumptions (in par- ticular that of no over-reporting), we must minimize u hℓ over I ℓ ∈ [0, I ℓ ].

Let M RS denote the marginal rate of substitution between consumption and income.

Given the standard assumptions on u (•, •), u hℓ (•, •) is concave with an interior maximum where M RS hℓ I -T ℓ , I/w h = w h . Now, from the rst-order condition (A1e), we have M RS ℓℓ (I ℓ -T ℓ ,

It follows from inequality (A2) and the concavity of u hℓ that I > I ℓ . Consequently, u hℓ is increasing over [0, I ℓ ], which in turn implies that it is minimized at I ℓ = 0. This also implies that the bracketed expression in equation ( A1d) is positive resulting in a positive solution for γ.

C Details of derivations for the example

In the laissez-faire, with c i = w i L i , each individual maximizes

with respect to L i . The rst-order condition reduces to w i -β/(1-L i ) = 0. Assuming w i -β > 0 for i = h, ℓ, this yields interior solutions:

First-best solution The Lagrangian expression for the maximization of α h u h + α ℓ u ℓ subject to the resource constraint is

The rst-order conditions for this problem are, for i = h, ℓ,

Substitute for µ from (A3) into (A4) and solve for L i to get

We also have, from (A3),

Observe also that at point a and all points to the left of it on the FB frontier, the FB is attainable under MS so that W F B -W M S = 0. The EL mechanism cannot then be optimal for low values of α ℓ corresponding to these allocations.