The Planning Solution in a Textbook Model of Search and Matching: Discrete and Continuous Time Robert Shimer∗ Department of Economics University of Chicago [email protected] February 23, 2004

1

Introduction

This note uses recursive methods to provide a simple characterization of the planner’s solution of the continuous time and discrete time version of the simplest Pissarides (2000) model. I show that the solutions are virtually identical, characterized by a constant vacancy-unemployment ratio θ∗ . In the continuous time model, employment adjusts monotonically to steady state, while it may oscillate in the discrete time model if the length of the time period is particularly long. These results appear to contradict the ﬁndings of a recent note by Bhattacharya and Bunzel (2003), which concludes that the planner’s solution in the discrete time model may exhibit explosive oscillatory cycles. Although the two papers take diﬀerent approaches to solving the same problem, both approaches are correct. The diﬀerence in results apparently reﬂects Bhattacharya and Bunzel’s (2003) failure to impose a necessary transversality condition in their characterization of the planner’s solution. ∗

I am grateful to the National Science Foundation and the Sloan Foundation for financial support.

1

2

Discrete Time

Following Bhattacharya and Bunzel (2003), consider a social planner who in each period chooses a sequence of vacancies levels vt > 0 and employment levels nt ∈ [0, 1] in order to maximize the present discounted value of output net of vacancy costs, max ∞

{vt ,nt+1 }t=0

∞ 

  β t ynt + z(1 − nt ) − cvt ,

(1)

t=0

where y > 0 is output per employed worker, z < y is output per unemployed worker, and c > 0 is the cost of a vacancy. Employment evolves according to nt+1 = (1 − s)nt + µ(θt )(1 − nt ),

(2)

where s ∈ (0, 1) is the separation rate, µ is the rate at which unemployed workers match, and θt ≡ vt /(1 − nt ) is the vacancy-unemployment ratio. I assume for simplicity that µ is strictly concave. Since µ is a probability, I also impose µ : R → [0, 1]. In addition, the planner takes the initial level of employment n0 as given. Rather than formulate this problem as a Lagrangian, I express the problem recursively. In addition, I think of the planner choosing the vacancyunemployment ratio θ rather than the vacancy level v; since n is predetermined, these choices are isomorphic. Let V (n) be the expected present value of output as a function of the beginning-of-period employment level n. This must solve    V (n) = max yn + z(1 − n) − cθ(1 − n) + βV (1 − s)n + µ(θ)(1 − n) . (3) θ

A standard argument based on Blackwell’s theorem ensures that there is a unique solution V to this functional equation. I conjecture that in fact V is a linear function, say V (n) = a0 + a1 n, and solve for the constants a0 and a1 . Given this conjecture, the ﬁrst order condition with respect to θ is −c + βa1 µ (θ) = 0.

(4)

Moreover, this condition is both necessary and suﬃcient, since µ is concave. In addition, the envelope theorem implies   (5) a1 = y − z + cθ + βa1 1 − s − µ(θ) 2

Eliminating a1 between these equations gives an implicit deﬁnition of θ:   y−z 1 − β 1 − s − µ(θ) + θµ (θ) = βµ(θ) c

(6)

Concavity of µ implies that the left hand side is increasing in θ and the right hand side is decreasing in θ. Thus there is at most one solution to this equation, the equilibrium vacancy-unemployment ratio θ∗ .1 Given this, equation (4) or (5) determines the value of a1 . Finally, go back to the Bellman equation to determine a0 . Matching constant coeﬃcients, I get   a0 = z − cθ + β a0 + a1 µ(θ) , which pins down a0 and veriﬁes the functional form of V . Given the constant value of θ = θ∗ , the remainder of the characterization of the planner’s solution is simple: employment evolves so as to satisfy equation (2). This is a linear ﬁrst order diﬀerence equation. The steady state employment level is µ(θ∗ ) . n∗ = s + µ(θ∗ ) If employment starts oﬀ away from steady state, it converges towards steady state. The speed of convergence is governed by the eigenvalue of this difference equation, 1 − s − µ(θ∗ ) ∈ (−1, 1). If this is positive, convergence is monotone. If it is negative, convergence is oscillatory, with employment jumping from above to below its steady state value in alternating periods. Obviously whether convergence is monotone depends on the period length; for short period lengths, both the separation and matching rates are small, and so the eigenvalue is positive. But if one takes the discrete time structure of this model seriously, it is possible to have non-monotone convergence.

3

Continuous Time

In continuous time, the planner solves  ∞   max ∞ e−rt yn(t) + z(1 − n(t)) − cv(t) dt {v(t),n(t)}t=0

(7)

0

1

Standard conditions are needed to ensure existence of an interior solution. See Bhattacharya and Bunzel (2003) for details.

3

subject to n(t) ˙ = µ(θ(t))(1 − n(t)) − sn(t).

(8)

Now the discount rate is r > 0. I ﬁnd it simplest to express this problem recursively again, using the state variable n:    (9) rW (n) = max yn + z(1 − n) − cθ(1 − n) + W  (n) µ(θ)(1 − n) − sn θ

Standard arguments again imply uniqueness of the solution to this functional equation. Again, I conjecture that the Bellman operator is linear, W (n) = b0 + b1 n. Given this conjecture, the ﬁrst order and envelope conditions are −c + b1 µ (θ) = 0   rb1 = y − z + cθ − b1 µ(θ) + s Eliminating b1 between these equations gives

y−z . (10) c Again, the left hand side is increasing in θ and the right hand side is decreasing, so there is at most one equilibrium vacancy-unemployment ratio θ∗ . I again ignore existence issues. As before, I can also calculate the exact values of b0 and b1 , verifying the functional form conjecture. In the continuous time model, the employment rate satisﬁes a linear ﬁrst order diﬀerential equation, with steady state µ(θ∗ ) . n∗ = s + µ(θ∗ ) r + s + µ(θ) − θµ (θ) = µ (θ)

I can verify that the eigenvalue of the linear diﬀerential equation, −µ(θ∗ ) − s, is negative, and so the unemployment rate adjusts monotonically to its steady state value. Since vacancies are proportional to 1 − n, vacancies also adjust monotonically to steady state.

References Bhattacharya, Joydeep, and Helle Bunzel (2003): “Dynamics of the Planning Solution in the Discrete-Time Textbook Model of Labor Market Search and Matching,” Economics Bulletin, 5(19), 1–10. Pissarides, Christopher (2000): Equilibrium Unemployment Theory. MIT Press, Cambridge, MA, second edn. 4

## The Planning Solution in a Textbook Model of ... - Semantic Scholar

Feb 23, 2004 - This note uses recursive methods to provide a simple characterization of the planner's solution of the continuous time and discrete time version of the simplest Pissarides (2000) model. I show that the solutions are virtually identical, characterized by a constant vacancy-unemployment ratio Î¸. â . In the.

#### Recommend Documents

Model Construction in Planning - Semantic Scholar
For all but the simplest domains, this technique has obvious deficiencies. ... programming might be a good trick to use, though, if we can develop a planner that can identify parts of a ... It might post a goal to visit the market to buy supplies.

Model Interoperability in Building Information ... - Semantic Scholar
Abstract The exchange of design models in the de- sign and construction .... that schema, a mapping (StepXML [9]) for XML file representation of .... databases of emissions data. .... what constitutes good modelling practice. The success.

A demographic model for Palaeolithic ... - Semantic Scholar
Dec 25, 2008 - A tradition may be defined as a particular behaviour (e.g., tool ...... Stamer, C., Prugnolle, F., van der Merwe, S.W., Yamaoka, Y., Graham, D.Y., ...

Model of dissipative dielectric elastomers - Semantic Scholar
Feb 3, 2012 - View online: http://dx.doi.org/10.1063/1.3680878. View Table of Contents: ... As a result of their fast response time and high energy density, dielectric ... transducer is an electromechanical system with two degrees of freedom.

Model-based Detection of Routing Events in ... - Semantic Scholar
Jun 11, 2004 - To deal with alternative routing, creation of items, flow bifurcations and convergences are allowed. Given the set of ... the tracking of nuclear material and radioactive sources. Assuring item ... factor in achieving public acceptance

Allocation Of Indivisible Goods: A General Model ... - Semantic Scholar
I.2.4 [Computing Methodologies]: Artificial Intelligenceâ ... INTRODUCTION. Allocation of ... neglected, and in artificial intelligence, where research has focused ...

SNIF-ACT: A Cognitive Model of User Navigation ... - Semantic Scholar
Applications of the SNIF-ACT Model. 6.2. Cognitive Models ..... The model also requires the specification of the attentional weight parame- ter Wj. We have ..... Parse the Interface Objects, Coded Protocol, and Event Log to deter- mine the next ...

SNIF-ACT: A Cognitive Model of User Navigation ... - Semantic Scholar
Users With Different Background Knowledge. APPENDIX A. THE ... SNIF-ACT. 357. 1. Internet use is estimated to be 68.3% of the North American population.

A Taxonomy of Model-Based Testing for ... - Semantic Scholar
ensure a repeatable and scientific basis for testing. ... is insufficient to achieve a desired level of test coverage, different test methods are ... The test specification can take the form of a model, executable model, script, or computer ..... Onl

A computational model of risk, conflict, and ... - Semantic Scholar
Available online 26 July 2007. The error likelihood effect ..... results of our follow-up study which revealed a high degree of individual ..... and Braver, 2005) as a value that best simulated the timecourse of .... Adaptive coding of reward value b

P*: A Model of Pilot-Abstractions - Semantic Scholar
internal to the P* Model, i.e., it is not known by or visible to. Resource Manager. SU. Application. Pilot. Resource. Pilot-Manager. 3) start pilot. SU. 4) submit CU. 2) submit pilot ... 1: P* Model: Elements, Characteristics and Interactions: The ma

A Taxonomy of Model-Based Testing for ... - Semantic Scholar
1. A Taxonomy of Model-Based Testing for Embedded. Systems from Multiple Industry Domains. Justyna Zander, Ina Schieferdecker, and Pieter J. Mosterman.

SNIF-ACT: A Cognitive Model of User Navigation ... - Semantic Scholar
beled with the text that includes cell, patient, dose, and beam. The user's cogni- tive task is to predict the .... clicking on a link, or clicking on the Back button to go back to the previous Web. 364. FU AND PIROLLI .... tracks user keystrokes, mo

A Monte Carlo tuned model of the flow in the ... - Semantic Scholar
Oct 20, 2001 - From our analyses it is concluded that the geothermal heatflux at NGRIP is .... rate âW0 can be calculated directly by assuming energy balance.

Variation of the Balanced POD Algorithm for Model ... - Semantic Scholar
is transformation-free, i.e., the balanced reduced order model ... over the spatial domain Î© = [0, 1] Ã [0, 1], with Dirichlet boundary ..... 9.07 Ã 100. 2.91 Ã 100. MA.

A Survey of the Bacteriophage WO in the ... - Semantic Scholar
cellular symbionts, which infect a wide range of arthropods and filarial ... The success of Wolbachia is best explained by the variety of phenotypes they induce ...

Variation of the Balanced POD Algorithm for Model ... - Semantic Scholar
is transformation-free, i.e., the balanced reduced order model is approximated directly ... one dimensional hyperbolic PDE system that has a transfer function that can be ... y)wy +b(x, y)u(t), over the spatial domain Î© = [0, 1] Ã [0, 1], with Diri

Pneumatics, Basic Level (Textbook) - Semantic Scholar
air service equipment is utilised to prepare the air before being applied to the control system. .... As a control element the directional control valve must deliver the re- ..... edge of the devices concerned and knowledge of the switching charac-.

Pneumatics, Basic Level (Textbook) - Semantic Scholar
Pneumatics has long since played an important role as a technology in .... Drainage points and exhaust outlets in the distribution system .... and the outlet port.

The economics of natural disasters in a developing ... - Semantic Scholar
We focus on Vietnam for several reasons: Vietnam experiences frequent weather-related natural disasters; not unlike other coastal ... geographical regions, the number of disasters and their frequency, the average deaths per disaster (as % of provinci

A SYMMETRIZATION OF THE SUBSPACE ... - Semantic Scholar
The Subspace Gaussian Mixture Model [1, 2] is a modeling ap- proach based on the Gaussian Mixture Model, where the parameters of the SGMM are not the ...

A SYMMETRIZATION OF THE SUBSPACE ... - Semantic Scholar
SGMM, which we call the symmetric SGMM. It makes the model ..... coln, J. Vepa, and V. Wan, âThe AMI(DA) system for meeting transcription,â in Proc.

Numerical solution to the optimal birth feedback ... - Semantic Scholar
Published online 23 May 2005 in Wiley InterScience ... 4 Graduate School of the Chinese Academy of Sciences, Beijing 100049, People's ... the degree of discretization and parameterization is very high, the work of computation stands out and ...