# Set-Asides in USDA Food Procurement Auctions

Ni Yan  WenTing Tao

(Stanford University)

###### Abstract

We study the partial and full set-asides and their implication for changes in
bidding behavior in first-price sealed-bid auctions in the context of United
States Department of Agriculture (USDA) food procurement auctions. Using five
years of bid data on different beef products, we implement weighted least
squares regression models to show that partial set-aside predicts decreases in
both offer prices and winning prices among large and small business bidders.
Full set-aside predicts a small increase in offer prices and winning prices
among small businesses. With these predictions, we infer that net profit of
small businesses is unlikely to increase when set-asides are present.

##  1 Introduction

Promoting U.S. small businesses has long been an important objective of the
U.S. government. Small Business Act of 1953 mandates that each federal agency
should create “an annual goal that presents, for that agency, the maximum
practicable opportunity for small business concerns”.11115 U. S. C.
§644(g)(1)(B) Through programs and rules that prioritize small businesses over
large ones, the U.S. government hopes to promote the overall growth and
competitiveness of various types of small businesses. This paper focuses on
the impact of partial and full set-asides policies favoring small businesses
in the U.S. Department of Agriculture (USDA) food procurement auctions.

Every year, the USDA purchases billions of pounds of agricultural products
from vendors across the country through first-price sealed-bid online
multiunit auctions. In each auction, the items being procured appear as
individual contracts that specify the product type, product quantity, delivery
destination and delivery window. Winners are responsible for completing
contracts (referred to as “items” hereinafter) to receive payment. Auctions
happen around once a week for each food category such as beef, milk, fruits,
etc. Partial or full set-asides are applied to product type, and guarantee
that a certain percentage of total quantity procured for that product is
awarded to small businesses. This percentage is subject to change based on
USDA’s assessment of small businesses’ production capacity.

This paper focuses on beef procurement auctions in the years 2015 to 2019.
Beef category is selected because we are able to identify a set of relatively
similar beef products that each fall into different set-aside percentages
across time and auctions. We include all the bids data available from the
USDA’s website, taking out years 2020 and 2021 to eliminate the potential
confounding effect of the Covid-19 pandemic. We quantify the association
between set-asides and the number of participants, offer prices and winning
prices using weighted least squares regressions with set-aside percentage,
bidder type, their interaction and other observables as covariates. Our
findings motivate the question of whether small businesses benefit from
partial set-aside policies used by USDA, as the existence of partial set-
asides is associated with a decrease in both offer prices and winning prices
among large and small businesses. Lower winning prices imply a smaller price
margin for small businesses, while the data suggests it’s unlikely that their
probability of winning can increase enough to compensate for price drop and
drive up their net profit. In full set-aside auctions, small businesses raise
their prices as expected, but the increase is small because there are few
large business competitors to begin with.

The paper is organized as follows. Section II reviews existing literature on
the topic of set-asides in procurement auctions. Section III describes in
detail the format of the USDA procurement auctions and how set-asides are
implemented. Section IV provides a simplified behavioral model for auctions of
homogeneous items with partial set-asides. Section V conducts weighted least
squares regressions of the number of bidders, offer prices and winning prices
using USDA beef auction data. Section VI discusses potential issues and
directions for future research. Section VII concludes.

##  2 Literature Review

Although set-asides policy is a popular topic in auction theory, most work has
been focusing on evaluating changes in revenue and social surplus resulting
from full set-asides, where large businesses are excluded from the auction.
Some other work compares the efficiency of set-asides and subsidies in the
form of discounted payments to small businesses. Opinions on the effectiveness
of set-asides are mixed and depend on the setting. Little work has been done
studying partial set-asides that allow large businesses into the auction but
put a constraint on their winnings. This paper aims to fill the gap of partial
set-asides policies in procurement auctions through empirical analysis.

Athey . (2011) compare the effect of set-asides and subsidies on revenue and
social surplus in U.S. Forest Service timber auctions. The timber auction
adopts either open competition between small and large bidders, or full set-
asides for small bidders. They estimate a parametric model to recover value
distribution from observed bids in auctions without set-asides, and build a
probabilistic model of bidder entry subject to entry cost and expected profit.
Counterfactual analysis shows that switching from set-asides to open
competition would introduce 1.7 large bidders and lose 1.9 small bidders, with
the fraction of sales won by small bidders dropping. Auction revenue and
bidder surplus increase by 6% and 21% respectively. As summarized in this
paper, understanding the impact of set-asides on participation of potential
bidders is a crucial step in the analysis of subsequent changes in bidding
strategy.

Nakabayashi (2013) studies full set-aside auctions in the context of Japanese
public construction projects focusing on the impact on government budgets,
taking into consideration changes in bidder participation. This paper
concludes that the set-aside programs increase small businesses’ participation
in procurement auctions by around 40%, which is shown to be high enough to
reduce government procurement costs.

The two papers above are both based on an environment with flexible entry, yet
the results are in different directions. It seems important to understand
features of the underlying industry before thinking about the effect of set-
asides. MacDonald . (2002) study the effect of an additional bidder on
competitive pricing in the USDA procurement auctions of flour, pasta and oil
in 1992 to 1996. Regressing winning prices on ingredient price, product,
quantity and competition, they find that an additional bidder is associated
with a much larger price drop when there are fewer competitors. We incorporate
this nonlinear effect of the number of bidders in our model and find similar
results. They also compare USDA realized prices with private sector wholesale
prices for comparable products and services, and find that USDA prices are
generally lower by 20-40%. The large price gap is a result of the USDA aiming
to find the lowest short-run marginal cost among vendors. This might account
for our observation of few large bidders but an abundance of small bidders in
our data. USDA auctions could be unattractive to large bidders because they
operate on a large scale and use more of long-term contracts. Inspired by
their analysis we include monthly wholesale beef price as a proxy for the
option to sell outside of the USDA auctions.

##  3 Description of USDA Food Procurement Auction and Set-Asides

The USDA spends around two billion dollars every year on two billion pounds of
domestically produced and processed food, including grain, meat, vegetable,
fruit, etc., to be delivered to schools, food banks, and households. About 400
million dollars is spent in the U.S. on beef products annually, mostly on
frozen ground beef and patties. The USDA’s Agricultural Marketing Service
(AMS) operates first-price sealed-bid multiunit auctions through their online
system WBSCM222Web-Based Supply Chain Management.. Only qualified vendors
registered in their system can participate in their online auctions. USDA
conducts a plant survey to evaluate the technical, production, and
transportation capabilities of applicants following its own product
specification and technical requirement that dictates food composition,
package size, and package material. This paper focuses on nine ground beef
products which we will discuss in more detail in Section V, and for these
products only 17 vendors ever occurred in a five-year span, suggesting that
bidder entry is quite limited and infrequent. Figure 1 shows that the number
of active bidders across time is quite stable.333We identify the earliest and
latest auction each vendor participated and count the time in between as their
active period. The surge at the beginning in Figure 1 is simply due to lack of
earlier data.

Figure 1: Number of potential bidders

Procurement announcements are first published in AMS newsrooms, followed by a
detailed solicitation letter posted on USDA’s website. Each solicitation
consists of a list of items that specifies a USDA product, the quantity in
pounds, delivery destination (city, state and zip code), and delivery window.
Qualified vendors submit an independent price per pound for each item, and any
maximum quantity constraint they have for certain product and delivery
window.444For example, if a vendor only has two trucks of ground beef and is
indifferent among all delivery windows, the vendor can bid for all delivery
windows and report a capacity constraint on at most two trucks for all
windows. To avoid confusion we use the word product to refer to the product
group that an item belongs to hereinafter.555A solicitation can involve
multiple products, and each product group can involve multiple items. Though
it is called product group, it is actually a very specific type of product
defined by USDA, e.g. BEEF 100% PTY 85/15 FRZ 2.0MMA CTN-40 LB, which means
100% frozen beef patty with fat content 85/15, packed in a 40-pound carton and
meeting the 2.0 credit level of meat/meat alternate. In beef procurement
auctions, which run weekly, there is usually one or two weeks for bidders to
submit their bids before the auction closes. The delivery window is usually a
two-week span, and winning bidders are responsible for delivering products to
specified central warehouses before they receive payment.

Each solicitation may contain one or multiple product types with different
quantities. Following FAR 52.219-6 Notice of Small Business Set-Aside, the
USDA implements a percentage set-aside policy, given that in the Animal
(except Poultry) Slaughtering industry (NAICS Code 311611) 1,000 employees or
less is the standard of small business. Set-aside percentage is announced for
the solicitation and is applied at the product type level, meaning that all
items of that product type are considered when calculating the percentage
quantity that is awarded to small businesses. For instance, consider a
solicitation where all items are 40,000 lbs. If there are five boneless fresh
beef items, four frozen ground beef items, and three beef patty items, with
set-aside percentages 0%, 50% and 100% respectively, then at least two frozen
ground beef items and all three beef patty items should be awarded to small
business bidders if feasible, and there’s no restriction on boneless fresh
beef items. In particular, applying the set-aside percentage at a product type
level guarantees that such a solicitation with three product types is
theoretically equivalent to three independent solicitations of one product
type each. AMS also has some discretion to cancel orders that end up with
unreasonably high prices. Because AMS’s price ceiling is not public
information, and cancellation due to high prices is rare in the data, these
online auctions are effectively first-price sealed-bid multiunit auctions
subject to linear constraints.

##  4 A Simplified Behavioral Model

To understand how partial set-asides affect the competition within and between
large and small bidders, let’s consider a simplified model that auctions one
product type with identical items. Suppose there are $M$ items, $N_{1}$ small
bidders and $N_{2}$ large bidders. A set-aside percentage of
$\alpha=\frac{M_{1}}{M}$ dictates that at least $M_{1}$ items go to small
bidders if feasible. Because items are identical, we focus on the pure
strategy that vendors bid one price for all items. Suppose the best response
functions of small and large bidders are $b_{1}(v_{1})$ and $b_{2}(v_{2})$
respectively where $v_{1}\overset{iid}{\sim}F_{1}$ and
$v_{2}\overset{iid}{\sim}F_{2}$. Assume that $b_{1}(\cdot)$ and $b_{2}(\cdot)$
are increasing and invertible, and their inverse are $c_{1}(\cdot)$ and
$c_{2}(\cdot)$ respectively.

The simplest case is a $\alpha$ set-aside auction, i.e. $M_{1}=\alpha M$, with
$N_{1}=2$ small bidders, $N_{2}=1$ large bidder and no capacity constraint or
price ceiling.666More than one small bidder guarantees that bidding at price
ceiling is a dominated strategy. If the large bidder has the lowest price,
then she gets any $(1-\alpha)M$ items and the cheaper small bidder gets the
remaining $\alpha M$ item. If the large bidder doesn’t have the lowest price,
then only the cheapest small bidder gets all $M$ items.

We compute the expected profit of a small bidder with value $v_{1}$ and price
p as the following:

$\displaystyle\pi_{1}(p,v_{1})$ $\displaystyle=(p-v_{1})\cdot
P(p<p_{other\,small})\cdot[\alpha M\cdot P(p>p_{large})+M\cdot P(p\leq
p_{large})]$ $\displaystyle=M\cdot(p-v_{1})\cdot
P(p<p_{other\,small})\cdot[\alpha\cdot P(p>p_{large})+(1-P(p>p_{large}))]$
$\displaystyle=M\cdot(p-v_{1})\cdot
P(p<b_{1}(v_{other\,small}))\cdot[1-(1-\alpha)\cdot P(p>b_{2}(v_{large}))]$
$\displaystyle=M\cdot(p-v_{1})\cdot
P(c_{1}(p)<v_{other\,small})\cdot[1-(1-\alpha)\cdot P(c_{2}(p)>v_{large})]$
$\displaystyle=M\cdot(p-v_{1})\cdot[1-F_{1}(c_{1}(p))]\cdot[1-(1-\alpha)F_{2}(c_{2}(p))]$

And the expected profit of a large bidder with value $v_{2}$ and price p is
the following:

$\displaystyle\pi_{2}(p,v_{2})$ $\displaystyle=(1-\alpha)\cdot
M\cdot(p-v_{2})\cdot P(p<both\;p_{small})$ $\displaystyle=(1-\alpha)\cdot
M\cdot(p-v_{2})\cdot P(p<b_{1}(v_{small}))^{2}$ $\displaystyle=(1-\alpha)\cdot
M\cdot(p-v_{2})\cdot P(c_{1}(p)<v_{small})^{2}$ $\displaystyle=(1-\alpha)\cdot
M\cdot(p-v_{2})\cdot[1-F_{1}(c_{1}(p))]^{2}$

Taking first order condition of a small bidder wrt. $p$ gives us:

$\displaystyle 0=[1-F_{1}(c_{1}(p)]\cdot[1-(1-\alpha)F_{2}(c_{2}(p))]+$
$\displaystyle\quad\quad(p-v_{1})\cdot\\{[-F_{1}(c_{1}(p))\cdot
f_{1}(c_{1}(p))\cdot c_{1}^{\prime}(p)]\cdot[1-(1-\alpha)\cdot
F_{2}(c_{2}(p))]$
$\displaystyle\quad\quad\quad\quad\quad\quad+[1-F_{1}(c_{1}(p))]\cdot(1-\alpha)[-F_{2}(c_{2}(p))\cdot
f_{2}(c_{2}(p))\cdot c_{2}^{\prime}(p)]\\}$ (1)

In the first order condition above, for any $v_{1}$, $p=b_{1}(v_{1})$ should
solve the equation. Equivalently, for any $p$, $v_{1}=b_{1}^{-1}(p)=c_{1}(p)$
also solves the equation (1).

Taking first order condition of a large bidder wrt. $p$ gives us:

$\displaystyle 0$ $\displaystyle=[1-F_{1}(c_{1}(p))]^{2}+(p-v_{2})\cdot
2\cdot[1-F_{1}(c_{1}(p))]\cdot[-f_{1}(c_{1}(p))\cdot c_{1}^{\prime}(p)]$
$\displaystyle 0$ $\displaystyle=1-F_{1}(c_{1}(p))+(p-v_{2})\cdot
2\cdot[-f_{1}(c_{1}(p))\cdot c_{1}^{\prime}(p)]$

where the term $1-F_{1}(c_{1}(p))$ can be factored out and dropped because
$1-F_{1}(c_{1}(p))=0$ would imply zero profit for the large bidder. With the
same reasoning as in the first order condition of small bidders, equation (2)
should hold for all p if we replace $v_{2}$ by $c_{2}(p)$.

Therefore, the solutions $c_{1}(\cdot)$ and $c_{2}(\cdot)$ are characterized
by the following first-order differential equation system:

$\displaystyle c_{1}(p)$
$\displaystyle=p-\frac{[1-F_{1}(c_{1}(p)]\cdot[1-(1-\alpha)F_{2}(c_{2}(p))]}{\\{[F_{1}(c_{1}(p)\cdot
f_{1}(c_{1}(p))\cdot c_{1}^{\prime}(p)]\cdot[1-(1-\alpha)F_{2}(c_{2}(p))]+}$
$\displaystyle\quad\quad\quad\;\;[1-F_{1}(c_{1}(p))]\cdot(1-\alpha)[F_{2}(c_{2}(p))\cdot
f_{2}(c_{2}(p))\cdot c_{2}^{\prime}(p)]\\}$ (3) $\displaystyle c_{2}(p)$
$\displaystyle=p-\frac{1-F_{1}(c_{1}(p))}{2f_{1}(c_{1}(p))\cdot
c_{1}^{\prime}(p)}$ (4)

Notice that $c_{1}(p)$ depends on $\alpha$ directly and $c_{2}(p)$ only
depends $\alpha$ through $c_{1}(p)$. $v_{2}=c_{2}(p)$ can be expressed with
$c_{1}(p)$ and it’s derivative, and
$\frac{1-F_{1}(c_{1}(p))}{2f_{1}(c_{1}(p))\cdot c_{1}^{\prime}(p)}$ is the
markup of large bidders. To do a thought experiment, suppose $F_{1}$, the
distribution of small bidders’ value, is uniform, then $f_{1}(c_{1}(p))$ is a
constant. If a change in $\alpha$ would result in an increase in both
$c_{1}(p)$ and $c_{1}^{\prime}(p)$ for every p, then the markup at every p
will decrease since the numerator falls, the denominator rises, and they are
both positive. It then follows that $c_{2}(p)$ also increases for every p.
Inverting $c_{1}(\cdot)$ and $c_{2}(\cdot)$ gives us the conclusion that under
such assumptions it’s possible that $b_{1}(v_{1})$ and $b_{2}(v_{2})$ move in
the same direction. In the next section we’ll use weighted least squares
models on USDA bids data to predict changes in bidders’ prices empirically.

##  5 Results

The USDA maintains a list of qualified vendors with their large or small
business statuses, and occasionally adjusts set-aside percentage for certain
product types. We obtained data of bid prices and awards of individual
auctions under the beef category on the USDA website from Jan 01, 2015 to Dec
31, 2019. There are 23 types of products auctioned in this period. We first
discard products that don’t involve a change in set-aside percentage which
leaves us with seven types of products. Because the exact same products
packaged in different sizes will be classified as different products by the
USDA, we add back two such products to enlarge the dataset, and control for
package size in regressions.777For example, BEEF FINE GROUND LFT OPT FRZ
PKG-40/1 LB and BEEF FINE GROUND LFT OPT FRZ CTN-40 LB only differ in package
size. The former is packed in forty 1-pound packages and the latter in four
10-pound packages. The data contains 128 auctions on 9 products, corresponding
to 13,192 items and 96,474 bids. Three set-aside percentages are involved: 0%,
50% and 100%, associated with 762, 12215 and 215 items respectively.888The 9
products are (1)BEEF 100% PTY 85/15 FRZ 2.0MMA CTN-40 LB (2)BEEF 100% PTY
90/10 FRZ 2.0MMA CTN-40 LB (3) BEEF PATTY LEAN FRZ CTN-40 LB (4) BEEF SPP PTY
85/15 FRZ 2.0 MMA CTN-40 LB (5) BEEF COARSE GROUND FRZ CTN-60 LB (6) BEEF FINE
GROUND FRZ CTN-40 LB (7) BEEF FINE GROUND FRZ PKG-40/1 LB (8) BEEF FINE GROUND
LFT OPT FRZ CTN-40 LB (9) BEEF FINE GROUND LFT OPT FRZ PKG-40/1 LB

Table 1 shows summary statistics of auctions and items grouped by set-aside
percentage. Small bidder pool and large bidder pool in Table 1 refer to the
number of small or large bidders that ever occurred in that set-aside group.
There are some differences in the size of bidder pool and the average number
of bidders because set-asides apply to product type level. Therefore we
control for the number of bidders and its quadratic term in all regressions.
Fifty percent set-aside is the most common in the data because it is related
to a product that the USDA procures frequently. We control for a variety of
product and item characteristics such as demand level, quantity, the number of
bidders, product type, product value, etc., to eliminate bias resulting from
differences among products because the set-aside percentage is not randomly
assigned to the products.

| SA=0% | SA=50% | SA=100%
---|---|---|---
number of auctions | 43 | 127 | 39
number of items | 762 | 12215 | 215
number of bids | 1659 | 94218 | 597
Small bidder pool | 4 | 14 | 8
Large bidder pool | 1 | 5 | 0
Average number of small bidders | 1.835 | 6.656 | 3.147
| (0.466) | (1.602) | (1.042)
Average number of large bidders | 0.507 | 1.788 | 0
| (0.533) | (1.082) | (0)
Mean Offer price | 2.966 | 2.587 | 2.708
| (0.593) | (0.354) | (0.374)
Mean Winning price | 2.742 | 2.469 | 2.577
| (0.468) | (0.332) | (0.322)
Mean Item Quantity | 32841 | 40707 | 38022
| (9088) | (4412) | (5086)
Exist in all years | Yes | Yes | Yes
Table 1: Summary Statistics

###  5.1 Number of Bidders

Figure 2 left panel draws a boxplot of the number of bids from large bidders
and from small bidders for each item and is grouped by set-aside percentage.
Figure 2 middle panel draws a jittered scatter plot of the share of quantity
won by large and small bidders in each auction and product type, also grouped
by set-aside percentage. Figure 2 right panel visualizes the percentage of
total quantities won by large and small bidders grouped by set-asides. First,
we look at products with no set-asides and see very few large and small
bidders. Their winning shares in each auction and product type are both spread
out, but small bidders win much more often and get 95% of total quantities in
general. Then for products with 50% set-asides, there are more large bidders
and a lot more small bidders. Their winning shares are pushed to different
sides of the 50% set-asides bar, with occasional exceptions due to lack of
small bidders’ offers. Small bidders continue to dominate the auction, winning
87% of total quantities. Lastly, we see a few small bidders for products with
100% set-asides, and they win all quantities unless they can’t supply that
much. In general, we see that small bidders win a large share of quantities in
any set-asides. It implies that even if it is the case that set-asides
increase the probability of a small bidder winning, such increase on average
cannot be large because small bidders are already doing well. In Section VI we
will link this idea with our findings on bidders’ price changes in Section 5.2
and 5.3 to comment on small businesses’ potential revenue change.

Figure 2: Number of bidders and Share of Winning

To quantify the effect of set-asides, we estimate a weighted least squares
model of the number of small or large bidders for each item on set-aside
percentage, bidder type, their interaction, and other covariates (equation 5).
For each item we count the number of large bidders and the number of small
bidders to generate two observations in the response variable. We include
bidder type as a covariate to indicate which type of bidder is being counted
in the response variable. Because the effect of set-aside is expected to be
nonlinear, set-aside percentage is considered as a categorical variable in all
regression models. We also control for demand level, which is computed as the
total quantity of the same product in million pounds requested in that
solicitation. Demand is taken into consideration because large orders may
attract more bidders who want a slice of the cake, and a quadratic term of
demand is included to capture potential non-linearity. Product and year999Year
is treated as a categorical variable in all regression models so that we don’t
assume a linear time trend. fixed effects are controlled for to capture the
intrinsic differences in competition level for different products, as well as
fluctuation in bidders’ activity. Lastly, an indicator variable SDVOSB
considers whether a 10% service disabled veteran owned small business set-
aside applies to the product.101010Unfortunately we do not have data on
service disabled veteran owned small businesses status. A web search suggests
that two vendors might be veteran-owned today, but historical records are
unavailable. Because 10% is relatively small, we decide to include an
indicator variable of SDVOSB in all regression models. SDVOSB is not additive
to the set-asides we are concerned about, in the sense that SDVOSB bidders
count toward both set-aside quota.

$\displaystyle\text{Num of bidders}\sim$ $\displaystyle\text{ Set-
Aside}+\text{Bidder Type}+\text{Set-Aside $\times$ Bidder Type}$
$\displaystyle+\text{Demand}+\text{Demand}^{2}+\text{Product}+\text{year}+\text{SDVOSB}$
(5)

For each observation in the sample, we use the quantity of that item as its
weight in the model. The quantity of most items is the weight of a standard
truckload, usually 42,000 lbs, 40,000 lbs or 38,000 lbs depending on the
product. However, some items in the data appears as less than a unit truckload
for administrative reasons111111For example, an auction may request half a
truck of ground beef to be delivered to one place and another half to a
different place in the same city. They would appear as two half-truck items in
the data, but in practice bidders are required to treat them as a bundle and
submit only one price. Using item quantity as weight perfectly “merges” them
into one standard truckload item., so we use quantity as weights to equally
represent every truckload. Table 2 reports the results using
heteroskedasticity-consistent robust standard errors.121212Coefficients on
commodity fixed effect is left out. The first two columns are results from
weighted least squares and the last two columns are those from ordinary least
squares without weighting. OLS coefficients are close to WLS coefficients, and
we’ll only discuss the latter in detail. The baseline of the interaction terms
is the number of large bidders in no set-asides setting. On average there are
1.4 more small bidders than large bidders for items without set-asides. Items
with a 50% set-aside are associated with a 1.3 decrease in the number of large
bidders and a 1.9 increase in the number of small bidders. Recall from Figure
2 that the median number of large and small bidders in this category is 1 and
6. Items with a 100% set-aside are associated with a 0.11 increase in the
number of small bidders. The smaller magnitude of absolute increase compared
to 50% set-asides is likely because the latter is more competitive to begin
with. All the coefficients are significant and the signs are intuitive.
Implementing set-asides discourages large bidders from participating likely
because there’s a lower chance of winning the auction and the set-asides
attract small bidders who are advantaged. Demand ranges from 0.038 to 13.272
million pounds. Within this range, demand predicts an increase in the number
of bidders of any type ranging from 0 to 1.1.

| WLS | OLS
---|---|---
| Coefficient | Robust Error | Coefficient | Robust Error
Constant | 0.6060 | (0.0353) | 0.6664 | (0.0344)
Small | 1.4050 | (0.0272) | 1.3373 | (0.0255)
SA50%, Large | -1.3251 | (0.0879) | -1.3592 | (0.0852)
SA50%, Small | 1.8804 | (0.0865) | 1.9113 | (0.0846)
SA100%, Small | 0.1142 | (0.0395) | 0.1582 | (0.0366)
Demand | 0.1506 | (0.0133) | 0.1479 | (0.0127)
Demand2 | -0.0050 | (0.0010) | -0.0046 | (0.0010)
SDVOSB | 0.4747 | (0.0696) | 0.4843 | (0.0689)
$n=26169$ | $R^{2}=0.8134$ | $R^{2}=0.8155$
Table 2: Number of Large or Small Bidders

###  5.2 Offer Price

Next we study the association between set-asides and offer prices submitted by
large and small bidders. We use a weighted least squares model of the log of
offer price regressed on set-aside percentage, bidder type, their interaction
and other covariates (equation 6). We include demand for the product and total
number of bidders for each item as covariates because prices should be
affected by demand and supply relationship. Although bidders do not know the
actual number of competitors when they submit their bids, they should have a
good estimate of it because auctions are highly repetitive and the pool of
qualified vendors is quite stable. Quadratic terms are used to capture non-
linearity.

Because the data spans five years and the market price of beef products
fluctuates, we introduce two other price data published by the USDA. One is
monthly data of the wholesale value of beef. Because the USDA auctions are
comparable to the wholesale process which also sells at large quantities to
retailers, we match the log of beef wholesale value with auction items by year
and month, and use it as a proxy for the opportunity cost to sell outside of
the USDA auctions. The other information is the 2018-2019 annual market price
of USDA products. As mentioned in the data section, we will consider USDA
products that only differ in package size as the same product. Considering
that package size affects cost, we use a product fixed effect, the log of 2019
USDA price, and a package indicator altogether to capture intrinsic price
differences among products. Last, we use vendor fixed effect to control for
the efficiency differences among bidders, and a SDVOSB indicator of service-
disabled veteran owned small business set-aside.

$\displaystyle\text{log(Offer Price)}\sim$ $\displaystyle\text{
SA}+\text{Bidder Type}+\text{SA $\times$ Bidder
Type}+\text{Demand}+\text{Demand}^{2}$
$\displaystyle+\text{nBid}+\text{nBid}^{2}+\text{log(Monthly Wholesale Beef
Price)}+\text{Product}$ $\displaystyle+\text{log(2019 USDA
price)}+\text{Vendor}+\text{Package}+\text{SDVOSB}$ (6)

Table 3 reports the results with heteroskedasticity-consistent robust standard
errors. OLS coefficients are close to WLS coefficients which we’ll discuss in
detail. Taking the exponential of the coefficients gives the corresponding
percentage change in offer price. A positive estimate of small bidder
indicator suggests that for items without set-asides, the average offer price
from small bidders is 6.5% higher than those from large bidders.
Interestingly, for items with 50% set-aside, the offer prices of large bidders
are expected to drop by 13.6% on average, suggesting that large bidders bid
more aggressively in reaction to their lower chance of winning. Similarly, 50%
set-aside predicts a larger drop of 21.7% in the offer prices of small bidders
on average. The decrease in offer prices among small bidders could be a
response to large bidders’ aggressive bidding. As we saw in the simplified
behavioral model, in a 50% set-aside setting only the cheaper small bidders
actually benefit from set-asides. In addition, extra uncertainty is introduced
to the allocation outcome in auctions with partial set-asides. Without set-
asides, a small bidder who wants to sell her only truck of beef to a close
destination only needs to be the lowest price for the targeted item. However,
with a 50% set-aside, this bidder wins if and only if the sum of her price and
other bidders’ low prices for all other items of the same product is the
cheapest for the USDA.

For items with a 100% set-aside, offer prices of small bidders increase by
3.2%, which is expected because there is no more competition from large
bidders. The relatively small magnitude is also understandable because there
are few large bidders compared to small bidders in USDA auctions. Total demand
for a product predicts a roughly linear increase in offer prices that ranges
from 0.06% to 12%. Number of bidders predicts a decrease in offer price as
expected, because competition will drive down prices in first price auctions.
A negative coefficient of the quadratic term implies that the impact of an
additional bidder gets weaker when there are more bidders to begin with, which
aligns with the results in MacDonald . (2002). The model predicts that adding
one more bidder for a typical item is expected to bring down the offer price
by about 3%. All the coefficients discussed above are significant.

| WLS | OLS
---|---|---
| Coefficient | Robust Error | Coefficient | Robust Error
Constant | -1.2210 | (0.0347) | -1.2399 | (0.0339)
Small | 0.0630 | (0.0173) | 0.0566 | (0.0163)
SA50%, Large | -0.1457 | (0.0200) | -0.1625 | (0.0187)
SA50%, Small | -0.2443 | (0.0092) | -0.2533 | (0.0083)
SA100%, Small | 0.0318 | (0.0069) | 0.0354 | (0.0068)
Demand | 0.0172 | (0.0005) | 0.0166 | (0.0005)
Demand2 | -0.0006 | (0.0000) | -0.0006 | (0.0000)
Number of bidders | -0.0310 | (0.0013) | -0.0309 | (0.0013)
Number of bidders2 | 0.0013 | (0.0001) | 0.0013 | (0.0001)
log(USDA price) | 1.6662 | (0.0301) | 1.7051 | (0.0299)
log(wholesale price) | 0.7936 | (0.0048) | 0.7947 | (0.0047)
SDVOSB | 0.0322 | (0.0016) | 0.0330 | (0.0016)
$n=96474$ | $R^{2}=0.3985$ | $R^{2}=0.4058$
Table 3: Log of Offer Price

###  5.3 Winning Price

In this section we study the association between set-asides and winning prices
among large and small bidders using the same weighted least squares model as
in equation 6. The only differences are that we use the log of winning price
as the response variable and restrict the data to winning bids.

Table 4 reports the results with heteroskedasticity-consistent robust standard
errors. Again, OLS coefficients are close to WLS coefficients which we’ll
discuss in detail. Taking the exponential of the coefficients gives the
corresponding percentage change in winning prices. Compared to Table 3, the
coefficient of small bidder indicator is now insignificant, suggesting that
for items without set-asides, small business winners and large business
winners win at the same price, which is an outcome of standard first price
auction. For items with a 50% set-aside, large winners and small winners are
still displaying price decreases, with small bidders dropping more. 50% set-
aside predicts a 7% and 12.5% decrease on average in large and small bidders’
winning prices respectively. These coefficients have a smaller magnitude than
those in the offer price model, potentially because there’s less room for
price adjustment on low prices and net profit drops faster at low prices. For
items with a 100% set-aside, small bidders’ winning prices are expected to
increase by 2.1%. Total demand for a product is associated with a roughly
linear increase in winning prices, ranging from 0.09% to 16.8%. Number of
bidders predicts a decrease in winning prices as expected, with a similar
nonlinear trend as in the offer price model. The model predicts that adding
one more bidder for a typical item is expected to bring down the winning price
by about 3%. All the coefficients discussed above are significant unless
otherwise noted.

| WLS | OLS
---|---|---
| Coefficient | Robust Error | Coefficient | Robust Error
Constant | -1.0725 | (0.0814) | -1.1516 | (0.0803)
Small | 0.0204 | (0.0210) | 0.0064 | (0.0183)
SA50%, Large | -0.0723 | (0.0284) | -0.0579 | (0.0243)
SA50%, Small | -0.1338 | (0.0234) | -0.1022 | (0.0192)
SA100%, Small | 0.0212 | (0.0090) | 0.0247 | (0.0090)
Demand | 0.0243 | (0.0013) | 0.0233 | (0.0013)
Demand2 | -0.0009 | (0.0001) | -0.0008 | (0.0001)
Number of bidders | -0.0324 | (0.0029) | -0.0305 | (0.0029)
Number of bidders2 | 0.0012 | (0.0002) | 0.0011 | (0.0002)
log(USDA price) | 1.3396 | (0.0816) | 1.4042 | (0.0820)
log(wholesale price) | 0.7956 | (0.0121) | 0.7949 | (0.0120)
SDVOSB | 0.0290 | (0.0023) | 0.0301 | (0.0022)
$n=12677$ | $R^{2}=0.4501$ | $R^{2}=0.4471$
Table 4: Log of Winning Price

##  6 Discussion

We have shown that 50% set-asides in USDA procurement auctions are associated
with a decrease in prices among large and small bidders. For small bidders in
particular, the price drop can be as large as 21.7% and 12.5% in their offer
prices and winning prices respectively, and it immediately follows that their
price margins will decrease by a larger percentage. Recall from Figure 2 right
panel that plots the share of quantities won by small bidders in total,
there’s little room for improvement regarding small businesses’ overall
probability of winning. Putting these evidences together, it’s very unlikely
that 50% set-asides would result in an increase in small bidders’ net profit
in general, but it’s still possible that some small businesses could benefit
at the cost of other small and large businesses.

A potential issue with our regressions is that the numbers of observations
associated with different product types are unbalanced, which may lead to
over-representation of certain products and under-representation of the
others. To resolve this concern, Appendix A replicates Table 2, 3 and 4 using
a different weight in the least squares model. The new weight is such that
within each product type, item weights are proportional to their quantity. We
then scale the weights in each product type by the total quantity of that
product, resulting in equal total weights for all product types. Standard
errors are larger but significance level do not change. Regression
coefficients change moderately in scale, but their sign and relative magnitude
are consistent with Table 2, 3 and 4, suggesting that we have effectively
controlled for differences among products and our predictions are not biased
by product heterogeneity.

Looking forward, there are several questions that can be explored in future
work. The first question is whether some small businesses, potentially the
cheaper and more developed ones, are benefiting from USDA’s set-aside rules at
the cost of sacrificing higher-cost small businesses. Set-asides focus on
selecting small businesses in the lower price quantile to meet a quota, which
might not benefit the most disadvantaged small businesses in practice. Flat
rate or percentage subsidy is a possible alternative to consider since the
extent to which small bidders benefit upon winning is relatively fixed.
Another question is whether bidders differ in bidding strategy not only
because of business type but also because some bidders are more experienced
and have more resources dedicated to implementing the optimal strategy.
Strategizing in such a complicated auction could be challenging to newcomers
and small firms. A question immediately follows that our models treat bidders
at all time equally, though it’s possible that they learn and adapt their
strategies through these highly repetitive sequential auctions. We hope to
engage in some of these questions in future research.

##  7 Conclusion

This paper studies the implications of small business set-asides policy in
USDA food procurement auctions, discerning partial set-asides and full set-
asides. Using five years of bid prices data on different beef products
procured by the USDA, we’ve shown that 50% set-aside is associated with a
decrease in offer prices and winning prices among both large and small
bidders. In a setting like the USDA where entry of new bidders is difficult,
applying set-asides has little power in attracting more small bidders, yet
large bidders are pushed by unfavorable policy to bid aggressively, and then
small bidders are also affected. In addition, partial set-asides increase the
uncertainty faced by all bidders because allocation of an individual item will
depend on all items of the same product type. In 100% set-asides, we do
observe a price increase among small bidders likely because competition is
relieved. But the magnitude is small because there are few potential large
competitors anyway. The impact of set-asides policies rely on various aspects
of the setting of the auction, and agencies need case-by-case investigation to
work out the optimal policy.

## References

  * Athey . (2011) athey2011Athey, S., Coey, D. Levin, J.  2011\.  Set-asides and subsidies in auctions Set-asides and subsidies in auctions.  American Economic Journal: Microecomics511-27.
  * MacDonald . (2002) macdonald2002MacDonald, JM., Handy, CR. Plato, GE.  2002\.  Competition and Prices in USDA Commodity Procurement Competition and prices in usda commodity procurement.  Southern Economic Journal691128-143.
  * Nakabayashi (2013) nakabayashi2013Nakabayashi, J.  2013\.  Small business set-asides in procurement auctions: An empirical analysis Small business set-asides in procurement auctions: An empirical analysis.  Journal of Public Economics10028-44.

##  Appendix A Replicate Table 2, 3 and 4 equally weighting product types

| WLS
---|---
| Coefficient | Robust Error
Constant | 0.3995 | (0.0322)
Small | 1.5193 | (0.0288)
SA50%, Large | -0.7710 | (0.0919)
SA50%, Small | 1.8050 | (0.0904)
SA100%, Small | 0.1864 | (0.0381)
Demand | 0.2259 | (0.0163)
Demand2 | -0.0083 | (0.0011)
SDVOSB | 0.4870 | (0.0584)
$n=26169$ | $R^{2}=0.8197$
Table 5: Replicating Number of Bidders Model (Table 2) | WLS
---|---
| Coefficient | Robust Error
Constant | -0.8150 | (0.0388)
Small | 0.0212 | (0.0109)
SA50%, Large | -0.1802 | (0.0142)
SA50%, Small | -0.2392 | (0.0089)
SA100%, Small | 0.0124 | (0.0067)
Demand | 0.0148 | (0.0007)
Demand2 | -0.0005 | (0.0000)
Number of bidders | -0.0324 | (0.0021)
Number of bidders2 | 0.0013 | (0.0001)
log(USDA price) | 1.3773 | (0.0344)
log(wholesale price) | 0.7357 | (0.0090)
SDVOSB | 0.0211 | (0.0018)
$n=96474$ | $R^{2}=0.4633$
Table 6: Replicating Log of Offer Price Model (Table 3) | WLS
---|---
| Coefficient | Robust Error
Constant | -1.2423 | (0.2137)
Small | -0.0099 | (0.0185)
SA50%, Large | -0.0527 | (0.0279)
SA50%, Small | -0.1058 | (0.0253)
SA100%, Small | 0.0197 | (0.0113)
Demand | 0.0246 | (0.0039)
Demand2 | -0.0011 | (0.0002)
Number of bidders | -0.0245 | (0.0069)
Number of bidders2 | 0.0009 | (0.0005)
log(USDA price) | 1.7384 | (0.2090)
log(wholesale price) | 0.6616 | (0.0390)
SDVOSB | 0.0157 | (0.0039)
$n=12677$ | $R^{2}=0.6129$
Table 7: Replicating Log of Winning Price Model (Table 4)


# Superfluidity in neutrino clusters

Maxim Dvornikov
Pushkov Institute of Terrestrial Magnetism, Ionosphere
and Radiowave Propagation (IZMIRAN),
108840 Moscow, Troitsk, Russia; and
Bogoliubov Laboratory of Theoretical Physics
Joint Institute for Nuclear Research,
141980 Dubna, Moscow region, Russia<EMAIL_ADDRESS>
###### Abstract

We study the formation of a superfluid condensate of neutrinos inside a
neutrino cluster. The attractive interaction between neutrinos is mediated by
a scalar boson which is lighter than a neutrino. We consider the appearance of
neutrino bound states consisting of particles with oppositely directed spins.
The gap equation for such a system is derived. We find that the relativistic
cosmic neutrino background can form the superfluid condensate inside a
neutrino cluster having realistic characteristics.

##  1 Introduction

The search of astronomical objects consisting of new particles, e.g., forming
dark matter, is a problem for modern astrophysics [1]. Star like objects
composed of bosons, e.g., of axions, which can be candidates for dark matter,
are discussed in Refs. [2, 3]. Virialized axions can form inhomogeneous
structures called miniclusters [4].

Neutrinos, especially sterile ones, can also contribute to dark matter [5].
The formation of spatially localized objects, called neutrino clusters,
consisting of neutrinos was studied in Refs. [6, 7]. Neutrinos are supposed in
Refs. [6, 7] to be confined in a cluster by the exchange of a light scalar
particle, which is beyond the standard model. Note that the possibility for
neutrinos to form bound states inside astrophysical objects even in frames of
the standard model was discussed in Ref. [8]. The neutrino trapping inside the
rapidly rotating matter of a neutron star was studied in Refs. [9, 10]. The
latter phenomenon is also supposed to take place in frames of the standard
model. The neutrino interaction by $Z$ and $W$ bosons with rotating background
fermions was considered in Refs. [9, 10] in the forward scattering
approximation.

The neutrino interaction through a scalar particle was shown, e.g., in Ref.
[11] to be attractive. Such interaction was suggested in Ref. [12] to lead to
the formation of bound states of neutrinos having opposite chiralities,
analogously to Cooper pairs of electrons in a metal. Assuming that the scalar
particle is the Higgs boson, the neutrino superfluidity was studied in Ref.
[12]. The superfluidity of neutrinos was found to take place in a neutron star
only if neutrinos are heavier than the Higgs boson. Thus, such neutrinos are
beyond the standard model. The possible phenomenological consequences of the
neutrino superfluidity are studied in Ref. [13].

Contrary to Ref. [12], we can consider the interaction between usual active
standard model neutrinos mediated by a light scalar boson to study the
superfluidity in a neutrino cluster discussed in Ref. [7]. The existence of
such light bosons was explored, e.g., in Ref. [14]. The study of the neutrino
superfluidity in a system with particles having analogous properties was
carried out in Ref. [15].

The present work is organized as follows. We start in Sec. 2 with the
description of the formation of the neutrino condensate. Then, in Sec. 3, we
derive the gap equation. The application of our results to the neutrino
superfluidity in a cluster is analyzed in Sec. 4. Finally, in Sec. 5, we
conclude.

##  2 Neutrino condensation

We start with the Lagrangian for the single neutrino $\nu$ interacting with
the scalar field $\phi$,

$\mathcal{L}=\bar{\nu}(\mathrm{i}\gamma^{\mu}\partial_{\mu}-m)\nu+\frac{1}{2}(\partial_{\mu}\phi\partial^{\mu}\phi-M^{2}\phi^{2})+g\phi\bar{\nu}\nu,$
(2.1)

where $m$ and $M$ are the masses of a neutrino and a scalar, and $g$ is the
coupling constant. We assume that $\nu$ is an electron neutrino for
definiteness. Writing down the field equation for $\phi$ and assuming that it
is independent of the coordinates, $\partial_{\mu}\phi=0$, we get that
$\phi=\frac{g}{M^{2}}\bar{\nu}\nu.$ Returning this $\phi$ back to Eq. (2.1),
we obtain the nonlinear neutrino Lagrangian in the form,

$\mathcal{L}=\bar{\nu}(\mathrm{i}\gamma^{\mu}\partial_{\mu}-m)\nu+\frac{g^{2}}{2M^{2}}(\bar{\nu}\nu)(\bar{\nu}\nu),$
(2.2)

where we include the additional combinatorial factor $1/2$ in the interaction
term.

Representing the neutrino bispinor as $\nu^{\text{T}}=(\varphi,\chi)$, where
$\varphi$ and $\chi$ are the two component spinors, we get that

$\bar{\nu}\nu=\frac{2m}{\varepsilon+m}\varphi^{\dagger}\varphi,$ (2.3)

where $\varepsilon=\sqrt{p^{2}+m^{2}}$ is the energy of a noninteracting
neutrino. If we neglect antineutrinos, we decompose $\varphi$ into the spin
states

$\varphi=\varphi_{+}+\varphi_{-},\quad\varphi_{\pm}=\frac{1}{\sqrt{\mathcal{V}}}\sum_{\mathbf{p}}u_{\pm}(\mathbf{p})b_{\pm}(\mathbf{p})e^{-i\varepsilon
t+i\mathbf{px}},$ (2.4)

where $u_{\pm}(\mathbf{p})$ are the basis spinors, $b_{\pm}(\mathbf{p})$ are
the anticommuting annihilation operators, and $\mathcal{V}$ is the
normalization volume. We can take that

$u_{+}(\mathbf{p})=\left(\begin{array}[]{c}1\\\ 0\end{array}\right),\quad
u_{-}(\mathbf{p})=\left(\begin{array}[]{c}0\\\ 1\end{array}\right),$ (2.5)

if we choose $z$ as the spin quantization axis.

The neutrino condensation in frames of the standard model mediated by the
Higgs boson exchange was studied in Ref. [12]. Chiral projections of neutrino
wavefunctions are inherent in the standard model. It was the motivation of
Ref. [12] to study the condensation of the chiral neutrino projections. In our
work, we consider the interaction mediated by the scalar field $\phi$ which is
beyond the standard model. That this why, as in the usual
Bardeen–Cooper–Schrieffer (BCS) superconductivity mechanism (see, e.g., Ref.
[16]), we suppose that the neutrino condensate is formed by particles with
opposite spins,

$\left\langle\varphi_{-a}(\mathbf{p})\varphi_{+b}(\mathbf{p})\right\rangle=\epsilon_{ab}D.$
(2.6)

where $a,b=1,2$ are the spinor indexes, $D$ is the scalar parameter to be
determined later, and $\epsilon_{ab}$ is the 2D Levi-Civita symbol.

Using Eqs. (2.3) and (2.4), as well as applying the Wick theorem, we transform
the term in the interaction Lagrangian to

$\displaystyle(\bar{\nu}_{\mathbf{p}}\nu_{\mathbf{p}})(\bar{\nu}_{\mathbf{p}}\nu_{\mathbf{p}})\to$
$\displaystyle\frac{8m^{2}}{\left(\varepsilon+m\right)^{2}}\epsilon_{ab}$
$\displaystyle\times\left[D^{*}\left(\varphi_{+a}\varphi_{-b}+\varphi_{-a}\varphi_{+b}\right)-D\left(\varphi_{+a}^{\dagger}\varphi_{-b}^{\dagger}+\varphi_{-a}^{\dagger}\varphi_{+b}^{\dagger}\right)\right].$
(2.7)

We use Eq. (2.7) in the interaction Hamiltonian
$H_{\mathrm{int}}=\int\mathrm{d}^{3}x\mathcal{H}_{\mathrm{int}}$, where
$\mathcal{H}_{\mathrm{int}}=-\mathcal{L}_{\mathrm{int}}$, which takes the
form,

$\displaystyle H_{\mathrm{int}}=$
$\displaystyle-\frac{4g^{2}}{M^{2}}\sum_{\mathbf{p}}\frac{m^{2}}{\left(\varepsilon+m\right)^{2}}$
$\displaystyle\times\Big{[}D^{*}e^{-2\mathrm{i}\varepsilon
t}\left(b_{+}(\mathbf{p})b_{-}(-\mathbf{p})-b_{-}(\mathbf{p})b_{+}(-\mathbf{p})\right)$
$\displaystyle+De^{2\mathrm{i}\varepsilon
t}\left(b_{-}^{\dagger}(\mathbf{p})b_{+}^{\dagger}(-\mathbf{p})-b_{+}^{\dagger}(\mathbf{p})b_{-}^{\dagger}(-\mathbf{p})\right)\Big{]},$
(2.8)

where we use Eqs. (2.4) and (2.5). The total Hamiltonian
$H=H_{0}+H_{\mathrm{int}}$ should contain the noninteracting term,

$H_{0}=\sum_{\mathbf{p}}(\varepsilon-\mu)\left[b_{+}^{\dagger}(\mathbf{p})b_{+}(\mathbf{p})+b_{-}^{\dagger}(\mathbf{p})b_{-}(\mathbf{p})\right],$
(2.9)

where $\mu$ is the chemical potential.

We suppose that $D=|D|e^{2\mathrm{i}\alpha}$, where $\alpha$ is the phase.
Making the Bogoliubov transformations

$\displaystyle c_{+}(\mathbf{p})$ $\displaystyle=e^{-\mathrm{i}\varepsilon
t-\mathrm{i}\alpha}\cos\frac{\theta}{2}b_{+}(\mathbf{p})+e^{\mathrm{i}\varepsilon
t+\mathrm{i}\alpha}\sin\frac{\theta}{2}b_{-}^{\dagger}(-\mathbf{p}),$
$\displaystyle c_{-}^{\dagger}(-\mathbf{p})$
$\displaystyle=-e^{-\mathrm{i}\varepsilon
t-\mathrm{i}\alpha}\sin\frac{\theta}{2}b_{+}(\mathbf{p})+e^{\mathrm{i}\varepsilon
t+\mathrm{i}\alpha}\cos\frac{\theta}{2}b_{-}^{\dagger}(-\mathbf{p}),$ (2.10)

and

$\displaystyle c_{-}(\mathbf{p})$ $\displaystyle=e^{-\mathrm{i}\varepsilon
t-\mathrm{i}\alpha}\cos\frac{\theta}{2}b_{-}(\mathbf{p})-e^{\mathrm{i}\varepsilon
t+\mathrm{i}\alpha}\sin\frac{\theta}{2}b_{+}^{\dagger}(-\mathbf{p}),$
$\displaystyle c_{+}^{\dagger}(-\mathbf{p})$
$\displaystyle=e^{-\mathrm{i}\varepsilon
t-\mathrm{i}\alpha}\sin\frac{\theta}{2}b_{-}(\mathbf{p})+e^{\mathrm{i}\varepsilon
t+\mathrm{i}\alpha}\cos\frac{\theta}{2}b_{+}^{\dagger}(-\mathbf{p}),$ (2.11)

where $c_{\pm}(\mathbf{p})$ are the new operators, we bring $H$ to the form,

$\displaystyle H=$
$\displaystyle\sum_{\mathbf{p}}\Big{\\{}\left[(\varepsilon-\mu)\cos\theta+\Delta\sin\theta\right]\left(c_{+}^{\dagger}(\mathbf{p})c_{+}(\mathbf{p})+c_{-}^{\dagger}(\mathbf{p})c_{-}(\mathbf{p})\right)$
$\displaystyle+\left[\Delta\cos\theta-\sin\theta(\varepsilon-\mu)\right]\left[c_{+}^{\dagger}(\mathbf{p})c_{-}^{\dagger}(-\mathbf{p})+c_{-}(-\mathbf{p})c_{+}(\mathbf{p})\right]$
$\displaystyle+\varepsilon-\mu)(1-\cos\theta)-\Delta\sin\theta\Big{\\}},$
(2.12)

where

$\Delta=\frac{8g^{2}m^{2}|D|}{M^{2}\left(\varepsilon+m\right)^{2}},$ (2.13)

is the magnitude of the energy gap.

Then, we take that

$\sin\theta=\frac{\Delta}{E},\quad\cos\theta=\frac{(\varepsilon-\mu)}{E},\quad\tan\theta=\frac{\Delta}{\varepsilon-\mu},$
(2.14)

where $E=\sqrt{(\varepsilon-\mu)^{2}+\Delta^{2}}$. In this situation, the
Hamiltonian in Eq. (2) takes the canonical form,

$\displaystyle H=$
$\displaystyle\sum_{\mathbf{p}}E\left(c_{+}^{\dagger}(\mathbf{p})c_{+}(\mathbf{p})+c_{-}^{\dagger}(\mathbf{p})c_{-}(\mathbf{p})\right),$
(2.15)

where one can see that $E$ is the quasiparticle energy. Note that we discard
the part of the Hamiltonian in Eq. (2.15) which does not contain operators.

##  3 Gap equation

To determine the gap in Eq. (2.13) we multiply Eq. (2.6) by $\epsilon_{ab}$,
average it over the ground state $\left\langle\dots\right\rangle_{\text{GS}}$,
and integrate over space. Eventually, we get that

$\displaystyle\mathcal{V}D=$
$\displaystyle-\frac{e^{2\mathrm{i}\alpha}}{4}\sum_{\mathbf{p}}\sin\theta\left\langle
c_{+}^{\dagger}(\mathbf{p})c_{+}(\mathbf{p})+c_{-}^{\dagger}(\mathbf{p})c_{-}(\mathbf{p})-1\right\rangle_{\text{GS}},$
(3.1)

where we use Eqs. (2.4), (2.5), (2.10), and (2.11). Then, we notice that

$\left\langle
c_{+}^{\dagger}(\mathbf{p})c_{+}(\mathbf{p})+c_{-}^{\dagger}(\mathbf{p})c_{-}(\mathbf{p})\right\rangle_{\text{GS}}=2n_{f},$
(3.2)

where $n_{f}=1/(e^{E/T}+1)$ is the Fermi-Dirac distribution of quasiparticles
and $T$ is the neutrino gas temperature. Here, we account for two
polarizations of quasiparticles. It is also convenient to express
$2n_{f}-1=-\tanh(E/2T)$.

Based on the the fact that

$\sin\theta=\frac{4g^{2}m^{2}|D|}{M^{2}\left(\varepsilon+m\right)^{2}}\frac{1}{\sqrt{(\varepsilon-\mu)^{2}+\Delta^{2}}},$
(3.3)

and replacing the sum over $\mathbf{p}$ with the 3D integration,
$\frac{1}{\mathcal{V}}\sum_{\mathbf{p}}=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}$,
we obtain the gap equation,

$\frac{2g^{2}}{M^{2}}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{m^{2}}{\left(\varepsilon+m\right)^{2}}\frac{\tanh(E/2T)}{\sqrt{(\varepsilon-\mu)^{2}+\Delta^{2}}}=1.$
(3.4)

Note that Eq. (3.4) contains the factor $\left(\varepsilon+m\right)^{2}$ in
the denominator of the integrand versus $\varepsilon^{2}$ in Ref. [12]. It is
the consequence of the condensation of spin states in our case rather than
chiral states considered in Ref. [12].

We suppose that the neutrino gas is degenerate, $T\ll E$. The difference
between the energy spectra of quasiparticles and unpaired neutrinos is the
most significant near the Fermi surface [16]. That is why we modify the factor
in the integrand

$\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{m^{2}}{\left(\varepsilon+m\right)^{2}}=N_{0}\mathrm{d}\varepsilon,\quad
N_{0}=\frac{p_{\mathrm{F}}^{2}v_{\mathrm{F}}m^{2}}{2\pi^{2}\left(\mu+m\right)^{2}},$
(3.5)

where $p_{\mathrm{F}}$ is the Fermi momentum and
$v_{\mathrm{F}}=\frac{\mathrm{d}p}{\mathrm{d}\varepsilon}=\frac{\mu}{p_{\mathrm{F}}}$
is the Fermi velocity. Note that $N_{0}$ is the constant factor and can be
removed out of the integral.

We change the variable $x=\varepsilon-\mu$ in the remaining integral. The
resulting range of integration is $m-\mu<x<\Lambda$, where $\Lambda$ is the
cut-off since the integral diverges at the upper limit. Thus, Eq. (3.4) takes
the form,

$GN_{0}\int_{0}^{\Lambda}\mathrm{d}x\frac{1}{\sqrt{x^{2}+\Delta^{2}}}=1,$
(3.6)

where $G=2g^{2}/M^{2}$. We assume that $\Lambda\gg\max(\Delta,|m-\mu|)$ in Eq.
(3.6). Based on Eq. (3.6), we find the energy gap,

$\Delta=2\Lambda e^{-x},\quad
x=\frac{\pi^{2}M^{2}(\mu+m)^{2}}{g^{2}m^{2}p_{\mathrm{F}}\mu}.$ (3.7)

The phase transition temperature can be found using the standard technique
(see, e.g., Ref. [16]) by solving the nonlinear equation,

$\int_{0}^{\infty}\mathrm{d}x\left[\frac{\tanh(x/2)}{x}-\frac{1}{\sqrt{x^{2}+(\Delta/T_{c})^{2}}}\right]=0.$
(3.8)

The numerical calculation gives one that $T_{c}\approx 0.57\Delta$.

##  4 Superfluidity of clustered neutrinos

The formation of clusters of neutrinos owing to the exchange by a new light
scalar boson was studied in Ref. [7]. The size of clusters can range from km
to Mpc. The maximal neutrino density can be $\sim 10^{9}\,\text{cm}^{-3}$ in
the center of a cluster. Such density corresponds to the neutrino mass
$m=0.1\,\text{eV}$, which is taken as a reference mass in Ref. [7]. It should
be noted that neutrinos in a cluster are considered as relativistic particles.
Thus, we can set $p_{\mathrm{F}}=\mu$ and $v_{\mathrm{F}}=1$. We take the
coupling constant $g=10^{-2}$ to form a cluster.

The chemical potential for the degenerate relativistic neutrino gas is
$\mu=(3\pi^{2}n_{\nu})$. Note that, here, we account for the two polarizations
of neutrinos since the interaction via the scalar boson mixes the neutrino
helicities even when particles are relativistic.

One can see in Eq. (3.7) that the gap depends on the cut-off $\Lambda$, which
is a free parameter. The cut-off is typically associated with the Debye
frequency in the BCS superconductivity. However, it is not applicable in our
case since neutrinos are electrically neutral particles. We can rely on
neutrons superfluidity in dense nuclear matter developed, e.g., in Ref. [17],
where $\Lambda=\lambda p_{\mathrm{F}},$where $\lambda$ is the fitting factor.
This factor is taken $\lambda=M/m$ in Ref. [17]. Since $|m-\mu|\approx
p_{\mathrm{F}}$ for relativistic neutrinos, we assume that $\lambda\gg 1$ to
safely set the lower limit in the integral to zero in Eq. (3.6).

The important quantity characterizing a superfluid state is the coherence
length [16], $\xi=\frac{v_{\mathrm{F}}}{\pi\Delta}$, which is
$\xi=(\pi\Delta)^{-1}$ for relativistic neutrinos. It is the geometrical size
of a Cooper pair of fermions. It is clear that $\xi$ should be smaller than
the length scale of a system.

The parameter $x$ in Eq. (3.7) is $x=\frac{\pi^{2}M^{2}}{g^{2}m^{2}}$ for
relativistic neutrinos. Assuming that [7] $\tfrac{M}{gm}=0.1$, we obtain that
$x\approx 0.1$. Taking the neutrino density [7]
$n_{\nu}=10^{9}\,\text{cm}^{-3}$, we get that $\mu=3.1\times
10^{3}\,\text{cm}^{-1}=713\,\text{K}$. Then,

$\displaystyle\Delta$ $\displaystyle=1.3\times
10^{3}\lambda\,\text{K}=5.6\times 10^{3}\lambda\,\text{cm}^{-1},$
$\displaystyle T_{c}$ $\displaystyle=7.4\times
10^{2}\lambda\,\text{K},\quad\xi=5.7\times 10^{-5}\lambda^{-1}\,\text{cm}.$
(4.1)

Even if we take that $\lambda=10$, we get that $T_{c}\gg
T_{\text{C}\nu\text{B}}$, where $T_{\text{C}\nu\text{B}}=1.95\,\text{K}$ is
the temperature of cosmic neutrino background [18]. The coherence length in
this case is $\xi<5.7\times 10^{-6}\,\text{cm}$, which is much shorter than
the typical size of a neutrino cluster predicted in Ref. [7]. It should be
noted that the neutrino energy corresponding to $T_{c}=7.4\times
10^{3}\,\text{K}$ is $\varepsilon=6.4\times 10^{-1}\,\text{eV}$, which is
greater than $m=0.1\,\text{eV}$ taken as a reference neutrino mass in Ref.
[7]. Thus neutrinos, forming the condensate, are relativistic. Greater values
of $\lambda$ increase the phase transition temperature allowing a neutrino gas
with higher temperatures to become superfluid.

Therefore, if a neutrino cluster is formed by relativistic cosmic neutrinos
and there is an interaction between neutrinos mediated by a light scalar
boson, the neutrino pairing and superfluidity is quite possible in such a
system.

##  5 Conclusion

In the present work, we have examined the possibility of the superfluidity
inside a neutrino cluster. We have considered a single neutrino eigenstate,
e.g., an electron neutrino with the mass $m=0.1\,\text{eV}$, which is close to
the experimental upper bound recently established in Ref. [19]. The attractive
interaction between neutrinos is mediated by a light scalar boson. We can take
that the coupling constant is $g=10^{-2}$ and the boson mass is
$M=10^{-4}\,\text{eV}$. Such boson properties are not ruled out by charged
pions and kaons decays studied in Ref. [14].

We have started in Sec. 2 with the description of the neutrino condensation.
Contrary to Ref. [12], we have assumed in Eq. (2.6) that the neutrino
condensate is formed by particles with oppositely directed spins like in the
usual BCS mechanism [16]. Then, in Sec. 3, we have derived Eq. (3.4) for the
energy gap which is slightly different from that in Ref. [12].

The application of the obtained results for the neutrino superfluidity in a
cluster, described in Ref. [7], has been developed in Sec. 4. We have obtained
that the relativistic neutrino gas with the temperature below $T_{c}=7.4\times
10^{3}\,\text{K}$ can form a superfluid condensate. The cosmic neutrino
background fits such a requirement. The coherence length in this case is much
shorter than the size of a neutrino cluster.

Thus, we conclude that the superfluidity of relativistic cosmic neutrinos is
quite possible in a neutrino cluster. The Earth can be situated inside such a
cluster, that is also quite probable accounting for the great cluster size
[7]. In this case one can try to check the existence of a superfluid
condensate in a future $\beta$-decay experiment [13].

## Acknowledgment

I am thankful to V. B. Semikoz for useful discussions.

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