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The evolving nature of the college wage premium
June 24, 2020
Abstract
Focusing on the signaling aspect of education, we show that the college wage
premium can be Ushaped in the share of the population with a college degree. This
prediction is consistent with empirical evidence from a range of countries. Moreover,
the equilibrium in our model is unique, which means that we are able to generate
empiricallytestable predictions linking income inequality and the premium enjoyed
by the college educated. Consequently, our model provides a framework for future
empirical studies.
Keywords: college wage premium; signaling; education beneﬁt.
JEL classiﬁcations: D82; I22; I28; J31.
1 Introduction
While some of us wistfully look back on our university days as the time of our life (and
others have never left), for most people the choice about whether or not to attend college
is a pragmatic one: is it an investment worth making? Will the returns from obtaining
a college degree in terms of the job opportunities and the extra wages it brings justify
its costs? This is a very real question for many, and particularly pertinent given the cost
of higher education and the current college debt crisis.1In this paper we explore the
relationship between the college wage premium – the extra earnings that college graduates
receive over their lifetimes – and the share of the population with a college education.
The observed relationship between the share of the population with a college education
and the college wage premium, both across country and time, is nuanced. In some countries
the relationship is positive (Australia, Finland, Korea and Switzerland to name a few),
whereas in other countries the relationship appears to be negative (such as Argentina).
Moreover, in the US the empirical evidence suggests a decline and subsequent rise in the
1The current accumulated student debt in the U.S. is nearly $1.6 trillion, estimated to rise to $2 trillion
by 2021 and to $3 trillion by 2040 (Johnson 2019).
1
college wage premium as the share of college educated increased over the 1970s and 80s.2
Established models of the college wage premium, in particular the literature following
Acemoglu (2002), rely on a skillsbiased technical change to explain an increase in any
premia for the college educated; essentially, skillsbiased technical change is a market size
eﬀect.3However, to accommodate both the decline and the rise in the college wage premium
one has to assume a signiﬁcant change in the production technology.
Here, following Spence (1973), we focus on the signaling aspect of education. Signaling
in higher education is not just a theoretical curiosity; Fang (2006), for instance, empirically
conﬁrms that signaling accounts for a signiﬁcant proportion of the college wage premium.
In our model, in the usual way, we assume that worker ability (productivity) is not directly
observed by employers, but that educational attainment is. Furthermore, as is standard
in labor economics, we assume lognormality of the distribution of abilities (productivity)
amongst workers in an economy. Under mild conditions we obtain a unique equilibrium in
which only those above a certain threshold of ability obtain a degree. In this setting, the
Ushaped relationship between the share of college educated and the college wage premium
arises quite naturally, without relying on exogenous shifts or structural changes. While the
factors other than signaling are clearly playing a role in the labor market, we are not aware
of another framework that can explain the Ushaped empirical patterns we are focusing on.
The intuition for the Ushape premium/share of the educated relationship is as follows.
When only the workers in the upper tail of the ability distribution are college educated, a
degree is a very informative signal and the resulting college wage premium is high. Con
versely, when only the bottom tail of the distribution does not have a degree, it is important
to signal that you are not from that tail, meaning that the college wage premium is again
high. However, in the intermediate case, when approximately half of the population obtains
a degree, the signal embedded in a degree is not that informative and the resulting college
wage premium is somewhat diminished. This leads to a Ushaped relationship between the
college wage premium and the share of college educated.4
In addition, our work highlights the role of the variance of the distribution of abilities.
Prior models of the college wage premium, e.g., Krugman (2000) or Hendel et al. (2005),
largely focus on a positive relationship between the premium and the share of college edu
cated. The Ushaped college wage premium obtained here implies that such a relationship
can only hold when the share of the educated in the population is large enough. We also
show that a higher variance of abilities in the population makes a positive premium/share
2See Figure 1 in Acemoglu (2002), Machin and McNally (2002) and Fernandez and Messina (2017) for
more on the empirical evidence of proportion college educated/college wage premium in various countries.
Other factors are also no doubt important. Hwang et al. (2013), for example, conducts an empirical
investigation of how unemployment rates aﬀect the humancapital returns to higher education.
3See Barros et al. (2011) for an empirical study of technical change and higher education policy in
France.
4The Ushape premium/share relationship does not hold for every distribution. A more elaborate
intuition that relies on the shape of the density of the lognormal distribution is given in Section 3.
2
relationship less likely. This observation is important in explaining the nuanced empirical
evidence from various countries presented in Section 3. Moreover, an advantage of our
model over existing signaling models of education is that it results in a unique equilib
rium. Our framework, consequently, generates empirically testable predictions, providing
potentially a workhorse model for future empirical studies of this topic.
2 The Model and the Equilibrium
There is a continuum of workers and at least two employers. Worker i’s ability xiis her
private information.5Each worker draws her xindependently from the same distribution
with cumulative distribution function (cdf) F. Following the labor economics literature,
we assume that Fis lognormal on [0,∞) with mean µand standard deviation σ.6
A worker can acquire zero or one unit of education, e∈ {0,1}. For the purposes of the
paper, we interpret e= 1 as receiving a college degree and not doing so as choosing e= 0.
Attaining e= 1 costs c(x) for a worker with ability x;e= 0 has a cost of 0. We assume that
c(x) is decreasing. Workers maximize their expected payoﬀs, U(x, e, w) = w(e)−e·c(x),
where wis the wage outlined below. U(x, e, w) satisﬁes the singlecrossing condition; if
type xis indiﬀerent between e= 1 and e= 0, any higher type strictly prefers being
educated, while any lower type prefers to stay uneducated (e= 0). As in Spence (1973),
education does not aﬀect productivity, but can serve as a signal of ability. Employers
observe each worker’s educational attainment and infer abilities using Bayes’ rule. Each
unit of the worker’s ability translates into one unit of output for her employer. The output
price is nomalized to 1. The expected proﬁt of an employer is then the expected ability of
her workers net of their wages w. The employers choose w(e) to maximize their expected
proﬁts, competing for workers a l´a Bertrand. We solve for the Perfect Bayesian Equilibrium
of this game.
Consider a threshold θ > 0 and the following strategies. Workers with x≥θacquire
education, so that e(xx≥θ) = 1. Conversely, workers with x < θ do not study, hence
e(xx<θ) = 0. Employers infer that e= 1 implies x≥θ, so they oﬀer wH(θ) = E[xx≥θ]
to workers with a degree (e= 1). The wage oﬀer to workers with e= 0 is wL(θ) =
E[xx≤θ]. Note that in equilibrium each worker is paid her (conditionally) expected
productivity and employers make zero expected proﬁts. Now we turn our attention to the
college premium function P(θ)≡wH(θ)−wL(θ). Incentive compatibility implies the
following proposition.
5Henceforth, for convenience, we will omit the subscript ion the worker’s ability.
6The assumption of lognormality of the unobservables in the wage equation is standard in labor eco
nomics. The literature on sample selection routinely associates these unobservable factors with abilities
known to the workers. While the econometrics literature has mainly been concerned with eliminating the
resulting “ability bias”, our focus is on the market (wage) response to the workers’ signaling choices. See
Appendix A for more details.
3
Proposition 1. The threshold θ∗and the strategies described above form a partially reveal
ing Perfect Bayesian Equilibrium if
c(θ∗) = P(θ∗).(1)
Proof. Suppose type xobtains a college education. Any higher type will receive the same
beneﬁt at a lower cost; consequently, they should also obtain education. If θ∗is the lowest
type to obtain a college education, and the employers infer the abilities and set the wages as
above, the beneﬁt of an education is P(θ∗).Clearly, type θ∗for which (1) holds is indiﬀerent
between obtaining a degree and not, while for any lower type the cost of education exceeds
its beneﬁt.
Remark 1. Under mild restrictions on the oﬀequilibrium beliefs, and if c(θ) and P(θ)
cross only once, the equilibrium given by the intersection θ∗is unique.7We provide plausible
restrictions on the beliefs and elaborate on uniqueness of equilibrium in Appendix A.
Importantly, the equilibrium college wage premium P(θ∗) depends on the share of
the population, 1 −F(θ∗), receiving a degree; this dependence is central to our study.
In signaling models with binary types and continuous choice of educational level, as in
Spence (1973), such dependence is absent.8In traditional models, in the unique equilibrium
that satisﬁes the intuitive criterion, the high ability type is educated whereas the low type is
not. The college wage premium is then just the (ﬁxed) diﬀerence between the productivities
of these two types. More recent models, for example Hendel et al. (2005), Balart (2016)
and Zheng (2019) incorporate credit constraints into signaling models with binary types.
Their models are crafted in such a way that the wage of college educated is constant, while
the wage of the uneducated group decreases with the share of college educated.
In our model an increasing share of college educated – that is, lowering θ∗– reduces
wages for both educated and uneducated workers. The way the college wage premium
changes with the share of educated depends on the relative rate of decrease of the wages of
these groups. As it turns out, the wages of the educated decrease at a faster rate when the
share of educated in the workforce is low, this produces the increase in the wage premium
at the upper tail of the ability distribution. In contrast, the wages of the uneducated
decrease faster when the share of the educated is high, this produces the decline in the
wage premium at the bottom tail of the ability distribution. The formal analysis and
intuitions are presented in the next section.
7P(θ) can be increasing or decreasing in equilibrium, however to guarantee the uniqueness of equilibrium
c(θ) always intersects P(θ) from above.
8One could also argue that a continuum of abilities rather than a binary choice in the signaling technology
is more empirically relevant. For example, Huang et al (2020) use a categorical measure of educational
attainment in their study of signaling eﬀects of education on lender default in the online lending market.
4
3 College Premium Function
This section explores the college premium function and provides some connections to related
empirical studies.9The following proposition ﬁrstly considers the characteristics of P(θ).
Proposition 2. The college premium function P(θ)is a continuous strictly quasiconvex
function. P(θ)strictly decreases for θ < ˆ
θ, reaches its minimum at ˆ
θ < med [x]and strictly
increases for θ > ˆ
θ.
Proof. See Appendix B.
Clearly both E[xx≥θ] and E[xx≤θ] are increasing in θ. The fact that their diﬀerence
P(θ) is Ushaped for the lognormal distribution for any values of µand σis nontrivial. Our
proof relies on Lipschitz Implicit Function Theorem from Border (2018). The intuition for
the Ushape of P(θ) relates to studies in reliability theory. Bagnoli and Bergstrom (2005)
explore the meanresiduallifetime function MLR (θ) = E[xx≥θ]−θ, and the mean
advantageoverinferiors function δ(θ) = θ−E[xx<θ]. Our P(θ) is a combination of
these two functions, explicitly P(θ) = MLR (θ) + δ(θ).
From Bagnoli and Bergstrom (2005) follows that δ(θ) is always increasing in our case.
The MLR (θ) function, on the other hand, is more complex. When xis lognormal with
mean µand variance σ2its density is logconcave for x≤exp (µ+ 1 −σ2) and logconvex
for x≥exp (µ+ 1 −σ2). From the arguments in Bagnoli and Bergstrom (2005), it follows
that MLR (θ) is increasing for x≥exp (µ+ 1 −σ2); in other words, M LR(θ) is increasing
where the density is logconvex. Hence, for θ≥exp (µ+ 1 −σ2), P(θ) is the sum of two
increasing functions. On the other hand, when θis low and the density is logconcave,
MLR (θ) could be decreasing. As P(θ) is the sum of a decreasing MLR (θ) and an in
creasing δ(θ), P(θ) could also be decreasing when θis low.
To provide some additional intuition, we plot an example of P(θ) in Figure 1, where
P(θ) is derived when xdistributed lognormally with mean µ= 10 and variance σ2= 1.
Figure 1 also shows c(θ), which is decreasing. Also note that the slope of c(θ) is always
more negative (steeper) than P(θ), even when P(θ) is negatively sloped. Their intersection
at θ∗, P (θ∗),forms a PBE, where workers with abilities x≥θ∗obtain education, and the
college wage premium is P(θ∗).To concentrate on the comparative statics of the college
wage premium, suppose that there is an exogenous decrease in the cost of education c(θ) for
every θ, so that the equilibrium θ∗shifts to the left (see Figure 1).10 The resulting college
wage premium ﬁrst decreases and reaches its minimum when θ∗=ˆ
θ. Any subsequent
decrease in c(θ) would result in an increase in the college wage premium.
9Function fis strictly quasiconvex if for any x, y,f(λx + (1 −λ)y)<max (f(x), f (y)) for any λ∈
(0,1).
10Costs of education could decrease due to the government subsidies and ﬁnancial aid, for example.
Technological progress is also likely to shift the c(θ) curve to the left, as it decreases the opportunity cost
of staying at college. Parttime and online options for higher education are likely to have a similar eﬀect.
5

6
0
P(θ∗)
c(θ)
P(θ)
θ∗
ˆ
θ
Figure 1: Equilibrium educational choices and college wage premium
The fact that signaling can lead to a Ushaped college wage premium is unique to
our model. Other signaling models of college wage premium focus on one side of the
premium/share relationship. Hendel et al. (2005) and Zheng (2019), for instance, use
models with two ability levels and credit constraints. In their models the college wage
premium rises with the share of the educated when ﬁnancial aid facilitates education of
poor students with high ability, as this decreases the expected productivity of those without
a degree. This holds in our model when θ < ˆ
θ, as illustrated in Figure 1. Proposition 2
shows, however, that the premium/share relationship can also work the other way.
To further tease out the implications of Proposition 2, ﬁx µand consider ˆ
θ(σ), where
ˆ
θ= arg minθP(θ) from Proposition 2. From the proof of Proposition 2 it follows that
ˆ
θ(σ) is a continuous function with limσ→0+ˆ
θ(σ) = med [x]. Figure 2 plots ˆ
θ(σ) for µ= 10
and σ∈(0,10). Importantly, ˆ
θ(σ) curve separates the regions with a positive/negative
relationship between the college wage premium and the share of the population with a
college education. From Figure 2, it is evident that higher values of σresult in a lower ˆ
θ,
increasing the region for which the premium/share relationship is negative. Recall that in
our model a higher share of the workforce receiving a college education means a lower θ.
When the share of college educated in the country is high, that is θ∗<ˆ
θ(σ), our model
predicts further increases in college wage premium. In contrast, when the educated share
is low – (θ∗>ˆ
θ(σ)) – the college wage premium will decrease as the college educated
franchise expands.
Associating higher variance of the ability distribution with higher income inequality,
our predictions can be reconciled with the empirical patterns observed in diﬀerent coun
tries. Machin and McNally (2007) report a positive relationship between the college wage
premium and the share of college educated in Australia, Finland, France, Korea, Sweden,
Switzerland, the Netherlands and the UK in 1997–2003. Consistent with our story, these
6
med[x]
∂P
∂θ >0
∂P
∂θ <0
ˆ
θ(σ)
θ
0
10
σ
Figure 2: Regions of increasing and decreasing college wage premium
are fairly egalitarian countries (hence low σ) with a high share of tertiary educated.11 Such
countries would be below the ˆ
θ(σ) curve in Figure 2.
In contrast, most Latin American countries have high income inequality and lower
share of college educated – 20% in Brazil and 33% in Chile, for example.12 Such economies
would be located above the ˆ
θ(σ) curve in Figure 2, where our model predicts a negative
premium/share relationship. This is consistent with the empirical evidence in Fernandez
and Messina (2017). These two aforementioned empirical studies report a positive pre
mium/share relationship in Germany and Czech Republic and a negative relationship in
Argentina. This highlights the nuanced interaction at play here, as the highest share of
college educated amongst these three countries is in Argentina. Our model explains the
apparent contradiction because income inequality in Argentina is much greater than in
these two European counterparts.
While this evidence is consistent with the predictions of our model, more econometric
studies are required. This is of particular importance given the public funding allocated to
the highereducation sector in many countries. With its unique equilibrium, our signaling
model provides empirically testable hypotheses that could be used as a framework for future
empirical research as we are able to sidestep issues of equilibrium selection.
Finally, the skillbiased technical change in Acemoglu (2002) increases the demand for
the college educated. Here, the beneﬁt of college education depends on the size of the college
educated cohort in the labor market. As in the skillbased technical change framework,
signaling can lead to an increase in the college wage premium following increased college
enrolment. This, of course, makes it diﬃcult to distinguish these two eﬀects empirically.
One way is to focus on the mean wage of the group without college education. This wage
will increase due to the skillbiased technical change, although not in proportion to the
mean wage of the educated group. Our signaling structure implies that the mean wage of
11See https://data.oecd.org/eduatt/populationwithtertiaryeducation.htm
12See Zoghbi et al. (2013) for an analysis of the eﬃciency of highereducation institutions in Brazil.
7
the noneducated group will decrease as the share of college educated increases. Indeed,
since the 1970s data from the United States suggests there has been a downward trend
in the mean wage of male high school graduates and dropouts – see Acemoglu and Autor
(2011) for instance. This observation is consistent with our signaling model, but not with
the skillbiased technical change theory. We leave more detailed empirical analysis to future
research.
4 Concluding remarks
In this paper we propose an explanation based on the signaling eﬀect of education for the
observed Ushaped relationship between the college wage premium and the share of the
population who are college educated. In doing so we touch on two key policy debates of
recent times – the role of the higher education sector and the issue of inequality. Our anal
ysis suggests that these two issues are linked. Expanding college enrolment has widespread
support, and governments in many countries actively assist university students via a vari
ety of schemes and interventions. College education is often seen as a path to both better
standards of living for the individuals themselves, and to better outcomes for the overall
economy. Our analysis, at very least, cautions that the impact of these policies depend
inextricably on a country’s existing stock of college graduates and existing wage inequality,
and that the desired welfare outcomes might not necessarily be forthcoming.
Appendix A
Signaling games usually involve discrete types (abilities) of the workers but allow a contin
uous choice of the signal, as well as a continuous action space for the employers.13 It is well
known that such a setup results in multiple (in fact, inﬁnitely many) separating equilibria,
as well as some pooling equilibria. Multiplicity of equilibria is due to the freedom with
which the oﬀequilibrium beliefs of the employers can be speciﬁed.14 When there is conti
nuity in the level of education a worker can choose, any level of education on some interval
can be supported as a separating equilibrium by a judicious choice of the oﬀequilibrium
beliefs.
Our model is diﬀerent. The educational (signal) choice is binary. Suppose, as outlined
in Section 2, each available signal is the equilibrium choice for a worker with some ability
level. Bayesian updating then pins down the equilibrium beliefs of the employers, and
there are no oﬀequilibrium beliefs as such. Consequently, expected payoﬀ maximization
requires that the wage schedules w(e) oﬀered are the workers’ expected productivities
13Like us, the setup of Fang (2006) includes a continuum of productivity types. As noted, his focus is
on distinguishing between the signaling and productivity enhancement eﬀects of education empirically.
14The large literature on equilibrium reﬁnements is surveyed in Sobel (2009).
8
given employer beliefs. This, in turn, determines the beneﬁt of obtaining education. Due
to the singlecrossing of the agents’ payoﬀs, the type space separates into the two sub
regions such that only the types above the threshold θ∗obtain education. Moreover, θ∗
satisﬁes (1) in Section 2. The number of such partially revealing equilibria is given by the
number of the intersections of the curves c(θ) and P(θ). If the intersection is unique, the
partially revealing equilibrium will also be unique. However, since c(θ) is decreasing and
P(θ) also has a decreasing section, it not impossible that the two curves overlap on some
interval. This would result in a continuum of equilibria, but clearly this case is nongeneric.
In contrast, in a “standard” signaling model with discrete types and a continuous action
space, a continuum of separating equilibria is a generic situation.
In our setting, the freedom with oﬀequilibrium beliefs can only be exploited to construct
the PBE with complete pooling: eis the same for every xand w=E[x]. There can be two
such equilibria, one with e= 1 and one with e= 0.Note that with P(0) = E[x]< c (0),
e= 1 for every x(every type is educated) is not a PBE. Given that P(θ) is quite ﬂat on
the left and c(θ) is decreasing, it is plausible that P(0) < c (0). The PBE with e= 0 for
every xcan be supported by the “unreasonable” belief that any deviant is a “low” enough
type. Suppose instead the belief is such that an oﬀequilibrium e= 1 is attributed to the
types x > ˜
θand leads to the wage ˜w=c(˜
θ) + E[x].Such a wage oﬀer is attractive only
for x≥˜
θ, since for them c(x)≤c(˜
θ).With such a reﬁnement and P(0) < c (0), the only
equilibrium in this model is given by Proposition 1. Using equilibrium reﬁnements would
allow for the construction of a unique separating equilibrium in the signaling model with
discrete types as well – see Sobel (2009) – however, the resulting college wage premium in
this case is given by the ﬁxed diﬀerence in the productivities of the types. The beneﬁt of
our formulation is that the equilibrium college wage premium, P(θ∗), depends on the share
of the population 1 −F(θ∗) obtaining a university degree in equilibrium.
The distribution of abilities. To isolate the eﬀects of signaling on college wage premium
we deliberately keep the setting as stylized as possible. One of the advantages is that
we can work with an empirically sound distribution of abilities rather than assuming two
ability levels, as is common in the extant literature. We assume that abilities are drawn
from the lognormal distribution; this is important for our results. Lognormality of the
unobservable components in the wage equation is a standard assumption in labor economics.
It is used in seminal works on sample selection into the labour market (Heckman, 1979)
and on migration (Borjas, 1987). At least since Griliches (1977) these unobservable factors
are associated with ability/productivity known to the agents, but not to the analysts.15
For the most part, the econometrics literature has been more concerned with eliminating
the resulting “ability bias” in estimation. Our focus is on the labor market response to
agents signaling their productivity via their educational choices, and the resulting eﬀect on
15Standard sample selection conditions, see equation (2) in Heckman (1979) or equation (3) in Bor
jas (1987) imply that an agent who makes a conscious decision to enter the labour market or to migrate
into a country has to know the realization of the unobservable factor.
9
the college wage premium.
Appendix B
Proof of Proposition 2
Proof. Introduce m= (ln θ−µ)/σ. Recall that for the lognormally distributed x, the
mean E[x] = exp µ+σ2
2and the median med [x] = exp µ. Then θ=med [x] corresponds
to m= 0 and θ=E[x] corresponds to m=σ/2.It is well known that
E[xx<θ] = E[x]·Φ (m−σ)
Φ (m)and E[xx≥θ] = E[x]·Φ (σ−m)
Φ (−m),(2)
where Φ and φstand for the cumulative distribution and density functions of the standard
normal distribution. Deﬁne B(m, σ)≡Φ(m)−Φ(m−σ)
Φ(m)(1−Φ(m)) , then
P(θ) = E[x]·B(m, σ).(3)
Clearly, P(θ) is increasing (decreasing) in θwhenever B(m, σ) is increasing (decreasing)
in m. Lemma 1 shows that B(m, σ) is increasing in mfor m≥0 for any σ > 0.Note that
m= 0 corresponds to θ=med [x].
Lemma 1. B(m, σ)is strictly increasing in mfor m≥0.
Proof. First, we show that B(m, σ) is strictly increasing in mfor m∈[0, σ/2]. Direct
computation provides Bm(0, σ) = 4 (φ(0) −φ(σ)) >0 for any σ > 0.Hence B(m, σ) is
strictly increasing at m= 0.Further, note that Φ (m) (1 −Φ (m)) is a symmetric function
with a max at m= 0,hence it is decreasing at any m≥0 (strictly at m > 0). Note also that
m≤σ/2 implies m−σ < 0 and m−σ ≥ m. Since φ(m) is itself a symmetric function
with a max at m= 0, φ (m)−φ(m−σ)≥0 for any m≤σ/2,with strict inequality
at m < σ/2.Therefore, Φ (m)−Φ (m−σ) is increasing in mfor m≤σ/2 (strictly at
m < σ/2).
Second, we show that B(m, σ) is strictly increasing in mfor m≥σ/2. Let h(m) =
φ(m)/Φ (m) denote the reverse hazard rate of the standard normal distribution. Introduce
another pair of functions
g(m) = h(m−σ)−h(m) and s(m) = h(−m)−h(σ−m).
It is known that the reverse hazard rate h(m) of the standard normal is a strictly decreasing
and strictly convex function. This implies i) that both g(m)>0 and s(m)>0 for
every m, and ii) that g(m) is (strictly) decreasing and s(m) is (strictly) increasing in m.
Straightforward diﬀerentiation produces
Bm(m, σ) = Φ (σ−m)
Φ (−m)s(m)−Φ (m−σ)
Φ (m)g(m).
10
Φ (σ−m)
Φ (−m)>Φ (m−σ)
Φ (m)for any m, recall (2). Note that m=σ/2 implies m−σ=−m,
hence s(m) = g(m). At every m > σ/2, s (m)> g (m),therefore Bm(m, σ)>0.
Next, we show that the extremum of B(m, σ) for a given σ, ˆm(σ),exists and is unique
for any σ∈[0,¯σ]. In addition, we show that ˆm(σ) is continuous. In view of Lemma 1
the unique extremum ˆm < 0 is then the minimum of B(m, σ).This implies that P(θ)
decreases in θfor θ < ˆ
θ < med [x] for any σ > 0 and µ; and P(θ) further increases in θfor
every θ > ˆ
θ.
Note that function B(m, σ) is continuously diﬀerentiable, and locating its extremal
points is equivalent to locating the extremal points of ln B(m, σ).The necessary condition
for an extremum of ln B(m, σ) for a given σis
L(σ, ˆm)≡φ( ˆm)−φ( ˆm−σ)
Φ ( ˆm)−Φ ( ˆm−σ)=φ( ˆm) (1 −2Φ ( ˆm))
Φ ( ˆm) (1 −Φ ( ˆm)) ≡R( ˆm).(4)
We treat (4) as an equation that deﬁnes function ˆm(σ) implicitly. Lipschitz Implicit
Function Theorem, see Theorem 6 in Border (2018), suggests that ˆm, the solution to (4)
exists for any σ∈[0,¯σ], moreover, such ˆmis unique. In addition, ˆm(σ) is a continuous
function. We further restate Border’s theorem and verify its conditions.
Border (2018) Theorem 6 applies to the solution ξ(m) of the equation g(ξ(m), m)=0.
Let Pbe a compact metric space. If a continuous function g:P× R → R satisﬁes
0< n ≤g(x, m)−g(y, m)
x−y≤N, (5)
for any x, y ∈Pand any m, then the solution ξ(m) exists, is unique and is a continuous
function of m.
In our setting g(ξ(m), m) is given by L(σ, ˆm)−R( ˆm), which is equal to the diﬀerence
between the left and the right hand sides of (4). Obviously this is a continuous function.
Direct computation provides
∂L (σ, ˆm)
∂σ =−φ( ˆm−σ)
Φ( ˆm)−Φ( ˆm−σ)φ( ˆm)−φ( ˆm−σ)
Φ( ˆm)−Φ( ˆm−σ)+ ˆm−σ.(6)
Clearly, Lσ(σ, ˆm) is a continuous function. We further prove that Lσis always positive.
Note that φ( ˆm−σ)
Φ( ˆm)−Φ( ˆm−σ)>0 for any σ > 0. Focus on the square bracket in (6); call it
Z(σ, ˆm).
L’Hopital’s rule implies
lim
σ→0
φ( ˆm)−φ( ˆm−σ)
Φ( ˆm)−Φ( ˆm−σ)+ ˆm−σ= 0.(7)
Introduce l(σ, ˆm) = φ( ˆm)−φ( ˆm−σ) and r(σ, ˆm) = (σ−ˆm) (Φ( ˆm)−Φ( ˆm−σ)) .
The above (7) implies limσ→0l(σ, ˆm)−r(σ, ˆm)=0.Direct computation provides
lσ(σ, ˆm) = −( ˆm−σ)φ( ˆm−σ) and
rσ(σ, ˆm) = Φ( ˆm)−Φ( ˆm−σ)−( ˆm−σ)φ( ˆm−σ).
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Thus, lσ(σ, ˆm)< rσ(σ, ˆm) for any σ > 0,and therefore l(σ, ˆm)< r (σ, ˆm) for any σ > 0.
This implies that Z(σ, ˆm)<0 for any σ > 0,which in turn implies that Lσ(σ, ˆm)>0 for
any ˆmand any σ > 0.Given that
L(x, ˆm)−L(y, ˆm) = Zx
y
Lσ(s, ˆm)ds,
inequality (5) is satisﬁed. Thus, the extremum of B(m, σ) for a given σ, ˆm(σ),exists and
is unique for any σ∈[0,¯σ].
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