Reservation Wage

8 Denote by U, the net present value to the worker from unemployed search,

9 where . The last equality follows from integration by parts and the second inequality reflects that any offer V < U is rejected.

10 The wage associated with a job that offers value V satisfies,

11 which by integration by parts can be written as,

12 Implicit in these expressions is the implication from the constant wage path in a job that the forward looking value of the job is constant at any tenure level. It is seen that the match wage is monotonically increasing in the utility promise of the job. Specifically,

13 The lowest acceptable wage, w(U), is also referred to as the reservation wage, w_R. It is given by,

14 It reflects that for a job to match the value of unemployed search, it must first of all provide the unemployed income stream. Second, if unemployed search is technologically superior to employed search, λ₀ > λ₁, it must also compensate the employed worker for the relative loss in search technology associated with employment. If employed search is technologically favored, λ₁ > λ₀, the reservation wage is reduced relative to b.

Steady State

15 Without loss of generality set the lower bound on the support of the offer distribution, V = U, and interpret λ₀ and λ₁ as the arrival rate of possibly acceptable offers and let F(·) be their distribution. Restrict attention to steady state. Denote by u the mass of unemployed agents. Let G(V) be the mass of agents employed with utility promise V or less. They are characterized by the steady state conditions,

16 Equation 2 states that u must be such that the outflow from the mass of unemployed equals the inflow. Equation 3 is again an equalization of inflows and outflows. In this case, the relevant group is that of employed workers employed at utility promise V or less, which has mass (1 – u)G(V). The right hand side states that the outflow from the group consists of job destruction which happens at rate δ, and when matches meet a better offer that results in the worker leaving the group, which happens at rate λ₁(1 – F(V)). The inflow into the group comes from unemployed workers who receive offers at or less than V.

17 It follows that in steady state the unemployment rate is given by u = δ/(δ + λ₀). Furthermore, the match value distribution is,

18 As λ₀ → ∞ unemployment limits to zero. As λ₁ → ∞ the mach distribution limits to a mass point at .

Only Rank Matters

19 A somewhat striking feature of this basic model of worker allocation is the fact that the basic mobility predictions are independent of the shape of the offer distribution. A notable implication is that the relative frequencies of job-to-job moves compared to moves out of or into unemployment are entirely independendent of the magnitude of the gains to upward movement in value hierarchy. To make this point explicit, define a job by its rank in the value distribution, R = F(V). By definition, R is distributed uniformly in the offer distribution. The rate at which an employed worker in a rank R job makes a job-to-job move is λ₁(1 – R). Denote by G_R(R) = δR/[δ + λ₁(1 – R)] the steady state match rank distribution. The aggregate job-to-job transition rate in steady state is then,

20 Some algebra yields,

21 which is purely a function of the employed offer arrival rate and the job destruction rate. Hornstein, Krusell and Violante [2011] features a version of this result. The rate at which workers move out of unemployment is simply λ₀ and the rate at which they are laid off into unemployment is δ. These three aggregate mobility statistics singlehandedly determine the mobility parameters (δ, λ₀, λ₁). US aggregate labor market data applied to this model results in λ₀/λ₁ estimates around 4-5. This is a rather stark and not easily understood difference. This is however more than anything just a reflection of the simplified mobility setup. In particular, it assumes that employed workers meet outside opportunities at a rate that is independent of the gains from them. As a result, employed workers keep on advancing up the value hierarchy according to λ₁ even as the gains to further advancement diminish. In order to understand the observed job-to-job movement rates, the model must reduce λ₁ accordingly. The result reflects a deeper issue in the model, that there is a disconnect between worker allocation to jobs and the relative value of those jobs. Workers cannot respond to incentives to reallocate beyond the simple acceptance and rejection of exogenously arriving job opportunities.

22 The search literature contains two prominent extensions where searchers respond to the magnitude of the gains to search: search intensity and directed search. In these models, equilibrium allocations are sensitive to the cardinality of match surplus. [1] In the following section, I detail the simple extension of the current framework with search intensity choice. Shimer [2005] provides an analysis of allocation in a directed search setting with a particular focus on the competitive search decentralization as in Moen [1997] and Shimer [1996]. Recent contributions such as Lentz, Maibom and Moen [2017], Sorkin [2018], and Lentz, Piyapromdee and Robin [2023] emphasize a multinomial Logit structure in search and/or job acceptance that also effectively introduce cardinality into the allocation mechanism.

Search Intensity

23 Extend the model framework with a search intensity choice so that the instantenous utility of a wage w and search intensity choice s is u(w, s) = w – c(s), where c(s) is increasing and convex. For now, assume c(0) = cʹ(0) = 0. A search intensity choice s delivers a Poisson arrival rate of meetings λs, where λ is the per unit of search arrival rate of offers. In the following, adopt the simplicification that λ₀ = λ₁ = λ. With this, the value of unemployed search is,

24 Maintain the assumption that employment contracts specify flat wage profiles. Furthermore, to be concrete assume that search intensity cannot be contracted upon and that the worker makes a search intensity choice at each instant. By implication, the firm delivers V through a wage w (V) that solves the equation,

25 The worker’s search choice in a job that delivers value V, s(V), solves the first order condition,

26 Note that , that is, at the highest value match, the gains to search are zero because no further advancement is possible, and therefore the worker chooses not to search. Differentiation by V yields that the search choice is decreasing in V,

27 It is immediately seen that the arrival rate of outside offers in a particular state, λs(V) now depends on the gains associated with upward mobility.

28 Denote by s₀ the optimal search choice when unemployed. The steady state equations for u and G(V) are in this case,

29 The steady state condition on G(V) is a first order Volterra integral equation which can be rewritten as,

30 With G(U) = 0, it numerically solves straightforwardly by forward recursion.

31 As an empirical model of labor mobility, relative to that of the second section, the endogenous search model has in addition both search cost function curvature and shape of offer distribution as variables to fit mobility patterns. Simple moments such as aggretate job-to-job, layoff, and job finding rates are more easily understood under a maintained assumption of equally efficient search whether employed or unemployed. Christensen et al. [2005] estimate the model and find a significant role for endogenous search intensity in the explanation of mobility patterns and equilibrium wage distributions.

Contractable Search Intensity

32 Foreshadowing efficiency discussions in the coming sections, employment contract restrictions in the above imply that the worker’s decision to leave one firm for another does not internalize the loss to the firm that is left behind. This manifests itself in a lower search intensity if it can be contracted upon. For this, consider explicitly the firm’s mechanism design problem. Again, restrict employment contracts to be tenure invariant, but allow a contract to specify both a wage and a search intensity. Assume a match produces output value p > b. The firm maximizes the net present value of future profits associated with a match subject to the promise to deliver value V,

33 Including the possibility of corner solutions, the optimal search intensity satisfies,

34 which implies a lower search intensity for a given value promise V than in the non-contractable search intensity case. The right hand side now reflects the joint gains to the current match associated with outside offers to the worker. The associated wage, w(V), will as a result be higher to compensate the worker for the lower contribution of the term in the worker’s value of the match.

Simple Unemployed Sequential Search

37 Here, I demonstrate results in Diamond [1971] in a simple unemployed sequential search setting. For this purpose, take the model in the second section and set λ₁ = 0. To this, add a firm side with a mass of vacant jobs. All jobs are identical with a flow output of p > b once matched with a worker. Denote by h(V) the conditional probability that given a meeting with a worker, the worker accepts the vacancy’s offer of V. It is in this case simple,

38 Denote by J(V) the firm’s net present value of future profits given a utility promise of V and the maintained assumption that employment contracts specify flat wage profiles.

39 Given the absence of on-the-job search, delivering value promise V implies a wage w(V),

40 It follows that wʹ(V) = r + δ > 0. The firm’s valuation of a match

41 is positive as long as w(V) ≤ p. The firm’s match value is decreasing in the worker value promise, Jʹ(V) = –1.

42 Assume value posting, meaning along with the vacancy, the firm also posts and commits to a value promise should the worker accept to match given a meeting. The optimal offer choice maximizes h(V)J(V). This is zero for any V < U and decreasing in V for any V ≥ U. Thus, as long as J(U) ≥ 0, all vacancies post U, and the offer distribution is a single mass point at U. This implies that the value of unemployed search to the worker is,

43 Thus, the equilibrium is characterized by a single wage w(U) = rU = b. Firm match value is J(U) = (p – b)/(r + δ).

44 The somewhat striking feature of this equilibrium is that this remains the case anywhere on a sequence as λ₀ limits to infinity. So, even as the economy is limiting to instant meetings with firms, workers are not paid their marginal product, p, but rather their reservation wage, w(U) = b. The result is sometimes referred to as the Diamond paradox—even with many firms and search costs limiting to zero, the price in the labor market does not converge to the competitive price. The result seems paradoxical in that the limit of the sequence where λ₀ goes to infinity is taken to be the competitive environment where workers face all vacancies simultaneously, and the competitive wage is p. However, anywhere on the sequence, the equilibrium has shifted entirely to the other side, giving full rent extraction to the firms at a wage of b.

45 There is, though, a qualitative difference between meeting vacancies sequentially, however infinitisimal the spacing, and that of meeting them simultaneously. Indeed, the key to understanding why pure equilibrium frictional wage dispersion arises in Burdett and Judd [1983] and in Burdett and Mortensen [1998] is that both environments create the occasional simultaneous comparison of two offers. In these environments as frictions limit to zero, the simultaneous meetings become more and more frequent delivering a sequence that smoothly limits to the competitive environment.

Pure Frictional Equilibrium Value Dispersion

46 Here, I am presenting the homogenous workers and firm case in Burdett and Mortensen [1998]. Take the setting of the previous subsection and reintroduce on-the-job search, λ₁ ≥ 0. Furthermore, simplify things and equalize search efficiency across employment states, λ₀ = λ₁ = λ. Workers are homogenous. So are vacancies. Specifically, all matches produce p > b. When a firm posts a vacancy they attach a value promise, V, to it. The firm chooses the value promise to maximize the value of the vacancy which is proportional to h(V)J(V), where h(V) is the acceptance probability conditional on having met a worker.

47 The introduction of on-the-job search introduces modifications to both the acceptance probability and the firm’s value of a match. Given the assumption of equally efficient search on and off the job, random search implies that it is equally likely to meet any worker in the economy. Thus, the probability that a randomly contacted worker is unemployed is simply the unemployment rate, u. As long as V ≥ U, such a worker accepts the offer.

48 The meeting with an employed worker is a different proposition. The employed worker will be simultaneously comparing the value of the existing job with that of the vacancy. The vacancy is accepted if it promises greater utility. G(V) is the probability that an employed worker is in a match that delivers value V or less. With this, the probability that a randomly contacted worker accepts the vacancy’s value promise of V is,

49 The match value to the firm from a match that delivers V to the worker is,

50 where w(V) is given by Equation 1, which implies,

51 Denote by V^p, J(V^p) = 0, which implies w(V^p) = p. This is the highest utility promise an employer would conceivably be willing to make to match with a worker.

52 The analysis imposes steady state given by Equations 2 and 3. This implies,

53 Any vacancy chooses to post V such that V ∈ argmax_Vʹ h(Vʹ)J(Vʹ). Since all vacancies are the same, in equilibrium it must then be that for all V₁,V₂ ∈ supp(F),

54 With a nod back to Diamond [1971], one question is whether the equilibrium offer distribution has genuine dispersion, that is, if there exist V₁,V₂ ∈ supp(F) such that V₁ ≠ V₂. Remember in the previous section, the offer distribution is a single mass point at U. It is a significant result, then, that with on-the-job search, if an equilibrium exists, the offer distribution cannot have any mass points. The argument goes as follows.

55 The question of mass points raises the issue of indifference events. To be specific and without loss of generality, assume that in case of indifference, a worker accepts the offer from a vacancy over the existing state, be it unemployment or an existing match.

56 In equilibrium, no firm will post a value offer V < U as it is surely rejected. Second, no firm will post value V > V^p as J(V) < 0 for any such offer. Denote by V = inf supp(F) and the lower and upper bounds of the offer distribution, respectively. It then must be that U ≤ V ≤ V ≤ V^P.

57 We can sharpen the characterization of these bounds a bit. Immediately establish that U < V^p. Assume to the contrary that U = V^p and consequently that the offer distribution is a point at V = V^p. But by Equation 1 this implies that w(V^p) = b contradicting the definition that J(V^p) = 0.

58 Thus, J(U) > 0. It then follows that V = U. Again, supppose to the contrary, V > U. It follows that J(U) > J(V), and since h(U) = h(V) = u, it follows that offering V = U dominates offering V = V, that is h(U)J(U) > h(V)J(V), which contradicts that V is an equilibrium offer. Hence, it must be that any V ∈ supp(F) has h(V)J(V) = h(U)J(U) > 0. And from this follows immediately that .

59 Next, consider the claim that the equilibrium offer distribution cannot have any mass points. Proof by contradiction: Assume to the contrary that there is some mass point . That is . The discontinuity in F results in a discontinuous jump down in J(·) in the point V coming from the right. As the firm offers V + ε, for some ε > 0, however small, the worker quit rate λ(1 – F(V + ε)) jumps down whereas the wage w(V) is smoothly increasing in V to the right. Thus, it must be that there exists some ε > 0 such that h(V + ε)J(V + ε) > h(V)J(V), contradicting that V is in the equilibrium support.

60 The broad idea of the proof is that if there is a mass point in the equilibrium distribution, then that mass point offer must be dominated by an ε-deviation up because the cost of delivering that extra value is smoothly increasing whereas the recruitment and retention gain jumps discretely. The assumptions about how workers behave in case of indifference dictate whether the argument is in terms of the hiring probability jumping or whether the retention rate jumps. Either way, the mass point offer turns out to be dominated and contradicts it as an equilibrium offering. Thus, the analysis has ruled out an equilibrium with any single market wage. Be it the competitive case or the reservation wage.

61 The equilibrium offer distribution also cannot have “gaps,” that is, for some pair V₁,V₂ ∈ supp(F) where without loss of generality V₁ < V₂ we have that F(V₁) = F(V₂). If such a gap exists one immediately obtains h(V₁)J(V₁) > h(V₂)J(V₂) because h(V₁) = h(V₂). This contradicts that V₂ is in the equilibrium support.

62 Thus, if an equilibrium offer distribution, F(V), exists it must display value dispersion. In this market equilibrium with perfectly identical workers and perfectly identical firms, the transaction price of labor nevertheless varies. Some workers are paid more than others for no other reason than simple chance. Burdett and Mortensen [1998] establish existence and uniqueness, and with that establish that in a frictional equilibrium, the law of one price need not hold. In fact, in this equilibrium it cannot hold.

63 The equilibrium offer distribution is the solution to the set of equations that for any ,

64 Numerically, this is a simple forward recursion problem. The assumption that r = 0, is helpful in finding an analytical solution to F(·).

65 As an empirical model, the equilibrium offer distribution in the pure dispersion case suffers from the problem that the density distribution of wages is increasing whereas the empirical distribution is roughly log normal looking. Thus, a serious empirical model of the labor market must consider significant extensions to the argument. The next section introduces amenities into the framework before the text moves on to heterogeneity across agents.

66 The equilibrium value distribution varies with λ/δ so that the model contains both the competitive and Diamond outcomes as special cases, for λ/δ → ∞ and λ/δ → 0, respectively. In both limits the offer distribution limits to a mass point, . For λ/δ → 0 (extreme friction), the limit is and in the other extreme with no friction, λ/δ → ∞, the limit is .

67 As a little aside, Burdett and Mortensen [1998] solve for the equilibrium analytically for the case where r = 0. In this case, the value offer distribution is uniform. Adopt the change of variable X = V – U, the equilibrium condition can be written as,

68 Differentiation with respect to X yields,

69 This delivers the result that the offer value distribution is uniform. The upper value bound is given by,

70 And so for ,

71 The wage is given by,

72 This has a max wage of,

73 Since w(X) is a concave function in X, it means the offer distribution in terms of wages has an increasing density. That is denote the wage offer distribution by H(w(X + U)) = F(X + U). Thus,

74 Stating in terms of w gives,

75 As can be seen, the wage offer density is increasing in the wage.

Value Posting with Amenities

76 So far, it has been assumed that firms can deliver value to workers by means of wages, only. In this section, I extend the environment of the previous section to include a non-wage amenity flow, produced by the firm and enjoyed by the worker, as part of the employment relationship. The treatment is based on Hwang, Mortensen and Reed [1998]. I maintain agent homogeneity and postpone the discussion of Roy model and compensating differentials considerations for the time being.

77 I assume that wages and amenities are normal goods for the worker and I impose a simple constant marginal cost to the firm from the delivery of an additional unit of the amenity. In this environment, frictional value dispersion implies a positive covariance between wages and amenities across jobs. The predictions of the theory of compensating differentials are in the opposite direction. If they are included in the model through agent heterogeneity, it does indeed become ambiguous about its predictions concerning the relationship between wages and amenities.

78 Assume the worker’s flow utility is u(a, w) = a^αw^β, where a is the flow amenity in the match and w is the wage. To be concrete assume earnings are turned into a consumption good, one-for-one. Assume that the net output of a match with amenity level a is p – a, and that the amenity level is under the control of the firm same as the wage. As a normalization, assume unemployment has income-amenity flow (b, a) = (0, 0). With this, the flow value of unemployed search is,

79 It is again assumed that employment contracts are constrained to be tenure invariant. So, a firm sets a wage-amenity combination so as to deliver a given utility promise, V. The firm does so in order to maximize the net present value of future profits from the match.

80 This immediately yields that an optimally designed employment contract has,

81 Substitute into the promise keeping constraint to obtain,

82 It is seen that a higher utility promise employment contract delivers on the promise by both higher wages and higher amenities.

83 The value posting equilibrium follows exactly the same construction as in the previous subsection using a firm match value,

84 where,

85 This compares to a cost function in the previous subsection of delivering utility promise V that could be written as,

86 for b = 0. Thus, the value posting equilibrium with amenities amounts to a minor modification of the firm’s cost function and I will not repeat it here.

87 Equilibrium features an equilibrium offer distribution F(·) with support . As worker advance up the value hierarchy they move from positions with low wages and low amenities toward positions that have both higher wages and higher amenities. That is, the cross section described by the steady state value distribution, G(V), wages and amenities are positively correlated. It reflects that worker preferences are such that combining wages and amenities such that is the cost efficient way of delivering value to the worker.

Value Posting with Tenure Conditional Contracts

88 This section considers the impact of allowing a broader class of employment contracts. The greater design flexibility allows the firm to address the joint inefficiency in the worker’s quit decision that was also touched upon in previous subsection “Contractable Search Intensity.” The section emphasizes the Burdett and Coles [2003] contribution showing that firms will want to design backloaded employment contracts to alleviate the joint inefficiency.

89 Another take away from the results in this section is the demonstrated sensitivity of the pure value dispersion result to employment contract assumptions. The results so far emphasize that value dispersion is a result of an environment where firms believe that there is a strictly positive probability of a direct pairwise comparison of offers and that the probability is less than one. The presence of on-the-job search is the key in Burdett and Mortensen [1998]. This section adds that the result is also sensitive to the kinds of contracts that can be written.

90 In service of this point, before tackling tenure variant contracts, consider the employment contract design where the contract can specify which offers the worker can accept. Specifically, denote by A(V), the value threshold such that outside offers above A(V) are accepted and offers below are rejected. In the environment where acceptance cannot be contracted upon, we have had that A(V) = V, reflecting the worker’s individually optimal acceptance decision.

91 Maintain for now that the contract must specify a constant pair, (w, A), so as to deliver value promise V to the worker,

92 where the promise keeping constraint uses

93 As a useful point of reference, define by M the most value a firm is willing to promise to the worker, J(M) = 0. It must involve w(M) = p and A(M) = M. Maintain that b = 0. It follows from the promise keeping constraint and definition of U, that M satisfies,

94 Turning to the design problem for a given value promise, V, the first order conditions deliver that,

95 The condition that A(V) = V + J(V) says that it is jointly efficient for the worker to accept any outside offer that fully compensates the current match for its destruction. This is an important side point that will be of relevance when the analysis moves to consider labor market equilibria where contracts are subject to renegotiation.

96 Using the definition of J(V), it follows that,

97 Therefore, it follows that J(V) + V = M.

98 In value posting equilibrium with homogenous agents, it must be that . Hence, for any V,

99 which means that the contract specifies that the worker is to reject any offer and will be paid a wage that compensates for the absence of outside offer option value. But, and here is the kicker, we are now back to an equilibrium that behaves like simple unemployed sequential search. And this equilibrium is characterized by an offer distribution that is a single mass point at U. Thus, meaning that rU = 0 and all wages in equilibrium are w(U) = 0. We are back to Diamond.

100 I present the previous example, not for empirical relevance, since it is characterized by the same lack of freedom of movement as that of slavery. [2] Rather, it is in the absence of this kind of mechanism that backloading of employment contracts obtain their relevance.

101 In an extreme form of backloading, the employer can present the risk neutral worker with a contract that “sells” the job to her. If the firm has promised a value V to the worker, in return for an initial value transfer from the worker to the firm of M – V, the worker will receive a wage of w = p from that point on. Therefore, all employed workers are employed at value M. No firm is willing to offer more than M and so it is understood in equilibrium that only unemployed workers will possibly accept offers. And so, again Diamond arises and the equilibrium is an offer distribution with a single mass point at U = 0. Thus, allowing wage contracts to be backloaded in combination with risk neutral agents eliminates pure wage dispersion again.

102 Enter risk averse agents. This forces the backloading to be smoother than the extreme where the worker buys the job up front. And this is the setup of Burdett and Coles [2003]. Assume agents take utility from wages according to flow utility u(w) where u(·) is increasing and concave. Normalize utility so that the unemployed income flow b is such that u(b) = 0. Assume agents cannot save or borrow. Employment contracts specify tenure conditional wage paths. The firm’s mechanism problem can now be written as,

103 where is the time rate of change in the utility promise to the worker. [3] The first order conditions for the problem imply,

104 By the envelope condition, differentiation of J(V) yields,

105 where γ is the Lagrange multiplier on the promise keeping constraint. Using the first order condition and some manipulation yields,

106 with strict inequality for . Thus, the optimal employment contract involves increasing wage and value promises over tenure. A firm may offer a particular value V at the outset of match or a match may arrive at that value at some point of tenure. From that point forward two such matches evolve identically. Burdett and Coles [2003] refer to this as a base line value-tenure profile. [4]

107 The steady state match distribution is characterized by,

108 The steady state hiring probability h(V) = u + (1 – u)G(V) therefore satisfies,

109 The equilibrium condition on F(·) remains that for any V ∈ supp(F), it must be that,

110 The equilibrium remains fully dispersed as long as u(·) is strictly concave.

111 The maximum utility promise is of particular use in the current setting because by Equation 8 and the intuition that at full rent extraction, the worker is now fully internalizing the cost of a quit and there is no longer a joint inefficiency in the contracting relationship. By this, . This gives us another condition for the equilibrium condition on F, that is,

112 While the wage remains an unknown, it nevertheless provides somewhat firmer ground, unlike J(U) which involves . Burdett and Coles [2003] adopt the simplification of r = 0 to obtain analytical expressions for much of the characterization of the equilibrium.

113 The backloading result in Burdett and Coles [2003] is in an immediate sense significant as it demonstrates a reason for why wages increase in tenure that is separate from standard arguments like human capital accumulation or learning. It informs how we may draw inference from the empirical relationship between tenure and wages.

114 The results also demonstrate that pure frictional dispersion is not merely an outcome of an environment where it is costly for agents to meet. Frictions must also extend to limitation to how firms and workers can write contracts to deal with existing joint inefficiencies. The basic inefficiency arises from the fact that frictional dispersion in Burdett and Mortensen [1998] relies on the presence of on-the-job search where the switching decision does not internalize the destruction of the current match. But there is no social value to worker reallocation in the homogenous agents environment. If reallocation is at all costly, contracting mechanisms that pull agents toward efficiency will tend to eliminate worker reallocation and with it, on-the-job search.

115 That said, on-the-job search is not essential. It is not how Burdett and Judd [1983] demonstrate pure frictional dispersion. They simply posit an environment where sometimes searching agents will receive two offers simultaneously. [5] The on-the-job search in Burdett and Mortensen [1998] is a particular argument for how it may come to pass that workers on occasion evaluate two offers simultaneously in a continuous time setting.

116 In the next section, the analysis moves to consider firm heterogeneity and how it is reflected in mobility and wage outcomes.

Value Posting with Firm Heterogeneity

117 The standard competitive market framework for wage determination states that a worker is paid her marginal productivity. Thus, the prominent role of human capital based theories of wage determination and differences in wages across workers. This is well explored elsewhere. This section is concerned with how firm side heterogeneity manifests itself in wages. In addition, the productive heterogeneity also serves an important ingredient to the study of allocation and reallocation since it introduces a social value to reallocation.

118 Assume vacancies differ in the productivity with which they employ a worker. Specifically, assume that the productivity of a given match is a random variable , where Γ is the CDF. Normalize the support of Γ to be unit interval [0, 1] and set b = 0. Maintain the assumption that λ₀ = λ₁ = λ. There is a mass m of firms and a unit mass of workers. A firm is characterized by its hiring process that produces matches with productivity p.

119 The equilibrium value distribution F(·) remains the distribution of value promises across the vacancies in equilibrium. A type p firm posts a value promise so as to maximize h(V)J(V | p) where J(V | p) is the value of match to a type p firm given a value promise V to the worker. As in previous subsection “Pure Frictional Equilibrium Value Dispersion,” assume that employment contracts specify a flat wage. That wage continues to be given by Equation 1. The firm’s value of a match is,

120 In equilibrium it must be that vacancy offer value is increasing in firm type. The proof is as follows. Consider two firm types p₁ < p₂. Furthermore, define V_i ∈ arg max_Vh(V)J(V | p_i) as equilibrium strategy for type i, i = 1, 2. It follows that,

121 where the first and third inequalities follow by definition of V_i and the second inequality follows by J(V | p) strictly increasing in p. First of all, this says that it must be that equilibrium vacancy value must be increasing in firm type. The inequalites also imply that,

122 Since the expression is monotonicaly increasing V it immediately follows that V₂ ≥ V_1.

123 For this continuous type case, Burdett and Mortensen [1998] demonstrate that the equilibrium is characterized by a one-to-one mapping between firm type p and the posted value, V(p). With discrete types, the equilibrium offer distribution support is the union of adjoining value intervals assigned to each firm type, where a particular type is indifferent between any value within its assigned equilibrium value interval.

124 Thus, the equilibrium offer distribution is F(V(p)) = Γ(p) and by implication, f(V) = γ(p)/Vʹ(p). The steady state condition remains as stated in Equations 2 and 3. In equilibrium, V(p) must satisfy for all p ∈ [0,1] the first order condition,

125 where h(V) is stated in Equation 5. Define worker match surplus by . This leads to,

126 which is a linear Volterra integral equation of the second kind in Vʹ(p). It solves by forward recursion from p = 0. From this, the value of unemployed search is recovered by,

127 And the value mapping is,

128 The firm productivity distribution Γ(p) is usually latent and now lends considerable freedom to the model toward fitting an empirical wage distribution. Adding discipline through observed firm productivity or profits does tend to challenge the model since firm profits are growing too fast in productivity. Mortensen [2003] explore this direction of the model extensively.

Firm Size in Burdett and Mortensen [1998]

129 While not the focus of the paper, it does contain a theory of firm size. It is worth paying attention to. Firms are assumed to be constant returns to scale. Firm size is constrained by the frictional process. Each firm is assumed to have a single vacancy process that generates meetings at rate η and in case of worker acceptance, a flow of workers into the firm. [6] The flow out is generated by the match destruction Poisson arrival process with rate δ and worker quits to higher productivity firms. To simplify notation denote by η(p) = ηh(V(p)) and . The firm’s labor force size is a stochastic birth-death process, which is a simple continuous time Markov-process where a firm’s size can only transition up or down by one unit in each arrival. The process is ergodic.

130 The inflows and outflows depend on the firm’s value promise. A higher V(p) results in a greater inflow through a higher acceptance rate and also a lower outflow through a reduced quit rate. Denote by m_i(p) the stationary distribution associated with the firm’s birth-death process in size. It is a productivity p firm’s probability of having a labor force size n unconditional on its current size. By definition . The stationary distribution is for any n ≥ 0 characterized by , implying an equalization of flow into and out of the mass,

131 This implies,

132 Use to obtain,

133 That is, a productivity p firm’s size is a Poisson distributed random variable with parameter η(p)/δ(p). In particular, this implies that the expected firm size of a productivity p firm is η(p)/δ(p) which is increasing in p. Thus, one obtains the prediction that firm size and average firm wage is positively related, which is robustly supported in firm data.

Homogenous Agents: A Tale of Two Prices

136 While empirically entirely irrelevant, it is useful to show the mechanisms in the model for the homogenous firm and worker case. Assume unemployed income flow is identical across all workers and normalized at zero, b = 0. Assume all matches produce some output value p > 0. Maintain equally efficient search across employment states, λ₀ = λ₁ = 0.

137 In the literal interpretation of Postel-Vinay and Robin [2002] above, there is no mention of value posting. Vacancies are assumed to be posted without any specified wage or value and the worker’s employment contract and value promise is arrived at after the parties have met and resolved outside options. It is however perfectly possible to also understand a vacancy as posted with a commitment to a value, which can be modified if both parties mutually agree to it. Thus, posting is more about what the firm can and cannot commit to. The extent to which a posted offer is cheap talk or not.

138 In the no-posting interpretation, when a vacancy meets an unemployed worker, it matches the worker’s outside option and offers an employment contract of V = U. In the posting interpretation, it is an equilibrium outcome that all vacancies post V = U since the vacancy is free to promise more in case it meets competition in the form of an existing firm and would not want to over promise in case it meets an unemployed worker. Either way, the value of unemployed search is that of unemployed sequential search,

139 Right away, note that this model in a fundamental sense invokes Diamond: unemployed workers receive none of the rents associated with job creation. It is a central feature. It is also fundementally responsible for the tractable transitional dynamics in the aggregate uncertainty setup in Robin [2011] and Lise and Robin [2017]. The modification of the model in Cahuc, Postel-Vinay and Robin [2006] can be seen as a response where bargaining is introduced to deliver rents to worker side search.

140 For now, simply assume that employment contracts are restricted to be constant wages until possibly renegotiated. This is no longer a restriction relative to tenure conditional contracts since the jointly inefficient worker quits are eliminated by the renegotiation and there is no longer a reason for backloading contracts. [7]

141 The promise keeping constraint on the wage w(V) for a given value promise V is thus,

142 where M is defined by J(M) = 0 implying that w(M) = p and, (r + δ)M = p. Therefore, the worker’s wage immediately upon leaving unemployment is, w(U) = –λp/(r + δ) < b = 0. If and when the worker meets an outside firm, the existing firm will match the outside opportunity and raise the wage to w(M) = p. There are two observed wages in this economy, w(U) and w(M). The first wage is associated with full rent extraction to the firm. The second is associated with full rent extraction to the worker. Workers have a value path that oscillates between two values, U and M. The employed wage in the first job out of unemployment, w(U) is lower than the unemployed income flow b in anticipation of the future higher wages that are realized when outside opportunities arrive. This will be a general point. Part of the value that a firm delivers to the worker is in how it increases the value of future vacancy meetings.

Heterogenous Firms

143 In the interest of simplicity, I maintain worker homogeneity and only add firm heterogeneity. It is straightforward to build worker productivity heterogeneity into the model and it clearly belongs in any serious empirical implementation that deals with wage dispersion. But for the purpose of understanding the interaction between wages and reallocation in the model, it is a distraction. Thus, assume firms differ in the productivity with which they employ workers, p ∈ [0, 1]. When a worker meets a vacancy, the productivity of the vacancy is distributed according to the CDF Γ(·). Denote by M(p) the most value a type p firm is willing to promise a worker. It is defined by J(M(p)| p) = 0, where J(V | p) is a productivity p firm’s match value given a value promise of V to the worker. Denote by F(M(p)) = Γ(p) the distribution of M(p) over vacancies in the economy. It is the offer distribution of the statistic that really matters in this version of the model.

144 It follows from the zero firm match value that w(M(p)| p) = p. A worker who is extracting full rents from a given employer has a zero net value from meeting new employers. Even if the new employer is more productive and the worker will move there, the new employer will only just match the current employer’s value promise. Hence, M(p) is,

145 by the assumption that b = 0 ⇒ U = 0.

146 The promise keeping constraint associated with w(V | p) for any U ≤ V ≤ M(p) is,

147 which by integration by parts yields,

148 It is common to do a change of variable and define q = M^–1(V) as the firm productivity type responsible for the worker’s current value promise (the second highest firm type the worker has met in the current employment spell),

149 If firm q delivers M(q) to the worker, it does so through a wage w = q. If a firm p > q delivers the same value to the worker it does so with a wage w < q, because with firm p the worker is expecting future value growth.

150 This feature received significant interest because it allows the model an understanding of why wages can decrease as the worker moves from one job to another, a not uncommon empirical phenomenon. As a practical matter, though, it has not turned out to be a particularly fruitful path in a broader context and more flexible mechanisms have been needed to understand wage dynamics across jobs. Rather, the model has the virtue of being able to accommodate very rich heterogeneity on both sides of the market while having an equilibrium solution that feels almost as simple as a partial equilibrium model. This has made it a popular choice in empirical studies of labor market outcomes that include frictional dynamics.

151 The model does have fairly rich wage dynamics. As with the wage posting model, values are monotonically increasing in the accumulation of meetings within an employment spell. In the value posting model this results in monotonically increasing wages across jobs and combined with a Burdett and Coles [2003] mechanism it also has the ability to explain within job wage growth. In the renegotiation framework, while values are monotonically increasing, wages need not rise monotonically within an employment spell since they can take occasional dips in between jobs, although the general trend will be upward. Within a job, wages are always increasing whenever an outside offer triggers a renegotiation but the outside firm is not more productive than the current firm.

152 The state of an employed worker is fully understood by the highest and second highest productivity firm types the worker has met within the current employment spell. Denote them by (q, p), where q ≤ p. The worker stays in this state until a meeting with an outside vacancy characterized by pʹ. If pʹ < q, the worker stays in current state (q, p). If pʹ ∈ [q, p], the worker moves to state (pʹ, p), which involves staying in the current job with a value increase from M(q) to M(pʹ) and wage increase. If pʹ > p then the worker moves to state (p, pʹ), which involves a job-to-job transition, a value increase from M(q) to M(p) and an ambiguous wage impact.

153 Denote by G(q | p) the mass of workers employed in type p firms at contract value M(q) or less. Furthermore denote the marginal g(p) = G(p | p) and . Steady state implies,

154 This implies,

155 where κ = λ/δ is the standard friction parameter.

156 Postel-Vinay and Robin [2002] consider the distribution of q within firm type a frictional contribution to wage and value variation, which is distributed according to Equation 11. The variation in firm wage due to going across firm types, they consider a firm heterogeneity contribution to wage and value variation. This decomposition should not be confused with a counterfactual where frictions are eliminated through κ → ∞, since both the firm and friction contributions as defined are eliminated in the limit. The paper also contains worker heterogeneity contributions to wages and values that are shown to be fully orthogonal to the variation considered so far.

157 The sequential auctions equilibrium is very simple to work with since there is no equilibrium feedback into worker and firm behavior. It delivers qualitatively similar wage dynamics as value posting, but solving for a Burdett and Coles [2003] equilibrium is just a considerably heavier enterprise. And this is without firm heterogeneity. With firm heterogeneity, Burdett and Coles [2010], solving for the tenure conditional employment contracts equilibrium is an achievement in itself.

158 If a worker leaves a productivity p firm for a more productive firm pʹ > p the sequential auctions framework above dictates that the worker’s new contract has value M(p) as a result of the old firm bidding up to the maximal value it is willing to offer. But why should an outgunned firm go to this trouble? There is indeed a time consistency issue, but it is in fact very much in the interest of an earlier version of firm p to credibly be able to tell the worker that it will help secure the worker the best possible terms of employment with a new employer. In fact, if the firm could bluff and somehow bid the outside firm up to M(pʹ), it would like to do so. However, in the absence of a commitment to employ the worker even at negative value, a promise of value V > M(p) would not be renegotiation proof and would be valued at whatever the renegotiation protocol dictates such an offer would be renegotiated to, at most M(p). Lentz [2014] state the contracting problem to depend on outside offers history where this point is made clear. By promising the value of the contract to be V = M(p) in case of an outside meeting with pʹ ≥ p, the worker achieves best possible terms in the move to pʹ. This maximizes the value of future meetings to the worker subject to renegotiation proofness and allows the firm to lower current wages and increase current profits.

Sequential Bargaining

159 The take-it-or-leave-it offer position that firms enjoy in the previous section has stark implications for rent sharing. Despite the positive social value associated with a worker’s reallocation from a lower to a higher productivity firm, there are no joint gains to a worker’s existing match from reallocation to a better firm. If there are costs to search on the worker side and such effort is essential for match creation, the model is at a loss in explaining why workers find jobs. The assumption also implies unattractive wages out of unemployment. The lowest wage in the economy is that of an unemployed worker who finds a job immediately with the best firm in the economy, w(0, 1).

160 Dey and Flinn [2005] and Cahuc, Postel-Vinay and Robin [2006] implement a bargaining version of the model with renegotiation which addresses some of these issues. The setup remains as in previous subsection “Heterogenous Firm.” The sole modification is in the value outcome to the worker when employed by a type p firm and an outside option of M(q), where q ≤ p. In the bargaining setup, it is assumed that the worker’s outcome is equivalent to the generalized Nash bargaining outcome V(q | p) = (1 – β)M(q) + βM(p), where β is the worker’s bargaining power. Notice that Postel-Vinay and Robin [2002] is the special case where β = 0. Dey and Flinn [2005] literally argue that the worker enters into bargaining with the more productive firm and that there is an understanding that the worker’s outside option is full surplus extraction with the less productive firm. Cahuc, Postel-Vinay and Robin [2006] argue a modified Rubinstein alternating offers game.

161 It remains the case that w(M(p) | p) = p. From the worker’s match value it follows that,

162 It follows from Equation 12 that,

163 Substitute back into Equation 12 to obtain,

164 The worker’s value of unemployed search is,

165 And so a fairly useful formulation of net worker value is,

166 M(p) and U can be solved for immediately from this.

167 With this one can move on to work out the wage w(q | p) for the type p firm dictated by promise keeping constraint for a given value promise of V = βM(p) + (1 – β)M(q), which is the contract value a worker has a result of accepting employment with type p and holding an outside option of M(q),

168 which can be re-written as,

169 Notice that Equation 10 appears as the special case of β = 0 in Equation 13. A higher β impacts how much wages may fall between jobs and as a flip side of that effect, it also reduces how much wage growth the model delivers within the job. β = 1 immediately kills all within job wage growth.

170 The rest of the model analysis, steady state conditions and mobility dynamics, is identical to that in previous subsection “Heterogenous Firm” which will not be repeated here. While not evident in this treatment of the model, the version of the model with renegotiation where workers extract rents from future matches, such as is the case with β > 0, equilibrium feedbacks on agent behavior can be considerably more complicated.

Tenure Conditional Contracts

171 Since job separations are jointly efficient once renegotiation is allowed, backloading incentives are absent in the exogenous arrivals model. However, once endogenous search intensity is introduced, another inefficiency is introduced and backloading once again becomes relevant. Lentz [2014] explores this argument for backloaded wages.

172 To illustrate the inefficiency, consider the sequential auctions version of the model. As emphasized earlier, there are no joint match gains from outside meetings and so the jointly efficient search intensity is zero. However, privately, as long as the worker is not extracting full rents from the current relationship, outside offers can deliver either a raise with the current firm, or the value equivalent of full surplus extraction with the current firm at a higher ranked firm. But whatever gains the worker realize, they are perfectly offset by a same size loss to the current employer.

173 Consider for simplicity a setup where the outside offer arrival rate is (1 + s)λ, where s ≥ 0 is chosen by the worker at cost c(s), where c(0) = cʹ(0) = 0. Let utility of wages by u(w) with u increasing and concave. Denote by J(V | p) the firm’s profit conditional on a value promise of V. The value of unemployed search to the worker remains U = 0 because any outside offer simply matches U, and so the optimal search choice when unemployed is s₀ = 0.

Contractable Search Intensity

174 It is a useful benchmark to first study the case where search intensity is contractable. It shows how the efficient contract implies no effort put into search. More importantly, it shows that when there is no joint inefficiency in search, the optimal contract is flat in its delivery of utility.

175 In the case where s is contractable, the full design problem is,

176 Let μ the Lagrange multiplier on the promise keeping constraint and ν the multiplier on the search intensity non-negativity constraint. The first order conditions on from w and choices are,

177 Finally, the first order condition associated with s is,

178 with complementary slackness, νs = 0. This implies,

179 By concavity of J(V | p) in V, the integrant for all M ∈ [V, M(p)]. Hence, for an interior solution for the search intensity, ν = 0, the right hand side of Equation 16 is negative, contradicting an interior solution for s(V | p). From this, it follows that,

180 By the envelope condition,

181 Hence, the optimal contract with contractable search intensity is flat, , and it dictates that the worker not search. By definition of M(p), J(M(p) | p) = 0 and therefore w(M(p) | p) = p. And from this, M(p) = u(p)/(r + δ). By change of variable, this means that the wage in the optimal contract with risk aversion and contractable search intensity is,

182 which limits to Equation 10 as preferences limit to risk neutrality.

183 This is all to say that in the sequential auctions model, there are no joint returns to search and therefore it sets search intensity to zero. Furthermore, the worker’s quit decision is jointly efficient which implies that the optimal contract is flat in respect of the worker’s preference for a smooth wage profile.

184 Bagger and Lentz [2019] emphasize search intensity as a sorting mechanism and do so in the Cahuc, Postel-Vinay and Robin [2006] bargaining framework where the jointly efficient search intensity is not zero. In this setup, the existing match benefits from meetings with higher willingness to pay outside firms as the bargaining technology allows the existing match to extract a share of the willingness to pay difference between the two firms. Thus, search would be strictly positive but the contract would remain flat.

Non-Contractable Search Intensity

185 Suppose the worker’s search choice is a hidden action, that is, it cannot be contracted upon. In this case, the optimal mechanism design problem is,

186 where an incentive compatibility constraint has been added to the problem which reflects that s(V | p) is constrained to be privately optimal to the worker.

187 The first order conditions associated with w and choices remain as in Equations 14 and 15. Let ν be the Lagrange multiplier on the incentive compatibility constraint. The first order condition for an optimal search intensity choice implies a shadow price on the constraint of,

188 By the envelope condition,

189 with strict inequality for V < M(p). [8]

190 Thus, the optimal contract has a backloaded utility promise which translates into backloaded wages. This point is emphasized in Lentz [2014]. The increase in the future utility promise adds value to the firm through the incentive compatibility constraint of . If the value promise increase is done sufficiently slowly, the worker’s aversion to non-smooth wage paths can be made small enough that the increased future utility promise can be offset enough in lower current wages to allow the firm a net benefit from a reduced future worker search intensity. Notice, the stark result in the risk neutral limit that firms will sell the job to the worker and all workers receive flow wages equal to their productivity. From a rent extraction point of view, the economy is as in Diamond [1971].

191 Thus, backloading is back. And again, in response to a joint inefficiency. In this case, the inefficiency is one where the worker will search too much since part of the private gains from search involve simple rent extraction from the current match. The firm responds to the efficiency through an increasing value promise path that reduces the ineffeciency relative to the flat utility promise path.