SciencesPo Paris
Master Economics and Public Policy — PhD Track
Steady-state Model of an Economy with Search
Frictions and Monopolistic Competition.
Margaux Luflade
Master’s dissertation
Supervisor: Etienne Wasmer
Steady-state Model of an Economy
with Search Frictions
and Monopolistic Competition.
May 21st, 2012
Margaux Luflade
Abstract. Recently, search and matching models have been used to investigate the
extent to which labor market variables can be impacted by the presence of frictions
on various markets. We build on Wasmer and Weil (2004) and introduce the need for
capital inputs in the firms’ production function. Introducing intermediate consumption leads us to specify the production function of final goods firms and therefore
the way intermediate goods interact in the production process. As the intermediate
goods market considered is not only frictional but also imperfectly competitive, a
contribution of this paper is to suggest a way to model a market cumulating both
types of market imperfections and see how they can interact. We use the suggested
framework to analyze how labor market variables, more particularly the equilibrium
wage and unemployment rate, can be impacted by strategies to reduce the level of
frictions or policies of deregulation. Finally, we extend the basic model by following
Wasmer (2009) and introducing frictions on the final goods market. In this context,
we take the opportunity to suggest an alternative interpretation of the final good
production function.
I am particularly grateful to Etienne Wasmer for the time and energy he spent supervising
my work, providing comments, advice and insights and trying to understand the details of
my (sometimes convoluted) reasoning.
Master EPP — SciencesPo 2012
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Introduction and literature review
Recently, search and matching models have been used to investigate the extent to which labor
market variables can be impacted by the presence of frictions on various markets. Building
on Pissarides’ (1990) equilibrium unemployment theory, Wasmer and Weil (2004) introduce
frictions on the credit market where entrepreneurs have to search for the funds they need to
hire a worker. They characterize a financial multiplier which stem from the interdependence
of markets, that is, from the fact that the respective states of the credit and labor market
mutually impact on each other. Wasmer (2009) adds a third layer of frictions in the life of the
firm by considering an imperfect goods market. Here, we suggest an alternative way to consider the impact on labor market variables of the existence of frictions on the goods market.
Instead of introducing frictions downstream from production we add them upstream, on the
intermediate goods market on which the (final good) firm needs to buy the capital inputs entering its production function. Intermediate consumption, defined as “the value of the goods
and services consumed as inputs by a process of production, excluding fixed assets whose
consumption is recorded as consumption of fixed capital”1 , indeed seems to constitute a large
part of the economic activity. In 2008, intermediate consumption represented 45%2 of the
total gross output for the US and 51% for France. As a comparison, GDP represented 54%
of gross output for these two countries. Besides, as the search for capital inputs constitutes
a pre-requisite to production, we expect the level frictions the intermediate goods market to
impact on labor market variables in a symmetric way as frictions on the credit market do
(rather than as frictions on the final goods market which are met after production).
Introducing intermediate consumption leads us to specify the production function of final
goods firms and therefore the way intermediate goods interact in the production process.
Further details will of course be provided below, but we point here that we make the assumption of imperfectly substitutable intermediate goods. In our model, the intermediate
goods market is therefore not only frictional but also imperfectly competitive (specifically
monopolistically competitive, à la Dixit and Stiglitz). From a purely theoretical point of
view, a contribution of this paper then is to suggest a way to model a market cumulating
both types of market imperfections and see how they can interact.
We proceed as follows. In the next subsections, we start by surveying the related literature, not only the search and matching models we have just mentioned but also the seminal
Dixit and Stiglitz’s (1977) model of monopolistic competition. In section 2, we present the
model, making precise the assumptions made about the intermediate goods market, and solve
for the equilibrium. In section 3, we analyze how labor market variables, more particularly
the equilibrium wage and unemployment rate, can be impacted by strategies to reduce the
level of frictions or policies of deregulation. In section 4, we extend the basic model by following Wasmer (2009) and introducing frictions on the final goods market. In this context,
we take the opportunity to suggest an alternative interpretation of the final good production
function.
1.1
Macroeconomics and steady-state models of multiple frictional markets
We start by surveying the steady-state models of frictional markets that we will build our
framework on. For each model, we make clear the assumptions and the logic underlying the
reasoning as this will allow a better understanding not only of what will be done in the next
section but also of the extent to which these results are impacted by the adjunctions we
1
2
OECD Glossary of Statistical Terms
OECD National Accounts Statistics (2010)
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make in our framework. However, we do not go into the details of the computations; indeed
those details can be found in the computations we make in the next section and which are
very similar to the ones met in the literature. Besides, although the papers we refer to also
develop extensions of the basic frameworks they introduce, we limit ourselves to presenting
the features which we will directly build on in the next section.
Before anything, let us define the concept of frictions. Basically, it translates the imperfection of the information available on markets. Specifically, it encompasses the fact that
willing-to-trade agents may not be perfectly aware of the market, for instance of the existence
of potential trade partners or of their location. In the case of the labor market, this lack
of information may be for example due to the fact that a worker may not live in the very
region where a position he would fit is available. Such spatial dispersal of potential trading
parties can be more generally evoked for any other markets. As a consequence, obtaining the
information they lack requires agents to spend time and possibly material resources, while
matching puts an end to the costly search process and thus creates a rent that has to be
shared among the contracting parties.
Labor market frictions. Pissarides (1990) suggests a general framework to model frictions on the labor market. Formally, frictions are modeled through a matching function mL
that gives the number of job positions filled per time period as a function of the number U of
unemployed workers prospecting on the labor market and the number V of vacancies posted
(which equals the number of firms looking for a worker as each firm is assumed to need a
single worker to produce): mL = mL (U, V). At each period, a prospective entrepreneur has
a probability q to meet a worker and fill his vacant position; and conversely, a worker has
a probability q̂ to find a job. These two probabilities are made consistent by the matching
,V)
technology. Indeed, the job-filling rate is actually given by mL (U
while the job-finding rate
V
mL (U ,V)
is equal to
. Denoting the tightness of the labor market (from the point of view of
U
V
the firm) by U =: θ yields
q̂ = q̂(θ) =
mL (U, V)
mL (U, V)
=
= θq(θ)
U
V UV
An important implication of this equation is that the job-filling and job-finding rates cannot
be simultaneously exogenous. It is thus enough to focus on the properties of q(θ). Intuitively,
an increase in θ translates a relative increase in the number of vacant positions with respect
to the number of unemployed workers. As a consequence, competition for workers is increased
among firms and a given entrepreneur is thus less likely to find a match. In other words, q is
a decreasing function of θ; the elasticity of the job-filling rate to the labor market tightness
θq 0 (θ)
is εq := − ∂ ∂lnlnq(θ)
θ = − q(θ) . This reasoning makes clear the existence of negative search
externatilities on the labor market: each searching firm (resp. worker) contributes to the
congestion of the labor market, making it more tight from the point of view of all other firms
(resp. workers), and thus contributes to their lower probability to find a match.
Besides these assumptions on job creation, it is supposed that jobs can be destroyed at
exogenous rate s. Such event sends both the entrepreneur and the worker back to the labor
market, where they start looking for new matches. From there, the law of motion of the
unemployment rate —denoted by u = u(t)— is clear:
u̇ = −θq(θ)u + s(1 − u)
On the one hand, each unemployed worker having probability θq(θ) to find a job, outflows
from the unemployment pool amount to the first term of the right hand side; on the other
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hand, 1 − u being the employment rate and each employed worker being fired with probability s, inflows into the unemployment pool are given by the second term. Note that this
implicitely assumes away growth of the labor force. The unemployment rate being constant
at equilibrium, u̇ = 0 so that
u=
s
s + θq(θ)
This equation decribes a decreasing relationship between u and θ which itself induces a decreasing relationship between the unemployment rate u and the number of vacancies V in
the economy. This relationship is called the Beveridge curve and, under the assumption that
there is a continuum of unit mass of workers, that is, that u = U, it is given by the equation
u = V q(s V ) obtained by substituting θ by its definition in the unemployment rate equation.
u
u
The solution of the model is based on an asset-value approach of employment and positions. The (per period) discounted value of a vacant position —denoted E1 by reference to
the first stage of the life of an entrepreneur— is equal to the expected gain from changing it
into a filled position net of the cost of posting the vacancy, say γ. The value E2 of a filled
position is equal to the profit made from production —P y − w if y denotes the amount of
output produced each period by the worker, P the unit price at which output is sold and w
the wage paid to the worker— net of the expected loss (negative gain) from an exogenous
destruction, i.e. from a return to vacancy. If firms are risk-neutral and r denotes the discount
rate, Bellman equations then write at equilibrium:
rE1 = −γ + q(θ)[E2 − E1 ]
(1)
rE2 = P y − w + s[E1 − E2 ]
(2)
The value U of unemployment to the worker is equal to the sum of unemployment benefits
T and the expected gain from finding a job. The value W of employment to the worker is
equal to the wage earned and paid by the firm net of the expected loss (negative gain) from
losing one’s job. Again, if workers are assumed to be risk-neutral, Bellman equations write
at steady state:
rU = T + θq(θ)[W − U ]
(3)
rW = w + s[U − W ]
(4)
A key assumption of the model then is free entry of entrepreneurs on the labor market. It
means that as long as expected profits are positive, new entrepreneurs will enter the market.
It thus implies that at equilibrium vacant positions have null value: E1 = 0, which in turn
yields a job-creation condition when substituting (2) into (1):
E2 =
γ
q(θ)
which equalizes the expected profit made by a filled position and the expected cost of a
1
is the average time it takes an
vacancy, as γ is the per-period cost of a vacancy while q(θ)
entrepreneur to find a worker. This job creation equation dictates the equilibrium value of the
labor market tightness θ. The existence of frictions of the labor market induces the creation
of a rent when a match is made and which is to be shared between the two contracting parties.
The wage is assumed to be determined through Nash bargaining between the firm and the
worker at the time they match. Letting α ∈ (0, 1) be the bargaing power of the worker
induces w = argmax{[W − U ]α [E2 − E1 ]1−α } or equivalently, as taking logs is a monotonic
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increasing transformation, w = argmax{α ln[W − U ] + (1 − α) ln[E2 − E1 ]}. The equilibrium
wage w∗ is thus given by the first-order condition
∂w [W − U ]
∂w [E2 − E1 ]
α
=0
+ (1 − α)
[W − U ]
[E2 − E1 ] |w∗
FOC then rewrites α(P y − w) = (1 − α)(w − T ) and gives the wage equation:
w = (1 − α)T + αP y
The equilibrium wage is thus obtained as a convex combination of revenues from production
on the one hand and unemployment benefits (which determine the value of the worker’s
outside option) on the other hand. The coefficient weighting revenues from production in
this combination corresponds to the worker’s bargaining power. Note that the worker’s
bargaining power also corresponds to the share of revenues from production that the firm
will leave to the worker.
Credit market frictions. Following the lead taken by Pissarides (1990), Wasmer and Weil
(2004) study the effects of credit market frictions on the labor market variables. Pissarides’
(1990) model is transformed by the assumption that entrepreneurs do not have funds on their
own and thus cannot finance by themselves their search on the labor market, which costs γ
at each period. They need to go to the credit market first and meet a financer; a new type of
agent is therefore introduced. The credit market is assumed to be frictional and is modeled
in the exact same way as the labor market discussed in the previous paragraph. A matching
technology mC gives the number of matches made at each time period on the credit market
as a function of the number N0 of entrepreneurs looking for funds and the number B0 of
unmatched bankers looking for a project to finance. The probability for an entrepreneur to
0 ,B0 )
0
find a financer is p = p(φ) = mC (N
where φ := N
N0
B0 is the tightness of the credit market
from the point of view entrepreneurs. Symmetrically to what was described for the labor
market, p decreases with φ and the probability for financers to find a match is p̂ = φp(φ).
At equilibrium, the value functions of the firm at the different stages of its life now write:
rE0 = −c + p(φ)[E1 − E0 ]
rE1 = −γ + γ + q(θ)[E2 − E1 ]
rE2 = P y − w − ρ + s[E0 − E2 ]
The first equation has no equivalent in Pissarides (1990) but is readily understandable from
the argument given in the previous paragraph. It gives the discounted value E0 of the
entrepreneur prospecting on the credit market as equal to the expected gain from a match
on the credit market net of the cost c of searching for a financer. While the labor market
search costs γ are viewed as the pecuniary costs of posting a vacancy (for instance to publicize
want ads in newspapers) and therefore require initial funds, credit market search costs are
rather understood as an effort to be made or a time to be spent (for instance to meet up
with potential financers and convince them of the worthiness of the project) which do not
need to be actually paid for but can be expressed in monetary terms by c. Once a match is
found on the credit market, the entrepreneur starts looking for a worker on the labor market.
Search costs are now financed by the banker: the entrepreneur receives funds γ from his
banker and immediately uses them to finance his search process. This operation translates
into the double term −γ + γ of the second Bellman equation which thus differ from (1). The
value of the firm on the labor market is now merely equal to the expected gain from a match.
The third equation differs from equation (2) by the negative term ρ which stands for the
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repayment the entrepreneur has to make to the bank once the profit stage has been reached.
Note also that when destruction of the firm occurs (at exogenous Poisson rate s) not only is
the entrepreneur/worker match dissolved but so is the entrepreneur/banker match, the firm
thus being sent back to the credit market stage.
The successive stages of the life of the bank are closely related to those of the life of the firm
since the credit market stage results in a one-to-one matching between entrepreneurs and
bankers. The value functions of the bank thus easily follow from those of the firm:
rB0 = −κ + φp(φ)[B1 − B0 ]
rB1 = −γ + q(θ)[B2 − B1 ]
rB2 = ρ + s[B0 − B2 ]
The discounted value of the bank at the credit market stage is equal to the expected gain from
a match on the credit market net of the cost κ of searching for a project to be financed. The
second equation gives the discounted value of the bank when associated to a firm prospecting
on the labor market. It is equal to the expected gain from the firm finding a match net of
the cost the bank has to pay to finance the firm’s prospects, that is, γ. Finally, when the
firm finds a worker, the firm enters the repayment stage in which the bank receives ρ from
the firm as long as the profit stage last —i.e. until the possible destruction of the firm, in
which case the banker starts looking for another project to finance.
The worker’s life cycle is not modified by the adjunction of the preliminary credit-market
stage to the life of the firm. The value functions corresponding to this agent are thus left
unchanged from equations (3) and (4).
The solution of the model partly proceeds just as in Pissarides (1990). The equilibrium
repayment and wage are assumed to be determined by Nash bargaining. Still denoting by
α the worker’s bargaining power in the wage bargaining, solving the Bellman equations and
setting the bank’s bargaining power in the repayment negociation to β yields:
ρ = β(P y − w) + (1 − β)(r + s)
γ
q(θ)
w = α(P y − ρ) + (1 − α)rU = αθ (P y − ρ) + (1 − αθ )T
r+s+θq(θ)
where αθ = α r+s+αθq(θ)
.
The interpretation of these expressions admits the same logic as in the previous paragraph.
The wage is a convex combination of the unemployment benefits T and the profit made by the
firm. This profit must be now understood as net of the repayment made to the bank, thus the
negative ρ term. A difference with respect to Pissarides (1990) lies in the coefficients in this
convex combination: it is now given by the term αθ which depends not only on the worker’s
bargaining power but also on the tightness of the labor market. It comes from the fact that
the value rU of the worker’s outside option depends on the labor market tightness: a tighter
labor market for the firms increases the value of this outside option and thus increases the
worker’s share in net profits.
Turning to the repayment, it is also a convex combination of two terms. The first corresponds
to the net-of-the-wage revenues made by the firm and that will then be shared between the
entrepreneur and the financer. The share of the bank in these revenues is given by its bargaining power β. The second term corresponds to the search costs financed by the financer
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when the firm was not making profit. This means that a share of the costs of frictions (determined by the entrepreneur’s bargaining power) ultimately remains supported by the financer.
As a consequence of the firm/worker interaction on the labor market being unchanged,
s
the equilibrium unemployment rate solves just as in Pissarides (1990) and writes u = s+θq(θ)
.
On the contrary, solving for the equilibrium tightness of the credit and labor markets proceeds differently. A remarkable feature of Wasmer and Weil (2004) is the block-recursiveness
of the solution for φ, that is, the fact that it can be determined independently from the
other variables of the model. This comes from the assumption of free entry of entrepreneurs
and bankers on the credit market. This implies that at equilibrium, E0 = 0 and B0 = 0.
Indeed, as long as E0 > 0 (resp. B0 > 0), that is, as long as potential entrepreneurs (resp.
financers) expect strictly positive profits, they enter the credit market and equilibrium is not
reached. Conversely, as soon as E0 < 0 (resp. B0 < 0), that is, as soon as prosepective entrepreneurs (resp. financers) expect strictly negative profits, they leave the credit market and
stop searching for a match to launch their project; thus equilibrium is not reached neither.
The credit-market-stage Bellman equations of the entrepreneur and financer then rewrite:
E1 =
c
p(φ)
and
B1 =
κ
φp(φ)
Next, using the first order condition of the repayment Nash bargaining problem, namely
(1 − β)[B1 − B0 ] = β[E1 − E0 ], straightforwardly gives the equilibrium value of φ as a
function of the credit market parameters:
φ=
1−βκ
β c
In Pissarides (1990), θ was derived from the job-creation equation, which in turn followed
from the assumption of free entry of firms on the labor market. In Wasmer and Weil (2004),
due to the obligation of entrepreneurs to go through the credit market and find a financer
prior to searching for a worker, there is no such free entry on the labor market anymore. The
assumption of free entry on the credit market can be used to determine θ though. Steadystate Bellman equations yields forward-looking values of E1 and B1 which can be equated to
the values obtained from the free-entry assumption:
c
q(θ)
Py − w
γ
= (1 − β)
−
p(φ)
r + q(θ) r + s
q(θ)
κ
q(θ)
Py − w
γ
=β
−
φp(φ)
r + q(θ) r + s
q(θ)
On the one hand, the first equation describes the set (EE) of all pairs (θ, φ) compatible
with the free entry condition for firms. It defines a decreasing relation between φ and θ.
Intuitively, an increase in the labor market tightness θ (from the point of view of the firm)
decreases the expected profit of the firm because in average the search costs on the labor
market will be paid for a longer period. To maintain the entry value of the firm equal to 0,
the higher expected search cost on labor market must be compensated by a lower expected
search cost on the credit market. The tightness of the credit market must then decrease from
the point of view of the firm. In other words, the increase in θ must be compensated along
the (EE)-curve by a decrease in φ.
On the other hand, the second equation describes the set (BB) of all pairs (θ, φ) compatible
with the free entry condition for banks. It defines an increasing relation between φ and θ.
Intuitively, an increase in the labor market tightness θ (from the point of view of the firm)
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forces the bank to pay the search costs on the labor market for a longer period in average.
To maintain the entry value of the bank equal to 0, the higher expected search cost on labor
market must be compensated by a lower expected search cost on the credit market. The
tightness of the credit market must then decrease from the point of view of the bank. In
other words, the increase in θ must be compensated along the (BB)-curve by an increase in
φ.
Under the condition that limθ→0 φBB (θ) =: φB < φE := limθ→0 φEE (θ), the two curves (EE)
and (BB) intersect in a single point (θ∗ , φ∗ ).
A highlighted earlier, one of the key consequences of the introduction of a frictional credit
market, which constitutes a compulsory prerequisite step for entrepreneurs who seek to hire
workers, is to suppress their free entry on the labor market. In turn, the number of vacancies
in the economy is less volatile than in Pissarides (1990) as the credit market acts as a filter
for the movements of entrepreneurs into and out of the labor market. In other words, the
number V of vacancies reacts less to changes in profits. This in turn implies the existence of
a financial multiplier : credit market frictions amplify the impact on θ of shocks on profits
as compared to what happens in Pissarides (1990) because the movements in profits are less
compensated by changes in V.
Goods market frictions. Wasmer (2009) considers a third layer of frictions in the life of
the firm: the goods market on which firms sell their product is now assumed to be frictional,
so that firms need to search for their consumers before being able to actually sell their products and make profits.
The assumptions about the credit and labor markets being basically the same as in Wasmer
and Weil (2004), we focus of the presentation of those regarding the goods market. As in
the previous models, frictions on the goods market are modeled through a matching function
mL = mL (N2 , W0 ) which gives the number of firm/consumer matches formed at each time
period as a function of the numbers N2 of productive firms looking for a sale opportunity and
N2
W0 of employed workers looking for a good to consume. Denoting by ξ = W
the tightness
0
of the goods market from the point of the firm allows deducing the matching rates λ(ξ) for
the firm and ξλ(ξ) for the prospective consumer such that λ0 (ξ) < 0.
Besides this one-to-one matching process between productive firms and future consumers, it
is assumed that consumers inelastically consume one unit of final good (per time period).
A rationale for such assumption is as follows: one can think of the final good as a capital
equipment good, such as a micro-wave, for which the first unit held is indeed useful and
valued by the consumer, while the marginal cost of holding a second unit would probably
exceed its marginal utility. Besides capital equipment, other examples of such goods may
be found in subscriptions (to cable TV, to wireless internet access, to a gym club, to public
transportation, etc.). As a consequence, the firm is assumed to produce y = 1 unit of good
per time period so that revenues from sales now simply write P , where the price P of the
good produced is bargained over by the firm and the consumer at the time they match.
From the point of view of the firm, the profit stage, in which the firm sells its production
to a consumer and repays the bank, is distinguished from an intermediary production stage,
in which the firm produces but cannot sell nor stock its product. This translates into the
steady-state Bellman equations:
rE0 = −c + p(φ)[E1 − E0 ]
rE1 = −γ + γ + q(θ)[E2 − E1 ]
rE2 = −w + w + λ(ξ)[E3 − E2 ] + s[E0 − E2 ]
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rE3 = P − w − ρ + s[E0 − E3 ] + τ [E2 − E3 ]
The first two equations are similar to those in Wasmer and Weil (2004) so that only the last
two ones require additional comments. The third equation describes the discounted value of
a firm who can produce (because it hired a worker in the previous step thanks to financial
help the bank) but has not find yet an opportunity to sell its product. The firm thus searches
for a consumer on the goods market and a may find a match with instantaneous probability
λ(ξ). As long as a match is not found, the firm does not make profits so the bank bears the
cost of the wage, thus the double term −w + w. During this production stage, the firm may
be subject to destruction at rate s. The fourth equation describes the discounted value of
a profit-making firm. As soon as it finds a consumer and starts selling its product, the firm
makes profit. The bank thus stops paying for the wage of the worker, which now is incumbent
upon the firm and starts receiving the repayment ρ from the firm. During this profit stage,
the firm may still be subject to destruction with instantaneous probability s. However, it
may also lose its sale opportunity at a rate τ standing for the versatility of the consumer.
The value functions for the bank are still symmetric from those for the firm:
rB0 = −κ + φp(φ)[B1 − B0 ]
rB1 = −γ + q(θ)[B2 − B1 ]
rB2 = −w + λ(ξ)[B3 − B2 ] + s[B0 − B2 ]
rB3 = ρ + s[B0 − B3 ] + τ [B2 − B3 ]
The workers/consumers’ side is now different from the previous frameworks.
rU = T + θq(θ)[W0 − U ]
rW0 = w + T + ξλ(ξ)[W1 − W0 ] + s[U − W0 ]
rW1 = w + T − P + (1 + Φ) + s[U − W1 ]
An important remark to be made before any comment is that Wasmer (2009) assumes every
worker receives a transfer T at each period. T should not be understood as the unemployment benefits anymore but as a common transfer to all unemployed and employed workers.
In section ??, when we extend our basic framework to a frictional (final) goods market at that
in Wasmer (2009), we will get rid of this assumption, sticking to benefits being transfers to
unemployed workers only for sake of continuity with the original framework built in section
2. Nevertheless, in the present literature review we stricly follow Wasmer (2009).
The first equation is readily understood from what has been stated in the previous paragraphs:
the value of unemployment to the worker equals the transfer received plus the expected gain
in a change of state. The second equation decribes the discounted value of employment without consumption to the worker. In this state, the worker is paid a wage w on top of the
universal transfer T . The firm in which he is employed being possibly subject to destruction,
he may lose his job with instantaneous probablity s, which would send him back to the unemployment pool. The employed worker also searches for a good to consume on the goods
market, which he finds with instantaneous probability ξλ(ξ). Finally, when the employed
worker consumes, his discounted value is given by the third equation. It corresponds to the
wage and transfer received plus the utility of consumption net of the price paid to purchase
the good. The worker may still lose his job at rate s which would imply him stop consuming. This last comment allows interpreting the versatility of the consumer: a consumer stops
consuming a good when he loses his job so that τ = s.
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Just as before, the repayment, the wage and now the price of the good are assumed
to be determing by Nash bargaining. Keeping the previous notations and letting δ be the
bargaining power of the consumer in the price negociation, equilibrium values write:
P =
1+Φ
1+
δ r+s+τ r+s+ξ
1−δ r+s r+s+λ+τ
1+Φ−P
−ξλ
+
w=
r + s + ξλ
1 + α θq(θ)
r+s
α
λP
θq(θ)
κ
c
1+
− (r + s + θq(θ))
+
r+s+λ+τ
r+s
φp(φ) p(φ)
1 + α θq(θ)
r+s
γ r + s + λ w w(1 − αλ )
ρ
P −w
+β
αθ
=β
+ +
r+s
r+s
q(θ)
λ
λ
r+s
1−α
where αλ =
r+s+λ
r+s+λ+τ .
Here again, the equilibrium wage is the weighted average of two terms accounting respectively
for the value of the outside option of the worker and the expected profit made by the firm
thanks to the worker’s labor. Let us look at the first term. Now, the value of the outside
option of the worker takes into account the fact no consumption is possible in the unemployment state. Also note that the transfer T does not appear anymore in the determination
of reservation wage. This is due to the fact that, as the transfer T is no more specific to
the unemployed workers but now made to both employed and unemployed workers, it does
constitutes a loss suffered by the worker when leaving the unemployment pool. Let us turn to
the second term of the wage equation. Profits now consist in the discounted revenues made
from sales net of the costs of search frictions. Finally, note that the weights in this equation
again do not only depend on the worker’s bargaining power but also on the tightness of the
labor market as it enters the determination of his reservation wage.
The repayment ρ follows the same logic as in Wasmer and Weil (2004). The financer receives
a share equal to his bargaining power the net-of-the-wage profits of the firm, and ultimately
goes on bearing a share of the costs he paid for while the firm was not making profits. However, these costs do not only consist in search costs, as it was the case in Wasmer and Weil
(2004). They also include the wage which the bank paid for while the productive firm was
looking for a sale opportunity. This corresponds to the term wλ in which λ1 is the expected
duration of the search for a consumer on the goods market.
Finally, the equilibrium price P of the final good is a fraction of the marginal utility enjoyed
when consuming the final good. The fraction is determined by the bargaining power δ of the
worker and the level of frictions (given by λ(ξ)) on the final goods market.
Due to the fact that the bank/firm interaction on the credit market and the worker/firm
interaction on the labor market are not modified as compared to Wasmer and Weil (2004),
the equilibrium values of the tightness of these markets are derived in the exact same way
κ
as in the previous framework. Free entry conditions imply φ = 1−β
β c and allow deriving
relationships between φ and θ at equilibrium
c
q(θ)
λ
P −w
γ
λ
w w(1 − αλ )
= (1 − β)
αλ
−
+ +
p(φ)
r + q(θ) r + s + λ
r+s
q(θ r + s + λ
λ
r+s
κ
q(θ)
λ
P −w
γ
λ
w w(1 − αλ )
= (1 − β)
αλ
−
+ +
φp(φ)
r + q(θ) r + s + λ
r+s
q(θ r + s + λ
λ
r+s
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which in turn yield the value for θ.
These two equations can be given the same interpretation as in the previous paragraph. The
only differences lie in the fact that when computing the right-hand-side profit to be expected
by the entrepreneur (resp. the financer) when entering the credit market, the existence of
frictions on the final goods market are taken into account for instance through an additional
time discounting due to the delay induced in profit-making.
The equilibrium rate is again given by its law of motion and happens to have the same
expression as in Pissarides (1990) and Wasmer and Weil (2004), still due to the fact that the
structure of the labor market, which drives the law of motion of u, is the same in the three
frameworks.
But beyond the law of motion of the unemployment pool (which is equivalent to the unemployment rate under the assumption that there is a continuum of unit mass of workers),
Wasmer (2009) uses the laws of motion of all stocks in the economy to derive the goods market tightness. The details of the reasoning will be given in section 4 (where as an extension to
our basic framework we also introduce frictions of the final goods market) but the key feature
is the following: due to the one-to-one matching operated on the labor market, the number
of employed workers (that is, prospective plus matched consumers) is equal to the number
of productive firms (that is, those which make profit plus those who are still looking for a
match on the goods market). Equating the stocks W0 + W1 (where W0 and W1 respectively
denote the numbers of prospective and matched consumers) and N2 + N3 (where N2 and
N3 respectively denote the numbers of prospective and matched productive firms) yields an
equation of the form g(ξ) = ξ. ξ is then solved as a fixed point of the constinuous function g
and ξ = 1 is found to be a solution.
1.2
Intermediate good firms and the monopolistically competitive intermediate goods market
Our main focus being on final goods firms and the labor market on which they recruit, we
adopt very simple modeling assumptions concerning intermediate goods firms. We think of
each intermediate good producer as consisting of a single entrepreneur who does not need to
hire additional labor on top of his own (nor buy intermediate goods) to produce. We also
assume he does not need to get financed before producing. In other words, contrary to a final
good producer, he does not need to go through the credit and labor markets before starting
production. Finally, we assume each intermediate good firm produces a differentiated good
which is only an imperfect substitute to the others. It can thus behave as a monopolistic competitor on the intermediate goods market. To analyze interactions on this market and derive
the equilibrium, we rely on Dixit and Stiglitz (1977) whose canonical model of monopolistic
competition markets allows capturing their basic properties.
There, monopolistically competitive behaviors arise from the existence of a group of n
goods, denoted i = 1, . . . , n that are imperfect substitutes among themselves and poor substitutes to the rest of the goods of the economy, which are aggregated into a single good 0
taken as the numéraire.
The consumers’ problem. Goods are consumed by agents assumed to have a separable
utility function U (x0 , V (x)) where x0 denotes the quantity of good 0 and x := (x1 , . . . , xn ) the
quantities of the imperfectly substituable goods that are consumed. Dixit and Stiglitz (1977)
study several examples of such function; we focus here on the one we will use later to model
the consumption by final
firms on
good
the intermediate goods market, namely the constant1
Pn
σ
σ
elasticity case: u = U x0 , [ i=1 xi ] . Consumers have income I and choose their consump-
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P
tion bundle by maximizing their utility under the the budget constraint x0 + ni=1 xi pi = I
where the pi ’s denote the prices of the goods. Denoting by µ the Lagrange multiplier associated to the constraint, the first-order conditions write

 1−σ
σ
n
X
σ

µpi = ∂xi U
xj
xσ−1
(∀i = 1, . . . , n)
i
j=1
This being true for all i = 1, . . . , n, combining FOCs yields
pj
=
pi
xj
xi
σ−1
⇔ xj = xi
pj
pi
1
σ−1
(∀i, j = 1, . . . , n)
(5)
Using the binding budget constraint gives
n
X
σ
σ−1
Pn
j=1 pj
xj p j = xi
= I ⇔ xi = I P
n
1
(∀i = 1, . . . , n)
σ
σ−1
j=1 pj
piσ−1
j=1
1
piσ−1
The compensated utility given by the consumption of the imperfect substitutes then is
σ

V (x) =  P
I
σ
n
σ−1
j=1 pj
n
X

1
σ
σ
σ−1
pi
 =I
n
X
σ
σ−1
! 1−σ
σ
pi
i=1
i=1
This allows defining the aggregate price index P :=
P
σ
n
σ−1
i=1 pi
σ−1
σ
as the amount of ex1
P
penditure needed to achieve the utility level V (x) = 1. Setting Y := ( ni=1 xσi ) σ as the
aggregate quantity index gives a relationship Y = I.P −1 between aggregate consumption and
aggregate price and allows writing the optimal (relative) consumption of good i as a function
of its relative price:
1
xi pi σ−1
=
Y
P
(6)
Elasticities and market shares. From equation (5) giving the relative demands of goods
i and j as a function of their prices, one can deduce the elasticity of substitution between
goods:
h 1 σ−1
xi
∂
ln
( ppji
∂ ln xj
1
=−
εsubst
:= −
=
(7)
i,j
1−σ
∂ ln ppji
∂ ln ppji
This being invariant across goods explains the reason why the utility function is called a
constant elasticity of substitution (CES) function.
From the consumer’s demand functions (6), one can deduce the elasticity of the consumption of any good i to its price. A priori, a change in pi may affect the consumption of good i
in two distinct ways. First, in a direct way, due to pi explicitely entering the expression of xi .
Second, in a indirect way: a change in pi may affect the aggregate level of price P, which not
only enters the expression of xi but also conditions the aggregate consumption level Y which
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can in turn impact xi as well. Consider the latter first. The impact of pi on the aggregate
price index is measured by:
h
P
σ i
n
σ−1
σ−1
p σ
∂
−
ln
p
p
i
j=1 j
σ
∂ ln P
i σ−1
−
=−
=
1
∂ ln pi
P
p
i
If all intermediate goods have prices of comparable magnitudes, the elasticity of the price
index P to the price of a given good i is of order n−1 . Assuming a sufficiently large number
n of goods then allows ignoring the indirect effect of pi on xi through P, and in turn through
Y. The price elasticity of good i then boils down to
εprice
i
1
ln Y + σ−1
[ln P − ln pi ]
∂ ln(xi )
1
:= −
=−
=
∂ ln(pi )
∂ ln(pi )
σ−1
(8)
As σ ∈ (0, 1), this elasticity is negative, meaning that the demand of good i decreases when
its price increases, keeping all other things constant.
Now, turning to the relationships between the group of imperfect substitutes and the rest of
the economy, let us call s(P) the budget share of Y: s(P)I = PY. The budget share of the
rest of the goods (taken as the numéraire) then is such that x0 = I −PY = I(1−s(P). Denote
σ0 (P) the elasticity of substitution between good 0 and the group of imperfect substitutes.
We have:
I(1−s(P))
d ln
s(P)
∂ ln xY0 (P)
I P
d [ln P + ln(1 − s(P)) − ln(s(P))]
σ0 (P) :=
=
=
∂ ln P
d ln P
d ln P
d ln(1 − s(P)) d ln(s(P))
=1+
−
(9)
d ln P
d ln P
The last term in the final expression corresponds to the elasticity of the market share s with
0 (P)
respect to aggregate price P: θ(P) := d ln(s(P))
= Ps
d ln P
s(P) . The second term also simplifies:
1
−s0 (P) 1−s(P)
Ps0 (P)
d ln(1 − s(P))
=
=
−
1
d ln P
1 − s(P)
P
Therefore, equation (9) also simplifies:
1 − σ0 (P) =
Ps0 (P)
Ps0 (P)
Ps0 (P)
1
−
=
=
θ(P)
1 − s(P)
s(P)
s(P)[1 − s(P)]
1 − s(P)
This last equation gives the elasticity of the budget share of Y as a function of the budget
share of good 0 and the elasticity of the demand in good 0 to the aggregate price P.
The producers’ problem. Each good i is produced by a firm i which maximizes its profit
pi xi −ci xi —where ci = c (∀i = 1, . . . , n) is the unit cost of production— given the consumers’
demand function (6). Substituting the demand function into the profit function and taking
the first-order condition with respect to the price3 (and recalling that, as shown above, the
effect on P and Y of a variation in pi is negligible) yields:
1
p 1
−1 1
1 pi σ−1
i σ−1
xi (pi ) + (pi − c)x0i (pi ) = Y
+ (pi − c)Y
=0
P
σ−1 P
P
3
Expressing the constraint in terms of demand as a function of price and taking derivatives with respect to
price, we implicitely assume Bertrand competition. However, similar results can be obtained when considering
Cournot competition.
The objective function of the firm remains the same and the constraint becomes
σ−1
pi = P xYi
. Assuming that the number of goods is large enough, a marginal change in xi , keeping all
other things constant, has no impact on aggregate demand Y and aggregate price P. Plugging the constraint
into the objective function and taking first-order conditions with respect to xi then yields the pricing rule
(10).
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Y
1
⇔ piσ−1
⇔
pi
P
1
σ−1
1−
Master’s dissertation — Margaux Luflade
1
−1
Y
1
1
σ−1
) = −c
1 pi
σ−1
σ − 1 P σ−1
!
(1 +
1
εprice
i
⇔
=c
pi =
c
σ
(10)
(11)
Due to imperfect substituability among goods i = 1, . . . , n, producers are in a position that
allow them to set their price above the marginal cost of production. The mark-up µ corresponds to the part of the price in excess to the marginal cost:
pi = (1 + µ)c ⇔ µ =
1−σ
>0
σ
Love of variety preferences. If the marginal cost is assumed to be constant across firms,
the pricing rule induces equality of prices across commodities, say pi = p̄ (= 1, . . . , n). In
turn, the goods are all demanded in the same quantity, denoted by x̄. Therefore, the aggregate
σ−1
σ
σ−1
1
σ
indexes then rewrite P = np̄ σ−1
= n σ p̄ and Y = n σ x̄ so that the compensated utility
1−σ
of the consumer is equal to V = In σ p̄−1 which is an increasing function of the number n of
goods: when all goods are sold at the same price, consumers have a preference for diversity
in the sense that they are better off spreading their consumption over the maximal number
of goods rather than consuming a reduced number of them.
Equilibrium number of firms. Given the free entry of firms on the market, profits are
driven to 0 at equilibrium. Thus, if a denotes the fixed cost of production, the marginal
firm entering the market must be just breaking even: (pn − cn )xn = a, or by symmetry:
(p̄ − c)x̄ = a for all firms. From this, one can get the equilibrium number n of imperfect
substitutes. This requires noting that by definition of the budget share:
σ−1
σ−1
¯
1
s(P)
s(n σ p)
σ ¯
σ
Y=I
⇔ n x̄ = I
⇔ nx̄ = I s(n p̄ p)
σ−1
P
n σ p̄
Now using the free-entry condition:
x̄ =
a
=
p̄ − c
c
1
σ
a
a σ
=
c1−σ
−1
(12)
which is positive as εprice
∈ (0, 1) and which allows rewriting the previous equation, we can
i
determine the equilibrium value n̄:
σ−1
¯
s(n̄ σ p)
a σ
I
=I
p̄n̄
c1−σ
Adaptation to our intermediate goods context. As hinted earlier, we use this framework to study the intermediate goods market. Final good firms will thus play the role of
consumers and will aim at maximizing their profit, which we shall describe with more details
later. Intermediate goods firms will play the role of producers and will adopt the same pricing
strategy as derived here.
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The basic model
2.1
2.1.1
Setting
Credit and labor markets
Our framework builds on Wasmer and Weil (2004) and relies on the exact same modeling assumptions as them. With regards to the credit market, we assume free entry of entrepreneurs
and financers and we go on modeling frictions through the credit-finding rate p(φ) so that
our credit market works exactly as theirs. Our labor market is also identical: frictions are
expressed through the job-finding rate q(θ) and the interaction between unemployed workers and firms looking for an employee are kept unchanged. The credit and labor markets
structures being unchanged, we expect to be able to solve for the tightnesses and the unemployment rate in the same way as suggested in the literature review. The prices on these
markets, namely the repayment ρ and the wage w, are assumed to be determined by Nash
bargaining between the contracting parts at the time they match. For simplicity, we assume
the price P of the final good is exogenous.
2.1.2
Introducing search on the intermediate goods market
In our economy, production by (final good) firms does not only require labor but also capital
in the form of intermediate goods. Capital inputs are purchased on the intermediate goods
market which is assumed to be frictional. For sake of simplicity, we assume a unilateral
search process: final good firms need to search for their intermediate goods suppliers while
intermediate goods firms do not have to actively search for customers. We also consider that
a final good firm may find a match on the intermediate goods market with instantaneous
probability b. We assume this Poisson rate b —the supplier-finding rate— to be constant as
the number of matches already made increases. To understand this, it is useful to look at
the alternative assumptions which could have been made. This rate could have been thought
of as decreasing with the number of matches already made: each additional match made
decreases by one unit the number of targetable suppliers so that if the number of suppliers
is not infinite, the probability to meet a new one decreases as well. On the contrary, the
supplier-finding rate could have been thought of as an increasing function of the number of
matches already made: as more and more suppliers have been searched for and met, the
final good firm gains experience in the search process; for instance, its administrative services
become more efficient in preparing contracts, its prospective services become more aware of
the structure of the intermediate goods market, have a larger network of contacts and thus
may find an additional supplier more quickly. One may then consider that both mechanisms
occur simultaneously and that their respective effects balance each other so that the supplierfinding rate remains roughly constant. An alternative justification is that some part of the
search process can be assumed to take in average the same amount of time for each contract
to be made, no matter how many have already been made in the past. This is for instance the
case of the administrative and legal procedures which involve the State services and which
are required for the establishment of a contract. If these steps are viewed as the main part
(in term of time consumption) of the search process, we can consider that the average duration of the search process is constant, no matter how much time is gained (by experience)
or lost (by an increased scarcity of potiential suppliers) in the other minor parts of the process.
Before giving more details about the final good firms’ search process on the intermediate
goods market, let us make some comments about the number of intermediate goods available
in the economy. We assume final good firms need to buy all intermediate goods available in
the economy to be able to start production. The number n of intermediate goods available is
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endogenous though. More precisely, as explained in Blanchard and Giavazzi (2003), two time
horizons may be considered. The short run is characterized by a fixed number of intermediate goods firms. As a consequence, when taking their decisions, final good firms (and their
associated bankers) can be thought of as taking the number of intermediate goods firms as
given. In the long run, assuming free entry of producers on the intermediate goods market
and given that each producer supplies a single differentiated intermediate good, the number
of intermediate goods available in the economy is determined by a zero-profit condition: as
long as profits from sales strictly exceed the fixed costs of production, new producers enter
the market and equilibrium is not reached. As the producers’ profits are given by the amount
of sales and the price at which they are made, the long-run equilibrium number of intermediate goods firms in the economy is therefore determined by the short-run equilibrium of the
intermediate goods market.
Let us now turn to the final good firms’ search process on the intermediate goods market. The large-firm version Pissarides’s (1990) model offers an alternative to the one-to-one
matching process described in many other models. Nevertheless, this static model considers productive firms which are already employing a given number of workers and only have
to maintain this number constant despite the fact that at each period each of their labor
matches may be destroyed at exogenous rate s; they do so by keeping on searching for new
matches, the number of which being chosen so as to exactly compensate the losses suffered.
The model does not include the stage at which the firm has to look for the initial workers who
allow starting production. As we are not only interested in the productive stage of final good
firms but in their prospective behavior on the intermediate goods market as well, we need
somehow to innovate regarding our final good firms looking for their initial capital inputs.
For sake of simplicity, we assume final good firms search for intermediate goods sequentially,
that is, one after another. Searching for an intermediate good costs χ per unit of time; this
cost is also assumed to be constant no matter how many intermediate goods have already
been found.
We further assume that there is no endogenous destruction of the matches between final
good firms and their intermediate goods suppliers. In other words, contracts established on
the intermediate goods market are signed once and for all (or can be automatically renewed
without cost nor frictions). This goes away from the modeling assumptions used in the large
firm version of Pissarides’s (1990) model. Indeed, it seems to us that it does not really make
sense to assume that contracts can be destroyed and must be replaced by others when a final
good firm needs to be matched to all intermediate goods firms: the destroyed matches would
then be replaced by the exact same matches as the only intermediate goods suppliers a final
good firm may want to be matched with are the ones it is not already (or not anymore)
matched with. A consequence is that only final good firms at stage 1, that is, looking for
their initial capital inputs, are searching on the intermediate goods market.
As a last point on final good firms and given what has just been explained, let us now
explicit their production function. The quantity of output produced per unit of time by the
employed worker is supposed to be given by the CES function
!1
n
σ
X
y = y(x1 , . . . , xn ) =
xσi
i=1
where σ ∈ (0, 1) is a parameter linked to the elasticity of substitution between capital inputs,
1
. Note that then, contrary to the models surveyed in section 1, the producwhich equals 1−σ
tivity y of the worker is not exogenously given but determined by the capital inputs acquired
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by the firm.
Now the situation of final good firms on the intermediate goods market has been made
clear, let us recall the assumptions we make about intermediate goods producers and that
we have already mentioned in section 1. As we are primarily interested in final good firms
and the labor market equilibrium, we take very simplifying assumptions regarding the supply
side of the intermediate goods market. Intermediate goods firms are thought of as single
entrepreneurs who do not need to hire labor (or buy intermediate goods) nor get financed
through the credit market to be able to produce. Consequences of this are that only final
good firms are present on the credit and labor markets and that there are only one type of
bankers and workers, all turned towards matching with final goods firms. This willingness
to simplify as much as possible the modeling of intermediate goods suppliers also motivates
our assumption of unilateral frictions on the intermediate goods market. Moreover, as explained in section 1, in the monopolistic competition context each firm produces a unique
and differentiated variety of intermediate good and set its price by maximizing its profits
given the demand curve it faces. We besides assume there is no exogenous destruction of
intermediate goods firms, contrary to what may happen to final good firms: for final good
firms, destruction corresponds to the exogenous break of the firm/worker match; this never
happens to intermediate goods firms which do not hire workers. Of course, alternative causes
of destruction of intermediate goods firms could be found, such as exhaustion of the natural
resources on which their production is based or prohibition of their output by law (for instance if the intermediate good is a chemical); but we rule them out here. As a consequence,
there is only one single stage for intermediate goods firms: on top of the fact that they do
not go through the credit market and labor market stages, they never leave the intermediate
goods market stage (which correspond to their profit stage) just as final good firms never
leave their profit stage as long as they do not exogenously loose their worker.
Finally, let us conclude this section by expliciting the tightness of the intermediate goods
market. From what has been explained in this section, the final good firms on the intermediate
goods market are those which are looking for the initial inputs that will then allow them to
start producting; denote their number by N1 . Facing them, all the n intermediate good
producers are present on the intermediate goods market. Tightness (from the point of view
of final goods firm) can then be written: ζ := Nn1 . The instantaneous probability b for a final
good firm to find a supplier then is in fact b = b(ζ). n being taken as given by final good
firms (and their associated bankers) when taking their decisions, so is ζ. As a consequence, ζ
(and n) will not be determined until the very last step of the solution procedure of the model.
To simplify notations, we denote the supplier-finding rate b as long as there is no need of ζ
explicitely appearing (ζ is reintroduced in subsection 2.3.6).
2.2
Bellman equations
Writing the value fonctions of the final good firms (and other types of agents) requires to make
clear the timing of the successive stages of their life. It is clear that the search for a financer
is a prerequisite to any other stage as it provides the funds needed to search and pay for the
labor and capital inputs. It is also clear that the production stage comes last as inputs are
needed to produce. However, under the assumption that the firm cannot look simultaneously
for all inputs, it is less clear whether it should look first for labor or capital inputs. A natural
answer is that the firm looks first for the type of inputs which is the less costly to be left
idle while looking for the other. Suppose the firm looks for the worker first. Once hired, he
will be left idle as long as all intermediate goods have not been found. With the notations
1
introduced in the previous section, each of the n intermediate goods being searched for b(ζ)
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periods in average, this would cost wn 1b to the firm since although he does not produce, the
employed worker must be paid a wage. On the contrary, suppose the firm looks first for
intermediate goods. If the inputs are perfectly storable, there is no cost in leaving them idle
while searching for the worker. Of course, one may think there is a cost of storing the goods:
for example, the entrepreneur may need to lend a warehouse to stock them; or (in agreement
with the assumption we will make below that the goods are actually purchased at the time
when the production is ready to start) to have the intermediate goods producers storing them
ε
until they are needed. In such case, storage costs q(θ)
to the entrepreneur, where ε denotes
the per-period storage costs. The assumption that seems more plausible to us is storage costs
being much lower than the wage to paid to the worker. As a consequence, we assume the
intermediate goods market stage occurs before the labor market stage in the life of the firm,
right after the credit market stage.
This allows writing the Bellman equations of the firm, which we comment below.
rE0 = −c + p(φ)[E1 − E0 ]

rE11 = −χ + χ + b[E12 − E11 ]




..



.
rE1k = −χ + χ + b[E1k+1 − E1k ]; k = 2, . . . , n − 1


..



.


 n
rE1 = −χ + χ + b[E2 − E1n ]
(13)
rE2 = −γ + γ + q(θ)[E3 − E2 ]
(15)
rE3 = P y − w −
n
X
xi pi − ρ + s[E0 − E3 ]
(14)
(16)
i=1
Equations (13), (15) and (16) are very similar to those of Wasmer and Weil (2004), with
the exception of intermediate goods entering the profit stage equation (16) through the third
and fourth terms. Indeed, in addition to the wage w and the repayment ρ to its bank, the
final good firm also has to pay for the intermediate goods entering production.
Yet, the main innovation here lies in the set of equations (14), representing the stage where
the final good firm looks for its initial inputs on the intermediate goods market, following
the process previously explained. E1k is the value of the firm at stage —denoted 1k — where
it searches for its k th input (k = 1, . . . , n). The per-period search costs on the intermediate
goods market amount to χ and are ultimately paid by the banker, thus the term −χ + χ
to show that the firm has to pay these costs but eventually benefits from a compensating
transfer from the bank. With probability b, a match is found.
As hinted in subsection 2.1.2, for sake of simplicity, we choose to model the search of
intermediate goods as a sequential process, that is, with the final good firm looking for
its inputs one after another. Besides, a natural solution to model the search process on
the intermediate goods market would have been to assume that the final good firm buys
each intermediate good right after the match with its producer is found. Equations (14)
would then have written: rE1k = −χ + χ + b.[E1k+1 − xk pk + xk pk − E1k ] for k < n, and
E1n = −χ + χ + b.[E2 − xn pn + xn pn − E1n ] for k = n, where once again the double term
−xk pk + xk pk indicates that the bank bears the financial cost of the inputs as long as the
final good firm has not reached profit stage. Substituting the value of E1n obtained from the
last equation of this version of system (14) into the equation for E1n−1 and repeating this
backward process up to the first equation of the system would then have yielded the value of
17
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
n
Pi=n b i Pi=n b i
b
E11 as a function of E2 : E11 = (−χ+χ)
+
(−x
p
+
x
p
)
+
E2 .
i
i
i
i
i=1
i=1
b
r+b
r+b
r+b
In this last equation, the symmetry among intermediate goods is broken by the second term of
the sum: the order in which the capital inputs are found matters as it affects their respective
expected discounted prices. Symmetry among intermediate goods allows for more tractability
when solving the model and for this reason we choose an alternative scenario to model the
acquisition of inputs. We assume that when a match is made on the intermediate goods
market, the final good firm and the intermediate good producer only enter into an agreement
establishing that the transaction will take place just before starting production, that is, once
all the intermediate goods and the worker have been found. All the intermediate goods are
then bought at the beginning of stage 3, as reflected by the third term of equation (16). Note
that, just as the payment of the wage to the worker, the purchase of intermediate goods occurs
at the profit stage and their cost is therefore never born by the bank4 . Iterative backward
P b i b n
resolution of E11 as a function of E2 then gives: E11 = (− χb + χb ). ni=1 r+b
+ r+b E2 ,
or more concisely:
n
b
1
E1 =
E2
(17)
r+b
Symmetrically to those for final good firms, the Bellman equations for bankers write:
rB0 = −κ + φp(φ)[B1 − B0 ]
(18)

rB11 = −χ + b[B12 − B11 ]




..



.
rB1k = −χ + b[B1k+1 − B1k ]; k = 2, . . . , n − 1


..



.


 n
rB1 = −χ + b[B2 − B1n ]
(19)
rB2 = −γ + q(θ)[B3 − B2 ]
(20)
rB3 = ρ + s[B0 − B3 ]
(21)
Again, equations (18), (20) and (21) are identical to those in Wasmer and Weil (2004) and
the difference lies in system (19). Just as the term −γ in (20), the term −χ in (19) reflects
the fact that the bank compensates the firm for its spendings as long as it does not make
profit. The evolution of the bank through the different search stages is determined by that of
the final good firm: the firm looking for its k th intermediate good finds it with instantaneous
probability b; and thus the bank is brought from stage 1k to the next —namely stage 1k+1 if
k < n and stage 2 if k = n— with same probability b.
We noted earlier that given that the intermediate goods are purchased at the beginning of
the profit stage, their cost is never directly born by the bank. The repayment ρ then should
not depend on the unit prices p1 , . . . , pn of the intermediate goods. On the contrary, we can
expect ρ to depend on the number n of varieties of intermediate goods that the final goods
firm will look for. Indeed, the number of varieties searched affects the duration of the search
4
In section 4, following Wasmer (2009), we will distinguish between a production stage, where production
has started but output cannot be sold yet as a consumer has not been found, and a profit stage where the firm
produces and makes profits from sales to a consumer. In this framework, as we will keep the assumption that
the bank bears all the costs as long as profits are not made by the firm, the bank will pay for labor and capital
inputs during the production stage. Intermediate goods, along with the wage, will then enter the repayment
to the bank that the firm has to make once the profit stage is reached.
18
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
process on intermediate goods market, the cost of which is born by the bank, and delays the
time at which the bank will finally access the repayment stage.
In this model, n is considered as taken as given by banks and final good firms: there are
n differentiated intermediate good producers and the final good firms buy inputs from each
of them. However, a possible extension would be to allow the final good producers to choose
how many varieties of capital inputs to buy among the numerous differentiated ones available
in the economy. With the number m of varieties purchased entering the repayment determination, one may then compare the optimal number m chosen depending on who choses it
among the final good firm, its associated banker or the pair they form. We considered studying this extension. However, as will be seen later, the equations determining the equilibrium
in the model are rather complex and solving the maximization problem yielding the optimal
number of intermediate goods to be purchased is far from being trivial. As a final comment
on this point, we would like to highlight a key difference between our economy and the one
of Dixit and Stiglitz (1977). As explained in section (1.2), a key feature of the Dixit-Stiglitz
economy is the consumers’ taste for variety. This result has been proved to directly stem
from the CES form of their utility function, and one may thus expect the same to apply to
the final good firms in our model since their production function is CES as well. However,
the goods market in Dixit and Stiglitz (1977) is implicitely assumed to be frictionless, which
implies that there is no cost in acquiring an additional variety. Indeed, to buy a quantity
xn+1 of a newly-available variety the consumer only needs to cut her spending on the other
goods by p̄.xn+1 (i.e. to evenly decrease the quantity of each already-consumed variety by
xn+1
xn ). Her total spending remains the same; her total consumption as well but the CES-form
implying a decreasing marginal utility for each given variety, it is immediate that a larger
utility is enjoyed when the same total quantity is spread over a larger range of varieties. On
the contrary, on our model, the imperfectly substituable goods are purchased on a frictional
market so that any additional variety must be searched for before being consumed. Due to
the CES production function, spreading a given amount of consumption over a larger range
of varieties still yields higher revenues. However there is now a trade-off between these additional revenues and the additional search costs. In presence of increasing returns to scale,
and under the assumption that the costs and duration of search are constant whatever the
number of varieties already purchased, the revenue-to-cost ratio of the marginal variety is a
decreasing function of the number of varieties already consumed. This allows us to expect
the existence of an optimal number of varieties from the point of view of the firm.
Let us go back to the bank’s Bellman equations. As before, we obtain the value of B11 as
a function of B2 :
i n
n χX
b
b
1
B1 = −
+
B2
(22)
b
r+b
r+b
i=1
Finally, turning to the workers’ side of the economy, which is not impacted in its modeling
by the introduction of the frictional intermediate goods market, the Bellman equations are
identical to those in Wasmer and Weil (2004):
rU = T + θq(θ)[W − U ]
(23)
rW = w + s[U − W ]
(24)
Solution of the Bellman equations. As a prerequisite to the determination of the parameters of interest, solving the Bellman equations yields the following expressions which will
be useful in the coming subsection.
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Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
The free entry condition of entrepreneurs on the credit market implies E0 = 0 and allows
solving equations (13) and (16):
E11 =
c
p(φ)
(25)
P
P y − w − ni=1 xi pi − ρ
E3 =
r+s
from which we deduce:
P
q P y − w − ni=1 xi pi − ρ
E2 =
r+q
r+s
P
n
q P y − w − ni=1 xi pi − ρ
b
1
E1 =
r+b
r+q
r+s
(26)
Symmetrically, the free entry condition of banks on the credit market implies B0 = 0 and
allows solving equations (18) and (21):
B11 =
κ
φp(φ)
(27)
ρ
r+s
from which we deduce:
q
γ
ρ
B2 =
− +
r+q
q
r+s
B3 =
n
B11 = −
χX
b
i=1
b
r+b
i
+
b
r+b
n
q
r+q
−
γ
ρ
+
q
r+s
(28)
Finally, looking at workers, equations (23) and (24) combined yield:
rU =
(r + s)T + θq(θ)w
r + s + θq(θ)
(29)
It is key to understand what this last equation exactly means before moving to the wage
determination. This equation can obviously be seen as determining the ex-post value of
unemployment, i.e. evaluated by an employed worker once she knows her wage. w entering
equation (29) then is the wage the employed worker, who looks backward at her former
unemployment situation, is actually paid. On the contrary, an unemployed worker does not
know the wage she will be paid once she she finds a job. To compute the value of her
unemployment she can only ground her expectations on the wage signalled by the market,
for instance the average wage. When considering the ex ante value of unemployment, the ‘w’
entering the expression of rU should in fact read w̄, where w̄ refers a constant signalled by
the market and potentially distinct from the actual wage w the worker will be paid when she
gets hired.
2.3
2.3.1
Solution of the model
Wage determination.
We assume the wage w is bargained over by the worker and the association bank/firm at the
time the worker and the firm match on the labor market. If α denotes the bargaining power
20
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
of the worker in this negociation, w is then solution of the problem: w = argmax{(W −
U )α (E3 + B3 − B2 − E2 )1−α }, or equivalently:
w = argmax{(1 − α). ln(W − U ) + α. ln(E3 + B3 − B2 − E2 )}.
On the one hand,
(16) + (21) ⇒ (r + s)[E3 + B3 ] = P y − w −
n
X
xi pi
i=1
−(n−1)
c
p(φ)
−(n−1) "
i #
n b
b
κ
χX
(22) and (27) ⇒ B2 =
+
r+b
φp(φ)
b
r+b
(17) and (25) ⇒ E2 =
b
r+b
i=1
so that: ∂w [E3 + B3 − E2 − B2 ] = ∂w [E3 + B3 ] =
1
.
− r+s
On the other hand, from Bellman equations (23) and (24), (r + s)[W − U ] = w − rU . As
explained at the end of the previous paragraph, as long as the wage is unknown to the future
worker, as it is the case during the bargaining process, rU should not be understood as a
function of the actual wage w but rather as a function of a given wage level w̄ signalled by
1
the market. Accordingly, ∂w rU = 0 so that ∂w [W − U ] = r+s
.
The first order condition then writes:
(1 − α)(W − U ) = α(E3 + B3 − E2 − B2 )
(30)
Plugging the expressions of [W − U ], [E3 + B3 ] and [E2 − B2 ] we have just obtained into this
last equation yields:
w = (1 − α)rU + α{P y −
n
X
xi pi − (r + s)K}
i=1
where K :=
b
r+b
−(n−1)
κ
φp(φ)
+
c
p(φ)
+
χ
b
Pn i=1
b
r+b
i will be interpreted right after the
computation is done.
At equilibrium, all workers (resp. firm/bank pairs) being identical, all bargains yield the
same wage, which then equal the wage level signalled by the market. We can then replace
rU by its value (29), and obtain:
"
!#
n
X
r+s
r + s + θq(θ)
w=
(1 − α)
T + α Py −
xi pi − (r + s)K
r + s + αθq(θ)
r + s + θq(θ)
i=1
r+s+θq(θ)
Setting as in Wasmer and Weil (2004) αθ := α r+s+αθq(θ)
> α > 0, which implies 1 − αθ =
r+s
(1 − α) r+s+αθq(θ) > 0 (and in turn αθ < 1), finally yields:
!
n
X
w = (1 − αθ )T + αθ P y −
xi pi − (r + s)K
(31)
i=1
1
A closer look at the different terms defining K allows interpreting it. First, p(φ)
(resp.
1
φp(φ) ) being the firm’s (resp. bank’s) expected duration of search on the credit market,
21
Master EPP — SciencesPo 2012
c
p(φ)
+
Master’s dissertation — Margaux Luflade
κ
φp(φ)
corresponds to the total expected cost of credit market frictions born by the
i
P
P b i
b
firm/bank association. Next, rewriting χb ni=1 r+b
= ni=1 χb r+b
in the expression
of K allows readily understanding the meaning of this sum: 1b being the expected duration
of search for each capital input on the intermediate goods market, χb corresponds to the
expected cost of finding one intermediate good. Then, reasoning from point of view of the
firm at the beginning of stage 11 : the instantaneous probability to find intermediate good
1 is b; the instantaneous probability to find intermediate good 2 is equal to the product of
the conditional instantaneous probability to find intermediate good 2 given that good 1 has
been found (i.e. b) and of instantaneous probability to find intermediate good 1 (i.e. b)
—thus it is equal to b2 . The same reasoning applies for the number periods over which each
intermediate good is time-discounted (by factor r + b). Inductive reasoning thus explains
why in the bracketed term of the expression of K the actual expected cost of finding the k th
k
n
b
intermediate good is χb r+b
. Finally, the multiplicative factor r+b
out of the brackets
b
simply accounts for time discounting. Therefore, K corresponds to the discounted sum of all
costs the firm/bank association had to bear at the time it enters the labor market due to the
presence of frictions in the economy.
The wage equation is of the same form as in Wasmer and Weil (2004) although some
terms differ. The wage w is a convex combination of the value of the worker’s outside option
(determined by the unemployment benefits T ) and the profit made by the firm/bank association thanks to her labor. More precisely, the profit
Pn corresponds to the revenues from sales
P y net of the cost of purchasing capital inputs ( i=1 xi pi ) and of the cost of searching for
them and for the formation of the firm/bank match (K). The share of profits going to the
worker is given, as in Wasmer and Weil (2004), by the term αθ which depends not only on
her bargaining power but also on the tightness of the labor market (and other parameters).
αθ also represents the share of the cost of frictions met by the firm/bank prior to entering
the labor market that is born by the worker: αθ K negatively enters the expression of w.
It is also important to note that w is independent from the repayment ρ negociated
between the bank and the entrepreneur.
2.3.2
Repayment determination.
ρ is bargained over by the final good firm and the bank at the time they match on the credit
market. If β denotes the bargaining power of the bank in this negociation, ρ is then solution
of the problem: ρ = argmax{(E11 − E0 )1−β (B11 − B0 )β }, or equivalently as taking the natural
logarithm is an increasing transformation:
ρ = argmax{(1 − β) ln(E11 − E0 ) + β ln(B11 − B0 )}
Partial derivatives ∂ρ [E11 − E0 ] and ∂ρ [B11 − B0 ] are obtained from equations (26) and (28)
and yield the first order condition:
(1 − β)(B11 − B0 ) = β(E11 − E0 )
(32)
Plugging (26) and (28) into this last equation finally yields:
(
P
−n X
i )
n P y − w − ni=1 xi pi
ρ
γ r+q
b
χ
b
=β
+(1−β)
+
(33)
r+s
r+s
q
q
r+b
b
r+b
i=1
The (discounted) equilibrium repayment ρ is the convex combination of two terms. The first
corresponds to the discounted future profits made by the firm/bank pair from production.
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Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
These profits amount to the revenues from sales net of the cost of purchasing capital and labor
inputs. The share of (net-of-the-wage) profits going to the bank is given by its bargaining
power β. The second term corresponds to the entrepreneur’s reimbursment of the costs born
by the bank to initiate production, namely the cost of searching for capital inputs and a
worker. As compared to the repayment equation in Wasmer and Weil (2004), the additional
burden of intermediate goods frictions is found in the second term of equation (33). It is
shared between the firm and the bank; more precisely, the higher the bargaining power of the
bank (i.e. the higher β) the higher is the share (1 − β) supported by the entrepreneur (i.e.
repayed to the bank through ρ).
2.3.3
Equilibrium tightness of the credit and labor markets.
The determination of the tightness φ of the credit market simply follows from equations (25)
and (27) obtained when solving the Bellman equations and from the first order condition (32)
of the repayment bargaining problem:
φ=
1−βκ
β c
(34)
The fact that the equilibrium tightness of the credit market tightness can be expressed as
a mere function of parameters of the credit market was already met in Wasmer and Weil
(2004) and Wasmer (2009), where it is coined as a block-recursiveness property. This property resulting only from free entry of entrepreneurs and financers on the credit market which
still holds in our framework, it is not surprising to meet it again.
The equilibrium pair (θ∗ , φ∗ ) can be determined as the intersection in the (θ, φ)-plane of
the respective free-entry loci of the firm and the bank.
On the one hand, the value E11 is given both by equations (25) and (26). Equating their
right-hand-sides and replacing ρ by its value determined in (33), we obtain:
"
P
n
i #
n P y − w − ni=1 xi pi γ
c
b
q
χX
b
= (1 − β)
−
−
(35)
p(φ)
r+b
r+q
r+s
q
b
r+b
i=1
and further replacing w by its value determined in (31), we finally get:
c
= (1 − β)
p(φ)
"
b
r+b
n
q
r+q
(1 − αθ )
Py −
Pn
γ
i=1 xi pi − T
+ αθ K −
r+s
q
n
χX
−
b i=1
b
r+b
i #
(36)
On the other hand, the value B11 is given both by equations (27) and (28). Equating their
right-hand-sides and replacing ρ by its value determined in (33), we obtain:
"
P
n
i #
n P y − w − ni=1 xi pi γ
b
q
χX
b
κ
=β
−
−
(37)
φp(φ)
r+b
r+q
r+s
q
b
r+b
i=1
and further replacing w by its value determined in (31), we finally get:
κ
=β
φp(φ)
"
b
r+b
n
q
r+q
(1 − αθ )
Py −
Pn
γ
i=1 xi pi − T
+ αθ K −
r+s
q
n
χX
−
b i=1
b
r+b
i #
(38)
As the interpretation of this set of equations has already been explained in the literature
review, we now just point at key points and diffferences with Wasmer and Weil (2004).
23
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
The first equation (EE) still describes the set of all pairs (θ, φ) compatible with the free
entry condition for firms and still defines a decreasing relation between φ and θ. Conversely,
the second equation (BB) still describes the set of all pairs (θ, φ) compatible with the free
entry condition for banks and still defines a decreasing relation between φ and θ. However,
the discounted profits computed to express this free entry condition (that is, the equality
between the cost of entering the credit market, on the left hand sides; and the expected profits
resulting from the future stages on the right hand sides) now incorporate (1) the additional
b n
time discounting ( r+b
) due to the time spend searching on the intermediate goods market;
P b i
and (2) the cost of this supplementary search χb i ( r+b
).
BB
Note that we still need the condition that limθ→0 φ (θ) =: φB < φE := limθ→0 φEE (θ), for
the two curves (EE) and (BB) intersect in a single point (θ∗ , φ∗ ).
2.3.4
Equilibrium on the intermediate goods market.
The final good firm decides on the quantities of inputs it will buy at the beginning of its search
process on the intermediate goods market, that is, when entering stage 11 . Indeed, it must
know which quantity of intermediate goods it will commit to buy before entering in agreement
with future suppliers. The objective function the final good firm aims at maximizing when
choosing its vector x = (xi )ni=1 of intermediate goods consumption is then the value of E11
as given by (26). It solves its maximization problem taking as given both the price vector
p = (pi )ni=1 of intermediate goods and the price P of the final good it will produce.
Differentiating once the function E11 (x) given by (26) with respect to xi gives
∂xi E11 (x) =
b
r+b
n−1 q
1
(1 − β)(1 − αθ )P ∂xi y − pi
r+qr+s

 1−σ
σ
n
X
σ−1 
σ
where ∂xi y = xi
xj
j=1
Differentiating a second time yields
∂x22 E11 (x) =
i
b
r+b
n−1
q
1
(1−β)(1−αθ )P ∂x22 y with ∂x22 y = −(1−σ)xσ−2
i
i
i
r+qr+s
X
j6=i

 1−2σ
σ
n
X
σ
σ
xj
xj
j=1
Given that (∀j) xj ≥ 0, ∂x22 y is negative for all positive values of xi , this proves that E11 is a
i
concave function of xi (∀i = 1, . . . , n) and the necessary and sufficient first-order conditions
for maximization are therefore: ∂xi E11 (x∗i ) = 0. Solving this condition yields:
x∗i =
p i
1
σ−1
P
y
(39)
where we recall that P is the price of the final good. Equation (39) being true for all
i ∈ {1, . . . , n}, then taking any pair (i, j) with i 6= j, gives the following relation:
xi
=
xj
pi
pj
1
σ−1
⇔ xi = xj
pi
pj
1
σ−1
(40)
This equation is also met in Dixit and Stiglitz (1977) where the following step towards eliciting
x∗i consists in substituting xj in the budget constraint of the consumers (see equation (5)).
Here, final goods firms are the counterpart to consumers in Dixit and Stglitz (1977); we
therefore use as a budget constraint the equality (36) obtained, after solving for ρ and w, as
a consequence of the free entry condition for entrepreneurs on the credit market. We use (40)
24
Master EPP — SciencesPo 2012
to see that
Pn
i=1 xi pi
Master’s dissertation — Margaux Luflade
=
xj
1
pjσ−1
σ
σ−1
Pn
i=1 pi
and replace it in (36). Elementary computational
steps lead to:
1
piσ−1
xi = P
n
σ
(P y − T + A1 )
(41)
σ−1
j=1 pj
where A1 = A1 (φ, θ, b) depends only on φ, θ, b and parameters of the model.5
Equation (41) gives the optimal quantity of intermediate good j demanded by the final good
firm as a function of its price pj and allows determining the equilibrium price of intermediate
good j. Indeed, the induced elasticity of demand of good j to its price is:
εprice
:= −
j
∂xj pj
∂ ln(xj )
1 xj p j
1
=−
=
=−
∂ ln(pj )
∂pj xj
σ − 1 p j xj
1−σ
(42)
Then, the pricing rule (10) presented in subsection 1.2 requires the intermediate goods producers to equate their marginal cost of production to the marginal revenue from their sales:
!
C
1
(43)
pj 1 − price = C ⇔ pj =
σ
εj
As in Dixit and Stiglitz (1977), the symmetry among intermediate goods is preserved, as the
equilibrium price is the same for all intermediate goods j = 1, . . . , n. Denote pIG = σC this
equilibrium price. It follows that the first term in (41) is equal to [n σC ]−1 and the quantity
demanded is thus the same for all intermediate goods. Denote this quantity by
−1
C
x∗ = n
(P y − T + A1 )
σ
1
This allows simplifying the expression of the aggregate quantity index into Y = n σ x∗ and
yields the final expression:
x∗ =
A1 − T
(44)
1
npIG − P n σ
One can check x∗ is a non-negative decreasing function ofhpIG . The complete
expression
of A1
s
κ
c
1
has be given in footnote. Letting r → 0, A1 tends to 1−αθ αθ φp(φ) − 1−β − α p(φ) − (1 − α)
i
χ
γ
b n − q . The sign of this expression is a priori ambiguous but given that all the substracted
elements in the bracketed term are positive, A1 can be expected to be negative. It is quite
small in magnitude though, as the exogenous destruction rate s is small. A1 being negative,
the whole numerator in (44) is negative; and we thus get a condition on intermediate goods
and final good prices necessary to induce a positive demand of intermediate goods (and then
1−σ
to induce production): x∗ > 0 ⇔ pIG < P n σ . As the elasticity of substitution (equal to
1
1−σ ) between goods increases, i.e. as σ increases and becomes closer to 1, the right hand side
decreases and become closer to P : the upper bound on pIG becomes tighter.
5
A1 (φ, θ, b) =
r+s
1−αθ
αθ K −
γ
q
−
b
r+b
−n
r+q
q
χ
b
Pn
i=1
25
b
r+b
i
+
1
c
1−β p(φ)
Master EPP — SciencesPo 2012
2.3.5
Master’s dissertation — Margaux Luflade
Equilibrium numbers of unemployed workers and vacancies
A stock-flow reasoning now allows determining the equilibrium numbers of final good firms,
banks, workers and intermediate goods firms at each stage; among them, of particular interest
are the equilibrium unemployment rate u and the equilibrium number of vacant positions V.
Let us start by considering the unemployment rate. If there is a continuum of mass 1 of
workers in the economy, if u denotes the number of unemployed workers and thus equals the
unemployment rate, we have the law of motion:
u̇ = −θq(θ)u + s(1 − u)
where Ẋ denotes the variation in X from one period to another. From one period to another,
the number of unemployed workers is on the one hand decreased by the number of unemployed who find a job, which occurs at rate θq(θ), and on the other hand increased by the
number of employed workers who loose their job, which occurs at rate s. At equilibrium, the
unemployment rate in constant, so that u̇ = 0 and thus:
u=
s
s + θq(θ)
(45)
This expression is identical to that found in Pissarides (1990) and Wasmer and Weil (2004).
This means that introducing a frictional intermediate goods market affects the unemployment
rate only through its effect on the labor market tightness.
Let N0 , N11 , . . . , N1n , V, and N2 denote the numbers of final goods firms respectively at
the credit market stage, the sucessive n intermediate goods market stages, the labor market
stage and the profit stage. Their laws of motion write:


N˙11 = −bN11 + p(φ)N0





N˙12 = −bN12 + p(φ)bN11




.

..



N˙ k = −bN k + p(φ)bN k−1 (∀k = 3, . . . , n − 1)
1
1
..


.





N˙1n = −bN1n + bN1n−1





V̇ = −q(θ)V + bN1n


Ṅ = −sN + q(θ)V
2
2
1
At equilibrium, the number of agents at each stage is constant so all these dotted variables
are equal to 0. Given that we have determined the equilibrium values u in equation (45) and
θ in equations (35) and (37), we can successively deduce:


V = θu



N = q(θ) V
2
s
n = N n−1 = · · · = N 1 = q(θ) V

N

1
1
1
b


N = b
0
p(φ)
For later use, let us denote the total number of final good firms on the intermediate goods
q(θ)
s
market by N1 := N11 + · · · + N1n = n q(bθ̄) V = n q(θ)
b θu = n b θ s+θq(θ) .
The equilibium number of banks at each stage is obtained proceeding very similarly as
with firms so we leave the explicitation of laws of motion for the appendix. Instead of re-doing
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computations, one can also notice that starting from stage 11 , the number of banks at each
stage is equal to the number of final good firms at the corresponding stage since they are
matched one-to-one on the credit market. The only stock that remains to be elicited then is
the number B0 of banks on the credit market at equilibrium; it is given by B0 = Nφ0 , with φ
determined by equation (34).
2.3.6
Equilibrium tightness of the intermediate goods market and equilibrium
number of intermediate goods producers
Now, turn to the determination of the equilibrium number n of intermediate goods firms in
the economy. As explained in the beginning of section 2, we have had a short-term reasoning
so far. We have taken the number n of intermediate goods producers as given. However,
in the long run this number is endogenously defined since producers are assumed to be able
to freely enter the market. This assumption of free entry of producers on the intermediate
goods market requires that at equilibrium profits must be driven to zero, that is, the profits
raised from sales exactly cover the fixed costs of production. Assuming these fixed costs do
not vary across firms and denoting them by a translates into:
(p − C)x = a ⇔ x =
a σ
.
C1−σ
(46)
Besides n, x∗ depends only on parameters of the model and endogenous quantities that have
been previously (or will be now) determined as function of n and parameters. Equation
(46) thus implicitely characterizes the equilibrium value n. No closed form can be obtained
from this equation though. To go a little further and see how the determination is made
let us make some simplifying assumption that allows for more tractability. Assume there is
q(θ)
no time discounting, that is, r = 0; this allows getting rid of terms such as r+q(θ)
or, more
b
interestingly, r+b
which is often raised at some power depending of n and thus considerably
complexifies the derivation of theh equilibrium
K =
i
value ofn. Under this assumption,
χ
χ
γ
κ
c
s
κ
1
c
φp(φ) + p(φ) + n b and A1 = 1−αθ αθ φp(φ) + αθ − 1−β p(φ) + (αθ − 1)n b − q(θ) . In this
expression of A1 , only the last two terms between the brackets implicitely depend on n.
Indeed, α, β, γ, κ, c, χ, and s are exogenous parameters and φ is a function of β, κ and c
only. On the contrary, θ and b are functions of n. For θ, this can be seen from equations (35)
and (37). For b, we need to return to the discussion we have had in subsection 2.1 about the
definition of b = b(ζ). According to this discussion and following Wasmer and Weil (2004) and
Wasmer (2009) who take an exponential form for the matching rates on the labor, credit and
final goods markets, we set b(ζ) = b0 ζ −b where ζ is the tightness of the intermediate goods
market (from the point of view of final good firms) and b0 and b are chosen parameters (the
latter corresponding to the elasticity of the supplier-finding rate with respect to the tightness
of the intermediate goods market). Under our simplifying assumptions,
s
n q(θ)
N1
q(θ)
s
q(θ)
s
b(ζ) θ s+θq(θ)
ζ=
=
=
θ
⇔ ζ 1−b =
θ
n
n
b0 ζ −b s + θq(θ)
b0 s + θq(θ)
⇔ ζ=
q(θ)
s
θ
b0 s + θq(θ)
1
1−b
(47)
The equilibrium tightness of the intermediate goods market can thus be obtained as a function
of parameters and of the (previsouly characterized) labor market tightness only.
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This last equation concludes the solution of the equilibrium. An equilibrium is indeed
is set (φ, θ, ζ, ρ, w, (xi )i , (pi )i ) describing the tightness and equilibrium price on each market
as well as the optimal demand function of the final good firms for intermediate goods. The
seven equations characterizing the equilibrium are summarized in appendix 5.1. They derive
not only from free-entry conditions and bargaining problems, as in Wasmer and Weil (2004)
or Wasmer (2009), but also from the individual profit maximization problems of intermediate
and final goods firms.
In the appendix are also recalled the equations which allow determining other quantities
of interest, such as the level of unemployment and the number of vacant positions in the
economy.
3
Selected comparative statics: frictions and regulation analysis
In this section, we look at the effects of a change in parameters on the equilibrium value of
labor market variables, namely the unemployment rate and the wage. In the first subsection,
we focus on parameters that command the level of frictions on the various markets of our
economy. In the second, we turn to parameters indicating the level of regulations of these
markets.
Due to the lack of tractability in the equilibrium equation, we simplify the analysis by assuming a 0 interest rate r = 0 (equilibrium equations under this assumption are provided in
Appendix (5.2) and the exponential form of matching rates p(φ) = p0 φ−p , q(θ) = q0 θ−q and
b(ζ) = b0 ζ −b . We also focus on the situation in which the number n of intermediate goods
available in the economy remains unchanged. Because the computations turn out to be cumbersome and not always to lead to unambiguous results, we limit ourselves to a qualitative
analysis here, giving only the key equations that allow understanding the propagation mechanisms. A more detailed version of the analysis, including formal computations, is provided
in Appendix 5.2.
In most cases, we will not be able to draw any general conclusion about the effect a the
change under scrutiny. However, we still present the analysis as it offers an interesting at the
way policy measures may propagate through the economy.
3.1
Effects of the level of frictions on the labor market equilibrium
Let us first consider the effects of decreasing frictions on the various markets of the economy.
As frictions translate the imperfection of the information available on markets, any policy
tending to improve the acquisition of information by agents may be considered as decreasing
frictions. In this regard, the development of communication technologies, such as the Internet, and their use for advertising or communication between potential trading partners may
be thought of as contributing to the reduction of market frictions. For instance, in the case of
the labor market, it allows a larger, quicker and cheaper diffusion of posted vacancies so that
recruiters and fitting workers can more easily enter in contact. As regards to the intermediate
goods market, they allow a better publicization of available inputs, in particular in the case
of raw materials which can only be produced or extracted in very localized areas but may be
needed in a larger geographic scale. Finally, concerning the credit market, they may enable
entrepreneurs to more efficiently convince financers of the viability of their project.
Given these intuitions, policy-makers interested in decreasing unemployment or driving the
labor market to set up a high-enough level of wage may want to invest in the development
of communication technologies. To conclude about the extent to which such strategy would
be efficient, one need to investigate further the mechanisms of propagation throughout the
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economy of the effects of a decrease in frictions.
Effects of intermediate goods market frictions. Specifically, let us consider a marginal
change in b0 . For instance, an increase in b0 , keeping the intermediate goods market tightness
constant, induces an increase in the instantaneous probability b for final good firms to find
a supplier on the intermediate goods market, that is, a decrease in the intermediate goods
market frictions.
Starting with the unemployment rate, the effect of a marginal change in b0 can be deduced
from equation (45)
(1 − q )q(θ) dθ
du
= −s
db0
(s + θq(θ))2 db0
The change in the unemployment rate after a variation in b0 is a consequence of the change
induced in the labor market tightness θ. In appendix, we show θ is positively impacted by
a change in b0 : as frictions on the labor market decrease (i.e. as b0 increases), the tightness
of the labor market increases. Intuitively, a decrease in intermediate goods market frictions
reduces the search costs for capital inputs and thus increases the profit of final good firms.
By free entry, since they expect higher profits, more entrepreneurs enter the credit market
and, as the tightness on this market given by (34) remains constant (their increased entries
being compensating by increased entries of bankers who also expect larger profits as they will
finance lower search costs and share an increased profit), more entrepreneurs arrive on the
labor market, where the tightness then increases.
du
The expression of db
then shows that as a direct consequence of the increase in the labor
0
market tightness, the unemployment rate decreases: due to the relatively larger number of
vacancies, the job-finding rate increases and the u decreases.
The impact of a marginal change in b0 on the equilibrium wage cannot be deduced as
straightforwardly as it was in the case of the unemployment rate. Indeed, from the wage
equation (31), we have:
1
1
dw
dαθ dx
dK
=
P n σ x − nxp − sK − T + αθ (P n σ − np)
−s
db0
db0
db0
db0
This indicates three chanels through which b0 impacts on the wage. One seems intuitively
clear: a change in the level of frictions on the intermediate goods market directly impacts
on the total cost of search frictions K the firm has to bear before entering the labor market;
as explained when we derived the wage equation (31), this cost K is shared between the
firm/bank and the worker through the wage. More specifically, as frictions on the intermediate
goods market decrease (i.e. as b0 increases), the financial burden of frictions K, which is partly
supported by the worker, decreases. This pushes the wage up.
The two others chanels are more indirect. First, due to the interconnection of markets, as
represented by equation (47), a change in the level of frictions on the intermediate goods
market induces a change in the labor market tightness, which in turn determines the share
of worker in production αθ and thus impacts on the wage. It can be shown that the effect of
b0 on αθ is a weaken consequence of the effect of b0 on θ, thus the share of workers in profits
increases as the frictions on the intermediate goods market decrease (i.e. b0 increases). This
is only due to the increase in the labor market tightness induced by the increase in b0 : the
competition for workers among final good firms being increased in the tighter labor market,
they are ready to set wages more profitable to workers.
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Finally, the wage depends on revenues from sales, thus on the quantity x of intermediate goods
used in production. As this quantity x depends on the profits the final good firm expects
from production, it also varies with θ, αθ and K and thus induces an additional (and more
complex) chanel of variation for w when b0 varies. The sign of dx
d0 ambiguous though. This
β
γ
κ
c
is due to the presence of the term φp(φ) + 1−β p(φ) − q(θ) whose sign is a priori unknown and
depends on the relative search costs and levels of frictions on the credit and labor markets.
A case where further conclusions could be drawn is if this ambiguous term happened to be
negative. This would correspond to a situation in which the (costs of) labor market frictions
are larger than the (costs of) credit market frictions. Then x would unambiguousy vary in
the opposite direction of θ: a decrease in the intermediate goods market frictions (thus an
increase in b0 ) would induce an increase in the quantity of intermediate goods consumed.
Even in this case, the global variation of w remains ambiguous though. On the one hand,
the wage is pushed up by the larger share of the worker in production and the lower costs of
frictions which are partly supported by the worker. On the other hand, the decrease in the
β
γ
κ
c
quantity of intermediate goods consumed (in the case where φp(φ)
+ 1−β
p(φ) − q(θ) < 0) tends
to decrease the quantity produced and thus the revenues from sales and the profits, which in
turn pushes the wage down.
Effects of labor market frictions. Now consider the consequences of a change in the
level of frictions on the labor market, namely an increase in q0 which, for each level of θ,
induces a higher job-filling rate q(θ) and can thus be nterpreted as a decrease in the labor
market frictions.
Again, let us start by analysing the change in the unemployment rate.
du
sθ1−q
=−
dq0
s + θq(θ)
This time, the change in u is directly induced by the change in q0 (as q0 directly enters the
unemployment rate equation through q(θ): a decrease in the labor market frictions (i.e. an
increase in q0 ) induces a decrease in the unemployment rate because the job-finding rate
θq(θ) = q0 θ1−q increases with q0 .
Turning to the wage,
1
1
dw
dαθ dx
dK
=
P n σ x − nxp − sK − T + αθ (P n σ − np)
−s
dq0
dq0
dq0
dq0
We recognize the three chanels of propagation identified in the previous case. The most direct
effect now comes from αθ which directly translates the change in the level of frictions on the
labor market: an increase in q0 increases the share of workers in profits and this contributes
to push the wage up.
On the contrary, the transmission of a change in the frictions through K is not direct anymore. Indeed, as explained earlier, K accounts for the cost of frictions on the credit and the
intermediate goods market and thus does not directly reflects labor market frictions. Nevertheless, due to the interdependence of markets (captured by dζ
d0 ) a decrease in the labor
market frictions (i.e. an increase in q0 ) increases the intermediate goods market tightness.
This is due to the increase in the number of entrepreneurs trying to launch their project as a
decrease in the labor market frictions implies lower search costs on the labor market and thus
possibly higher profits. In turn, the rate b(ζ) at which final goods firms find their suppliers
on the intermediate goods market decreases and this increases the cost of frictions K. This
pushes the wage down.
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Given the opposite effects of the variations in αθ and K on w, we already know the change in
the wage will be a priori unknown. The variation of x also adds some ambiguity as its sign
is again unknown. This is due to the presence of same term as in case of intermediate goods
market frictions.
Effects of credit market frictions. Finally, let us consider an increase in p0 which, given
that φ is fixed as soon as the credit market parameters β, c and κ are left unchanged, increases
the probability for entrepreneurs to find a financer and thus can thought of a representing a
decrease in the credit market frictions.
The change in the unemployment rate is given by:
(1 − q )q(θ) dθ
du
= −s
dp0
(s + θq(θ))2 dp0
As in the case of intermediate goods market frictions, the effect of an increase in p0 propagates
dθ
to u through θ. As shown in appendix, dp
is determined by taking derivatives in the (EE)
0
dx
equation; in particular, it depends on dp0 which once again suffers an ambiguity issue. If the
ambiguous term is thought of as being negative (as was previouly analyzed in the previous
paragraphs) then θ is a decreasing function on p0 : as frictions on the credit market decrease
(i.e. p0 increases), the labor market tightness decreases because less entrepreneurs enter the
credit market and thus happen to arrive on the labor market afterwards.
This would in turn increase the unemployment rate.
Turning to the wage,
1
1
dw
dαθ dx
dK
=
P n σ x − nxp − sK − T + αθ (P n σ − np)
−s
dp0
dp0
dp0
dp0
The same ambiguity issue arises since the terms involved are the same. In the case where
θ decreases as p0 increases, the wage is on the one hand pushed down by a decrease in the
share of workers in profits (due to the fact that as the labor market becomes more tight to
workers, they are ready to accept a lower share in profits); and on the other hand pushed
down by the decrease in the cost K of frictions. Additional ambiguity is brought by the
unknown direction of variation in x.
3.2
Effects of regulation and deregulation
In this subsection, following Blanchard and Giavazzi (2003), we look at special comparative
statics which can be analyzed as effects of deregulation policies. Specifically, we focus on the
change in the unemployment rate and the wage following two examples of policies, namely a
labor market deregulation policies leading to a decrease in the worker’s bargaining power; and
intermediate goods market deregulation policies which induce an increase in the elasticity of
substitution between intermediate goods.
Effects of a decrease in the worker’s bargaining power. α can be thought of as deriving from institutions which increase the workers’ bargaining power, such as the right to
strike or the existence of powerful unions.
The variation in the unemployment rate is given by:
(1 − q )q(θ) dθ
du
= −s
dα
(s + θq(θ))2 dα
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The same ambiguity problem arises as before, which in turn prevent drawing a definitive
conclusion concerning the direction of variation of θ. Again, we can infer consequences on
β
γ
κ
c
the case in which φp(φ)
+ 1−β
p(φ) − q(θ) is negative. Before explaining the conclusions in this
dθ
θ
case, we underline an interesting point. dα
through (EE) requires computing dα
dα which, in
dθ
appendix, we show is equal to Aα,2 + Aα,1 dα where the Aα,2 > 0 accounts for the direct effect
on αθ of a change in α while Aα,1 > 0 weights the change in αθ due to the change in the
labor market tightness, in turn induced by the original variation in α.
β
γ
κ
c
dθ
Going back to the case in which φp(φ)
+ 1−β
p(φ) − q(θ) is negative, dα is negative: a decrease
in the worker’s bargaining power induces an increase in the labor market tightness.
The increase in the labor market tightness θ would in turn induce a decrease in the unemdu
ployment rate, as seen from the expression of dα
. In terms of employment, the workers would
then benefit from a decrease in α.
Now turning to the wage,
1
1
dw
dx
dαθ dK
=
P n σ x − nxp − sK − T + αθ (P n σ − np)
−s
dα
dα
dα
dα
The ambiguity arises again in the case of the wage and cannot be resolved. First, due to
dθ
< 0 implies the
the change in αθ : the direct effect of a change in α is positive, but dα
indirect effect is negative. In other words, as a consequence of a decrease in their bargaining
power, the workers’ share in profit would on the one hand decrease (as a direct consequence
of their lower bargaining power) and on the other hand be pushed up due to the tighter labor
market, that is, to the increased competition among firms for workers. The second source of
dx
ambiguity for the change in w is can be found in dα
whose sign in turn depends on that of
dαθ
dα .
Effects of a change in the elasticity of substitution between intermediate goods.
Let us start by writing the equations showing the variations in u and w:
(1 − q )q(θ) dθ
du
= −s
dα
(s + θq(θ))2 dσ
1
dw
dαθ =
P n σ x − nxp − sK − T + αθ
dσ
dσ
1
dx d(P n σ − np)
dK
+
x−s
(P n − np)
dα
dσ
dα
!
1
σ
The variation in unemployment follows the same logic as in the previous paragraphs. However, the second equation suggests the variation in w is more complex though because a
variation in σ induces an additional change in profits as compared to the previous cases. As
can be seen in appendix, trying conclude about the global variation of w is even more cumbersome than in the previous paragraphs, although it follows the same strategy. Therefore,
here, we will limit ourselves to the analysis of the new mechanism that arises in this case,
namely the one due to the change in the generated profits.
Let us start by noting that an increase in σ ∈ (0, 1) induces an increase in the elasticity of
substitution between intermediate goods and thus decreases the monopoly power of intermediate goods producers. As a consequence their mark up decreases and so does the price of
the intermediate goods. As compared to the other changes in parameters we have considered
so far, it is the first time the price of intermediate goods varies; this is due to the pricing rule
of intermediate goods producers: the price they set depends only on their marginal cost of
production and the elasticity of substitution between goods. This is the first chanel through
1
which the final goods firms’ profits (namely P n σ x−pnx ) are affected by a change in σ: keeping
all other things constant, the cost of purchasing the quantity x of inputs is lower. The second
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1
chanel comes from the first part P n σ x of the profit expression: an increase in σ decreases the
total amount of production y reached with a given bundle x on intermediate goods, which
decreases the sales and thus the revenues. The global change in profits generated by the consumption of capital inputs x is thus a priori unclear. Beyond this direct effects of a change
in σ on inputs prices and productivity, we have similar indirect effects as in the previous
paragraphs through θ and x.
4
4.1
Extension: frictional final goods market
Setting: re-thinking the production function
We now turn to an extension of our basic framework. In line with Wasmer (2009), we now
consider a frictional final good market. We follow the model proposed in this paper, that
is, of a one-to-one matching process between productive firms looking for a consumer and
employed workers ready to consume on the final good market. It is assumed that consumers
inelastically consume one unit of final good (per time period); a rationale underlying is this
assumption has been provided in section 1. The utility yielded by the consumption of one
unit of the final good is 1 + Φ. As a consequence of inelastic consumption and one-to-one
matching, final good firms inelastically produce one unit of good (per time period). This
brings us to reinterpret the production function of final good firms that we introduced in the
basic model. In the initial version, the quantities of intermediate goods purchased determine
the quantity of output. This quantity is now fixed to 1; we therefore propose a reasoning in
terms of quality of the final good. An increase in the quality of the output entails an increase
in the utility of the consumer, thus we propose the following modeling:
Φ(x) =
n
X
!
1
1−σ
xσi
(48)
i=1
We proceed as in the previous sections: we write the Bellman equations and, while commenting them, give more details about the different stages modeled.
4.2
Bellman equations
Concretely, considering a frictional final goods market corresponds to adding a final fifth
stage in the model of the evolution process of the final good firms (and banks). The Bellman
equations corresponding to the first three stages of the life of the firm —search for a banker, for
intermediate goods and for a employee— are left unchanged to (13), (14) and (15). Equation
(16) is replaced by:
rE3 = −w −
n
X
xi pi + w +
i=1
rE4 = P − w −
n
X
n
X
xi pi + λ(ξ)[E4 − E3 ] + s[E0 − E3 ]
(49)
i=1
xi pi − ρ + τ [E3 − E4 ] + s[E0 − E4 ]
(50)
i=1
In stage 3, the firm produces the unstorable final good but cannot sell it yet. It searches for
a consumer on the final goods market, where it founds a match with probability λ = λ(ξ). ξ
corresponds to the tightness of the final goods market (from the point of view of the firms),
defined as the ratio of the number of prospective firms on the final goods market to the numN2
ber of unmatched consumers, respectively denoted N2 and W0 : ξ := W
. Note that as there
0
is a continuum of unit mass of workers-consumers, if u denotes the unemployment rate and
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W1 the number of matched consumers, we have 1 = u + W0 + W1 . As long as the productive
firm remains unmatched, it does not make profit and its production costs are reimbursed by
the bank. As in the basic model, at this stage the firm can be subject to destruction, which
happens at exogenous Poison rate s. When a consumer is found, the firm finally enters the
profit stage. It sells its production at unit price P , starts repaying the bank ρ and needs
to pay production costs. It may still be subject to destruction at rate s. In addition, the
match with the consumer may be broken with instantaneous probability τ , interpreted as the
versatility of the consumer by Wasmer (2009). If such event occurs, the firm goes back to
the final goods market to look for a new match, while we assume the bank starts financing
production costs again.
Symmetrically for banks, equations (18), (19) and (20) are kept while (16) is replaced by:
rB3 = −w −
n
X
xi pi + λ(ξ)[B4 − B3 ] + s[B0 − B3 ]
(51)
i=1
rB4 = ρ + τ [B3 − B4 ] + s[B0 − B4 ]
(52)
The interpretation of these equations is symmetric to those of the firms. In the basic model,
since supply and demand instantaneously meet on the final goods market, the bank starts
being repayed by the firm right when the worker is found and production starts; here, it
must wait one stage more for the firm to make profits. During this stage 3, it pays for the
firm’s production costs and is subjected to the same evolution as the firm: with instantaneous probability s it suffers the destruction of the firm which sends it back to the entry
stage; with probability λ(ξ) it benefits from the firm-consumer match being formed to enter
the repayment stage. There, it gets ρ from the firm until the firm is either destroyed or left
by the versatile consumer.
Finally, let us have a look on the workers-consumers’ side. As announced in section 1, we
follow our basic framework and assume T is the unemployment benefits, and not a universal
transfer paid to all workers independently of thei employment status, as in Wasmer (2009).
Equation (23) corresponding to the unemployed worker’s search for a job is left unchanged
as only the consumption phase is modified here. Assuming only employed workers can look
for a good and consume, so that equation (24) is replaced by:
rW0 = w + ξλ(ξ)[W1 − W0 ] + s[U − W0 ]
(53)
rW1 = w − P + (1 + Φ) + s[U − W1 ]
(54)
Once she gets a job and is paid a wage w, the consumer starts looking for a consumption
good on the final goods market, which she founds with probability ξλ(ξ), complementary to
the firm’s probability to find a consumer. In the event a good is found, she buys one unit of
it at price P and enjoys utility 1 + Φ. As she is a worker herself employed in a final good
firm, no matter whether she is consuming or still looking for a match on the goods market,
the consumer is affected by the same job-destruction rate s as productive firms. From this,
Wasmer (2009) interprets further the versatility of the consumer: she stops consuming in the
event she looses her job —thus τ = s.
Solving Bellman equations. As computation proceeds just as in section 2, based on
free entry of entrepreneurs and capitalists on the credit market, we only provide here the
equations that will later be of direct use. Further details can be found in appendix.
c
E11 =
p(φ)
34
Master EPP — SciencesPo 2012
E11
=
b
r+b
n
Master’s dissertation — Margaux Luflade
P
P − w − ni=1 xi pi − ρ
λ
q
αλ
r+qr+s+λ
r+s
(55)
r+s+λ
is taken from Wasmer (2009) and interpreted as an indicator
where the ratio αλ := r+s+λ+τ
of the fluidity of the final goods market from the point of view of the firms: it varies between
0 and 1, increasing both as frictions on the final goods markets λ1 or as the versatility τ of
the consumers decrease, that is, as consumers become easier to locate and less likely to stop
consuming.
On the banks’ side:
B11 =
B11 =
−
χ
b
κ
φp(φ)
b
r+b
n X
i=1
n
b
r+b
!#
n
X
ρ
1
1 − αλ
αλ
− (w +
−
xi p i )
r+s
r+s
λ
"
q
γ
λ
− +
r+q
q
r+s+λ
i=1
i
(56)
Equations (55) and (56) can be compared to the analogous (26) and (28). Looking at the E11
λ
equations, the difference lies on two additional multiplicative terms, r+s+λ
and αλ , both accounting for the additional discounting and expectation-taking of profits due to the existence
of frictions on the final goods market and to the possibility for firms to loose their customers
which delay and introduce uncertainty over the firm’s access to the profit stage.
Turning to the B11 equations, recall that (28) expresses B11 as the discounted value of the repayment net of the discounted costs of search on the labor and intermediate goods markets which
are born by the bank. In (56), we meet again the two additional discounting/expectationtaking terms which translate the fact that delays and uncertainty over the firm’s access to
the profit stage induce the same delays and uncertainty over the bank’s access to the repayment stage. On top of being more discounted, the repayment ρ is taken net of the expected
costs of purchasing capital and labor inputs which the bank will have to bear as long as the
productive firm has not met a consumer whom to sell the output.
Finally, looking at workers, equations (53) and (54) combined yield:
W1 − W0 =
1+Φ−P
r + s + ξλ(ξ)
(57)
In line with the comment made for equation (29) in the basic model, it is important to see
that, from the point of view of the newly-matched consumer who is bargaining the price P of
the final good and who aims at maximizing her expected surplus W1 − W0 , the ‘P ’ entering
equation (57) is not the (still indetermined) future price she will have to pay but should
rather be understood as a signal P̄ sent by the market.
4.3
4.3.1
Solution of the model
Repayment, wage and price determination.
As it is the only innovation from the basic model presented in section 2, we develop the
final goods price determination in details while reporting only the key results concerning
repayment and wage fixation. Further details can be found in appendix sections 5.3.
35
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
Final good price determination. P is bargained over between the consumer on the
one hand and the firm/bank association on the other hand at the time of the match on
the final goods market. It is solution of the log-transformed Nash bargaining equation P =
argmax{δ ln(W1 − W0 )(1 − δ) ln(B4 + E4 − B3 − E3 )} where δ ∈ (0, 1) denotes the bargaining
power of consumers. First-order condition writes:
δ
∂P [W1 − W0 ]
∂P [B4 + E4 − B3 − E3 ]
+ (1 − δ)
=0
W1 − W0
B4 + E4 − B3 − E3
On the one side, using equation (??) ⇒ (r + s)[W1 − W 0] = w − P + (1 + Φ) + sU − (r + s)W0
where as explained above, U and W0 do not depend on the still indetermined price P . Thus
1
∂P [W1 − W0 ] = − r+s
.
On the other
side, using the equations provided in Appendix ??, B4 + E4 − B3 − E3 =
P
P − w − ni=1 xi pi − (r + s)(B3 + E3 ) where (B3 + E3 ) does not depend on the still unknown
1
P , thus ∂P [B4 + E4 − B3 − E3 ] = r+s+τ
.
FOC then rewrites:
δ
P
[B4 + E4 − B3 − E3 ] = (1 − δ)(r + s)[W1 − W0 ]
r+s+τ
Using [W1 − W0 ] =
1+Φ−P
r+s+ξλ(ξ)
and [B4 + E4 − B3 − E3 ] =
P
r+s+λ+τ
finally yields:
1+Φ
P =
1+
(58)
δ r+s+τ r+s+ξλ(ξ)
1−δ r+s r+s+λ(ξ)+τ
As in Wasmer (2009), the equilibrium price P of the final good is a fraction of the marginal
utility enjoyed when consuming the final good. The fraction is determined by the bargaining
power δ of the worker and the level of frictions (given by λ(ξ)) on the final goods market.
However, note that in the present framework the marginal utility Φ of the consumer, which
P positively depends on, translates the quality of the final good which is in turn determined
through the production function (48) by the quantity of intermediate goods used by the final
good firm. Through Φ, the equilibrium price of the final goods thus indirectly depends on the
quantity of intermediate goods (and their variety). On may note that b does not explicitely
appear here. However, it still affects P through its effect on ξ and xi (i = 1, . . . , n). Note that
consumers are also affected by the existence of frictions on the intermediate goods market
due to the fact that they are also workers whose wage, as given by equation (??), depends
on the cost of these frictions, as measured by K.
Wage determination. w is bargained over between the workers and the firm/bank association at the time of the match on the labor market. It is solution to: w = argmax{(1 −
α) ln(W − U ) + α ln(E3 + B3 − B2 − E2 )}. Partials can be computed using equations provided
1
1
in appendix: ∂w [W0 − U ] = r+s
and ∂w [E3 + B3 − B2 − E2 ] = − r+s
. First-order condition
thus writes:
α[E3 + B3 − B2 − E2 ] = (1 − α)[W0 − U ]
and yields
X
1+Φ−P
λP
w=α
−
xi pi − (r + s)K + (1 − α) rU − ξλ(ξ)
r+s+λ+τ
r + s + ξλ(ξ)
This intermediate step allows recognizing the familiar for on the wage equation: it is a
convex combination of the value of the outside option, given by rU and now taking in into
account that no comsumption is possible in the unemployment state; and of the expected
36
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
future profits of the firm.
As shown in the appendix, rU = T
w=
αθ0
α
+ θq(θ) 1−α
h
λP
r+s+λ+τ
−w−
P
i
xi pi − (r + s)K and thus
X
λP
1+Φ−P
0
−
xi pi − (r + s)K + (1 − αθ ) T − ξλ(ξ)
r+s+λ+τ
r + s + ξλ(ξ)
1+θq(θ)
1+θq(θ)
where αθ0 = α 1+αθq(θ)
, which implies 1 − αθ0 = (1 − α) 1+αθq(θ)
so that αθ0 ∈ (0, 1).
Here again, the equilibrium wage is the convex combination of two terms accounting, as in
the basic framework, for the outside option of the worker and the expected profit made by the
firm thanks to the worker’s labor. However, their expressions differ from the wage equation
(31) elicited in section 2. Indeed, the value of the outside option of the worker is now given
by the difference between the unemployment benefits T she gets when she does not work and
the expected net (of-the-price) marginal utility she could get is she were working (and thus
potentially consuming). As for the profits made by the firm and shared with the worker, they
now consist in the expected revenues made from sales net of the certain costs of purchasing
capital inputs and of search frictions: as in the basic framework, at the time it hires a worker
and negotiates the wage, the firm has already searched for and purchased capital inputs; on
the contrary, as the production and profit stages are not confounded anymore, sales are still
uncertain.
Repayment determination. ρ is bargained over by entrepreneurs and capitalists at the
time they match on the credit market. It is solution to: ρ = argmax{(1 − β) ln(B11 −
1
B0 ) + β ln(E
1−
E0 )}. Partials can be computed using
equations provided in Appendix ??:
∂ρ [B11 ] =
b
r+b
n
then writes (1 −
q
αλ
λ
1
r+q r+s+λ r+s and ∂ρ [E1 ]
β)B11 = βE11 and yields:
=−
b
r+b
n
q
αλ
λ
r+q r+s+λ r+s .
First-order condition
P
P
P − w − ni=1 xi pi
w + ni=1 xi pi r + s + λ γ
+ (1 − β) (1 − αλ )
+
+
r+s
r+s
λ
q
P
−n
i )
n w + ni=1 xi pi
b
r+qr+s+λχX
b
+
(59)
λ
r+b
q
λ
b
r+b
ρ
αλ
= βαλ
r+s
i=1
The repayment ρ follows the same logic as in Wasmer (2009). On the one hand, the financer
receives a share equal to his bargaining power the profits of the firm, understood as net of the
wage and the purchasing cost of capital inputs. On the other hand, he is reiumbursed for a
shar 1 − β the costs he paid for while the firm was not making profits. As in Wasmer (2009)
and contrary to Wasmer and Weil (2004), these costs do not only consist in search costs: they
also include the wage and the capital inputs which the bank paid for while the productive firm
P b i
was looking for a consumer. Note also the presence of the term χb ni=1 r+b
accounting
for the intermediate goods market search cost.
4.3.2
Equilibrium tightness of the credit, labor and final goods markets
As in the basic model, free entry of firms and banks on the credit markets yields the equilibκ
rium tightness of the credit market: φ = 1−β
β c . Besides, a system of two free-entry loci can
also be derived to yield a relationship between the equilibrium φ and θ. These are obtained
as in the basic model. First, we equate the two expressions found for E11 (resp. B11 ), that is,
one coming from applying the free entry to the E0 and B0 Bellman equation; and the other
37
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
resulting from the iterative solution of all others Bellman equations. Second, we substitute ρ
by its value just determined in the previous paragraph.
c
= (1 − β)
p(φ)
κ
=β
(φ)
(
(
b
r+b
b
r+b
n
n
q
r+q
q
r+q
P
P
i )
n λ
αλ P − w −
xi p i
γ
w+
xi p i
χX
b
−
−
−
r+s+λ
r+s
q(θ)
r+s+λ
b i=1 r + b
P
P
i )
n λ
αλ P − w −
xi p i
γ
w+
xi pi
χX
b
−
−
−
r+s+λ
r+s
q(θ)
r+s+λ
b i=1 r + b
These two equations can again be given the same interpretation of free entry loci for the
firm and the bank. The new terms met here only account for the additional costs and time
discounting induced by the existence of frictions on the final goods market.
Let us turn to the final goods market tightness. Solving for ξ can be done in the same way
as in Wasmer (2009). The introduction of the intermediate goods market does not modify
the structure of final goods market and the same result can thus be obtained.
As before, let N0 , N11 , . . . , N1n , V, and N2 denote the numbers of final goods firms respectively
at the credit market stage, the sucessive n intermediate goods market stages, the labor market
stage and the production stage. Let also N3 be the number of final goods firm at the profit
stage (i.e. producing and matched with a consumer). Their laws of motion write:
 ˙
N11 = −bN11 + p(φ)N0





N˙12 = −bN12 + p(φ)bN11




..



.


 ˙k
k−1
k


(∀k = 3, . . . , n − 1)
N1 = −bN1 + p(φ)bN1
..
.





N˙1n = −bN1n + bN1n−1





V̇ = −q(θ)V + bN1n





Ṅ2 = −(s + λ(ξ))N2 + q(θ)V



Ṅ3 = −sN3 + λ(ξ)N2
There are only two differences with the stock/flow equations we had in the basic model. The
first consists in the adjunction of the last equation: at each time period, the number N3 of
profit-making final good firms on the one hand decreases due to some of the profit-making
firms being destroyed (at rate s); and on the other hand increases due to the fact that some
of the productive firms which were still prospecting on the final goods market finally find a
consumer (at rate λ(ξ)). The second difference lies in the modification of the Ṅ2 equation to
account for the fact there is now another stage to be reached after the production stage: with
instantaneous probability λ(ξ) these firms may find a consumer and reach the profit stage.
On the worker’s side, if u denotes the number of unemployed workers (or, equivalently
the unemployment rate as there is a continuum of mass 1 of workers) and W0 and W1 denote
respectively the numbers of prospective and matched consumers, we have the laws of motion:


u̇ = −θq(θ)u + s(1 − u)
Ẇ0 = −(ξλ(ξ) + s)W1 + θq(θ)u


Ẇ1 = −sW1 + ξλ(ξ)W0
The law of motion of the unemployment rate is identical as that of the basic framework.
The other two equations translate the two stages of the employed worker’s life. At each time
period, an unemployed worker may find a position with probability θq(θ) and thus increases
38
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
the number W0 of employed workers who do not consume. But this number is also decreased
by the amount of workers who leave the goods-market stage, that is, those who lose their
job and are sent back to the labor-marker stage (which occurs at rate s) and those who find
a good to be consumed (which occurs at rate ξλ(ξ)). As a consequence, the number W1
of consuming workers is increased by those ξλ(ξ)W0 who found a match on the final goods
market. The only reason why workers would leave this consumption stage is the exogenous
destruction of the firm they are employed in (with rate s).
All these dotted variables being zero at equilibrium, we get the equilibirium stocks:

V = θu




q(θ)


N2 = λ(ξ)+s V
N3 = λ(ξ)
s N2


n

N1 = N1n−1 = · · · = N11 =




b
N0 = p(φ)
q(θ)
b V



u =
s
s+θq(θ)
θq(θ)
W0 = ξλ(ξ)+s
u


W = ξλ(ξ) W
0
1
s
The same reasoning as in Wasmer (2009) can be applied: given the one-to-one matching
that occurs on the labor market, the number of employed workers equals the number of
productive firms. In turn, employed workers consist in prospective and matched consumers
while productive firms gather not only those which actually make profit but also those which
look for a consumer. Thus: 1 − u = W0 + W1 = N2 + N3 . Now, seeing that N2 + N3 =
ξλ(ξ)
(1 + λ(ξ)
s )N2 and W0 + W1 = (1 + s )W0 , we must have:
s + λ(ξ)
s + ξλ(ξ)
N2
s + ξλ(ξ)
N2 =
W0 ⇔ ξ :=
=
s
s
W0
s + λ(ξ)
ξ can then be solved as a fixed point of the continuous function g : z → s+zλ(z)
s+λ(z) and ξ = 1
thus arises as a solution. This implies that λ = λ(ξ) = ξλ(ξ), that is, prospective consumers
find a good to consume at the same rate as productive firms find a consumer to purchase
their product.
As in the basic framework, the as stock-flow analysis also allows eliciting the equilibrium
number V of vacancies, the intermediate goods market tightness ξ as well as any other quantity
of interest.
4.3.3
Equilibrium on the intermediate goods market
Here again, we proceed as in the basic model: after checking the profit E11 is a concave
function of xi (done in appendix), we solve for the optimal xi . This amounts to solving the
first order condition ∂xi = 0. Detailled computations are provided in appendix. We obtain:

 (1 −
xi = pi
αθ0 )
h
ξλ(ξ)
αλ AP + λ+r+s
λ(r+s) r+s+ξλ(ξ) (1
λ+r+s
(1 − αθ0 ) λ(r+s)
39
1
 σ−1  σ−1
σ
− AP )
X


xσj 

i
j
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
where AP ∈ (0, 1) is a constant stemming from the price equation (58). This being true for
all j = 1, . . . , n,
xi
=
xj
5
5.1
pi
pj
1
σ−1
X
so that
xj X
xi p i =
1
σ−1
pj
i
σ
piσ−1
i
Appendix
Appendix to the basic model: summary of the equilibrium equations
• Wage equation (31) w = (1 − αθ )T + αθ P y −
• Repayment equation (33)
ρ
r+s
n
=β
Pn
i=1
xi pi − (r + s)K
P
o
P y−w− n
i=1 xi pi
r+s
• Equilibrium credit market tightness (34) φ =
+ (1 − β)
γ
q
+
r+q
q
b
r+b
−n
χ
b
Pn
b
r+b
i=1
b
r+b
i 1−β κ
β c
• (EE)–(BB) equations (36), (38)
–
c
p(φ)
–
κ
φp(φ)
= (1 − β)
=β
b
r+b
b
r+b
n
n
q
r+q
q
r+q
P
P y− n
i=1 xi pi −T
(1 − αθ )
+ αθ K −
r+s
(1 − αθ )
P
P y− n
i=1 xi pi −T
r+s
+ αθ K −
γ
q
γ
q
−
χ
b
−
χ
b
Pn
Pn
i=1
i=1
b
r+b
i i • Intermediate goods pricing equation (43) pj = C(1 + Σ) = piG (∀j = 1, . . . , n)
A1 −T
• Demand for intermediate goods (44) x =
1
npIG −P n σ
• Equilibrium unemployment rate (45) u =
s
s+θq(θ)
• Equilibrium stocks/flows equations
– V = θu
– N2 =
q(θ)
V
s
– N1n = N1n−1 = · · · = N11 =
– N0 =
q(θ)
V
b
b
p(φ)
• Equilibrium number of intermediate good producers (46) x =
• Equilibrium intermediate goods market tightness (47) ζ =
a σ
C 1−σ
q(θ)
s
θ s+θq(θ)
b0
r = 0 and b(ζ) = b0 ζ −b
With the short-hand notations:
−(n−1) κ
b
+
• K = r+b
φp(φ)
c
p(φ)
+
χ
b
Pn
i=1
b
r+b
i r+s+θq(θ)
• αθ = α r+s+αθq(θ)
• A1 =
r+s
1−αθ
αθ K −
γ
q(θ)
−
b
r+b
−n
r+q
q
χ
b
Pn
i=1
40
b
r+b
i
+
1
c
1−β p(φ)
1
1−b
under the assumptions that
Master EPP — SciencesPo 2012
5.2
Master’s dissertation — Margaux Luflade
Appendix to section 3
Equilibrium equations under the assumption r = 0.
1
• Wage equation (31) w = (1 − αθ )T + αθ (n σ P − np)x − sK
n
P
gamma
χ
• Repayment equation (33) ρ = β P y − w − n
+ n b(ζ)
i=1 xi pi + (1 − β)s
q(θ)
• Equilibrium credit market tightness (34) φ =
1−β κ
β c
• (EE)–(BB) equations (36), (38)
P
h
i
P y− n
c
i=1 xi pi −T
– p(φ)
= (1 − β) (1 − αθ )
+ αθ K − γq − n χb
s
P
h
P y− n
κ
i=1 xi pi −T
– φp(φ)
= β (1 − αθ )
+ αθ K − γq − n χ ]
s
• Intermediate goods pricing equation (43) pj = C(1 + Σ) = piG (∀j = 1, . . . , n)
• Equilibrium intermediate goods market tightness (47) ζ =
q(θ)
s
θ s+θq(θ)
b0
1
1−b
under the assumption b(ζ) =
b0 ζ −b
With the short-hand notations:
• K=
κ
φp(φ)
+
c
p(φ)
+ n χb
s+θq(θ)
s
• αθ = α s+αθq(θ) = 1 − (1 − α) s+αθq(θ)
• A1 =
s
1−αθ
h
αθ K −
γ
q(θ)
− n χb −
1
c
1−β p(φ)
i
5.2.1 Effects of the level of frictions on the labor market equilibrium
Effects of intermediate goods market frictions Specifically, let us consider a marginal change in
b0 . For instance, an increase in b0 , keeping the intermediate goods market tightness constant, induces an increase in the
instantaneous probability b for final good firms to find a supplier on the intermediate goods market, that is, a decrease
in the intermediate goods market frictions.
Starting with the unemployment rate, the effect of a marginal change in b0 can be deduced from equation (45)
du
(1 − q )q(θ) dθ
= −s
db0
(s + θq(θ))2 db0
dθ
This requires to elicit db
, which translates the interconnection of markets: the level of frictions on the intermediate
0
goods market contributes to determining the equilibrium tightness of the labor market. This can be shown from
sθq(θ)
equations (47) or (35). In particular, rewriting (47) b0 = 1−b
implies
ζ
(s+θq(θ))
db0
(1 − q )q(θ)
= 1−
(s + (1 − s)θq(θ))
b (s + θq(θ))2
dθ
ζ
θq 0 (θ)
q(θ)
∈ (−1, 0) is the elasticity of the job-filling rate with respect to the labor market tightness. The
−1
dθ
0
right-hand side of the last equation being positive, db
= db
> 0: as frictions on the labor market decrease
dθ
where q =
0
(i.e. as b0 increases), the tightness of the labor market increases. Intuitively, a decrease in intermediate goods market
frictions reduces the search costs for capital inputs and thus increases the profit of final good firms. By free entry, since
they expect higher profits, more entrepreneurs enter the credit market and, as the tightness on this market given by
(34) remains constant (their increased entries being compensating by increased entries of bankers who also expect larger
profits as they will finance lower search costs and share an increased profit), more entrepreneurs arrive on the labor
market, where the tightness then increases.
du
The expression of db
then shows that as a direct consequence of the increase in the labor market tightness, the un0
employment rate decreases: due to the relatively larger number of vacancies, the job-finding rate increases and the u
decreases.
The impact of a marginal change in b0 on the equilibrium wage cannot be deduced as straightforwardly as it was
in the case of the unemployment rate. Indeed, from the wage equation (31), we have:
1
1
dαθ dx
dK
dw
=
P n σ x − nxp − sK − T + αθ (P n σ − np)
−s
db0
db0
db0
db0
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Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
Before looking more precisely at the differential terms entering this equation, let us give an intuition of it. There are
three chanels through which b0 impacts on the wage. One seems intuitively clear: a change in the level of frictions
on the intermediate goods market directly impacts on the total cost of search frictions K the firm has to bear before
entering the labor market; as explained when we derived the wage equation (31), this cost K is shared between the
firm/bank and the worker through the wage. The two others chanels are more indirect. First, due to the interconnection
of markets, as represented by equation (47), a change in the level of frictions on the intermediate goods market induces
a change in the labor market tightness, which in turn determines the share of worker in production αθ and thus impacts
on the wage.
Then, the wage depends on revenues from sales, thus on the quantity x of intermediate goods used in production. As
this quantity x depends on the profits the final good firm expects from production, it also varies with θ, αθ and K and
thus induces an additional (and more complex) chanel of variation for w when b0 varies.
−b
ζ
dK
= −nχ b(ζ)
More formally, let us first consider db
2 < 0: as frictions on the intermediate goods market decrease (i.e.
0
as b0 increases), the financial burden of frictions K, which is partly supported by the worker, decreases. This pushes
the wage up.
θ
indicates how the share of the worker in production varies when the level of frictions on the intermediate
Then, dα
db0
goods market varies. It is given by:
dαθ
s
dθ
dθ
= α(1 − α)(1 − q )
= Aα,1
db0
(s + αθq(θ))2 db0
db0
with
Aα,1 =
dαθ
s
= α(1 − α)(1 − q )
>0
db0
(s + αθq(θ))2
dθ
From the value of db
computed earlier, the share of workers in profits increases as the frictions on the intermediate
0
goods market decrease (i.e. b0 increases). As shown here, this is only due to the increase in the labor market tightness
induced by the increase in b0 : the competition for workers among final good firms being increased in the tighter labor
dθ
market, they are ready to set wages more profitable to workers. All the terms multiplying db
in the right-hand side of
0
θ
<
the equation lying between 0 and 1 implies dα
db0
weaken consequence of the effect of b0 on θ.
Finally, let us look at
b0 .
dx
s
=
db0
1 − αθ
=
s
1 − αθ
dx
,
db
and confirms the intuition that the effect of b0 on αθ is only a
which gives the variation in the demand of any intermediate good induced by a variation in
dαθ
db0
dαθ
db0
dθ
db0
K−
γ
χ
1
c
−n −
q(θ)
b
1 − β p(φ)
β
γ
c
κ
+
−
φp(φ)
1 − β p(φ)
q(θ)
+ αθ
+
dK
γ q 0 (θ) dθ
χ ζ −b
+
+n
db0
q(θ) q(θ) db0
b(ζ) b(ζ)
χ ζ −b
γ q 0 (θ)
+ (1 − αθ )n
q(θ) q(θ)
b(ζ) b(ζ)
dθ
db0
β
γ
κ
c
Though, the sign φp(φ)
+ 1−β
− q(θ)
is ambiguous and depends on the relative search costs and levels of frictions on
p(φ)
the credit and labor markets. A case where further conclusions could be drawn is if this ambiguous term were negative.
This would correspond to a situation the (costs of) labor market frictions are larger than the (costs of) credit market
dx
frictions. Then db
would unambiguousy be negative: a decrease in the intermediate goods market frictions (thus an
0
increase in b0 would induce an increase in the quantity of intermediate goods consumed
Even in this case, the global variation of w remains ambiguous though. On the one hand, the wage is pushed
up by the larger share of the worker in production and the lower costs of frictions which are partly supported by
the worker. On the other hand, the decrease in the quantity of intermeditae goods consumed (in the case where
β
γ
κ
c
+ 1−β
− q(θ)
< 0) tends to decrease the quantity produced and thus the revenues from sales and the profits,
φp(φ)
p(φ)
which in turn pushes the wage down.
Effects of labor market frictions. Now consider the consequences of a change in the level of frictions
on the labor market, namely an increase in q0 which, for each level of θ, induces a higher job-filling rate q(θ) and can
thus be nterpreted as a decrease in the labor market frictions.
Again, let us start by analysing the change in the unemployment rate.
du
sθ1−q
=−
dq0
s + θq(θ)
This time, the change in u is directly induced by the change in q0 (as q0 directly enters the unemployment rate equation
through q(θ): a decrease in the labor market frictions (i.e. an increase in q0 ) induces a decrease in the unemployment
rate because the job-finding rate θq(θ) = q0 θ1−q increases with q0 .
Turning to the wage,
1
1
dw
dαθ dx
dK
=
P n σ x − nxp − sK − T + αθ (P n σ − np)
−s
dq0
dq0
dq0
dq0
42
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
We recognize the three chanels of propagation identified in the previous case. However, it is αθ which now directly
sθ −q
θ
= α(1 − α) (s+αθq(θ))
translates the change in the level of frictions on the labor markets: dα
2 > 0: the decrease in
dq0
the labor market frictions (i.e. the increase in q0 ) causes an increase in the worker’s share in profits which pushes the
wage up. On the contrary, the transmission of a change in the frictions through K is not direct anymore. Indeed, as
explained earlier, K accounts for the cost of frictions on the credit and the intermediate goods market and thus does
not directly reflects labor market frictions. Nevertheless, due to the interdependence of markets (captured by ddζ ) a
0
decrease in the labor market frictions (i.e. an increase in q0 ) increases the intermediate goods market tightness. This
is due to the increase in the number of entrepreneurs trying to lauch their project as a decrease in the labor market
frictions implies lower search costs on the labor market and thus possibly higher profits. In turn, the rate b(ζ) at which
final goods firms find their suppliers on the intermediate goods market decreases and this increases the cost of frictions
b0 (ζ) dζ
dζ
1
θ 1−q s
dK
= −nχ b(ζ)2 dq
with b0 (ζ) < 0 and dq
= 1−
ζ −1 b (s+θq(θ))
K. This pushes the wage down. Formally, dq
2 > 0.
0
0
0
b
0
Given the opposite effects of the variations in αθ and K on w, we already know the change in the wage will be a priori
unknown. The variation of x also adds some ambiguity as its sign is again unknown. This is due to the presence of
same term as in case of intermediate goods market frictions. Indeed,
dx
s
=
dq0
1 − αθ
=
s
1 − αθ
dαθ
dq0
dαθ
dq0
K−
γ
χ
1
c
−n −
q(θ)
b
1 − β p(φ)
κ
β
c
γ
+
−
φp(φ)
1 − β p(φ)
q(θ)
+
+ αθ
dK
γ θ−q
b0 (ζ) dζ
+
+ nχ
dq0
q(θ) q(θ)
b(ζ)2 dq0
γ θ−q
b0 (ζ) dζ
+ (1 − αθ )nχ
q(θ) q(θ)
b(ζ)2 dq0
Effects of credit market frictions. Finally, let us consider an increase in 0 which, given that φ is fixed
as soon as the credit market parameters β, c and κ are left unchanged, increases the probability for entrepreneurs to
find a financer and thus can thought of a representing a decrease in the credit market frictions.
The change in the unemployment rate is given by:
(1 − q )q(θ) dθ
du
= −s
dp0
(s + θq(θ))2 dp0
As in the case of intermediate goods market frictions, the effect of an increase in p0 propagates to u through θ. Deriving
du
du
du
is not as easy as db
and dq
were in the previous cases though. It is done by taking derivatives with respect to
dp0
0
0
p0 in (EE). This gives:
−
cφ−p
= (1−β)
p(φ)2
(
−
dαθ
dp0
1
P n σ x − npx − T − sK
s
!
+
1
dx
1 − αθ
q 0 (θ) θ.
b0 (ζ) dζ
(P n σ − np)
+ αθ K + γ
+ nχ
2
s
dp0
q(θ) dp0
b(ζ)2 p0
)
This in turns requires writing:
dαθ
θ
= Aα,1 .
dp0
dp0
dK
φ−p
=−
dp0
p(φ)2
where Aα,1 > 0 as been defined previously
κ
+c
φ
− nχ
b0 (ζ) dζ
b(ζ)2 p0
dθ
dζ
1
s(1 − q )q(θ)
dθ
1
s(1 − q )q(θ)
=
ζ −1
(s+(1−s)θq(θ))
= Aζ
with Aζ =
ζ −1
(s+(1−s)θq(θ)) > 0
dp0
1 − b
bo(s + θq(θ))2
dp0
dp0
1 − b
bo(s + θq(θ))2
s
dαθ
γ
χ
1
c
s
q 0 (θ) dθ
b0 (ζ) dζ
1 cφ−p
dx
dK
=
K
−
−
n
−
+
α
+
γ
+
nχ
+
θ
dp0
(1 − αθ )2 dp0
q(θ)
b(ζ)
1 − β p(φ)
1 − αθ
dp0
q(θ)2 dp0
b(ζ)2 p0
1 − β p(φ)2
So that the differentiated (EE) equation rewrites:
−
1 cφ−p
dθ
=
1 − β p(φ)2
dp0
1
−(1 + P n σ − np)
Aα,1 K −
φ−p
p(φ)2
γ
χ
1
c
b0 (ζ)
q 0 (θ)
−n
−
+ (1 − αθ )nχ
Aζ + γ
2
2
q(θ)
b(ζ)
1 − β p(φ)
b(ζ)
q(θ)
1
− αθ
1−β
c − αθ
κ
φ
so that
h
i −p
1
1
φ
κ
c
(1 + P n σ − np)αθ φ
+ c + (P n σ − np) 1−β
dθ
p(φ)2
h
i
=
b0 (ζ)
q 0 (θ)
γ
χ
1
c
dp0
Aα,1 K − q(θ)
− n b(ζ)
− 1−β
+ (1 − αθ )nχ b(ζ)2 Aζ + γ q(θ)2
p(φ)
The numerator is positive while the sign on denominator is not known for sure. In the case (previously analyzed in the
previous paragraphs) that the bracketed expression is negative, then the denominator is negative and θ is a decreasing
43
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
function on p0 : as frictions on the credit market decrease (i.e. p0 increases), the labor market tightness decreases
because less entrepreneurs enter the credit market and thus happen to arrive on the labor market afterwards.
This would in turn increase the unemployment rate.
Turning to the wage,
1
1
dw
dαθ dx
dK
=
P n σ x − nxp − sK − T + αθ (P n σ − np)
−s
dp0
dp0
dp0
dp0
Given the previous equations, no clear conclusion can be drawn about the change in the wage. Nevertheless, the
following analysis can be made. In the case where θ decreases as p0 increases, the wage is on the one hand pushed down
by a decrease in the share of workers in profits (due to the fact that as the labor market becomes more tight to the
workers they are ready to accept a lower share in profits): and on the other hand pushed down by the decrease in the
cost K of frictions. Additional ambiguity is brought by the unknown direction of variation in x.
5.2.2 Effects of regulation and deregulation
Effects of a decrease in the worker’s bargaining power.
The variation in the wage is given
by:
(1 − q )q(θ) dθ
du
= −s
dα
(s + θq(θ))2 dα
Again, u varies in the opposite direction as θ, whose variation is in turn determined by taking derivatives with respect
to α in (EE).
0=
dαθ
dα
1
"
−
P n σ x − pnx − sK − T
s
As in the previous paragraph, eliciting
dθ
dα
#
+ αθ
dK
q 0 (θ) dθ
b0 (ζ) dζ
+γ
+ nχ
2
dα
q(θ) dα
b(ζ)2 dα
requires determining the following derivates:
1 − αθq(θ)
s
dθ
dθ
dαθ
= αθ
+ α(1 − α)(1 − q )
= Aα,2 + Aα,1
dα
α
(s + αθq(θ))2 dα
dα
1−αθq(θ)
where Aα,1 > 0 has been previously defined and Aα,2 := αθ
> 0. This reveals the to chanels through which
α
αθ is impacted by a change in α: Aα,1 > 0 accounts for the direct effect on αθ of a change in α while Aα,2 > 0 weights
the change in αθ due to the change in the labor market tightness, in turn induced by the original variation in α.
dK
b0 (ζ) dζ
= −nχ
dα
b(ζ)2 dα
dζ
dα
dθ
= Aζ dα
with Aζ > 0 as previously defined. Note that then the last three terms of the differentiated (EE)
b0 (ζ)
q 0 (θ)
dθ
equation rewrite (1 − αθ )nχ b(ζ)2 Aζ + γ q(θ)2 dα
, where the bracketed term is negative as b0 (ζ) < 0 and q 0 (θ) < 0.
!
dαθ
s
γ
χ
1
c
s
dK
q 0 (θ) dθ
b0 (ζ)
dx
=
K
−
−
n
−
+
α
+
γ
+
nχ
θ
dα
dα (1 − αθ )2
q(θ)
b(ζ)
1 − β p(φ)
1 − αθ
dα
q(θ)2 dα
b(ζ)2 dζ
where
dα
=
dαθ
s
γ
χ
1
c
s
b0 (ζ)
q 0 (θ) dθ
Aα,2 + Aα,1
K−
−n
−
+
(1 − αθ )nχ
Aζ + γ
2
2
2
dα
(1 − αθ )
q(θ)
b(ζ)
1 − β p(φ)
1 − αθ
b(ζ)
q(θ)
dα
So finally, the differentiated (EE) equation rewrites:
0=
Aα,2 + Aα,1
where Z = − P n
1
σ
dαθ
dα
x−npx−sK−T
s
1
b0 (ζ)
q 0 (θ) dθ
Z + 1 + P n σ − np
A
+
γ
(1 − αθ )nχ
ζ
b(ζ)2
q(θ)2 dα
h
1
+ P n σ − np K −
γ
q(θ)
− n χb −
1
c
1−β p(φ)
i
This yields:
dθ
Aα,1 Z
=−
1
b0 (ζ)
q 0 (θ)
dα
Aα,2 Z + 1 + P n σ − np (1 − αθ )nχ b(ζ)2 Aζ + γ q(θ)2
1
The sign of Z is ambiguous for the same reason met in the previous subsection. On the one hand, P n σ x−nxp−sK −T >
1
0 and P n σ − np > 0 are required conditions for positive production by final good firms. On the other hand, the sign
γ
β
γ
1
c
κ
c
of the last term K − q(θ)
− n χb − 1−β
= φp(φ)
+ 1−β
− q(θ)
is ambiguous and depends on the relative search
p(φ)
p(φ)
44
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
costs and levels of frictions on the credit and labor markets. As a consequence, the sign of
Again, we can infer consequences on the case in which
κ
φp(φ)
+
β
c
1−β p(φ)
−
γ
q(θ)
dθ
dα
is a priori unknown.
is negative. Then Z would be negative
dθ
itself and dα
negative: a decrease in the worker’s bargaining power would then induce an increase in the labor market
tightness.
The increase in the labor market tightness θ would in turn induce a decrease in the unemployment rate, as seen from
du
.
the expression of dα
However, the sign ambiguity arises again when turning to the wage. First, due to the change in αθ : as pointed
dθ
< 0 implies the indirect effect is negative. In other words,
earlier the direct effect of a change in α is positive, but dα
as a consequence of a decrease in their bargaining power, the workers’ share in profit would on the one hand decrease
(as a direct consequence of their lower bargaining power) and on the other hand be pushed up due to the tighter labor
market, that is, to the increased competition among firms for workers. The second source of ambiguity for the change
dx
θ
in w can be found in dα
whose sign in turn depends on those of dα
. This can be seen from the equation:
dα
1
1
dαθ dx
dK
dw
=
P n σ x − nxp − sK − T + αθ (P n σ − np)
−s
dα
dα
dα
dα
θ
Note that the expression of dα
allows separating the direct effects from the general equilibrium effects on the wage of
dα
a change in the workers’ bargaining power:
1
1
dw
= Aα,1 P n σ x − nxp − sK − T + Aα,2 P n σ x − nxp − sK − T + αθ
dα
1
dx
dK
(P n σ − np)
−s
dα
dα
Effects of a change in the elasticity of substitution between intermediate goods.
Let us start by writing the equations showing the variations in u and w:
(1 − q )q(θ) dθ
du
= −s
dα
(s + θq(θ))2 dσ
(60)
1
dw
dαθ =
P n σ x − nxp − sK − T + αθ
dσ
dσ
1
1
(P n σ − np)
dx
d(P n σ − np)
dK
+
x−s
dα
dσ
dα
!
(61)
The variation in unemployment follows the same logic as in the previous paragraphs. The variation in w is more complex
though because a variation in σ induces an additional change in profits as compared to the previous cases. Let us start
by noting that an increase in σ ∈ (0, 1) induces an increase in the elasticity of substitution between intermediate goods
and thus decreases the monopoly power of intermediate goods producers. As a consequence their mark up decreases
and so does the price of the intermediate goods:
C
dp
=− 2
dσ
σ
As compared to the other change in parameters we have considered so far, it is the first time the price of intermediate
goods varies; this is due to the pricing rule of intermediate goods producers: the price they set depends only on their
marginal cost of production and the elasticity of substitution between goods. This is the first chanel through which the
1
final goods firms’ profits (namely P n σ x − pnx) are affected by a change in σ: keeping all other things constant, the
1
cost of purchasing the quantity x of inputs is lower. The second chanel comes from the first part P n σ x of the profit
expression:
1
dn σ
ln(n) 1
= − 2 nσ
dσ
σ
An increase in σ decreases the total amount of production y reached with a given bundle x on intermediate goods,
which decreases the sales and thus the revenues. The global change in profits generated by the consumption of a given
bundle x of capital inputs is thus a priori unclear:
h
i
1
d P n σ x − pnx
dσ
=−
1
1 ln(n)n σ − C
σ2
To study the variations in θ and x induced by a change in σ, one can proceed as in the previous paragraphs taking
derivatives with respect to σ in both sides of equation (EE):
1
dαθ P n σ x − nxp − T − sK
0=−
+(1−αθ )
dσ
s
1
dx
(P n σ
dσ
1
dp
n σ
− np) + x −P ln
n − n dσ
σ2
s
45
+αθ
dK
q 0 (θ) dθ
b0 (ζ) dζ
+γ
+nχ
dσ
q(θ)2 dσ
b(ζ)2 dσ
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
and sbstituting in this expression the following differential terms:
dαθ
dθ
= Aα,2
dσ
dσ
dK
b0 (ζ) dζ
= −nχ
dσ
b(ζ)2 dσ
dx
=
dσ
dA1
dσ
where
dθ
dζ
= Aζ
dσ
dσ
1
1
dp
n σ
np − P n σ − (A1 − T ) n dσ
+ P ln
n
σ2
1 2
np − P n σ
where
dK
dA1
s
dαθ
γ
χ
1
c
s
q 0 (θ) dθ
b0 (ζ) dζ
=
K−
−n −
+
αθ
+γ
+ nχ
2
2
2
dσ
(1 − αθ ) dσ
q(θ)
b
1 − β p(φ)
1 − αθ
dσ
q(θ) dσ
b(ζ) dσ
5.3
Appendix to the extension with a frictional final goods market
Further details on solving Bellman equations
Solution proceeds just as in section 2 by iteratively
plugging each equation into the previous one. Firms’ side:
E4 = αλ
P −w−
E3 =
i=1
xi p i − ρ
r+s
r+s+λ
r+s+λ+τ
where αλ :=
Pn
; and from which we deduce:
λ
E4 ,
r+s+λ
E2 =
q
E3
r+q
and
E11 =
b
r+b
n
E2
Banks’ side:
B4 =
αλ ρ + (αλ − 1)(w +
r+s
Pn
i=1
xi p i )
which allows obtaining the values:
B3 =
P
w+ n
λ
i=1 xi pi
B4 −
,
r+s+λ
r+s
B2 =
Workers-consumers’ side:
1
1+Φ−P
θq(θ)
+w+s
,
W1 =
1 − ξλ(ξ) r + s + ξλ(ξ)
r + θq(θ)
Wage bargaining
q
r+q
B3 +
W0 = W1 −
γ
q
and B11 = −
1+Φ−P
r + s + ξλ(ξ)
i n
n χX
b
b
+
B2
b i=1 r + b
r+b
and U =
The first order condition to the bargaining problem is: (1−α)
0.
θq(θ)
r + θq(θ)
T
+ W0
θq(θ)
∂ [B3 +E3 −B2 −E2 ]
∂w [W0 −U ]
+α wB +E
W0 −U
3
3 −B2 −E2
On the one hand, (r + s)[W0 − U ] = w + ξλ(ξ)[W1 − W0 ] − rU , where rU does not depend on the still unknown future
1+Φ−P
1
wage and [W1 − W0 ] = r+s+ξλ(ξ)
does not depend on w neither. Thus ∂w [W0 − U ] = r+s
.
On the other hand, Bellman equations allow writing (r + s + λ)[B3 + E3 ] = −w −
P
B4 + E4 = r+s+λ+τ
+ B3 + E3 , this yields to
(r + s)[B3 + E3 ] = −w −
n
X
i=1
xi p i +
Pn
i=1
xi pi + λ[B4 + E4 ]. With
λP
r+s+λ+τ
where P does not depend on w as can be seen from equation (58).
Moreover,
B2 +E2 =
b
r+b
−n "
B11 +
i # −n
−n "
i #
n n χX
b
b
b
c
κ
χX
b
+
E11 =
+
+
=K
b i=1 r + b
r+b
r+b
p(φ)
φp(φ)
b i=1 r + b
46
=
Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
which does not depend on w.
1
Therefore, ∂w [B3 + E3 − B2 − E2 ] = − r+s
.
FOC then rewrites: α[B3 +E3 −B2 −E2 ] = (1−α)[W0 −U ], which can be further transformed using the expressions
of [W0 − U ] and [B3 + E3 − B2 − E2 ] previously stated: as given in the main text
w=α
X
λP
1+Φ−P
−
xi pi − (r + s)K + (1 − α) rU − ξλ(ξ)
r+s+λ+τ
r + s + ξλ(ξ)
α
α
Then, noting that rU = T +θq(θ)[W0 −U ] = T +θq(θ) 1−α
[B3 +E3 −B2 −E2 ] = T +θq(θ) 1−α
we get the final expression given in the main text:
w = α0θ
h
λP
r+s+λ+tau
−w−
P
xi pi − (r + s)K
X
1+Φ−P
λP
−
xi pi − (r + s)K + (1 − α0θ ) T − ξλ(ξ)
r+s+λ+τ
r + s + ξλ(ξ)
1+θq(θ)
1+θq(θ)
where α0θ = α 1+αθq(θ) , which implies 1 − α0θ = (1 − α) 1+αθq(θ) so that α0θ ∈ (0, 1).
Repayment bargaining
Given the derivatives provided in the main text and the expressions for E11 and B11
given by equations (55) (56), the first order condition to the bargaining problem rewrites:
(1 − β)
=β
b
r+b
b
r+b
n
n
q
r+q
αλ ρ − (1 − αλ )(w +
λ
r+s+λ
r+s
Pn
i=1
xi pi )
−
P
n
n w+ n
γ
χX
b
i=1 xi pi
−
−
q
r+s+λ
b i=1 r + b
P
P −w− n
q
λ
i=1 xi pi − ρ
αλ
r+q r+s+λ
r+s
which simplifies to the repayment equation given in the main text:
αλ
ρ
= βαλ
r+s
w+
Pn
i=1
xi p i
λ
P
P
P −w− n
w+ n
r+s+λγ
i=1 xi pi
i=1 xi pi
+ (1 − β) (1 − αλ )
+
+
r+s
r+s
λ
q
+
b
r+b
−n
i
n r+q r+s+λχ X
b
q
λ
b i=1 r + b
)
(62)
References
Blanchard O.J. and F. Giavazzi (2003), Macroeconomic Effects of Regulation
and Deregulation on the Goods and Labor Markets, Quarterly Journal of Economics
118(3): 879–908.
Brakman S. and B.J. Heijdra (2004), The Monopolistic Competition Revolution in Retrospect, Cambridge: Cambridge University Press.
Chamberlin, E.H (1956), The Theory of Monopolistic Competition: a Reorientation of the Theory of Value, 7th edition, Cambridge, MA: Harvard University
Press.
Dixit A.K. and J.E. Stiglitz (1977), Monopolistic Competition and Optimum
Product Diversity, American Economic Review 67(3): 297–308.
Green H.A.J. (1964), Aggregation in Economic Analysis — An Introductory
Survey, Princeton, NJ: Princeton University Press.
47
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Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
OECD (2010), “Detailed National Accounts: Value added and its components
by activity”, OECD National Accounts Statistics (database), doi: 10.1787/data-00006-en
(Accessed on May, 19th 2012)
Pissarides C. (1990), Equilibrium Employment Theory, Oxford and Cambridge:
Basil Blacwkell.
Wasmer E. (2009), A Steady-State Model of a Non-Walrasian Economy with
Three Imperfect Markets, IZA Working Papers 5758.
Wasmer E. and N. Petrosky-Nadeau (2011), Macroeconomic Dynamic in a
Model of Goods, Labor and Credit Market Frictions, IZA Working Paper 5763.
Wasmer E. and P. Weil (2004), The Macroeconomics of Labor and Credit Market Imperfections, American Economic Review 94(4): 944–963.
Ziesemer T. (2005), Monopolistic Competition and Search Unemployment,
Metroeconomica 56(3): 334–359.
Contents
1 Introduction and literature review
1.1 Macroeconomics and steady-state models of multiple frictional markets . . . .
1.2 Intermediate good firms and the monopolistically competitive intermediate
goods market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
10
2 The basic model
2.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Credit and labor markets . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Introducing search on the intermediate goods market . . . . . . . . . .
2.2 Bellman equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Solution of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Wage determination. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Repayment determination. . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Equilibrium tightness of the credit and labor markets. . . . . . . . . .
2.3.4 Equilibrium on the intermediate goods market. . . . . . . . . . . . . .
2.3.5 Equilibrium numbers of unemployed workers and vacancies . . . . . .
2.3.6 Equilibrium tightness of the intermediate goods market and equilibrium
number of intermediate goods producers . . . . . . . . . . . . . . . . .
14
14
14
14
16
20
20
22
23
24
26
3 Selected comparative statics: frictions and regulation analysis
3.1 Effects of the level of frictions on the labor market equilibrium . . . . . . . .
3.2 Effects of regulation and deregulation . . . . . . . . . . . . . . . . . . . . . .
28
28
31
48
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Master EPP — SciencesPo 2012
Master’s dissertation — Margaux Luflade
4 Extension: frictional final goods market
4.1 Setting: re-thinking the production function . . . . . . . . . . . . . . . . .
4.2 Bellman equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Solution of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Repayment, wage and price determination. . . . . . . . . . . . . .
4.3.2 Equilibrium tightness of the credit, labor and final goods markets .
4.3.3 Equilibrium on the intermediate goods market . . . . . . . . . . .
5 Appendix
5.1 Appendix to the basic model: summary of the equilibrium equations .
5.2 Appendix to section 3 . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Effects of the level of frictions on the labor market equilibrium
5.2.2 Effects of regulation and deregulation . . . . . . . . . . . . . .
5.3 Appendix to the extension with a frictional final goods market . . . .
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