# Properties of 208Pb predicted from the relativistic equation of state in the
full Dirac space

Hui Tong  Department of Physics, Chongqing University, Chongqing 401331, China
Strangeness Nuclear Physics Laboratory, RIKEN Nishina Center, Wako, 351-0198,
Japan  Jing Gao  School of Physics, Nankai University, Tianjin 300071, China
Chencan Wang  School of Physics and Astronomy, Sun-Sat-Sen University, Zhuhai
519082, China  Sibo Wang<EMAIL_ADDRESS>Department of Physics, Chongqing
University, Chongqing 401331, China

###### Abstract

Relativistic Brueckner-Hartree-Fock (RBHF) theory in the full Dirac space
allows one to determine uniquely the momentum dependence of scalar and vector
components of the single-particle potentials. In order to extend this new
method from nuclear matter to finite nuclei, as a first step, properties of
208Pb are explored by using the microscopic equation of state for asymmetric
nuclear matter and a liquid droplet model. The neutron and proton density
distributions, the binding energies, the neutron and proton radii, and the
neutron skin thickness in 208Pb are calculated. In order to further compare
the charge densities predicted from the RBHF theory in the full Dirac space
with the experimental charge densities, the differential cross sections and
the electric charge form factors in the elastic electron-nucleus scattering
are obtained by using the phase-shift analysis method. The results from the
RBHF theory are in good agreement with the experimental data. In addition, the
uncertainty arising from variations of the surface term parameter $f_{0}$ in
the liquid droplet model is also discussed.

##  I Introduction

Exploring the equation of state (EOS) of neutron rich matter has attracted
considerable attentions in the field of nuclear physics and astrophysics,
particularly the poorly determined density dependence of the symmetry energy
Lattimer and Prakash (2000); Li  _et al._ (2008a); Oertel  _et al._ (2017);
Burgio  _et al._ (2021). In terrestrial laboratories, the thickness of the
neutron skin in neutron-rich atomic nuclei, e.g., 208Pb, has been identified
as an ideal laboratory observable to constrain the EOS of neutron rich matter
during the last decades Alex Brown (2000); Centelles  _et al._ (2009); Roca-
Maza  _et al._ (2011a); Reed  _et al._ (2021). The neutron skin thickness is
defined as the difference between the neutron ($r_{n}$) and proton ($r_{p}$)
root-mean-square (rms) radii: $\Delta r_{np}=r_{n}-r_{p}$. Currently, precise
data on $r_{p}$ are available. In contrast, as neutrons are uncharged, the
determination on $r_{n}$ still suffer from uncontrolled uncertainties due to
hadron dynamics which requires model assumptions to deal with the strong force
Thiel  _et al._ (2019). A model-independent method to probe the neutron
densities was also proposed, i.e., parity-violating electron scattering
Donnelly  _et al._ (1989); Horowitz  _et al._ (2001). Recently, an improved
value for the neutron skin thickness of 208Pb has been reported by the updated
Lead Radius EXperiment (PREX-II): $\Delta r_{np}=0.283\pm 0.071$ fm Adhikari
_et al._ (2021). By using the strong correlation between $\Delta r_{np}$ and
the slope of the symmetry energy $L$ calculated from several sets of
relativistic energy density functionals, a new value of $L=106\pm 37$ MeV was
reported and it is larger than those from other theoretical and experimental
estimations Reed  _et al._ (2021). This result provides a challenge to our
present understanding of the nuclear matter EOS, particularly the density
dependence of symmetry energy.

In theoretical studies, the predictions from the density functionals span a
fairly wide range of neutron skin thickness for 208Pb, since the isovector
channels are loosely constrained in the fitting procedures Roca-Maza  _et al._
(2011b). For _ab initio_ methods, calculating such a heavy nucleus directly is
difficult due to the huge computational cost. Recently, $\Delta
r_{np}=0.14-0.20$ fm for 208Pb was predicted by the nonrelativistic _ab
initio_ calculations from the chiral effective field theory (EFT) with
realistic two- and three-nucleon forces Hu  _et al._ (2022). In the
relativistic framework, Relativistic Brueckner-Hartree-Fock (RBHF) theory is
one of the most successful _ab initio_ theories based on realistic two-nucleon
forces only Brockmann and Machleidt (1990). Comparing to the nonrelativistic
Brueckner-Hartree-Fock (BHF) theory with two-nucleon forces only, the RBHF
theory improves the description considerably, both for the nuclear matter and
finite nuclei Shen  _et al._ (2017, 2019). Due to considerable numerical
difficulties, the investigation of 208Pb within a full solution in the
relativistic framework is not available. The present fully self-consistent
RBHF theory has been accomplished only for the neutron skin thickness in the
medium-mass nucleus 48Ca Shen  _et al._ (2018). To extend the RBHF theory from
the nuclear matter to finite nuclei in previous studies, most investigations
were based on the local density approximation (LDA) Negele (1970) and the
liquid droplet model Oyamatsu  _et al._ (1998). LDA aims to find an effective
interaction which can reproduce the nuclear matter properties from the RBHF
theory, while the key point of the liquid droplet model is to combine an
energy functional based on the semiempirical mass formula with the EOS from
RBHF theory Alonso and Sammarruca (2003); Sammarruca and Liu (2009);
Sammarruca (2016); Sammarruca and Nosyk (2016).

One of the most essential ingredient in the RBHF calculations for nuclear
matter properties is to identify the single-particle potentials of the
nucleons. However, to avoid the complexity of extracting self-consistently the
scalar and vector components of the single-particle potentials from the $G$
matrix, the momentum-independence approximation method and the projection
method are usually introduced in the Dirac space with positive-energy states
(PESs) only Brockmann and Machleidt (1990); Gross-Boelting  _et al._ (1999);
Schiller and Müther (2001). These two frequently used approximations lead to
contradictory results for the isospin dependence of the Dirac mass Ulrych and
Müther (1997). Therefore, the mapping from nuclear matter to finite nuclei is
far from unique because of the uncertainties of the RBHF theory in nuclear
matter Brockmann and Toki (1992); Fritz  _et al._ (1993); Fuchs  _et al._
(1995); Ulrych and Müther (1997); Shen  _et al._ (1997); Ma and Liu (2002);
van Dalen and Müther (2011).

Recently, by considering the PESs and negative-energy states (NESs)
simultaneously, a self-consistent RBHF calculation is developed in the full
Dirac space Wang  _et al._ (2021, 2022a). In this case, the momentum
dependence for the scalar and vector components of the single-particle
potentials are determined uniquely, since the matrix elements of single-
particle potential operator can be decomposed in the full Dirac space. The
RBHF theory in the full Dirac space has been successfully applied to study the
symmetric nuclear matter (SNM) Wang  _et al._ (2021, 2022b), asymmetric
nuclear matter (ANM) Wang  _et al._ (2022a); Tong  _et al._ (2022) and neutron
star properties Tong  _et al._ (2022); Wang  _et al._ (2022c). Especially,
this new method has clarified the long-standing controversy about the isospin
dependence of the Dirac mass in the RBHF calculations of ANM Wang  _et al._
(2022a).

With the success of RBHF theory in the full Dirac space in nuclear matter, it
is interesting and timely to extend this method to study finite nuclei. As a
first step, the EOS from the RBHF theory in the full Dirac space and a liquid
droplet model are used to study the properties of 208Pb. To further compare
the charge densities predicted from the RBHF theory in the full Dirac space
with the experimental charge densities, the differential cross sections (DCS)
and the electric charge form factors in the elastic electron-nucleus
scattering are discussed by using the phase-shift analysis method.

In Sec. II, the theoretical framework of the RBHF theory in the full Dirac
space, the liquid droplet model, and the phase-shift analysis method are
introduced. The calculated results and discussions are presented in Sec. III.
Finally, a summary is given in Sec. IV.

##  II Theoretical framework

###  II.1 Relativistic Brueckner-Hartree-Fock theory in the full Dirac space

To describe the single-particle motion of a nucleon in nuclear matter from the
RBHF theory, the essential point is to use the Dirac equation,

$\left\\{\bm{\alpha}\cdot\bm{p}+\beta\left[M+\mathcal{U}_{\tau}(\bm{p})\right]\right\\}u_{\tau}(\bm{p},s)=E_{\bm{p},\tau}u_{\tau}(\bm{p},s),\quad\tau=n,p,$
(1)

where $\bm{\alpha}$ and $\beta$ are the Dirac matrices, $M$ is the nucleon
mass, $\bm{p}$ and $E_{\bm{p},\tau}$ are the momentum and single-particle
energy, $s$ and $\tau$ are the spin and isospin. The single-particle potential
$\mathcal{U}_{\tau}$ in the infinite, uniform nuclear matter can be decomposed
in its Lorentz form Serot and Walecka (1986):

$\mathcal{U}_{\tau}(\bm{p})=U_{S,\tau}(p)+\gamma^{0}U_{0,\tau}(p)+\bm{\gamma\cdot\hat{p}}U_{V,\tau}(p),$
(2)

where $U_{S,\tau}$ is the scalar potential, $U_{0,\tau}$ and $U_{V,\tau}$ are
the timelike and spacelike parts of the vector potential.
$\hat{\bm{p}}=\bm{p}/|\bm{p}|$ is the unit vector parallel to the momentum
$\bm{p}$. By introducing the following effective quantities,
$\bm{p}^{*}_{\tau}=\bm{p}+\hat{\bm{p}}U_{V,\tau}(p)$,
$M^{*}_{\bm{p},\tau}=M+U_{S,\tau}(p)$, and
$E^{*}_{\bm{p},\tau}=E_{\bm{p},\tau}-U_{0,\tau}(p)$, the solutions of Eq. (1)
in the full Dirac space are

$u_{\tau}(\bm{p},s)=\
\sqrt{\frac{E_{\bm{p},\tau}^{*}+M_{\bm{p},\tau}^{*}}{2M_{\bm{p},\tau}^{*}}}\begin{bmatrix}1\\\
\frac{\bm{\sigma}\cdot\bm{p}^{*}_{\tau}}{E_{\bm{p},\tau}^{*}+M_{\bm{p},\tau}^{*}}\end{bmatrix}\chi_{s}\chi_{\tau},~{}~{}~{}v_{\tau}(\bm{p},s)=\
\gamma^{5}u_{\tau}(\bm{p},s),$ (3)

where $u_{\tau}$ and $v_{\tau}$ are in-medium baryon spinors with positive and
negative energies, $\chi_{s}$ and $\chi_{\tau}$ are the spin and isospin
doublets.

The baryon spinor can be calculated exactly once the scalar and vector
components of the single-particle potentials are determined. To this end,
three matrix elements of the single-particle potential $\mathcal{U}_{\tau}$ in
the full Dirac space are defined as Anastasio  _et al._ (1981); Poschenrieder
and Weigel (1988),

$\displaystyle\Sigma^{++}_{\tau}(p)=$ $\displaystyle\
\bar{u}_{\tau}(\bm{p},1/2)\mathcal{U}_{\tau}(\bm{p})u_{\tau}(\bm{p},1/2)=U_{S,\tau}(p)+\frac{E^{*}_{\bm{p},\tau}}{M^{*}_{\bm{p},\tau}}U_{0,\tau}(p)+\frac{p^{*}_{\tau}}{M^{*}_{\bm{p},\tau}}U_{V,\tau}(p),$
(4a) $\displaystyle\Sigma^{-+}_{\tau}(p)=$ $\displaystyle\
\bar{v}_{\tau}(\bm{p},1/2)\mathcal{U}_{\tau}(\bm{p})u_{\tau}(\bm{p},1/2)=\frac{p^{*}_{\tau}}{M^{*}_{\bm{p},\tau}}U_{0,\tau}(p)+\frac{E^{*}_{\bm{p},\tau}}{M^{*}_{\bm{p},\tau}}U_{V,\tau}(p),$
(4b) $\displaystyle\Sigma^{--}_{\tau}(p)=$ $\displaystyle\
\bar{v}_{\tau}(\bm{p},1/2)\mathcal{U}_{\tau}(\bm{p})v_{\tau}(\bm{p},1/2)=-U_{S,\tau}(p)+\frac{E^{*}_{\bm{p},\tau}}{M^{*}_{\bm{p},\tau}}U_{0,\tau}(p)+\frac{p^{*}_{\tau}}{M^{*}_{\bm{p},\tau}}U_{V,\tau}(p).$
(4c)

When $\Sigma^{++}_{\tau}$, $\Sigma^{-+}_{\tau}$, and $\Sigma^{--}_{\tau}$ are
determined, the single-particle potentials in Eq. (2) can be obtained uniquely
from

$\displaystyle U_{S,\tau}(p)=$ $\displaystyle\
\frac{\Sigma^{++}_{\tau}(p)-\Sigma^{--}_{\tau}(p)}{2},$ (5a) $\displaystyle
U_{0,\tau}(p)=$ $\displaystyle\
\frac{E^{*}_{\bm{p},\tau}}{M^{*}_{\bm{p},\tau}}\frac{\Sigma^{++}_{\tau}(p)+\Sigma^{--}_{\tau}(p)}{2}-\frac{p^{*}_{\tau}}{M^{*}_{\bm{p},\tau}}\Sigma^{-+}_{\tau}(p),$
(5b) $\displaystyle U_{V,\tau}(p)=$ $\displaystyle\
-\frac{p^{*}_{\tau}}{M^{*}_{\bm{p},\tau}}\frac{\Sigma^{++}_{\tau}(p)+\Sigma^{--}_{\tau}(p)}{2}+\frac{E^{*}_{\bm{p},\tau}}{M^{*}_{\bm{p},\tau}}\Sigma^{-+}_{\tau}(p).$
(5c)

Therefore, the matrix elements for the positive-energy solutions, the elements
coupling positive- with negative-energy solutions given in Eq. (3) and those
for the negative-energy solutions, i.e., $\Sigma^{++}_{\tau}$,
$\Sigma^{-+}_{\tau}$, and $\Sigma^{--}_{\tau}$ should be calculated
simultaneously. These three matrix elements can be evaluated through the
effective $NN$ interactions $G$ matrix in the full Dirac space

$\displaystyle\Sigma^{++}_{\tau}(p)=$ $\displaystyle\
\sum_{s^{\prime}\tau^{\prime}}\int^{k^{\tau^{\prime}}_{F}}_{0}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{M^{*}_{\bm{p}^{\prime},\tau^{\prime}}}{E^{*}_{\bm{p}^{\prime},\tau^{\prime}}}\langle\bar{u}_{\tau}(\bm{p},1/2)\bar{u}_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})|\bar{G}^{++++}(W)|u_{\tau}(\bm{p},1/2)u_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})\rangle,$
(6a) $\displaystyle\Sigma^{-+}_{\tau}(p)=$ $\displaystyle\
\sum_{s^{\prime}\tau^{\prime}}\int^{k^{\tau^{\prime}}_{F}}_{0}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{M^{*}_{\bm{p}^{\prime},\tau^{\prime}}}{E^{*}_{\bm{p}^{\prime},\tau^{\prime}}}\langle\bar{v}_{\tau}(\bm{p},1/2)\bar{u}_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})|\bar{G}^{-+++}(W)|u_{\tau}(\bm{p},1/2)u_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})\rangle,$
(6b) $\displaystyle\Sigma^{--}_{\tau}(p)=$ $\displaystyle\
\sum_{s^{\prime}\tau^{\prime}}\int^{k^{\tau^{\prime}}_{F}}_{0}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{M^{*}_{\bm{p}^{\prime},\tau^{\prime}}}{E^{*}_{\bm{p}^{\prime},\tau^{\prime}}}\langle\bar{v}_{\tau}(\bm{p},1/2)\bar{u}_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})|\bar{G}^{-+-+}(W)|v_{\tau}(\bm{p},1/2)u_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})\rangle,$
(6c)

where $k_{F}^{\tau^{\prime}}$ specifies the Fermi momentum for nucleon
$\tau^{\prime}$. $\bar{G}$ is the antisymmetrized $G$ matrix, where $\pm$ in
the superscript denotes PESs or NESs.

In the RBHF theory, the in-medium covariant Thompson equation Brockmann and
Machleidt (1990) is one of the most widely used equations to derive the $G$
matrix in nuclear matter,

$\begin{split}G_{\tau\tau^{\prime}}(\bm{q}^{\prime},\bm{q}|\bm{P},W)=&\
V_{\tau\tau^{\prime}}(\bm{q}^{\prime},\bm{q}|\bm{P})+\int\frac{d^{3}k}{(2\pi)^{3}}V_{\tau\tau^{\prime}}(\bm{q}^{\prime},\bm{k}|\bm{P})\\\
&\times\frac{M^{*}_{\bm{P}+\bm{k},\tau}M^{*}_{\bm{P}-\bm{k},\tau^{\prime}}}{E^{*}_{\bm{P}+\bm{k},\tau}E^{*}_{\bm{P}-\bm{k},\tau^{\prime}}}\frac{Q_{\tau\tau^{\prime}}(\bm{k},\bm{P})}{W-E_{\bm{P}+\bm{k},\tau}-E_{\bm{P}-\bm{k},\tau^{\prime}}}G_{\tau\tau^{\prime}}(\bm{k},\bm{q}|\bm{P},W),\end{split}$
(7)

where $\tau\tau^{\prime}=nn,~{}pp$ or $np$, $W$ is the starting energy,
$V_{\tau\tau^{\prime}}$ denotes a realistic bare $NN$ interaction and Bonn
potentials Machleidt (1989) are used here. $\bm{P}$ is the center-of-mass
momentum and $\bm{k}$ is the relative momentum of the two interacting
nucleons. $\bm{q},\bm{k}$ and $\bm{q}^{\prime}$ are the initial, intermediate,
and final relative momenta of the two nucleons scattering in nuclear matter,
respectively. $Q_{\tau\tau^{\prime}}(\bm{k},\bm{P})$ is the Pauli operator,

$Q_{\tau\tau^{\prime}}(\bm{k},\bm{P})=\left\\{\begin{array}[]{cl}1&\quad|\bm{P}+\bm{k}|>k_{F}^{\tau}~{}\textrm{and}~{}|\bm{P}-\bm{k}|>k_{F}^{\tau^{\prime}},\\\
0&\quad\textrm{otherwise},\end{array}\right.$ (8)

which prevents $NN$ scattering into occupied states in the nuclear medium.

Through equations (1), (5), (6), and (7), the $G$ matrix is self-consistently
calculated with the single-particle potentials in the standard RBHF iterative
procedure. When the iteration has converged, the binding energy per nucleon in
nuclear matter can be calculated by

$\begin{split}E/A=&\
\frac{1}{\rho}\sum_{s,\tau}\int^{k^{\tau}_{F}}_{0}\frac{d^{3}p}{(2\pi)^{3}}\frac{M^{*}_{\bm{p},\tau}}{E^{*}_{\bm{p},\tau}}\langle\bar{u}_{\tau}(\bm{p},s)|\bm{\gamma}\cdot\bm{p}+M|u_{\tau}(\bm{p},s)\rangle-M\\\
&\
+\frac{1}{2\rho}\sum_{s,s^{\prime},\tau,\tau^{\prime}}\int^{k^{\tau}_{F}}_{0}\frac{d^{3}p}{(2\pi)^{3}}\int^{k^{\tau^{\prime}}_{F}}_{0}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{M^{*}_{\bm{p},\tau}}{E^{*}_{\bm{p},\tau}}\frac{M^{*}_{\bm{p}^{\prime},\tau^{\prime}}}{E^{*}_{\bm{p}^{\prime},\tau^{\prime}}}\\\
&\
\times\langle\bar{u}_{\tau}(\bm{p},s)\bar{u}_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})|\bar{G}^{++++}(W)|u_{\tau}(\bm{p},s)u_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})\rangle.\end{split}$
(9)

###  II.2 From nuclear matter to finite nuclei

To connect the nuclear matter EOS to properties of finite nuclei, the energy
of a nucleus is written in terms of a volume, a surface, and a Coulomb term
motivated by a liquid droplet model Oyamatsu  _et al._ (1998); Alonso and
Sammarruca (2003),

$\begin{split}e(Z,A)=&\int d^{3}rE/A(\rho_{n}(r),\rho_{p}(r))\rho(r)+f_{0}\int
d^{3}r|\nabla\rho(r)|^{2}\\\
&+\frac{e^{2}}{4\pi\epsilon_{0}}(4\pi)^{2}\int_{0}^{\infty}dr^{\prime}r^{\prime}\rho_{p}(r^{\prime})\int_{0}^{r^{\prime}}drr^{2}\rho_{p}(r),\end{split}$
(10)

where the volume term is calculated directly from the EOS in the RBHF theory
with the density $\rho(r)=\rho_{n}(r)+\rho_{p}(r)$. The constant $f_{0}$ is
the surface energy parameter. The proton and neutron density distributions are
parameterized as standard Thomas-Fermi distributions

$\rho_{i}(r)=\frac{a_{i}}{1+e^{(r-R_{i})/d_{i}}}\quad(i=n,~{}p),$ (11)

where the free parameters of radius $R_{i}$ and diffuseness $d_{i}$ are
extracted by minimization of the energy, while $a_{i}$ is obtained by
normalizing the proton (neutron) distribution to $Z(N)$,

$\int d^{3}r\rho_{i}(r)=\begin{cases}A-Z,&i=n,\\\ Z,&i=p.\end{cases}$ (12)

The neutron skin thickness is defined as

$\begin{split}\Delta r_{np}&=r_{n}-r_{p},\end{split}$ (13)

where $r_{n}$ and $r_{p}$ are the rms radii of the neutron and proton density
distributions.

###  II.3 Phase-shift analysis method

Phase-shift analysis method is used to calculate the DCS and the electric
charge form factors for elastic electron scattering from the charge densities
obtained by the RBHF theory in the full Dirac space. To obtain the DCS of the
elastic electron scattering, one needs to solve the Dirac equation

$[\bm{\alpha}\cdot\bm{p}+\beta m+V(r)]\psi(\bm{r})=E\psi(\bm{r}),$ (14)

where $E$ and $\bm{p}$ are the energy and momentum of the incident electrons,
$m$ is the rest mass of the electron, and $V(r)$ is the Coulomb potential
between the electron and the nucleus. Since $V(r)$ is assumed to be spherical
Bjorken and Drell (1965),

$V(r)=-{e^{2}\over 4\pi\epsilon_{0}}\int
d^{3}r^{\prime}{\rho_{\mathrm{ch}}(r^{\prime})\over|\bm{r}-\bm{r}^{\prime}|},$
(15)

with $\rho_{\mathrm{ch}}(r)$ being the charge distribution. The wave function
$\psi(\bm{r})$ can be expressed by a series of spherical spinors with definite
angular momenta,

$\psi(\bm{r})=\frac{1}{r}\begin{bmatrix}P(r)\Omega_{\kappa,m_{j}}(\theta,\phi)\\\
iQ(r)\Omega_{-\kappa,m_{j}}(\theta,\phi)\end{bmatrix},$ (16)

where $\Omega$ are the spherical spinors, $\kappa=(l-j)(2j+1)$ is defined as
the relativistic quantum number, $j$ and $l$ are the total and orbital angular
momentum quantum numbers. The radial wave functions $P(r)$ and $Q(r)$ satisfy
the following coupled differential equations:

$\displaystyle\frac{dP}{dr}=-\frac{\kappa}{r}P(r)+[E-V(r)+2m]Q(r),$ (17a)
$\displaystyle\frac{dQ}{dr}=-[E-V(r)]P(r)+\frac{\kappa}{r}Q(r),$ (17b)

The spin-up $\delta_{l}^{+}$ and spin-down $\delta_{l}^{-}$ phase shifts for
the partial wave with orbital angular momentum $l$ can be obtained by solving
the coupled radial equations (17) with the asymptotic behavior. Then one can
calculate the direct scattering amplitude

$f(\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}[(l+1)(e^{2i\delta_{l}^{+}}-1)+l(e^{2i\delta_{l}^{-}}-1)]P_{l}(\cos\theta)$
(18)

and the spin-flip scattering amplitude

$g(\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}[e^{2i\delta_{l}^{-}}-e^{2i\delta_{l}^{+}}]P_{l}^{1}(\cos\theta),$
(19)

where $k$ is the wave number of the projectile electron, $P_{l}(\cos\theta)$
and $P_{l}^{1}(\cos\theta)$ are Legendre polynomials and associated Legendre
functions. The DCS for elastic electron scattering off nucleus can be given by

$\frac{d\sigma}{d\Omega}=|f(\theta)|^{2}+|g(\theta)|^{2}.$ (20)

##  III Results and discussion

Table 1:  Parameters of the density distributions for 208Pb calculated by the RBHF theory in the full Dirac space.  Method | Potential | $f_{0}$ | $a_{n}$ | $R_{n}$ | $d_{n}$ | $a_{p}$ | $R_{p}$ | $d_{p}$
---|---|---|---|---|---|---|---|---
(MeV$\cdot$fm5) | (fm-3) | (fm) | (fm) | (fm-3) | (fm) | (fm)
Full Dirac Space | Bonn A | 60 | 0.096 | 6.62 | 0.60 | 0.065 | 6.55 | 0.56
70 | 0.095 | 6.62 | 0.65 | 0.064 | 6.54 | 0.61

The RBHF calculation is performed for the nuclear matter EOS in the full Dirac
space, where the bare nucleon-nucleon ($NN$) interaction is adopted as
relativistic potential Bonn A Machleidt (1989). With potential Bonn A, The
empirical saturation properties of SNM are well reproduced from the RBHF
calculations in the full Dirac space Wang  _et al._ (2021). From the EOS
calculated by the RBHF theory and the energy functional based on the
semiempirical mass formula Oyamatsu  _et al._ (1998); Alonso and Sammarruca
(2003), the two-parameter Fermi functions for neutron and proton densities of
208Pb are obtained. The parameters for $a_{i}$, $R_{i}$, $d_{i}$ are listed in
Table 1, where $i=n,p$. It should be noted that the constant $f_{0}$ in Eq.
(10) from the surface term is typically obtained by fitting to $\beta$-stable
nuclei and found to be about 60–70 MeV$\cdot$fm5 Oyamatsu  _et al._ (2010). In
order to study how this uncertainty impacts the corresponding predictions for
the properties of 208Pb, Table 1 shows the results obtained with $f_{0}$ = 60
MeV$\cdot$fm5 and those with $f_{0}$ = 70 MeV$\cdot$fm5. It can be seen that
these two values of $f_{0}$ lead to different parameters, especially for
$d_{i}$.

Figure 1:  (Color online) Neutron and proton density distributions for 208Pb
calculated by the RBHF theory in the full Dirac space with the potential Bonn
A. The solid lines and dashed lines are obtained by using $f_{0}=60$
MeV$\cdot$fm5 and $f_{0}=70$ MeV$\cdot$fm5, respectively.

In Fig. 1, the radial dependence of the neutron and proton densities for 208Pb
are calculated by the RBHF theory in the full Dirac space with the potential
Bonn A. The values of central densities from $f_{0}=60$ MeV$\cdot$fm5 are
larger than $f_{0}=70$ MeV$\cdot$fm5 for both neutron and proton densities. In
contrast, the densities from $f_{0}=70$ MeV$\cdot$fm5 are larger than those
from $f_{0}=60$ MeV$\cdot$fm5 when the radius becomes larger than $6.8$ fm.

Table 2:  Neutron radii, proton radii, and neutron skins calculated by the RBHF theory in the full Dirac space.  Method | Potential | $f_{0}$ | $r_{n}$ | $r_{p}$ | $\Delta r_{np}$
---|---|---|---|---|---
(MeV$\cdot$fm5) | (fm) | (fm) | (fm)
Full Dirac Space | Bonn A | 60 | 5.59 | 5.48 | 0.11
70 | 5.67 | 5.55 | 0.12

From the density distributions obtained by the RBHF theory in the full Dirac
space, the neutron radii $r_{n}$, proton radii $r_{p}$, and neutron skins
$\Delta r_{np}$ for 208Pb can be extracted. In Table 2, the results for
$r_{n}$, $r_{p}$, and $\Delta r_{np}$ are 5.59, 5.48, and 0.11 fm from
$f_{0}=60$ MeV$\cdot$fm5, as compared to 5.67, 5.55, and 0.12 fm from
$f_{0}=70$ MeV$\cdot$fm5. One can see that the predictions from the latter one
are even larger. Recently, an accurate measurements of the neutron skin of
208Pb from PREX-II is available Adhikari  _et al._ (2021). Combined with the
previous measurement Abrahamyan  _et al._ (2012); Horowitz  _et al._ (2012),
the extracted neutron skin is $\Delta r_{np}=0.283\pm 0.071$ fm. Furthermore,
a value of the slope of the symmetry energy $L=(106\pm 37)$ MeV was reported
through exploiting the strong correlation between the neutron skin of 208Pb
and $L$ from a specific class of relativistic energy density functionals Reed
_et al._ (2021). It should be noted that this result systematically
overestimates current limits from both theoretical approaches and experimental
measurements and suggests that the EOS at the typical densities found in
atomic nuclei is stiff. It can be also found that the neutron skin from PREX-2
is larger than those predicted by the RBHF theory in the full Dirac space.
This indicates that the EOS from the RBHF theory at the typical densities is
rather soft. In contrast, the neutron skin from the RBHF theory are close to
other empirical values, such as $\Delta r_{np}=0.15\pm 0.08$ fm and $\Delta
r_{np}=0.14\pm 0.10$ from pionic probes Friedman (2012), $\Delta
r_{np}=0.18\pm 0.05$ fm Garcia-Recio  _et al._ (1992) from the pionic atom
potentials with varying radial parameters of the neutron distributions, and
$\Delta r_{np}=0.18\pm 0.05$ fm Tsang  _et al._ (2012) from symmetry energy
constraints. Therefore, the next generation of terrestrial experiments and
astronomical observations are necessary to improve our understanding of the
EOS.

Figure 2:  (Color online) Energy per nucleon $E/A$ for 208Pb and the charge
radius $r_{c}$ calculated by the RBHF theory in the full Dirac space, in
comparison with the results obtained by the BHF theory Bu  _et al._ (2016)
with the Argonne V18 or Bonn B two-nucleon potentials plus their corresponding
microscopic three-nucleon forces, the RBHF theory within the momentum-
independence approximation Alonso and Sammarruca (2003), and the EFT including
only two-neutron forces at N4LO Sammarruca (2016). The solid black star
represents the experimental data.

In Fig. 1, the densities of point protons and neutrons are obtained without
considering that protons and neutrons are composite particles with extended
size. In order to compare with the experimental signature of the density in
208Pb by elastic electron scattering, the charge density is calculated by
convolution of the proton density with a Gaussian form factor Negele (1970)

$\rho_{\rm ch}(r)=\frac{1}{a\sqrt{\pi}}\displaystyle\int
dr^{\prime}r^{\prime}\rho_{p}(r)\left[e^{-(r-r^{\prime})^{2}/a^{2}}/r-e^{-(r+r^{\prime})^{2}/a^{2}}/r\right],$
(21)

where $a=\sqrt{2/3}\langle r^{2}\rangle_{p}^{1/2}$ is the proton size for
spherically symmetric density distributions Yao  _et al._ (2012). Furthermore,
the charge radii $r_{c}$ can be calculated from the charge density $\rho_{\rm
ch}(r)$.

Figure 2 shows the energy per nucleon $E/A$ for 208Pb and the charge radius
$r_{c}$ calculated by the RBHF theory in the full Dirac space, in comparison
with the BHF theory Bu  _et al._ (2016), the RBHF theory within the momentum-
independence approximation Alonso and Sammarruca (2003), and the EFT
Sammarruca (2016). The solid black star indicates the experimental data. As
for the RBHF theory in the full Dirac space, the prediction from $f_{0}$ = 60
MeV$\cdot$fm5 is much closer to the experimental data than those from $f_{0}$
= 70 MeV$\cdot$fm5, since the magnitude of surface term in Eq. (10) is
proportional to $f_{0}$ and the latter one introduces more repulsion in the
present model. Similar results also apply to chiral EFT. To specify the
importance of the RBHF calculations in the full Dirac space, the prediction
from the RBHF theory within the momentum-independence approximation with
$f_{0}$ = 70 MeV$\cdot$fm5 is also shown Alonso and Sammarruca (2003). It can
be seen that the full Dirac space gives a larger radius than the momentum-
independence approximation, while the magnitude of the energy is smaller. In
order to further improve the accuracy of the results from the RBHF theory in
the full Dirac space, the mapping from nuclear matter to finite nuclei in a
more microscopic way will be treated carefully in the near future, such as LDA
Negele (1970). One can note that the energy per nucleon $E/A$ from chiral EFT
agree well with the experimental data Sammarruca (2016), since the EOS for the
symmetric nuclear matter was obtained from empirically determined values of
characteristic constants in homogeneous matter at saturation and subsaturation
Alam  _et al._ (2014). In addition, the predictions from the RBHF theory in
the full Dirac space with only two-nucleon potential are compatible with those
from the BHF theory with the Argonne V18 (AV18) Wiringa  _et al._ (1995) and
Bonn B Machleidt (1989) two-nucleon potentials plus their corresponding
microscopic three-nucleon forces Li  _et al._ (2008b); Li and Schulze (2008).
However, the parameter $f_{0}$ in the BHF theory is optimized by fitting the
binding energies of 16O, 40Ca, 48Ca, 90Zr, 114Sn, and 208Pb and the values
$f_{0}=$ 35 (50) MeV$\cdot$fm5 for the AV18 (Bonn B) potential are
substantially smaller than 60–70 MeV$\cdot$fm5 Oyamatsu  _et al._ (2010). It
is obvious that the energy will be far away from the experimental data if
$f_{0}=$ 60–70 MeV$\cdot$fm5 Oyamatsu  _et al._ (2010) were used in the BHF
theory.

Figure 3:  (Color online) Elastic DCS for electron-nucleus scattering in 208Pb
as a function of the scattering angle $\theta$ at the electron beam energies
502 MeV. The results from the RBHF theory in the full Dirac space are compared
with the measured DCS (Exp) Friar and Negele (1973).

The comparison between the charge densities from the RBHF theory in the full
Dirac space and the experimental charge densities shall be connected with the
discussion of the DCS and the electric charge form factors in the elastic
electron-nucleus scattering, which are the quantities measured in real
experiments. Several theoretical methods have been used to study the DCS and
the electric charge form factors, such as the plane-wave Born approximation,
the eikonal approximation, and the phase-shift analysis method Baker (1964);
Yennie  _et al._ (1954). In this work, the DCS are calculated by the phase-
shift analysis method which has been widely used in elastic electron-nucleus
scattering off both stable and unstable nuclei Antonov  _et al._ (2005);
Sarriguren  _et al._ (2007); Roca-Maza  _et al._ (2008, 2013). This method
takes into account the Coulomb distortion effect and agrees well with the
experimental scattering data in a broad scattering energy range for both light
and heavy nuclei Chu  _et al._ (2009).

Figure 3 shows the DCS for elastic electron scattering in 208Pb as a function
of the scattering angle $\theta$ at the electron beam energies 502 MeV. For
$f_{0}$ = 60 MeV$\cdot$fm5, the results from the RBHF theory give a good
description of the experimental data Friar and Negele (1973) at all scattering
angles. However, in the case of 70 MeV$\cdot$fm5, the experimental data can
only be similarly reproduced at small scattering angles, up to the second
diffraction minimum. The DCS curve obtained from the RBHF theory using $f_{0}$
= 70 MeV$\cdot$fm5 has the same shape as that from $f_{0}$ = 60 MeV$\cdot$fm5.
The positions of the diffraction minima and maxima calculated from these two
curves are almost the same, but the DCS curve from $f_{0}$ = 70 MeV$\cdot$fm5
is shifted downward as a whole, especially at large scattering angles. The
deviations for the DCS predicted by the RBHF theory between the two values of
$f_{0}$, and the discrepancies with respect to experimental data, become more
prominent with the increasing of the scattering angles.

Figure 4:  (Color online) Squared charge form factor for 208Pb as a function
of the momentum transfer $q$ at the electron beam energies 502 MeV.

The electric charge form factor $F(q)$ is a very important quantity to
characterize the elastic electron-nucleus scattering which can be used to
analyse the effect of the finite size of the nucleus. At a given beam energy,
the squared charge form factor $|F(q)|^{2}$ is obtained as

$|F(q)|^{2}=\frac{{d\sigma}/{d\Omega}}{{d\sigma_{M}}/{d\Omega}},$ (22)

where the momentum transfer $q$ is related to the scattering angle $\theta$ in
the laboratory frame by $q=2E\sin(\theta/2)$, ${d\sigma}/{d\Omega}$ is the DCS
calculated from the RBHF theory in the full Dirac space and
${d\sigma_{M}}/{d\Omega}$ is the Mott DCS.

In Fig. 4, the squared charge form factor $|F(q)|^{2}$ for the nuclei 208Pb at
the electron beam energies 502 MeV are depicted as functions of the momentum
transfer $q$ calculated by the RBHF theory in the full Dirac space. The
experimental data of $|F(q)|^{2}$ can be well reproduced in the low-momentum
transfer regions by the RBHF theory for both $f_{0}$ = 60 MeV$\cdot$fm5 and
$f_{0}$ = 70 MeV$\cdot$fm5, while the discrepancies appear between the
theoretical predictions from the RBHF theory and the experimental data at
large-momentum transfer regions, as is expected from the previous analysis of
the DCS at small scattering angles in Fig. 3. This indicates that the RBHF
theory describe differently the central region of the experimental charge
density. Comparing to $|F(q)|^{2}$ obtained from the RBHF theory by using
$f_{0}$ = 70 MeV$\cdot$fm5, the positions of the diffraction minima and maxima
calculated from $f_{0}$ = 60 MeV$\cdot$fm5 are almost the same, but the
results from $f_{0}$ = 60 MeV$\cdot$fm5 shifted upward as a whole and became
much closer to the experimental data, especially at large-momentum transfer
regions.

##  IV Summary

The computation of a heavy nucleus, e.g, 208Pb, has been out of reach for
relativistic _ab initio_ theory. Relativistic Brueckner-Hartree-Fock (RBHF)
theory provides a promising approach to achieve this goal. Comparing with the
momentum-independence approximation method and the projection method in the
RBHF theory, one of the most significant advantage for the RBHF theory in the
full Dirac space is that the momentum dependence of scalar and vector
components of the single-particle potentials can be determine uniquely. In
order to apply this new method to study finite nuclei, as a first step,
properties of 208Pb are explored by using the microscopic equation of state
for asymmetric nuclear matter and a liquid droplet model. The neutron and
proton density distributions, the binding energy and the charge radius are
calculated by minimization of the energy. As for $f_{0}$ = 60 (70)
MeV$\cdot$fm5 in the surface term of the liquid droplet model, the neutron
radii, the proton radii and the neutron skin thickness are 5.59 (5.67) fm,
5.48 (5.55) fm and 0.11 (0.12) fm, respectively. To further compare the charge
densities predicted from the RBHF theory in the full Dirac space with the
experimental charge densities, the differential cross sections and the
electric charge form factors in the elastic electron-nucleus scattering are
obtained by using the phase-shift analysis method. The results from the RBHF
theory are in good agreement with the experimental data. This work will
further motivate us to study finite nuclei in a more microscopic way based on
the RBHF theory in the full Dirac space in the near future, such as the local
density approximation (LDA).

###### Acknowledgements.

We thank Dr. Qiang Zhao and Di Wu for helpful discussions. This work was
partly supported by the National Natural Science Foundation of China (NSFC)
under Grants No. 12205030 and No. 12147102, the Fundamental Research Funds for
the Central Universities under Grants No. 2020CDJQY-Z003 and No.
2021CDJZYJH-003, the MOST-RIKEN Joint Project “ _Ab initio_ investigation in
nuclear physics”. Part of this work was achieved by using the supercomputer
OCTOPUS at the Cybermedia Center, Osaka University under the support of
Research Center for Nuclear Physics of Osaka University.

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# The Market for Lemons and the
Regulator’s Signalling Problem

Roy Long University of Chicago Economics
Original version July 2022. This version with major revisions December 2023

(December 2023)

###### Abstract

The Market for Lemons is a classic model of asymmetric information first
studied by Nobel Prize economist George Akerlof. It shows that information
asymmetry between the seller and buyer may result in market collapse or some
sellers leaving the market. “Lemons” in the used car market are cars of poor
quality. The information asymmetry present is that the buyer is uncertain of
the cars’ true quality. I first offer a simple baseline model that illustrates
the market collapse, and then examine what happens when regulation, ie. a DMV
is introduced to reveal (signal) the true car quality to the buyer. The effect
on the market varies based on the assumptions about the regulator. The central
focus is on the DMV’s signal structure, which can have interesting effects on
the market and the information asymmetry. I show that surprisingly, when the
DMV actually decreases the quality of their signal in a well constructed way,
it can substantially increase their profit. On the other hand, this negatively
effects overall welfare.

##  1 Baseline Model

In this article, we will review the famous Market for Lemons by Akerlof. It is
a simple model that shows how information asymmetry between buyer and seller
can have very interesting effects on a market. Then we will introduce a
certain third party and study the implications of information design.

“Lemons” in a market are essentially bad goods of low quality. Akerlof showed
that information asymmetries in the market can cause the quality of goods to
degrade until only “lemons” are left.

We use a car market for our setting as in Akerlof’s model. The information
asymmetry here is that the quality of the car is known to the sellers (car
dealers) but not to buyers. We can represent the quality of a used car by a
hidden type $\theta\in[0,1],$ where $\theta=0$ means a busted car and
$\theta=1$ means in new condition.

Payoffs for the buyer and seller are given by:

Market for lemons no sale sale seller $\pi\theta$ $p$ buyer $0$ $\theta-p$

where $\pi<1$ indicates that the seller is somewhat tired of the car and it
has less utility for them (especially the longer they’re unable to sell the
car), though $\pi$ is not too small, so we assume $\pi>\frac{1}{2}.$ And $p$
is the price of the car determined endogenously by the market.

Now each seller has a type represented by their $\theta\in[0,1].$ Suppose all
sellers are on the market. Then we note that the buyer accepts $p$ only if
$p\leq E(\theta)=\frac{1}{2}$ under uniform prior. But then given
$p=\frac{1}{2},$ seller gets net gain of $p-\pi\theta=\frac{1}{2}-\pi\theta,$
so she’ll only sell if $\theta\leq\frac{1}{2\pi}<1.$

Now since only the sellers with $\theta\leq\frac{1}{2\pi}$ will sell, the
buyer will only buy if $p\leq
E\left(\theta\mid\theta\leq\frac{1}{2\pi}\right)=\frac{1}{4\pi}.$ But by
similar reasoning then the seller will only sell if
$\frac{1}{4\pi}-\pi\theta\geq 0,$ or
$\theta\leq\frac{1}{4\pi^{2}}<\frac{1}{2\pi}.$

This reasoning can easily be repeated recursively, and it follows by induction
at the $n$-th step that the only car dealers left willing to sell are
$\theta\leq\frac{1}{(2\pi)^{n}}$ and buyers will only buy if
$\theta\leq\frac{1}{2(2\pi)^{n}}$ but clearly as $n\to\infty$ the
$\frac{1}{(2\pi)^{n}}\to 0,$ so in fact only $\theta=0,$ the lemons will sell.
Do we have a market anymore? Apparently not.

This kind of reasoning is called unraveling to the bottom, or retrograde
analysis or working backwards. When this was first proposed, it had profound
implications, because it showed that under reasonable assumptions and
information asymmetry, a market could suffer market failure.

What is a proposed solution? There are indeed market solutions on both supply
and demand side. On the supply side, sellers may provide a warranty to assuage
buyers’ concerns about quality uncertainty, but this may be costly for sellers
with low quality cars. The sellers may also increase the proportion of higher
quality cars, so the buyer’s prior belief of buying a good car is higher.
However this may not be so easy to achieve for a particular seller type. On
the demand side, buyers may seek third party quality specialists to examine
the cars, thereby removing the information asymmetry. We consider a third
alternative, a regulation based solution to create a different kind of
incentive in the market.

Let us assume the DMV, a regulator, will assist the buyer in determining the
true value of the seller’s car quality, thereby mitigating the effects of the
information asymmetry.

Suppose the DMV offers certificates that disclose quality for free. (We will
re-evaluate this assumption later.) Each seller then chooses whether to
disclose $\theta$ or not. Thus, the buyer is willing to pay for

  * •

$w=\theta$ if $\theta$ is disclosed, and

  * •

$w=E(\theta\mid\text{undisclosed})=\widehat{\theta}$ if $\theta$ is not
disclosed.

However, note that given $\widehat{\theta},$ the seller types
$\theta\geq\widehat{\theta}$ will disclose, and sellers with
$\theta<\widehat{\theta}$ are better off not disclosing. But this means that
$\widehat{\theta}=E(\theta\mid\text{undisclosed})=E(\theta\mid\theta<\widehat{\theta})=\frac{\widehat{\theta}}{2},$
which solves $\widehat{\theta}=0.$ Thus, only the lemons won’t disclose.

We can easily understand this substantively with a similar recursive argument.
Suppose some sellers $\theta$ disclose and some don’t disclose. Consider a
nondisclosing seller with $\theta$ on the high end of the types that don’t
disclose. To the buyer, her type can only be evaluated as best as
$E(\theta\mid\text{undisclose}),$ which is strictly lower than her true type
$\theta$. In this case the seller is essentially being lumped in with the low
$\theta$’s that are also not disclosing, and that is not good for her. So that
means actually the seller should disclose. So then more types will disclose,
and we can keep repeating this argument until all but the $\theta=0$ will
disclose, because as long as multiple types of $\theta$ are not disclosing
then some $\theta$ in that group is being disadvantaged by the information
asymmetry and the rest of the nondisclosing group.

Rather remarkably the mere introduction of a DMV completely reverses the
market failure from no one willing to sell except the lemons to everyone
willing to sell except the lemons.

This suggests that regulation, ie. a DMV, may solve the problem. However, the
DMV probably won’t give the seller $\theta$ for free because it probably costs
them money to do an inspection etc. Or the DMV also wants to make a profit.

Suppose the DMV charges a fee of $c>0$ for certificates. Then each seller
chooses whether to buy a certificate. The certificate will display the car’s
true quality $\theta$ to the buyer. Sellers that don’t buy the certificate
will still have their $\theta$ concealed. However, an important implication of
this is that the buyer is aware that some sellers on the market may disclose
and others won’t and can reason about this when evaluating a seller.

This means that the buyer is willing to pay for

  * •

$w=\theta$ if $\theta$ is disclosed and

  * •

$w=E(\theta\mid\text{undisclosed})=\widehat{\theta}$ if $\theta$ is not
disclosed.

Given $\widehat{\theta},$ the seller is better off disclosing if

$\theta-c\geq\widehat{\theta}$

and better off not disclosing if

$\theta-c<\widehat{\theta}.$

Thus,

$\widehat{\theta}=E(\theta\mid\text{undisclosed})=E(\theta\mid\theta<\widehat{\theta}+c)=\frac{\min\left(\widehat{\theta}+c,1\right)}{2},$

which solves $2\widehat{\theta}-\min\left(\widehat{\theta}+c,1\right)=0$ ie.
$\min\left(\widehat{\theta}-c,1-2\widehat{\theta}\right)=0,$ which means
$\widehat{\theta}=\min\left(c,\frac{1}{2}\right).$ If $c>\frac{1}{2},$ then
$\widehat{\theta}=\frac{1}{2}$ and
$\theta-c<1-c<\frac{1}{2}=\widehat{\theta},$ so no one will disclose. If
$c\leq\frac{1}{2},$ then $\widehat{\theta}=c,$ and the ones that disclose are
$\theta\geq c+c=2c.$

Given this, the DMV maximizes their profit of $P(c):=c(1-2c)$ for
$c\leq\frac{1}{2}.$ This is maximized at $c=\frac{1}{4},$ with DMV’s profit
being $\frac{1}{8}$ and only car dealers with $\theta\geq\frac{1}{2}$
disclose.

We see here the DMV is maximizing under a tradeoff. The DMV wants more sellers
to buy the certificate, but by making the certificate’s cost higher, fewer
sellers will buy it. So by charging the fee for the certificate, the DMV is
not able to ensure all types will disclose.

##  2 Regulation with Noisy Signalling

We now turn to our main focus, the DMV’s signalling problem. Can the DMV
cheat? What if the DMV gives the seller a noisy signal? Realistically, the
DMV’s signal can have some error in it. Problems might arise in the inspection
or they might do a poor job at inspecting the car. Suppose the DMV is profit-
motivated and seeks to maximize its profit, with welfare being only a
secondary concern. It may also be costly for the DMV to do a full inspection
of the car to determine its true type $\theta.$ Instead, they might shirk and
do a low level inspection, revealing a noisy signal, or maybe they might even
try to give the seller a fake signal completely. This is likely not beneficial
for social welfare. Can the DMV still prevent the market failure with a lower
quality signal and get away with it, achieving a pareto efficient outcome? Can
the DMV even improve on their total profit (more than their profit from
sending the true $\theta$ signal)? These are some interesting questions that
motivate our study of the DMV’s signalling problem.

We begin by stating several possible information structures and discussing
them briefly.
First signal structure: The DMV gives a signal $\theta+\varepsilon$ where
$\varepsilon$ is drawn from a distribution with cdf $F$ and density centered
at $0.$ This model is realistic, for example, letting $\varepsilon$ be
standard normal noise. This signal can be interpreted as the DMV attempting to
perform an accurate inspection of the car, but the result may have some error
in it.

Second signal structure: DMV gives $\theta$ with probability $p$ and gives an
uninformative signal with probability $1-p,$ possibly just drawn randomly from
$[0,1],$ ie. $s\sim U(0,1).$ This model is interesting as there is again noise
in the signal, but unlike the first signal structure, the signal is not
centered around $\theta.$ Here the DMV is choosing a lottery over two signal
structures - giving a perfect signal ($s=\theta$) and an uninformative signal
($s\sim U(0,1)$ independent of $\theta$). It may not be clear why such a
signal structure may be beneficial to the DMV compared to a perfect signal
$(p=1),$ but it turns out that the randomized arbitrary signal in the $s\sim
U(0,1)$ is not as innocuous as upon first glance. Given the particular market
structure and how the buyers view different car sellers under the information
asymmetry, the inclusion of this noise can create an interesting effect on the
market. This signal structure leads to new considerations for sellers in their
decision to disclose or not. In particular, lower seller types may have
greater incentive to disclose. This signal structure will be the main topic of
this paper.

Third signal structure: DMV gives a signal drawn from some distribution $G$
with support containing $[0,1]$ independent of $\theta.$ This is a totally
uninformative signal. Such a signal can be interpreted as the DMV does not
perform any inspection of the car at all and completely fabricates an
arbitrary quality for the seller to display to the buyer. Notice that this is
equivalent to the $p=0$ signal of the second As expected, this cannot be
sustained. However, to gain intuition about the model and the signal
structure, we will begin with this “completely fake” signal which provides no
information about the true car quality and discuss how the market responds to
it formally.

Let us consider how the market responds to the signal $s\sim U(0,1)$
regardless of what $\theta$ is. Under such a completely uninformative signal,
a buyer that sees a seller without displayed quality is willing to pay for
$E(\theta\mid\text{undisclosed}),$ whereas if he sees some seller with
disclosed quality $s$ he is willing to pay for

$E(\theta\mid\text{disclosed},s)=E(\theta\mid\text{disclosed})$

as $s$ provides no information about the car’s actual quality. Given this, a
seller (of any type) will choose to disclose if

$E(\theta\mid\text{disclosed})-c\geq E(\theta\mid\text{undisclosed})$

Since this decision criteria is the same for all seller types (given the
decisions of the other sellers to disclose or not), it follows that the
sellers must either all disclose or all undisclose. The latter is not a
reasonable equilibrium given that no information is conveyed to the buyer (as
there is no nontrivial partition into disclose/undisclosed) and all sellers
suffer a cost of $c,$ so we conclude that no types will disclose, as expected.
Thus, the market failure is not prevented.

Although this simple analysis yields the trivial result that we expected - no
one will pay for a completely uninformative signal - it does highlight the
central incentive for a seller to disclose. There is a tradeoff between the
cost incurred for disclosing by purchasing the DMV’s certificate and the
strategic “cost” of having the quality hidden from the buyer. Intuitively, if
the signal is informative, then higher types will disclose while lower types
will not disclose - even if the signal is noisy. Thus, if a seller decides to
not disclose the car quality, her perceived quality to the buyer may be lower
as she will be grouped with other, lower quality sellers that also do not
disclose.

Now we turn to our central focus, the second signalling information structure,
where the DMV charges $c$ and gives $\theta$ with probability $p>0$ and with
probability $1-p$ gives random $r\sim U(0,1)$ drawn uniformly, independent of
$\theta.$ We say that this is the DMV’s signal $s.$
Then the buyer is willing to pay for:

  * •

$w=E(\theta\mid
s=\varnothing)=E(\theta\mid\text{undisclosed})=\widehat{\theta}$ if $\theta$
is not disclosed.

  * •

$w=p\theta+(1-p)E(\theta\mid\text{disclosed})$ if $s=\theta$ is disclosed,
because if the disclosed value is $\theta,$ with probability $p$ it is the
true $\theta$ and with probability $1-p$ it is a fake signal and in that case
the expected value of $\theta$ is $E(\theta\mid\text{disclosed}).$ More
formally, if the buyer observes signal $s,$ he is willing to pay for

$\displaystyle E(\theta\mid s)$ $\displaystyle=\text{Pr}(\text{true
signal})E(\theta\mid s,\text{true signal})+\text{Pr}(\text{fake
signal})E(\theta\mid s,\text{fake signal})$ $\displaystyle=p\cdot s+(1-p)\cdot
E(\theta\mid\text{disclosed})$

by total expectation.

We also have by total expectation

$\displaystyle E(\theta)=\frac{1}{2}$
$\displaystyle=\text{Pr}(\text{undisclosed})E(\theta\mid\text{undisclosed})+\text{Pr}(\text{disclosed})E(\theta\mid\text{disclosed})$
$\displaystyle=q\widehat{\theta}+(1-q)E(\theta\mid\text{disclosed})$

And so
$E(\theta\mid\text{disclosed})=\frac{\frac{1}{2}-q\widehat{\theta}}{1-q}$
where $q$ is the probability $\theta$ is not disclosed.

Let $w(\theta)$ be the buyer’s willingness to pay if he sees a seller with
disclosed quality $\theta.$ Then we have

$w(\theta)=p\theta+(1-p)E(\theta\mid\text{disclosed})=p\theta+(1-p)\cdot\frac{\frac{1}{2}-q\widehat{\theta}}{1-q}$

Let $W(\theta_{0})$ then be the expected profit (buyer’s expected willingness
to pay) for seller of type $\theta_{0}$ if she discloses. With probability $p$
the $\theta_{0}$ is accurately displayed to the buyer, in which case the buyer
is willing to pay $w(\theta_{0}),$ and with probability $1-p$ some random
$\theta\sim U(0,1)$ is drawn uniformly in which case the buyer is willing to
pay $w(\theta).$
Thus we have

$\displaystyle W(\theta_{0})$
$\displaystyle=pw(\theta_{0})+(1-p)\int_{0}^{1}w(\theta)\,d\theta$
$\displaystyle=p\left(p\theta_{0}+(1-p)\cdot\frac{\frac{1}{2}-q\widehat{\theta}}{1-q}\right)+(1-p)\int_{0}^{1}\left(p\theta+(1-p)\cdot\frac{\frac{1}{2}-q\widehat{\theta}}{1-q}\right)\,d\theta$

So type $\theta_{0}$ will

  * •

disclose if

$W(\theta_{0})-c>\widehat{\theta}$

  * •

not disclose if

$W(\theta_{0})-c\leq\widehat{\theta}$

Clearly $W(\cdot)$ is strictly increasing in $\theta_{0}$ as $\frac{\partial
W}{\partial\theta_{0}}=p^{2}>0$

It follows that the car dealers still employ a cutoff strategy with
$\theta\in[0,q]$ not disclosing, and so $\widehat{\theta}=\frac{q}{2},$ which
means

$E(\theta\mid\text{disclosed})=\frac{\frac{1}{2}-q\widehat{\theta}}{1-q}=\frac{1+q}{2}=\frac{1+2\widehat{\theta}}{2}.$

Thus

$\displaystyle W(\theta_{0})$
$\displaystyle=p\left(p\theta_{0}+(1-p)\frac{1+2\widehat{\theta}}{2}\right)+(1-p)\int_{0}^{1}\left(p\theta+(1-p)\frac{1+2\widehat{\theta}}{2}\right)\,d\theta$
$\displaystyle=(1-p)\frac{1+2\widehat{\theta}}{2}+p\cdot
p\theta_{0}+(1-p)\int_{0}^{1}p\theta\,d\theta$
$\displaystyle=(1-p)\frac{1+2\widehat{\theta}}{2}+p^{2}\theta_{0}+(1-p)\frac{p}{2}$
$\displaystyle=(1-p)\frac{1+p+2\widehat{\theta}}{2}+p^{2}\theta_{0}$

Now solving:

$(1-p)\frac{1+p+2\widehat{\theta}}{2}+p^{2}\theta_{0}-c>\widehat{\theta}$

ie.

$(1-p)(1+p+2\widehat{\theta})+2p^{2}\theta_{0}-2c>2\widehat{\theta}$
$1-p^{2}+2p^{2}\theta_{0}-2c>p\cdot 2\widehat{\theta}$
$2p^{2}\theta_{0}>p\cdot 2\widehat{\theta}+p^{2}-1+2c$
$\theta_{0}>\frac{p\cdot 2\widehat{\theta}+p^{2}-1+2c}{2p^{2}}$

So the types that don’t disclose are:

$\theta\leq\frac{p\cdot 2\widehat{\theta}+p^{2}-1+2c}{2p^{2}}$

And so we have

$\widehat{\theta}=E\left(\theta\,\,\middle|\,\,\theta\leq\frac{p\cdot
2\widehat{\theta}+p^{2}-1+2c}{2p^{2}}\right)=\min\left(\frac{p\cdot
2\widehat{\theta}+p^{2}-1+2c}{4p^{2}},\,\frac{1}{2}\right)$

Solving:

$4p^{2}\widehat{\theta}=p\cdot 2\widehat{\theta}+p^{2}-1+2c$
$2p(1-2p)\widehat{\theta}=1-p^{2}-2c$
$\widehat{\theta}=\min\left(\frac{1-p^{2}-2c}{2p(1-2p)},\,\frac{1}{2}\right)$

We have the constraint:

$q=\frac{1-p^{2}-2c}{p(1-2p)}\in[0,1]$

And types that do disclose are $\theta_{0}>2\widehat{\theta}=q$ Now we are
maximizing:

$P=c(1-2\widehat{\theta})=c\left(1-\frac{1-p^{2}-2c}{p(1-2p)}\right)$

Thus, the DMV solves the constrained optimization problem:

$\max_{c,\,p}\hskip 2.84544ptc\left(1-\frac{1-p^{2}-2c}{p(1-2p)}\right)$

subject to

$0\leq\frac{1-p^{2}-2c}{p(1-2p)}\leq 1,\hskip 14.22636ptp\in[0,1]$

We first consider $p>1/2.$ Clearly this is a concave problem in $c$ for any
fixed $p=\overline{p}$ as

$c\left(1-\frac{\overline{p}^{2}-2c}{\overline{p}(1-2\overline{p})}\right)=c\left(1+\frac{\overline{p}^{2}}{\overline{p}(2\overline{p}-1)}-\frac{2c}{\overline{p}(2\overline{p}-1)}\right)$

where $\frac{1}{\overline{p}(2\overline{p}-1)}>0$ so it is a downward opening
sloping quadratic in $c.$
Thus, the FOC (for fixed $p$) is

$[c]:\hskip
14.22636pt1-\frac{1-p^{2}-2c}{p(1-2p)}+c\left(\frac{2}{p(1-2p)}\right)=0$
$p(1-2p)-(1-p^{2})+4c=0$ $c=\frac{1-p+p^{2}}{4}$

Then profit is

$P^{*}(p)=\left(\frac{1-p+p^{2}}{4}\right)\left(1-\frac{1-p^{2}-2\cdot\left(\frac{1-p+p^{2}}{4}\right)}{p(1-2p)}\right)$

And constraint is

$\frac{1-p^{2}-2\cdot\left(\frac{1-p+p^{2}}{4}\right)}{p(1-2p)}\geq 0$

$\frac{1-p^{2}-\tfrac{1}{2}(1-p+p^{2})}{p(1-2p)}=\frac{2-2p^{2}-(1-p+p^{2})}{2p(1-2p)}=\frac{1+p-3p^{2}}{2p(1-2p)}\geq
0$

Thus, $1+p-3p^{2}\leq 0\implies p\geq p_{min}:=\frac{1+\sqrt{13}}{6}\approx
0.7676$

For $p>p_{min}$ then there is an interior solution.

At $p=p_{min}$ all types except lemons $(\theta=0)$ will disclose.

We can simplify profit:

$\displaystyle P^{*}(p)$
$\displaystyle=\left(\frac{1-p+p^{2}}{4}\right)\left(1-\frac{1+p-3p^{2}}{2p(1-2p)}\right)$
$\displaystyle=\left(\frac{1-p+p^{2}}{4}\right)\left(\frac{p-1-p^{2}}{2p(1-2p)}\right)$
$\displaystyle=\frac{(1-p+p^{2})^{2}}{8p(2p-1)}$

As expected, when $p=1$ we have $P^{*}(1)=1/8$ which is the baseline with no
signal manipulation.

At $p=p_{min}$ the profit is:

$P^{*}\left(\frac{1+\sqrt{13}}{6}\right)=\frac{11-\sqrt{13}}{36}\approx
0.2054$

Note that this already greatly exceeds the DMV’s basline profit of
$P^{*}(1)=0.125$ under a perfect signal. Now if $1/2<p<p_{min}$ then the DMV
chooses $c$ so that $\widehat{\theta}=0$ is binding, so we have a corner
solution, ie. all types except the lemons $(\theta=0)$ disclose to maximize
profit.

Thus we have

$\displaystyle q=\frac{1-p^{2}-2c}{p(1-2p)}$ $\displaystyle=0$ $\displaystyle
c=\frac{1-p^{2}}{2}$

And we get

$P^{*}(p)=c\cdot(1-0)=\frac{1-p^{2}}{2}$

Now let us consider $p<1/2.$ In this case, $P_{p}(c)$ is an upward facing
parabola with minimum at $c=\frac{1-p+p^{2}}{4}.$ And the constraint is
$0\leq\frac{1-p^{2}-2c}{p(1-2p)}\leq 1$ and $1-2p>0$ so rewrite this as

$0\leq 1-p^{2}-2c\leq p-2p^{2}$

which rearranges to

$\frac{1-p+p^{2}}{2}\leq c\leq\frac{1-p^{2}}{2}$

But note that

$\frac{1-p+p^{2}}{4}\leq\frac{1-p+p^{2}}{2}<\frac{1-p^{2}}{2}$

and thus clearly $P_{p}$ is maximized at $c=\frac{1-p^{2}}{2}$ as $P_{p}$ is
increasing on $(\frac{1-p+p^{2}}{4},\infty)$

Plugging this in we get

$P^{*}(p)=\frac{1-p^{2}}{2}\left(1-\frac{1-p^{2}-2\cdot\frac{1-p^{2}}{2}}{p(1-2p)}\right)=\frac{1-p^{2}}{2}$

And so for $p<1/2$ the DMV again charges sufficiently low cost to induce all
types except lemons $(\theta=0)$ to disclose.

We can now summarize the optimal DMV profit $P^{*}(p)$ for all $p\in[0,1]$ as
follows:

$P^{*}(p)=\begin{dcases}\hskip
5.69046pt\frac{(1-p+p^{2})^{2}}{8p(2p-1)},&\hskip 8.5359ptp\geq p_{min}\\\
\hskip 5.69046pt\frac{1-p^{2}}{2},&\hskip 8.5359pt0<p<p_{min}\\\ \hskip
5.69046pt0,&\hskip 8.5359ptp=0\end{dcases}$

The DMV’s profit is increasing with decreasing $p,$ ie. by lowering the
informativeness of the signal the DMV can make more overall. In fact the DMV
can achieve any profit less than $(1-0^{2})/2=1/2$ by choosing sufficiently
low values of $p.$ However, if the signal is completely uninformative ($p=0$)
then the DMV makes no profit as no seller will purchase the certificate.
Recall that in the baseline model with perfect information the DMV made profit
$P=1/8.$

Thus the DMV can achieve almost quadruple the profit compared to the perfect
signal in the baseline model. This is a surprising and compelling result. By
degrading the quality of the signal, the DMV can actually increase its overall
profit, even by charging the same amount $c$ per seller for any type. By
introducing uncertainty into the signal, the DMV can actually encourage lower
$\theta$ types to “disclose,” because there is some possibility that their
revealed $s$ is higher than their true $\theta$ when the DMV gives the fake
signal. Indeed, in the baseline model, only the sellers with types
$\theta>1/2$ disclose. And if $s\sim U(0,1)$ then $E(s)=1/2,$ so a seller of
type $\theta<1/2$ has an incentive to disclose with this possibility of
misrepresenting its $\theta$ to a higher type.

So by degrading the quality of the signal this way, the DMV incentivizes the
lower type sellers to disclose and can charge more per seller. This continues
until the DMV can incentivize all types except lemons ($\theta=0$) to
disclose, which maximizes profit.

We also have the optimal cost that the DMV charges,

$c^{*}(p)=\begin{dcases}\hskip 5.69046pt\frac{1-p+p^{2}}{4},&\hskip
8.5359ptp\geq p_{min}\\\ \hskip 5.69046pt\frac{1-p^{2}}{2},&\hskip
8.5359pt0<p<p_{min}\\\ \hskip 5.69046pt0,&\hskip 8.5359ptp=0\end{dcases}$

Note that $c^{*}$ is increasing on $(p_{min},1)$ and decreasing on
$(0,p_{min}).$
We see that for $p>p_{min},$ there is a tradeoff between lowering the cost per
seller to attract more sellers (lower types) to disclose, and the profit made
per seller that discloses. As $p$ decreases and the signal becomes less
informative, lower types become willing to disclose, but the DMV continues to
charge lower costs. At the threshold $p=p_{min},$ the DMV is able to capture
the entire market at optimum, with all types except lemons disclosing. When
the DMV further decreases $p,$ it can then begin to raise the cost again while
still getting all non-lemons to disclose.

Perhaps the most surprising result is that even for $p<1/2,$ the DMV continues
to make more profit by further decreasing the quality of the signal, and
charging even more per seller. For such $p,$ the DMV has
$P^{*}(p)=c^{*}(p)=\frac{1-p^{2}}{2}$ and all types except lemons ($\theta=0$)
disclose. This can be understood as follows: When the DMV continues to degrade
the quality of its signal, this increases the lower types ($\theta<1/2$)
willingness to pay, because the probability of getting a fake signal, $1-p,$
increases, and the fake signal is on average better than the true $\theta$
being revealed. Thus, the lower types are wiling to pay more to receive the
certificate. The higher types, on the other hand, suffer from the lower signal
quality, but have no choice and still must disclose, because the alternative
is to be grouped in with the undisclosed types, which are the very low types,
ie. lemons. So they are willing to pay the higher cost of disclosing, even as
the signal quality continues to degrade. This is the reason why the car
dealers still employ a cutoff strategy for any $p>0$ signal structure.

At optimum, the DMV charges cost $c$ equal to the lemons’ reservation, just
high enough to make the lemons indifferent and not disclose. Then the DMV is
able to incentivize all higher types to disclose.

We observe, however, that by lowering the signal quality, overall welfare is
clearly worse. The overall quality of disclosed $\theta$ is lower and buyers
may be misled by fake signals. When $p<1/2,$ even though every seller is on
the market (which is pareto efficient), the DMV is cheating and giving a fake
signal more than half the time.

In practice, it is unrealistic for the regulator to provide signals with such
poor quality. However, in this model note that any nonzero value of $p<1$
achieves strictly greater profit for the DMV than the profit from the baseline
perfect signal. Even an almost perfect signal $p=1-\epsilon$ achieves higher
profit than $p=1.$ Thus, the DMV can exploit the information asymmetry in the
market and increase its profit with imperfect signals.

Acknowledgements

I would like to thank Profs. Scott Gehlbach and Zhaotian Luo at the UChicago
Political Science department (and Political Economy program) for teaching me
game theory and formal modelling in the social sciences in the SSI Formal
Theory sequence during my first year. Prof. Luo discussed the classic Market
for Lemons information asymmetry example with the baseline model (perfect
signal) in his class and I am grateful to have discussed my generalizations
with him. I also enjoyed seeing the Market for Lemons again this year in the
ECON 201 class taught by Prof. Ivan Kwok.

References

[1] Akerlof George (1970) The market for ‘Lemons’: quality uncertainty and the
market mechanism. Quarterly Journal of Economics, 84:488–500