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Models for Dynamic Macroeconomics$

Fabio-Cesare Bagliano and Giuseppe Bertola

Print publication date: 2004

Print ISBN-13: 9780199266821

Published to Oxford Scholarship Online: January 2005

DOI: 10.1093/0199266824.001.0001

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Coordination and Externalities in Macroeconomics

Coordination and Externalities in Macroeconomics

Chapter:
(p.170) 5 Coordination and Externalities in Macroeconomics
Source:
Models for Dynamic Macroeconomics
Author(s):

Fabio-Cesare Bagliano (Contributor Webpage)

Giuseppe Bertola

Publisher:
Oxford University Press
DOI:10.1093/0199266824.003.0005

Abstract and Keywords

Focusses on the macroeconomic effects of strategic interactions among optimizing agents in several contexts. Trading externalities are introduced in a non‐monetary economy and the possibility of multiple equilibria is discussed. A search model of money is then presented, which formalizes the use of money as a medium of exchange. Finally, a model of the labour market characterized by a process of search on the part of workers and firms is studied, and the dynamics of frictional unemployment, vacancies, and wages are discussed.

Keywords:   frictional unemployment, money as a medium of exchange, multiple equilibria, search models, strategic interactions

As we saw in Chapter 4, externalities play an important role in endogenous growth theory. Many recent contributions have explored the relevance of similar phenomena in other macroeconomic contexts. In general, aggregate equilibria based on microeconomic interactions may differ from those mediated by the equilibrium of a perfectly competitive market in which agents take prices as given. If every agent correctly solves her own individual problem, taking into consideration the actions of all other agents rather than the equilibrium price, then nothing guarantees that the resulting equilibrium is efficient at the aggregate level. Uncoordinated “strategic” interactions may thus play a crucial role in many modern macroeconomic models with micro foundations.

In this chapter we begin by considering the relationship between the externalities that each agent imposes on other individuals in the same market and the potential multiplicity of equilibria, first in an abstract trade setting (Section 5.1) and then in a simple monetary economy (Section 5.2). (The appendix to this chapter describes a general framework for the analysis of the relationship between externalities, strategic interactions, and the properties of multiplicity and efficiency of the aggregate equilibria.) Then we study a labor market characterized by a (costly) process of search on the part of firms and workers. This setting extends the analysis of the dynamic aspects of labor markets of Chapter 3, focusing on the flows into and out of unemployment. Attention to labor market flows is motivated by their empirical relevance: even in the absence of changes in the unemployment rate, job creation and job destruction occur continuously, and the reallocation of workers often involves periods of frictional unemployment. The stylized “search and matching” modeling framework introduced below is realistic enough to offer empirically sensible insights, reviewed briefly in the “Further Readings” section at the end of the chapter. We formally analyze determination of the steady state equilibrium in Section 5.3 and the dynamic (p.171) adjustment process in Section 5.4. Finally, Section 5.5 characterizes the efficiency implications of externalities in labor market search activity.

5.1 Trading Externalities and Multiple Equilibria

This section analyzes a basic model where the nature of interactions among individuals creates a potential for multiple equilibria. These equilibria are characterized by different levels of “activity” (employment, production) in the economy. The model presented here is based on Diamond (1982a) and features a particular type of externality among agents operating in a given market: the larger the number of potential trading partners, the higher the probability that an agent will make a profitable trade (trading externality). Markets with a high number of participants thus attract even more agents, which reinforces their characteristic as a “thick” market, while “thin” markets with a low number of participants remain locked in an inferior equilibrium.

5.1.1 Structure of the model

The economy is populated by a high number of identical and infinitely lived individuals, who engage in production, trade, and consumption activities.

Production opportunities are created stochastically according to a Poisson distribution, whose parameter a defines the instantaneous probability of the creation of a production opportunity. At each date t 0, the probability that no production opportunity is created before date t is given by e a(tt 0) (and the probability that at least one production opportunity is created within this time interval is thus given by 1−ea(tt 0)). This probability depends only on the length of the time interval tt 0 and not on the specific date t 0 chosen. The probability that a given agent receives a production opportunity between t 0 and t is therefore independent of the distribution of production prior to t 0.39

All production opportunities yield the same quantity of output y, but they differ according to the associated cost of production. This cost is defined by a random variable c, with distribution function G(c) defined on cc<0, where c represents the minimum cost of production. Trade is essential in the model, because goods obtained from exploiting a production opportunity cannot be consumed directly by the producer. This assumption captures in a stylized way (p.172) the high degree of specialization of actual production processes, and it implies that agents need to engage in trade before they can consume. At each moment in time, there are thus two types of agent in the market:

  1. 1. There are agents who have exploited a production opportunity and wish to exchange its output for a consumption good: the fraction of agents in this state is denoted by e, which can be interpreted as a “rate of employment,” or equivalently as an index of the intensity of production effort.

  2. 2. There are agents who are still searching for a production opportunity: the corresponding fraction 1-e can be interpreted as the “unemployment rate.”

Like production opportunities, trade opportunities also occur stochastically, but their frequency depends on the share of “employed” agents: the probability intensity of arrivals per unit of time is not a constant, like the a parameter introduced above, but a function b(e), with b(0)=0 and b′(e)>0. The presence of a larger number of employed agents in the market increases the probability that each individual agent will find a trading opportunity. This property of the trading technology is crucial for the results of the model and its role will be highlighted below.

Consumption takes place immediately after agents exchange their goods. The instantaneous utility of an agent is linear in consumption (y) and in the cost of production (−c), and the objective of maximizing behavior is

V = E [ i = 1 ( e r t i c + e r ( t i + τ i ) y ) ] ,
where r is the subjective discount rate of future consumption, the sequence of times {t i} denotes dates when production takes place, and {τi} denotes the interval between such dates and those when consumption and trade take place. Since production and trade opportunities are random, both {t i} and {τi} are uncertain, and the agent maximizes the expected value of discounted utility flows.

To maximize V, the agent needs to adopt an optimal rule to decide whether or not to exploit a production opportunity. This decision is based on the cost that is associated with each production opportunity or, equivalently, on the effort that a producer needs to exert to exploit the production opportunity. The agent chooses a critical level for the cost c*, such that all opportunities with a cost level equal to cc* are exploited, while those with a cost level c>c* are refused.

To solve the model, we need to determine this critical value c* and the dynamic path of the level of activity or “employment” e.

(p.173) 5.1.2 Solution and characterization

To study the behavior of the economy outlined above, we first derive the equations that describe the dynamics of the level of activity (employment) e and the critical value of the costs c* (the only choice variable of the model).

The evolution of employment is determined by the difference between the flow into and out of employment. The first is equal to the fraction of the unemployed agents that receive and exploit a production opportunity: this fraction is equal to (1−e)aG(c*). The flow out of employment is equal to the fraction of employed agents who find a trading opportunity and who thus, after consumption, return to the pool of unemployed. This fraction is equal to eb(e). The assumption b′(e)>0 that was introduced above now has a clear interpretation in terms of the increasing returns to scale in the process of trade. Calculating the elasticity of the flow out of employment eb(e) with respect to the rate of employment e, we get

ε = 1 + e b ( e ) b ( e ) ,
which is larger than one if b′(e)>0 (implying increasing returns in the trading technology). In other words, a higher rate of activity increases the probability that an employed agent will meet a potential trading partner.

Given the expressions for the flows into and out of employment, we can write the following law of motion for the employment rate:

e ˙ = ( 1 e ) a G ( c * ) e b ( e ) .

In a steady state of the system the two flows exactly compensate each other, leaving e constant. The following relation between the steady-state value of employment and the critical cost level c* therefore holds:

( 1 e ) a G ( c * ) = e b ( e ) d e d c * | = ( 1 e ) a G ( c * ) b ( e ) + e b ( e ) + a G ( c * ) > 0.

A rise in c* increases the flow into employment, since it raises the share of production opportunities that agents find attractive, and thus determines a higher steady-state value for e, as depicted in the left-hand panel of Figure 5.1. For points that are not located on the locus of stationarity, the dynamics of employment are determined by the effect of e on ė: according to (5.1), a higher value for e reduces ė, as is also indicated by the direction of the arrows in the figure.

In order to determine the production cost below which it is optimal to exploit the production opportunity, agents compare the expected discounted value of utility in the two states: employment (the agent has produced the good and is

                   Coordination and Externalities in Macroeconomics

Fig 5.1. Stationarity loci for e and c *

(p.174) searching for a trading partner) and unemployment (the agent is looking for a production opportunity with sufficiently low cost). The value of the objective function in the two states is denoted by E and U, respectively. These values depend on the path of employment e and thus vary over time; however, if we limit attention to steady states for a moment, then E and U are constant over time (Ė=0 and U ˙ = 0 ). The relationships that tie the values of E and U can be derived by observing that the flow utility from employment (rE) needs to be equal to utility of consumption y, which occurs with probability b(e), plus the expected value of the ensuing change from employment to unemployment:
r E = b ( e ) y + b ( e ) ( U E ) .

There is a clear analogy with the pricing of financial assets (which yield periodic dividends and whose value may change over time), if we interpret the left-hand side of (5.3) as the flow return (opportunity cost) that a risk-neutral investor demands if she invests an amount E in a risk-free asset with return r. The right-hand side of the equation contains the two components of the flow return on the alternative activity “employment”: the expected dividend derived from consumption, and the expected change in the asset value resulting from the change from employment to unemployment. This interpretation justifies the term “asset equations” for expressions like (5.3) and (5.4).

Similarly, the flow utility from unemployment comprises the expected value from a change in the state (from unemployment to employment) which occurs with probability aG(c*) whenever the agent decides to produce; and the expected cost of production, equal to the rate of occurrence of a production (p.175) opportunity a times the average cost (with a negative sign) of the production opportunities that have a cost below c* and are thus realized. The corresponding asset equation is therefore given by

r U = a G ( c * ) ( E U ) a c ¯ c * c d G ( c ) = a c ¯ c * ( E U c ) d G ( c ) ,
where G ( c * ) c ¯ c * d G ( c ) .

Equations (5.3) and (5.4) can be derived more rigorously using the principle of dynamic programming which was introduced in Chapter 1. In the following we consider a discrete time interval Δt, from t=0 to t=t 1, and we keep e constant. Moreover, we assume that an agent who finds a production opportunity and returns to the pool of unemployed does not find a new production opportunity in the remaining part of the interval Δt. Given these assumptions, we can express the value of employment at the start of the interval as follows:

E = 0 t 1 b e b t e r t y d t + e r Δ t [ e b Δ t E + ( 1 e b Δ t ) U ] ,
where the dependence of b on e is suppressed to simplify notation. The first term on the right-hand side of (5.5) is the expected utility from consumption during the interval, which is discounted to t=0. (Remember that e bt defines the probability that no trading opportunity arrives before date t.) The second term defines the expected (discounted) utility that is obtained at the end of the interval at t=t 1. At this date, the agent may be either still “employed,” having not had a chance to exchange the produced good (which occurs with probability ebΔt), or “unemployed,” after having traded the good (which occurs with complementary probability 1−e bΔt).40 Solving the integral in (5.5) yields
E = b b + r ( 1 e ( r + b ) Δ t ) y + e r Δ t [ e b Δ t E + ( 1 e b Δ t ) U ] = b r + b y + e r Δ t ( 1 e b Δ t ) 1 e ( r + b ) Δ t U .

Taking the limit of (5.6) for Δt→0 and applying l’Hôpital's rule to the second term, so that

lim Δ t 0 r e r Δ t ( 1 e b Δ t ) + b e r Δ t e b Δ t ( r + b ) e ( r + b ) Δ t = b r + b
(p.176) we get the asset equation for E which was already formulated in (5.3):
E = b r + b y + b r + b U r E = b y + b ( U E ) .

Similar arguments can be used to derive the second asset equation in (5.4). The critical value c* is set in order to maximize E and U.

In the optimum, therefore, the following first-order conditions hold:

E c * = U c * = 0.

The derivative of the value of “unemployment” with respect to the threshold cost level c* can be obtained from (5.4) using Leibnitz's rule,41

d d b a b f ( z ) d z = f ( b ) .

In our case, f(z)=(EUz)(dG/dz). Differentiating (5.4) with respect to c* and equating the resulting expression to zero yields

r U c * = a ( E U c * ) G ( c * ) = 0 c * = E U .

In words, whoever is unemployed (searching for a production opportunity) is willing to bear a cost of production that is at most equal to the gain, in terms of expected utility, from exploiting a production opportunity to move from unemployment to “employment.” Now, subtracting (5.4) from (5.3), we get

r ( E U ) = b ( e ) y b ( e ) ( E U ) a G ( c * ) ( E U ) + a c ¯ c * c d G ( c ) .

Using (5.8) we can now derive the equation for the stationary value of c*, which expresses c* as a function of e. Writing

E U = c * = b ( e ) y + a c ¯ c * c d G ( c ) r + b ( e ) + a G ( c * ) ,
(p.177) rearranging to
b ( e ) y + a c ¯ c * c d G ( c ) = ( r + b ( e ) + a G ( c * ) ) c * ,
and differentiating, we find that the slope of the locus of stationarity (5.9) is
d c * d e | = b ( e ) ( y c * ) r + b ( e ) + a G ( c * ) .

The sign of this derivative is positive since y>c* (agents accept only those production possibilities with a cost below the value of output) and b′(e)>0. Notice also that if e=0 no trade ever takes place. (There are no agents with goods to offer.) In this case, agents are indifferent between employment and unemployment and there is no incentive to produce: c*=EU=0. Finally, if we assume that b″(e)<0, one can show that d 2 c*/de 2<0. Hence, the function that represents the locus of stationarity is strictly concave, and the locus of stationarity, which is drawn in the right-hand panel of Figure 5.1, starts in the origin and increases at a decreasing rate. The positive sign of dc*/de|ċ*=0 implies that there exists a strategic complementarity between the actions of individual agents. The concept of strategic complementarity is formally introduced in the appendix to this chapter. Intuitively, it implies that the actions of one agent increase the payoffs from action for all other agents; expressed in terms of the model studied here, the higher the fraction of employed agents, the more likely each individual agent will find a trading partner. This induces agents to increase the threshold for acceptance of production opportunities. At the aggregate level, therefore, the optimal individual response implies a more than proportional increase in the level of activity. To determine the dynamics of c*, we need to remember that the equilibrium relations (5.3) and (5.4) are obtained on the basis of the assumption that E and U are constant over time. In general, however, these values will depend on the path of employment e. In that case, we need to add the terms ė=ė∂E(·)/∂e and U ˙ = e ˙ U ( ċ ) / e to the right-hand sides of (5.3) and (5.4), respectively, yielding:

r E ( ċ ) = E ( ċ ) e e ˙ + b ( e ) ( y E ( ċ ) + U ( ċ ) )
r U ( ċ ) = U ( ċ ) e e ˙ + a c ¯ c * ( E ( ċ ) U ( ċ ) c ) d G ( c ) .

In terms of asset equations, Ė and U ˙ represent the “capital gains” that, together with the flow utility, give the “total returns” rE and rU. Now, subtracting (5.2) (p.178) from (5.11), and noting from (5.7) that

c ˙ * = E ˙ U ˙ = ( E ( ċ ) e U ( ċ ) e ) e ˙ ,
we can derive the expression for the dynamics of c*:
c ˙ * = r c * b ( e ) ( y c * ) + a c ¯ c * ( c * c ) d G ( c ) .

Moreover, if we assume that ċ*=0, we obtain exactly (5.9). Since

c ˙ * c * = r + b ( e ) + a G ( c * ) > 0 ,
the response of ċ* to c* is positive, as shown by the direction of the arrows in Figure 5.2. We are now in a position to analyze the possible equilibria of the economy, and we can make the interpretation of individual behavior in terms of the strategic complementarity more explicit.

First of all, given the shape of the two loci of stationarity, there may be multiple equilibria. The origin (c*=e=0) is always an equilibrium of the system. In this case the economy has zero activity (shut-down equilibrium). If there are more equilibria, then we may have the situation depicted in Figure 5.2. In this case there are two additional equilibria: E 1, in which the economy has a low level of activity, and E 2, with a high level of activity.

                   Coordination and Externalities in Macroeconomics

Fig 5.2. Equilibria of the economy

(p.179) Graphically, the direction of the arrows in Figure 5.2 implies that the system can settle in the equilibrium with a high level of activity only if it starts from the regions to the north-east or the south-west of E 2. As in the continuous-time models analyzed in Chapters 2 and 4, the dynamics are therefore characterized by a saddlepath. Also drawn in the figure is a saddlepath that leads to equilibrium in the origin; finally, there is an equilibrium with low (but non-zero) activity. For a formal analysis of the dynamics we linearize the system of dynamic equations (5.1) and (5.13) around a generic equilibrium ( ( e ¯ , c ¯ * ) . ).

In matrix notation, this linearized system can be expressed as follows:

( e ˙ c ˙ * ) = ( ( a G ( c ¯ * ) + b ( e ¯ ) + e ¯ b ( e ¯ ) ) ( 1 e ¯ ) ( a G ( c ¯ * ) b ( e ¯ ) ( y c ¯ * ) r + b ( e ¯ ) + a G ( c ¯ * ) ) ( e e ¯ c * c ¯ * ) ( α β γ δ ) ( e e ¯ c * c ¯ * ) , where α , γ < 0 ; β , δ > 0.

If in a given equilibrium the curve ė=0 is steeper than ċ*=0, then this equilibrium is a saddlepoint, as in the case of E 2. Formally, we need to verify the following condition:

det ( α β γ δ ) = α δ β γ < 0.

This can be rewritten as

α β > γ δ ,
where -α/β is the slope of the curve ė=0 and -γ/δ is the slope of the curve ˙*=0. In contrast, at E 1 the relationship between the steepness of the two curves is reversed and the determinant of the matrix is positive. Such an equilibrium is called a node. The trace of the matrix is α+δ=rb′(ē): whether its node is negative or positive depends on its sign. This in turn depends on the specific values of r and ē and on the properties of the function b(·). The existence of a strategic complementarity, arising from the trading externality implied by the assumption that b′(e)>0, has thus resulted in multiple equilibria.

A low level of employment induces agents to accept only few production opportunities (c* is low) and in equilibrium the economy is characterized by a low level of activity. If, on the contrary, employment is high, each agent will accept many production opportunities and this allows the economy to maintain an equilibrium with a high level of activity. Finally, it is important to note that agents' expectations play a crucial role in the selection of the equilibrium. Looking at point e 0 in Figure 5.2, it is clear that there exist values (p.180) of e for which the economy can either jump to the saddlepath that leads to the “inferior” equilibrium (the origin), or to the one that leads to the equilibrium with a high level of activity. Which of these two possibilities is actually realized depends on the beliefs of agents. If agents are “optimistic” (i.e. if they expect a high level of activity and thus a convergence to the equilibrium at E 2), then they choose a value of c* on the higher saddlepath, while if they are “pessimistic” (and anticipate convergence to the origin), they choose a point on the lower saddlepath.

5.2 A Search Model of Money

The stylized Diamond model of the previous section represents a situation where heterogeneous tastes and specialization in production force agents to trade in order to consume. Unlike Robinson Crusoe, the economic agents of the model cannot consume their own production: in the original article, Diamond (1982a) outlines how the economic decisions and interactions of his model could be applicable to a tropical island where a religious taboo prevents each of the natives from eating fruit he has picked. And, since trade occurs on a bilateral basis, rather than in a competitive auctioneered market, the economy's general equilibrium cannot be viewed as a representative-agent welfare maximization problem of the type that is sometimes discussed in terms of Robinson Crusoe's activities in undergraduate microeconomics textbooks.

The insights are qualitatively relevant in many realistic settings. In particular, whenever trade does not occur simultaneously in a frictionless centralized market, a potential role arises for a “medium of exchange”—an object that is accepted in a trade not to be directly consumed or used in production, but only to be exchanged in future trades. It would certainly be inconvenient for the authors of this book to carry copies of it into stores selling groceries they wish to consume, hoping that the owner might be interested in learning advanced macroeconomic techniques. In reality, of course, authors and publishers exchange books for money, and money for groceries. So, money's medium-of-exchange role facilitates exchanges of goods and, ultimately, consumption. The model presented in this section, a simplified version of that in Kiyotaki and Wright (1993), formalizes the use of money as a medium of exchange. As in the Diamond model of the previous section, strategic interaction among individuals is crucial in determining the equilibrium outcome. Moreover, different equilibria (characterized by different degrees of acceptability of money in the exchange process) may arise, depending on the particular traders’ beliefs: again, agents’ expectations are self-fulfilling.

(p.181) 5.2.1 The structure of the economy

Consider an economy populated by a large number of infinitely lived agents. There is also a large number of differentiated and costlessly storable consumption goods, called commodities, coming in indivisible units. Agents differ as to their preferences for commodities: each individual “likes” (and can consume) only a fraction 0<x>1 of the available commodities. The same exogenous parameter x denotes the fraction of agents that like any given commodity. Production occurs only jointly with consumption: when an agent consumes one unit of a commodity in period t, he immediately produces one unit of a different good, which becomes his endowment for the next period t+1. The utility obtained from consumption, net of any production cost, is U>0. As in Diamond's model of Section 5.1, we assume that commodities cannot be consumed directly by the producer: this motivates the need for agents to engage in a trading activity before being able to consume.

In the economy, besides commodities, there is also a certain amount of costlessly storable fiat money, coming in indivisible units as well as the commodities. Fiat money has two distinguishing features: it has no intrinsic value (it does not yield any utility in consumption and cannot be used as a production input), and it is inconvertible into commodities having intrinsic worth. Initially, an exogenously given fraction 0 < M < 1 of the agents are each endowed with one unit of money, whereas 1−M are each endowed with one unit of a commodity.

We can now describe how agents in the economy behave during any given period t, in which a fraction M of them are money holders and a fraction 1−M are commodity holders.

  • A money holder will try to exchange money for a consumable commodity. For this to happen, two conditions must jointly be fulfilled: (i) she must meet an agent holding a commodity she “likes” (since only a fraction x of all commodities can be consumed by each agent), and (ii) the commodity holder must be willing to accept money in exchange for the consumption good. Only when these two conditions are met does trade take place: the money holder exchanges her unit of money for a commodity that she consumes enjoying utility U; she then immediately produces one unit of a different commodity (that she “dislikes”), and will start the next period as a commodity holder. If, on the contrary, trade does not occur, she will carry money over to the next period.

  • A commodity holder will also try to exchange his endowment for a commodity he “likes.” For this to happen, he must meet another commodity holder and both must be willing to trade (i.e. each agent must “like” the commodity he would receive in the exchange). Exchanges of commodities (p.182) for commodities occur only if they are mutually agreeable, and therefore both goods are consumed after trade.42 It is also possible that a commodity holder meets a money holder who “likes” his particular commodity; if trade takes place, then the agent starts the next period as a money holder.

The artificial economy here described highlights the different degree of acceptability of commodities and fiat money. Each consumption good will always be accepted in exchange by some agents, whereas money will be accepted only if agents expect to trade it in the future in exchange for consumable goods.

A final assumption concerns the meeting technology generating the agents' trading opportunities. Agents meet pairwise and at random; in each period an agent meets another with a constant probability 0>β≤1.

5.2.2 Optimal strategies and equilibria

Each agent chooses a trading strategy in order to maximize the expected discounted utility from consumption. A trading strategy is a rule allowing the agent to decide whether to accept a commodity or money in exchange for what he is offering (either a commodity or money). The optimal trading strategy is obtained by solving the utility maximization problem, taking as given the strategies of other traders: this is the agent's optimal response to other traders’ strategies. When all optimal strategies are mutually consistent, a Nash equilibrium configuration arises. We focus attention on symmetric and stationary equilibria, that is, on situations where all agents follow the same time-invariant strategies. In equilibrium, agents exchange commodities for other commodities only when both traders can consume the good they receive, whereas fiat money is used only if it has a “value.” Such a value depends on its acceptability, which is not an intrinsic property of money but is determined endogenously in equilibrium.

The agent's strategy is defined by the following rule of behavior: when a meeting occurs, the agent accepts a commodity only if he or she “likes” it (then with probability x), and he or she accepts money in exchange with probability π when other agents accept money with probability Π. The agent must choose π as the best response to the common strategy of other agents, Π. To this end, at the beginning of period t he or she compares the payoffs (in terms of expected utility) from holding money and from holding a commodity, which we call V M(t) and VC(t) respectively.

(p.183) For a money holder, the payoff is equal to

V M ( t ) = 1 1 + r { ( 1 β ) V M ( t + 1 ) + β [ ( 1 M ) x Π ( U + V C ( t + 1 ) ) + ( 1 ( 1 M ) x Π ) V M ( t + 1 ) ] } ,
where r is the rate of time preference. If a meeting does not occur (with probability 1−β) the agent will end period t holding money with a value VM(t+1), whereas if a meeting does occur (with probability β) she will end the period with an expected payoff given by the term in square brackets on the right-hand side of (5.15). If the agent meets a commodity holder who is offering a good that she “likes” and is willing to accept money, the exchange can take place and the payoff is the sum of the utility from consumption U and the value of the newly produced commodity VC(t+1). This event occurs with probability (1−M)xΠ. With the remaining probability, 1−(1−M)xΠ, trade does not take place and the agent's payoff is simply VM(t+1).

For a commodity holder, the payoff is

V C ( t ) = 1 1 + r { ( 1 β ) V C ( t + 1 ) + β [ ( 1 M ) x 2 U + + M π χ V M ( t + 1 ) + ( 1 M π χ V C ( t + 1 ) ] } .

Again, the term in square brackets gives the expected payoff if a meeting occurs and is the sum of three terms. The first is utility from consumption U, which is enjoyed only if the agent meets a commodity holder and both like each other's commodity (a “double coincidence of wants” situation), so that a barter can take place; the probability of this event is (1−M)x 2. The second term is the payoff from accepting money in exchange for the commodity, yielding a value VM(t+1): this trade occurs only if the agent is willing to accept money (with probability π) and meets a money holder who is willing to receive the commodity he offers (with probability Mx). The third term is the payoff from ending the period with a commodity, which happens in all cases except for trade with a money holder, so occurs with probability 1−Mπx.

To derive the agent's best response, we focus on equilibria in which all agents choose the same strategy, whereby π=Π, and payoffs are stationary, so that VM(t)=V M(t+1)≡ VM and VC(t)=V C(t+1)≡ VC. Using these properties in (5.15) and (5.16), multiplying by 1/(1+r), and rearranging terms we get

r V M = β { ( 1 M ) x Π U + ( 1 M ) x Π ( V c V M ) } ,
r V C = β { ( 1 M ) x 2 U + M x Π ( V M V C ) } .

Expressed in this form, (5.17) and (5.18) are readily interpreted as asset valuation equations. The left-hand side represents the flow return from investing in a risk-free asset. The right-hand side is the flow return from holding either (p.184) money or a commodity and includes the expected utility from consumption (the “dividend” component) as well as the expected change in the value of the asset held (the “capital gains” component). Finally, subtracting (5.17) from (5.18), we obtain

V C V M = β ( 1 M ) x U r + β x π ( x Π ) .

The sign of VCVM depends on the sign of the difference between the degree of acceptability of commodities (parameterized by the fraction of agents that “like” any given commodity x) and that of money (Π). Consequently, the agents’ optimal strategy in accepting money in a trade depends solely on Π.

  • If Π < x, money is being accepted with lower probability than commodities. Then VC > VM, and the best response is never to accept money in exchange for a commodity: π=0.

  • If Π > x, money is being accepted with higher probability than commodities. In this case V C < VM, and the best response is to accept money whenever possible: π=1.

  • Finally, if Π=x, money and commodities have the same degree of acceptability. With VC=VM, agents are indifferent between holding money and commodities: the best response then is any value of π between 0 and 1.

The optimal strategy π=π (Π) is shown in Figure 5.3. Three (stationary and symmetric) Nash equilibria, represented in the figure along the 45° line where π=Π, are associated with the three best responses illustrated above:

  1. (i) A non-monetary equilibrium (Π=0): agents expect that money will never be accepted in trade, so they never accept it. Money is valueless (VM=0) and barter is the only form of exchange (point A).

                       Coordination and Externalities in Macroeconomics

    Fig 5.3. Optimal πΠ response function

  2. (p.185)
  3. (ii) A pure monetary equilibrium (Π=1): agents expect that money will be universally acceptable, so they always accept it in exchange for goods (point C).

  4. (iii) A mixed monetary equilibrium (Π = x): agents are indifferent between accepting and rejecting money, as long as other agents are expected to accept it with probability x. In this equilibrium money is only partially acceptable in exchanges (point B).

The main insight of the Kiyotaki–Wright search model of money is that acceptability is not an intrinsic property of money, which is indeed worthless. Rather, it can emerge endogenously as a property of the equilibrium. Moreover, as in Diamond's model, multiple equilibria can arise. Which of the possible equilibria is actually realized depends on the agents’ beliefs: if they expect a certain degree of acceptability of money (zero, partial or universal) and choose their optimal trading strategy accordingly, money will display the expected acceptability in equilibrium. Again, as in Diamond's model, expectations are self-fulfilling.

5.2.3 Implications

The above search model can be used to derive some implications concerning the agents' welfare and the optimal quantity of money.

Welfare

We can now compare the values of expected utility for a commodity holder and a money holder in the three possible equilibria. Solving (5.17) and (5.18) with Π=0, x, and 1 in turn, we find the values of VC i and VM i, where the superscript i = n,m,p denotes the non-monetary, the mixed monetary, and the pure monetary equilibria associated with Π = 0,x,1 respectively. The resulting expected utilities are reported in Table 5.1, where K≡ (β(1−M)xU/r)>0.

Table 1.1.

Π

VC i

VM i

(1)

(2)

(3)

0

κχ

0

κ

κχ

κχ

1

K χ r + β ( ( 1 M ) χ + M ) r + β x > K χ

K r + β χ ( ( 1 M ) χ + M ) r + β x > K χ

(p.186) Some welfare implications can be easily drawn from the table. First of all, the welfare of a money holder intuitively increases with the degree of acceptability of money. In fact, comparing the expected utilities in column (3), we find that V M n < V M m < V M p .

Further, in the pure monetary equilibrium (third row of the table) money holders are better off than commodity holders: VC p<VM p. Holding universally acceptable money guarantees consumption when the money holder meets a commodity holder with a good that she “likes”: trade increases the welfare of both agents and occurs with certainty. On the contrary, a commodity holder can consume only if another commodity holder is met and both like each other's commodity: a “double coincidence of wants” is necessary, and this reduces the probability of consumption with respect to a money holder.

Exercise 49 Check that, in a pure monetary equilibrium, when a money holder meets a commodity holder with a good she “likes” both agents are willing to trade.

Finally, looking at column (2) of the table, we note that a commodity holder is indifferent between a non-monetary and a mixed monetary equilibrium, but is better off if money is universally acceptable, as in the pure monetary equilibrium:

V C n < V C m < V C p .

Summarizing, the existence of universally accepted fiat money makes all agents better off. Moreover, moving from a non-monetary to a mixed monetary equilibrium increases the welfare of money holders without harming commodity holders. Thus, in general, an increase in the acceptability of money (Π) makes at least some agents better off and none worse off (a Pareto improvement).

Optimal quantity of money

We now address the issue of the optimal quantity of money from the social welfare perspective. The amount of money in circulation is directly related to the fraction of agents endowed with money M; we therefore consider the possibility of choosing M so as to maximize some measure of social welfare. A reasonable such measure is an agent's ex ante expected utility, that is the expected utility of each agent before the initial endowment of money and commodities is randomly distributed among them. The social welfare criterion is then

W = ( 1 M ) V C + M V M .

The fraction of agents endowed with money can be optimally chosen in the three possible equilibria of the economy. First, we note that, in both the non-monetary and the mixed monetary equilibria, money does not facilitate the (p.187) exchange process (thus making consumption more likely); it is then optimal to endow all agents with commodities, thereby setting M=0. In the pure monetary equilibrium, social welfare Wp can be expressed as

W p = ( 1 M ) V C p + M V M p = K ċ [ M + x ( 1 M ) ] = β U r ( 1 M ) [ M x + ( 1 M ) x 2 ] ,
where we used the definition of K given above. Maximization of Wp with respect to M yields the optimal quantity of money M*:
W p M = β U r x [ ( 1 2 x ) 2 M * ( 1 x ) ] = 0 1 2 x = 2 M * ( 1 x ) M * = 1 2 x 2 2 x .

Since 0≤ M* ≤ 1, for x ≥ ½ we get M*=0. When each agent is willing to consume at least half of the commodities, exchanges are not very difficult and money does not play a crucial role in facilitating trade: in this case it is optimal to endow all agents with consumable commodities. Instead, if x < ½, fiat money plays a useful role in facilitating trade and consumption, and the introduction of some amount of money improves social welfare (even though fewer consumable commodities will be circulating in the economy). From (5.22) we see that, as x → 0, M* → ½, as shown in the left-hand panel of Figure 5.4.

                   Coordination and Externalities in Macroeconomics

Fig 5.4. Optimal quantity of money M * and ex ante probability of consumption P

(p.188) To further develop the intuition for this result, we can rewrite the last expression in (5.21) as follows:

r W p = U ċ β ( 1 M ) [ M x + ( 1 M ) x 2 ] ,
where the left-hand side is the “flow” of social welfare per period and the right-hand side is the utility from consumption U multiplied by the agent's ex ante consumption probability. The latter is given by the probability of meeting an agent endowed with a commodity, β (1−M), times the probability that a trade will occur, given by the term in square brackets. Trade occurs in two cases: either the agent is a money holder and the potential counterpart in the trade offers a desirable commodity (which happens with probability Mx), or the agent is endowed with a commodity and a “double coincidence of wants” occurs (which happens with probability (1−M)x 2). The sum of these two probabilities yields the probability that, after a meeting with a commodity holder, trade will take place. The optimal quantity of money is the value of M that maximizes the agent's ex ante consumption probability in (5.23). As M increases, there is a trade-off between a lower probability of encountering a commodity holder and a higher probability that, should a meeting occur, trade takes place. The amount of money M* optimally weights these two opposite effects. The behavior of the consumption probability (P) as a function of M is shown in the right-hand panel of Figure 5.4 for two values of x (0.5 and 0.25) in the case where β=1. The corresponding optimal quantities of money M* are 0 and 0.33 respectively.

5.3 Search Externalities in the Labor Market

We now proceed to apply some of the insights discussed in this chapter to labor market phenomena. While introducing the models of Chapter 3, we already noted that the simultaneous processes of job creation and job destruction are typically very intense, even in the absence of marked changes in overall employment. In that chapter we assumed that workers' relocation was costly, but we did not analyze the level or the dynamics of the unemployment rate. Here, we review the modeling approach of an important strand of labor economics focused exactly on the determinants of the flows into and out of (frictional) unemployment. The agents of these models, unlike those of the models discussed in the previous sections, are not ex ante symmetric: workers do not trade with each other, but need to be employed by firms. Unemployed workers and firms willing to employ them are inputs in a “productive” process that generates employment, a process that is given a stylized and very tractable (p.189) representation by the model we study below. Unlike the abstract trade and monetary exchange frameworks of the previous sections, the “search and matching” framework below is qualitatively realistic enough to offer practical implications for the dynamics of labor market flows, for the steady state of the economy, and for the dynamic adjustment process towards the steady state.

5.3.1 Frictional unemployment

The importance of gross flows justifies the fundamental economic mechanism on which the model is based: the matching process between firms and workers. Firms create job openings (vacancies) and unemployed workers search for jobs, and the outcome of a match between a vacant job and an unemployed worker is a productive job. Moreover, the matching process does not take place in a coordinated manner, as in the traditional neoclassical model. In the neoclassical model the labor market is perfectly competitive and supply and demand of labor are balanced instantaneously through an adjustment of the wage. On the contrary, in the model considered here firms and workers operate in a decentralized and uncoordinated manner, dedicating time and resources to the search for a partner. The probability that a firm or a worker will meets a partner depends on the relative number of vacant jobs and unemployed workers: for example, a scarcity of unemployed workers relative to vacancies will make it difficult for a firm to fill its vacancy, while workers will find jobs easily. Hence there exists an externality between agents in the same market which is of the same “trading” type as the one encountered in the previous section. Since this externality is generated by the search activity of the agents on the market, it is normally referred to as a search externality. Formally, we define the labor force as the sum of the “employed” workers plus the “unemployed” workers which we assume to be constant and equal to L units. Similarly, the total demand for labor is equal to the number of filled jobs plus the number of vacancies. The total number of unemployed workers and vacancies can therefore be expressed as uL e vL, respectively, where u denotes the unemployment rate and v denotes the ratio between the number of vacancies and the total labor force. In each unit of time, the total number of matches between an unemployed worker and a vacant firm is equal to mL (where m denotes the ratio between the newly filled jobs and the total labor force). The process of matching is summarized by a matching function, which expresses the number of newly created jobs (mL) as a function of the number of unemployed workers (uL) and vacancies (vL):

m L = m ( u L , v L ) .

The function m(·), supposed increasing in both arguments, is conceptually similar to the aggregate production function that we encountered, for (p.190) example, in Chapter 4. The creation of employment is seen as the outcome of a “productive process” and the unemployed workers and vacant jobs are the “productive inputs.” Obviously, both the number of unemployed workers and the number of vacancies have a positive effect on the number of matches within each time period (mu > 0,mv > 0). Moreover, the creation of employment requires the presence of agents on both sides of the labor market (m(0,0) = m(0,vL) = m(uL,0) = 0). Additional properties of the function m(·) are needed to determine the character of the unemployment rate in a steady-state equilibrium. In particular, for the unemployment rate to be constant in a growing economy, m(·) needs to have constant returns to scale43. In that case, we can write

m = m ( u L , v L ) L = m ( u , v ) .

The function m(·) determines the flow of workers who find a job and who exit the unemployment pool within each time interval. Consider the case of an unemployed worker: at each moment in time, the worker will find a job with probability p = m(·)/u. With constant returns to scale for m(·), we may thus write

m ( u , v ) u = m ( 1 , v u ) p ( θ ) , an increasing function of θ v u .

The instantaneous probability p that a worker finds a job is thus positively related to the tightness of the labor market, which is measured by θ, the ratio of the number of vacancies to unemployed workers44. An increase in θ, reflecting a relative abundance of vacant jobs relative to unemployed workers, leads to an increase in p. (Moreover, given the properties of m, p″ (θ)<0.) Finally, the average length of an unemployment spell is given by 1/p(θ), and thus is inversely related to θ. Similarly, the rate at which a vacant job is matched to a worker may be expressed as

m ( u , v ) v = m ( 1 , v u ) u v = p ( θ ) θ q ( θ ) ,
a decreasing function of the vacancy/unemployment ratio. An increase in θ reduces the probability that a vacancy is filled, and 1/q(θ) measures the average (p.191) time that elapses before a vacancy is filled.45 The dependence of p and q on θ captures the dual externality between agents in the labor market: an increase in the number of vacancies relative to unemployed workers increases the probability that a worker finds a job (∂ p(·)/∂ v>0), but at the same time it reduces the probability that a vacancy is filled (∂ q(·)/∂ v>0).

5.3.2 The dynamics of unemployment

Changes in unemployment result from a difference between the flow of workers who lose their job and become unemployed, and the flow of workers who find a job. The inflow into unemployment is determined by the “separation rate” which we take as given for simplicity: at each moment in time a fraction s of jobs (corresponding to a fraction 1−u of the labor force) is hit by a shock that reduces the productivity of the match to zero: in this case the worker loses her job and returns to the pool of unemployed, while the firm is free to open up a vacancy in order to bring employment back to its original level. Given the match destruction rate s, jobs therefore remain productive for an average period 1/s. Given these assumptions, we can now describe the dynamics of the number of unemployed workers. Since L is constant, d ( u L ) / d t = u ˙ L and hence

u ˙ L = s ( 1 u ) L p ( θ ) u L u ˙ = s ( 1 u ) p ( θ ) u ,
which is similar to the difference equation for employment (5.1) derived in the previous section. The dynamics of the unemployment rate depend on the tightness of the labor market θ: at a high ratio of vacancies to unemployed workers, workers easily find jobs, leading to a large flow out of unemployment.46 From equation (5.28) we can immediately derive the steady-state relationship between the unemployment rate and θ:
u = s s + p ( θ ) .

Since p′(·) > 0, the properties of the matching function determine a negative relation between θ and u: a higher value of θ corresponds to a larger flow of newly created jobs. In order to keep unemployment constant, the unemployment rate must therefore increase to generate an offsetting increase in the flow

                   Coordination and Externalities in Macroeconomics

Fig 5.5. Dynamics of the unemployment rate

(p.192) of destroyed jobs. The steady-state relationship (5.29) is illustrated graphically in the left-hand panel of Figure 5.5: to each value of θ corresponds a unique value for the unemployment rate. Moreover, the same properties of m(·) ensure that this curve is convex. For points above or below u ˙ = 0 , the unemployment rate tends to move towards the stationary relationship: keeping θ constant at θ0, a value u>u 0 causes an increase in the flow out of unemployment and a decrease in the flow into unemployment, bringing u back to u 0. Moreover, given u and θ, the number of vacancies is uniquely determined by v = θu, where v denotes the number of vacancies as a proportion of the labor force. The picture on the right-hand side of the figure shows the curve u ˙ = 0 in (v,u)-space. This locus is known as the Beveridge curve, and identifies the level of vacancies v 0 that corresponds to the pair (θ0,u 0) in the left-hand panel. In the sequel we will use both graphs to illustrate the dynamics and the comparative statics of the model. At this stage it is important to note that variations in the labor market tightness are associated with a movement along the curve u ˙ = 0 , while changes in the separation rate s or the efficiency of the matching process (captured by the properties of the matching function) correspond to movements of the curve u ˙ = 0 . For example, an increase in s or a decrease in the matching efficiency causes an upward shift of u ˙ = 0 . Equation (5.29) describes a first steady-state relationship between u and θ. To find the actual equilibrium values, we need to specify a second relationship between these variables. This second relationship can be derived from the behavior of firms and workers on the labor market.

5.3.3 Job availability

The crucial decision of firms concerns the supply of jobs on the labor market. The decision of a firm about whether to create a vacancy depends on the expected future profits over the entire time horizon of the firm, which (p.193) we assume is infinite. Formally, each individual firm solves an intertemporal optimization problem taking as given the aggregate labor market conditions which are summarized by θ, the labor market tightness. Individual firms therefore disregard the effect of their decisions on θ, and consequently on the matching rates p(θ) and q(θ) (the external effects referred to above). To simplify the analysis, we assume that each firm can offer at most one job. If the job is filled, the firm receives a constant flow of output equal to y. Moreover, it pays a wage w to the worker and it takes this wage as given. The determination of this wage is described below. On the other hand, if the job is not filled the firm incurs a flow cost c, which reflects the time and resources invested in the search for suitable workers. Firms therefore find it attractive to create a vacancy as long as its value, measured in terms of expected profits, is positive; if it is not, the firm will not find it attractive to offer a vacancy and will exit the labor market. The value that a firm attributes to a vacancy (denoted by V) and to a filled job (J) can be expressed using the asset equations encountered above. Given a constant real interest rate r, we can express these values as

r V ( t ) = c + q ( θ ( t ) ) ( J ( t ) V ( t ) ) + V ˙ ( t ) ,
r J ( t ) = ( y w ( t ) ) + s ( V ( t ) J ( t ) ) + j ( t ) ,
which are explicit functions of time. The flow return of a vacancy is equal to a negative cost component (−c), plus the capital gain in case the job is filled with a worker (J − V), which occurs with probability q(θ), plus the change in the value of the vacancy itself V ˙ . Similarly, (5.31) defines the flow return of a filled job as the value of the flow output minus the wage (y − w), plus the capital loss (V − J) in case the job is destroyed, which occurs with probability s, plus the change in the value of the job J ˙ .

Exercise 50 Derive equation (5.31) with dynamic programming arguments, supposing that J ˙ = 0 and following the argument outlined in Section 5.1 to obtain equations (5.3) and (5.4).

Subtracting (5.30) from (5.31) yields the following expression for the difference in value between a filled job and a vacancy:

r ( J ( t ) V ( t ) ) = ( y w ( t ) + c ) [ s + q ( θ ( t ) ) ] ( J ( t ) V ( t ) ) + ( j ( t ) V ˙ ( t ) ) .

Solving equation (5.32) at date t 0 for the entire infinite planning horizon of the firm, we get

J ( t 0 ) V ( t 0 ) = t 0 ( y w ( t ) + c ) e t 0 t [ r + s + q ( θ ( τ ) ) ] d τ d t ,
(p.194) where we need to impose the following transversality condition:
lim T [ J ( T ) V ( T ) [ e t 0 T ( r + s + q ( θ ( τ ) ) ) d τ = 0.

Equation (5.33) expresses the difference between the value of a job and the value of a vacancy as the value of the difference between the flow return of a job (yw) and that of a vacancy (−c) over the entire time horizon, which is discounted to t 0 using the appropriate “discount rate.” Besides on the real interest rate, this discount rate also depends on the separation rate s and on the tightness of the labor market via q(θ). Intuitively, a higher number of vacancies relative to unemployed workers decreases the probability that a vacant firm will meet a worker. This reduces the effective discount rate and leads to an increase in the difference between the value of a filled job and a vacancy. Moreover, θ may also have an indirect effect on the flow return of a filled job via its impact on the wage w, as we will see in the next section.

Now, if we focus on steady-state equilibria, we can impose V ˙ = J ˙ = 0 in equations (5.30) and (5.31). Moreover, we assume free entry of firms and as a result V = 0: new firms continue to offer vacant jobs until the value of the marginal vacancy is reduced to zero. Substituting V = 0 in (5.30) and (5.31) and combining the resulting expressions for J, we get

J = c / q ( θ ) J = ( y w ) / ( r + s ) } y w = ( r + s ) c q ( θ ) .

Equation (5.30) gives us the first expression for J. According to this condition, the equilibrium value of a filled job is equal to the expected costs of a vacancy, that is the flow cost of a vacancy c times the average duration of a vacancy 1/q(θ). The second condition for J can be derived from (5.31): the value of a filled job is equal to the value of the constant profit flow yw. These flow returns are discounted at rate r + s to account for both impatience and the risk that the match breaks down. Equating these two expressions yields the final solution (5.34), which gives the marginal condition for employment in a steady-state equilibrium: the marginal productivity of the worker (y) needs to compensate the firm for the wage w paid to the worker and for the flow cost of opening a vacancy. The latter is equal to the product of the discount rate r + s and the expected costs of a vacancy c/q(θ).

This last term is just like an adjustment cost for the firm's employment level. It introduces a wedge between the marginal productivity of labor and the wage rate, which is similar to the effect of the hiring costs studied in Chapter 3. However, in the model of this section the size of the adjustment cost is endogenous and depends on the aggregate conditions on the labor market. In equilibrium, the size of the adjustment costs depends on the unemployment rate and on (p.195) the number of vacancies, which are summarized at the aggregate level by the value of θ. If, for example, the value of output minus wages (y−w) increases, then vacancy creation will become profitable (V > 0) and more firms will offer jobs. As a result, θ will increase, leading to a reduction in the matching rate for firms and an increase in the average cost of a vacancy, and both these effects tend to bring the value of a vacancy back to zero.

Finally, notice that equation (5.34) still contains the wage rate w. This is an endogenous variable. Hence the “job creation condition” (5.34) is not yet the steady-state condition which together with (5.29) would allow us to solve for the equilibrium values of u and θ. To complete the model, we need to analyze the process of wage determination.

5.3.4 Wage determination and the steady state

The process of wage determination that we adopt here is based on the fact that the successful creation of a match generates a surplus. That is, the value of a pair of agents that have agreed to match (the value of a filled job and an employed worker) is larger than the value of these agents before the match (the value of a vacancy and an unemployed worker). This surplus has the nature of a monopolistic rent and needs to be shared between the firm and the worker during the wage negotiations. Here we shall assume that wages are negotiated at a decentralized level between each individual worker and her employer. Since workers and firms are identical, all jobs will therefore pay the same wage.

Let E and U denote the value that a worker attributes to employment and unemployment, respectively. The joint value of a match (given by the value of a filled job for the firm and the value of employment for the worker) can then be expressed as J + E, while the joint value in case the match opportunity is not exploited (given by the value of a vacancy for a firm and the value of unemployment for a worker) is equal to V + U. The total surplus of the match is thus equal to the sum of the firm's surplus, JV, and the worker's surplus, EU:

( J + E ) ( V + U ) ( J V ) + ( E U ) .

The match surplus is divided between the firm and the worker through a wage bargaining process. We take their relative bargaining strength to be exogenously given. Formally, we adopt the assumption of Nash bargaining. This assumption is common in models of bilateral negotiations. It implies that the bargained wage maximizes a geometric average of the surplus of the firm and the worker, each weighted by a measure of their relative bargaining strength. In our case the assumption of Nash bargaining gives rise to the following optimization problem:

max w ( J V ) 1 β ( E U ) β ,
(p.196) where 0≤ β ≤ 1 denotes the relative bargaining strength of the worker. Given that the objective function is a Cobb–Douglas one, we can immediately express the solution (the first-order conditions) of the problem as:
E U = β 1 β ( J V ) E U = β [ ( J V ) + ( E U ) ] .

The surplus that the worker appropriates in the wage negotiations (EU) is thus equal to a fraction β of the total surplus of the job.

Similar to what is done for V and J in (5.30) and (5.31), we can express the values E and U using the relevant asset equations (reintroducing the dependence on time t):

r E ( t ) = w ( t ) + s ( U ( t ) E ( t ) ) + E ˙ ( t )
r U ( t ) = z + p ( θ ) ( E ( t ) U ( t ) ) + U ˙ ( t ) .

For the worker, the flow return on employment is equal to the wage plus the loss in value if the worker and the firm separate, which occurs with probability s, plus any change in the value of E itself; while the return on unemployment is given by the imputed value of the time that a worker does not spend working, denoted by z, plus the gain if she finds a job plus the change in the value of U. Parameter z includes the value of leisure and/or the value of alternative sources of income including possible unemployment benefits. This parameter is assumed to be exogenous and fixed. Subtracting (5.39) from (5.38), and solving the resulting expression for the entire future time horizon, we can express the difference between the value of employment and unemployment at date t 0 as

E ( t 0 ) U ( t 0 ) = t 0 ( w ( t ) z ) e t 0 t [ r + s + p ( θ ( τ ) ) ] d τ d t .

As in the case of firms, apart from the real interest rate r and the rate of separation s, the discount rate for the flow return of workers depends on the degree of labor market tightness via its effect on p(θ). A relative abundance of vacant jobs implies a high matching rate for workers, and this tends to reduce the difference between the value of employment and unemployment for a given wage value.

There are two ways to obtain the effect of variations θ on the wage. Restricting attention to steady-state equilibria, so that E ˙ = U ˙ = 0 , we can either derive the surplus of the worker EU directly from (5.38) and (5.39), or we can solve equation (5.40) keeping w and θ constant over time:

E U = w z r + s + p ( θ ) .

According to (5.41), the surplus of a worker depends positively on the difference between the flow return during employment and unemployment (wz) and negatively on the separation rate s and on θ: an increase in the ratio of (p.197) vacancies to unemployed workers increases the exit rate out of unemployment and reduces the average length of an unemployment spell. Using (5.41), and noting that in steady-state equilibrium

J V = J = y w r + s ,
we can solve the expression for the outcome of the wage negotiations given by (5.37) as
w z r + s + p ( θ ) = β 1 β y w r + s .

Rearranging terms, and using (5.34), we obtain the following equivalent expressions for the wage:

w z = β [ ( y + c θ w ) + ( w z ) ]
w = z + β ( y + c θ z ) .

Equation (5.42) is the version in terms of flows of equation (5.37): the flow value of the worker's surplus, i.e. the difference between the wage and alternative income z, is a fraction β of the total flow surplus. The term yw + cθ represents the flow surplus of the firm, where cθ denotes the expected cost savings if the firm fills a job. Moreover, the wage is a pure redistribution from the firm to the worker. If we eliminate the wage payments in (5.42), we obtain the flow value of the total surplus of a filled job y + cθ − z, which is equal to the sum of the value of output and the cost saving of the firm minus the alternative costs of the worker. Finally, equation (5.43) expresses the wage as the sum of the alternative income and the fraction of the surplus that accrues to the worker.

It can easily be verified that the only influence of aggregate labor market conditions on the wage occur via θ, the ratio of vacancies to unemployed workers. The unemployment rate u does not have any independent effect on wages. The explanation is that wages are negotiated after a firm and a worker meet. In this situation the match surplus depends on θ, as we saw above. This variable determines the average duration of a vacancy, and hence the expected costs for the firm if it continued to search.

The determination of the equilibrium wage completes the description of the steady-state equilibrium. The equilibrium can be summarized by equations (5.29), (5.34), and (5.43) which we shall refer to as BC (Beveridge curve), JC (job creation condition), and W (wage equation):

u = s s + p ( θ ) ( B C )
y w = ( r + s ) c q ( θ ) ( J C )
w = ( 1 β ) z + β ( y + c θ ) ( W )
                   Coordination and Externalities in Macroeconomics

Fig 5.6. Equilibrium of the labor market with frictional unemployment

(p.198) For a given value of θ, the wage is independent of the unemployment rate. The system can therefore be solved recursively for the endogenous variables u, θ, and w. Using the definition for θ, we can then solve for v. The last two equations jointly determine the equilibrium wage w and the ratio of vacancies/unemployed θ, as is shown in the left-hand panel of Figure 5.6. Given θ, we can then determine the unemployment rate u, and consequently also v, which equates the flows into and out of unemployment (the right-hand panel of the figure).

This dual representation facilitates the comparative static analysis, which is intended to analyze the effect of changes in the parameters on the steady-state equilibrium. (Analysis of transitional dynamics is the subject of the next section.) In some cases, parameter changes have an unambiguous effect on all of the endogenous variables. This is true for instance in the case of an increase in unemployment benefits, a component of z, or an increase in the relative bargaining strength of workers β: the only effect of these changes is an upward shift of W which causes an increase in the wage and a reduction in θ. This reduction, along the curve BC, is accompanied by an increase in u and a reduction in v.

In other cases the effects are more complex and not always of unambiguous sign. Consider, for example, the effects of the following two types of shock which may be at the root of cyclical variations in overall unemployment. The first is an “aggregate” disturbance. This is represented by a variation in the productivity of labor y which affects all firms at the same time and with the same intensity. The second shock is a “reallocative” disturbance, represented by a change in the separation rate s. This shock hits individual firms independently of the aggregate state of the economy (captured by labor productivity y).

A reduction in y moves both JC and W downwards. This results in a reduction of the wage but has an ambiguous effect on θ. However, formal analysis (p.199) (which is required in the exercise below) shows that in a stationary equilibrium θ also decreases; since the curve BC does not shift, the unemployment rate must increase while the number of vacancies v is reduced. In the case of a reallocative shock, we observe an inward shift of JC along W. This results in a joint decrease of the wage and the labor market tightness θ, as in the case of the aggregate shock. At the same time, however, the curve BC shifts to the right. Hence, while the unemployment rate increases unambiguously, it is in general not possible to determine the effect on the number of vacancies. In reality, however, v appears to be procyclical, and this suggests that aggregate shocks are a more important source of cyclical movements in the labor market than allocative shocks.

Exercise 51 Derive formally, using the system of equations formed by (5.44), (5.45), and (5.46), the effects on the steady-state levels of w, θ, u, and v of a smaller labor productivity (Δ y > 0) and of a higher separation rate (Δ s > 0).

5.4 Dynamics

Until now, all the relationships we derived referred to the steady-state equilibrium of the system. In this section we will analyze the evolution of unemployment, vacancies, and the wage rate along the adjustment path toward the steady-state equilibrium.

The discussion of the flows into and out of unemployment in the previous section has already delivered the law of motion for unemployment. This equation is repeated here (stressing the time dependence of the endogenous variables):

u ˙ ( t ) = s ( 1 u ( ( t ) ) p ( θ ) ( t ) ) u ( t ) .

The dynamics of u are due to the flow of separations and the flow of newly created jobs resulting from the matches between firms and workers. The magnitude of the flow out of unemployment depends on aggregate labor market conditions, captured by θ, via its effect on p(·). Outside a steady-state equilibrium, the path of θ will influence unemployment dynamics in the economy. Moreover, given the definition of θ as the ratio of vacancies to unemployed workers, this will also affect the value of the labor market tightness. In order to give a complete description of the adjustment process toward a steady-state equilibrium, we therefore need to study the dynamics of θ. This requires an analysis of the job creation decisions of firms.

(p.200) 5.4.1 Market tightness

At each moment in time firms exploit all opportunities for the profitable creation of jobs. Hence in a steady-state equilibrium, as well as along the adjustment path, V(t)=0∀t, and outside a steady-state equilibrium V ˙ ( t ) = 0 , t . . The value of a filled job for the firm can be derived from (5.30) and (5.31). From the first equation, setting V ( t ) = V ˙ ( t ) = 0 , we get

J ( t ) = c q ( θ ( t ) ) .

Equation (5.48) is identical to the steady-state expression derived before. Firms continue to create new vacancies, thereby influencing θ, until the value of a filled job equals the expected cost of a vacancy. Since entry into the labor market is costless for firms (the resources are used to maintain open vacancies), equation (5.48) will hold at each instant during the adjustment process. Outside the steady-state, the dynamics of J needs to satisfy difference equation (5.31), with V(t) = 0:

J ˙ ( t ) = ( r + s ) J ( t ) ( y w ( t ) ) .

The solution of (5.49) shows that the value J(t) depends on the future path of (expected) wages. Besides that, J(t) also depends on labor productivity, the real interest rate, and the rate of separation, but all these variables are assumed to be constant:

J ( t ) = t ( y w ( t ) ) e ( r + s ) τ d τ .

Wages are continuously renegotiated. Outside steady-state equilibrium the surplus sharing rule (5.37) with V(t) = 0 therefore remains valid:

E ( t ) U ( t ) = β 1 β J ( t ) .

Outside the steady-state E, U, and J may vary over time, but these variations need to ensure that (5.51) is satisfied. Hence, we have

E ˙ ( t ) U ˙ ( t ) = β 1 β J ˙ ( t ) .

The dynamics of J are given by (5.49), while the dynamics of E and U can be derived by subtracting (5.39) from (5.38):

E ˙ ( t ) U ˙ ( t ) = [ r + s + p ( θ ( t ) ) ] ( E ( t ) U ( t ) ) ( w ( t ) z ) .

Equating (5.52) to (5.53), and using (5.49) and (5.48) to replace J ˙ and J, we can solve for the level of wages outside steady-state equilibrium as:

w ( t ) = z + β ( y + c θ ( t ) z ) .
                   Coordination and Externalities in Macroeconomics

Fig 5.7. Dynamics of the supply of jobs

(p.201) The wage is thus determined in the same way both in a steady-state equilibrium and during the adjustment process. Moreover, given the values for the exogenous variables, the wage dynamics depends exclusively on changes in the degree of labor market tightness, which affects the joint value of a productive match.

We are now in possession of all the elements that are needed to determine the dynamics of θ. Differentiating (5.48) with respect to time, where by definition q(θ) ≡ p(θ)/θ, we have

J ˙ ( t ) = c p ( θ ( t ) ) c θ ( t ) p ( θ ( t ) ) p ( θ ( t ) ) 2 θ ˙ ( t ) = c p ( θ ( t ) ) [ 1 η ( θ ( t ) ) ] θ ˙ ( t ) ,
where 0 < η(θ) < 1 (defined above) denotes the elasticity of p(θ) with respect to θ. To simplify the derivations, we henceforth assume that η(θ) = η is constant (which is true if the matching function m(·) is of the Cobb–Douglas type). Substituting (5.49) for J ˙ and using the expression J = c(θ/p(θ)), we can rewrite equation (5.55) as
θ ˙ ( t ) c p ( θ ( t ) ) ( 1 η ) = ( r + s ) c θ ( t ) p ( θ ( t ) ) ( y w ( t ) ) .

Finally, substituting the expression for the wage as a function of θ from (5.54), the above law of motion for θ can be written as

θ ˙ ( t ) = r + s 1 η θ ( t ) p ( θ ) ( t ) ) c ( 1 η ) [ ( 1 β ) ( y z ) β c θ ( t ) ] .

Changes in θ depend on (in addition to all the parameters of the model) only the value of θ itself. The labor market tightness does not in any independent way depend on the unemployment rate u. In (θ,u)-space the curve θ ˙ = 0 can thus be represented by a horizontal line at θ ¯ , which defines the unique steady-state equilibrium value for the ratio between vacancies and unemployed. This is illustrated in the left-hand panel of Figure 5.7. Once we have determined θ ¯ , (p.202) we can determine, for each value of the unemployment rate, the level of v that is compatible with a stationary equilibrium. For instance, in the case of u 0 this is equal to v 0.

Besides that, equation (5.57) also indicates that, for points above or below the curve θ ˙ = 0 , θ tends to move away from its equilibrium value. Formally, one can show this by calculating47

θ ˙ θ | = ( r + s ) + p ( θ ) 1 η β > 0
from (5.57). The apparently “unstable” behavior of θ is due to the nature of the job creation decision of firms. Looking at the future, firms' decisions on whether to open a vacancy today are based on expected future values of θ. For example, if firms expect a future increase in θ resulting from an increase in the number of vacant jobs, they will anticipate an increase in future costs to fill a vacancy. As a result, firms have an incentive to open vacancies immediately in anticipation of this increase in cost. At the aggregate level, this induces an immediate increase in v (and in θ) in anticipation of further increases in the future. Hence, there is an obvious analogy between the variations of v and the movement of asset prices which we already alluded to when we interpreted (5.30) and (5.31) as asset equations: expectations of a future increase in price cause an increase in current prices.

As a result of forward-looking behavior on the part of firms, both v and θ are “jump” variables. Their value is not predetermined: in response to changes in the exogenous parameters (even if these changes are expected in the future and have not yet materialized), v and θ may exhibit discrete changes. The unemployment rate, on the other hand, is a “predetermined” or state variable. The dynamics of the unemployment rate are governed by (5.47), and u adjusts gradually to changes in θ, even in case of a discrete change in the labor market tightness. An unanticipated increase in v and θ leads to an increase in the flow out of unemployment, resulting in a reduction of u. However, the positive effect of the number of vacancies on unemployment is mediated via the stochastic matching process on the labor market. The immediate effect of an increase in θ is an increase in the matching rate for workers p(θ), and this translates only gradually in an increase in the number of filled jobs. The unemployment rate therefore will start to decrease only after some time.

The aggregate effect of the decentralized decisions of firms (each of which disregards the externalities of its own decision on aggregate variables) consists of changes

                   Coordination and Externalities in Macroeconomics

Fig 5.8. Dynamics of unemployment and vacancies

(p.203) in the degree of labor market tightness θ and, as a result, in changes in the speed of adjustment of the unemployment rate. The dynamics of u are therefore intimately linked to the presence of the externalities that characterize the functioning of the labor market in the search and matching literature.

5.4.2 The steady state and dynamics

We are now in a position to characterize the system graphically, using the differential equations (5.47) and (5.57) for u and θ. In both panels of Figure 5.8 we have drawn the curves for θ ˙ = 0 and u ˙ = 0 . Moreover, for each point outside the unique steady-state equilibrium we have indicated the movement of θ and u.

As we have seen in the analysis of dynamic models of investment and growth theory, the combination of a single-state variable (u) and a single jump variable (θ) implies that there is only one saddlepath that converges to the steady-state equilibrium (saddlepoint).48 Since the expression for θ ˙ = 0 does not depend on u, the saddlepath coincides with the curve for θ ˙ = 0 : all the other points are located on paths that diverge from the curve θ ˙ = 0 and never reach the steady-state, violating the transversality conditions. Hence, as a result of the forward-looking nature of the vacancy creation decisions of firms, the labor

                   Coordination and Externalities in Macroeconomics

Fig 5.9. Permanent reduction in productivity

(p.204) market tightness θ will jump immediately to its long-run value and remain there during the entire adjustment process.

Let us now analyze the adjustment process in response first to a reduction in labor productivity y (an aggregate shock) and then to an increase in the rate of separation s (a reallocative shock). Figure 5.9 illustrates the dynamics following an unanticipated permanent reduction in productivityy > 0) at date t 0. In the left-hand graph, the curve θ ˙ = 0 shifts downward while u ˙ = 0 does not change. In the new steady-state equilibrium (point C) the unemployment rate is higher and labor market tightness is lower. Moreover, from the right-hand graph, it follows that the number of vacancies has also decreased. The figure also illustrates the dynamics of the variables: at date t 0 the economy jumps to the new saddlepath which coincides with the new curve θ ˙ = 0 . Given the predetermined nature of the unemployment rate, the whole adjustment is performed by v and θ, which make a discrete jump downwards as shown by B in the two graphs. From t 0 onwards both unemployment and the number of vacancies increase gradually, keeping θ fixed until the new steady-state equilibrium is reached.

The permanent reduction in labor productivity reduces the expected profits of a filled job. Hence, from t 0 onwards firms have an incentive to create fewer vacancies. Moreover, initially the number of vacancies v falls below its new equilibrium level because firms anticipate that the unemployment rate will rise. In future it will therefore be easier to fill a vacancy. As a result, firms prefer to reduce the number of vacancies at the beginning of the adjustment process, increasing their number gradually as the unemployment rate starts to rise.

Finally, the reduction in labor productivity also reduces wages, but this reduction is smaller than the decrease in y. Since the labor market immediately jumps to a saddlepath along which θ (t) is constant, equation (5.54) implies that

                   Coordination and Externalities in Macroeconomics

Fig 5.10. Increase in the separation rate

(p.205) the wage w(t) is constant along the whole adjustment process. The short-run response of the wage is thus equal to the long-run response, which is governed by (5.34). According to this equation, the difference y-w is proportional to the expected cost of a vacancy for the firm. This cost depends on the average time that is needed to fill a vacancy, which diminishes when v and θ fall in response to a productivity shock. Hence, in this version of the model productivity changes do not imply proportional wage changes. (On this point see exercise 51 at the end of the chapter.)

A similar adjustment process takes place in the case of a (unanticipated and permanent) reallocative shock Δs > 0, as shown in Figure 5.10. However, in this case u ˙ = 0 is also affected. This curve shifts to the right, which reinforces the increase in the unemployment rate, but has an ambiguous effect on the number of vacancies. (The figure illustrates the case of a reduction in v.) Finally, let us consider the case of a temporary reduction of productivity: agents now anticipate at t 0 that productivity will return to its higher initial value at some future date t 1. Given the temporary nature of the shock, the new steady-state equilibrium coincides with the initial equilibrium (point A in the graphs of Figure 5.11. At the time of the change in productivity, t 0, the immediate effect is a reduction in the number of vacancies which causes a discrete fall in θ. However, this reduction is smaller than the one that resulted from a permanent change, and it moves the equilibrium from the previous equilibrium A to a new point B′. From t0 onwards, the unemployment rate and the number of vacancies increase gradually but not at the same rate: as a result, their ratio θ increases, following the diverging dynamics that leads towards the new and lower stationary curve θ ˙ = 0 . To obtain convergence of the steady-state equilibrium at A, the dynamics of the adjustment need to bring θ to its equilibrium level at t 1 when the shock ceases and productivity returns to its previous level

                   Coordination and Externalities in Macroeconomics

Fig 5.11. A temporary reduction in productivity

(p.206) (point B″). In fact, convergence to the final equilibrium can occur only if the system is located on the saddlepath, which coincides with the stationary curve for θ, at date t 1. After t 1 the dynamics concerns only the unemployment rate u and the number of vacancies v, which decrease in the same proportion until the system reaches its initial starting point A.

The graph on the right-hand side of Figure 5.11 also illustrates that cyclical variations in productivity give rise to a counter-clockwise movement of employment and vacancies around the Beveridge curve. This is consistent with empirical data for changes in unemployment and vacancies during recessions, which are approximated here by a temporary reduction in productivity.

5.5 Externalities and efficiency

The presence of externalities immediately poses the question of whether the decentralized equilibrium allocation is efficient. In particular, in the previous sections it was shown that firms disregard the effect of their private decisions on the aggregate labor market conditions when they are deciding whether or not to create a vacancy. In this section we analyze the implications of these external effects for the efficiency of the market equilibrium and compare the decentralized equilibrium allocation with the socially efficient allocation.

To simplify the comparison between individual and socially optimal choices, we reformulate the problem of the firm so far identified with a single job, allowing firms to open many vacancies and employ many workers. Moreover, we also modify the production technology and replace the previous linear production technology with a standard production function with decreasing marginal (p.207) returns of labor. Let N i denote the number of workers of firm i. The production function is then given by F(Ni), with F′(·) > 0 and F″ (·) < 0. The case F″(·) = 0 corresponds to the analysis in the previous sections, while the case of decreasing marginal returns to labor corresponds to our analysis in Chapter 3.

The employment level of a firm varies over time as a result of vacancies that are filled and because of shocks that hit the firm and destroy jobs at rate s. The evolution of Ni is described by the following equation, where we have suppressed the time dependence of the variables to simplify the notation:

N ˙ i = q ( θ ) X i s N i ,
where Xi represents the number of vacancies of a firm and is the control variable of the firm. Each vacancy is transformed into a filled job with instantaneous probability q(θ), which is a function of the aggregate tightness of the labor market. In deciding Xi firms take θ as given, disregarding the effect of their decisions on the aggregate ratio of vacancies to unemployed. More specifically, we assume that the number of firms is sufficiently high to justify the assumption that a single firm takes the level of θ as an exogenous variable.

The problem of the representative firm is therefore

max X i 0 [ F ( N i ) w N i c X i ] e r t d t ,
subject to the law of motion for employment given by (5.58). Moreover, we assume that the firm takes the wage w as given and independent of the number of workers that it employs.

The solution can be found by writing the associated Hamiltonian,

H ( t ) = [ F ( N i ) w N i c X i + λ ( q ( θ ) X i s N i ) ] e r t
(where λ is the Lagrange multiplier associated with the law of motion for N), and by deriving the first-order conditions:
H X i = 0 λ c q ( θ ) ,
H N i = d ( λ ( t ) e r t ) d t F ( N i ) w = ( r + s ) λ λ ˙ ,
lim t e r t λ N i = 0.

Equation (5.61) implies that firms continue to create vacancies until the marginal profits of a job equal the marginal cost of a vacancy (c/q(θ)). This condition holds at any moment in time and is similar to the condition for job creation (5.48) derived in Section 5.4.1. The Lagrange multiplier λ can therefore be interpreted as the marginal value of a filled job for the firm, which we denoted by J in the previous sections. The dynamics of λ are given by (5.62), (p.208) which in turn corresponds to equation (5.49) for J ˙ . Finally, equation (5.63) defines the appropriate transversality condition for the firm's problem.

In what follows we consider only steady-state equilibria. Combining (5.61) and (5.62) and imposing λ ˙ = 0 , we get

F ( N i * ) w = ( r + s ) c q ( θ )
where Ni* denotes the steady-state equilibrium employment level of the firm. The optimal number of vacancies, Xi*, can be derived from constraint (5.58), with Ṅ = 0:
q ( θ ) X i * = s N i * X i * = s q ( θ ) N i *

Hence, if all firms have the same production function and start from the same initial conditions, then each firm will choose the same optimal solution and the ratio of filled jobs to vacancies for each firm will be equal to the aggregate ratio:

X i * N i * = v 1 u .

This completes the characterization of the decentralized equilibrium.

We now proceed with a characterization of the socially efficient solution. For simplicity we normalize the mass of firms to one. X and N therefore denote the stock of vacancies and of filled jobs, at both the aggregate level and the level of an individual firm. Since the relations

θ v u = v L L N = X L N
hold true, aggregate labor market conditions as captured by θ are endogenous in the determination of the socially efficient allocation. The efficient allocation can be found by solving the following maximization problem:
max X 0 [ ( F N ) z N c X ] e r t d t ,
subject to the condition
N ˙ = q ( X L N ) X s N .

The bracketed expression in (5.67) denotes aggregate net output. The first term (F(N)) is equal to the output of employed workers. From this we have to subtract the flow utility of employed workers (zN), and the costs of maintaining the vacancies (cX). The wage rate does not appear in this expression because it is a pure redistribution from firms to workers: in the model considered here, distributional issues are irrelevant for social efficiency. The important point to note is that the effect of the choice of X on the aggregate conditions on the (p.209) labor market is explicitly taken into account: the ratio θ is expressed as X/(LN) and is not taken as given in the maximization of social welfare.

The problem is solved using similar methods as for the case of the problem of individual firms. Constructing the associated Hamiltonian and deriving the first-order conditions for X and N (with μ as the Lagrange multiplier for the dynamic constraint) yields

H X = 0 μ = c q ( θ ) θ + q ( θ )
H N = d ( μ ( t ) e r t ) d t F ( N ) z = ( r + s q ( θ ) θ 2 ) μ μ ˙ .

Explicit consideration of the effects on θ introduces various differences between the above optimality conditions and the first-order conditions of the individual firm, (5.70) and (5.69). First of all, comparing (5.61) with the corresponding condition (5.61) shows that individual firms tend to offer an excessive number of vacant jobs compared with what is socially efficient (recall that q′ < 0). The reason for this discrepancy is that firms disregard the effect of their decisions on the aggregate labor market conditions.

Moreover, from the marginal condition for N (5.70), it follows that the “social” discount rate associated with the marginal value of a filled job μ contains an additional term, −q′(θ)θ2 > 0, which does not appear in the analogous condition for the individual firm (5.62). That is, an increase in the number of employed workers diminishes the probability that the firm will hire additional workers in the future. Equation (5.70) correctly reflects this dynamic aspect of labor demand, which tends to reduce the marginal value of a filled job in a steady-state equilibrium (in which μ ˙ = 0 ). Hence, also from this perspective the decentralized decisions of firms result in an excessive number of vacancies compared with the social optimum. Finally, comparing the left-hand side of equations (5.62) and (5.70) reveals that the individual conditions contain the value of productivity net of the wage w, while in the condition for social efficiency the value of productivity is net of the opportunity cost z. Hence, for the same value of θ, individual firms attribute a lower “dividend” to filled jobs since wz, and firms thus tend to generate an insufficient number of vacancies. This last effect runs in the opposite direction to the two effects discussed above, and this makes a comparison between the two solutions—the individual and the social—interesting. The socially optimal solution may coincide with the corresponding decentralized equilibrium if the wage determination mechanism “internalizes” the externalities that private agents ignore. However, in the model that we have constructed, wages are determined after a firm and a worker meet. Hence, although the wage is perfectly flexible, it cannot perform any allocative function.

(p.210) Nonetheless, we can determine the conditions that the wage determination mechanism needs to satisfy for the decentralized equilibrium to coincide with the efficient solution. For this to occur, the marginal value of a filled job in the social optimum, which is given by (5.69), needs to be equal to the marginal value that the firm and the worker attribute to this job in the decentralized equilibrium. The latter is equal to the value that a firm and a worker attribute to the joint surplus that is created by a match. Since firms continue to offer vacancies until their marginal value is reduced to zero (V = 0), the condition for efficiency of a decentralized equilibrium is

μ = J + E U ,
where E and U, introduced in Section 5.3, denote the value that a worker attributes to the state of employment and unemployment, respectively.

Using (5.69), (5.48), and (5.51) we can rewrite (5.71) as:

c q ( θ ) θ + q ( θ ) = 1 1 β c q ( θ ) ,
where β denotes the relative bargaining strength of the worker. From (5.72), we obtain
β = q ( θ ) θ q ( θ ) [ η ( θ ) 1 ] , β = 1 η ( θ ) ,
where η(θ) and 1 − η(θ) denote the elasticity of the matching probability of a worker p(θ) and the average duration of a vacancy 1/q(θ) with respect to θ. Since β is constant, condition (5.73) can be satisfied only if the matching function has constant returns to scale with respect to its arguments v and u. This condition is satisfied for a matching function of the Cobb–Douglas type
m = m 0 v η u 1 η , 0 < η < 1.

It is easy to verify that (5.74) has the following properties:

p ( θ ) = m 0 θ η , q ( θ ) = m 0 θ η 1 , 1 q ( θ ) = 1 m 0 θ 1 η .

The constant parameter η represents both the elasticity of the number of matches m with respect to the number of vacancies v, and the elasticity of p(θ) with respect to θ, while 1 − - η denotes the elasticity of m with respect to u and also the elasticity of the medium duration of a vacancy, 1/q(θ), with respect to θ.

Returning to efficiency condition (5.73), we can thus deduce that, if the average duration of a vacancy strongly increases with an increase in the number of vacancies (i.e. if 1 − η is relatively high), there is a strong tendency for firms (p.211) to exceed the efficient number of vacancies. Only a relatively high value of β, which implies high wage levels, can counterbalance this effect and induce firms to reduce the number of vacancies. When β = 1 − η, these two opposing tendencies exactly offset each other and the decentralized equilibrium allocation is efficient. For cases in which β ≠ 1 − η, there are two types of inefficiency:

  1. 1. if β < 1 − η firms offer an excessive number of vacancies and the equilibrium unemployment rate is below the socially optimal level;

  2. 2. if β 1 − η wages are excessively high because of the strong bargaining power of workers and this results in an unemployment rate that is above the socially efficient level.

In sum, in the model of the labor market that we have described here we cannot make a priori conclusions about the efficiency of the equilibrium unemployment rate. Given the complex externalities between the actions of firms and workers, the properties of the matching function and the wage determination mechanism are crucial to determine whether the unemployment rate will be above or below the socially efficient level.

This appendix presents a general theoretical structure, based on Cooper and John (1988), which captures the essential elements of the strategic interactions in the models discussed in this chapter. We will discuss the implications of strategic interactions in terms of the multiplicity of equilibria and analyze the welfare properties of these equilibria.

Consider a number I of economic agents (i = 1,…,I), each of which chooses a value for a variable ei∈ [0,E] which represents the agent's “activity level,” with the objective of maximizing her own payoff σ (ei,e ii), where e−i represents (the vector of) activity levels of the other agents and λi is an exogenous parameter which influences the payoff of agent i. Payoff function σ(·) satisfies the properties σii < 0 and σiλ > 0. (This last assumption implies that an increase in λ raises the marginal return of activity for the agent.)

If all other agents choose a level of activity ē, the payoff of agent i can be expressed as σ (ei,ē,λi)≡ V(ei,ē). In this case the optimization problem becomes

max e i V ( e i , e ¯ ) ,
from which we derive
V 1 ( e i * , e ¯ ) = 0
where V 1 denotes the derivative of V with respect to its first argument, ei. First-order condition (5.A2) defines the optimal response of agent i to the activity level of all
                   Coordination and Externalities in Macroeconomics

Fig 5.12. Strategic interactions

(p.212) other agents: ei* = ei*ē). Moreover, using (5.A1), we can also calculate the slope of the reaction curve of agent i:
d e i * d e ¯ = V 12 V 11 m 0 , if V 12 m 0.

By the second-order condition for maximization, we know that V 11 < 0; the sign of the slope is thus determined by the sign of V 12(ei,ē). In case V 12 > 0, we can make a graphical representation of the marginal payoff function V 1(ei,ē) and of the resulting reaction function ei*(ē). The left-hand graph in Figure 5.12 illustrates various functions V 1, corresponding to three different activity levels for the other agents: ē = 0, ē = e, and = E.

Assuming V 1(0,0) > 0 and V 1(E,E) < 0 (points A and B) guarantees the existence of at least one symmetric decentralized equilibrium in which e = ei*(e), and agent i chooses exactly the same level of activity as all other agents (in this case V 1(e,e) = 0 and V 11(e,e) = 0). In Figure 5.12 we illustrate the case in which the reaction has a positive slope, and hence V 12 > 0, and in which there is a unique symmetric equilibrium.

In general, if V 12(ei,ē) > 0 there exists a strategic complementarity between agents: an increase in the activity level of the others increases the marginal return of activity for agent i, who will respond to this by raising her activity level. If, on the other hand, V 12(e i,ē) < 0, then agents' actions are strategic substitutes. In this case agent i chooses a lower activity in response to an increase in the activity level of others (as in the case of a Cournot duopoly situation in which producers choose output levels). In the latter (p.213) case there exists a unique equilibrium, while in the case of strategic complementarity there may be multiple equilibria.

Before analyzing the conditions under which this may occur, and before discussing the role of strategic complementarity or substitutability in determining the characteristics of the equilibrium, we must evaluate the problem from the viewpoint of a social planner who implements a Pareto-efficient equilibrium.

The planner's problem may be expressed as the maximization of a representative agent's welfare with respect to the common strategy (activity level) of all agents: the optimum that we are looking for is therefore the symmetric outcome corresponding to a hypothetical cooperative equilibrium. Formally,

max e V ( e , e ) ,
from which we obtain
V 1 ( e * , e * ) + V 2 ( e * , e * ) = 0.

Comparing this first-order condition49 with the condition that is valid in a symmetric decentralized equilibrium (5.A2), we see that the solutions for e* are different if V 2(e*,e*) ≠ 0. In general, if V 2(e i,ē) > (<)0, there are positive (negative) spillovers. The externalities are therefore defined as the impact of a third agent's activity level on the payoff of an individual.

A number of important implications for different features of the possible equilibria follow from this general formulation.

  1. 1. Efficiency Whenever there are externalities that affect the symmetric decentralized equilibrium, that is when V 2(e,e) ≠ 0, the decentralized equilibrium is inefficient. In particular, with a positive externality (V 2(e,e) < 0), there exists a symmetric cooperative equilibrium characterized by a common activity level e′ < e.

  2. 2. Multiplicity of equilibria As already mentioned, in the case of strategic complementarity (V 12 > 0), an increase in the activity level of the other agents increases the marginal return of activity for agent i, which induces agent i to raise her own activity level. As a result, the reaction function of agents has a positive slope (as in Figure 5.12). Strategic complementarity is a necessary but not a sufficient condition for the existence of multiple (non-cooperative) equilibria. The sufficient condition is that dei*/ > 1 in a symmetric decentralized equilibrium. If this condition is satisfied, we may have the situation depicted in Figure 1.3, in which there exist three symmetric equilibria. Two of these equilibria (with activity levels e 1 and e 3) are stable, since the slope of the reaction curves is less than one at the equilibrium activity levels, while at e 2 the slope of the reaction curve is greater than one. This equilibrium is therefore unstable.

  3. 3. Welfare If there exist multiple equilibria, and if at each activity level there are positive externalities (V 2(ei,ēe) > 0 ∀ ē), then the equilibria can be ranked. Those

                       Coordination and Externalities in Macroeconomics

    Fig 5.13. Multiplicity of equilibria

    (p.214) with a higher activity level are associated with a higher level of welfare. Hence, agents may be in an equilibrium in which their welfare is below the level that may be obtained in other equilibria. However, since agents choose the optimal strategy in each of the equilibria, there is no incentive for agents to change their level of activity. The absence of a mechanism to coordinate the actions of individual agents may thus give rise to a “coordination failure,” in which potential welfare gains are not realized because of a lack of private incentives to raise the activity levels.

    Exercise 52 Show formally that equilibria with a higher ē are associated with a higher level of welfare if V 2(ei,ē) > 0. (Use the total derivative of function V(·) to derive this result.)

  4. 4. Multipliers Strategic complementarity is necessary and sufficient to guarantee that the aggregate response to an exogenous shock exceeds the response at the individual level; in this case the economy exhibits “multiplier” effects. To clarify this last point, which is of particular relevance for Keynesian models, we will consider the simplified case of two agents with payoff functions defined as V 1 ≡ σ1(e 1,e 21) and V 2 ≡ σ2(e 1,e 22), respectively. All the assumptions about these payoff functions remain valid (in particular, V 13 1 ≡ σ13 1 > 0). The reaction curves of the two agents are derived from the following first-order conditions:

    V 1 1 ( e 1 * , e 2 * , λ 1 ) = 0 ,
    V 2 2 ( e 1 * , e 2 * , λ 2 ) = 0.

We now consider a “shock” to the payoff function of agent 1, namely dλ1 > 0, and we derive the effect of this shock on the equilibrium activity levels of the two agents, e 1* and e 2*, and on the aggregate level of activity, e 1 * + e 2 *. Taking the total derivative of the above system of first-order conditions (5.A6) and (5.A7), with dλ2 = 0, and (p.215) dividing the first equation by V 11 1 and the second by V 22 2, we have:

d e 1 * + ( V 12 1 V 11 1 ) d e 2 * + ( V 13 1 V 11 1 ) d λ 1 = 0 , ( V 21 2 V 22 2 ) d e 1 * + d e 2 * = 0.

The terms V 12 1/V 11 1 and V 21 2/V 22 2 represent the slopes, with opposing signs, of the reaction curves of the agents which we denote by ρ (given that the payoff functions are assumed to be identical, the slope of the reaction curves is also the same). The term V 13 1/V 11 1 represents the response (again with oppositing signs) of the optimal equilibrium level of agent 1 to a shock λ1. In particular, keeping e 2 * constant, we have

V 1 1 ( e 1 * , e 2 * , λ 1 ) = 0 e 1 * λ = V 13 1 V 11 1 > 0.

We can thus rewrite the system as follows:

( 1 ρ ρ 1 ) ( d e 1 * d e 2 * ) = ( e 1 * λ 1 * ) d λ 1 ,
which yields the following solution:
d e 1 * d λ 1 = 1 1 ρ 2 e 1 * λ 1
d e 2 * d λ 1 = ρ 1 ρ 2 e 1 * λ 1 = ρ d e 1 * d λ 1 .

Equation (5.A8) gives the total response of agent 1 to a shock λ1. This response can also be expressed as

d e 2 * d λ 1 = e 1 * λ 1 = ρ d e 2 * d λ 1 .

The first term is the “impact” (and thus only partial) response of agent 1 to a shock affecting her payoff function; the second term gives the response of agent 1 that is “induced” by the reaction of the other agent. The condition for the additional induced effect is simply ρ ≠ 0. Moreover, the actual induced effect depends on ρ and de 2 */dλ1, as in (5.A9), where de 2 */dλ1 has the same sign ρ: positive in case of strategic complementarity and negative in case of substitutability. The induced response of agent 1 is therefore always positive.

This leads to a first important conclusion: the interactions between the agents always induce a total (or equilibrium) response that is larger than the impact response. In particular, for each ρ ≠ 0, we have

d e 1 * d λ 1 > e 1 * λ 1 .

(p.216) For the economy as a whole, the effect of the disturbance is given by

d ( e 1 * + e 2 * ) d λ 1 = ( 1 1 ρ 2 + ρ 1 ρ 2 ) e 1 * λ 1 = 1 1 ρ e 1 * λ 1 = ( 1 + ρ ) d e 1 * d λ 1 .

The relative size of the aggregate response compared with the size of the individual response depends on the sign of ρ: if ρ > 0 (and limiting attention to stable equilibria for which ρ < 1), then aggregate response is bigger than individual response. Strategic complementarity is thus a necessary and sufficient condition for Keynesian multiplier effects.

Exercise 53 Determine the type of externality and the nature of the strategic interactions for the simplified case of two agents with payoff function (here expressed for agent 1) V 1(e 1,e 2) = e 1 α e 2 α − e1 (with 0 < 2α < 1). Furthermore, derive the (symmetric) decentralized equilibria and compare these with the cooperative (symmetric) equilibrium.

Review Exercises

Exercise 54 Introduce the following assumptions into the model analyzed in Section 5.1:

  1. (i) The (stochastic) cost of production c has a uniform distribution defined on [0,1], so that G(c) = c for 0 ≤ c ≤ 1.

  2. (ii) The matching probability is equal to b(e) = b · e, with parameter b < 0.

    • (a) Determine the dynamic expressions for e and c* (repeating the derivation in the main text) under the assumption that y < 1.

    • (b) Find the equilibria for this economy and derive the stability properties of all equilibria with a positive activity level.

Exercise 55 Starting from the search model of money analyzed in Section 5.2, suppose that carrying over money from one period to the next now entails a storage cost, c > 0. Under this new assumption,

  1. (a) Derive the expected utility for an agent holding a commodity (VC) and for an agent holding money (VM), and find the equilibria of the economy.

  2. (b) Which of the three equilibria described in the model of Section 5.2 (with c = 0) always exists even with c > 0? Under what condition does a pure monetary equilibrium exist?

Exercise 56 Assume that the flow cost of a vacancy c and the imputed value of free time z in the model of Section 5.3 are now functions of the wage w (instead of being (p.217) exogenous). In particular, assume that the following linear relations hold:

c = c 0 w , z = z 0 w

Determine the effect of an increase in productivity (Δ y > 0) on the steady-state equilibrium.

Exercise 57 Consider a permanent negative productivity shocky < 0) in the matching model of Sections 5.3 and 5.4. The shock is realized at date t1, but is anticipated by the agents from date t 0 < t 1 onwards. Derive the effect of this shock on the steady-state equilibrium and describe the transitional dynamics of u, v, and θ.

Exercise Consider the effect of an aggregate shock in the model of strategic interactions for two agents introduced in Appendix A5. That is, consider a variation in the exogenous terms of the payoff functions, so that dλ1 = dλ2 = dλ >0, and derive the effect of this shock on the individual and aggregate activity level.

Further Reading

The role of externalities between agents that operate in the same market as a source of multiplicity of equilibria is the principal theme in Diamond (1982a). This article develops the economic implications of the multiplicity of equilibria that have a Keynesian spirit. The monograph by Diamond (1984) analyzes this theme in greater depth, while Diamond and Fudenberg (1989) concentrate on the dynamic aspects of the model. Blanchard and Fischer (1989, chapter 9) offer a compact version of the model that we studied in the first section of this chapter. Moreover, after elaborating on the general theoretical structure to analyze the links between strategic interactions, externalities, and multiplicity of equilibria, which we discussed in Appendix A5, Cooper and John (1988) offer an application of Diamond's model. Rupert et al. (2000) survey the literature on search models of money as a medium of exchange and present extensions of the basic Kiyotaki–Wright framework discussed in Section 5.2.

The theory of the decentralized functioning of labor markets, which is based on search externalities and on the process of stochastic matching of workers and firms, reinvestigates a theme that was first developed in the contributions collected in Phelps (1970), namely the process of search and information gathering by workers and its effects on wages. Mortensen (1986) offers an exhaustive review of the contributions in this early strand of literature.

Compared with these early contributions, the theory developed in Section 5.3 and onwards concentrates more on the frictions in the matching process. Pissarides (2000) offers a thorough analysis of this strand of the literature. In this literature the base model is extended to include a specification of aggregate demand, which makes the interest rate endogenous, and allows for growth of the labor force, two elements that are not considered in this chapter. Mortensen and Pissarides (1999a, 1999b) provide (p.218) an up-to-date review of the theoretical contributions and of the relevant empirical evidence.

In addition to the assumption of bilateral bargaining, which we adopted in Section 5.3, Mortensen and Pissarides (1998a) consider a number of alternative assumptions about wage determination. Moreover, Pissarides (1994) explicitly considers the case of on-the-job search which we excluded from our analysis. Pissarides (1987) develops the dynamics of the search model, studying the path of unemployment and vacancies in the different stages of the business cycle. The paper devotes particular attention to the cyclical variations of u and v around their long-run relationship, illustrated here by the dynamics displayed in Figure 5.11. Bertola and Caballero (1994) and Mortensen and Pissarides (1994) extend the structure of the base model to account for an endogenous job separation rate s. In these contributions job destruction is a conscious decision of employers, and it occurs only if a shock reduces the productivity of a match below some endogenously determined level. This induces an increase in the job destruction rate in cyclical downturns, which is coherent with empirical evidence.

The simple Cobb–Douglas formulation for the aggregate matching function with constant returns to scale introduced in Section 5.3 has proved quite useful in interpreting the evidence on unemployment and vacancies. Careful empirical analyses of flows in the (American) labor market can be found in Blanchard and Diamond (1989, 1990), Davis and Haltiwanger (1991, 1992) and Davis, Haltiwanger, and Schuh (1996), while Contini et al. (1995) offer a comparative analysis for the European countries. Cross-country empirical estimates of the Beveridge curve have been used by Nickell et al. (2002) to provide a description of the developments of the matching process over the 1960–99 period in the main OECD economies. They find that the Beveridge curve gradually drifted rightwards in all countries from the 1960s to the mid-1980s. In some countries, such as France and Germany, the shift continued in the same direction in the 1990s, whereas in the UK and the USA the curve shifted back towards its original position. Institutional factors affecting search and matching efficiency are responsible for a relevant part of the Beveridge curve shifts. The Beveridge curve for the Euro area in the 1980s and 1990s is analysed in European Central Bank (2002). Both counter-clockwise cyclical swings around the curve of the type discussed in Section 5.4 and shifts of the unemployment–vacancies relation occurred in this period. For example, over 1990–3 unemployment rose and the vacancy rate declined, reflecting the influence of cyclical factors; from 1994 to 1997 the unemployment rate was quite stable in the face of a rising vacancy rate, a shift of the Euro area Beveridge curve that is attributable to structural factors.

Not only empirically, but also theoretically, the structure of the labor force, the geographical dispersion of unemployed workers and vacant jobs, and the relevance of long-term unemployment determine the efficiency of a labor market's matching process. Petrongolo and Pissarides (2001) discuss the theoretical foundations of the matching function and provide an up-to-date survey of the empirical estimates for several countries, and of recent contributions focused on various factors influencing the matching rate.

(p.219) The analysis of the efficiency of decentralized equilibrium in search models is first developed in Diamond (1982b) and Hosios (1990), who derive the efficiency conditions obtained in Section 5.5; it is also discussed in Pissarides (2000). In contrast, in a classic paper Lucas and Prescott (1974) develop a competitive search model where the decentralized equilibrium is efficient.

References

Bibliography references:

Bertola, G., and R. J. Caballero (1994) ‘Cross-Sectional Efficiency and Labour Hoarding in a Matching Model of Unemployment,‘ Review of Economic Studies, 61, 435–456.

Blanchard, O. J., and P. Diamond (1989) ‘The Beveridge Curve,‘ Brookings Papers on Economic Activity, no. 1, 1–60.

—— —— (1990) ‘The Aggregate Matching Function,‘ in P. Diamond (ed.), Growth, Productivity, Unemployment, Cambridge, Mass.: MIT Press, 159–201.

—— and S. Fischer (1989) Lectures on Macroeconomics, Cambridge, Mass.: MIT Press.

Contini, B., L. Pacelli, M. Filippi, G. Lioni, and R. Revelli (1995) A Study of Job Creation and Job Destruction in Europe, Brussels: Commission of the European Communities.

Cooper, R., and A. John (1988) ‘Coordinating Coordination Failures in Keynesian Models,‘ Quarterly Journal of Economics, 103, 441–463.

Davis, S., and J. Haltiwanger (1991) ‘Wage Dispersion between and within US Manufacturing Plants, 1963–86,‘ Brookings Papers on Economic Activity, no. 1, 115–200.

—— —— (1992) ‘Gross Job Creation, Gross Job Destruction and Employment Reallocation,‘ Quarterly Journal of Economics, 107, 819–864.

—— —— and S. Schuh (1996) Job Creation and Destruction, Cambridge, Mass.: MIT Press.

Diamond, P. (1982a) ‘Aggregate Demand Management in Search Equilibrium,‘ Journal of Political Economy, 90, 881–894.

—— (1982b) ‘Wage Determination and Efficiency in Search Equilibrium,‘ Review of Economic Studies, 49, 227–247.

—— (1984) A Search-Equilibrium Approach to the Micro Foundations of Macroeconomics, Cambridge, Mass.: MIT Press.

—– and D. Fudenberg (1989) ‘Rational Expectations Business Cycles in Search Equilibrium,‘ Journal of Political Economy, 97, 606–619.

European Central Bank (2002) ‘Labour Market Mismatches in Euro Area Countries,‘ Frankfurt: European Central Bank.

Hosios, A. J. (1990) ‘On the Efficiency of Matching and Related Models of Search and Unemployment,‘ Review of Economic Studies, 57, 279–298.

Kiyotaki, N., and R. Wright (1993) ‘A Search-Theoretic Approach to Monetary Economics,‘ American Economic Review, 83, 63–77.

Lucas, R. E., and E. C. Prescott (1974) ‘Equilibrium Search and Unemployment,‘ Journal of Economic Theory, 7, 188–209.

(p.220) Mortensen, D. T. (1986) ‘Job Search and Labor Market Analysis,‘ in O. Ashenfelter and R. Layard (eds.), Handbook of Labor Economics, Amsterdam: North-Holland.

—— and C. A. Pissarides (1994) ‘Job Creation and Job Destruction in the Theory of Unemployment,‘ Review of Economic Studies, 61, 397–415.

—— —— (1999a) ‘New Developments in Models of Search in the Labor Market,‘ in O. Ashenfelter and D. Card (eds.), Handbook of Labor Economics, vol. 3, Amsterdam: North-Holland.

—— —— (1999b) ‘Job Reallocation, Employment Fluctuations and Unemployment,‘ in J. B. Taylor and M. Woodford (eds.), Handbook of Macroeconomics, Amsterdam: North-Holland.

Nickell S., L. Nunziata, W. Ochel, and G. Quintini (2002) ‘The Beveridge Curve, Unemployment and Wages in the OECD from the 1960s to the 1990s,‘ Centre for Economic Performance Discussion Paper 502; forthcoming in P. Aghion, R. Frydman, J. Stiglitz, and M. Woodford (eds.), Knowledge, Information and Expectations in Modern Macroeconomics: In Honor of Edmund S. Phelps, Princeton: Princeton University Press.

Petrongolo B., and C. A. Pissarides (2001) ‘Looking into the Black Box: A Survey of the Matching Function,‘ Journal of Economic Literature, 39, 390–431.

Phelps, E. S. (ed.) (1970) Macroeconomic Foundations of Employment and Inflation Theory, New York: W. W. Norton.

Pissarides, C. A. (1987) ‘Search, Wage Bargains and Cycles,‘ Review of Economic Studies, 54, 473–483.

—— (1994) ‘Search Unemployment and On-the-Job Search,‘ Review of Economic Studies, 61, 457–475.

—— (2000) Equilibrium Unemployment Theory, 2nd edn. Cambridge, Mass.: MIT Press.

Rupert P., M. Schindler, A. Shevchenko, and R. Wright (2000) ‘The Search-Theoretic Approach to Monetary Economics: A Primer,‘ Federal Reserve Bank of Cleveland Economic Review, 36(4), 10–28.

Notes:

(39) The stochastic process therefore has the Markov property and is completely memoryless. In a more general model, a may be assumed to be variable. The function a(t) is known as the hazard function.

(40) Since we limit attention to steady-state outcomes in which e is constant, E and U are also constant over time. As a result, there is no difference between the values at the beginning and at the end of the time interval.

(41) In general, the definition of an integral implies

d d x a ( x ) b ( x ) f ( z ; x ) d z = a ( x ) b ( x ) f ( z ; x ) x d z + b ( x ) f ( b ( x ) ) a ( x ) f ( a ( x ) )
(Leibnitz's rule). Intuitively, the area below the curve of f(·) and between the points a(·) and b(·) is equal to the integral of the derivative of f(·) over the interval. Moreover, an increase in the upper limit increases this area in proportion to f(b(x)), while an increase in the lower limit decreases it in proportion to f(a(x)).

(42) The introduction of an arbitrarily small transaction cost paid by the receiver can rule out the possibility that an agent agrees to receive in a trade a commodity he cannot consume.

(43) Empirical studies of the matching technology confirm that the assumption of constant returns to scale is realistic (see Blanchard and Diamond, 1989, 1990, for estimates for the USA).

(44) As in the previous section, the matching process is modeled as a Poisson process. The probability that an unemployed worker does not find employment within a time interval dt is thus given by e p(θ)dt. For a small time interval, this probability can be approximated by 1−p(θ)dt. Similarly, the probability that the worker does find employment is 1−e p(θ)dt, which can be approximated by p(θ)dt.

(45) To complete the description of the functions p and q, we define the elasticity of p with respect to θ as η(θ). We thus have: η(θ)=p′(θ)θ/p(θ). From the assumption of constant returns to scale, we know that 0≤η(θ)≤1. Moreover, the elasticity of q with respect to θ is equal to η(θ)−1.

(46) To obtain job creation and destruction “rates,” we may divide the flows into and out of employment by the total number of employed workers, (1−u)L. The rate of destruction is simply equal to s, while the rate of job creation is given by p(θ)[u/(1−u)].

(47) Note that this derivative is computed at a steady-state equilibrium point (on the θ ˙ = 0 locus). Hence, we may use (5.34), and replace yw with (r+s)(cθ/p(θ)) to obtain the expression in the text.

(48) Formally, we can determine the saddlepoint nature of the equilibrium by evaluating the linearized system (5.47) and (5.57) around the steady-state equilibrium point ( u ¯ , θ ¯ ) , yielding

( ) = ( ( s + P ( θ ¯ ) ) u ¯ p ( θ ¯ ) 0 ( r + s ) + p ( θ ¯ ) 1 η β ) ( u u ¯ θ θ ¯ ) .
The pattern of signs in the matrix is [ 0 + ] Thus, the determinant is negative, confirming that the equilibrium is a saddlepoint.

(49) The second-order condition that we assume to be satisfied is given by V11(e*,e*)+2V12(e*,e*)+V22(e*,e*)<0. Furthermore, in order to ensure the existence of a cooperative equilibrium, we assume that V1(0,0)+V2(0,0)>0, V1(E,E)+V2(E,E)<0, which is analogous to the restrictions imposed in the decentralized optimization above.