Kien Pham

Ronan Sauleau
email: ronan.sauleau@univ-rennes1.fr

Erwan Fourn
email: erwan.fourn@insa-rennes.fr

Fatimata Diaby
email: fatimata.diaby@cea.fr

Antonio Clemente
email: antonio.clemente@cea.fr

Laurent Dussopt
email: laurent.dussopt@cea.fr

Metal-Only Transmitarray Based on C-Shaped Slot

Keywords: metal-only transmitarray, Ka-band, Satcom

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I. INTRODUCTION

Transmitarrays antennas (TAs) are mature and eligible antenna technologies for many applications demanding high gain and relatively wide bandwidth. Thanks to its spatially-fed configuration in transmission mode, TAs do not suffer from feed blockage as observed in reflectors or reflectarray antennas. In addition, TAs exhibit lower insertion loss as compared to phased arrays whose beam forming network is complex and relatively lossy at higher frequency. Many different types of TA prototypes have been demonstrated in the past with high gain (> 30 dBi), broad bandwidth, dual-band operation and/or dual-polarization, beam-switching and steering capabilities, etc. Among those, most of the corresponding TA demonstrators are fabricated using printed circuit board (PCB) technology; the associated fabrication cost contributes to a major part of the total cost of the antenna and leads to rather expensive solutions for mass product applications. This motivates a recent interest on a very lowcost solutions based on TAs without dielectric substrates, also referred to as metal-only transmitarray antennas (MOTAs).

Several publications illustrate this trend and have validated such concepts in X-band [START_REF] Abdelrahman | Transmitarray antenna design using cross slot elements with no dielectric substrate[END_REF]- [START_REF] Rahmati | High-efficient wideband slot transmitarray antenna[END_REF] and Ku-band [START_REF] Liu | A high efficiency transmitarray antenna using double split rings slot elements[END_REF]- [START_REF] Wu | High-gain dualband transmitarray[END_REF]. Our objective is to demonstrate a low-cost transmitarray for ground terminals operating in the Satcom downlink K-band (17.7 -21.2 GHz). The proposed unit-cell is based on a C-shaped slot frequency selective surface operating as a band pass filter. It is described in Section II. An example of 20-by-20 element TA is designed and its performance is presented in Section III.

II. UNIT-CELL FOR METAL-ONLY TRANSMITARRAY

A. Geometry

The three-dimensional geometry of the proposed unit-cell is depicted in Fig. 1. This cell is based on a C-shaped slot frequency selective surface, with a periodicity of 7 mm, which is less than half of a wavelength at 18.75 GHz. The metal layer thickness is 500 µm, and the material is chosen to be conductive, e.g. copper, aluminum, etc. A four-layer stack-up configuration is selected to cover a phase-shift range of 360°. The layer separation is about quarter wavelength (4 mm) at 18.75 GHz. The bandpass response of this unit-cell can be shifted by changing the inner radius r i of the C-shaped slot, as shown in Fig. 1. In this study, the outer slot radius is fixed and is equal to 3.3 mm.

B. Performance

The metal-only unit-cell (MOUC) has been characterized numerically by full-wave simulations using Ansys HFSS v.15 with periodic boundary conditions. The main results in transmission are presented in Fig. 2 for four frequency points in the down-link band. The unit-cell exhibits a phase-shift of more than 360° (Fig. 2b) when the value of the inner radius r i varies between 1.5 mm and 2.7 mm. It is interesting to see that the phase curves are almost parallel; this indicates the wide bandwidth properties of the proposed unit-cell. The transmission magnitude at 19 GHz is better than -3 dB and higher degradations are observed at the band edges (18 GHz and 21 GHz) (Fig. 2a).

III. EXAMPLE OF METAL-ONLY TRANSMITARRAY

The proposed MOUCs have been used to design several MOTA prototypes of various sizes. All prototypes radiate at boresight at K-band. We describe here the main performance of one of these prototypes. It has been designed at 19 GHz and contains 400 MOUCs arranged in a 20-by-20 element square lattice. The directivity and gain are plotted in Fig. 3 as a function of F/D, assuming no insertion loss and ideal phase compensation at 19 GHz. The TA is illuminated by a 10-dBi feed horn. The optimal F/D maximizing the antenna gain is equal to 0.86, equivalent to a focal length of 122 mm. The computed phase distribution across the transmitarray aperture at 19 GHz is depicted in the inset of Fig. 3 for this optimal position of horn.

The radiation patterns have been computed using array theory [START_REF] Pham | Dual-band transmitarrays with dual-linear polarization at Kaband[END_REF] and HFSS simulations. The E-plane patterns are represented in Fig. 4. The gain reaches 24 dBi and crosspolarization level below -40 dB in all elevation angles.

IV. CONCLUSIONS AND PERSPECTIVES

Metal-only unit-cells and transmitarrays have been introduced for very low cost applications, particularly for Satcom ground terminals at down-link band. The corresponding unit-cells are based on pass-band C-shaped frequency selective surfaces. Fabrication of the prototypes is under progress and detailed experimental results will be presented during the conference (fabrication techniques, assembly, and performance in radiation). The proposed concept can be scaled for other applications (e.g., up-link Satcom in Ka-band, outdoor flat antenna panels for 5G access points and backhaul links in Ka-band and V-band).

Fig. 1 .

 1 Fig. 1. Geometry of the proposed MOTA unit-cell based on C-shaped slot frequency selective surface. Dimensions are in mm.

Fig. 2 .

 2 Fig. 2. Varition of the transmission coefficients in magnitude (a) and phase (b) of the metal-only unit-cell as a function of the inner diameter ri, for four frequency points in the down-link band.

Fig. 3 .

 3 Fig. 3. Computed-directivity and gain of 20-by-20 MOTA for various F/D values at 19 GHz. Inset: phase layout for the optimal F/D value (0.86).

Fig. 4 .

 4 Fig. 4. Simulated co-polarized gain components of the MOTA at 19 GHz.

This works is supported by the French National Research Agency (ANR) in the project named TRANSMIL (TRANSmitarray antenna at MILlimeterwave) under Grant ANR-14-CE28-0023.


Kien Pham

Ronan Sauleau
email: ronan.sauleau@univ-rennes1.fr

Erwan Fourn
email: erwan.fourn@insa-rennes.fr

Fatimata Diaby
email: fatimata.diaby@cea.fr

Antonio Clemente
email: antonio.clemente@cea.fr

Laurent Dussopt
email: laurent.dussopt@cea.fr

Polarization Control of a Metal-Only Transmitarray Unit-Cell

Keywords: metal-only transmitarray, Ka-band, Satcom

This paper presents a metal-only transmitarray unit-cell exhibiting polarization twisting which allows reducing side lobe level of transmitarray radiation. This dominant capability is proposed and illustrated by two versions of a metalonly unit-cell consisting of four or three identical layers separated by quarter-wavelength spacers at 18.75 GHz (4 mm). The unit-cell is based on C-shaped slot, and the phase variation is achieved by varying the inner radius of the slot. The linear polarization is twisted by gradually rotating consecutively each layer with the same rotation angle.

I. INTRODUCTION

Transmitarray antennas (TAs) are alternatives to traditional antenna types like reflectors or phased-arrays, for future highgain applications thanks to its spatially-fed configuration and flat radiating aperture. Many TA prototypes have been demonstrated in the past by using printed circuit board (PCB) technology; however the associated fabrication cost may remain expensive for commercial extremely low-cost applications. This motivates the current strong interest in cheap solutions called metal-only transmitarray antennas (MOTAs). Successful experiments have been demonstrated in X-band [START_REF] Abdelrahman | Transmitarray antenna design using cross slot elements with no dielectric substrate[END_REF] and Ku-band [START_REF] Wu | High-gain dualband transmitarray[END_REF] with good radiation characteristics.

Polarization manipulation is presented in several transmitarray prototypes realized by PCB fabrication. The main key point of these configurations is based on via-hole structures in which the receiving and transmitting layers are separated by a common ground plane, and they interact each other by a via, as described in [START_REF] Pan | A dual linearly polarized transmitarray element with 1-bit phase resolution in x-band[END_REF], [START_REF] Di-Palma | Circularly-polarized reconfigurable transmitarray in Kaband with beam scanning and polarization switching capabilities[END_REF]. Therefore the polarization at each side of the unit-cell depends on their patch geometries and orientation, which can lead to different polarization directions. In other prototypes, polarization manipulation is implemented by stacking patches with variable length and width to convert linear to circular polarization by differentiating 90°-transmitted phase between orthogonal linear polarized (LP) waves (e.g. [START_REF] Tian | Circularly polarized transmitarray antenna using low-profile dual-linearly polarized elements[END_REF]). When using MOTAs, there is no substrate, and microstrip patches cannot stand alone without support; hence, implementation of this capability is restricted. We introduce here a new technique to twist the LP incident wave to arbitrary LP transmitted wave with capability of beam forming in specific direction.

The unit-cell is based on a wideband C-shaped slot frequency selective surface exhibiting spectral bandpass filter. The rotation technique is applied for each layer to change the polarization at radiating side. In this work we illustrate polarization control of low-cost MOTAs in the Satcom downlink K-band (17.7-21.2 GHz). The technique to control polarization is described in Section II. The numerical results are presented in Section III with polarization conversion from y-direction to x-or 45°-inclined direction.

II. POLARIZATION CONTROL TECHNIQUE

Twisting polarization orthogonally between receiving and transmitting side of transmitarray is an effective way to reduce the side lobe level [START_REF] Pham | Dual-band transmitarrays with dual-linear polarization at Kaband[END_REF] when part of it is due to spill-over radiation from feeding horn [START_REF] Pham | Design of wideband dual linearly polarized transmitarray antennas[END_REF]. The three-dimensional geometry of the proposed unit-cell is depicted in Fig. 1 for a particular case illustrating polarization conversion from ydirection to 45°-inclined direction. The unit-cell proposed in this paper is an evolution of the linearly-polarized C-slot unitcell described in [START_REF] Pham | Metal-only transmitarray based on C-shaped slot[END_REF]. The unit-cell edge is 7 mm, and the metal layer thickness equals 500 µm. A multi-layer configuration is selected to provide a phase-shift with a separation of 4 mm between two consecutive layers (see Fig. 1a for the four-layer design and Fig. 1b for the three-layer one). The transmission phase is shifted by varying the inner radius r i of the C-shaped slot, as shown in Fig. 1 with fixed outer radius of 3.3 mm.

For the 4-layer case (Fig. 1a), polarization control is obtained by rotating uniformly the C-shaped slots of three layers (layers #2, #3 and #4) consecutively by an equal amount of angle so that the field radiated by the last layer (layer #4 in Fig. 1) is polarized in the desired direction. This is applicable to any unit-cell in stacked configuration with at least three layers.

III. PARAMETRIC STUDY

The metal-only unit-cells (MOUC) have been characterized numerically by full-wave simulations using Ansys HFSS v.15 with periodic boundary conditions. The MOUCs are excited by two Floquet ports at the bottom (Port 1) and top (Port 2) to compute the two-port scattering matrix. Demonstration of polarization twisting is provided for two cases: (i) x-polarized and (ii) diagonally-polarized outgoing wave. In all cases the incident wave is polarized along y-axis. In the first case, for the four-layer MOUC (Fig. 1a), the rotation angle is 15° between each layer, while in the second case, this angle equals is 30°. For the three-layer design (Fig. 1b), rotation angles equal 22.5° and 45° for cases (i) and (ii) respectively. The transmission coefficients for case (i) is denoted by while for case (ii) it is derived as follows

          yy S xy S y S 21 21 2 1 45 21
The transmission coefficients at 19 GHz are depicted in Fig. 2 and Fig. 3 for cases (i) and (ii), respectively. In both cases, the unit-cells exhibit a phase-shift of 360° with insertion loss below 1.5 dB when the inner radius varies between 1.8 mm and 2.8 mm or from 1.6 mm to 2.8 mm in case (i) or case (ii), respectively. As nearly-linear transmission phase curves are observed, the proposed unit-cells are rather robust with respect to fabrication uncertainties (around 50 µm by laser machine). For both polarization scenarios, the characteristics of the MOTA unit-cell are very promising and suitable for realization of MOTAs. This technique is also eligible for three-layer unit-cell in both cases; nevertheless, phase range is reduced to around 270° with transmission magnitude better than -3 dB at 19 GHz (see Fig. 4, transmission coefficients in the case (i) for x-polarized output wave). The similar behavior is also observed in the case (ii) for the three-layer MOUC.

IV. CONCLUSIONS AND PERSPECTIVES

Metal-only unit-cells based on C-shaped slot have demonstrated a capability of linear polarization manipulation. This characteristic is very convenient to obtain desired polarization direction for a low-cost transmitarray solution. This unit-cell is possibly scalable to adapt many frequency bands to serve plenty of applications (5G Satcom links, frontand backhaul links, etc…).

Fig. 1 .

 1 Fig. 1. MOTA unit-cell converting y-polarized incident wave to 45°-inclined polarized outgoing wave: (a) Four-layer design, and (b) three-layer design.

Fig. 2 .

 2 Fig. 2. Transmission coefficients in magnitude and phase as a function of the inner diameter ri for x-polarization of the four-layer MOUC.

Fig. 3 .

 3 Fig. 3. Transmission coefficients in magnitude and phase versus the inner diameter ri for 45°-inclined polarization for four-layer MOUC.

Fig. 4 .

 4 Fig. 4. Transmission coefficients in magnitude and phase as a function of the inner diameter ri for x-polarization of the three-layer MOUC.

This works is supported by the French National Research Agency (ANR) in the project named TRANSMIL (TRANSmitarray antenna at MILlimeterwave) under Grant ANR-14-CE28-0023.


Valentin Jouvanceau
email: valentin.jouvanceau@cnrs.fr

Quantitative Easing and Excess Reserves

Keywords: Quantitative Easing, Excess Reserves, Transmission Channels, Securitization Crisis JEL codes: E32, E44, E52, E58, G01 Quantitative Easing, Excess Reserves, Transmission Channels, Securitization Crisis. JEL Classification: E32, E44, E52, E58, G01

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Introduction

A lot remains unknown about the effects of Quantitative Easing (QE), from the size of the effects that it had, to the transmission channels through which programs worked. At the onset of the Great Recession (GR thereafter), the FED lowered dramatically the effective fund rate until it hit the zero lower bound. While engaging in large rescue programs providing exceptional liquidity facilities in the discount window and in the interbank market 1 , the FED also announced the first of its QE asset purchase programs. 2 QE appeared as a potentially good solution for several reasons. To start with, QE could be qualitative and remove toxic assets from the capital markets. Further, QE could reduce long-term nominal yields and then flatten the yield curve. Finally, QE could offer an additional degree of freedom for banks to reallocate portfolios, roll over or bail-out debts and resort to recapitalization. Nevertheless, QE and the liquidity rescue programs had a dramatic effect on the size of the FED's balance sheet (see Figures 8 and9 in Appendix A). In particular, the expansion of the liabilities has been covered by the issuance of deposits; that is, interest-bearing excess reserves (see Figures 10 and

11 in Appendix A). 3

Hence, by what avenues excess reserves transmit to the real economy? In other words:

To what extents such maturity transformation (swaps of excess reserves for assets) affects macroeconomic variables? So far in the theoretical literature, the absence of excess reserves limits/overlooks the identification of the transmission channels of QE, as will be clear later. We address this issue by using a general equilibrium model that incorporates the two most popular types of financial frictions in the theoretical literature, namely banking frictions and collateral constraints. Our model is equipped to quantify the relative importance of three theoretical transmissions channels of QE programs, the portfolio balance, bank lending (credit) and asset prices channels.

Our contributions are twofold.

First, we provide a more realistic way of modeling the implementation of QE programs. In their seminal papers, [START_REF] Gertler | A model of unconventional monetary policy[END_REF][START_REF] Gertler | Qe 1 vs. 2 vs. 3...: A framework for analyzing large-scale asset purchases as a monetary policy tool[END_REF] (GK) design QE as a public intermediation 1 Discount window programs: Term Auction Facility (TAF), Primary Dealer Credit Facility (PDCF), and Term Securities Lending Facility (TSLF). Market liquidity facilities: Commercial Paper Funding Facility (CPFF), Asset-Backed Commercial Paper Money Market Mutual Fund Liquidity Facility (AMLF), Money Market Investor Funding Facility (MMIFF), and the Term Asset-Backed Securities Loan Facility (TALF). See more information in https://www.federalreserve.gov/monetarypolicy/bst.htm 2 Forward guidance was also a great part of the unconventional monetary instruments. Often defined as Delphic and Odyssean forward guidance, these explicit commitments to the future stance of monetary policy consistently paced the monetary announcements during the ZLB period. 3 The interest rate on required reserves (IORR) and the interest rate on excess reserves (IOER) were settled by the Emergency Economic Stabilization Act of 2008 in the supply of credit: the central bank elastically issues short-term liabilities subject to negligible efficiency costs to supply private capital at a lower interest rate. We offer an alternative, and include reserve requirements in the balance sheet of bankers. In addition, we impose limits to arbitrage to cause an excess return in reserves. In [START_REF] Curdia | The central-bank balance sheet as an instrument of monetarypolicy[END_REF], the nominal interest rate is always greater or equal to the interest rate on reserves. Here, we assume the opposite, in line with the practice of the FED. Then, we design QE programs as the exogenous supply of excess reserves. This way of modeling QE programs is much closer to the actual QE programs than any alternative in the literature.

Second, we disentangle the different transmission channels of QE, not only with the introduction of excess reserves, but also by having various financial frictions. The model includes banking frictions in the shadow banking system, banking frictions on commercial banks à la GK, and a collateral constraint in the housing market in the spirit of [START_REF] Guerrieri | Collateral constraints and macroeconomic asymmetries[END_REF]. The latter basically sets an upper borrowing limit for impatient households. In their framework, impatient agents borrow directly from patient households. In our model, we impose a private financial intermediation in the funding process. Doing so not only directly constrains the housing consumption of impatient agents by the market value of their collateral, but also indirectly constrains it through the moral hazard problem that bankers face.

For the sake of reality towards the episode of GR, we take into account a shadow banking system à la [START_REF] Meeks | Shadow banks and macroeconomic instability[END_REF] that provides ABS after the securitization of loan bundles.

Shadow banks support the supply of credit in the economy by providing high-quality collateral to bankers, and the mechanism is in line with the observed countercyclicality of the size of the shadow banking system (see Figure 12 in Appendix A). In many contributions that include financial frictions, the GR is modeled as a negative capital quality shock. We adopt a more realistic mechanism and consider a fall in the quality of ABS. In this greater risk environment, bankers proceed to portfolio reallocations that lead them to reduce the supply of loans, and cut the demand for ABS. As a consequence, shadow banks engage in fire sales of securitized assets.

The latter is important in terms of results. Indeed, we find that our model successfully replicates key features of the GR when driven by a negative shock on the quality of ABS.

In this greater risk environment, we are able to capture an endogenous mechanism within the financial system that causes ABS to be procyclical, reconciling theory with the data. We demonstrate that in such environment, a public intermediation à la GK limits the understanding of the transmission channels of QE programs because it bypasses the endogenous portfolio reallocations, and neglects the endogenous response of the financial system to the program. In addition, it greatly overstates -roughly by a factor of 1 to 10 -the macroeconomic effects of QE programs. The main reason is that QE à la GK overlooks the relative changes in the balance sheets. Finally, we find that most macroeconomic effects generated by QE programs are channeled through the wealth effect -asset prices. In this paper, we find that the portfolio balance channel of QE programs is always present, working as the first stage of a multi-stage rocket. But on top of the portfolio balance channel, the model allows to gauge the relative contribution of the credit channel and the asset prices channel. The introduction of excess reserves is crucial because it allows to trigger changes in the relative supply of assets; an essential transmission mechanism of QE within the financial system.

While the model performs reasonably well in replicating the GR, it is not its main focus. Indeed, we abstract from important New-Keynesian features such as wage rigidities, important non-linearities like the ZLB or the occasionally binding constraints. However, our results do not qualitatively suffer from these lacks, since the addition of such features would rather amplify the magnitude of the effects of QE programs, making our evaluation here a lower-bound estimate.

In the literature, the transmission channels of QE programs are still being discussed. Indeed, the economic literature mostly focused on the quantitative effects of QE programs empirically (see [START_REF] Krishnamurthy | The effects of quantitative easing on interest rates: channels and implications for policy[END_REF] or [START_REF] Weale | What are the macroeconomic effects of asset purchases?[END_REF] among others).4

In most cases, empirical contributions find that QE programs have very large effects on financial conditions and the real economy. However, these results are in stark contrast with the relative neutrality of QE suggested by the theoretical literature. The neutrality assertion is embraced by the monetarist's views of [START_REF] Curdia | The central-bank balance sheet as an instrument of monetarypolicy[END_REF] and [START_REF] Williamson | New monetarist economics: Understanding unconventional monetary policy[END_REF][START_REF] Williamson | Scarce collateral, the term premium, and quantitative easing[END_REF], unless financial markets are segmented or if there are financial frictions. For instance, under the preferred-habitat assumption [START_REF] Modigliani | Innovations in interest rate policy[END_REF], [START_REF] Vayanos | A preferred-habitat model of the term structure of interest rates[END_REF]) and/or according to [START_REF] Tobin | A general equilibrium approach to monetary theory[END_REF]'s portfolio balance theory of the term structure of interest rates, QE should be effective/relevant. In the former, markets are segmented by the preferences of investors over asset maturities and yields. For investors to be willing to invest in other segments, the incurred risk must at least be offset by the expected return in the asset. In this context, QE fosters demand for imperfect substitutes because of the change in the supply of the targeted asset, and QE programs reduce the yields of close substitutes. In the latter, a financial crisis raises private and public spreads over risk-less assets because of financial frictions, and QE programs help alleviate those. QE was also shown to have large effects in models of financial frictions: in GK a premium on risky assets emerges as the result of bankers diverting a fraction of the total assets intermediated. The friction limits the arbitrage in financial markets among assets, and any exogenous drop in asset prices tightens the balance sheet constraints. This drop produces an endogenous financial accelerator mechanism, that in turn freezes the supply of credit by private intermediaries.

The paper is organized as follows. Section 2 presents the baseline model. Section 3 investigates the effects of securitization in the transmission of a GK negative capital quality shock, and investigates the effects of a fall in the quality of ABS. It also quantifies the effects of QE programs, modeled either as in GK (public credit intermediation), or as a rise in excess reserves.

Section 4 concludes.

The Baseline Model

Our baseline model is a generalization of the seminal paper of GK. Extensions capture key features of the housing market and the shadow banking system to allow for a better understanding of the origins of the GR -lining-up with usual narratives -and a better understanding of the effects of QE programs.

In the model, patient households hold financial assets and acquire housing. They fund impatient households and firms in their investment projects. However, patient workers are unable to perfectly enforce payments from borrowers, and resort to commercial banks. Commercial banks are financial intermediaries specialized in lending activities. Due to financial frictions, bankers are limited in their actions. Bankers must hold sufficient amounts of high-quality collateral to meet their balance sheet constraints. Shadow banks trade loan bundles for high-quality ABS.

This section offers details about the respective behavior of commercial and shadow banks, households, producers, and the central bank.

Commercial banks

Commercial banks offer multiple services to three types of agents. First, they grant loans to intermediate good producers. Second, they facilitate government expenditures by holding longterm government bonds. Last, they provide mortgage loans to impatient households. Each bank is owned and managed by a fraction of patient households, namely, bankers. Their assets are backed by the issuance of deposits and net worth (equity). However, financial frictions limits the amount of funds patient households are willing to deposit. Accordingly, a rational banker would circumvent frictions by entirely relying on equity. We then bound the expected lifetime of bankers with an i.i.d probability σ to create a rotation within the workforce. The financial frictions arise from a typical moral hazard problem à la GK in which bankers can be shady and divert a fixed fraction of their assets. Depositors demand that bankers pledge a fraction of each asset, and the composition of the bankers balance sheet shapes the relationship of trust between both parties. Bankers hold a variety of assets, that may differ in their degrees of pledgeability. In particular, primary loans induce strong suspicion in the eyes of patient households. Bankers then must hold high-quality collateral such as ABS.

The asset side of the individual balance sheet of a banker (superscript c) is made of the remainder of primary loans s c t , ABS a c t , long-term government bonds b c t and mortgage loans to impatient households m c t . Liabilities consist of deposits d c t and net worth n c t . The balance sheet identity writes:

Q t s c t + q a t a c t + q b t b c t + q h t m c t = d c t + n c t (1)
in which lower case symbols represent individual quantities. Let be Q t , q a t , q b t and q h t the respective market prices of assets. The net worth evolves as:

n c t = R s t Q t-1 s c t-1 + R a t q a t-1 a c t-1 + R b t q b t-1 b c t-1 + R t q h t-1 m c t-1 -R t d c t-1 (2) = (R s t -R t )Q t-1 s c t-1 + (R a t -R t )q a t-1 a c t-1 + (R b t -R t )q b t-1 b c t-1 + R t n c t-1
where R s t is the real stochastic return on primary loans, R a t on ABS and R b t on long-term government bonds. The return on deposits is the real interest rate R t . In the interest of parsimony we do not design financial frictions in the housing market. In other words, mortgage loans are riskless, hence their cost in the real interest rate.

Going deeper into details, the timeline of a banker's decisions proceeds as follows: In the dawn of t, a banker issues deposits, grants loans to firms and impatient households, funds the government and receives ABS. She then makes a choice between two options: Be honest or shady. An honest banker accumulates net worth until all stakeholders receive their payoffs in t + 1. Conversely, a dubious banker sells a fraction of her assets to outside investors. For the sake of discretion, the amount of fraud cannot exceed an upper bound θ c . Her home household enjoys the booty in period t + 1. Unfortunately, this larceny leads the bank to a fragile position. Indeed, rational depositors stop funding the banks in this context. As a consequence, bankers are in danger of bankruptcy. Both parties benefit from reaching an agreement, that takes the form of an incentive compatibility constraint, that states that the discounted franchise value of the bank must be at least as large as a fraction of diverted assets:

V c t ≥ θ c (Q t s c t + Ξq a t a c t + ∆q b t b c t ) (3)
in which Ξ and ∆ are weights related to the assets' pledgeability. By fixing

θ c > θ c ∆ > θ c Ξ ∈ [0,1],
we assume that ABS have the highest collateral quality. In line with [START_REF] Meeks | Shadow banks and macroeconomic instability[END_REF], it is quite natural to think that securitization mitigates risk through diversification. In addition, ABS increase the liquidity of the underlying assets. It is also reasonable to define a higher collateral value for long-term government bonds than private loans, public issuers being less likely to default than private issuers. The franchise value of a bank V c t depends on the sum of accumulated net worth conditional on the death and survival probabilities:

V c t-1 = E t-1 Λ t-1,t ((1 -σ)n c t + σV c t )) (4)
which is a standard Bellman equation. Variable Λ t-1,t is the stochastic discount factor derived from the patient households program. The banker's objective is then to maximize the franchise value (4) subject to the incentive constraint (3), the balance sheet identity (1) and the law of motion of net worth (2). Using the method of undetermined coefficients, we guess and later verify that the bank's franchise value V c t is linear in the time-varying coefficients µ c t , µ a t , µ b t , η c t . Accordingly, we conjecture that the value function is in period t:

V c t = µ c t Q t s c t + µ a t q a t a c t + µ b t q b t b c t + η c t n c t (5)
Hereafter, insert the particular solution (5) into the recurrence relation (4) and differentiate using the method of Lagrange multipliers. Let λ c t be the associated Lagrangian multiplier of the incentive constraint (3). The differentiations yield the following first-order necessary conditions, respectively for s c t , a c t , b c t , λ c t :

(1 + λ c t )µ c t = λ c t θ c (6) µ a t = µ c t Ξ (7) µ b t = µ c t ∆ (8) Q t s c t + Ξq a t a c t + ∆q b t b c t = φ c t n c t (9)
where,

φ c t = η c t θ c -µ c t ( 10
)
The LHS of equation ( 6) shows the marginal benefit from expanding what we call primary loans. The RHS indicates the marginal cost of fraud. Due to the fact 1 > ∆ > Ξ, the holdings of ABS and government bonds are marginally less attractive than loans. In this respect, as long as µ c t < θ c , it is always profitable for bankers to divert assets. The latter is also a necessary condition for the incentive constraint to bind, otherwise, bankers would indefinitely supply loans to offset frictions. In other words, the incentive constraint imposes that the holding of assets is curbed by a maximum equity multiplier φ c t (see equation ( 10)). Hence, as long as the incentive constraint binds, the size of a banker balance sheet is endogenously constrained by fluctuations in its net worth. The latter also induces an equality between the gains from larceny θ c φ c t and the costs of losing franchise value µ c t φ c t + η c t . At the end of period t + 1, a shady banker entirely consumes her net worth with probability (1 -σ). In this case the marginal gain of net worth is then one. Conversely, an honest banker ensures the continuity of her services. Hence, either the marginal gain from larceny or the marginal cost of losing franchise value apply. Thereby, the weighted shadow value of a banker net worth is defined by:

Ω c t+1 = 1 + σ(θ c φ c t+1 -1) (11)
Finally, the time-varying coefficients linear in s c t , a c t , b c t , n c t satisfy the Bellman equation with:

5 µ c t = E t Λ t,t+1 Ω c t+1 R s t+1 -R t+1 (12) µ a t = E t Λ t,t+1 Ω c t+1 R a t+1 -R t+1 (13) µ b t = E t Λ t,t+1 Ω c t+1 R b t+1 -R t+1 (14)
η c t = E t Λ t,t+1 Ω c t+1 R t+1 (15)
which implies that financial frictions induce discounted excess returns. To give the reader a preview of the model mechanics, imagine an exogenous financial crash that lowers the price of capital. The market value of securities is reduced, hence the bankers' net worth falls. Assets tighten in proportion to the equity multiplier. In addition, equation ( 6) indicates that a tightening in the constraint substantially increases the marginal gain from expanding primary loans; that is, the marginal cost of fraud. The latter then raises the excess return in loans and the cost of capital. In a nutshell, an exogenous crash in asset prices is in turn amplified by the endogenous tightening in the size of the balance sheet. This feedback effect further reduces the lending activities of the commercial banks.

Shadow banks

The shadow banking sector is composed by competitive shadow banks in the exclusive charge of the securitization of primary loans. A shadow bank purchases loan bundles in a secondary market to create high-quality ABS. Each shadow bank is owned and managed by a fraction of patient households, namely, brokers. The existence of shadow banks is motivated by the 5 Appendix B provide details for commercial and shadow banks problems obligation of banks to hold high-quality collateral. Hence, the entire stock of ABS is sold to bankers. In this way, the flow of ABS is confined within the financial system. However, financial frictions limit trade between bankers and brokers. A broker survives with probability σ and exits her occupation with probability (1 -σ). Some brokers are tempted to divert a fraction of their assets. Therefore, bankers demand that brokers produce a sufficient amount of ABS.

The individual balance sheet of a broker (superscript b) comprises securitized assets s b t , ABS a b t and net worth (equity) n b t with:

Q t s b t = q a t a b t + n b t (16)
in which the accumulated net worth follows:

n b t = R s t Q t-1 s b t-1 -R a t q a t-1 a b t-1 = (R s t -R a t )Q t-1 s b t-1 + R a t n b t-1 (17)
For the sake of parsimony, the real returns and market prices in the same class of assets are equivalent across markets.6

The timeline of a broker's lifetime is as follows. At the beginning of t, a broker purchases loan bundles and transforms those into ABS. She then chooses to be honest or shady. An honest broker pursues securitization until the final payoffs in t + 1. A dubious broker diverts a fraction of her securitized assets. She fixes her greediness to an upper bound θ b to avoid drawing attention. The related booty goes to her home household at the end of t + 1. Unfortunately, the future of the shadow bank is at risk due to this embezzlement. Indeed, bankers are perfectly familiar with this ploy being faced with such a dilemma themselves. In this situation, bankers decide to cut offering loan bundles. However, bankers are in need for high-quality collateral to obtain funds from depositors. Thus, they agree to the condition that the discounted franchise value of the shadow bank must be at least as large as the amount of fraud. The resulting incentive constraint writes:

V b t ≥ θ b Q t s b t (18)
in which the franchise value V b t is the sum of accumulated net worth conditional on the exiting and survival probabilities:

V b t-1 = E t-1 Λ t-1,t (1 -σ)n b t + σV b t ) (19)
The broker's objective is to maximize the franchise value ( 19) subject to the incentive constraint (18), the balance sheet identity ( 16) and the law of motion of net worth (17). Using the method of undetermined coefficients, the particular solution is linear in the time-varying coefficients ν b t , η b t . Accordingly, we conjecture that the value function satisfies:

V b t = ν b t Q t s b t + η b t n b t (20)
Then substitute the particular solution (20) into the Bellman equation ( 19). Let λ b t be the associated Lagrangian multiplier of the incentive constraint (18). By differentiating, the firstorder necessary conditions for, s b t and λ b t are:

(1 + λ b t )ν b t = λ b t θ b (21) Q t s b t = φ b t n b t (22)
where,

φ b t = η b t θ b -ν b t ( 23
)
The LHS of equation ( 21) is the marginal gain from increasing the securitization activity. The RHS indicates the marginal cost of embezzlement. As long as ν b t < θ b , it is always profitable for brokers to divert assets. This is a necessary condition for the incentive constraint to bind, otherwise, brokers would extend securitization to infinite amounts to offset frictions. Hence, an isomorphic equity multiplier φ b t applies to the brokers balance sheets (see equation ( 23)). The binding constraint induces an equality between the gains from diverting securitized assets θ b φ b t and the costs of losing the franchise value µ b t φ b t + η b t . As for bankers, a shady broker entirely consumes her net worth at the end of period t + 1 with probability (1 -σ). In this situation, the marginal gain of net worth is one. An honest broker builds on equity with probability σ until t + 1. In this case, either the marginal gain from embezzlement or the marginal cost of squeezing the franchise value apply. The weighted shadow value of a broker net worth is thus:

Ω b t+1 = 1 + σ(θ b φ b t -1) (24)
The time-varying coefficients in s b t and n b t satisfy the Bellman equation according to:

µ b t = E t Λ t,t+1 Ω b t+1 R s t+1 -R a t+1 ( 25
)
η b t = E t Λ t,t+1 Ω b t+1 R a t+1 (26)
The above system of equations provides a general description of this financial system. In the preceding example of the exogenous crash in the price of capital, we highlighted the feedback effect in the absence of shadow banks. Henceforth, both net worth of bankers and brokers are seriously impaired by the fall in the price of capital. Indeed, primary securitized loans are similarly market priced. The incentive constraints tighten and feed the downturn.

Aggregation

By definition, the components of the equity multipliers φ c t and φ b t are identical across commercial and shadow banks. Thus, the sum of the individual balance sheet constraints yield the following aggregate identities:

Q t S c t + Ξq a t A c t + ∆q b t B c t = φ c t N c t (27) Q t S b t = φ b t N b t (28)
The same principle applies to the net worth of bankers and brokers. The aggregate net worth consist of the retained earnings of survivors and the initial provision for newcomers: 7

N c t = σ (R s t -R t )Q t-1 S c t-1 + (R a t -R t )q a t-1 A c t-1 + (R b t -R t )q b t-1 B c t-1 + R t N c t-1 + χ c (29) N b t = σ (R s t -R a t )Q t-1 S b t-1 + R a t N b t-1 + χ b
Let Z t be the gross profits per unit of capital as will be clear later. The parameter δ represents the depreciation rate of capital. Accordingly, the real return on primary loans is:

R c t+1 = Z t+1 + (1 -δ)Q t+1 Q t ξ t+1 (30)
where ξ t+1 is an exogenous capital quality shock à la GK. Accordingly, the excessive payoff of capital corresponds to the sum of gross profits and the market value of capital stock net depreciation. The ABS are derivative products of primary loans, hence their real return is:

R a t+1 = Z t+1 + (1 -δ)q a t+1 q a t ξ t+1 (31)
Finally, the long-term government bonds return a periodic unit of currency. Let P t be the aggregate price level of the economy. Thus, the real return on long-term government bonds is:

R b t+1 = 1 Pt + q b t+1 q b t (32) 7 χ c = ω c Q S +

Households and Production

There is a unit mass of households in two different groups, namely, patient and impatient households. In the group of patient households, people switch occupations each period t.

A fixed proportion f is a worker while the remaining part is either a banker or a broker.

Conversely, impatient households can not work for financial intermediaries. The utility flows for each type of household are then given by:

u t = E t ∞ i=0 β i Γ c log(C t+i -ε c C t+i-1 ) + jΓ h log(H t+i -ε h H t+i-1 ) - 1 1 + η L 1+η t ( 33
)
u t = E t ∞ i=0 (β ) i Γ c log(C t+i -ε c C t+i-1 ) + jΓ h log(H t+i -ε h H t+i-1 ) - 1 1 + η L 1+η t (
C t + D h t + q h t (H t -H t-1 ) + Q t S h t + 1 2 κ(S h t -Sh ) 2 + q b t B h t + 1 2 κ(B h t -Bh ) 2 = W t L t + T t + R t D h t-1 + R s t Q t-1 S h t-1 + R b t q b t-1 B h t-1 + Π t ( 35
)
where W t stands for the real wage, T t denotes lump-sum taxes and Π t defines dividends from the ownership of commercial banks, shadow banks and capital producing firms. Patient households accumulate deposits D c t subject to the moral hazard problem exposed earlier. They also purchase corporate debt and long-term government bond, subject to potential transaction costs.

They operate frictionless arbitrage when the convex transaction cost κ is zero. Conversely, they hold assets to their steady-state levels ( Sh , Bh ) when the cost tends towards infinity. The impatient households budget constraint is:

C t + q h t (H t -H t-1 ) + R t M h t-1 = W t L t + M h t ( 36
)
meaning that impatient agents must borrow amounts M h t from commercial banks to secure housing consumption H t . Following [START_REF] Guerrieri | Collateral constraints and macroeconomic asymmetries[END_REF], we introduce a collateral constraint on housing demand:

M h t ≤ γ m M h t-1 + (1 -γ m )χq h t H t (37)
where γ m > 0 governs the inertia of the borrowing upper bound χ. This constraint states that some (1 -γ m ) impatient agents can roll over debts.9 For simplicity, housing is in a fixed supply.

In this context, housing is a durable good that serves as a collateral for consumption. Hence, in trouble times, the fall in housing prices is further amplified by the constraint and largely contributes to the drop in consumption.10

The production sector comprises different entities that produce the final output of the econ-

Y m t = A t (ξ t K t ) α L (1-α)(1-ϕ) t L (1-α)ϕ t ( 38
)
where ϕ is the relative wage share of impatient households.11 Variable A t is an exogenous productivity measure. After production, firms need to invest in new capital for the next period.

Hence, they borrow by issuing state-contingent claims in the capital market. Securities are backed by primary and securitized loans or are in the form of corporate debts in the bond market. Accordingly, the equilibrium condition on the capital market writes:

Q t K t+1 = Q t (S c t + S b t + S h t ) (39)
Under perfect competition, firms earn zero profits state by state, hence the marginal gain of capital is:

Z t = P m t α Y m t K t (40)
in which P m t is the relative intermediate output price. The accumulation process of capital consists of the leftover capital stock net depreciation and the investment in new units of capital, as follows:

K t+1 = ξ t+1 (1 -δ)K t + I t (41)
Note that firms are not directly exposed to financial frictions. However, they indirectly face borrowing constraint due to the relationship between commercial banks and patient households.

Therefore, the level of the cost of capital is higher than in a standard new-Keynesian model.

Capital producers are in charge of the creation of the new unit of capital. After production, they sell it to the intermediate good producers at the market price Q t . The production of capital is subject to adjustment costs. Thus, the profit maximization program is given by:

max E t ∞ i=1 Λ t,t+i Q i I i -1 + f I i I i-1 I i (42)
which by optimization yields to a standard Tobin's Q ratio between the assets' market value and the replacement cost:

Q t = 1 + f I t I t-1 + I t I t-1 f I t I t-1 -E t Λ t,t+1 I t+1 I t 2 f I t+1 I t (43)
A continuum of j different retailers of a unit mass closes the production sector. The following CES aggregator function yields to the final output:

Y t = 1 0 Y ε-1 ε jt dj ε ε-1 (44)
Following sticky prices principle, we introduce nominal rigidities à la Calvo (1983). Accordingly, retailers change the price of the final output with probability (1 -γ). Otherwise, retailers index prices to the inflation rate π t . The optimal reset price P * t is then conditional on these probabilities with:

max E t ∞ i=0 γ i β i Λ t,t+i P * t P t+i i k=1 (1 + π t+k-1 ) γ p -P m t+i Y jt+i (45)
The first-order necessary condition writes:

∞ i=0 γ i β i Λ t,t+i P * t P t+i i k=1 (1 + π t+k-1 ) γ p - 1 1 -1/ε P m t+i Y jt+i = 0 (46)
for which we derive the aggregate price level dynamic by using the law of large numbers:

P t = (1 -γ) (P * t ) 1-ε + γ π γ p t-1 P t-1 1-ε 1 1-ε (47)

Market clearing and Equilibrium conditions

The market clearing conditions for primary loans, ABS, long-term government bonds, housing, mortgage loans and deposits are:

K t+1 = S c t + S b t + S h t A c t = A b t B = B c t + B h t H t + H t = 1 M h t = M c t D h t = D c t (48)
The aggregate resource constraint is:

Y t = C t + C t + 1 + f I t I t-1 I t + Ḡ (49)
The government budget constraint writes:

Ḡ + R b t B = T t + B (50)
Last, the central bank conducts standard monetary policy following a Taylor rule of the form:

i t = ī + κ π π t + κ y (log Y t -log Y * t ) + t ( 51
)
where ī is the steady-state level of the nominal interest rate, Y * t is the flexible-price output and t is an exogenous monetary shock.

Experiments

In this section we explore the ability of the model to mimic features of the GR and QE episodes.

As stated before, our main goal is not quantitative but qualitative. Indeed, for the sake of parsimony, we ignore standard new-Keynesian features such as wage rigidities, and abstract from any securitization process in the housing market. In addition, we abstract from occasionally binding constraints, that would certainly refine the results. However, we do believe that the addition of these features would strengthen rather than reduce our key findings, and leave these refinements for further research.

The first two experiments consist in highlighting the comovements between primary and securitized loans. In the literature, much of the empirical and theoretical evidences support a countercyclical nature of the shadow banking system (see empirical evidences in [START_REF] Nelson | Do contractionary monetary policy shocks expand shadow banking?[END_REF] and [START_REF] Meeks | Shadow banks and macroeconomic instability[END_REF] among others). The Figure 12 in Appendix A depicts the cyclical relationship between the sum of business non-financial corporate and non-corporate business loans by depository institutions and securitized other loans and advances by issuers of ABS.12 Securitized loans are mostly countercyclical and highly volatile until 2004. A clear break appears after 2004 (see Figure 13 in Appendix A for further details). This change coincides with the contractionary US monetary policy of the summer 2004 in response to the concerns about housing prices. Usually, securitization helps support the supply of credit by the banking sector. Issuers of ABS are mainly highly leveraged bankruptcy-remote companies that hold pools of assets. Pools cover a large range of assets such as consumer credits, Treasury securities and mortgages, and business loans. Their liabilities essentially consist of the issuance of pass-through securities, namely ABS. In other words, the interests, the principal payments and the amortization pass to the investors through the securities. In the end, these entities are inclined to cover more risk because they are only intermediaries between borrowers and investors. After securitization, the assets are more liquid and more marketable. Nevertheless, ABS are associated to higher risk. Indeed, if interest rates increase, borrowers are more likely to default which induces a great drop in the market value of the ABS. On the contrary, if interest rates decrease, borrowers are more likely to refinance debts which causes a fall in the interests of the ABS. In our view, the former case is a plausible explanation of the procyclical spiral in the credit market for firms between 2005Q1 and 2009Q1. Indeed, the traditional sector has incentives to expand securitization in such a period of asset bubbles and deflating interest rates. Thus, they benefit from the decrease in the cost of capital and the soaring in asset prices by clearing the risk out of their balance sheets. But there is a point where the explosion of the assets bubble and defaults inevitably cause the disruption of this mechanism. The latter is the main interest of our simulated experiments. In this respect, Figure 13 in Appendix A displays the cyclical components in both business and securitized loans in deviation from their levels in 2007Q4. Their joint rise in the beginning of the recession are certainly related to the strong inertia in the credit market for firms. In other words, the downturn in the real economy leads the crash in the market. To convince the reader, we expose the model to an exogenous degradation of the collateral value of ABS and find an intrinsic procyclical spiral.

As exposed earlier, the remainder of the exercises serves to dissociate over the transmission channels of QE. To do so, we refine the standard specification of QE by introducing banking reserve requirements and the supply of excess reserves to the model.

Calibration

The parameters of the baseline model are calibrated along the values of GK, [START_REF] Meeks | Shadow banks and macroeconomic instability[END_REF] and the estimates of [START_REF] Guerrieri | Collateral constraints and macroeconomic asymmetries[END_REF]. 13 These values are adjusted to a quarterly frequency. They mostly reflect economic conditions before the GR in the US.

The estimates of [START_REF] Guerrieri | Collateral constraints and macroeconomic asymmetries[END_REF] are: the discount factors for patient β and impatient households β , the capital share α, the capital depreciation rate δ, the habit parameters in consumption ε c and housing ε h , the weight for housing preferences j, the wage share between patient and impatient workers ϕ, the maximum loan-to-value ratio χ, the labor disutility η and the factor of inertia γ m . The inverse elasticity of investment η i , the price rigidity parameters γ and γ p and the Taylor rule coefficients κ π , κ X , ρ, are taken from GK.

The parameters related to the financial system are set to hit the following data targets: The average equity capital to total assets for commercial banks is 9.5 between 2000Q1 and 2007Q1. 15 However, in the model, the moral hazard problem overstates risk within the financial system, hence fluctuations in asset prices. In that respect, GK argue that values between 4 and 6 compensate the absence of the risk-sharing components between borrowers and commercial banks. We follow their lead and fix the steady-state value for the bankers' equity multiplier φc to 4. The two parameters θ b , ξ b dedicated to the shadow banks are adjusted to match a broker's equity multiplier of 8. For the same above reasons, this value reflects half of the average of investment banks' leverage ratio reported in GK and Gertler and Kiyotaki (2015).

The portfolio adjustment cost κ ensures that steady-state returns are equivalent along the 13 Readers are reminded that the main contribution of this paper is not quantitative, hence the estimation of parameters is not of high interest. In addition, most of the parameter values are adjusted to match data targets. 14 This value reflects an average pre-crisis ABS spread over swap rates for high-quality securitized assets such as credit card, equipment or mortgages, see [START_REF] Meeks | Shadow banks and macroeconomic instability[END_REF] for further details. 15 This figure is obtained using time series for the Total Equity Capital for Commercial Banks in United States and Total Assets, All Commercial Banks in the F.R.E.D database.

Financial crisis experiments

Figures 1 and 14 in Appendix A, report the impulse response functions (IRF) to an exogenous capital quality shock à la GK. This shock either captures a huge depreciation of the price of capital or a fall in the quality of capital. In this context, we compute the IRF of the model with and without shadow banks to quantify the marginal contribution of the securitization. The results indicate that the securitization weakly amplifies the macroeconomic fluctuations. This result can be understood along the following lines. The decline in the market price of primary and securitized loans deteriorates the net worth of bankers and brokers. Brokers' balance sheet constraint explodes due to their high degree of leverage. As a consequence, brokers proceed to fire sales of assets to dampen losses, and cease supplying ABS, which raises their market price. The leverage constraint also leads bankers to experience losses. The marginal gains from diverting primary loans shoots up. Hence, bankers roll over debts by purchasing cheaper primary securities, which further diminish the demand for deposits. The shock causes a sharp reduction in the supply of ABS, which serves as a buffer mechanism inside the financial system.

The latter is particularly visible in the response of bankers' net worth. However, this buffer mechanism can not entirely offset the contraction in investment and output.

While this GK exercise is widely used in the literature to mimic the effects of the GR, it can not generate the procyclical spiral in the credit market for firms that actually characterized the GR. In the next experiment, we look at the effects of a direct disruption in the securitization mechanism, that actually captures this saliant feature of the GR.

The next experiment lowers the collateral value of ABS by exogenously increasing the weight Ξ in the bankers balance sheet constraints (see Equation ( 3)). In the interest of gauging the quantitative properties of the baseline model, the shock targets the 2009Q2 detrendred drop of real GDP (see Figure 13 in Appendix A). 16 Figure 2 shows the matching of the IRF. 17

The exogenous change in the weight parameter Ξ distorts the optimal condition in Equation ( 7). The marginal benefit from diverting ABS suddenly jumps to a huge level, and bankers swiftly cut their demand for ABS. In response, the market price of ABS falls and triggers a big contraction in the net worth of bankers. Bankers then sell large amounts of assets in return for short-term liabilities, reflecting a greater discipline. As a consequence, the net worth of brokers falls.

The procyclicality spiral applies when bankers are intrinsically subject to a greater risk 16 The persistence parameter of the shock is 0.96 in order to match a persistent decline in real GDP. We loop over the variance parameter of the shock to minimize the distance between the data target and the deviation in output in the model. 17 exposure; that is, when the collateral quality of ABS shrinks. Along the lines of [START_REF] Gertler | Financial crises, bank risk exposure and government financial policy[END_REF], bankers adopt more discipline in the presence of a riskier environment. Indeed, the Figure 17 in Appendix A displays evidence that macroeconomic fluctuations are more pronounced in a safer environment. In our view, the baseline model generates convincing qualitative results and adequate quantitative properties. Thus, we believe that the model is a suitable environment to study for the theoretical transmission channels of QE programs.

Quantitative Easing experiments

In order to perform QE experiments, we impose some modification to the baseline model, and first introduce QE shocks à la GK. A more realistic QE experiment is conducted after this first-pass analysis. To this end, a balance sheet of the central bank is introduced to the model.

The monetary authority finances its operations by issuing costless short-term debts D g t , bought by patient households. The balance sheet of the central bank then writes:

D g t = Q t S g t + q a t A g t + q b t B g t ( 52
)
which is simply zero in the absence of asset purchases. The equilibrium conditions in the financial markets must now read:

K t+1 = S c t + S b t + S h t + S g t ( 53
)
A c t + A g t = A b t (54) B = B c t + B h t + B g t ( 55
)
The consolidated government budget constraint is now modified:

Ḡ + R b t B = T t + B + (R s t -R t )Q t-1 S g t-1 + (R a t -R t )q t-1 A g t-1 + (R b t -R t )q b t-1 B g t-1 (56)
In a baseline scenario, we only allow the central bank to purchases assets in the primary market; that is, in the balance sheet of bankers.18 These asset purchases are driven by exogenous firstorder disturbances, as follows: Figures 3 and4 display similar results in our model. 19 Unsurprisingly, a purchase of primary loans is marginally the most effective policy because these assets have, by definition the greatest leverage effect. Hence, QE à la GK is almost transmited one for one through the rise in credit supply (see the equilibrium relations ( 53), ( 54) and ( 55)), abstracting from the relative changes in the supply of assets. This mechanism artificially overestimates the impacts of QE, as will be clear later. Including banking reserve requirements is more realistic. In addition, the introduction of the supply of excess reserves allows us to monitor the portfolio reallocations within the balance sheets of financial intermediaries. Indeed, it will make it possible to assess the implications of an excess return on both types of reserves, something that the GK mechanism overlooks. The addition of reserve requirements changes the optimization program of a representative banker.

S g t = ψ c t (S c t (+S b t + S h t )), A g t = ψ a t (A c t (+A b t )), B g t = ψ b t ( Bc (+B h t )), ψ i t = ρ i ψ i t-1 + ε i t , i = {c, a, b}. (57
The balance sheet identity and the law of motion of net worth become:

Q t s c t + q a t a c t + q b t b c t + q h t m c t + q rr t rr c t = d c t + n c t ( 58
)
n c t = R s t Q t-1 s c t-1 + R a t q a t-1 a c t-1 + R b t q b t-1 b c t-1 + R t q h t-1 m c t-1 + R rr t q rr t-1 rr c t-1 -R t d c t-1 (59) = (R s t -R t )Q t-1 s c t-1 + (R a t -R t )q a t-1 a c t-1 + (R b t -R t )q b t-1 b c t-1 + (R rr t -R t )q rr t-1 rr c t-1 + R t n c t-1
where q rr t is a market price for banking reserves, determined by the general equilibrium. The market price for banking reserves will help to dissociate the different transmission channels of QE, as will be clearer later. Realistically, the baseline (first) scenario defines the excess return on both, requirements and excess reserves, as an upper target of the nominal interest rate. In this scenario, the excess return on reserves is given by:

Rrr t = υ × i t (60)
where υ sets the upper limit. In the second scenario (market rate), banking reserve requirements pay a fixed coupon as follows:

R rr t+1 = ς + q rr t+1 q rr t (61)
Banking reserves are defined as a fixed fraction of households deposits: 21

rr c t = ϕ rr d c t ( 62
)
where the requirement ratio ϕ rr is up to 10%. 22 We also modify the incentive constraint to allow bankers to divert a fraction of reserve requirements, which is a way for us to model a 21 ς is set so that the steady-state level of the market price of reserves is one. 22 This reserve requirement ratio corresponds to the ratio imposed to the highest tranche of depository institutions. Ref: https://www.federalreserve.gov/monetarypolicy/reservereq.htm. limits to arbitrage. 23 We adjust the parameter Θ to match the average interest spread of 10 basis points between the IORR and the effective FFR (see Figure 11 in Appendix A). The banker's incentive constraint and the linear guessed solutions of undertermined coefficients are modified and become:

V c t ≥ θ c (Q t s c t + Ξq a t a c t + ∆q b t b c t + Θq rr t rr c t ) (63)
V c t = µ c t Q t s c t + µ a t q a t a c t + µ b t q b t b c t + µ r t q rr t rr c t + η c t n c t ( 64
)
which yields an additional first-order condition, and a slight change in the incentive constraint.

The time-varying coefficient µ r t ; that is, the marginal cost of diverting reserves is:

µ r t = µ c t Θ (65)
Q t s c t + Ξq a t a c t + ∆q b t b c t + Θq rr t rr c t = φ c t n c t (66) µ r t = E t Λ t,t+1 Ω c t+1 R rr t+1 -R t+1 (67)
where the second equation ( 66) is the new balance sheet equation. 24 Aggregating among bankers, the law of motion of net worth is now:

N c t = σ (R s t -R t )Q t-1 S c t-1 + (R a t -R t )q a t-1 A c t-1 + (R b t -R t )q b t-1 B c t-1 (68)
+ (R rr t -R t )q rr t-1 RR rr t-1 + R t N c t-1 + χ c (69)
We close the model by the QE program, designed as an excess supply of reserves driven by first-order disturbances: 25

ER g t = υ c t Q t S g t + υ a t q a t A g t + υ b t q b t B g t ( 70
)
υ i t = ρ i υ i t-1 + ε i t , i = {c, a, b}. (71)
The consolidated government budget constraint and the equilibrium condition for total reserves are:

Ḡ + R b t B + R rr t T R g t = T t + (R rr t -R t )ER g t + B + T R g t (72) T R g t = RR c t + ER g t (73)
Before we compare the results obtained with this design of QE program through excess reserves to the GK specification of QE programs, we consider the effects of an excess reserves QE experiment on each type of asset purchases in the baseline framework (first scenario). Figures 6 and7 reveal that QE programs have similar effects regardless of the type of asset purchases.

The mechanism proceeds as follows: The liabilities of the central bank is exogenously expanded with excess reserves. Reserve requirements are mechanically diminished due the equilibrium relation in equation ( 73). Consequently, in any case, QE here, exerts a relaxing leverage effect proportional to the weight Θ in the balance sheet of bankers (see equation ( 66)). We believe that this decrease in reserve requirements is analogous to a purchase of assets in the balance sheet of bankers. Here the loosening of constraints is thus indirect. The latter entails a leveraging process. As a result, bankers take more risks and supply more primary loans.

These reallocations dampen the demand of bankers for ABS, and brokers suffer from this shortage. However, their net worth is partially protected by the soaring in the market price of capital. From a macroeconomic perspective, the increase in asset prices causes wealth effects for households and firms. In addition, both also benefit from the sharp fall in the cost of capital, 24 χ c = ω c Q Sc + qa Āc + qb Bc + qrr RR rr 25 The persistence parameters ρ i ,i = {c, a, b} have similar values as in the GK-like QE experiments; that is 0.96. The market price of reserves q rr t has to be removed in each of these equations to obtain the first scenario; that is, the upper target of the nominal interest rate. and from the increase in the collateral value of housing. In brief, QE is clearly effective in fostering financial conditions in the traditional sector. Nevertheless, the macroeconomic effects of QE are extremely small quantitatively in comparison of the huge increase in the supply of excess reserves.

Given the small effects of a QE program modeled as an excess of reserves, we compare our results to an equivalent QE program with a mechanism à la GK. Figures 18 and19 in Appendix A displays a program of ABS purchases under both specifications. 26 We remind the reader that the ABS purchase is the less effective purchase in a GK specification. The results show that the fluctuations are significantly greater with a GK mechanism than when QE is modeled as an excess of reserves. The way QE programs are modeled is thus crucial to determine the overall qualitative and quantitative effects of QE programs, and mechanisms à la GK can seriously overstate their importance. In addition, we compute gradual changes in the weight Θ that as a reminder, symbolizes the steady-state level in the excess return of reserves. Our goal is to find a value for Θ that matches the deviation in output under both specifications. Figure 20 in Appendix A displays the results. A simple back of the envelope calculation indicates that Θ must be roughly 5 for output to get close to the target -the output effect obtained with a GK mechanism. The latter would imply a steady-state excess return on reserves of 500 basis points, very far from empirically realistic values. 27 These elements indicate that the macroeconomic effects of QE programs might be significantly overestimated in the literature, that mostly relies on GK mechanisms. 28 Let us now focus on a potential drawback of the modeling of QE as an excess of reserves in the first scenario, namely the fact that it is rather tricky to identify the relative importance of the various transmission channels of QE. The portfolio balance channel -changes in the composition of portfolios of financial intermediaries -unquestionably triggers fluctuations within the financial system in the above experiment. In addition, the reduction in the cost of capital and the rise in primary securities can be thought of as a credit channel. Finally, the rise in asset prices generates strong wealth effects. Unfortunately however, the first (baseline) scenario

27 Such a large steady-state spread also leads to equilibrium indeterminacy issues. 28 see Jouvanceau ( 2019) for an empirical discussion on this issue.

of QE modeled as an excess of reserves does not help distinguish the different channels at play.

Our second scenario addresses this issue.

In the second scenario, we separate the transmission channels of a QE program modeled as an excess of reserves. To do so, we switch to the market rate scenario. The idea is to write a model in which asset prices exhibit more dynamics and affect the financial system more significantly.

On the one hand, larger macroeconomic fluctuations would be supportive evidence of the asset prices effects of QE programs (wealth effect). On the other hand, smaller fluctuations would indicate that the credit channel dominates. The results in Figures 21 and22 in Appendix A support the view that wealth effects are more important. Indeed, the portfolio reallocations are qualitatively similar to the first scenario. However, the fluctuations in asset quantities are dramatically reduced, while deviations in output and investment are of close magnitude. In addition, the baseline and market scenario also exhibit differences in the persistence of the effects. The latter are mostly explained by the sluggish rises in capital, that alters the bank lending transmission channel -this is certainly at play but is dominated in magnitude by the wealth effects. Finally, as in the GK vs. excess reserves comparison exercise, we play with the value of Θ. All of these experiments support the view that the relative changes in asset prices clearly alter bankers' decisions. On the one hand, the portfolio rebalancing effects of QE alleviate financial frictions and foster financial conditions in the traditional sector. On the other hand, aggregate demand is slightly stimulated by wealth effects, that in turn boost households' consumption and private investment.

Conclusion

In this paper, we show that financial frictions are useful to understand the GR and the effects of QE programs. We extend the seminal paper of GK and refine the financial sector to include securitization (in the form of ABS and a shadow banking system) and housing, with a collateral constraint that affects households consumption. These elements help explain the procyclical credit market spiral in the wake of the GR. In this context, we demonstrate that a greater exposure to risk triggers discipline in bankers' decisions of portfolio reallocations. Shadow and commercial bankers accumulate losses which further and endogenously magnify the size of the subsequent economic downturn.

We then conduct QE experiments. Our first contribution is to model QE programs as being implemented through an excess of reserves, closer to the actual implementation schemes. We show that implementation matters for the effects of QE programs: the financial and macroeconomic effects are clearly dampened when QE is implemented through an excess of reservesdespite huge amounts of simulated excess reserves -compared to an simplified implementation scheme à la GK. Our second contribution is to allow for multiple transmission channels of QE programs, and gauge their relative importance. Our assumptions allow for a portfolio balance channel, a credit channel and an asset prices (wealth) channel of QE. Our results suggest that QE programs are mainly effective through wealth effects; that is, through the asset prices channel. Finally, our qualitative results are robust to quantitative refinements. The values of Θ are less flexible in this scenario, to satisfy Blanchard-Kahn conditions. However, the macroeconomic effects are much larger. Accordingly, the QE program seems to be effective mostly through the portfolio balance channel and wealth effects for private agents.

Appendix B -Model Equations

The set of equations for the baseline model without excess reserves is:

• Patient households: (R s t+1 -R t+1 ) κ (5.B)

C t + D h t + q h t (H t -H t-1 ) + Q t S h t + 1 2 κ(S h t -Sh ) 2 + q b t B h t + 1 2 κ(B h t -Bh ) 2 = W t L t + T t + R t D h t-1 + R s t Q t-1 S h t-1 + R b t q b t-
B h t = Bh + Λ t,t+1
(R b t+1 -R t+1 ) κ (6.B) q h t u c,t = u h,t + βE t q h t+1 u c,t+1 (7.B)

• Impatient households:

C t + q h t (
N c t = σ (R s t -R t )Q t-1 S c t-1 + (R a t -R t )q a t-1 A c t-1 + (R b t -R t )q b t-1 B c t-1 + R t N c t-1
+ ω c Q S + qa Āc + qb Bc (34.B)

Q t S c t + q a t A c t + q b t B c t + q h t M c t = D c t + N c t (35.B) Ω c t+1 = (1 -σ) + σ(µ c t φ c t + η c t ) (36.B)
• Shadow banks:

ν b t = E t Λ t,t+1 Ω b t+1 R s t+1 -R a t+1
(37.B)

η b t = E t Λ t,
N b t = σ (R s t -R a t )Q t-1 S b t-1 + R a t N b t-1 + ω b Q Sb (41.B) Q t S b t = q a t A b t + N b t (42.B
)

Ω b t+1 = (1 -σ) + σ(ν b t φ b t + η b t ) (43.B)
• Market clearing and miscellaneous: • Shadow banks:

K t+1 = S c
Given the balance sheet identity ( 16) and the evolution of net worth (17), broker's optimization problem comes down to a Bellman equation. Using the method of undetermined coefficients, we guess that the solution is linear in the time-varying coefficients:

V b t = ν b t Q t s b t + η b t n b t (16.C)
Now insert the guessed solution into the Bellman equation then maximize the objective function subject to the incentive constraint (18). Using a Lagrangian multiplier λ b t , the problem writes:

L = (1 + λ b t ) ν b t Q t s b t + η b t n b t -λ b t θ b Q t s b t (17.C)
The first-order conditions associated with s b t , λ b t are:

(1 + λ b t )ν b t = λ b t θ b (18.C) Q t s b t = φ b t n b t , if λ b t > 0 (19.C)
where, Then, using the law of motion of a broker net worth (17):

ν b t Q t s b t + η b t n b t = E t Λ t,t+1 Ω b t+1 n b t+1 ν c t Q t s b t + η b t n b t = E t Λ t,t+1 Ω b t+1 ((R s t+1 -R a t+1 )Q t s b t + R a t+1 n b t ) (24.C)
which gives the solutions:

ν b t = E t Λ t,t+1 Ω b t+1 R s t+1 -R a t+1
(25.C)

η b t = E t Λ t,t+1 Ω b t+1 R a t+1 (26.C)

  qa Āc + qb Bc and χ b = ω b Q Sb , in which the bars indicate the steady-state values.

  the survival probability σ matches an expected lifetime for bankers and brokers of roughly ten years. The parameters θ c , ξ c , ∆ and Ξ are specific to the commercial banks and are adjusted to target data on the excess returns ( Rs -R), ( Ra -R), ( Rb -R) and the equity multiplier φc . The excess return on primary loans ( Rs -R) corresponds to the average Moody's AAA corporate bond yield relative to yield on 10-year Treasury constant maturity between 1990Q1 and 2007Q4; that is, 100 basis points. According to estimates from[START_REF] Meeks | Shadow banks and macroeconomic instability[END_REF], we fix the excess return on ABS ( Ra -R) to 25 basis points.14 The excess return on long-term government bonds ( Rb -R) is of 50 basis points, in line with the average estimates of the term premium on a 10-year zero coupon bond byAdrian et al. (2013

  Figures 15 and 16 in Appendix A provide other key IRF.

Figure 1 :

 1 Figure 1: Capital quality shock

Figure 2 :

 2 Figure 2: Securization crisis

  ) in which blue colored variables refer to the interventions of the central bank into the balance sheet of brokers and in the bond market. By relaxing the degrees of interventions, we can easily disentangle how each type of purchase, in each market, is effective in supporting the real economy. In their studies, GK and Meeks et al. (2017) find that the effectiveness of asset purchases is an increasing function of the value of the asset weights in the incentive constraints.

Figure 5

 5 Figure5allows for variations in market interventions, and shows that the effects of QE programs depending on the type of asset are similar, regardless of the market on which the intervention takes place. For instance, the impact of primary securities purchases S c in the primary market are equivalent to the effects of a purchases of households securities S h in the bond market. 20 As such, this way of designing QE intervention clearly shuts down some of the theoretical transmission channels of QE. In particular, it remains silent on the implications of QE programs within the financial system, as it does not trigger changes in the relative supply of assets. These shortcomings are overturned by refining the design of QE intervention, introducing banking reserves in the model and using the supply of excess reserves to implement QE programs. This way of designing QE programs in a DSGE model with financial frictions is both more realistic with respect to the actual implementation of QE programs, and allows for more refined understanding of the different channels through which QE affects the financial system, and in fine the real economy.

Figure 3 :

 3 Figure 3: Quantitative Easing: public intermediation

Figure 4 :

 4 Figure 4: Quantitative Easing: public intermediation, cont.

Figure 5 :

 5 Figure 5: Quantitative Easing: interventions on different markets

Figure 6 :

 6 Figure 6: Quantitative Easing through Excess Reserves

Figure 7 :

 7 Figure 7: Quantitative Easing through Excess Reserves, cont.

  Figure 23 in Appendix A emphasizes that macroeconomic fluctuations are clearly increasing in the fluctuations in asset prices.

Figure 10 :

 10 Figure 10: Share of excess reserves in total assets, for all Federal Reserve banks

Figure 11 :

 11 Figure 11: Main interest rates for the conduct of conventional monetary policy in the US

Figure 12 :

 12 Figure 12: Cyclical components of business loans and ABS

Figure 13 :

 13 Figure 13: US detrended data from 2007Q4 to 2017Q4

Figure 14 :

 14 Figure 14: Capital quality shock, cont.

Figure 15 :

 15 Figure 15: Securitization crisis, cont.

Figure 16 :

 16 Figure 16: Securitization crisis, cont.

Figure 17 :

 17 Figure 17: Capital quality shock, changes in Ξ

Figure 18 :

 18 Figure 18: Public intermediation (GK) vs Excess Reserves

Figure 19 :

 19 Figure 19: Public intermediation (GK) vs Excess Reserves, cont.

Figure 20 :

 20 Figure 20: First scenario (fixed rate), changes in the value of Θ

Figure 21 :

 21 Figure 21: Excess Reserves, different scenarios

Figure 22 :

 22 Figure 22: Excess Reserves, different scenarios, cont.

Figure 23 :

 23 Figure 23: Second scenario (market rate), changes in the value of Θ

  (18.C), one can rewrite the guessed solution as:V b t = ν b t φ b t n b t + η b t n b t (21.C)which in turn can be plugged into the Bellman equation as follows:ν b t Q t s b t + η b t n b t = E t Λ t,t+1 (1 -σ) + σ(

  omy. Competitive intermediate good producers are one of those. Intermediate producers sell goods to retailers by using raw capital and labor in the two groups of households. A standard Cobb-Douglas function captures the production of intermediate goods:

  H t -H t-1 ) + R t M h t-1 = W t L t + M h = u h ,t + β E t q h t+1 u c ,t+1 + u c ,t λ h t (1 -γ m )χq h -λ h t )u c ,t = β E t R t+1 -γ m λ h t+1 u c ,t+1(12.B)where λ h t is the multiplier of the borrowing constraint (37) normalized by the marginal utility of consumption of impatient households u c ,t .J t = Y t + E t βγΛ t,t+1π

				R c t+1 =	Z t+1 + (1 -δ)Q t+1 Q t	ξ t+1	(17.B)
				R a t+1 =	Z t+1 + (1 -δ)q a t+1 t q a	ξ t+1	(18.B)
	• Capital producers:
				K t+1 = ξ t+1 (1 -δ)K t + I t	(19.B)
	Q t = 1 + f	I t I t-1	+	I t I t-1	f	I t I t-1		-E t Λ t,t+1	I t+1 I t	2	f	I t+1 I t	(20.B)
	• Final goods producers:
								Y m t = D t Y t	(21.B)
	D t = γD t-1 π -γ p ε t-1 π ε t + (1 -γ)	1 -γπ 1 -γ γ p (1-γ) t-1	π γ-1 t	-ε 1-γ	(22.B)
			F t = Y t P m t + E t βγΛ t,t+1	π -γ p ε t t+1 π -ε	F t+1	(23.B)
													γ p (1-ε)
													t π	t+1 (1-ε)	J t+1	(24.B)
							π * t =	ε ε -1	F t J t	π t	(25.B)
				π 1-ε t	= γπ	γ p (1-ε) t-1	+ (1 -γ)(π * t ) 1-ε	(26.B)
							i t = R t+1 π t+1	t	(8.B) (27.B)
	• Commercial banks:		M h t = γ m M h t-1 + (1 -γ m )χq h t H t	(9.B)
				W t u c ,t = u L ,t t = E t Λ t,t+1 Ω c µ c t+1 R s t+1 -R t+1	(10.B) (28.B)
	q h t u c ,t • Intermediate goods producers: µ a t = E t Λ t,t+1 Ω c t+1 R a t+1 -R t+1 µ b t = E t Λ t,t+1 Ω c t+1 R b t+1 -R t+1 η c t = E t Λ t,t+1 Ω c t+1 R t+1 φ c t = η c t θ c -µ c t Q t S c t + Ξq a t A c t + ∆q b t B c t = φ c t N c t	(29.B) (30.B) (31.B) (32.B) (33.B)
			Y m t = A t (ξ t K t ) α L	(1-α)(1-ϕ) t	L t (1-α)ϕ	(13.B)
							Z t = P m t α	Y m t K t	(14.B)
				W t = P m t (1 -α)(1 -ϕ)	Y m t L t	(15.B)
					W t = P m t (1 -α)ϕ	Y m t L t	(16.B)

t (11.B)

(1

  t = ī + κ π π t + κ y (log Y t -log Y * t ) + t (53.B)Then, using the law of motion of a banker net worth (2):

	µ c t Q t s c t + µ a t q a t a c t + µ b t q b t b c t + η c t n c t = E t Λ t,t+1 Ω c t+1 n c t+1
	µ c t Q t s c t + µ a t q a t a c t + µ b t q b t b c t + η c t n c t = E t Λ t,t+1 Ω c t+1 ((R s t+1 -R t+1 )Q t s c t + (R a t+1 -R t+1 )q a t a c t
	+ (R b t+1 -R t+1 )q b t b c t + R t+1 n c t )	(11.C)
	which gives the solutions:
	µ c t = E t Λ t,t+1 Ω c t+1 R s t+1 -R t+1	(12.C)
	µ a t = E t Λ t,t+1 Ω c t+1 R a t+1 -R t+1	(13.C)
	µ b t = E t Λ t,t+1 Ω c t+1 R b t+1 -R t+1	(14.C)
	η c t = E t Λ t,t+1 Ω c t+1 R t+1			(15.C)
	t + S b t + S h t + S g t		(44.B)
	A c t + A g t = A b t			(45.B)
	B = B c t + B h t + B g t		(46.B)
	H t + H t = 1			(47.B)
	M h t = M c t			(48.B)
	D h t = D c t			(49.B)
	Ḡ + (R b t -1) B = T t + (R s t -R t )Q t-1 S g t-1 + (R a t -R t )q a t-1 A g t-1 + (R b t -R t )q b t-1 B g t-1	(50.B)
	R b t+1 =	1 Pt + q b t+1 t q b		(51.B)
	Y t = C t + C t + 1 + f	I t I t-1	I t	(52.B)

i

For further empirical literature on QE see[START_REF] Borio | Unconventional monetary policies: a re-appraisal[END_REF].

It is conceivable to relax this assumption by adding an intermediation between bankers and brokers such as a special purpose vehicle or impose transaction costs.

The factors Γ c , Γ c , Γ h , Γ h scale consumption and housing so that steady-state values of consumption and housing do not depend on habit factors. Hence,Γ c = (1 -ε c )/(1 -βε c ), Γ c = (1 -ε c )/(1 -β ε c ), Γ h = (1 -ε h )/(1 -βε h ), Γ h = (1 -ε h )/(1 -β ε h )

This specification is based on findings in[START_REF] Guerrieri | Collateral constraints and macroeconomic asymmetries[END_REF], and implies that housing prices lead changes in the level of debt.

In our framework, the collateral constraint is always binding for practical reasons. The latter means that we do not capture the asymmetric effects of housing booms and busts à la[START_REF] Guerrieri | Collateral constraints and macroeconomic asymmetries[END_REF]. In the absence of slackness of the constraint, housing booms will then overfeed consumption. However,[START_REF] Guerrieri | Collateral constraints and macroeconomic asymmetries[END_REF] find that exogenous shocks that are not directly related to housing prices and/or consumption cause limited asymmetries. Accordingly, our qualitative results are relatively immune.

When ϕ = 0, the weight of impatient households in production goes to zero.

Other loans and advances by issuers of ABS are non-financial business loans securitized by depository institutions and finance companies and syndicated loans to non-financial corporate businesses. Ref: https: //www.federalreserve.gov/apps/fof/Guide/z1_tables_description.pdf

Superscript g indicates central bank related variables. Unlike GK, we ignore efficiency costs per unit of asset purchases, as these costs have negligible implications for the overall dynamics and therefore are of interest only for welfare analysis.

In the baseline scenario, we match the increase in the total assets of Federal Reserve banks as a percentage of GDP, i.e. 7.59% between

2008Q3 and 2008Q4 to investigate the macroeconomic implications of QE 20 This result applies even with changes in the transaction costs κ.

Some empirical evidences of such frictions are documented in[START_REF] Martin | Federal Reserve Tools for Managing Rates and Reserves[END_REF].

We report ABS purchases because responses are closer in magnitude for both specifications. Indeed, other asset purchases with a mechanism à la GK, such as primary loans and long-term government bonds, are by far more effective, as already shown in Figures3 and 4.

capital markets; in other words, credit and bond markets. In doing so, borrowers will most likely turn to the bond market to bypass financial frictions during bust periods. In addition, the bond market represents a large fraction of the capital market according to the steady-state values Sh / K and Bh / B. Those are in line with the calibration of GK as well as the shares of long-term government bonds and public expenditure to output ( B/ Ȳ , Ḡ/ Ȳ ). Parameter values are summarized in Table 1 below Given the balance sheet identity (1) and the evolution of net worth (2), banker's optimization problem comes down to a Bellman equation. Using the method of undetermined coefficients, we guess that the solution is linear in the time-varying coefficients:

Now insert the guessed solution into the Bellman equation then maximize the objective function subject to the incentive constraint (3). Using a Lagrangian multiplier λ c t , the problem writes:

The first-order conditions associated with s c t , a c t , b c t , λ c t are:

where,

Using the FOCs (4.C), (5.C) and (6.C), step by step, one can rewrite the guessed solution as:

which in turn can be plugged into the Bellman equation as follows:

Define the shadow price of a banker net worth as:

Appendix D -Data


Kien Pham

Ronan Sauleau

Erwan Fourn

Fatimata Diaby

Antonio Clemente

Laurent Dussopt

Fatimata Diaby # §

K/Ka-Band Transmitarray Antennas Based on Polarization Twisted Unit-Cells

Keywords: Transmitarray antenna, dual-band, Ka-band, discrete lens antenna. Receiving array Transmiting array

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Transmitarrays are promising solutions for many applications at millimeter wave frequencies [START_REF] Abbaspour-Tamijani | A millimeter-wave bandpass filter-lens array[END_REF]- [START_REF] Palma | Circularly-polarized transmitarray with sequential rotation in Ka-band[END_REF]. The working principle of a transmitarray consists in collimating the incident wave from a feed horn into a desired direction. A transmitarray can be decomposed into three main building blocks, namely a receiving layer to collect the incident wave from the feed horn, a transmitting layer to radiate the wave to free space, and a phase-shifter used to compensate the different path lengths from the feed horn to the unit-cells (see Fig. 1). To implement successfully a transmitarray, the unitcell must ideally offer total transmission in the band of interest with capability to control the transmission phase.

Various technologies of multi-band antenna arrays have been proposed over the last decades. Like reflectarray antennas (RAs) [START_REF] Guo | Single-layered broadband dual-band reflectarray with linear orthogonal polarizations[END_REF], transmitarray antennas (TAs) are in this flow. Some preliminary demonstrations at the unit-cell level have been proposed in X-, K-and Ka-bands [START_REF] Chaloun | Active transmitarray submodule for K/Ka band satcom applications[END_REF]- [START_REF] Naseri | Dual-band circularly polarized transmit-array unit-cell at X and K bands[END_REF]. A dualband unit-cell based on MMIC technology is introduced in [START_REF] Chaloun | Active transmitarray submodule for K/Ka band satcom applications[END_REF] with up-and down-link patches arranged in aperiodic array.

Moreover, several passive dual-band unit-cells operating in circular polarization and based on rotation technique to control the transmission phase have been proposed. As reported in [START_REF] Hasani | Dualband 20/30 GHz circularly polarized transmitarray for SOTM applications[END_REF], five identical layers of rotated cut-rings printed on four 0.784-mm-thick RT 5880 substrates are combined to obtain 360° phase variation at each frequency band. In [START_REF] Naseri | Dual-band circularly polarized transmit-array unit-cell at X and K bands[END_REF], a combination of a cut-ring with a thin microstrip patch is proposed to vary simultaneously the phase-shift at two frequencies by rotation technique. A dual-band metal-only transmitarray operating in X-band with high efficiency (37.2 % at 11 GHz and 34 % at 12.5 GHz) and dual-polarization for two close bands has been presented in [START_REF] Bagheri | Dual-band, dualpolarised metallic slot transmitarray antenna[END_REF]. However the layout is not optimal in each band because the average frequency of 11 GHz and 12.5 GHz is used in the design. Better performance and independency of each frequency are proposed in [START_REF] Wu | High-gain dualband transmitarray[END_REF] at Ku-band. Although high efficiency is obtained, the achieved bandwidth is quite narrow and no steering capability is studied due to large periodicity of unitcell (0.90 at 18 GHz). Recently, a dual-band Satcom transmitarray at Ka-band has been presented in [START_REF] Matos | High gain dual-band beam-steering transmit array for satcom terminals at Ka-band[END_REF] with complex stack-up configuration (6 substrates and 7 metallic layers) and the unit-cell exhibits dependent phase variation between two frequencies (20 GHz and 30 GHz). To overcome the above-mentioned limitations, we introduce here a simple and easy-to-design dual-band unit-cell operating in the 18.8 -21 GHz band (down link) and 27.5 -31 GHz band (up-link) for satellite communications in K/Ka band [START_REF] Pearson | Next generation mobile SATCOM terminal antennas for a transformed world[END_REF]. We will show that transmission phase of the unitcell can be varied independently in each frequency band without interfering the frequency response in the other band. The unit-cell architecture is very simple and is based on a three metal layer design. The frequency responses of the proposed unit-cell are discussed in Section II. Then, we compare in Section III the radiation performance of two sets of TAs, one dual-band TA built from the proposed dual-band unit-cell, and two single-band TAs operating either in the down-link band or in the up-link one. Finally, conclusions are drawn in Section IV.

II. NUMERICAL STUDY OF UNIT-CELLS

The proposed unit-cell relies on two design principles: polarization twisting and frequency scalability. Its dimensions are optimized separately in each frequency band using Floquet port and periodic boundary conditions in Ansys HFSS version 15. All frequency responses have been computed assuming a Floquet illumination under normal incidence.

A. Designs and Characteristics

Fig. 2 represents the 3D view of the single-band (SB) unitcell in a periodicity of 6 mm. It is made of narrow microstrip patches printed on the receiving and transmitting layers on two identical dielectric substrates (Rogers Duroid RT6002, hs = 0.762 mm, εr = 2.94, and tanδ = 0.0012). Both patches are connected by a metalized via hole (diameter of 200 µm). A circular aperture with a diameter of 800 µm is used to isolate this via hole from the ground plane. A thin film (Arlon CuCLAD 6700, εr = 2.35, and tanδ = 0.0025) is used to bond the two substrates.

This unit-cell operates in linear polarization. The metallized via connects the receiving and transmitting patches; it is located at the center of the unit-cell and divides each patch into two unbalanced parts. Thanks to this via working as a reference point for the electric field, the polarization can be changed either at the receiving or transmitting layer by rotation of the patches. In the current work, the receiving layer is rotated by 180° so that the unitcell exhibits a 1-bit transmission phase quantization (180° phase resolution), whereas the transmitting layer is rotated by 90° to rotate the field polarization. Consequently, contribution of spill-over radiation on the co-polarization pattern is significantly reduced. The SB unit-cell is able to operate either in the up-or down-link bands by scaling the dimensions of the radiating patches. Their length and width equal 3 mm/4.65 mm and 0.5mm/0.65mm for the up-/down links respectively. Fig. 3 represents the 3D view of the dual-band (DB) unitcell with a square periodicity of 6 mm. It is a combination of the interleaved SB unit-cells operating individually in the uplink and down-link. In this figure, the up-link unit-cell is plotted in blue and located at the center, while the down-link unit-cell is shown in red.

B. Frequency Responses

The frequency responses of the up-link unit-cell alone and in presence of surrounding down-link unit-cells are compared in Fig. 4. We can notice in Fig. 4(a) that the reflection and transmission coefficients in magnitude are the same for both phase states, and for the SB and DB designs. More precisely, the -10 dB impedance bandwidth spans from 27.7 GHz to 30.3 GHz (2.6 GHz or 9% around 29 GHz). The insertion loss is low over the fractional bandwidth, e.g. about 0.4 dB at 29 GHz. The superimposing of SB and DB unit-cell in transmission phase over frequency is also observed (Fig. 4b). Moreover, Fig. 4(b) also shows that the unit-cell exhibits a 180° phase shift over the whole bandwidth with a maximum phase variation of +/-2°. Finally, similar conclusions can be drawn for the down-link band (results not shown here for brevity), with a -10dB reflection bandwidth of 1.8 GHz (from 18.8 GHz to 20.6 GHz), equivalent to 9.2% around 19.5 GHz. These very attractive features lead to a simplified design procedure because only one phase state needs to be considered, e.g. 0°-state (UC0). In addition, as the frequency responses are exactly the same for the SB and DB designs, this implies that mutual coupling between up-and down-link unitcells is extremely low. Therefore, the DB transmitarray can be designed easily first by designing separately two SB transmitarrays, and second by interleaving them, as done at the unit-cell level (Fig. 3).

III. DUAL-BAND TRANSMITARRAY ANTENNAS BASED ON POLARIZATION TWISTED UNIT-CELLS

Based on the DB unit-cell described in Section II, we design here a DB TA, and compare its performance to those of two SB TAs (one operating in the up-link, and one in the down-link). The DB TA works in dual-linear polarization, one polarization for each band, e.g. y-polarized for the up-link band and x-polarized for the down-link. It is illuminated by two separate pyramidal feed horns. In the DB scenario, two feed horns are placed at 79 mm (focal distance) away from the radiating panel. The up-and down-link horns (in yellow and orange in Fig. 5, respectively) are placed in offset positions (+/-10 mm from z-axis) in the xz-plane.

The TAs have been simulated using hybrid finite element boundary integral method (FE-BI) of HFSS as illustrated in Fig. 5 for the DB TA. They radiate a pencil beam at broadside.

A. Transmitarray Designs

The up-and down-link transmitarrays layouts are represented in Fig. 6. The up-link array is made of 400 unitcells arranged in 20-by-20 square matrix while the down-link one consists of a 21-by-21 matrix (441 unit-cells). The layouts of single-and dual-band TAs have been generated using our in-house hybrid analysis tool [START_REF] Kaouach | Wideband low-loss linear and circular polarization transmitarray in V-band[END_REF]- [START_REF] Palma | Circularly-polarized transmitarray with sequential rotation in Ka-band[END_REF], [START_REF] Pham | Design of wideband dual linearly-polarized transmitarray antennas[END_REF]. Thanks to the very low mutual coupling between the SB unitcells, the TA design procedure first starts with the definition of the single-band layouts for the up-and down links separately. Then, the DB layout is simply obtained by interleaving both SB layouts.

B. Radiation Performance

The radiation patterns simulated at the center frequencies of each band (29 GHz and 19.5 GHz for the up-and downlink bands, respectively) are represented in Figs. 7 and8.

The co-polarization and cross-polarization components at 29 GHz are almost superimposed in both principal planes (Eand H-planes) for single-band (SB) and dual-band (DB) TA (see Fig. 7). The side lobe level remains below -20 dB while the cross-polarization level is kept under -24 dB for all elevation angles. Similar conclusions can be drawn at 19.5 GHz (Fig. 8).

These results clearly demonstrate independency between both bands, and thus validate the interleaving concept with polarization twisted unit-cells. This concept has been also validated numerically for antenna beams pointing in offset directions (results not shown for brevity purposes).

IV. CONCLUSION

Dual-band unit-cells and transmitarray antennas have been proposed for applications in K/Ka-bands. The proposed concept consists in interleaving two very weakly coupled single-band transmitarrays sharing the same radiating aperture and operating in linear orthogonal polarization. Comparison between the single-band and dual-band designs confirms the relevance of the proposed designs. Thanks to this technique, the design procedure is simplified significantly since the layout is optimized individually for each band.

ACKNOWLEDGMENT

This works is supported by The French National Research Agency (ANR) in the project named TRANSMIL (TRANSmitarray antenna at MILlimeter-wave) under Grant ANR-14-CE28-0023.

Fig. 1 .

 1 Fig. 1. Geometry of a transmitarray with its unit-cell.

Fig. 2 .

 2 Fig. 2. 3D view of the single-band (SB) unit-cell.

Fig. 3 .

 3 Fig. 3. 3D view of the dual-band (DB) unit-cell. The up-link and down-link patches are shown in blue and red respectively. Note that the lattice size is the same in both bands.

Fig. 4 .

 4 Fig. 4. Frequency responses of the single-band (SB) and dual-band (DB) unitcells in the up-link band. (a) Magnitude of reflection and transmission coefficients for the 0°/180°-phase states of the unit-cell (UC0/UC1). (b) Transmission phase for the 0°/180°-phase states of the unit-cells (UC0/UC1).

Fig. 5 .

 5 Fig. 5. Simulation configuration for SB and DB TAs using in HFSS.FE-BI.

Fig. 6 .

 6 Antenna layouts (phase states of the unit-cells, in deg.) for a beam pointing at broadside computed at 29 GHz for the up-link (a), and at 19.5 GHz for down-link (b).

Fig. 7 .

 7 Co-and cross-polarization components computed in E-plane (a) and H-plane (b) for single-band (SB) and dual-band (DB) transmitarray antennas at 29 GHz.

Fig. 8 .

 8 Co-and cross-polarization components computed in E-plane (a) and H-plane (b) for single-band (SB) and dual-band (DB) transmitarray antennas at 19.5 GHz.

  -pol DB, co-pol SB, cross-pol DB, cross-pol