Diamond–Mortensen–Pissarides Model

Diamond–Mortensen–Pissarides Model is the canonical continuous-time and discrete-time macroeconomic framework for analyzing frictional unemployment, job creation, job destruction, and decentralized wage determination in markets characterized by costly search and bilateral matching. Developed through a series of foundational papers by Peter A. Diamond (1982), Dale T. Mortensen (1982), and Christopher A. Pissarides (1985, 2000)—for which the three economists shared the 2010 Nobel Memorial Prize in Economic Sciences—the model replaces the frictionless Walrasian auctioneer of neoclassical labor economics with an aggregate matching technology that brings unemployed job seekers and open firm vacancies together at finite Poisson arrival rates. By embedding decentralized bilateral bargaining (generalizing the Rubinstein bargaining model and axiomatic Nash bargaining) into a dynamic general equilibrium environment, the Diamond–Mortensen–Pissarides (DMP) framework provides rigorous microfoundations for the empirical Beveridge curve, structural unemployment rates, labor market tightness fluctuations, and the welfare properties of search externalities under the Hosios efficiency condition.

In a frictionless competitive labor market, unemployed workers and firms with open positions find each other instantaneously at a market-clearing wage, leaving zero involuntary or frictional unemployment in equilibrium. In reality, workers possess heterogeneous skills, geographic locations, and preferences, while firms have distinct job requirements and screening protocols. Recruiting a worker requires posting a vacancy, reviewing applications, and conducting interviews; finding a job requires searching across job boards, submitting applications, and waiting for offers.

The DMP model summarizes these decentralized information and coordination frictions through an aggregate matching function M(U, V), which maps the stock of unemployed workers U and the stock of open job vacancies V into a flow of newly formed employment relationships per unit of time:

M = M(U, V) = m₀ U^η V^{1 − η}

where m₀ > 0 denotes aggregate matching efficiency (reflecting recruiting technology, geographic mobility, and skill mismatch) and η ∈ (0, 1) represents the elasticity of matching with respect to unemployment. The matching function is assumed to be strictly increasing, concave in both arguments, and homogeneous of degree one (constant returns to scale), with M(0, V) = M(U, 0) = 0.

Because M(U, V) exhibits constant returns to scale, all transition rates in the labor market depend only on the ratio of vacancies to unemployment, defined as labor market tightness θ = V / U:

1. Vacancy-Filling Rate (q(θ)): The Poisson rate at which an open vacancy meets an unemployed worker is q(θ) = M(U, V) / V = M(θ⁻¹, 1) = m₀ θ^−η. Because q′(θ) < 0, a tighter labor market (higher θ) makes it harder for firms to fill open positions due to congestion among competing recruiters.
2. Job-Finding Rate (f(θ)): The Poisson rate at which an unemployed worker finds a job is f(θ) = M(U, V) / U = θ q(θ) = m₀ θ^{1 − η}. Because f′(θ) > 0, a tighter labor market increases the hazard rate at which unemployed workers transition into employment.

Normalizing the total labor force to unity (L = E + U = 1) and assuming that existing employment matches dissolve at an exogenous Poisson separation rate s > 0, the continuous-time law of motion for the unemployment rate u is governed by the differential equation:

du / dt = s(1 − u) − f(θ)u

Setting du / dt = 0 yields the steady-state Beveridge curve, which defines an inverse locus between the unemployment rate u and the vacancy rate v (since f(θ) = m₀(v / u)^{1 − η}):

u* = s / [s + f(θ*)] = s / [s + m₀(v* / u*)^{1 − η}]

Movements along the Beveridge curve reflect cyclical variations in aggregate labor demand and tightness θ, whereas outward shifts of the Beveridge curve reflect structural declines in matching efficiency m₀ or surges in the separation rate s.

In the baseline Pissarides (2000) formulation, time is continuous, agents are risk-neutral, and future cash flows are discounted at the real interest rate r > 0. Each active employment match produces flow output y > 0 (labor productivity) and pays the worker a real wage w. Maintaining an open vacancy entails a flow recruiting cost κ > 0 per unit of time, while an unemployed worker receives flow non-market income z < y (encompassing unemployment insurance benefits, home production, and the monetized value of leisure).

Let J denote the present discounted value to a firm of a filled job, V_f the present discounted value of an open vacancy, W the present discounted value to a worker of being employed, and U_w the present discounted value of being unemployed. The four continuous-time Hamilton–Jacobi–Bellman (HJB) asset value equations are:

rV_f = −κ + q(θ)(J − V_f)

rJ = y − w + s(V_f − J)

rU_w = z + f(θ)(W − U_w)

rW = w + s(U_w − W)

Under free entry into vacancy creation, firms post new vacancies until the expected capital value of an open vacancy is driven to zero in equilibrium (V_f = 0). Substituting V_f = 0 into the first two asset equations yields two fundamental relationships:

J = κ / q(θ) and J = (y − w) / (r + s)

Equating these two expressions for the value of a filled job gives the Job Creation Curve (JC):

(y − w) / (r + s) = κ / q(θ)

The left-hand side represents the expected present value of net profit flows generated by an active match over its expected duration 1 / (r + s); the right-hand side represents the expected capitalized recruiting cost required to fill a vacancy (the flow cost κ multiplied by the expected duration of a vacancy 1 / q(θ)). Because q(θ) is decreasing in θ, the Job Creation Curve slopes downward in (θ, w) space: higher wages compress firm profit margins (y − w), requiring a shorter expected vacancy duration (lower θ) to justify vacancy posting.

When an unemployed worker and a firm with a vacancy meet, they do not face a competitive spot wage because dissolving the match forces both parties to return to costly, time-consuming search. Consequently, an active match generates a positive total match surplus S:

S = (J − V_f) + (W − U_w) = J + W − U_w

This bilateral monopoly rent is divided between the worker and the firm via generalized axiomatic Nash bargaining. Letting β ∈ (0, 1) denote the worker's bargaining power (and 1 − β the firm's bargaining power), the negotiated wage w maximizes the weighted Nash product:

max_w (W − U_w)^β (J − V_f)^{1 − β}

Since ∂W / ∂w = −∂J / ∂w = 1 / (r + s), the first-order condition requires the worker to receive a fraction β of the total match surplus, and the firm to receive a fraction 1 − β:

W − U_w = βS and J = (1 − β)S

Combining the four HJB equations with free entry (V_f = 0) and the surplus-sharing rule yields the Wage Curve (WC):

w = (1 − β)z + β(y + κθ)

The negotiated wage is a weighted average of the worker's flow outside option z and the total flow surplus generated by the match, which consists of direct productivity y plus the average recruiting cost κθ that the economy saves per unemployed worker because the match is already formed. Because a higher θ increases the job-finding rate f(θ) and improves the worker's threat point U_w during bargaining, the Wage Curve slopes upward in (θ, w) space.

The unique steady-state DMP equilibrium is determined recursively:

1. Tightness and Wage (θ*, w*): The intersection of the downward-sloping Job Creation Curve and the upward-sloping Wage Curve determines equilibrium labor market tightness θ* and the real wage w* via the fundamental surplus equation:

(1 − β)(y − z) / (r + s) = κ / q(θ*) + *βκθ** / (r + s)

1. Unemployment (u*): Given θ*, the Beveridge curve determines the steady-state unemployment rate u* = s / [s + f(θ*)].
2. Vacancies (v*): Equilibrium vacancies follow from v* = θ* u*.

Unlike a competitive Arrow–Debreu economy, a decentralized search market is not automatically Pareto efficient because individual firms and workers fail to internalize two opposing search externalities when deciding whether to enter the market:

• Congestion Externality (Negative): When an additional firm posts a vacancy, it lowers the vacancy-filling rate q(θ) for all other recruiting firms. Similarly, an additional unemployed worker lowers the job-finding rate f(θ) for all other job seekers.
• Thick-Market Externality (Positive): When an additional firm posts a vacancy, it raises the job-finding rate f(θ) for all unemployed workers. Similarly, an additional unemployed worker raises the vacancy-filling rate q(θ) for all recruiting firms.

Arthur J. Hosios (1990) demonstrated that a benevolent social planner who chooses labor market tightness θ to maximize the present discounted value of aggregate net output (output y(1 − u) plus home production zu minus total vacancy costs κθu) equates the marginal social benefit of a vacancy to its marginal cost. Comparing the planner's Euler equation with the decentralized equilibrium equation reveals that the decentralized DMP equilibrium achieves constrained social efficiency if and only if the Hosios condition holds:

β = η = −θ q′(θ) / q(θ)

That is, the decentralized market internalizes both congestion and thick-market externalities only when the worker's Nash bargaining share β equals the elasticity of the matching function with respect to unemployment η. If β > η, workers capture an excessive share of match surplus, wages are too high, firms post too few vacancies, and equilibrium unemployment is inefficiently high (u* > u^SP). Conversely, if β < η, firms over-invest in vacancy creation, generating excessive congestion and inefficiently low unemployment (u* < u^SP).

While the DMP model successfully accounts for long-run structural relationships, Robert Shimer (2005) documented a major quantitative challenge known as the Shimer puzzle: under standard parameter calibrations (where β ≈ η ≈ 0.5 and flow unemployment income z ≈ 0.4y), the canonical DMP model generates less than 10% of the empirical business-cycle volatility observed in U.S. unemployment u, vacancies v, and tightness θ in response to realistic fluctuations in aggregate labor productivity y.

The root of the Shimer puzzle lies in the flexibility of Nash-bargained wages. When productivity y rises during an economic expansion, tightness θ begins to rise, which rapidly strengthens the worker's outside option U_w. Through the Wage Curve w = (1 − β)z + β(y + κθ), wages absorb almost the full productivity gain, leaving the firm's net flow profit (y − w) nearly unchanged and muting the incentive to post new vacancies.

Subsequent macroeconomic research has resolved the Shimer puzzle through several structural refinements:

• Real Wage Rigidity (Hall, 2005): Replacing period-by-period Nash bargaining with sticky wages or social norms within the bargaining set prevents wages from absorbing productivity shocks, amplifying vacancy and unemployment volatility to match empirical data.
• Alternating-Offer Credible Bargaining (Hall & Milgrom, 2008): Modeling wage negotiations via a Rubinstein bargaining model where the threat point during bargaining is temporary delay rather than permanent match breakdown insulates the negotiated wage from cyclical swings in the aggregate job-finding rate f(θ).
• High Non-Market Opportunity Cost (Hagedorn & Manovskii, 2008): Calibrating the flow value of non-work z close to productivity y (e.g., z ≈ 0.955*y*, accounting for home production and leisure alongside low worker bargaining power β ≈ 0.05) makes the percentage change in firm surplus (y − w) highly elastic with respect to small changes in y.
• Endogenous Job Destruction (Mortensen & Pissarides, 1994): Allowing match-specific productivity y ε to experience idiosyncratic shocks causes recessions to trigger endogenous separation waves below a reservation productivity threshold ε_d, capturing both job creation and job destruction dynamics.

Why does unemployment exist in equilibrium in the DMP model?

Equilibrium unemployment persists in the DMP model because matching workers and jobs takes time (f(θ) < ∞) and existing jobs continually experience exogenous or endogenous separations (s > 0). Even when the aggregate number of vacancies equals the number of unemployed workers (θ = 1), informational, geographic, and skill frictions prevent instantaneous pairing, generating a positive steady-state frictional unemployment rate u* = s / [s + f(θ*)].

What is the difference between a movement along the Beveridge curve and a shift of the curve?

A movement along a fixed Beveridge curve occurs when aggregate demand or productivity y changes: during a boom, firms post more vacancies (v rises) and unemployment falls (u falls) along a downward-sloping locus. An outward shift of the Beveridge curve occurs when structural matching efficiency m₀ deteriorates (for example, due to occupational mismatch) or the job separation rate s rises, causing higher unemployment for any given level of vacancies.

How does the DMP model differ from the Shapiro–Stiglitz efficiency wage model?

In the Shapiro–Stiglitz Theory, unemployment arises from asymmetric information and moral hazard inside the firm: employers pay wages above market clearing so that the penalty of becoming unemployed deters workers from shirking. In the DMP model, unemployment arises from search and matching frictions required to form employment relationships, and wages are determined through bilateral bargaining over match rents rather than unilateral efficiency-wage setting.

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2. Mortensen, D. T., & Pissarides, C. A. (1994). Job creation and job destruction in the theory of unemployment. The Review of Economic Studies, 61(3), 397–415. https://doi.org/10.2307/2297896
3. Pissarides, C. A. (1985). Short-run equilibrium dynamics of unemployment, vacancies, and real wages. The American Economic Review, 75(4), 676–690. https://doi.org/10.1257/aer.101.4.1092
4. Hosios, A. J. (1990). On the efficiency of matching and related models of search and unemployment. The Review of Economic Studies, 57(2), 279–298. https://doi.org/10.2307/2297382
5. Shimer, R. (2005). The cyclical behavior of equilibrium unemployment and vacancies. The American Economic Review, 95(1), 25–49. https://doi.org/10.1257/0002828053828572