Risky Matching
Hector Chade∗and Ilse Lindenlaub†
July 2015
Abstract
Risky matching is pervasive in many markets. It refers to situations where agents make an
investment under uncertainty before entering a matching market. A prominent example is an
individual who can invest in education before going on the labor market. He faces uncertainty
about how skilled he turns out, and also about the prevailing aggregate conditions at the time
of hiring. We develop a general framework to study risky matching problems. Our focus is
on the equilibrium comparative statics with respect to changes in risk. We provide conditions
on primitives for a stochastically better or riskier matching market to induce more agents
to invest, and show how this affects the equilibrium assignment and wages. Many economic
problems are applications of our model. We show how it can account for several stylized facts
of labor and marriage markets, and on entrepreneurs seeking funding from venture capitalists.
Keywords. Matching, Sorting, Risk, Endogenous Distributions, Changes in Risk.
∗
Arizona State University, hector.chade@asu.edu
NYU, ilse.lindenlaub@nyu.edu
We are grateful to Katka Borovickova, Alejandro Manelli, and Eddie Schlee, as well as seminar participants at
Collegio Carlo Alberto, and 2015 SED–Warsaw, for their helpful comments and suggestions.
†
1
Introduction
In numerous matching markets, agents engage in pre-match investments of uncertain returns.
Consider an individual after completing high school, who decides whether to go to college before
matching with a firm on the labor market. At the investment stage, there is uncertainty about the
conditions that will prevail in the labor market upon graduation, which can range from boom to
recession. Moreover, college education has its own idiosyncratic risks such as college completion
risk, and uncertainty about graduating with honors or simply just making it. When deciding
whether to go to college, agents form expectations about these multiple risks and compare the
expected utility of the alternative choices. We argue that changes in these risks may help explain
some of the puzzling evidence on educational trends of US men: Despite a steady increase in the
skill premium, growth in college attainment has stagnated. At the same time, the variability of the
permanent income component has declined. We show in this paper why agents make pre-match
investments to shield themselves against this income risk or, in turn, why they abstain from it if
risk diminishes. Our theory provides a novel link between these two facts.
This paper develops a general framework of risky matching: It involves a matching problem
and, before matching, one side of the market faces an investment choice that involves multiple
risks. By treating agents’ investment in their characteristics as a choice under uncertainty, we endogenize the underlying distributions and focus on how agents alter their investment upon changes
in risk. We then analyze how this affects the matching market equilibrium, i.e. allocation and
wages. Our model cannot only shed light on why US male college attainment has stagnated during a period where earnings volatility has fallen. It can also help explain why women have chosen
to get more education in the face of larger marriage market uncertainty and why entrepreneurs,
during the Great Recession, have selected into riskier projects that yield higher returns.
The following is a simple overview of our model. There is a large number of risk averse agents
on one side of the market and the same number of risk-neutral agents on the other side, who both
are heterogeneous in their attributes. There are two stages. In the first one, the investment stage,
the risk averse agents decide whether to invest in a payoff-relevant attribute. Investment is costly,
and agents are ex-ante heterogeneous in the investment cost. The benefit of investing is that
the agent’s attribute is drawn from a better distribution in the first-order stochastic dominance
sense (FOSD). At the beginning of the second stage, the matching stage, there is a realization
of an aggregate shock that affects the amount of output of each prospective match. Then both
sides match pairwise in a frictionless market. We provide results for both, the transferable utility
(TU) case where agents can transfer utility at a constant rate, and the strict nontransferable
utility (strict NTU) case where agents split the surplus in a prespecified way. These are standard
benchmarks in the matching literature since Becker (1973).
After analyzing equilibrium existence, uniqueness, and stability, we delve into the issue of
1
equilibrium changes in risk. More precisely, we analyze under what conditions on the primitives
of the model - risk preferences and properties of the match output function - a stochastically
better (FOSD) or riskier (increase in risk, IR, a là Rothschild and Stiglitz (1970)) aggregate
shock increases the number of agents who choose to invest in their attributes. We think of this
as the natural comparative statics result since it is the most intuitive one in applications: Agents
have more incentives to invest in their attributes either to take advantage of a stochastically better
shock or to shield themselves from a riskier environment.
For a FOSD shift in the aggregate shock, we find that the result holds (i.e. more agents will
invest) if there are enough complementarities in production or wages, and if agents are not too
risk averse. The intuition relies on the presence of both substitution and income effects which
go in opposite directions. Complementarities between attribute and shock in the match output
function (which implies similar properties for wages) provide agents with more incentives to invest
(substitution effect). This is because wages are increasing in the shock and more so for agents
with higher attributes - something that investment yields stochastically. In turn, when agents
are risk averse higher wages lead to a lower marginal utility of income (income effect), thereby
reducing the incentives to invest in their attribute.
As with the FOSD shift, the corresponding result for an IR shift relies on two forces (complementarities of the shock and agents’ attribute in the wage function, as well as the agents’ attitude
towards risk) but it is more subtle. If agents are sufficiently prudent, then a riskier shock pushes
more agents to invest in their attribute. As is well-known, prudence implies downside risk aversion
and triggers precautionary actions to insure against bad realizations of a shock. Similarly here, a
riskier shock induces sufficiently prudent agents to engage in precautionary investment. How large
prudence needs to be depends on curvature and complementarity properties of the wage function
that we will explain in detail.
These comparative statics hold for any wage function with the specified properties. We choose
to close the model through a frictionless matching market. For this case, there are several natural
classes of primitives (i.e., utility and match output functions) that satisfy our conditions for
monotone comparative statics. We also analyze the equilibrium effects of changes in risk on
the matching of agents with different characteristics, i.e., the matching function, and on wages.
Regarding matching, we show that any risk-averse agent matches with a partner of poorer quality
after a change in risk that triggers more investment (FOSD or IR). This is due to increased
competition for partners with better attributes from the risk neutral side. Regarding wages, the
effects are more ambiguous. The (expected) wage function can go up or down under a FOSD
shift, and it goes down only under stringent conditions after an IR shift in the aggregate shock.
We then demonstrate the usefulness of our framework through a variety of applications for
the labor, marriage and entrepreneurs’ market and show that they can be analyzed in a unified
matching framework with pre-match investments where agents adjust their investment decisions
2
upon changes in risk. Finally, we discuss some variations of the model, including the extension
to the case in which both sides make pre-matching investments, illustrated with an application to
the well-known phenomenon of directed technological change.
The paper is mainly related to the large static matching literature initiated by Becker (1973),
Shapley and Shubik (1972), Gale and Shapley (1962), and recently generalized by Legros and
Newman (2007) (see Chade, Eeckhout, and Smith (2014) for a survey), which has been extensively
applied to labor and marriage markets. To the standard matching framework, we add a pre-match
investment stage, a feature that is also included in Cole, Mailath, and Postlewaite (2001), Mailath,
Samuelson, and Postlewaite (2013), Peters and Siow (2002), and Noldeke and Samuelson (2014).
What distinguishes our paper is the presence of multiple risks (the investment return is uncertain
and so is the state of the matching market that will prevail). To our knowledge, analyzing the
equilibrium effects of changes in risks (FOSD and IR) in a matching setting is novel.
Given our focus on comparative statics of risks, the paper is also related to the large literature
in economics of uncertainty that assesses the effects of changes in risks on decision problems (see
Gollier (2004) for an extensive survey of the literature and the references therein). Unlike this
literature, the multiple risks involved in our model affect agents’ expected utility through the wage
function, which is an equilibrium object, and they enter into that function in a nonadditive way.
Both features, equilibrium and nonadditive risks, not only significantly complicate the analysis
but they also enrich the economics of the comparative statics results.
Our applications relate to papers from several fields. Our labor market application is motivated
by changing trends in US educational attainment for men (Castro and Coen-Pirani (2014) or
Athreya and Eberly (2013)) and by changes in the volatility of permanent versus transitory
earnings (Blundell, Pistaferri, and Preston (2008)). Our marriage market application relates
to Chiappori, Iyigun, and Weiss (2009) who also analyze educational choices before marriage
market entry but in a different setting and with different focus. The one on the entrepreneurs’
market relies on stylized facts documented by Navarro (2014) who argues that the increase in
macroeconomic volatility during the Great Recession induced entrepreneurs to invest in riskier
projects that yield a higher return. Finally, our application on directed technological change
relates to numerous papers in that area, pioneered by Acemoglu (1998) and Acemoglu (2002). Our
contribution is to show that all these applications can be analyzed in a unified matching framework
with pre-match investments where agents adjust their investment decisions upon changes in risk.
The next section introduces the model. Section 3 describes the equilibrium properties. The
main results, the equilibrium comparative statics of risk, are in Sections 4 and 5. Section 6 contains
additional comparative statics results. In Section 7, we provide four economic applications of the
model. Section 8 concludes. Omitted proofs are in the Appendix.
3
2
The Model
We normalize to one the measure of each of the two sides of the market. One population consists
of risk neutral agents, heterogeneous in an attribute y that is distributed with a continuously
differentiable cumulative distribution (cdf) G and density g.1 The other population consists of
risk averse agents with von Neumann-Morgestern utility function for income given by u, which
is three times continuously differentiable, with u0 > 0, u00 ≤ 0 (this covers also the risk neutral
case), and u000 ≥ 0. Initially, these agents differ in a characteristic θ, which is wlog uniformly
distributed and determines an agent’s investment cost c(θ). The function c is continuous and
strictly decreasing in θ, with c(1) = 0. To rule out uninteresting cases and simplify the exposition
we will also assume that limθ→0 c(θ) = +∞. If an agent with characteristic θ invests and obtains
income w, then her payoff is u(w) − c(θ); if she does not invest it is simply u(w). Moreover,
investment allows the agent to draw an attribute x from a cdf H1 and density h1 ; without
investment, it is drawn from H0 or h0 with the same support.2 These distributions are ordered
by strict FOSD: H1 (x) < H0 (x) for all x. Unless we explicitly state otherwise, the support of all
the random variables in the model is the unit interval [0, 1].
After the investment stage agents match pairwise in a decentralized market. If an agent with
attribute y matches with one with x, they produce output f (x, y, α), where α is an aggregate shock
that affects production and whose realization is revealed before matching takes place. The function
f is nonnegative, three times continuously differentiable, strictly increasing in each argument,
and has positive cross partial derivatives (i.e., fxy , fxα , and fyα are positive). It also satisfies
f (0, y, α) = f (x, 0, α) = 0. These are more assumptions on f than we need in most instances but
they suffice to cover all of our cases. The shock is distributed with cdf L(·|t) indexed by t ∈ [0, 1].
Notice that agents face two risks, a controllable risk via investment and a background risk α.
Changes in t will be the focus of our comparative statics exercises, and we examine two standard
textbook cases of changes in risk: FOSD and IR shifts in the distribution of the shock α.
Regarding the division of the match output, we study the standard polar cases analyzed in
matching models (Becker (1973)): the transferable utility case (TU) where agents can transfer
utility at a constant rate among them, and the strictly non transferable utility case (strict NTU)
where the division is exogenously given (wlog, we will assume that output is shared equally).
Notice that since the shock is revealed before matching takes place, it follows that there is no
uncertainty when agents match, and hence the matching stage is a fairly standard assignment
game with TU or marriage problem with strict NTU. We will discuss this issue further below.
1
For simplicity, we assume differentiable cdfs but it will be clear that the main insights apply more generally.
The common support assumption is made just for simplicity. Simlarly, the results go through for discrete
distributions of x, although with more cumbersome notation. Finally, we assume that agents do not have any level
of this attribute before the draw. This is an innocuous assumption; we could instead assume that all agents start
with an initial level of the attribute equal to a constant s and then there is a draw of x that adds to this level, so
that the ex post characteristic is s + x. Wlog, we have set s ≡ 0 for all agents.
2
4
In short, the timing of the model is as follows: first, the risk averse agents decide whether
to invest; then their attributes are drawn from the corresponding distributions; after that the
aggregate shock is realized; finally, matching takes place in a decentralized market.
Let a : [0, 1] → {0, 1} be a measurable function, where a(θ) = 0 means that an agent with cost
c(θ) does not invest, and a(θ) = 1 means that she does. Given a, the resulting distribution of
attribute x ∈ [0, 1] is H(·, a), which is a mixture of H1 and H0 with weights given by the measure
of agents that invests and one minus this measure, respectively. Formally,
!
Z
H(x, a) =
Z
dθ H1 (x) +
1−
!
dθ H0 (x).
{θ:a(θ)=1}
{θ:a(θ)=1}
Let µ(·, a) : [0, 1] → [0, 1] be a measurable and measure-preserving function that assigns
each agent with attribute x to one with attribute y = µ(x, a). For each α ∈ [0, 1], denote
by w(·, α, a) : [0, 1] → R+ the measurable function that determines the wage w(x, α, a) that a
worker with type x obtains when the realization of the shock is α and the education investment
decision in stage one is given by a. An equilibrium is a triple (a, µ, w) such that, given (w, µ), the
investment decision of each agent is optimal, i.e., the function a satisfies a(θ) = 1 if and only if
U1 (w, t) − c(θ) ≥ U0 (w, t), where
Z
1Z 1
Ui (w, t) =
u(w(x, α, a))dHi (x)dL(α|t) i = 0, 1;
0
(1)
0
and for any measurable a (hence H(·, a)) and any realization of α, (w, µ) clears the market
at the matching stage. Under TU this means that (w, µ) is a Walrasian equilibrium of the
matching market, so each agent takes w as given and optimally chooses the partner given by
µ, and w ensures that this is the case. In the strict NTU case wages are exogenously given by
w(x, α, a) = f (x, µ(x, a), α)/2 and each agent optimally selects the partner given by µ.
Remarks. We here discuss some of the assumptions of the model and how to relax them.
First, notice that we make assumptions on f that ensure positive sorting under both TU and
strict NTU. This is made for both definiteness and realism in some of the economic applications
of our framework. But one can easily modify the assumptions and obtain instad negative sorting
in the second stage. None of our main insights changes with this variation.
Second, we assume that the shock α is realized after investment but before matching takes
place. This strikes us as a natural timing for the applications of our the model: when, say, agents
invest in their education, they are not apprised of the market conditions under which they will
enter the labor market. Equally important, we want to capture in a simple way situations where
the shock (or part of it) is uninsurable, and this would not be the case under TU if it is realized
after the match (it will still be uninsurable in the strict NTU case, or if we add some friction such
5
as moral hazard under which matching becomes NTU, e.g., Legros and Newman (2007)).
Third, we assume that only one side is risk averse. This dovetails nicely with our applications
on the labor and financial markets but may be less so with the one on the marriage market. But
barring some technical difficulties such as using a more sophisticated sorting condition (Legros
and Newman (2007)), the insights are robust to the assumption that the other side is risk averse
too. A simple variation along these lines is to assume a one-sided or partnership model where
there is only one population of agents, all of them risk averse, who each decide whether to invest.
This version is simpler to analyze and less interesting since µ = x for all x under positive sorting.
Thus µ is independent of a, a crucial channel through which equilibrium effects play a role in our
model. We will elaborate further on this extension below in a couple of footnotes.
Fourth, we assume that investment is a binary decision. Barring a more complicated existence
proof, the comparative statics results also hold under continuous investment a ∈ [0, 1] and H that
shifts in FOSD as a increases. If the cost of investing a for an agent with θ is c(θ, a), and c is
submodular in (θ, a), then a will be increasing in θ and the same results obtain.
Finally, we assume that only one side invests. Although this is a plausible assumption for
most of our applications, we will also provide one (directed technological change) that illustrates
how this framework can be profitably extended to the case of two-sided investment.
3
Equilibrium Analysis
We solve the equilibrium problem backwards, starting from the second stage. Consider any
subgame that starts in the second stage given an investment strategy a (so that H(·, a) is the cdf
of x) and a realization of the shock α. Since f is supermodular in (x, y), the optimal matching
under TU exhibits positive sorting. The same holds under strict NTU as f is strictly increasing
in x and y. That is, the optimal matching function µ solves H(x, a) = G(µ(x, a)) and thus
µ(x, a) = G−1 (H(x, a)), which is strictly increasing in x as positive sorting demands.
It is also well-known that the optimal matching can be decentralized either as a Walrasian
equilibrium under TU or as the unique stable matching under strict NTU. Indeed, assume TU
and let w(x, α, a) be the wage or price (share of the output) of an agent with attribute x in the
subgame indexed by (α, a). Then an agent with attribute y who takes w as given solves
max f (x, y, α) − w(x, α, a).
x
The first-order condition of this problem is fx (x, y, α) = wx (x, α, a), and hence in equilibrium
fx (x, µ(x, a), α) = wx (x, α, a), which yields the following equilibrium wage function:
Z
w(x, α, a) = w(0, α, a) +
x
Z
fx (s, µ(s, a), α)ds =
0
0
6
x
fx s, G−1 (H(s, a)) , α ds,
(2)
where we have used the optimal µ and the fact that if x is zero output is zero and so is w(0, α, a).
In short, µ(x, a) = G−1 (H(x, a)) and a wage function given by (2) describe the Walrasian equilibrium of the subgame (α, a). The equilibrium payoff for a matched pair (x, y) in this subgame
is u(w(x, α, a)) for the agent with attribute x (minus the investment cost if she invested) and
f (µ−1 (y, α, a), y, α) − w(µ−1 (y, α, a), α, a) for the agent with attribute y.
Similarly, under strict NTU, µ(x, a) = G−1 (H(x, a)) and a wage function given by w(x, a) =
f (x, µ(x, a), α)/2 is the unique stable matching of the subgame (α, a).
Having pinned down equilibrium behavior in the matching stage, let us now consider the
investment stage. Consider an agent with investment cost c(θ). Given the wage she anticipates in
the matching stage, she will invest if and only if U1 (w, t)−U0 (w, t) ≥ c(θ). Notice that the left side
is independent of θ from the point of view of the agent (since she takes (w, µ) as given). Moreover,
since w is strictly increasing in x in both the TU and strict NTU cases, and H1 dominates H0 in
the strict FOSD sense, it follows that
1
Z
Z
1
u(w(x, α, a))dH0 (x),
u(w(x, α, a))dH1 (x) >
0
0
and hence U1 (w, t)−U0 (w, t) > 0. As a result, there exists a unique threshold θ∗ ∈ (0, 1) such that
a(θ) = 1 if and only if θ ≥ θ∗ , where θ∗ solves U1 (w, t) − U0 (w, t) = c(θ∗ ) (recall the properties of
c). Hence, in any equilibrium, the investment strategy a has the following functional form:
(
a(θ) =
1 if θ ≥ θ∗
0 if θ < θ∗
Notice that the function a is completely specified by the threshold θ∗ . Thus, with a slight
abuse of notation we can write H(·, θ∗ ), µ(·, θ∗ ), and w(·, α, θ∗ ), where
H(x, θ∗ ) = (1 − θ∗ )H1 (x) + θ∗ H0 (x)
µ(x, θ∗ ) = G−1 ((1 − θ∗ )H1 (x) + θ∗ H0 (x)) ,
and in the TU case
∗
Z
w(x, α, θ ) =
x
fx s, G−1 ((1 − θ∗ )H1 (s) + θ∗ H0 (s)) , α ds,
0
while in the strict NTU case
w(x, α, θ∗ ) = 0.5f x, G−1 ((1 − θ∗ )H1 (x) + θ∗ H0 (x)) , α .
Consequently, we can replace Ui (w, t) by Ui (θ∗ , t), i = 0, 1, where
7
(3)
(4)
Ui (θ∗ , t) =
Z
0
1 Z 1
u (w(x, α, θ∗ )) dHi (x) dL(α|t).
0
Now equilibrium existence reduces to finding a threshold θ∗ that satisfies U1 (θ∗ , t) − U0 (θ∗ , t) =
c(θ∗ ). To see this, notice that once we solve for θ∗ , the induced function a yields H(·, θ∗ ) and thus
a Walrasian equilibrium given by µ(·, θ∗ ) and w(·, α, θ∗ ) for any realization of α. This determines
U1 (w, t) − U0 (w, t) in the education stage and, by construction, it is higher than c(θ) for θ ≥ θ∗
and lower otherwise, thereby rationalizing H(·, θ∗ ) and completing the equilibrium construction.
The function U1 (θ∗ , t) − U0 (θ∗ , t) is positive and continuous for all θ∗ ∈ [0, 1]. Since c diverges
to infinity when θ∗ goes to zero and it is zero at θ∗ = 1, it follows that there is at least one solution
to U1 (θ∗ , t) − U0 (θ∗ , t) = c(θ∗ ) where U1 − U0 crosses c from below, and also that any solution to
this equation is interior. We have thus proved that an equilibrium exists; moreover, the argument
reveals that all equilibria are interior (i.e., θ∗ ∈ (0, 1)).3
Nothing precludes the existence of multiple equilibria.4 Although it will become clear below
that equilibrium uniqueness is an ancillary issue for our purposes, for completeness we seek some
sufficient conditions that guarantee the existence of a unique equilibrium. From the properties of
c, it follows that equilibrium is unique if U1 − U0 is increasing in θ∗ , and this holds if
Z
1
u(w(x, α, θ∗ ))dH1 (x) −
0
Z
1
u(w(x, α, θ∗ ))dH0 (x)
(5)
0
is increasing in θ∗ . Simple integration by parts reveals that (5) equals
1
Z
(H0 (x) − H1 (x)) u0 (w(x, α, θ∗ ))wx (x, α, θ∗ )dx.
(6)
0
Differentiating (6) with respect to θ∗ we obtain
Z
1
(H0 − H1 ) u00 (w) wx wθ∗ + u0 (w) wxθ∗ dx,
(7)
0
where we have omitted the arguments to simplify the notation. By FOSD, the first term in the
integrand is positive, but the second term can be positive or negative.
One would think that it should be obvious that an agent has more incentives to invest in
her attribute the lower is the measure of agents who invest, i.e. that U1 − U0 is increasing in
θ∗ . However, this intuitive conjecture is generally not true. The problem is that there are two
If we do not assume limθ→0 c(θ) = ∞, then there could be an equilibrium with θ∗ = 0, i.e., everyone invests.
In the one population model described at the end of the previous section, µ = x implies that U1 − U0 is
independent of θ∗ and hence U1 − U0 = c(θ∗ ) pins down the unique equilibrium of the model. The presence of θ∗
in µ and thus in w in our model makes uniqueness more difficult to hold.
3
4
8
opposing effects. On the one hand the wage function exhibits complementarities in (x, θ∗ ) (e.g.,
under TU). Then an increase in wages is particularly high for agents with high x, thus increasing
the incentives to invest. On the other hand, this effect is mitigated by an income effect, as a
higher wage (triggered by higher x) reduces the marginal utility of income. Only if the original
effect dominates this income effect, the intuitive conjecture holds.
It is apparent from the integrand in (7) that if agents are risk neutral and wxθ∗ ≥ 0, then
equilibrium is unique. By continuity, this holds also if risk aversion is sufficiently small. Under
Rx
TU, wxθ∗ = 0 fxy µθ∗ ≥ 0 for all x. Therefore, under TU the equilibrium is unique if agents’ risk
aversion is sufficiently small. We summarize the results so far:
Proposition 1 An equilibrium exists, and all equilibria are interior, i.e., θ∗ ∈ (0, 1). Moreover,
under TU it is unique if agents’ absolute risk aversion is sufficiently small.
The situation is more complex under strict NTU since wxθ∗ = 0.5[fxy µθ∗ + fyy µx µθ∗ + fy µxθ∗ ,
which can be positive or negative depending on x. It is easy, however, to construct examples with
uniqueness when risk aversion is small. For example, one can show that this is indeed the case
if G(y) = y, H0 (x) = x, H1 (x) = x2 , and f (x, y, α) = αxy for all values of α, x, and y, and the
constant absolute risk aversion utility function (CARA) u(w) = e−Rw /R for all w, where R > 0
is the coefficient of absolute risk aversion.
To close the discussion on uniqueness, notice that the condition that U1 − U0 be increasing
in θ∗ ensures uniqueness for all cost functions c, which is a demanding requirement. Without
this, it is not hard to construct examples with a unique equilibrium for a class of cost functions
c. For instance, under the same assumptions on distributions and match output as above, plus
u(w) = log w for all w, one can show that U1 − U0 is strictly decreasing and convex in θ∗ and
thus equilibrium is unique for any linear or concave c (which violates the divergence to infinity
assumption), as well as for sufficiently convex c.
4
Comparative Statics
We now turn to the focus of our analysis, namely, the equilibrium comparative statics of risk.
Our goal is to provide sufficient conditions on primitives under which a stochastically better or
riskier aggregate shock induces a larger measure of agents to invest, which is the result that fits
nicely with the economic applications that we describe in detail below.
There is a general thread underlying the intuition of our results. Our sufficient conditions
hinge upon a trade-off between complementarities in production and the intensity of attitudes
towards risk. First, when the shock becomes stochastically better, then more agents invest in
their attributes if risk aversion is not too large relative to the complementarities between shock
and attribute in production. We will show that in that case, the income effect triggered by
9
U1 − U0
U1 − U0
c
c
θ∗
θ∗
Figure 1: A. Unique equilibrium. B. Multiple equilibria.
larger wage due to a better shock, will be dominated by a suitably defined substitution effect,
thus incentivizing investment. Second, when the shock becomes riskier, a sufficient degree of
prudence (relative to the complementarities of shock and attribute in production) induces more
precautionary investment into the attribute to cushion a possibly more severe shock realization.5
Since we are looking at comparative statics in equilibrium, it is important to emphasize that
our results do not depend on having a unique equilibrium. Indeed, the results apply to all
equilibria where ‘U1 − U0 crosses c from below’ (we know at least one exists), see Figure 1.B.
Equilibria with the ‘crossing from below’ property are stable in a natural sense: if θ∗ is to the left
of the crossing point, then U1 − U0 < c and θ∗ will tend to go up since some agents will not find
it optimal to invest. The opposite happens for values of θ∗ above the crossing point, hence the
asserted stability property holds. By contrast, an analogous analysis shows that equilibria where
the crossing occurs from above are unstable, captured by Figure 2.A.
Before we get to the detailed analysis of the sought-after comparative statics results, it is
instructive to go over the main thrust of the argument. Consider the marginal agent who is
indifferent between investing or not. That is, for this agent U1 (θ∗ , t) − U0 (θ∗ , t) = c(θ∗ ).6 To
R1
simplify the notation, let J(x, t) ≡ 0 u(w(x, α, θ∗ ))dL(α|t), which can be interpreted as the
5
Eeckhoudt, Gollier, and Schlesinger (1996) provide characterization results for first and second order stochastic
dominance shifts when the controlable and background risks are additive. The general way in which they enter our
problem precludes the use of their nice results.
6
Note in passing that this condition can be interpreted as the first-order condition of the following planner’s
R θ∗
R1
problem that allocates agents to investment choices: maxθ∗ θ∗ (U1 (θ, t) − c(θ)) dθ − 0 U0 (θ, t)dθ.
10
U1 − U0
U1 − U0 − c
θ∗
c
θ∗
Figure 2: A. Two stable equilibria at the extremes, one unstable equilibrium in between. B.
Comparative Statics of U1 − U0 with respect to t.
agent’s ‘derived utility function’ for attribute x. Simple algebra reveals that
∗
Z
∗
1
Z
J(x, t)dH1 (x) −
U1 (θ , t) − U0 (θ , t) =
J(x, t)dH0 (x)
0
0
Z
1
1
Jx (x, t) (H0 (x) − H1 (x)) dx,
=
0
where the second equality follows from integration by parts. Notice that the marginal agent will
have more incentives to invest as t increases if U1 − U0 goes up, and this is equivalent to
Z
1
Jxt (x, t) (H0 (x) − H1 (x)) dx ≥ 0.
0
Consequently, a sufficient condition for more agents to invest when t increases is that Jx is
increasing in t for all (x, t). Indeed, this condition is necessary if we want the result to hold for
all Hi , i = 0, 1 such that H1 (x) ≤ H0 (x) for all x. If, say, Jxt < 0 on an interval of positive
measure, then it is easy to construct an example where H0 and H1 only differ on that interval
and the comparative statics result fails. Now, the explicit expression for Jx is
Z
Jx (x, t) =
1
u0 (w(x, α, θ∗ ))wx (x, α, θ∗ )dL(α|t),
(8)
0
and the monotonicity of Jx with respect to t depends on the type of shift that we apply to L. We
consider the standard textbook changes in risk: first-order stochastic dominance (FOSD) shift
and increase in risk (IR) shift. For FOSD, Jx is increasing in t if the integrand in (8) is increasing
11
in α, and for IR if it is convex in α.7
The argument above makes it clear that it suffices to analyze the behavior of U1 − U0 as a
function of t for any given threshold θ∗ . If U1 − U0 is increasing in t (see Figure 2.B), then more
agents invest in response to the shift in risk. To simplify the notation we will frequently set
v(x, α) ≡ u(w(x, α, θ∗ )) and derive properties of v. Therefore, we will show below under which
conditions vx = u0 (w)wx is increasing in α (for FOSD shift) and convex in α (for IR shift).
We first provide conditions on preferences and the wage function that yields the desired results.
This simplifies the presentation and highlights the main forces underlying the results. But since
the wage function is endogenous, we then provide conditions on the match output function that
deliver the assumed properties of the wage function for both the TU and the strict NTU cases.
4.1
FOSD Shift in the Shock Distribution
Consider first the case in which t shifts L in the sense of FOSD, that is, the aggregate shock
becomes stochastically better and thus it is more likely that match output will be higher for each
pair. Formally, a family of cdfs {L(α|t)}t∈[0,1] is ordered by FOSD if L(α|·) is decreasing in t for all
α; equivalently, the expectation of any increasing function of α is increasing in t. As mentioned,
the question is: Under what conditions will the measure of agents who invest increase?
It is important to note that the answer to this question is not a forgone conclusion. Indeed, it
is easy to come up with examples where a FOSD shift reduces the measure of agents who invest.
For instance, assume that f (x, y, α) = αxy, wages are bounded below by a positive number
(e.g., attributes are distributed on [x, 1], x > 0, instead of on [0, 1]), and CARA utility, i.e.,
u(w) = −e−Rw . Then the analysis below reveals that a FOSD shift in L decreases the measure
of agents who invest if R is sufficiently large.
We now provide conditions under which a positive answer to this question obtains. By definition of FOSD and by the analysis leading to equation (8), the condition for the comparative
statics result is that vx ≡ u0 (w)wx increases in α, i.e., vxα ≥ 0.8 Formally,
7
A complementary way to grasp the results below is by setting up the problem each θ solves (given θ∗ ) as follows:
Z 1Z 1
max
u(w(x, α, θ∗ ))dHa (x)dL(α|t) − ac(θ).
a∈{0,1}
0
0
Then the conditions just described ensure that the objective function is supermodular in (a, t, θ) and hence by
standard monotone comparative arguments a is increasing in t for each θ, and in θ for each t. For a contrast, the
standard portfolio problem analyzed extensively in the economics of uncertainty literature is of the form
Z 1Z 1
max
u(ax + α)dH(x)dL(α).
a∈[0,1]
0
0
where either H or L is indexed by t. The differences help explain why our conditions are more involved (we need
to account for the endogenous wage function that is nonlinear in the risks).
8
The cross-partial vxα (for an arbitrary v of two risks (x, α)) has appeared in other contexts in the economics of
uncertainty literature. Eeckhoudt, Rey, and Schlesinger (2007) interpret the property vxα ≥ 0 as the agent being
correlation loving: she prefers a 50-50 lottery ((x, α); (x − δ, α − λ)) to the lottery ((x − δ, α); (x, α − λ)), where δ
12
vxα = u00 (w)wα wx + u0 (w)wxα
1 wxα w
0
− R(w) ≥ 0
= u (w)wα wx
w wα wx
(9)
(10)
and this holds if and only if R(w) ≤ (1/w)(wxα w/wx wα ). Notice that if w is log-supermodular in
(x, α), then wxα w/wx wα ≥ 1.9 We thus have the following result:
Proposition 2 A FOSD shift in L increases the measure of agents who invest if either
(i) u is linear and w is supermodular in (x, α); or
(ii) u00 < 0 and R(w) ≤ 1/w, and w is log-supermodular in (x, α).
To gain intuition into this result, first focus on (9). This condition is nonnegative if the
complementarities of attribute x and shock α in the wage are large enough compared to the
decrease in marginal utility that stems from a higher wage. We interpret the second term of (9) as
a substitution effect where some agents switch from not investing to investing, incentivized by the
complementarities between x and now a stochastically higher α. In turn, the first term resembles
an income effect where the better shock increases all wages and therefore reduces marginal utility
of income. If the substitution effect dominates the income effect, our result obtains. Expression
(10) restates both effects as a bound on the coefficient of absolute risk aversion. Proposition 2
provides conditions on the utility and wage functions that lead to the result. Finally, notice that
the special case of risk neutrality is much easier, as it only relies on the substitution effect.
We still need to show that the condition on w in the proposition is not vacuous. The following
corollary derives a class of problems where w satisfies the log-supermodularity condition and thus
a FOSD shift increases the equilibrium number of agents who invest.
Corollary 1 A FOSD shift in L increases the measure of agents who invest under TU if either
(i) u is linear in w; or
(ii) u00 < 0 and R(w) ≤ 1/w for all w, and fx is log-supermodular in (x, α) and in (y, α).
A FOSD shift in L increases the measure of agents who invest under strict NTU if either
(iii) u is linear in w; or
(iv) u00 < 0 and R(w) ≤ 1/w for all w, and f is log-supermodular in (x, α) and in (y, α).
The explanation is simple (see Appendix A.1). Under TU, wxα > 0 and thus if R(w) = 0 for all
w then the result follows solely from complementarities of f in (x, α). And if 0 < R(w) ≤ 1/w
and λ are positive constants. In our context, the cross partial vxα ≥ 0 of v(x, α) = u(w(x, α)) is complicated by
the presence of the endogenous nonlinear wage function, whose behavior we need to account for.
9
Recall that a twice continuously differentiable function z is supermodular in (x, y) if zxy ≥ 0, and it is logsupermodular if log z is supermodular or, equivalently, if zxy z − zx zy ≥ 0.
13
for all w, then the conditions on fx imply that w exhibits stronger complementarities, i.e., it is
log-supermodular in (x, α). A similar explanation holds for the NTU case with f instead of fx .
The conditions in Corollary 1 are satisfied in many natural settings. It is easy to verify that
this is the case with a multiplicative match output function f (x, y, α) = αxy, and utility function
u that exhibits constant relative risk aversion (CRRA)
(
u(w) =
w1−σ −1
1−σ
if σ > 0 & σ 6= 1
log w
if σ = 1,
with the risk aversion parameter σ less than one, for then R(w) = σ/w and wxα w/wx wα = 1.10
4.2
IR Shift in the Shock Distribution
Consider now the case in which t shifts L in the sense of IR, that is, the aggregate shock becomes
Rα
riskier. Formally, a family of cdfs {L(α|t)}t∈[0,1] is ordered by IR if 0 Lt (s|t)ds ≥ 0 for all α and
R1
t, and 0 Lt (s|t)ds = 0 (the mean remains constant). Equivalently, the expectation of any convex
function of α is increasing in t. We seek conditions under which the measure of agents who invest
increases when there is an IR shift in the distribution of the shock.
As with the FOSD shift, we stress that this is not an obvious result. For instance, it will be
clear below that a multiplicative f and a CRRA utility function (with risk aversion parameter
less than one) lead to a decrease in the measure of agents who invest in face of an IR shift.
We now explore under what conditions the comparative statics result holds, i.e. conditions
such that U1 − U0 increases with an IR shift in the distribution of the shock. If both decrease
(increase) with an IR shift, what we need for the desired comparative statics result is that the
decrease (increase) in U1 be smaller (larger) than that in U0 .
Using our v(x, α) notation, equation (8) and the discussion thereafter reveal that it suffices
that vx = u0 (w)wx be convex in α or, equivalently, vxαα ≥ 0 for all (x, α).11
To understand the condition of the ‘third derivative’ of v, consider J(x, t) =
R
v(x, α)dL(α|t)
for any given x. That is, fix an agent at the stage when x has been realized but before the IR shift
in L. For the sake of argument, let v be concave in α, i.e., vαα ≤ 0. Then an increase in t will
reduce the expected utility (after x has been realized), but less so for higher values of x if Jxt ≥ 0,
and this holds if vααx ≥ 0.12 That is, with a higher x the IR shift in L hurts the agent less, which
10
Regarding the counterexample at the beginning of the section, notice that if u is CARA, wages are bounded
away from zero, and f (x, y, α) = αxy, then the condition reduces to 1/w − R, which is negative if R is large enough,
and then the measure of agents who invest decreases with a FOSD shift.
11
Eeckhoudt, Rey, and Schlesinger (2007) interpret the property vxαα ≥ 0 (for any given v of two risks) as the
agent being cross prudent in x: she prefers a 50-50 lottery ((x + , α); (x, α − λ)) to ((x, α); (x + , α − λ)), where λ
is a positive constant and a zero mean random variable.
12
Notice that vxαα ≥ 0 holds if absolute risk aversion with respect to α is decreasing in x, since
2
∂(−vαα (x, α)/vα (x, α))/∂x ≤ 0 ⇔ vααx vα − vαα
≥ 0, and thus it is necessary that vααx ≥ 0. A similar argument applies if the worker is risk loving with respect to lotteries over α.
14
is why he invests. And since investment yields stochastically higher levels of x, if follows that an
IR shift in L provides agents with more incentives to invest. The same is true if v is convex in α,
for then an agent is risk loving with respect to lotteries over α and even more so if x is higher.
To fully understand the effects embedded in vxαα ≥ 0, we need to zoom in. Simple differentiation and manipulation yields
vxαα = u000 (w)wx wα2 + 2u00 (w)wα wαx + u00 (w)wx wαα + u0 (w)wααx
1 2wαx w wαα w
0
2
+ wααx ≥ 0,
= u (w) R(w)wx wα P (w) −
+
w wx wα
wα2
(11)
where P (w) ≡ −u000 (w)/u00 (w) is the coefficient of absolute prudence associated with w. As
with FOSD, this comparative statics result hinges on both agents’ attitudes towards risk and
properties of the wage function, which ultimately depend on the match output function f and
the distributions of attributes H and G via the matching function.
√
√
To understand the following result, recall that a function z is -submodular ( -supermodular)
√
in (x, y) if z is submodular (supermodular) in (x, y) (Eeckhout and Kircher (2010)), and this is
equivalent to zxy z − 0.5zx zy ≤ 0 (zxy z − 0.5zx zy ≥ 0).
Proposition 3 If either of the following conditions holds, then an increase in the riskiness of the
shock increases the measure of agents who choose to invest in attribute x.
(i) u is linear in w and wααx ≥ 0; or
(ii) P (w) ≥ 3/w for all w, w is log-submodular in (x, α) and log-concave in α, with wααx ≥ 0; or
√
(iii) P (w) ≥ 2/w for all w, w is -submodular in (x, α) and log-concave in α, with wααx ≥ 0.
The proof is straightforward and we provide it in the text. Part (i) follows immediately by inspection of (11), since in this case R(w) = 0 for all w and the first term in the square brackets vanishes.
Consider (ii). By log-submodularity and log-concavity, wxα w/wx wα ≤ 1 and wαα w/wα2 ≤ 1, and
hence the first term is nonnegative if P (w) ≥ 3/w for all w. Coupled with wxαα ≥ 0, the result
follows. Regarding (iii), square-root submodularity implies that 2wxα w/wx wα ≤ 1, and we can
weaken the condition on absolute risk prudence to be P (w) ≥ 2/w for all w.
As with the FOSD shift, this result relies on two forces, one related to complementarities of x
and α in the wage function and the other to the agents’ attitude towards risk. We first consider
the risk neutral case, thereby shutting down the first term in (11). Then, a riskier shock induces
more agents to invest if wx is convex in α. This implies that the riskier shock causes an increase
in the mean wage, which is more pronounced for agents with high x (see Appendix A.4).
If instead agents are risk averse, then the first term of (11) plays a central role. The difference
to the FOSD shift is that here prudence, not risk aversion, is the driving force. If agents are
sufficiently prudent, then a riskier shock induces more agents to invest in their attribute. As is
well-known, prudence implies downside risk aversion and triggers precautionary actions to insure
15
against bad realizations of a shock. Similarly here, a riskier shock induces sufficiently prudent
agents to engage in precautionary investments. How large prudence needs to be depends on curvature and complementarity properties of the wage function. If the wage function is convex in α,
a riskier shock generates more upside compared to downside risk. Since prudent agents particularly dislike downside risk (but do not mind upside risk as much), only prudent enough agents
would make the precautionary investment in their attributes. Moreover, if the wage function is
complementary in x and α, a riskier shock exposes agents with high x even more to this risk
than those with low x (since for high x the difference in payoffs between high and low shock
is much larger), reducing the incentives for precautionary investment. Both of these properties
of the wage function imply that only sufficiently prudent agents would invest, which is why our
sufficient conditions rely on submodularity and concavity of the wage function.
Our next task is to provide classes of utility functions and match output functions that lead
to these conditions on attitudes toward risk and the equilibrium wage function.
Regarding the utility function, our result holds for the standard ones used in economic applications. Indeed, consider the CRRA u. Then it is straightforward to show that P (w) = (σ +1)/w.
Hence, σ = 2 or 3 satisfy the condition in Proposition 3. For another u that satisfies the condition
on P , consider the CARA case u(w) = −eRw /R, R > 0. Then P (w) = R, and if R is large enough
and w is bounded away from zero, then first term of (11) is positive.
It should be clear below that the conditions in the strictly risk averse case are only sufficient,
since they ensure that each of the two terms in (11) is nonnegative. But we will see that they are
easy to interpret from primitives in some natural classes of problems.
The first class of problems for which the comparative statics result holds is straightforward.
Corollary 2 Let f be such that fxαα ≥ 0 if TU or fαα ≥ 0 if strict NTU. If either u is linear or
if R(w) is sufficiently small for all w, then an increase in the riskiness of the shock increases the
measure of agents who choose to invest.
To see this, notice that the first term in (11) either vanishes or is close to zero and thus dominated
Rx
by the second term, which is nonnegative with TU if fxαα ≥ 0 as wxαα = 0 fxαα , and with NTU
if fαα ≥ 0 as wxαα = 0.5fαα . For a simple example, let u(w) = −e−Rw /R, f (x, y, α) = η(α)xy, η
strictly increasing and strictly convex. Then the corollary holds if R > 0 is sufficiently small.
Beyond the case of small risk aversion, we have the following result for the strict NTU:
Corollary 3 Assume strict NTU. If either of the following conditions hold, then an increase in
the riskiness of the shock increases the measure of agents who choose to invest.
(i) P (w) ≥ 3/w for all w, f is log-submodular in (x, α) and in (y, α), convex and log-concave in
α, with fααx ≥ 0; or
(ii) P (w) ≥ 2/w for all w, f is
√
-submodular in (x, α) and in (y, α), convex and log-concave in
α, with fααx ≥ 0.
16
The conditions on f ensure that wages are log-submodular in (x, α) and log-concave in α (see
Appendix A.2), and thus the first term in (11) is nonnegative if P (w) is large enough for all w,
while the last condition in (i)–(ii) ensure that the last term is nonnegative.
An analogous general result is not available under TU, since log-submodularity is not preserved
under integration and thus we cannot directly map such a property of f into properties of w. This
difficulty notwithstanding, we provide two results for the TU case that focus on canonical classes
commonly used in economic applications.
The first class is the oft-used case where output is multiplicatively separable in the shock.
Corollary 4 Assume TU and let f (x, y, α) = η(α)z(x, y), with η strictly increasing, convex, and
log-concave in α, and z strictly increasing in each argument and strictly supermodular in (x, y).
If u is such that P (w) ≥ 3/w, then an increase in the riskiness of the shock increases the measure
of agents who choose to invest.
Here is the proof. Under the conditions on η and z, the last term of (11) is positive (it is given by
Rx
Rx
wxαα = η 00 zx > 0), and simple algebra reveals that wxα w/wx wα = zx α 0 zx /αzx 0 zx = 1 and
thus the expression in parentheses in the first term is nonnegative if and only if
1
P (w) ≥
w
η 00 (α)η(α)
2+
.
η 02 (α)
If in turn η is log-concave in α, then it suffices that P ≥ 3/w since the last term is less than one.
For a closed form example, if η(α) = αn , then P (w) ≥ (1 + ((n − 1)/n))/w yields the comparative
statics result. So when n = 2 the bound is P (w) ≥ 2.5/w, and when n = 1 the last term in (11)
is zero and it suffices that P (w) ≥ 2/w for the result to hold.
The second class for the TU case is defined by a bilinear match output function f (x, y, α) =
ax + αxy + by (e.g., see Anderson and Smith (2010)).
Corollary 5 Assume TU and f (x, y, α) = ax + αxy + by. If either of the following conditions
hold, then an increase in the riskiness of the shock increases the measure of agents who invest.
(i) a ≤ 0, and P (w) ≥ 2/w for all w; or
(ii) a ≥ 0, x = y > 0, and P (w) ≥ κ/w, where κ = 1/y > 0.
The proof is in the Appendix but the logic of the result is straightforward. First, notice that this
match output function implies wαα = wxαα = 0. Hence, the comparative statics result ensues if
and only if P (w) ≥ (1/w)(2wxα w/wx wα ). Now, if a ≤ 0, one can show that w is log-submodular
in (x, α), and part (i) follows. And if a ≥ 0 then w is log-supermodular; but if w is bounded away
from zero, then there is an upper bound for the ratio wxα w/wx wα , and this yields part (ii).
17
5
Implications for Matching and Wages
Recall that an equilibrium consists not only of the threshold θ∗ whose behavior we extensively
analyzed above, but also of the matching function µ and the wage function w. What are the
effects on µ and w of a FOSD or IR shift in the distribution of the shock?
Under either shift, the effect on µ is unambiguous when our comparative statics result holds:
it decreases for every x as the following expression shows (see equation (4) and recall θt∗ ≤ 0)
∂µ(x, θ∗ )
H0 (x) − H1 (x) ∗
=
θ ≤ 0.
∂t
g(µ(x, θ∗ )) t
This is due to the improvement in the distribution H = (1 − θ∗ )H1 + θ∗ H0 as more people invest,
reinforcing the competition for agents with better y. As a result, this lowers the characteristic
y = µ(x, θ∗ ) with whom x matches in equilibrium.
We are also interested in how the discussed shifts in risk affect the wage function w. We will
focus on the change in wages with a change in risk after the realization of the attribute x. At
R
that stage, the expected wage of an agent with x is w(x, α, θ∗ )dL(α|t). Therefore
∂
R
w(x, α, θ∗ )dL(α|t)
=
∂t
Z
w(x, α, θ∗ )dLt (α|t) +
Z
wθ∗ (x, α, θ∗ )θt∗ dL(α|t).
(12)
The effect of a change in risk on the expected wage can be decomposed into two parts. The first
term in (12) is the direct effect of a change in the distribution of the shock α on the expected
wage. The second term representes the indirect effect on it through the matching function.
For the FOSD shift, the direct effect is positive (given θ∗ , wages tend to increase under a
stochastically better shock) but the indirect effect is negative (wages tend to decrease because
the matching function decreases when more agents invest), making the overall effect on wages
ambiguous. The net effect depends on how steep the cost function c is. If it decreases sufficiently
fast, then the effect of a change in risk on investment (i.e., the effect on θ∗ ) will be small, leading
to unambiguously higher wages in face of a stochastically better shock.
For the IR shift, the indirect effect of an increase in investment plays out the same way but
the direct effect depends on the curvature of the wage function in α. If convex, then the direct
effect of an IR shift on wages is positive while it is negative if concave. Therefore, the two effects
reinforce each other if the wage function is concave in α, thereby leading to unambiguously lower
expected wages for all agents with attribute x as a consequence of a riskier shock. For example,
expected wages decrease for all agents under TU if f (x, y, α) = αxy and u is CRRA with σ = 2.13
13
The indirect effect is absent in the one population model mentioned at the end of Section 2, since µ = x implies
w(x, α, θ∗ ) = f (x, x, α)/2 in both the TU and strict NTU cases, so wages are independent of θ∗ . Hence, the effect
of changes in risk on wages is unambiguous in this variation of the model.
18
6
Changes in G and H1
We have focused on the effects of changes in the distribution of α since it is the most interesting
comparative statics exercise in our model, and it fits nicely with our applications. But one can
also analyze the implications of changes in G or Hi , i = 0, 1, and we do so now for completeness.
Since we assume that H1 dominates H0 is FOSD, we analyze how the equilibrium changes upon
a FOSD shift in G and H1 (i.e., either the distribution of characteristics on the risk neutral side
improves, or the distribution of attributes if an agent invests becomes stochastically better than
before, e.g., the quality of education improves). Given that α plays a minor role for these results,
we will henceforth ignore it and assume that α = 1.
Consider first a change in G. To this end, let G(·|t) be the distribution of y, and assume that
G decreases in t (FOSD). Given any θ∗ , matching is now given by
G(µ(x, t, θ∗ )|t) = θ∗ H0 (x) + (1 − θ∗ )H1 (x) ⇒ µ(x, t, θ∗ ) = G−1 (θ∗ H0 (x) + (1 − θ∗ )H1 (x)|t).
An increase in t decreases G(µ|t) and thus increases µ, so µt ≥ 0. This is intuitive: if the
distribution of y improves, there is less competition for those with high y and hence any agent
with attribute x is matched to a weakly better partner. Then wt ≥ 0, and differentiation reveals
that wxt ≥ 0 if TU, and also if strict NTU when f is convex in y and µ is supermodular in (x, t).
R1
Following then the same steps as in Section 4 we obtain that U1 − U0 = 0 (H0 − H1 )vx dx,
and this expression increases in t if vx increases in t or, equivalently,
00
0
vxt = u (w)wx wt + u (w)wxt
1
≥ 0 ⇒ R(w) ≤
w
wxt w
wx wt
.
(13)
Thus, under this condition, which has a similar intuition as the one in Section 4 based on substitution and income effects, a FOSD shift in G increases the measure of agents who invest. If the
sign is reversed, we obtain the opposite comparative statics result.
Things are more complex if the difference in H0 and H1 becomes larger, i.e., if we let H(·|t)
be indexed by t and a shift in t improves H1 in FOSD sense. Now there are two effects, a direct
one via H1 and an indirect one via µ, which is an equilibrium object. To see this, notice that now
both terms in the integrand of U1 − U0 above, H0 − H1 and vx , depend on t. Therefore
∂(U1 − U0 )
=
∂t
Z
0
1
∂H1
−
∂t
Z
1
(H0 − H1 ) vxt dx.
vx dx +
0
The first term is nonnegative by FOSD. The second term, however, has an ambiguous sign. It is
nonnegative and reinforces the first term if vxt ≥ 0, which occurs when R(w) ≥ (1/w)(wxt w/wx wt )
(H1 decreasing in t implies that µt ≤ 0 and hence wt ≤ 0, which is why the sign in condition (13)
reverses). Under this condition then U1 − U0 increases in t and more people invest after a FOSD
19
shift in H1 . But if the reverse inequality holds, then vxt ≤ 0 and the result is ambiguous (the two
terms in the derivative of U1 − U0 have opposite signs).
7
Applications
Labor Markets. Despite a strong increase in the skill premium since the beginning of the 80s, the
growth in educational attainment of U.S. men has not been keeping up pace (e.g. Goldin and
Katz (2008)). At the same time, there has been a shift in the composition of earnings variability
with the variance of the permanent income component decreasing relative to the variance of the
transitory component, which agents insure more easily through borrowing and saving (Blundell,
Pistaferri, and Preston (2008)). We argue that the incentives to acquire education not only
depend on average returns but, crucially, on earnings variability in permanent income. For when
earnings are very volatile agents are encouraged to obtain education for self-insurance, and the
opposite is true when earnings are more stable, as in recent years. (Blundell, Pistaferri, and
Preston (2008) show that educated workers can insure better against permanent income shocks
while education does not seem to matter for the insurability of transitory shocks.) Our model
can rationalize that workers use education as a precautionary investment against riskier earnings
shocks, thereby providing a unified explanation for the observed facts of the slowing growth in
educational attainment and the decline in earnings’ variability in permanent income.
To fit the application with our model, we give the model the following interpretation. There is
a large number of risk averse workers and a large number of firms that are heterogeneous in their
productivity. There are two stages: an education investment stage and a matching stage. In the
first stage workers simultaneously decide whether to invest in education. Education is costly, and
workers are ex-ante heterogeneous in the investment cost. The benefit of obtaining an education
is that a worker’s skill is drawn from a stochastically better distribution. At the beginning of
the second stage, workers observe their skills and also the realization of an aggregate shock on
production. Technology exhibits complementarities in the attributes of the matched worker-firm
pair, so workers and firms match in a positively assortative way in this frictionless market.
The decision to acquire education not only depends on the worker’s initial ability θ that
determines his cost of education. It crucially depends on the distribution of α. Under the
conditions in Section 4.2, a riskier shock induces more agents to invest in education. As mentioned,
education plays the role of a precautionary investment that provides self-insurance against bad
realizations of the shock. An inverse IR shift in α (which is how we capture the observed decline
in volatility) reduces the role of education for self-insurance and thus the incentives to obtain
education, which aligns well with the stagnating educational attainment observed in the data.
In a companion paper (Chade and Lindenlaub (2015), in progress), we explore the insurance
role of education in labor markets in detail. We develop a dynamic version of our model in which
20
workers have multi-dimensional skill bundles. Investment in education improves the skill bundle.
Crucially, if one of the skills is hit by a shock, educated workers can rely on a stronger second skill
as a source of income. This diversification through education strengthens its insurance role and
is not present in the current model where workers have a single skill. We then bring the dynamic
multi-dimensional framework to the data to quantify the degree of prudence necessary to match
the slow down in educational attainment in the face of declining volatility in permanent earnings.
Marriage Markets. A recent interesting phenomenon is the drastic change in the dating/marriage
market technology, triggered by the advancement of social media and changes in the gender composition at both school and work. At the same time, the educational attainment of women has
significantly increased. Our model can connect these two seemingly unrelated facts.
We focus here on the educational choices of women as their educational attainment has undergone huge changes over the last decades, showing a strong increase. In contrast, as mentioned
above the growth of men’s educational attainment has considerably slowed down, which is why
we assume that they are the side of the market with ex-ante fixed characteristics.
An important feature of this application is how to interpret the aggregate shock α. The
marriage market underwent substantial changes in matching and meeting technologies of potential
spouses. First, increasingly gender-integrated workplaces now create greater opportunities for
men and women to meet. Second, a related shock comes from the now balanced sex-ratios
at universities. Third, the internet with its dating platforms drastically changed the marriage
market’s matching technology (see Stevenson and Wolfers (2007)). An attractive interpretation
of the shock is thus the recent advancements in social media and gender composition changes at
work that both facilitate interactions but also foster marital disruption.
This type of shock may have two distinct effects on marriage: On the one hand, this shock
enlarges agents’ choice sets and allows them to screen partners more thoroughly, improving the
match quality and reducing the probability of splitting up. On the other hand, divorce becomes
more likely because married individuals meet new potential partners at a higher rate, increasing
the likelihood of breaking up.14 In other words, the improvements in the matching technology
encourages some sort of on-the-match search. We model this increased chance of extreme marriage
outcomes (match of high quality versus divorce) through an increase in the riskiness of the shock
that affects marriages.
The model is the same as before with the following interpretation. There are two stages,
investment decision and marriage market. Women face a choice to invest in an attribute, for
instance, education, that makes them more attractive on the marriage market. After this investment choice, a shock to marriage output is realized. Women then enter the marriage market,
14
One way to model this is to assume that v(x, α) = p(α)u(w(x, θ∗ )). That is, α determines the probability p(α)
that marriage materializes and each partner enjoys their marital output (and with complementary probability, each
obtains zero). Then vxαα ≥ 0 if p is convex in α.
21
where they match with men in an assortative way. Under the conditions of Section 4.2, an IR shift
in the distribution of the shock, induces more women to obtain education. Our model therefore
contributes to explain the recent increase in women’s educational attainment, based on differential
risk attitudes between men and women as well as changes in the marriage market technology.
Entrepreneurs’ Markets. During the Great Recession, the US economy experienced a significant increase in economic volatility, reflected, for instance, by a larger volatility of equity returns.
At the same time, firms selected into riskier projects with higher expected returns (Navarro
(2014)). We will offer an interpretation of our model, based on matching between entrepreneurs
and venture capitalists, to link these two stylized facts.15
Consider a market where one side consists of entrepreneurs, each of them having a potential
idea whose quality is θ ∈ [0, 1]. On its own, an entrepreneur’s idea will develop into a marketable
start up of quality x ∼ H0 . But at a cost c(θ), an idea of initial quality θ will turn into a start
up of increased quality x ∼ H1 . To succeed, any start up requires funding. There is a large
population of risk neutral venture capitalists on the other side of the market who differ in their
‘liquidity’ y, and who match with the entrepreneurs to form partnerships (each venture capitalist
invests in one and only one start up). After x is realized but before funding takes place, there is
an aggregate shock that affects the market and thus the profits that each partnership can obtain.
We want to know how an increase in the riskiness of the shock, a metaphor for the increased
economic volatility during the Great Recession, affects investment in better quality start ups.
To address this question with our model in a simple way, we use the example of a multiplicative
match output function f (x, y, α) = α2 xy and assume that the condition on prudence in Corollary
4 holds. Then, an IR shift in the distribution of the aggregate shock induces more entrepreneurs
to upgrade the quality of their projects. This increase in the measure of entrepreneurs with
high quality projects then maps into properties of the matching and payoff functions. Every
entrepreneur will match with a venture capitalist of lower liquidity and the expected returns
(wage function) increases when the cost function is sufficiently steep so that the equilibrium
effects of a change in α do not offset its direct effects on production (see Section 5). Moreover,
under the same assumption of a sufficiently steep cost function, one can show that the variance
of returns (wage function) increases, and this effect is more pronounced for entrepreneurs who
upgraded their idea (i.e., those who draw a stochastically higher x). In other words, high-x
projects are riskier than low-x ones. To see the effect on the variance of returns formally, consider
the variance of an entrepreneur before the realization of the shock but after the realization of his
2
Rx
project’s characteristic x: it is given by Var[w(x, α, θ∗ )] = E[α4 |t] 0 µ(s, θ∗ )ds .
15
An alternative explanation for a shift towards risk-taking entrepreneurship is that the gender composition of
entrepreneurs has significantly shifted towards men during the crisis (Thebaud and Sharkey (2014)). Much evidence
suggests that men have been shown to be more risk takers than women (Bertrand (2011)), which is why this change
in gender composition would lead to riskier start up activity than before the crisis.
22
An IR shift in the distribution of the shock α affects the variance as follows:
∂Var[w(x, α, θ∗ )]
∂E[α4 |t]
=
∂t
∂t
Z
x
∗
µ(s, θ )ds
2
4
Z
+ 2E[α |t]
0
x
Z
µ(s, θ )ds
0
∗
x
∗
µθ∗ (s, θ )ds θt∗
0
where the first term is positive by the IR shift and the second term vanishes when the cost
function is sufficiently steep (see Section 5). Notice that the increase in variance is larger for
those entrepreneurs who invest since they tend to have a larger x. In sum, our simple mechanism
can account for the two conjoined facts in the data: the increase in economic volatility and the
increase in investment into projects with higher expected but more dispersed returns.
Directed Technological Change. A vivid debate has emerged on the relation between technological change and labor market outcomes. It is uncontroversial that skill-biased technological
change affects individuals’ investment into human capital since more sophisticated technologies
require workers to have more skills to operate them. But it is also natural that firms’ investment
into technology responds to changes in skill supply – a phenomenon that has been named directed
technological change (e.g., Acemoglu (1998) and Acemoglu (2002)). We present in this application
a tractable extension of our model to two-sided investment, with workers investing into skills and
firms into technology, which captures the phenomenon of directed technological change.
We interpret the risk-averse side of the market as workers who choose whether to invest in
education in order to improve their skills x. On the other side of the market, there are firms that
face the choice of investing in technology, in which case they draw their y from a stochastically
better distribution. To capture the phenomenon of directed technological change, we focus on
the case in which both investments are strategic complements, meaning that more investment into
skills triggers more investment into technology and vice versa. We are interested in how changes
in the risk of the aggregate shock α affect both workers’ and firms’ investments. In particular,
when do FOSD and IR shifts in the shock lead to more investment?
Since this is a non-trivial extension of our model, we now describe the setting in more detail
but place most of the technical details in the Appendix. Let y ∼ G1 if an agent invest and y ∼ G0
without investment, G1 (y) ≤ G0 (y) for all y, and assume ξ ∼ U[0,1] indexes the investment cost
of firms. Intuitively, there will be a threshold ξ ∗ such that risk neutral firms invest if and only
if ξ ≥ ξ ∗ . An equilibrium is a fourtuple (θ∗ , ξ ∗ , µ, w) satisfying the usual best response and
market clearing properties, with µ and w functions of (x, θ∗ , ξ ∗ , α). The analysis of the matching
stage is as before, except that µ does not have an explicit functional form, but it is increasing
in x and in θ∗ , and decreasing in ξ ∗ . Equilibrium existence reduces to finding (θ∗ , ξ ∗ ) such that
U1 (θ∗ , ξ ∗ , t)−U0 (θ∗ , ξ ∗ , t) = c(θ∗ ) for the risk-averse workers and V1 (θ∗ , ξ ∗ , t)−V0 (θ∗ , ξ ∗ , t) = c(ξ ∗ )
for the firms, where Vi , i = 0, 1 are the analogues of Ui , i = 0, 1, for the risk neutral side.
We now discuss some conditions under which we can sign our comparative statics results.
Assume that U1 − U0 is decreasing in ξ ∗ and so is V1 − V0 in θ∗ . Notice that U1 − U0 = c defines
23
a ‘best response’ function θ∗ = φ(·, t) that is increasing in ξ ∗ . Similarly, V1 − V0 = c defines
ξ ∗ = ψ(·, t) increasing in θ∗ . Intuitively, investment decisions are strategic complements (i.e., an
increase in investment on one side increases the incentives on the other). An equilibrium is stable
if φ0 ψ 0 < 1 (i.e., ψ is steeper than φ on [0, 1]2 with θ∗ on the vertical axis).
Take any stable equilibrium. Then both an IR and a FOSD shift in α shift ψ and φ downward.
If condition (11) holds, these shifts decrease both the equilibrium θ∗ and ξ ∗ (See Appendix A.5
for details). This leads to the natural comparative statics results that we extensively discussed
above. First, under a FOSD shift in α, all agents in the market want to take advantage of this
shift and the underlying complementarities between the shock and their attributes. They do so by
investing more, and the presence of strategic complementarities reinforces the actions of workers
and firms. Second, if workers are prudent enough, then they invest more in education in the
face of a riskier shock. Since investments are strategic complements, firms take advantage of the
now more educated work force by upgrading their technology as well, meaning they direct their
technological change towards the new skill supply.
8
Concluding Remarks
In many economic situations, agents invest in a payoff-relevant characteristic before entering a
matching market. This invariably takes place under a considerable amount of uncertainty, both
about the final quality of the characteristic and the conditions that will prevail at the matching
stage. For instance, individuals invest in education without knowing how skilled they will turn
out to be or what the aggregate state of the economy is when they enter the labor market.
Intuitively, besides the complementarities in production, the individual’s attitude toward risk
play a crucial role in this decision. This paper develops a general matching model with pre-match
risky investment. After deriving the equilibrium of the model, we analyze the effects of changes in
risk. Focusing on the standard FOSD and IR shifts in an aggregate shock, we provide conditions
under which more agents invest when the shock is stochastically better or riskier. For instance,
we show that under an IR shift, more agents invest if they are sufficiently prudent to trigger a
precautionary investment motive. We also describe how the other equilibrium objects, matching
and wages, change with these shifts in risk. To illustrate the usefulness of the model, we apply it
to labor, marriage, and entrepreneur markets, as well as to directed technological change, and we
shed light on some interesting stylized facts in each of these applications.
Our model overcomes a major shortcoming of standard assignment models, which assume that
agents do not adjust their characteristics in the face of aggregate changes in risk. We therefore
believe that this tractable model can serve as a building block for richer frameworks – e.g., with
mutidimensional heterogeneity and dynamic considerations – that are suitable for empirical work.
24
A
Appendix
A.1
Proof of Corollary 1
Parts (i) and (iii) are immediate since the assumptions on f imply that w is supermodular in
(x, α) in both the TU and strict NTU cases, and hence vxα (x, α) ≥ 0 for all (x, α).
Consider (ii). Rewrite the wage function in the TU case as follows:
w(x, α, θ∗ ) =
Z
1
I[0,x] (s)fx (s, µ(s, θ∗ ), α)ds.
(14)
0
It is well-known (Karlin and Rinott (1980)) that w is log-spm in (x, α) if the integrand is logsupermodular in (x, s, α) for any θ∗ , and this holds if I is log-supermodular in (x, s) and fx is
log-supermodular in (s, α).
It is easy to verify that fx is log-supermodular in (s, α) if and only if
(fxxα fx − fxα fxx ) + (fxyα fx − fxα fxy ) µx ≥ 0,
which holds if the parentheses are nonnegative, i.e., if fx is log-supermodular in (x, α) and (y, α).
Similarly, if we take any two pairs (s, x), (s0 , x0 ), it is easy to check that
I[0,x∨x0 ] (s ∨ s0 )I[0,x∧x0 ] (s ∧ s0 ) ≥ I[0,x] (s)I[0,x0 ] (s0 ),
and thus I is log-supermodular in (x, s).
Hence, w is log-spm in (x, α) and thus wxα w/wx wα ≥ 1. It follows that R(w) ≤ (1/w)(wxα w/wx wα )
if R(w) ≤ 1/w. Therefore, under the conditions assumed in the proposition, a FOSD shift in L
increases U1 − U0 and this decreases the equilibrium value of θ∗ . Hence, the measure of agents
investing increases, thereby completing the proof of (ii).
Regarding (iv), it is straightforward to show that w(x, α, θ∗ ) = f (x, µ(x, θ∗ ), α)/2 is logsupermodular in (x, α) if f is log-supermodular in (x, α) and in (y, α). The rest of the proof
follows along the same line of case (ii) and it is omitted.
A.2
Proof of Corollary 3
We only prove that the conditions in f in part (i) provide the needed conditions in w. And since
the proof of part (ii) is analogous, we omit it. Recall that w(x, α) = f (x, µ(x, θ∗ ), α)/2. Then
easy algebra reveals that w is log-submodular in (x, α) if and only if
wxα w − wx wα = (fxα f − fx fα ) + (fyα f − fy fα )µx ≤ 0,
and this holds if f is log-submodular in (x, α) and in (y, α). Similarly, w is convex and
25
log-concave in α if and only if wαα = fαα /2 ≥ 0 and wαα w/wα2 = fαα f /fα2 ≤ 1.
A.3
Proof of Corollary 5
The wage function is given by w(x, α) =
Rx
∗
0 (a + αµ(s, θ ))ds = ax + α
Rx
0
µ(s)ds. Hence
Rx
Z x
axµ(x, θ∗ ) + αµ(x, θ∗ ) 0 µ(s)ds
wxα w
∗
Rx
µ(s)ds
≶ 0.
= Rx
≶
1
⇔
a
xµ(x,
θ
)
−
wx wα
a 0 µ(s)ds + αµ(x, θ∗ ) 0 µ(s)ds
0
Since µ is strictly increasing in x, it follows that xµ(x, θ∗ )−
Rx
0
(15)
µ(s, θ∗ )ds > xµ(x, θ∗ )−xµ(x, θ∗ ) =
0. Therefore, if a ≤ 0 then w is log-submodular in (x, α) and part (i) follows.
Assume instead that a ≥ 0. Then w is log-supermodular in (x, α). Since the left side of (15)
is of the form (A + B)/(C + B), A > C, and it is decreasing in B, it follows that
Rx
axµ(x, θ∗ ) + αµ(x, θ∗ ) 0 µ(s)ds
xµ(x, θ∗ )
Rx
Rx
Rx
<
.
a 0 µ(s)ds + αµ(x, θ∗ ) 0 µ(s)ds
0 µ(s)ds
If x > 0 and y > 0, then
xµ(x, θ∗ )
µ(1, θ∗ )
1
Rx
<
= .
∗
µ(x, θ )
y
x µ(s)ds
Setting κ = 1/y completes the proof of part (ii).
A.4
Proof of Supermodularity of the Expected Wage
We prove an assertion made in Section 4.2, that the mean wage is supermodular in (x, α) under
the assumptions of Proposition 3. To use calculus, let s be an index for the family {H(·|s)}s∈[0,1] ,
where higher s implies better H in the FOSD sense. Then the mean of the wage function is
∗
1Z 1
Z
w(x, α, θ∗ )dH(x|s)dL(α|t)
Z 1
Z 1
=
w(1, α, θ∗ ) −
wx (x, α, θ∗ )H(x|s)dx dL(α|t)
E[w(α, x, θ )|s, t] =
0
0
0
0
where the second equality follows from integration by parts. We want to know if the cross-partial
of the expected wage with respect with s and t is positive. Differentiating with respect to s yields
∂E[w(x, α, θ)|s, t]
=
∂s
Z
1 Z 1
∗
wx (x, α, θ )(−Hs (x|s))dx dL(α|t) ≥ 0.
0
(16)
0
Since Hs ≤ 0 by strict FOSD, (16) is increasing in t if wxαα ≥ 0, which proves the assertion.
26
A.5
Two-Sided Investment
We now sketch the equilibrium of the model with two-sided investment and justify the assertions
made in the direct techological change application. Our goal is not to derive a full blown solution
to this extension, which is beyond the scope of the paper, but to provide enough detail for
the application at hand. An equilibrium is a four-tuple (θ∗ , ξ ∗ , µ, w) such that, given (w, µ),
the investment decision of each agent is optimal, i.e., θ∗ and ξ ∗ satisfy U1 (θ∗ , ξ ∗ , t) − c(θ∗ ) =
U0 (θ∗ , ξ ∗ , t) and V1 (θ∗ , ξ ∗ , t) − c(ξ ∗ ) = V0 (θ∗ , ξ ∗ , t), respectively, where
∗
Z
∗
1Z 1
Ui (θ , ξ , t) =
0
Vi (θ∗ , ξ ∗ , t) =
Z
0
0
1Z 1
u(w(x, α, θ∗ , ξ ∗ ))dHi (x)dL(α|t) i = 0, 1
(f (µ−1 (y, ξ ∗ , θ∗ ), y, α) − w(µ−1 (y, ξ ∗ , θ∗ ), α, θ∗ , ξ ∗ ))dGi (y)dL(α|t) i = 0, 1,
0
and for any (θ∗ , ξ ∗ ) and any realization of α, (w, µ) clears the market at the matching stage.
We solve the equilibrium problem backwards. Notice that for any subgame after (θ∗ , ξ ∗ , α)
the solution to the matching problem is exactly the same as before, except that now G(y, ξ ∗ ) =
ξ ∗ G1 (y) + (1 − ξ ∗ )G0 (y). Hence, µ is implicitly and uniquely defined by
ξ ∗ G0 (µ(x, ξ ∗ , θ∗ )) + (1 − ξ ∗ )G1 (µ(x, ξ ∗ , θ∗ )) = θ∗ H0 (x) + (1 − θ∗ )H1 (x).
Now there is no explicit solution for µ but we can still derive the main properties (increasing in x,
increasing in θ∗ , and decreasing in ξ ∗ ). The rest of the analysis for the second stage is as before
for both the TU and NTU cases, except for the presence of ξ ∗ in µ.
Equilibrium existence reduces to finding thresholds θ∗ and ξ ∗ that satisfy the indifference
conditions above. Consider U1 (θ∗ , ξ ∗ , t)−c(θ∗ ) = U0 (θ∗ , ξ ∗ , t). It defines a ‘best response’ function
θ∗ = φ(ξ ∗ , t). Similarly, V1 (θ∗ , ξ ∗ , t) − c(ξ ∗ ) = V0 (θ∗ , ξ ∗ , t) defines a ‘best response’ function
ξ ∗ = ψ(θ∗ , t).16 Hence, an equilibrium is a fixed point of this system.
Proving existence now is involved since this is a system of highly nonlinear equations, and it
is not the focus here. For the sake of our application, we assume an equilibrium exists and focus
on the comparative statics. To shorten notation, we respectively write the two equations as
z(θ∗ , ξ ∗ , t) = c(θ∗ )
(17)
m(θ∗ , ξ ∗ , t) = c(ξ ∗ ).
(18)
We make the following additional assumptions (which we could also state in terms of primitives). First, we assume that there is a unique θ∗ for each ξ ∗ , which we know requires zθ∗ ≥ 0.
Similarly, there is a unique ξ ∗ for each θ∗ , which requires mξ∗ ≥ 0. Second, we assume that the
16
These are not really best response functions since they already incorporate the wage function in it, whereas
each individual agent has no power over equilibrium thresholds.
27
two investments are strategic complements: An increase in ξ ∗ decreases U1 − U0 (if fewer firms
invest, this gives workers less incentives to invest), so that zξ∗ ≤ 0. Similarly, an increase in θ∗
decreases V1 − V0 (if fewer workers invest, then incentives are lower for firms to invest), so that
mθ∗ ≤ 0. Finally, we assume that z and m are increasing in t.
We seek the result that an increase in t (FOSD or IR shift) decreases both θ∗ and ξ ∗ . Total
differentiation of system 17-18 gives:
θt∗ =
ξt∗ =
(zθ∗
(zθ∗
−zt (mξ∗ − c0 (ξ ∗ )) + mt zξ∗
− c0 (θ∗ ))(mξ∗ − c0 (ξ ∗ )) − mθ∗ zξ∗
zt mθ∗ − mt (zθ∗ − c0 (θ∗ ))
.
− c0 (θ∗ ))(mξ∗ − c0 (ξ ∗ )) − mθ∗ zξ∗
(19)
(20)
Notice that both numerators of (19) and (20) are nonpositive. Thus, the natural comparative
static result requires that both denominators are positive.
What drives the sign of the denominators is the relative slope of the two best-response functions. The slope of z = c on the (θ∗ , ξ ∗ ) space is ∂θ∗ /∂ξ ∗ = −zξ∗ /(zθ∗ − c0 (θ∗ )) ≥ 0. In turn,
the slope of m = c is ∂θ∗ /∂ξ ∗ = −(mξ∗ − c0 (ξ ∗ ))/mθ∗ ≥ 0. Now, the denominator of the above
equations is positive if and only if the slope of the ‘best response’ function of the workers is flatter
than that of the firms. One way to ensure the right slopes (flatter z = c) is to multiply the cost
function by λ > 0, so that the marginal cost is also multiplied. Then by making λ large we make
z = λc as flat as we want and m = λc as steep as we want.
28
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