Volume 29, Issue 3. Equilibrium in Matching Models with Employment Dependent Productivity

Transcription

1 Volume 29, Iue 3 Equilibrium in Matching Model with Employment Dependent Productivity Gabriele Cardullo DIEM, Faculty of Economic, Univerity of Genoa Abtract In a tandard earch and matching framework, the labor market preent friction while in the competitive product market the demand i infinitely elatic.to have a more realitic framework, ome model abandon the aumption of infinite elaticity and conider a two-tier productive cheme in the good market. In thi paper, I etablih the condition that are ufficient for the exitence and the uniquene of a teady-tate equilibrium for thi kind of model. I alo notice that ome tandard aumption about the production and matching technology (a Cobb-Dougla function) do not fulfill uch condition and o may hinder the exitence of an equilibrium. I am indebted to the Aociate Editor Etienne Lehmann for everal inightful remark. I alo thank David de la Croix, Fabien Potel-Vinay, Frank Portier, Bruno Van der Linden, and the participant to the Matagne workhop for ueful dicuion and comment. I acknowledge financial upport from the Belgian Federal Government (Grant PAI P5/21, ``Equilibrium theory and optimization for public policy and indutry regulation''). The uual diclaimer applie. Citation: Gabriele Cardullo, (2009) ''Equilibrium in Matching Model with Employment Dependent Productivity'', Economic Bulletin, Vol. 29 no.3 pp Submitted: Apr Publihed: September 18, 2009.

2 1 Introduction In a tandard earch and matching model, the labor market preent friction, wherea in the perfectly competitive product market the demand i infinitely elatic, o that an increae in upply doe not affect the equilibrium price (Mortenen and Piaride 1999, and Piaride 2000). In order to introduce a more realitic framework for the good market, ome cholar (e.g. Joeph, Pierrard, and Sneeen 2004, and Pierrard 2005, and Cahuc and Zylberberg 2004, p ) abandon the aumption of an infinite elaticity of demand and conider a two-tier productive cheme. Different type of worker (uually, low-killed and highkilled one) are hired in intermediate good ector. Such good face a decreaing demand from a final repreentative firm that produce the unique conumption good 1. Paper of thi kind make often ue of numerical imulation and cant attention i paid to analytical propertie. Thi paper take a different tand. I conider a implified framework in which there are only two intermediate ector and I look for the condition under which a (unique) teadytate equilibrium exit. Uniquene i guaranteed by the aumption of contant return to cale (henceforth, CRS) in the final good function, wherea a ufficient condition for the exitence concern the difference between the marginal productivity and the income received when unemployed. If it i poitive a the level of labor market tightne in both ector tend to zero, then an equilibrium exit. Since thi condition i not fulfilled by impoing a Cobb-Dougla technology both in the matching and in the production function, introducing uch a functional form may hinder the exitence of an equilibrium. 2 The Model 2.1 Production Technology Aume an economy with one final good (the numeraire), two intermediate good ector and two type m and n of infinitely-lived and rik-neutral worker. The good market are perfectly competitive. Each producer of an intermediate good hire only one type of worker. Moreover, every m-killed (repectively, n-killed) employee produce one unit of the intermediate good m (rep. n). So E i (i {m, n}) denote both the amount of the i intermediate good produced and the number of employee in the i-th ector. The final good production function exhibit CRS and i written a: Y = F(E m, E n ), with F > 0 and 2 F 2 < 0, i {m, n}. (1) The two input are p-ubtitute ( 2 F E m E n > 0). 2 Let p i denote the real price of the intermediate good i. Cot minimization in the final ector lead to p i = F(E m, E n )/, with i {m, n}. Further, the value of home production i denoted by b i > 0. 1 Acemoglu (2001) ha alo contructed a imilar model, but with one deciive difference: worker are identical ex ante and can be employed in high-paid or low-paid job. 2 F I alo aume one Inada condition: lim E 0 i = +. Impoing a condition a E i + i uele, ince in thi model the upper bound of both input are given by the labor force that ha a poitive finite value. 2

3 2.2 Search Technology The model i developed in teady tate. Time i continuou and r denote the dicount factor. Each type of worker can be either unemployed or be employed in hi ector. The labor market i perfectly egmented, meaning that every i-type worker can be hired only by firm in the i ector. The matching function i written repectively M i = m(u i, V i ), with U i being the number of unemployed people and V i the number of job vacancie in ector i. It i aumed to be increaing, concave and homogeneou of degree 1. Labor market tightne i defined a θ i V i U i. The job filling rate i q(θ i ) M i /V i = m( 1 θ i, 1), q (θ i ) < 0, wherea the job finding rate i equal to α(θ i ) M i /U i = θ i q(θ i ), with α (θ i ) > 0. 3 At an exogenou rate φ i a match i detroyed. In teady tate, the tock of individual in each poition are contant. With an exogenou ize of the labor force, L i, the employment level i given by: E i = E i (θ i ) α(θ i ) φ i + α(θ i ) L i, i {m, n}. (2) Notice that E i(θ i ) > 0 and, from the condition in footnote 3, lim θi + E i (θ i ) = L i. 2.3 Vacancy Supply and Wage-Setting Curve Once a worker find a firm with a vacant job, a urplu of the match arie. The Nah bargaining olution i aumed in order to plit the urplu, wherea a zero profit condition i impoed on the demand ide of the market. The characterization of the model i tandard. Hence, I directly preent the equilibrium equation (obtained by merging the o-called vacancy upply and wage-etting curve) and refer to Piaride (2000, chapter 1) for the primitive condition under which they are derived: G i (θ i, θ j ) (1 β i ) [ ] F (E i (θ i ), E j (θ j )) b i k i ( r + φi q(θ i ) + β i θ i ) = 0, (3) with i {m, n}, i j, k i i the flow cot of poting a vacancy in unit of final good, and β i repreent worker bargaining power. Function G i = 0 i the equilibrium condition in labor market i and depend on θ j only through the marginal productivity F. Differentiating G i with repect to θ i, I obtain: dg i dθ i = A i + B i, (4) with [ ] A i k i (r + φ i ) q (θ i ) q(θ i ) β 2 i < 0 and B i (1 β i ) 2 F 2 E i (θ i) < 0, i {m, n}. I alo differentiate G i with repect to E j : dg i 2 F C i, j = (1 β i ) E j dθ j E (θ j) > 0 j with i, j {m, n}, i j. (5) 3 Moreover, I impoe that lim θi 0 q(θ i ) = + and lim θi + α(θ i ) = +. 3

4 3 Equilibrium 3.1 Conditional Equilibrium in each market The firt exitence reult conit in howing under which aumption there exit a θ i uch that G i (θ i, θ j ) = 0 hold, conditional on θ j. Note that dg i dθ i < 0 and lim θi + G i (θ i, θ j ) for the condition impoed in footnote 3 and the limit behavior of equation (2). Thi reult hold for any value of θ j [0, + ). If lim θi 0 G i (θ i, θ j ) > 0 θ j [0, + ), I could ue the intermediate value theorem to prove the exitence of a conditional equilibrium. However, I can compute uch a limit only for value of θ j that are not cloe to zero 4. In fact, a both θ i and θ j (and conequently E i and E j ) tend to 0, the input of the marginal productivity take an indeterminate form 5 0 and it i not poible to acertain the ign of thi 0 expreion6. To rule out thi poibility, two alternative aumption are needed. Lemma 1 ummarize the reult. Lemma 1 There alway exit a θ i (0, + ) that olve G i (θ i, θ j ) = 0 θ j [0, + ), i, j {n, m}, i j if, alternatively,: 1. lim θi 0 E i = E i (θ i ) = 0 and F (E i, E j ) > b i θ i = θ j = 0) with i, j {m, n}, i j. θ i and θ j [0, + ) (except for 2. lim θi 0 E i = E i (θ i ) = ǫ i > 0, and F (E i = ǫ i, E j = ǫ j ) > b i with i, j {m, n}, i j.. Proof. CASE 1 Since lim θi 0 E i = E i (θ i ) = 0, i {m, n}, the et θ i [0, + ) and E i [0, L i ] are repectively the domain and the range of the function E i (θ i ). Conider the term inide the quare bracket in (3) a θ j = 0; it i poitive, decreaing in θ i (0, + ), and tend to F (L i, E j = 0) b i > 0 a θ i +. The econd term in (3) i a ray tarting from the origin and that tend to + a θ i +. So, there exit a θ i [0, + ) uch that G i (θ i, θ j = 0) = 0. The ame reaoning can be applied for any θ j > 0. CASE 2 Since lim θi 0 E i = E i (θ i ) = ǫ i > 0, the domain and range of the function E i (θ i ) become repectively θ i [0, + ) and E i [ǫ i, L i ], i {m, n}. Impoing F (E i = ǫ i, E j = ǫ j ) > b i implie that lim θi 0 G i (θ i, θ j ) > 0 θ j 0. I can apply the intermediate value theorem and conclude that there exit a θ i [0, + ) uch that G i = 0 θ j [0, + ). 4 In thi cae, it tend to infinity for the Inada condition in footnote (2) and if lim θi 0 E i = E i (θ i ) = 0. 5 Recall that the firt derivative of CRS function can be expreed in term of the ratio of the two input. 6 On the contrary, there are no difficultie a θ j +. Thi i becaue lim θj + E j = L j and F take a poitive finite value a E j = L j. So there exit a θ i > 0 that olve G i (θ i, θ j + ) = 0. 4

5 Some Example CASE 1: CES production function with > 1. Conider a CES production function with > 1: Y = [ E 1 m + E 1 n ] 1 where F (E i, E j = 0) = 1 i, j {n, m}, i j. For Lemma 1 (CASE 1), impoing b i < 1 i ufficient to enure the exitence of a conditional equilibrium. CASE 2: CES matching function with > 1. Conider a CES matching function: M i = [V 1 i + U 1 i ] 1 and α(θ i ) = [ ] θ 1 1 i + 1 i {n, m}. ) If > 1, lim θi 0 E i = E i (θ i ) = L i 1+φ i. Impoing F L i L 1+φ i, j 1+φ j > b i, i, j {n, m}, i j, i ufficient to enure the exitence of a conditional equilibrium. Cobb-Dougla technology. ( A Cobb-Dougla production function Y = aene γ m 1 γ doe not belong to CASE 1. In fact, F = γy/e i = 0 a E j = 0, i, j {n, m}, i j. Similarly, with a Cobb-Dougla matching function M i = av γ i U1 γ i, α(θ i ) = aθ γ i, and lim θi 0 E i = E i (θ i ) = 0, o condition of CASE 2 are not fulfilled. Impoing a Cobb-Dougla formulation both in the the production and in the matching technology implie that the conditional equilibrium in market i may not be atified for value of θ j cloe to 0. A we will ee in the next paragraph, thi may caue the abence of the general equilibrium. 3.2 General Equilibrium I firt apply the implicit function theorem. Uing (4) and (5), I get: dθ i dθ j Gi =0 = C i,j A i + B i > 0 with i {m, n}, i j. (6) G i = 0 define a monotonouly increaing relationhip in (θ i, θ j ) pace. At the general equilibrium, the value of labor market tightne olve the following ytem: { Gn (θ n, θ m ) = 0 (7) G m (θ m, θ n ) = 0 Propoition 1 preent the reult. Propoition 1 If the condition in Lemma 1 are atified, a teady tate equilibrium exit and i unique. Proof. I denote θ m = g m (θ n ) the explicit function of G m (θ m, θ n ) = 0, and θ m = g n (θ n ) the explicit function of G n (θ n, θ m ) = 0. Both function are monotonically increaing. Let alo θ n = gm 1 (θ m ) and θ n = gn 1 (θ m ) denote their invere function. 5

6 EXISTENCE For the condition impoed in Lemma 1, there alway exit a θ m (0, + ) that olve G m (θ m, θ n = 0) = 0. Thi i tantamount to writing that θ m = g m (θ n ) ha a poitive intercept in the vertical axi, that I denote with χ m. Similarly, there alway exit a θ n (0, + ) that olve G n (θ m = 0, θ n ) = 0, implying that θ m = g n (θ n ) ha a poitive intercept in the horizontal axi, that I denote with χ n. Moreover, a noticed in footnote 6, there exit a θ m (0, + ) that olve G m (θ m, θ n + ) = 0. So, lim θn + g m (θ n ) = Ψ m R +. Since there alo exit a θ n (0, + ) that olve G n (θ m +, θ n ) = 0, one get that lim θm + gn 1(θ m) = Ψ n R +. The domain and the range of θ m = g m (θ n ) are repectively [0, + ) and [χ m, Ψ m ]. The domain and the range of θ m = g n (θ n ) are repectively [χ n, Ψ n ] and [0, + ). Since both function are monotonouly increaing, they mut interect at leat once (ee Figure 1). UNIQUENESS I define H(θ n ) g m (θ n ) g n (θ n ). If H(θ n ) i a monotonic function in the neighborhood of the equilibrium teady-tate, the equilibrium i unique. Let θ m and θ m denote the equilibrium level of tightne, H (θ n ) = g m (θ n ) g n (θ n ) < 0 i a ufficient condition for the uniquene of the equilibrium. Thi implie: dθ m dθ n Gn(θ n,θ m) =0 > dθ m. (8) dθ n Gm(θ m,θ n)=0 From (6), one derive: dθ m dθ n Gn(θ n,θ m )=0 = B n + A n C n, m (9) dθ m = dθ n Gm(θ C m, n (10) m,θ n )=0 Bm + A m I multiply the numerator of (9) by the denominator of (10) and the numerator of (10) with the denominator of (9). I get four poitive term on the LHS and only one poitive term on the RHS. For (8) to hold, the four poitive term on the LHS mut be greater than the term on the RHS. One of the term on the LHS i: B mb n = (1 β m )(1 β n ) 2 F E m 2 (E(θ n), E(θ m)) 2 F E n 2 (E(θ m), E(θ n)) E (θ m)e (θ m) (11) The poitive term on the RHS i: [ Cn, m C m, n = (1 β 2 F m)(1 β n ) (E(θm E m E ), E(θ m )) n ] 2 E (θ m )E (θn ). (12) Expreion (11) and (12) are equal becaue of the Euler formula for function with ( ) contant return to cale, that i 2 F 2 F E 2 n E 2 m = 2 F 2. E n E m Then, inequality (8) i verified. The equilibrium i unique. 4 Final Remark A Cobb-Dougla technology both in the matching and in the production function may hinder the exitence of a teady-tate equilibrium. The reaon i that the function 6

7 θ m = g m (θ n ) and θ m = g 1 n (θ n ) may not exit for value of repectively θ n and θ m cloe to 0, point 1 of the exitence proof cannot be etablihed, and the two curve may not cro each other. Figure 2 illutrate thi cae. Thi i not a rare poibility. Conider a tandard parametrization uch a the one performed by Cahuc and Zylberberg (2004, page 623) for thi kind of model. The unit of time correpond to one year, r = 0.05, β i = 0.5, φ i = 0.15, h i = 0.1 i. The matching function i Cobb-Dougla with a parameter γ i = 0.5 i. Contrarily to Cahuc and Zylberberg (that conider a CES production function with elaticity of ubtitution equal to 1.5), I impoe a Cobb-Dougla formulation even in the production technology. A n i an efficiency parameter for worker in market n and it i equal to 1.5. For value of b m 0.9 and b n 0.95, no equilibrium exit 7. Keeping the Cobb-Dougla formulation and impoing either b i = ρ i w i with 0 < ρ i < 1 or b i = 0 i {n, m} would not prevent from the abence of an equilibrium. The reaon i that, even under thee hypothee, the olution of the equation G(θ i, θ j = 0) = 0 i indefinite the exitence proof cannot be made 8. The reult of Propoition 1 hould be een a a warning about the ue of a tandard Cobb-Dougla technology in model of thi kind. Reference Acemoglu, D. (2001) Good Job veru Bad Job Journal of Labor Economic 19, Cahuc, P. and A. Zylberberg (2004) Labor Economic MIT Pre: Cambridge (MA). Joeph, G., Pierrard, O., and Sneeen, H. (2004) Job Turnover, Unemployment and Labor Market Intitution Labour Economic 11, Mortenen, D. and C. Piaride (1999) New Development in Model of Search in the Labor Market, in Handbook of Labor Economic,Volume 3B by O. Ahenfelter and D. Card, Ed., Elevier Science Publiher (North-Holland): Amterdam. Pierrard, O. (2005) Impact of Selective Reduction in Labor Taxation Applied Economic Quarterly 51, Piaride, C. (2000) Equilibrium Unemployment Theory MIT Pre: Cambridge (MA). 7 The file containing the numerical exercie i available on requet. 8 In other term, the two function may exit for value of θ j 0 (ince a b i = 0 or b i = ρ i w i, it i ufficient that the marginal productivity i lightly poitive for lim θi 0 G i (θ i, θ j ) to be poitive) but are till not defined along the axi (i.e. a θ j = 0). One could not rule out the cae that the only poible point of interection of the two curve i the origin, that however doe not belong to the domain of g m (θ n ) and gn 1 (θ m ). See Figure 2 on the right. 7

8 Figure 1: Exitence and uniquene of the equilibrium. Figure 2: No equilibrium. On the Left: A θ n 0 (rep. θ m 0 ), the function g m (θ n ) (rep. g n (θ n ) ) i not defined. On the Right: At the origin, the function g m (θ n ) and g n (θ n ) are not defined. 8