Giulio Coluccia

Aline Roumy

Enrico Magli

EXACT PERFORMANCE ANALYSIS OF THE ORACLE RECEIVER FOR COMPRESSED SENSING RECONSTRUCTION

Keywords: Compressed Sensing, Oracle Receiver, Wishart Matrix

A sparse or compressible signal can be recovered from a certain number of noisy random projections, smaller than what dictated by classic Shannon/Nyquist theory. In this paper, we derive the closed-form expression of the mean square error performance of the oracle receiver, knowing the sparsity pattern of the signal. With respect to existing bounds, our result is exact and does not depend on a particular realization of the sensing matrix. Moreover, our result holds irrespective of whether the noise affecting the measurements is white or correlated. Numerical results show a perfect match between equations and simulations, confirming the validity of the result.

INTRODUCTION

Compressed sensing (CS) [START_REF] Donoho | Compressed sensing[END_REF][START_REF] Candès | Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?[END_REF] has emerged in past years as an efficient technique for sensing a signal with fewer coefficients than dictated by classic Shannon/Nyquist theory. The hypothesis underlying this approach is that the signal to be sensed must have a sparse -or at least compressible -representation in a convenient basis. In CS, sensing is performed by taking a number of linear projections of the signal onto pseudorandom sequences. Therefore, the acquisition presents appealing properties. First, it requires low encoding complexity, since no sorting of the sparse signal representation is required. Second, the choice of the sensing matrix distribution is blind to the source distribution.

Several different techniques can be used to reconstruct a signal from CS measurements. Often, for performance assessment, the ideal oracle receiver, i.e., a receiver with perfect knowledge of the signal sparsity support, is considered as a benchmark. But even for this ideal receiver, only upper and lower performance bounds are available. For example, in [START_REF] Eldar | Compressed Sensing: Theory and Applications[END_REF] a bound depending on a particular realization of the This work has been supported by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement n 279848 sensing matrix was derived. This bound represents a worstcase scenario since it depends on the maximum norm of the noise vector. An average (over noise) bound was presented in [START_REF] Davenport | The pros and cons of compressive sensing for wideband signal acquisition: Noise folding vs. dynamic range[END_REF] for white noise and in [START_REF] Laska | Regime change: Bitdepth versus measurement-rate in compressive sensing[END_REF] for correlated noise. Both bounds depend on the Restricted Isometry Property (RIP) constant of the sensing matrix, a parameter taking different values from realization to realization of the sensing matrix and whose evaluation represents a combinatorial complexity problem. Even if there exist classes of matrix respecting the RIP with a certain constant with high probability, this would give a probabilistic result, restricted to a specific class of sensing matrices. Moreover, note that [START_REF] Laska | Regime change: Bitdepth versus measurement-rate in compressive sensing[END_REF] overestimates the reconstruction error giving a result which depends on the maximum eigenvalue of the noise covariance matrix. Other results can be found in [START_REF] Arias-Castro | On the fundamental limits of adaptive sensing[END_REF] and [START_REF] Candès | How well can we estimate a sparse vector?[END_REF].

In this paper, we present the exact average performance of the oracle receiver. The average is taken over noise distribution but also over the sensing matrix distribution, and does not depend on the RIP constant of a specific sensing matrix (or family of sensing matrices), but only on system or signal parameters. Using some recent results about Wishart random matrix theory [START_REF] Cook | On the mean and variance of the generalized inverse of a singular wishart matrix[END_REF], we show that the performance depends, apart from system parameters, on the variance of the noise, only, and not on its covariance. Hence, our result can be applied both to systems where measurements are corrupted by either white or correlated noise.

BACKGROUND

Compressed Sensing

In the standard CS framework, introduced in [1, 2], the signal x ∈ R N ×1 , having a K-sparse representation in some basis Ψ ∈ R N ×N , i.e.: x = Ψθ, θ 0 = K, K N , can be recovered by a smaller vector of noisy linear measurements y = Φx + z, y ∈ R M ×1 and K < M < N , where Φ ∈ R M ×N is the sensing matrix and z ∈ R M ×1 is the vector representing additive noise such that z 2 < ε, by solving the 1 minimization with inequality constraints

θ = arg min θ θ 1 s.t. ΦΨθ -y 2 < ε (1)
and x = Ψ θ, known as basis pursuit denoising, provided that M = O(K log(N/K)) and that each submatrix consisting of K columns of ΦΨ is (almost) distance preserving [3, Definition 1.3]. The latter condition is the Restricted Isometry Property (RIP). Formally, the matrix ΦΨ satisfies the RIP of order K if ∃δ K ∈ (0, 1] such that, for any θ with θ 0 ≤ K:

(1 -δ K ) θ 2 2 ≤ ΦΨθ 2 2 ≤ (1 + δ K ) θ 2 2 , (2)
where δ K is the RIP constant of order K. It has been shown in [START_REF] Baraniuk | A simple proof of the restricted isometry property for random matrices[END_REF] that when Φ is an i.i.d. random matrix drawn from any subgaussian distribution and Ψ is an orthogonal matrix, ΦΨ satisfies the RIP with overwhelming probability.

Wishart Matrices

Let x i be a zero-mean Gaussian random vector with covariance matrix Σ. Collect n realizations of x i as rows of the n×p matrix X. Hence, X T X is distributed as a p-dimensional Wishart matrix with scale matrix Σ and n degrees of freedom [START_REF] Press | Applied Multivariate Analysis: Using Bayesian and Frequentist Methods of Inference[END_REF]:

W = X T X ∼ W p (Σ, n) .
When n > p, W can be inverted. The distribution of W -1 is the Inverse Wishart, whose distribution and moments were derived in [START_REF] Rosen | Moments for the inverted wishart distribution[END_REF]:

W -1 ∼ W -1 p Σ -1 , n .
On the other hand, when n < p, W is rank-deficient, hence not invertible. Its Moore-Penrose pseudoinverse W † follows a generalized inverse Wishart distribution, whose distribution is given in [START_REF] Díaz-García | Distribution of the generalised inverse of a random matrix and its applications[END_REF] and mean and variance were recently derived in [8, Theorem 2.1], under the assumptions that p > n + 3 and Σ = I.

PERFORMANCE OF THE ORACLE RECEIVER

System model

Consider the vector x = Ψθ ∈ R N . The nonzero components of the K-sparse vector θ are modeled as i.i.d. centred random variables with variance σ 2 θ . The vector x is observed through a smaller vector of noisy Gaussian measurements defined as the vector y ∈ R M such that y = Φx + z, where the sensing matrix Φ ∈ R M ×N , with M < N , is a random matrix with i.i.d. entries drawn from a zero-mean Gaussian distribution with variance σ 2 Φ and z ∈ R M ×1 , representing the noise, is drawn from a zeromean multivariate random distribution with covariance matrix Σ z .

We remark here that in our analysis we consider measurements affected both by white noise, i.e., the case where Σ z = I, like thermal noise or quantization noise deriving from uniform scalar quantizer in the high-rate regime, as well as correlated noise, like the one affecting measurements quantized using vector quantization or the noise at the output of a low-pass filter.

Error affecting the oracle reconstruction

We now evaluate the performance of CS reconstruction with noisy measurements. The performance depends on the amount of noise affecting the measurements. In particular, the distortion xx 2 2 is upper bounded by the noise variance up to a scaling factor [START_REF] Candès | Stable signal recovery from incomplete and inaccurate measurements[END_REF][START_REF] Candès | The restricted isometry property and its implications for compressed sensing[END_REF] xx

2 2 ≤ c 2 ε 2 ,
where the constant c depends on the realization of the measurement matrix, since it is a function of the RIP constant. Since we consider the average 1 performance, we need to consider the worst case c and this upper bound will be very loose [3, Theorem 1.9].

Here, we consider the oracle estimator, which is the estimator knowing exactly the sparsity support Ω = {i|θ i = 0} of the signal x.

Let U Ω be the submatrix of U obtained by keeping the columns of ΦΨ indexed by Ω, and let Ω c denote the complementary set of indexes. The optimal reconstruction is then obtained by using the pseudo-inverse of U Ω , denoted by U † Ω :

   θ Ω = U † Ω y := U T Ω U Ω -1 U T Ω y θ Ω c = 0 (3) x = Ψ θ (4)
For the oracle estimator, upper and lower bounds depending on the RIP constant can be found, for example in [START_REF] Davenport | The pros and cons of compressive sensing for wideband signal acquisition: Noise folding vs. dynamic range[END_REF] when the noise affecting the measurements is white and in [START_REF] Laska | Regime change: Bitdepth versus measurement-rate in compressive sensing[END_REF] when the noise is correlated. Unlike [START_REF] Davenport | The pros and cons of compressive sensing for wideband signal acquisition: Noise folding vs. dynamic range[END_REF][START_REF] Laska | Regime change: Bitdepth versus measurement-rate in compressive sensing[END_REF], in this paper the average performance of the oracle, depending on system parameters only, is derived exactly. Relations with previous work will be thoroughly described in section 3.2.1.

As we will show in the following sections, the characterization of the ideal oracle estimator allows to derive the reconstruction RD functions with results holding also when non ideal estimators are used. Theorem 1. Let x and y be defined as in section 3.1. Assume reconstruction by the oracle estimator, when the support Ω of x is available at the receiver. The average reconstruction error of any reconstruction algorithm is lower bounded by that of the oracle estimator that satisfies

E x -x 2 2 = K M (M -K -1) Tr (Σ z ) σ 2 Φ ( 5
)
Proof. We derive a lower bound on the achievable distortion by assuming that the sparsity support Ω of x is known at the decoder.

Hence,

E x -x 2 2 = E θ -θ 2 2 = E θ Ω -θ Ω 2 2 (6) = E U † Ω z 2 2 (7) = E z T E (U Ω U T Ω ) † z (8)
The first equality in [START_REF] Arias-Castro | On the fundamental limits of adaptive sensing[END_REF] follows from the orthogonality of the matrix Ψ, whereas the second one follows from the assumption that Ω is the true support of θ. [START_REF] Candès | How well can we estimate a sparse vector?[END_REF] comes from the definition of the pseudo-inverse, and (8) from the equality

U † T Ω U † Ω = (U Ω U T Ω )
† and from the statistical independence of U and z. Then, if M > K + 3,

E x -x 2 2 = E z T K M (M -K -1) 1 σ 2 Φ I z (9) = K M (M -K -1) Tr (Σ z ) σ 2 Φ ( 10
)
where ( 9) comes from the fact that, since M > K, U Ω U T Ω is rank deficient and follows a singular M -variate Wishart distribution with K degrees of freedom and scale matrix σ 2 Φ I [START_REF] Díaz-García | Distribution of the generalised inverse of a random matrix and its applications[END_REF]. Its pseudo-inverse follows a generalized inverse Wishart distribution, whose distribution is given in [START_REF] Díaz-García | Distribution of the generalised inverse of a random matrix and its applications[END_REF] and the mean value is given in [8, Theorem 2.1], under the assumption that M > K + 3. Note that the condition M > K + 3 is not restrictive since it holds for all K and M of practical interest. It can be noticed that the distortion of the oracle only depends on the variance of the elements of z and not on its covariance matrix. Therefore, our result holds even if the noise is correlated (for instance if vector quantization is used). As a consequence, we can apply our result to any quantization algorithm or to noise not resulting from quantization. Note that, if the elements of z have the same variance, (5) reduces to

E x -x 2 2 = K M -K -1 σ 2 z σ 2 Φ (11)

Relations with previous work

The results obtained in Theorem 1 provide a twofold contribution with respect to results already existing in literature about the oracle reconstruction. First, they are exact and not given as bounds. Second, they do not depend on parameters which cannot be evaluated in practical systems, e.g., the RIP constant of the sensing matrices. For example, in [START_REF] Eldar | Compressed Sensing: Theory and Applications[END_REF] the following worst-case upper bound was derived

x -x 2 2 ≤ 1 1 -δ 2K z 2 2 , (12)
which depends on a particular realization of the sensing matrix, since it depends on its RIP constant δ 2K , and is very conservative, since it is function of the maximum 2 norm of the noise vector. An average evaluation (over noise) was given in [START_REF] Davenport | The pros and cons of compressive sensing for wideband signal acquisition: Noise folding vs. dynamic range[END_REF]Theorem 4.1] where the performance of the oracle receiver with measurements affected by white noise was derived

K 1 + δ K σ 2 z ≤ E z x -x 2 2 ≤ K 1 -δ K σ 2 z ( 13
)
but still the equation depends on the RIP constant of the sensing matrix and hence, on a particular realization. The result of ( 13) was generalized in [START_REF] Laska | Regime change: Bitdepth versus measurement-rate in compressive sensing[END_REF] to correlated noise

E z x -x 2 2 ≤ K 1 -δ K λ max (Σ z ) , (14)
where Σ z is the covariance matrix of z and λ max (•) represents the maximum eigenvalue of the argument. Hence, ( 14) represents an even looser bound, since the contribution of the noise correlation is upper bounded by using its biggest eigenvalue. Finally, the results of Theorem 1 can help to generalize related results, e.g., the Rate-Distortion performance of systems based on Compressed Sensing. See for example [15, section III.C], where a lower bound is derived, or [START_REF] Coluccia | Operational Rate-Distortion Performance of Single-Source and Distributed Compressed Sensing[END_REF], where the exact RD performance is derived.

NUMERICAL RESULTS

In this section, we show the validity of the results of Theorem 1 by comparing the equations to the results of simulations. Here and in the following sections, signal length is N = 512 with sparsity K = 16. M = 128 measurements are taken. The nonzero elements of the signal are distributed as N (0, 1). The sparsity basis Ψ is the DCT matrix. The sensing matrix is composed by i.i.d. elements distributed as zero-mean Gaussian with variance 1/M . The noise vector is Gaussian with zero mean, while the covariance matrix depends on the specific test and will be discussed later. The reconstructed signal x is obtained using the oracle estimator. A different realization of the signal, noise and sensing matrix is drawn for each trial, and the reconstruction error, evaluated as E xx 2 2 , is averaged over 1,000 trials.

White noise

In this first experiment, the measurement vector y is corrupted by white Gaussian noise, i.e., z ∼ N p (0, σ 2 z I M ). Fig. 1 shows the comparison between the simulated reconstruction error and [START_REF] Rosen | Moments for the inverted wishart distribution[END_REF]. It can be easily noticed that the match between simulated and theoretical curve is perfect. As a term of comparison, we plot also the upper and lower bounds of (13), for δ K = 0 (ideal case) and δ K = 0.5. It can be noticed that for δ K = 0 the two bounds match and are close to the simulated curve, but even the upper bound is lower than the real curve. Instead, for δ K = 0.5 the two bounds are almost symmetric with respect to the realistic curve but quite far from it. The conclusion is that bounds in the form of ( 13) are difficult to use due to the lack of knowledge of the RIP constant. Even if the sensing matrix belongs to a class where a probabilistic expression of the RIP constant exists, like the ones in [START_REF] Vershynin | Non-asymptotic random matrix theory[END_REF], a specific value depending on system parameters only is usually difficult to obtain since it depends on constants whose value is unknown or hard to compute. Tests with generic diagonal Σ z have also been run, confirming a perfect match with (5).

Uniform scalar quantization

A practical application of the white noise case is a system where the measurement vector is quantized using an uniform scalar quantizer with step size ∆. In this case, equation [START_REF] Rosen | Moments for the inverted wishart distribution[END_REF] is very handy because it is well known that in the high-rate regime the quantization noise can be considered as uncorrelated and its variance is equal to ∆ 2 12 . In Fig. 2, we plot the reconstruction error of the oracle from quantized measurements vs. the step size ∆. It can be noticed that the match between simulations and proposed equation is perfect in the high-rate regime, i.e., when the step size gets small. z ρ |i-j| and ρ = 0.9, 0.999.

Correlated noise

We also report in Fig. 3 the results obtained reconstructing with the oracle receiver the measurements corrupted by correlated noise. In particular, the i, j-th element of the noise covariance matrix will be given by (Σ z ) i,j = σ 2 z ρ |i-j| . The correlation coefficient takes the values of ρ = 0.9 and 0.999. We compare the simulations with (5) and with the upper bound of ( 14), for δ K = 0 (ideal case) and δ K = 0.5. First, it can be noticed from Fig. 3 that simulations confirm the result that the performance of the oracle does not depend on noise covariance but only on its variance. This is shown by the fact that simulations for ρ = 0.9 overlap the ones for ρ = 0.999, and both match [START_REF] Laska | Regime change: Bitdepth versus measurement-rate in compressive sensing[END_REF], confirming the validity of Theorem 1 even in the correlated noise scenario. Second, Fig. 3 shows that the upper bounds of ( 14) highly overestimate the real reconstruction error of the oracle, even for the ideal δ K = 0 case. This can be explained by considering that in [START_REF] Candès | The restricted isometry property and its implications for compressed sensing[END_REF], for the chosen correlation model, λ max tends to σ 2 z M when ρ tends to 1.

CONCLUSIONS AND FUTURE WORK

In this paper, we derived the closed-form expression of the average performance of the oracle receiver for Compressed Sensing. Remarkably, this result is exact, and does not depend on the RIP constant or the noise covariance matrix. We showed that the theoretical results perfectly match the ones obtained by numerical simulations. This represents a significant improvement with respect to existing results, which consist in bounds depending on parameters that are hardly available.

As a future activity, this work can be extended to non ideal receivers, with a mismatched knowledge of the signal sparsity pattern. In that case, the performance will depend both on the noise affecting the signal and on the number of misestimated position in the sparsity pattern.

Fig. 1 .

 1 Fig. 1. Oracle reconstruction error. Simulations vs. Theorem 1. N = 512, K = 16, M = 128. White noise: Σz = σ 2 z IM .

Fig. 2 .

 2 Fig. 2. Oracle reconstruction error. Simulations vs. Theorem 1. N = 512, K = 16, M = 128. Measurements quantized with Uniform Scalar Quantizer with step size ∆.

1 Fig. 3 .

 13 Fig. 3. Oracle reconstruction error. Simulations vs. Theorem 1. N = 512, K = 16, M = 128. Correlated noise: (Σz)i,j = σ 2z ρ |i-j| and ρ = 0.9, 0.999.

The average performance is obtained averaging over all random variables i.e. the measurement matrix, the non-zero components θ and noise, as for example in[START_REF] Laska | Regime change: Bitdepth versus measurement-rate in compressive sensing[END_REF].


Alice Fabre
email: alice.fabre@univ-amu.fr.

Stéphane Pallage
email: pallage.stephane@uqam.ca.

Christian Zimmermann
email: zimmermann@stlouisfed.org.

Steve Ambler

Philippe De Donder

Jean-Denis Garon

Universal Basic Income versus Unemployment Insurance

Keywords: Universal basic income, Idiosyncratic shocks, Unemployment insurance, Heterogeneous agents, Moral hazard JEL classification : E24, D7, J65

In this paper we compare the welfare effects of unemployment insurance (UI) with an universal basic income (UBI) system in an economy with idiosyncratic shocks to employment. Both policies provide a safety net in the face of idiosyncratic shocks. While the unemployment insurance program should do a better job at protecting the unemployed, it suffers from moral hazard and substantial monitoring costs, which may threaten its usefulness. The universal basic income, which is simpler to manage and immune to moral hazard, may represent an interesting alternative in this context. We work within a dynamic equilibrium model with savings calibrated to the United States for 1990 and 2011, and provide results that show that UI beats UBI for insurance purposes because it is better targeted towards those in need.

Introduction

In recent years, universal basic income policies have been proposed by economists and political scientists as credible alternatives to existing social programs. The idea is simple:

Provide citizens with a minimum allowance without means-testing that would give everyone the means to live on a basic level with dignity. A recent example is a constitutional initiative launched in October 2013 in Switzerland. 1 Advocates of universal basic income [UBI] can be found all over the political spectrum and in many academic circles, according to the "rainbow coalition" image provided by Anthony Atkinson [START_REF] Atkinson | Public Economics in Action: The Basic Income/Flat Tax Proposal[END_REF]. From economists James Meade, Milton Friedman, James Tobin, and Anthony Atkinson to philosophers John Rawls and Philippe Van Parijs, the concept has received much support over the years, for a wide range of reasons: alleviating poverty, simplifying the social protection system, providing additional freedom, etc. James Meade was one of the first to voice the idea of a social dividend back in 1935 [START_REF] Meade | Outline of an economic policy for a labour government[END_REF][START_REF] Meade | Agathotopia: The Economics of Partnership[END_REF]. Juliet Rhys-Williams in the 1940s and then Milton Friedman in the 1960s reintroduced the idea in the form of a negative income tax [START_REF] Rhys-Williams | Something to look forward to[END_REF][START_REF] Friedman | The case for a negative income tax[END_REF], and Tobin offered the concept of a credit income tax (Tobin, 1966;Tobin, Pechman, & Mieszkowski, 1967). 2 Friedman's interest in UBI was motivated by the fact that free markets did not guarantee that everyone could meet basic needs. Seeking ways for society to correct for this result, he saw efficiency gains in the simplicity of the program. Milton Friedman argued that there were too many problems with existing social programs and that they should be replaced by a unique program, i.e. a negative income tax [START_REF] Frank | The other Milton Friedman: a conservative with a social welfare program[END_REF]. [START_REF] Rawls | A Theory of Justice[END_REF] agreed with Friedman on that point. UBI programs vary in scope according to the perspective of their advocates. While Philippe Van Parijs highlights its universal, unconditional aspect (Van Parijs, 1991), An-1 See, for instance, [START_REF] Müller | Débat sur le revenu de base: libération de la Suisse ou attaque contre l'Etat social[END_REF]. This debate has contributed to similar questionning in North America and in Europe [START_REF] Lowrey | Switzerland's proposal to pay people for being alive[END_REF]Van Parijs, 2013, for example).

2 Although the UBI policy and the negative income tax can theoretically achieve the same result, there are slight differences between the two: the negative income tax is distributed at the end of the fiscal year to those capable of filling a tax return. Technically, UBI is paid upfront to all adult members of society.

Tobin's credit income tax is quite close to the negative income tax, yet it may differ in its administration. It inspired the "Demogrant" proposal of the US democratic presidential candidate George McGovern (defeated in 1971 by Richard Nixon). It should be noted Martin Luther King also promoted the idea of a guaranteed income for all Americans to reduce poverty [START_REF] King | Where Do We Go from Here: Chaos or Community? Harper and Row[END_REF].

thony Atkinson views it rather as a participation income program [START_REF] Atkinson | Public Economics in Action: The Basic Income/Flat Tax Proposal[END_REF]. Van Parijs (2000Parijs ( , 2004) ) thinks of the basic income as a way to provide everyone with the same minimum income, but also as a way to avoid much of the moral hazard of other social programs. For him, UBI certainly reduces poverty, but it is also motivated by ethical reasons: beneath this premise is a sense of social justice and the social reponsibility of a free state for its members. For freedom to be meaningful, a free society must provide everyone with the proper means to exert that freedom (Van Parijs, 1991). 3 Herbert Simon also made a statement in favor of a UBI policy, praising its justice while highlighting its possible disincentives (Simon, 2000). In the economic literature, UBI is sometimes viewed as a way to help the unemployed (Van der Linden, 2002;[START_REF] Lehmann | Évaluation de la mise en place d'un système d'allocation universelle en présence de qualifications hétérogènes : le rôle institutionnel du salaire minimum[END_REF]Van der Linden, 2004). [START_REF] Drèze | Can varying social insurance contributions improve labour market efficiency?[END_REF] shows for instance that a basic income policy could be a good alternative to the reduction of social security contributions (RSSC) if associated with some deregulation of the labor market. There is currently an important debate in Europe on the appropriateness of using UBI as a way to overcome poverty and to replace the existing grant systems (see Vanderborght (2013), for a summary of the political arguments in favor of this system). In a very interesting paper, Van der Linden (2004) uses a dynamic general equilibrium model to analyze the effect of UBI policies on welfare and employment. He shows that the UBI policy reduces the equilibrium unemployment rate in an environment with risk-neutral agents. The actual comparison between UBI and unemployment insurance [UI] in a framework with self-insurance via savings, moral hazard, and monitoring costs, in the spirit of Friedman's criticism, has not been investigated.

In this paper, we focus on the welfare comparison between a UBI policy and the often-

3
In this sense, basic income can meet the requirement of justice and freedom advocated by Sen [START_REF] Sen | Equality of what?[END_REF][START_REF] Sen | What do we want from a theory of justice?[END_REF](Sen, , 2009)), which implies that people should be given sufficient capabilities to exercise their individual responsibility and choose what they want to do with them. We should note a controversy between Rawls and Van Parijs on the unconditionality of basic income. Although he considers some leisure to be a basic need, [START_REF] Rawls | Political Liberalism[END_REF] argued that a social policy should not subsidize those whose life is devoted to leisure.

The example used by both authors is that of surfers in Malibu, California.

criticized yet resilient UI. The latter does clearly suffer from several serious flaws, such as tax distortion, the lower savings it generates, moral hazard, and disincentives to work.

Designed to offer temporary support to those affected by involuntary unemployment, UI requires important monitoring of beneficiaries. Do they actually look for employment with reasonable effort? [START_REF] Krueger | Job search and unemployment insurance: new evidence from time use data[END_REF] find that the time spent searching is on average a few minutes a day. Do they refuse offers and still try to collect unemployment benefits?

Monitoring makes the management of an unemployment insurance program costly. In the state of Oregon, for example, the cost of running the unemployment agency is about $500 per unemployed person per year. This is the type of cost some want to avoid with this simpler policy. Furthermore, since monitoring is never perfect, a proportion of individuals who refuse employment offers manage to go undetected. Moreover, it is common in some countries to disguise quits as lay-offs [START_REF] Barros | Os incentivos adversos dos programas de seguro-desemprego e do FGTS no Brasil[END_REF]. Hence, the burden of financing an unemployment insurance system can be substantial. ). When observing that he has the chance to work (s t = 1), the agent decides whether or not to accept the offer.

The agent who elects to work is paid an income normalized to 1.

While model agents face a borrowing constraint, they have access to a storage technology that allows them to accumulate assets to smooth consumption against potential future adverse employment shocks. We call m t the asset available to the agent at time t. The budget constraint of an agent in period t can be written as

m t+1 + c t = m t + y d t , (1)
where c t is the agent's consumption at time t, m t+1 is his future asset, and y d t is his disposable income. Normalizing an agent's productivity to 1, we can write the disposable income as one of three possible types depending on the work status and eligibility for unemployment benefits:

y d t =          (1 -τ )(1 + ω) if he works (1 -τ )(θ + ω) if he collects UI benefits (1 -τ )ω otherwise (2)
with τ a proportional income tax rate, θ the fraction of labor income an eligible unemployed agent receives as income replacement from the UI agency, and ω a UBI. The UBI policy is characterized by a constant periodic transfer to all agents.

Eligibility for unemployment benefits is restricted to those agents without a job offer. The unemployment insurance agency's monitoring of applicants, however, may be imperfect. We model moral hazard by allowing a fraction π of agents who refuse job offers to succeed in fraudulently collecting unemployment benefits.

Unemployment insurance and basic income transfers are financed with a proportional income tax. The endogenous tax rate, τ , is such that the government balances its budget.

Unlike the case for basic income, which is given to all, with no questions asked and no strings attached, there is a cost to the provision of unemployment benefits. The necessary -albeit imperfect -monitoring of applicants involves a management cost of the unemployment insurance agency. Let λ represent the management cost per unemployed per period, regardless of the size of benefits. This cost is taken into account in the government's budget balancing rule below.

Individual agents have preferences over consumption, c t , and leisure, l t , every period, which can be represented by the following utility function:

u(c t , l t ) = [c 1-σ t l σ t ] 1-γ -1 1 -γ , ( 3
)
where γ is the degree of risk aversion of the agent and σ is the elasticity of substitution between consumption and leisure. We assume for simplicity that labor is indivisible, i.e., an agent works either a fixed proportion h of his unit time endowment or does not work at all.

Agents are infinitely lived. They maximize the expected present-value of infinite streams of utility, subject to the above budget constraint:

max E ∞ t=0 β t u(c t , l t ), (4)
with β ∈ [0, 1), a discount factor.

This problem is recursive and can be written in the form of a Bellman equation [START_REF] Bellman | The theory of dynamic programming[END_REF], where we drop time subscripts and use prime symbols for future states. The binary decision to work is denoted by x, with x = 1 if a job offer is accepted and 0 otherwise. For an agent with asset m and a job offer (s = 1), the Bellman equation is therefore:

V (s = 1, m) = max x x max m u((1 -τ )(1 + ω) + m -m , 1 -h) + s p(s |1)V (s , m ) +(1 -x)(1 -π) max m u((1 -τ )ω + m -m , 1) + s p(s |1)V (s , m )] +(1 -x)π max m u((1 -τ )(θ + ω) + m -m , 1) + s p(s |1)V (s , m )] .
For an agent of asset m without a job offer (s = 0), the Bellman equation is:

V (s = 0, m) = max m u((1 -τ )(θ + ω) + m -m , 1) + s p(s |0)V (s , m )] .
Once Bellman equations are solved and optimal decision rules extracted, we can construct the distribution of agents f (s, m). We then obtain a series of useful aggregate statistics. The number of workers, for example, is

L = m f (1, m)x(1, m)dm. (5)
The number of voluntary unemployed is thus

U V = m f (1, m)[1 -x(1, m)]dm. ( 6
)
The number of involuntarily unemployed is simply U I = p(0|1) + p(0|0). Hence the government's budget constraint can be written as

τ [L + ω + θ(πU V + U I )] = ω + θ(πU V + U I ) + λ(1 -L). (7)
In the sections to come, we investigate separately the optimal UI policy and the optimal UBI policy for calibrations of the model to the United States at two different time periods.

We then turn to the optimal mix of UI and UBI.

Solution technique and equilibrium definition

Solving the agents' problem implies finding, for every possible level of assets and lottery outcome, the optimal labor, consumption, and savings decisions of the agent, as well as the value function V (s, m). Since there is no closed-form solution for the problem just displayed, we use numerical methods. After properly parametrizing the model, we make use of the fact that the value function is a contraction mapping. Applying Banach's fixed-point theorem [START_REF] Kolmogorov | Introductory Real Analysis[END_REF], given a policy vector (θ, ω) and a tax rate τ , we iterate on the value function from an arbitrary starting point. This ensures convergence to the unique value function that solves the Bellman equation. We then extract optimal decision rules, given the policy vector, and construct the corresponding stationary distribution of agents, f * (s, m). We then verify if the government balances its budget for the given tax rate. When needed, we adjust the tax rate and start over until the government budget is balanced.

A steady-state equilibrium in this economy is a choice of labor, consumption, and savings

for every individual at every state of the world (s, m); a distribution of agents over states f * ; a policy vector (θ, ω), and a flat tax rate τ , such that (1) all agents maximize their objective function given the policy vector, (2) the distribution of agents is stationary, and (3) the government balances its budget.

When assessing the social desirability of a series of policy vectors, we use a utilitarian social welfare criterion and compare the average value in the steady states corresponding to each policy vector.

Parametrization

To conduct numerical simulations, we need to quantify all model parameters to reflect the specific conditions of the United States as closely as possible. We do so for two different time periods: in the 1990s when unemployment risk in the United States was low and in 2011, when unemployment risk was much higher and more persistent in the aftermath of the 2008 crisis and comparable to some European labor markets. This parametrization enables us to compare two policies, unemployment insurance (UI) and universal basic income (UBI), in the context of shocks of very different amplitudes.

We set the length of a period to six weeks, the longest that can accommodate the unemployment durations of the 1990s, as in [START_REF] Hansen | The role of unemployment insurance in an economy with liquidity constraints and moral hazard[END_REF] and [START_REF] Pallage | Voting on unemployment insurance[END_REF], and set the discount factor, β, to 0.995, corresponding to a 4% annual real interest rate.

The parameters of the utility function are standard in the optimal UI literature. We set the risk aversion parameter, γ, to 2.5 and the elasticity of substitution between consumption and leisure, σ, to 0.67 as in [START_REF] Kydland | Time to build and aggregate fluctuations[END_REF] and [START_REF] Hansen | The role of unemployment insurance in an economy with liquidity constraints and moral hazard[END_REF].

Since we have normalized a worker's income to 1, we can interpret all quantitative results in terms of GDP per worker. We consider that an individual works for 45% of his available time when employed, as in [START_REF] Hansen | The role of unemployment insurance in an economy with liquidity constraints and moral hazard[END_REF]. The monitoring costs introduced represent the costs of running the UI agency; while we do not have data for the whole of the United States, we have a reliable figure for Oregon, a state that exhibits features similar to the national US labor market [START_REF] Pallage | Unemployment Accounts: A Better Way to Protect the Unemployed[END_REF]. The cost of monitoring and administration in that state was $500 per unemployed worker in 2010. 5 Based on this figure, we infer a monitoring cost of λ = 0.0048, i.e., the ratio of $500 and the 2010 US GDP per employed person, $104,257 (OECD Economic Outlook, 2010). We assume that the relative monitoring cost is the same in 1990 and 2011.

Shocks in the labor market match the Markov transition probabilities we have computed using the US unemployment rates and unemployment durations provided in Table 1. The probability p(1|0) to exit from unemployment -i.e., to receive an offer if previously without an offer -is the inverse of the average unemployment duration. Using Bayes laws, we can compute the remaining transition probabilities. Table 2 provides the resulting numbers.

Quantitative results

We want to compare the socially optimal universal basic income (UBI) and the socially optimal unemployment insurance (UI) in 1990 and in 2011 in terms of average welfare. We do so by comparing steady-state outcomes for various policy parameter values, and we compare their respective effects on welfare, voluntary unemployment, and asset accumulation.

According to our results (Tables 3 and4), both policies can make individuals better off than a self-insurance mechanism (θ = 0 or ω = 0) and are sustainable at relatively low tax rates.

Table 3 and Figure 1 present the UI results for the 1990 and 2011 United States labor markets, assuming a success rate of shirkers (π) of 20%6 and a monitoring cost, λ, of 0.0048 per unemployed person. Based on the average utility of agents in our model of the 2011 labor market, the optimal replacement ratio in the UI policy is 25%. This number is not far from the observed coverage once you take into account eligibility requirements. As is standard in this literature, as the replacement ratio θ increases, average assets diminish and the number of voluntarily unemployed (shirkers) rises. The optimal replacement rate is higher in 2011 than in 1990, reflecting the higher riskiness of the 2011 labor market (higher unemployment rate and average unemployment duration). This also implies a stronger tendency for agents to accumulate assets in the steady state. Average welfare is clearly higher in the less risky 1990 economy.

Table 4 and Figure 2 are the UBI counterparts to Table 3 andFigure 1. Moral hazard is no longer relevant as every agent in the economy is eligible for UBI benefits whether employed or unemployed; consequently there is no need to monitor agents and, hence, the monitoring cost is zero. Our results indicate that the average welfare is maximized in 1990 and 2011

for UBI allowances of, respectively, 1.25% and 2.25% of a worker's usual labor income. A UBI program is clearly sustainable, but its socially optimal form implies very small transfers that are equivalent to a yearly supplement of about $2,000 in 2011, before tax. We do not observe large numbers of agents quitting employment in the neighborhood of the optimal UBI policy. But quitters quickly become a problem as the UBI transfer is increased. For a 5% income supplement, for instance, the voluntary unemployment rate would be close to 4%

under both the 1990 and 2011 calibrations. Of course, with fewer agents employed, the tax rate required to sustain the policy becomes rapidly large and induces more quitting behavior.

Unlike in Van der Linden (2004), the UBI policy thus raises the actual unemployment rate.

The likelihood of quitting rises exponentially in response to increases in UBI benefits.7 This explains why the socially optimal UBI policy cannot be very generous.

We now want to compare both policies. As previously noted, both UI and UBI programs are significant improvements over the laissez-faire, i.e., self-insurance shown in the first lines of Tables 3 and4. The UBI policy around its optimum may appear as a credible alternative to UI in both periods. However, for the shirking success probability considered in Table 3 (π = 0.20) and the monitoring cost calibrated to Oregon, the optimal UI policy is socially more desirable, as a comparison of average welfare across Tables 3 and4 suggests. For both calibrations, UI is clearly a better insurance mechanism against idiosyncratic employment shocks than UBI, and the inefficiency caused by monitoring costs and cheating behavior do not offset this fact in our simulations. In fact, for society to switch from UI to UBI would require a transfer corresponding to 0.02% and 0.18% of consumption in 1990 and 2011, respectively.8 Given the small benefits, such small consumption equivalence numbers should not be surprising. Indeed, self-insurance can at least partially offset what either of the policies does not offer through its lack of generosity. Figure 3 gives a good idea of the respective impacts of UI and UBI on all variables in the 2011 calibration. A similar figure could be produced for the 1990 calibration with somewhat smaller effects.

Why doesn't UBI measure up to UI? While proponents of the UBI policy stress the insurance factor and the reduced administrative and monitoring cost, the main argument against it has always been that it could significantly reduce the labor supply. We find that this latter effect is very strong, to the point that UBI in the optimal results barely provides any transfer and thus provides little insurance. UI is socially preferred because it does provide insurance -that is, a conditional transfer -and transfers are not "wasted" on people who do not need them (employed workers). Moral hazard under UI does not seem to be enough of a problem to overturn this in our calibrations of the US labor market in 1990

and 2011.

Of course, it is a standard result in the optimal unemployment insurance literature-as visible in Table 5-that increased moral hazard drives down the optimal replacement rate.

Similar results were reported by [START_REF] Hansen | The role of unemployment insurance in an economy with liquidity constraints and moral hazard[END_REF], Wang & Williamson (1996, 2002), [START_REF] Hopenhayn | Optimal unemployment insurance[END_REF][START_REF] Hopenhayn | Optimal unemployment insurance[END_REF]Pallage &[START_REF] Pallage | Voting on unemployment insurance[END_REF]. We show here that society, on average, does dislike moral hazard, but not as much as the implicit free riding that comes with UBI in our model economies.9 In fact, it would take shirking success probabilities or monitoring costs substantially larger than our base scenario in Table 3 for UBI to be socially preferred to UI. Table 5 shows that moral hazard needs to be quite high in both economies for society to favor UBI (more than 50% of shirkers succeeding in collecting UI benefits in 2011 and 30% in 1990). Similarly, setting shirking success at π = 0.20, Table 6 shows that for monitoring costs in the neighborhood of that calibrated from Oregon, UI dominates UBI in terms of the average welfare it provides. The table also shows how monitoring costs affect the average well-being and the degree of optimal UI generosity, as measured by the replacement ratio θ. We restrict ourselves in Table 6 to reasonable values of monitoring costs and illustrate their impact on average utility. As we increase monitoring costs beyond those values, however, we eventually reach a level in which UBI starts dominating UI on the social preference scale.

In the space of shirking success π and monitoring costs λ, Figure 4 provides society's indifference frontiers between both polices for the job market calibrations of 1990 and 2011.

For every point east of either frontier, UBI is socially preferred to UI for the relevant period.

The higher riskiness of labor markets in 2011 pushes the indifference frontier to the east of that in 1990. For every level of shirking success, society in 2011 is willing to tolerate higher monitoring costs to keep its unemployment insurance or, alternatively, for given monitoring costs, society is willing to tolerate higher levels of shirking success. The switch to UBI requires more extreme values of shirking success or monitoring costs than in 1990.

We also tested whether a combination of UI and UBI would be socially preferred. It turns out that the optimal combination of both policies implies UBI transfers of zero. Figure 5 illustrates this for the 2011 calibration. The universal basic income makes it easier for agents to shirk. In an economy with moral hazard and idiosyncratic shocks, shirking is costly as it imposes a higher tax burden on those who finance the social policy. The addition of a UBI component to a UI policy thus never improves average welfare for reasonable values of the parameters. One way to think of why UI so clearly dominates UBI is that UI is simply better at targeting support where it is needed most. UBI has the major disadvantage of requiring much higher (distorting) taxes to sustain the same level of support for the needy, while wasting most of its payments on those with low marginal utility of consumption. Clearly UBI, even if it is sustainable for the economy, is not a good insurance mechanism against shocks.

The model economy we consider here is obviously a very crude approximation and could benefit from some bells and whistles. We want to argue that such additions are not necessary, as they would only reinforce our results that UI generally dominates UBI. For example, suppose we add heterogeneity with regard to the employment lottery: some have lower employability as is typical for lower-skilled workers. These would prefer the targeted help from UI. The most-skilled workers would prefer UI as well, as they do not need the income support from UBI (low marginal utility of consumption) and would reduce their labor supply due to the distorting taxes. Just as in the benchmark case we presented, UBI would end up contributing very little. UBI would help those who cannot work at all. But it is easy to argue that they would be better helped with a program specifically addressing this inability to work, as is the case in many countries.

One could also argue that our UI policy is oversimplified. For example, almost every UI system in the world has a finite eligibility period or a cap for unemployment generosity.

Even when unemployed, one has to satisfy some additional criteria to qualify. And the level of benefits is often adjustable to the needs of the claimant, such as the family situation or capacity to overcome an unemployment spell, through the means of an asset test. Adding all these features would enhance the targeting ability of UI and thus increase its advantage over UBI.

We should note that in countries in which it is difficult for the unemployment agency to distinguish between quits and lay-offs, moral hazard is likely much higher than in the base scenario considered here for the United States. In such case, the relevant value for π is closer to 1. As Figure 4 shows, UBI may dominate UI policies in such a scenario. A relevant example may be Brazil according to [START_REF] Barros | Os incentivos adversos dos programas de seguro-desemprego e do FGTS no Brasil[END_REF].

13

In this paper, we provide a quantitative comparison of an optimal universal basic income policy and an optimal unemployment insurance program in an economy with idiosyncratic shocks. While an unemployment insurance program is better equipped to respond to employment shocks, it suffers from moral hazard and is costly to manage. Monitoring costs are not trivial and add to the social cost of administering the policy. So does moral hazard, by allowing a fraction of agents to abuse the unemployment insurance program. A universal basic income policy, however, has no such social costs and is very simple to manage.

We test the conjecture that a universal basic income may, under moral hazard and in the presence of monitoring costs, perform better. We conduct a wide variety of experiments and compare the social desirability of these policies.

Our results show that an optimal UBI is feasible. Nevertheless, the UI policy is socially robust to the introduction of UBI in the presence of idiosyncratic shocks. It takes empirically implausible monitoring costs and shirking success probabilities for the optimal basic income policy to dominate in terms of welfare the unemployment insurance policy in the economy calibrated to the 2011 United States labor market. With idiosyncratic shocks of smaller amplitude (such as those characterizing the 1990 US labor market dynamics), UBI may represent a more reasonable alternative, even if the UI policy remains socially preferred.

The superiority of UI is anchored in its ability to help those who are most in need. Even a very crude UI system with no asset tests and indefinite eligibility is able to easily beat UBI, which distributes funds blindly and must be financed through distortionary taxation.

Of course, this paper limits the study of universal basic income to its comparison with a UI policy in a particular model. It does not claim to have the last word on UBI systems.

Other paths could be explored. We could consider different skills among the population; in this case we have argued that UBI is likely to reinforce the dominance of UI. One could study the impact of transitions. Indeed, we have only compared steady states so far. Transitions can induce large costs in the short-term that can outweigh long-term advantages. Given that the status quo in industrialized countries is typically a UI policy, it is very unlikely that a transition to UBI would prove to be beneficial overall. Note: Self-insurance represents the case without UBI and without an unemployment insurance system, ω = θ = 0. Note: Self-insurance represents the case without UBI and without an unemployment insurance system, ω = θ = 0.

  Figure 1:

  Figure 2:

Figure 5 :

 5 Figure 5:

  Alaska provides one version of basic income, through the Permanent Fund passed in 1976

(Alaska Constitution, Article IX, Section 15), with the very special case of sharing dividends from an Alaskan natural resource, oil. Some field experiments, close to what a basic income could be, were tried in the 1970s in the United States and Canada; see

[START_REF] Harris | The guaranteed income movement of the 1960s and 1970s[END_REF]

for a review.

Table 1 :

 1 US unemployment statistics

		1990	2011
	unemployment rate	6%	9%
	unemployment duration 12 weeks 36 weeks
	Source: Bureau of Labor Statistics (2010-2011)

Table 2 :

 2 Transition probabilities

			1990
	p(1|0) p(0|0) p(1|1) p(0|1)
	0.5	0.5	0.9892	0.06
			2011
	p(1|0) p(0|0) p(1|1) p(0|1)
	0.1667 0.8333 0.9835 0.0165

Table 3 :

 3 UI with π=0.2 and λ=0.0048Note: In the table, average welfare is computed as the weighted sum of households' value function at the steady state corresponding to the given UI policy. The tax rate presented guarantees a balanced budget for the chosen policy. The socially optimal UI replacement ratio, θ, is identified with a . All statistics are aggregated from equilibrium households' decisions.

				1990
	θ	τ	Welfare	Inv.unempl Vol.unempl. Assets
	0	0	-110.7749	0.06	0.0000	2.82
	0.05 0.0035 -110.7328	0.06	0	2.20
	0.10 0.0070 -110.6931	0.06	0.0091	1.80
	0.15* 0.0108 -110.6909	0.06	0.0209	1.46
	0.20 0.0151 -110.7360	0.06	0.0352	1.16
	0.25 0.0200 -110.8332	0.06	0.0518	0.91
	0.30 0.0256 -110.9870	0.06	0.0697	0.68
	0.35 0.0318 -111.1998	0.06	0.0890	0.50
	0.40 0.0386 -111.4552	0.06	0.1077	0.36
				2011
	θ	τ	Welfare	Inv.unempl Vol.unempl. Assets
	0	0	-112.8519	0.09	0.0000	7.27
	0.05 0.0054 -112.6159	0.09	0.0023	5.31
	0.10 0.0107 -112.3922	0.09	0.0111	4.12
	0.15 0.0163 -112.2290	0.09	0.0226	3.16
	0.20 0.0225 -112.1350	0.09	0.0370	2.36
	0.25* 0.0292 -112.0963	0.09	0.0519	1.70
	0.30 0.0366 -112.1332	0.09	0.0689	1.16
	0.35 0.0449 -112.2727	0.09	0.0887	0.72
	0.40 0.0541 -112.4968	0.09	0.1097	0.39

Table 4 :

 4 Universal Basic IncomeNote: In the table, average welfare is computed as the weighted sum of households' value function at the steady state corresponding to the given UBI policy. The tax rate presented guarantees a balanced budget for the chosen policy. The socially optimal UBI, ω, is identified with a . All statistics are aggregated from equilibrium households' decisions.

				1990
	ω	τ	Welfare	Inv.unempl Vol.unempl. Assets
	0	0	-110.7749	0.06	0.0000	2.82
	0.01	0.0105 -110.7585	0.06	0.0000	2.61
	0.0125* 0.0131 -110.7581	0.06	0.0018	2.58
	0.0150 0.0158 -110.7596	0.06	0.0039	2.55
	0.0175 0.0184 -110.7629	0.06	0.0064	2.52
	0.02	0.0210 -110.7647	0.06	0.0077	2.49
	0.03	0.0315 -110.7921	0.06	0.0168	2.37
	0.05	0.0526 -110.9219	0.06	0.0396	2.15
	0.10	0.1073 -111.7223	0.06	0.1078	1.68
	0.20	0.2249 -115.6647	0.06	0.2507	1.00
	0.30	0.3569 -124.6774	0.06	0.3994	0.53
				2011
	ω	τ	Welfare	Inv.unempl Vol.unempl. Assets
	0	0	-112.8519	0.09	0.0000	7.2739
	0.01	0.0109 -112.7925	0.09	0.0022	6.59
	0.0125 0.0136 -112.7834	0.09	0.0035	6.49
	0.0150 0.0163 -112.7774	0.09	0.0057	6.39
	0.0175 0.0190 -112.7705	0.09	0.0066	6.29
	0.02	0.0217 -112.7689	0.09	0.0093	6.20
	0.0225* 0.0244 -112.7626	0.09	0.0102	6.11
	0.0250 0.0271 -112.7669	0.09	0.0134	6.02
	0.0275 0.0298 -112.7633	0.09	0.0143	5.93
	0.03	0.0325 -112.7714	0.09	0.0179	5.84
	0.05	0.0541 -112.8397	0.09	0.0360	5.21
	0.10	0.1100 -113.5896	0.09	0.1011	3.90
	0.20	0.2305 -117.7240	0.09	0.2422	2.08
	0.30	0.3639 -126.9929	0.09	0.3856	0.95

Table 5 :

 5 Policy comparisons for different levels of shirking success, with λ = 0.0048

				1990
		θ or ω	τ	Welfare	Inv.unempl Vol. unempl. Assets
	Opt UBI	0.0125 0.0131 -110.7581	0.06	0.0018	2.58
	Opt UI (π=0)	0.70	0.0431 -110.0127	0.06	0	0.00
	Opt UI (π=0.1)	0.30	0.0206 -110.5311	0.06	0.0259	0.72
	Opt UI (π=0.2)	0.15	0.0108 -110.6909	0.06	0.0209	1.46
	Opt UI (π=0.3)	0.05	0.0036 -110.7456	0.06	0.0043	2.20
	Opt UI (π=0.5) 0.0250 0.0019 -110.7765	0.06	0.0019	2.44
	Opt UI (π=1)	0.01	0.0009 -110.7952	0.06	0	2.61
	Self-insurance	n.a.	0	-110.7749	0.06	0	2.82
				2011
		θ or ω	τ	Welfare	Inv.unempl Vol. unempl. Assets
	Opt UBI	0.0225 0.0244 -112.7626	0.09	0.0102	6.11
	Opt UI (π=0)	0.65	0.0608 -110.2378	0.09	0	0
	Opt UI (π=0.1)	0.55	0.0608 -111.2316	0.09	0.0716	0.25
	Opt UI (π=0.2)	0.25	0.0292 -112.0963	0.09	0.0519	1.70
	Opt UI (π=0.3)	0.15	0.0180 -112.4240	0.09	0.0405	3.12
	Opt UI (π=0.5) 0.0750 0.0093 -112.6666	0.09	0.0260	4.64
	Opt UI (π=1)	0.02	0.0027 -112.8323	0.09	0.0093	6.24
	Self-insurance	n.a.	0	-112.8519	0.09	0	7.27

Table 6 :

 6 Policy comparisons for different levels of monitoring costs with π=0.20

				1990
		θ or ω	τ	Welfare	Inv.unempl Vol. unempl. Assets
	Opt UBI	0.0125 0.0131 -110.7581	0.06	0.0018	2.58
	Opt UI (λ=0)	0.15	0.0104 -110.6402	0.06	0.0209	1.46
	Opt UI (λ=0.001)	0.15	0.0105 -110.6529	0.06	0.0209	1.46
	Opt UI (λ=0.0048)	0.15	0.0108 -110.6909	0.06	0.0209	1.46
	Opt UI (λ=0.01)	0.10	0.0073 -110.7393	0.06	0.0091	1.20
	Self-insurance	n.a.	0	-110.7749	0.06	0	2.82
				2011
		θ or ω	τ	Welfare	Inv.unempl Vol. unempl. Assets
	Opt UBI	0.0225 0.0244 -112.7626	0.09	0.0102	6.11
	Opt UI (λ=0)	0.25	0.0284 -112.0027	0.09	0.0521	1.70
	Opt UI (λ=0.001)	0.25	0.0286 -112.0226	0.09	0.0521	1.70
	Opt UI (λ=0.0048)	0.25	0.0292 -112.0963	0.09	0.0519	1.70
	Opt UI (λ=0.01)	0.25	0.0300 -112.1992	0.09	0.0517	1.70
	Self-insurance	n.a.	0	-112.8519	0.09	0	7.27

Another paper that quantitatively investigates the effects of UBI policies isVan der Linden (2002).The author verifies how UBI affects unemployment in a dynamic general equilibrium model of a unionized economy and shows that some basic income schemes lower the steady-state unemployment rate; in this context, the dynamic adjustment of such reforms can be Pareto-improving.

Computed from data provided by Christina Martin of the Cascade Policy Institute, Portland, Oregon.

It is commonly agreed that 20% is a reasonable number for π, the proportion of agents refusing job offers who nevertheless manage to collect UI benefits[START_REF] Pallage | Heterogeneous labor markets and the generosity towards the unemployed: an international perspective[END_REF]. We provide results for other levels of shirking success in Table5.

Quitting is not necessarily bad for well-being, as quitters enjoy leisure, but it drives up the tax rate very fast.

The transfer is computed as the increase in consumption at all states needed to equate the average value function under the optimal UBI policy and under the optimal UI policy.

Interestingly, moral hazard is less detrimental for the UI system with large shocks (2011 case) than with small shocks (1990 case). With large shocks, a UI system is always preferred to a self-insurance system, even with high moral hazard (see Table5); with smaller shocks, it is not that definite.

The views official positions of the Federal Reserve Bank of St. Louis, the Federal Reserve System, or the Board of Governors.

§ The first draft of this paper was written when Fabre was visiting ESG UQAM.