Equilibrium Wage Dispersion: An Example

International Monetary Fund

doi:10.5089/9781451862799.001.A001

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Equilibrium Wage Dispersion: An Example

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Contributor Notes

Author’s E-Mail Address: damien.gaumont@free.fr; mschindler@imf.org; rwright@econ.upenn.edu

Publication Date:: 01 Jan 2006

eISBN:: 9781451862799

Language:: English

Keywords:: WP

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Search models with posting and match-specific heterogeneity generate wage dispersion. Given K values for the match-specific variable, it is known that there are K reservation wages that could be posted, but generically never more than two actually are posted in equilibrium. What is unknown is when we get two wages, and which wages are actually posted. For an example with K = 3, we show equilibrium is unique; may have one wage or two; and when there are two, the equilibrium can display any combination of posted reservation wages, depending on parameters. We also show how wages, profits, and unemployment depend on productivity.

Abstract

I. Introduction

Wage dispersion—a deviation from the law of one price in the labor market—is a subject of long-standing theoretical and empirical interest in economics.² In Gaumont, Schindler, and Wright (2005), hereinafter referred to as GSW, we discuss several models of wage dispersion in search equilibrium with wage posting. Models based on ex ante homogeneous agents but ex post heterogeneous matches are shown to have advantages over earlier specifications based on ex ante heterogeneity (e.g., Albrecht and Axell, 1984, or Diamond, 1987). In particular, they do not “unravel” with the introduction of small but positive search costs. Although they do admit deviations from the law of one price, however, models with ex post heterogeneity are bound by the law of two prices (Curtis and Wright, 2004).

To explain this, suppose there are K possible realizations of the match-specific random variable. Then there are K distinct reservation wages, say w_k, k = 1, …K, and no profit-maximizing firm would post anything but one of these w_k. One can show that in a given equilibrium, generically, no more than two of them actually are posted. What is unknown is when we get two wages, as opposed to a single wage; and when we get two, which reservation wages are posted–the two highest, the highest and the lowest, two consecutive wages, or another combination. Here we present an example with K = 3 and characterize the outcome. We show there is always a unique equilibrium, which will have one wage or two, depending on parameters. Then we show that when the equilibrium has two wages, again depending on parameters, these can be either w₁ and w₂, w₂ and w₃, or w₃ and w₁.

This example is instructive because it helps us understand how and when wage dispersion happens, and exactly what kinds of wage dispersion can arise. It is appealing that we have a unique equilibrium, and the economic structure of this equilibrium is intuitively reasonable and simple. It does take some effort, however, to solve the example. In this paper, we show how to do it.

II. The Model

There is a [0, 1] continuum of firms and a [0, L] continuum of workers. Time is continuous. All agents live forever, are risk neutral, and discount at rate r. Each firm has a constant returns technology with labor as the only input and productivity y. Firms with vacancies contact workers at rate γ, and unemployed workers contact firms at rate α; there is no on-the-job search. For our purposes, it makes sense to set L = 1, so that the arrival rates α and γ are effectively pinned down exogenously, helping to keep the analysis simple.³ Matches end at exogenous rate δ. Firms post wages to maximize expected profit, given other firms’ wages and worker behavior.

Workers are ex ante homogeneous but matches are ex post heterogeneous. Thus, when a worker contacts a firm he draws at random c ∈ {c₁, …, c_K}, where c is the per period cost to taking the job, with c₁ < c₂ < … < c_K < y, and the probability of c = c_j is λ_j. For example, c could be the cost of commuting. Generally there is also an opportunity cost b to taking a job, incorporating leisure, home production, etc. To reduce notation, normalize b = 0. Also, we assume that c is permanent for the duration of the match.⁴

Let W_j(w) be the value to having a job with wage w and c = c_j, and U the value of unemployed search. Clearly, conditional reservation wage strategies are optimal: given c = c_j, accept a job iff w ≥ w_j, where W_j(w_j) = U. Notice w_j+1 > w_j. Hence there can be at most K wages posted since, as is completely standard, no firm would post anything other than one of the reservation wages: a firm posting w ∈ (w_j, w_j+1) could reduce w to w_j and make more profit per worker without changing the set of workers who accept. Let θ_j denote the fraction of firms posting w_j, $\sum_{j} θ_{j} = 1$ .

A special case of this is the well-known result of Diamond (1971) that arises when K = 1: with homogeneous matches, all firms post w₁ = c₁. A problem with that model is that when there is any cost to search, no matter how small, the market will shut down since workers get no surplus from employment at w = c₁. The same is true when there are ex ante heterogeneous workers, say K distinct types with different (but fixed) values of c. The highest c workers get no surplus from employment, so they drop out, and so on, and so the market “unravels” and shuts down. This is why we study models with ex post heterogeneity; in these models, as long as θ₁ < 1, workers get gains from search (e.g. he may get offer w > w₁ and draw c = c₁).

Bellman’s equations for a worker are

\begin{matrix} r U = α \sum_{j = 1}^{K} λ_{j} \sum_{i = j}^{K} θ_{i} [W_{j} (w_{i}) - U] & (1) \end{matrix}

and

\begin{matrix} r W_{j} (w) = w - c_{j} + δ [U - W_{j} (w)] . & (2) \end{matrix}

In words, (1) says that he contacts firms at rate α, draws c = c_j with probability λ_j, and accepts if the posted wage is w_i ≥ w_j, which occurs with probability θ_i. Given w is acceptable, (2) says that an employed worker gets w − c_j until the match ends, which occurs at rate δ. Using W_j(w_j)= U, we have

\begin{matrix} w_{j} = c_{j} + r U . & (3) \end{matrix}

Expected profit for a firm posting a vacancy at w_j is

\begin{matrix} Π_{j} = \frac{γ ρ_{j} (y - w_{j})}{r + δ} = \frac{γ ρ_{j} (y - c_{j} - r U)}{r + δ}, & (4) \end{matrix}

where γ is the arrival rate of workers, $ρ_{j} = \sum_{h = 1}^{j} λ_{h}$ is the probability a random worker accepts, and we use (3) to substitute for w_j in terms of U. As we said, no firm posts anything other than one of the K reservation wages. Following Curtis and Wright (2004), one can strengthen this to show that generically there are no more than two wages posted.

^{Proposition 1}

For generic parameter values, we can have θ_h > 0 for at most two values of h.

Proof.

Suppose θ_i > 0, θ_j > 0, θ_k > 0 for distinct i, j, and k. Then Π_i = Π_j = Π_k = max {Π₁, …, Π_K}. Hence, g_i(U) = g_j(U) = g_k(U), where from (4)

g_{h} (U) \equiv ρ_{h} (y - c_{h}) - r ρ_{h} U .

For generic parameter values, there does not exist a solution U to g_i(U) = g_j(U) = g_k(U). ■

In GSW we studied the case K = 2. We showed there always exists a unique equilibrium, which may or may not entail wage dispersion. If y is small, all firms post w₁ = c₁; if y is big all firms post w₂ ∈ (c₂, y); and if y is intermediate, a fraction post w₁ ∈ (c₁, w₂) while the rest post w₂ ∈ (c₂, y). For other values of K, although we know there can be no more than two wages posted, we do not know when there are two, as opposed to one. And when there are two, we also do not know which of the two reservation wages they will be.

III. The Example

Consider K = 3. As the only wages posted are in {w₁, w₂, w₃}, we write W_ij = W_i(w_j) for the value of employment at reservation wage w_j when a worker draws c_i. Then (1) and (2) reduce to

\begin{array}{l} r U = α θ_{2} λ_{1} (W_{12} - U) + α θ_{3} [λ_{1} (W_{13} - U) + λ_{2} (W_{23} - U)] \\ r W_{i j} = w_{j} - c_{i} + δ (U - W_{i j}) . \end{array}

Here we use the result that a worker who draws w = w_j and c = c_j gets no surplus from the match (in equilibrium he still accepts). Using w_j = c_j + rU,

\begin{matrix} r U = η θ_{2} λ_{1} (c_{2} - c_{1}) + η θ_{3} λ_{1} (c_{3} - c_{1}) + η θ_{3} λ_{2} (c_{3} - c_{2}), & (5) \end{matrix}

where η = α/(r + δ). Also, (4) reduces to $\begin{array}{l} Π_{j} = \frac{γ}{r + δ} \sum_{i = 1}^{j} λ_{i} (y - w_{j}) \end{array}$ .

By Proposition 1, at least one θ_j = 0, so there are exactly 6 possible equilibria as listed in Table 1. We now give conditions determining when each equilibrium exists. We give these conditions in two ways: as restrictions on y, which are relatively easy and facilitate comparison with earlier work (e.g. the results reported in the last paragraph of Section 2); and as restrictions on λ = (λ₁, λ₂), which provide a nice graphical representation of the equilibrium set. To begin, it will be useful to define the following:

Table 1.

Possible Equilibria

\begin{array}{l} {\underline{y}}_{1} = η λ_{1} (c_{2} - c_{1}) + \frac{c_{3} - (λ_{1} + λ_{2}) c_{2}}{1 - λ_{1} - λ_{2}} and {\bar{y}}_{1} = {\underline{y}}_{1} + η (λ_{1} + λ_{2}) (c_{3} - c_{2}) \\ {\underline{y}}_{2} = \frac{c_{3} - λ_{1} c_{1}}{1 - λ_{1}} and {\bar{y}}_{2} = {\underline{y}}_{2} + η [λ_{1} (c_{3} - c_{1}) + λ_{2} (c_{3} - c_{2})] \\ {\underline{y}}_{3} = \frac{(λ_{1} + λ_{2}) c_{2} - λ_{1} c_{1}}{λ_{2}} and {\bar{y}}_{3} = {\underline{y}}_{3} + η λ_{1} (c_{2} - c_{1}) \end{array}

^{Equilibrium 1:}

θ₁ = 1. This case implies rU = 0 by (5); hence w_j = c_j and equilibrium profit is

Π_{1} = \frac{γ}{r + δ} λ_{1} (y - c_{1}) .

Given all firms post w = w₁ = c₁, no firm wants to deviate and post w₂ iff Π₂ ≤ Π₁ and no firm wants to post w₃ iff Π₃ ≤ Π₁, where

\begin{array}{l} Π_{2} = \frac{γ}{r + δ} (λ_{1} + λ_{2}) (y - c_{2}) \\ Π_{3} = \frac{γ}{r + δ} (y - c_{3}) . \end{array}

Algebra implies Π₂ ≤ Π₁ iff

y \leq {\underline{y}}_{3}

and Π₃ ≤ Π₁ iff

y \leq {\underline{y}}_{2}

. The corresponding conditions in λ-space are given by

\begin{matrix} \begin{matrix} λ_{1} \geq {\tilde{λ}}_{1} \equiv \frac{y - c_{3}}{y - c_{1}} \end{matrix} \\ \begin{matrix} λ_{2} \leq ℓ_{1} (λ_{1}) \equiv \frac{c_{2} - c_{1}}{y - c_{2}} λ_{1} . & (6) \end{matrix} \end{matrix}

This gives necessary and sufficient conditions for equilibrium 1.

^{Equilibrium 2:}

θ₂ = 1. Given θ₂ =1, rU = ηλ₁(c₂ − c₁) by (5), and hence w_j = c_j + rU = c_j + ηλ₁(c₂ − c₁). Using this,

\begin{array}{l} Π_{1} = \frac{γ}{r + δ} λ_{1} (y - c_{1} - r U) \\ Π_{2} = \frac{γ}{r + δ} (λ_{1} + λ_{2}) (y - c_{2} - r U) \\ Π_{3} = \frac{γ}{r + δ} (y - c_{3} - r U) . \end{array}

No firm wants to deviate and post w₁ iff Π₁ ≤ Π₂, which holds iff $y \geq {\bar{y}}_{3}$ , and no firm will deviate and post w₃ iff Π₃ ≤ Π₂, which holds iff $y \leq {\underline{y}}_{1}$ . In λ-space, these conditions on y can be expressed as

\begin{array}{l} λ_{1} < {\hat{λ}}_{1} \equiv \frac{y - c_{2}}{η (c_{2} - c_{1})} \\ λ_{2} \geq ℓ_{2} (λ_{1}) \equiv \frac{(1 - λ_{1}) [y - η λ_{1} (c_{2} - c_{1})] - c_{3} + λ_{1} c_{2}}{y - η λ_{1} (c_{2} - c_{1}) - c_{2}} \\ λ_{2} > ℓ_{3} (λ_{1}) \equiv \frac{λ_{1} (c_{2} - c_{1})}{y - c_{2} - η λ_{1} (c_{2} - c_{1})} . & (7) \end{array}

The properties of the ℓ functions are given below, but we need some properties of ℓ₃ now to conclude the following: although ${\bar{y}}_{3} \leq y \leq {\underline{y}}_{1}$ is also satisfied if the above three inequalities are all reversed, this case can be ignored because $λ_{1} > {\hat{λ}}_{1}$ implies ℓ₃ (λ₁) < 0. Thus, (7) gives necessary and sufficient conditions for equilibrium 2.

Lemma 1

ℓ₃(λ₁) goes through the origin, is strictly increasing, strictly convex and positive if $λ_{1} < {\hat{λ}}_{1}$ , and strictly concave and negative if $λ_{1} > {\hat{λ}}_{1}$ , with a discontinuity at $λ_{1} = {\hat{λ}}_{1}$ .

Proof.

ℓ₃ (0) = 0 is obvious. The first derivative is $ℓ_{3}^{'} = \frac{(y - c_{2}) (c_{2} - c_{1})}{{[y - c_{2} - η λ_{1} (c_{2} - c_{1})]}^{2}} > 0$ . The second derivative is $ℓ_{3}^{″} = \frac{2 η (c_{2} - c_{1})}{{[y - c_{2} - η λ_{1} (c_{2} - c_{1})]}^{3}}$ which is positive if $λ_{1} < {\hat{λ}}_{1}$ and negative otherwise. ■

^{Equilibrium 3:}

θ₃ = 1. Given θ₃ = 1 we can solve for rU, w_j, and Π_j, and check that no firm will deviate iff

y \geq {\bar{y}}_{1}

and

y \geq {\bar{y}}_{2}

. The first condition

y \geq {\bar{y}}_{1}

can be written as a quadratic in λ₂ for a given λ₁, say

Q (λ_{2}) = A λ_{2}^{2} + B λ_{2} + C \geq 0

, where

\begin{array}{l} A = η (c_{3} - c_{2}) \\ B = - y + c_{2} + η λ_{1} (c_{3} - c_{1}) - η (1 - λ_{1}) (c_{3} - c_{2}) \\ C = (1 - λ_{1}) y - c_{3} + λ_{1} c_{2} - η (1 - λ_{1}) λ_{1} (c_{3} - c_{1}) . \end{array}

It is easy to see that Q(λ₂) is convex and Q(λ₂) < 0 at λ₂ = 1 − λ₁. Hence, Q(λ₂) has two real roots that depend on λ₁, say ℓ⁻(λ₁) and ℓ⁺(λ₁), one on each side of 1 − λ₁. Since only λ₂ ≤ 1 − λ₁ is relevant, we conclude that Q(λ₂) ≥ 0 iff

λ_{2} \leq ℓ^{-} (λ_{1}) = \frac{- B - \sqrt{B^{2} - 4 A C}}{2 A} .

The second condition $y \geq {\bar{y}}_{2}$ is equivalent to

λ_{2} \leq ℓ_{4} (λ_{1}) = \frac{(1 - λ_{1}) y - c_{3} + λ_{1} c_{1} - η (1 - λ_{1}) λ_{1} (c_{3} - c_{1})}{η (1 - λ_{1}) (c_{3} - c_{2})} .

Hence, equilibrium 3 exists iff

\begin{matrix} λ_{2} \leq \min {ℓ^{-} (λ_{1}), ℓ_{4} (λ_{1})} . & (8) \end{matrix}

The description above exhausts the single-wage equilibria. By inspection of the y-cutoffs, these cases are mutually exclusive, so there cannot be multiple single-wage equilibria. We now consider two-wage equilibria.

^{Equilibrium 4:}

θ₁, θ₂ > θ₃ = 0. This equilibrium requires Π₂ = Π₁, an equality that can be solved for

θ_{2} = \frac{λ_{2} y - (λ_{1} + λ_{2}) c_{2} + λ_{1} c_{1}}{η λ_{1} λ_{2} (c_{2} - c_{1})} .

Notice θ_{2 ∈} (0, 1) iff

y \in ({\underline{y}}_{3}, {\bar{y}}_{3})

, which is equivalent to λ₂ > ℓ₁ (λ₁) and

λ_{2} < ℓ_{3} (λ_{1}) if λ_{1} < {\hat{λ}}_{1}; λ_{2} > ℓ_{3} (λ_{1}) if λ_{1} > {\hat{λ}}_{1} .

No firm wants to deviate and post w₃, rather than either w₁ or w₂, iff ⁵

λ_{2} \geq ℓ_{5} (λ_{1}) = \frac{λ_{1} (1 - λ_{1}) (c_{2} - c_{1})}{c_{3} - λ_{1} c_{1} - (1 - λ_{1}) c_{2}} .

Hence, equilibrium 4 exists iff

\begin{matrix} λ_{2} < ℓ_{3} (λ_{1}) if λ_{1} < {\hat{λ}}_{1}; and λ_{2} > \max {ℓ_{5} (λ_{1}), ℓ_{1} (λ_{1})} . & (9) \end{matrix}

^{Equilibrium 5:}

θ₁, θ₃ > θ₂ = 0. This requires Π₃ = Π₁, which can be solved for

θ_{3} = \frac{(1 - λ_{1}) y - c_{3} + λ_{1} c_{1}}{λ_{1} (c_{3} - c_{1}) + λ_{2} (c_{3} - c_{2})} \frac{1}{(1 - λ_{1}) η} .

Hence θ₃ ∈ (0, 1) iff

y \in ({\underline{y}}_{2}, {\bar{y}}_{2})

, which is equivalent to

λ_{1} < {\tilde{λ}}_{1}

and λ₂ > ℓ₄(λ₁). No firm wants to deviate and post w₂ iff λ2 ≤ ℓ₅(λ₁). Hence equilibrium 5 exists iff

\begin{matrix} λ_{1} < {\tilde{λ}}_{1}, λ_{2} > ℓ_{4} (λ_{1}), and λ_{2} \leq ℓ_{5} (λ_{1}) . & (10) \end{matrix}

^{Equilibrium 6:}

θ₂, θ₃ > θ₁ = 0. This requires Π₃ = Π₂, which can be solved for

θ_{2} = - \frac{(1 - λ_{1} - λ_{2}) Ψ - c_{3} + (λ_{1} + λ_{2}) c_{2}}{η (1 - λ_{1} - λ_{2}) (λ_{1} + λ_{2}) (c_{3} - c_{2})},

where Ψ = y − ηλ₁ (c₃ − c₁) − ηλ₂ (c₃ − c₂). Observe that the denominator in this expression is the quadratic Q (λ₂) defined in the discussion of equilibrium 3. Hence, we can write θ₂ ∈ (0, 1) iff $y \in ({\underline{y}}_{1}, {\bar{y}}_{1})$ , which is equivalent to $λ_{1} < {\hat{λ}}_{1}$ , λ₂ > ℓ⁻(λ₁), and λ₂ < ℓ₂(λ₁). Actually, the condition θ < 1 is also satisfied if $λ_{1} > {\hat{λ}}_{1}$ and λ₂ > ℓ₂ (λ₁), but the following lemma shows that this can never be satisfied in the relevant region.

Lemma 2

$ℓ_{2} (0) = \frac{y - c_{3}}{y - c_{2}} \in (0, 1)$ ; ℓ₂(λ₁) → –∞ as $λ_{1} \to {\hat{λ}}_{1}$ from below; ℓ₂(λ₁) → ∞ as $λ_{1} \to {\hat{λ}}_{1}$ from above; ℓ₂(λ₁) > 1 − λ₁ iff $λ_{1} > {\hat{λ}}_{1}$ ; ℓ₂(λ₁) → 1 − λ₁ from above as λ₁ → ∞; ℓ₂(λ₁) → 1 − λ₁ from below as λ₁ → −∞; ℓ₂(λ₁) > ℓ⁻(λ₁).

Proof.

The first parts involve straightforward analysis. Proving ℓ₂(λ₁) → 1 −λ₁ is equivalent to proving ℓ₂(λ₁)/(1 − λ₁) → 1, which follows from l’Hôpital’s rule. For the last part, observe that ∂θ₂/∂λ₁ > 0; hence, as we increase λ₁ for any given λ₂, we hit the threshold at which θ₂ = 0 before we hit the threshold at which θ₂ =1. This means the ℓ₂(λ₁) curve lies above the ℓ⁻ (λ₁) curve in (λ₁, λ₂) space. ■

Finally, no firm wants to deviate and post w₁ iff λ₂ ≥ ℓ₅(λ₁). Hence equilibrium 6 exists iff

\begin{matrix} λ_{1} < {\hat{λ}}_{1}, λ_{2} > ℓ^{-} (λ_{1}), λ_{2} < ℓ_{2} (λ_{1}) and λ_{2} \geq ℓ_{5} (λ_{1}) . & (11) \end{matrix}

The completes our analysis of every case in Table 1. For each candidate equilibrium 1-6 we provide necessary and sufficient conditions for existence in terms of y and also λ. We can summarize the results as follows:

^{Proposition 2}

Generically equilibrium exists and is unique. If λ₂ < ℓ₅ (λ₁) then: equilibrium 1 exists iff $y \leq {\underline{y}}_{2}$ ; equilibrium 5 exists iff $y \in ({\underline{y}}_{2}, {\bar{y}}_{2})$ ; and equilibrium 3 exists iff $y \geq {\bar{y}}_{2}$ .

If λ₂ > ℓ₅ (λ₁) then: equilibrium 1 exists iff $y \leq {\underline{y}}_{3}$ ; equilibrium 4 exists iff $y \in ({\underline{y}}_{3}, {\bar{y}}_{3})$ ; equilibrium 2 exists iff $y \in ({\bar{y}}_{3}, {\underline{y}}_{1})$ ; equilibrium 6 exists iff $y \in ({\underline{y}}_{1}, {\bar{y}}_{1})$ ; and equilibrium 3 exists iff $y \geq {\bar{y}}_{1}$ .

Proof.

There are two generic cases, λ₂ < ℓ₅ (λ₁) and λ₂ > ℓ₅ (λ₁). In the former case it is clear that for all y there is a unique equilibrium. The same is true in the latter case once one recognizes that ${\bar{y}}_{3} < {\underline{y}}_{1}$ in the case where λ₂ > ℓ₅ (λ₁). ■

In order to present the results graphically, we move to λ-space, and make use of conditions (6)-(11). To do so, we first describe some more properties of the ℓ_j functions used in these conditions. Proofs are omitted where obvious.

Lemma 3

ℓ⁻(0) ∈ (0, 1). There is a unique $λ_{1}^{0} \in (0, 1)$ such that $ℓ^{-} (λ_{1}^{0}) = 0$ .

Proof.

For the first part, note that λ₁ = 0 implies Q(0) > 0 and Q(1) < 0, and since Q(λ₂) has two real roots ℓ⁻(0) and ℓ⁺(0), one on each side of 1 − λ₁, we conclude ℓ⁻(0) ∈ (0, 1). The second part is shown by noting that λ₁ = 1 implies Q(0) < 0, which in turn implies ℓ⁻(1) < 0. Convexity of Q then implies the existence of a unique $λ_{1}^{0} \in (0, 1)$ such that $ℓ^{-} (λ_{1}^{0}) = 0$ . ■

Lemma 4

ℓ₅ (λ₁) is concave, lies below λ₂ =1 − λ₁, and goes through (0, 0) and (1, 0).

Lemma 5

ℓ₄ (λ₁) is monotonically decreasing with ℓ₄(0) = (y − c₃)/η(c₃ − c₂) > 0.

Lemma 6

ℓ₁ (λ₁) is a linearly increasing function with ℓ₁(0) = 0 and slope $ℓ_{1}^{'} (λ_{1}) = \frac{c_{2} - c_{1}}{y - c_{2}} > 0$ .

Lemma 7

ℓ⁻(λ₁), ℓ₅(λ₁) and ℓ₄(λ₁) intersect at the same point $λ_{1}^{a}$ .

Proof.

This follows from considering the corresponding functions in y-space, ${\bar{y}}_{1}$ and ${\bar{y}}_{2}$ , where it is easy to show that ${\bar{y}}_{1} = {\bar{y}}_{2}$ iff λ₂ = l₅. ■

Lemma 8

ℓ₅(λ₁), ℓ₂(λ₁) and ℓ₃(λ₁) intersect at the same point $λ_{1}^{b} < {\hat{λ}}_{1}$ .

Proof.

That the three functions intersect at the same point $λ_{1}^{b}$ follows from simple algebra. Then $λ_{1}^{b} < {\hat{λ}}_{1}$ follows from the properties of ℓ₂ (λ₁) which imply that if ℓ₂ intersects with another function within the simplex, it must be at some $λ_{1} < {\hat{λ}}_{1}$ . ■

Lemma 9

ℓ₅(λ₁) and ℓ₁(λ₁) intersect at $λ_{1}^{c} = {\tilde{λ}}_{1}$ .

Lemma 10

$λ_{1}^{a} < λ_{1}^{b} < λ_{1}^{c}$

Proof.

Note first that ℓ₃ (λ₁) > ℓ₁ (λ₁) for all λ₁ > 0, and ℓ₃ (λ₁), ℓ₁ (λ₁) < ℓ₅ (λ₁) for small λ₁, which follows from noting that ℓ₃(0) = ℓ₁(0) = 0 and $ℓ_{3}^{'} (0) = ℓ_{1}^{'} (λ_{1}) = \frac{c_{2} - c_{1}}{y - c_{2}} < ℓ_{5}^{'} (0) = \frac{c_{2} - c_{1}}{c_{3} - c_{2}}$ . This implies $λ_{1}^{b} < λ_{1}^{c}$ . To show that $λ_{1}^{a} < λ_{1}^{b}$ , we claim that ℓ⁻(λ₁) < ℓ₂(λ₁) for all λ₁ such that ℓ⁻ (λ₁) > ℓ₅ (λ₁). Suppose not. Then if ℓ⁻(λ₁) > ℓ₂(λ₁) for some λ₁ such that ℓ⁻(λ₁) > ℓ₅ (λ₁), equilibria 2 and 3 coexist, and we would contradict the uniqueness part of Proposition 2; and if ℓ⁻(λ₁) > ℓ₂(λ₁) for all λ₁ such that ℓ⁻(λ₁) > ℓ₅ (λ₁), we would contradict the existence part. Consequently, ℓ⁻(λ₁) intersects ℓ₅(λ₁) at a smaller λ₁ than does ℓ₂(λ₁). ■

Lemma 11

ℓ⁻(λ₁) < ℓ₄(λ₁) for λ₂ > ℓ₅(λ₁) and ℓ^–(λ₁) > λ₄(λ₁) for λ₂ <ℓ₅(λ₁).

Proof.

It must be true that ℓ⁻(λ₁) < ℓ₄(λ₁) iff ℓ₄(λ₁) > ℓ₅ (λ₁), otherwise we violate the existence or uniqueness part of Proposition 2. ■

Given these properties we can draw the ℓ-functions, as shown in Figure 1 for two sets of parameter values. It is now a simple matter to fill in the different regions generated by the ℓ-functions with the equilibrium that exists in each case. In terms of economics, the results are quite intuitive. Consider, for example, the case of λ₁ close to 1. Then we get a single-wage equilibrium, all firms post the lowest reservation wage w₁, which maximizes profit per worker, and does not reduce the hiring probability too much as λ₁ is big. As λ₂ becomes big we get equilibrium where all firms post w₂, and as λ₃ becomes big we get wage equilibrium where all firms post w₃, because firms are willing to sacrifice profit per worker to keep the hiring probability from falling too much.

Figure 1.

Equilibrium Regions

Citation: IMF Working Papers 2006, 019; 10.5089/9781451862799.001.A001

When we are in a region between those with a single-wage equilibrium, we get wage dispersion; for example, between the regions where all firms post w₁ and where all firms post w₂, some firms post w₁ and others w₂. The figure illustrates two key points. First, two-wage equilibria are not especially rare. Second, when two-wage equilibria exist, they may entail any combination of w₁, w₂ and w₃. Of course, these wages are themselves endogenous — they depend on the equilibrium as well as parameters. Table 2 lists wages in each equilibrium, including those that are not posted; note that in each case, consistent with (3), we have w_j = c_j + rU.

Table 2.

Wages

In Figure 2, we plot the average wage $\bar{ω}$ , profit $\bar{Π}$ , unemployment u, and the fraction of firms posting each wage, as functions of productivity y, leaving explicit calculations as an exercise.⁶ There are two panels, corresponding to the two cases in Proposition 2: λ₂ < ℓ₅ (λ₁) and λ₂ > ℓ₅ (λ₁). This figure shows the intuitive result that higher productivity must benefit either firms or workers in terms of wages or profit, but interestingly, never at the same time: $\bar{Π}$ is constant in single-wage equilibria and $\bar{ω}$ is constant in two-wage equilibria. Also, u is constant in single-wage equilibria and decreasing in y in two-wage equilibria.⁷

Figure 2.

Selected Equilibrium Variables as a Function of y

Citation: IMF Working Papers 2006, 019; 10.5089/9781451862799.001.A001

IV. Conclusion

We analyzed in detail the case of K = 3 in a model with wage posting and ex post heterogeneity. We found that a unique equilibrium exists which may or may not exhibit wage dispersion. Also, any pair of reservation wages may be posted. We think the results teach us something interesting about endogenous wage dispersion and search theory more generally.

References

Albrecht, James, and Bo Axell, 1984, “An Equilibrium Model of Search Unemployment,” Journal of Political Economy, Vol. 92, pp. 824–40.
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Curtis, Elisabeth, and Randall Wright, 2004, “Price Setting, Price Dispersion, and the Value of Money: or, the Law of Two Prices,” Journal of Monetary Economics, Vol. 51, pp. 1599–1621.
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Diamond, Peter A., 1971, “A Model of Price Adjustment,” Journal of Economic Theory, Vol. 3, pp. 156–68.
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Diamond, Peter A., 1987, “Consumer Differences and Prices in a Search Model,” Quarterly Journal of Economics, Vol. 102, pp. 429–36.
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Gaumont, Damien, Martin Schindler, and Randall Wright, 2005, “Alternative Theories of Wage Dispersion,” IMF Working Paper 05/64 (Washington: International Monetary Fund); also forthcoming in Contributions to Economic Analysis: Structural Models of Wage and Employment Dynamics, ed. by Henning Bunzel and others (Amsterdam: Elsevier North-Holland).
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Mortensen, Dale T., 2003, Wage Dispersion (Cambridge, United Kingdom: Zeuthen Lecture Book Series).
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Rogerson, Richard, Robert Shimer, and Randall Wright, 2005, “Search Theoretic Models of the Labor Market: A Survey,” forthcoming in the Journal of Economic Literature.
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Damien Gaumont is a Professor of Economics at the Université Panthéon-Assas (Paris II); Martin Schindler is an Economist in the Financial Studies Division of the IMF Research Department; and Randall Wright is a Professor of Economics at the University of Pennsylvania. The authors thank the Federal Reserve Bank of Cleveland, Paris II, and the National Science Foundation for research support, as well as seminar participants at the University of Pennsylvania, Université du Québec à Montréal, Georgetown University, and the IMF Institute for comments.

See Mortensen (2003) or Rogerson, Shimer, and Wright (2005) for extended discussions and references.

This is because, with L = 1, the ratio of unemployed workers to vacancies is always 1. See GSW for details concerning the arrival rates, and how to solve for them in equilibrium, in generalized versions of the model.

In GSW we also consider the case where employed workers draw a new c each period.

There is no corresponding condition in terms of y: for equilibria 4-6, the no deviation conditions depend only on λ₂ versus ℓ₅(λ₁).

The only variable we have not defined is unemployment which, as is standard, evolves according to $\dot{u} = (1 - u) δ - u α \sum_{i = 1}^{3} θ_{i} \sum_{j = 1}^{i} λ_{j}$ , so that in steady-state $u = \frac{δ}{δ + α \sum_{i = 1}^{3} θ_{i} \sum_{j = 1}^{i} λ_{j}} .$ .

The shapes in the figure are general with the exception of the relative position of $\bar{Π}$ and $\bar{ω}$ . In general, we can have $\bar{ω} < \bar{Π}$ or vice versa, although for small y we must have $\bar{ω} > \bar{Π}$ and for very large y we must have $\bar{ω} < \bar{Π}$ . Put differently, for very small (large) y workers extract a higher (lower) share of the surplus than firms.

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Equilibrium Wage Dispersion: An Example

Author:

International Monetary Fund

Volume/Issue:: Volume 2006: Issue 019

Publisher:: International Monetary Fund

ISBN:: 9781451862799

ISSN:: 1018-5941

Pages:: 16

DOI:: https://doi.org/10.5089/9781451862799.001

Keywords
I. Introduction
II. The Model
III. The Example
IV. Conclusion
References

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Figure 1.

Equilibrium Regions

View raw image

Figure 2.

Selected Equilibrium Variables as a Function of y

Equilibrium	Wages w_j for j = 1, 2, 3
1. θ₁ = 1	w_j = c_j
2. θ₂ = 1	w_j = c_j + ηλ₁(c₂ − c₁)
3. θ₃ = 1	w_j = c_j + ηλ₁(c₃−c₁) + ηλ₂(c₃ − c₂)
4. θ₁θ₂ > 0	$w_{j} = c_{j} + y + \frac{λ_{1} c_{1} - (λ_{1} + λ_{2}) c_{2}}{λ_{2}}$
5. θ₁θ₃ > 0	$w_{j} = c_{j} + y + \frac{λ_{1} c_{1} - c_{3}}{1 - λ_{1}}$
6. θ₂θ₃ > 0	$w_{j} = c_{j} + y + \frac{(λ_{1} + λ_{2}) c_{2} - c_{3}}{1 - λ_{1} - λ_{2}}$

Equilibrium	Wages w_j for j = 1, 2, 3
1. θ₁ = 1	w_j = c_j
2. θ₂ = 1	w_j = c_j + ηλ₁(c₂ − c₁)
3. θ₃ = 1	w_j = c_j + ηλ₁(c₃−c₁) + ηλ₂(c₃ − c₂)
4. θ₁θ₂ > 0	$w_{j} = c_{j} + y + \frac{λ_{1} c_{1} - (λ_{1} + λ_{2}) c_{2}}{λ_{2}}$
5. θ₁θ₃ > 0	$w_{j} = c_{j} + y + \frac{λ_{1} c_{1} - c_{3}}{1 - λ_{1}}$
6. θ₂θ₃ > 0	$w_{j} = c_{j} + y + \frac{(λ_{1} + λ_{2}) c_{2} - c_{3}}{1 - λ_{1} - λ_{2}}$

Equilibrium	Wages w_j for j = 1, 2, 3
1. θ₁ = 1	w_j = c_j
2. θ₂ = 1	w_j = c_j + ηλ₁(c₂ − c₁)
3. θ₃ = 1	w_j = c_j + ηλ₁(c₃−c₁) + ηλ₂(c₃ − c₂)
4. θ₁θ₂ > 0	$w_{j} = c_{j} + y + \frac{λ_{1} c_{1} - (λ_{1} + λ_{2}) c_{2}}{λ_{2}}$
5. θ₁θ₃ > 0	$w_{j} = c_{j} + y + \frac{λ_{1} c_{1} - c_{3}}{1 - λ_{1}}$
6. θ₂θ₃ > 0	$w_{j} = c_{j} + y + \frac{(λ_{1} + λ_{2}) c_{2} - c_{3}}{1 - λ_{1} - λ_{2}}$

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II. The Model

III. The Example

IV. Conclusion

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