Efficiency Wage, Bargained Wage and Search ...

Efficiency Wage, Bargained Wage and Search Frictions in a Two-Sector General Equilibrium Model By Titas Kumar Bandopadhyay Bagnan College, Bagnan, Howrah, West Bengal, INDIA

Abstract: The main purpose of this paper is to make integration between the job-matching theory and the efficiency-wage theory in a two-sector general equilibrium set-up. We consider two types of labour: high-skill and low-skill. We assume unemployment of both types of labour. We find that frictional wage cannot be determined within the firm if job destruction rate is exogenous. However, it may be determined within the firm if such rate is inversely related to the worker’s efficiency. Our comparative static results show that under reasonable condition trade reforms may lower aggregate unemployment in both cases where job destruction is exogenous as well as endogenous. On the other hand, labour market reforms may lead to ambiguous effects or does not have any effect on aggregate unemployment when job-destruction is inversely related to work’s efficiency. This is quite interesting and hardly found in the standard literature on trade and unemployment. Jel Code: C30, C78, J64. Key Words: efficiency-wage, job-matching, general equilibrium.

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Address of correspondence: Titas Kumar Bandopadhyay. 24/1 B.B.Halder Lane, Shibpur, Howrah-711102, West Bengal,India. E-mail: [email protected] This is financed by the University Grant Commission, INDIA. 1

1. Introduction: In less developed countries though we find a large pool of low-skill labour, there also exists high- skill labour, especially in the service sector which has been growing after globalization. Thus, both high-skill as well as low-skill labour co-exists in many less developed economies like India. This skill differences also causes wage-gap in theses countries. The most striking feature of the high-skill labour market is the job-search and job-match. In this labour market labourers search better jobs and at the same time firms also search better workers. Production starts only when they are matched with each other. Job-searching is a costly and time-consuming process. Thus, when workers are matched with the firms, search costs are saved from both ends and this gives rise to match surplus which can be distributed between workers and firm through wage setting rule. The most widely used wage setting rule is the Nashbargaining solution. This is the basic idea of the job-matching theory as developed by DiamondMortensen and Pissarides (DMP called hereafter, 2000).The DMP (2000) model is a one-sector job-matching model and this model explains the effects of different labour policies on equilibrium rate of frictional unemployment. Other notable works on this front are Diamond (1982a, 1982b, 1984), Pissarides (1979, 1984, 1985a, 1985b, 1986, 2000), Mortensen (1987, 2011), Mortensen and Pissarides (1994, 1998, 1999) etc. All these models explain the existence of frictional unemployment where matching plays the central role to determine frictional unemployment. The job-matching model has been extended by Albrecht et al. (1984), Cole and Rogerson (1989), Andolfatto (1996), Acemoglu and Shimer (1999), Shimer (2004, 2005), Sheng and Xu (2007), Zenou (2008), Hall (1979,), Balanchard and Diamond (1994), Smith and Zenou (2003), Steven (2002), Gariboldi and Moen (2010), Helpman and Redding (2009), Hagedorn et al. (2008), Fujita et al. (2009), Bosch et al. (2008), Bown and, Merkl and Snower (2009), Moscarini (2005), Shimer and Smith (2000), Zaharieva ( 2010), Dolado et al. (2009), etc. All these works either extend the DMP (2000) model or use the DMP (2000) framework to investigate the parametric effects on frictional unemployment. The existence of frictional unemployment can be explained with the help of match friction and on the other hand normal unemployment can be explained in terms of efficiency wage theory. The earliest version of the efficiency wage theory is the ‘Consumption efficiency hypothesis’ of Leibenstein (1957, 1958). According to this hypothesis, efficiency of a worker depends positively on the level of his consumption. This is widely observed in less developed countries 1 where level of consumption per worker is very low. Assuming consumption per worker is a stable function of wage income per worker then we can reasonably link worker’s efficiency to the wage rate that he receives. 1

See Bose (1996) 2

The intuition behind the efficiency wage theory is that an employer desires to pay a worker efficiency wage instead of profit maximizing wage because worker’s efficiency can be maintained in production activity. In the efficiency wage model, a firm generally minimises unit labour cost and in equilibrium, efficiency wage is set where the elasticity of efficiency function 2

with respect to wage is equal to unit. This is the standard Solow-elasticity condition. This implies that efficiency wage is constant. A firm will not pay lower wage to the worker even if there exists excess supply of labour at the efficiency wage. So, the labour market does not clear which leads to normal/ involuntary unemployment in the economy. This is the implication of the efficiency wage theory to the labour market.3

In this paper we integrate job-matching model of DMP (2000) and efficiency wage model of Leibenstein (1957, 1958) in a simple two-sector general equilibrium set-up. We consider two types of unemployment normal and frictional. Normal unemployment is related to the low-skill labour where as frictional unemployment is associated to the high-skill labour. We also assume workers’ efficiency depends positively on the wage rates in both sectors. However, non-frictional wage is determined exclusively by the efficiency wage theory but the frictional wage rate may be determined by the Nash-bargaining theory. Like DMP (2000) model, we consider both cases where job destruction is exogenous as well as endogenous. We investigate the effects of trade 4

reforms and labour market reforms on sectoral as well as aggregate unemployment in a small open economy when job destruction is exogenous as well as endogenous. Our results show that in both of the two cases trade reforms may reduce total unemployment under reasonable condition. However, labour market reforms may give different results in two cases. Such reforms may lower total unemployment when job destruction is exogenous, but it may lead to either no effects or ambiguous effects on aggregate unemployment if job destruction is endogenous. Thus, 2

See Solow (1979). Note that the simple efficiency wage theory has been developed by Stiglitz and Shapiro (1984) where a worker’s efficiency depends on both wage rate and unemployment rate. It has been further developed by ‘Fair wage hypothesis by Agell and Lundborg (1992, 1995) where such efficiency depends on the functional distribution of income. 4 Here, we use a reduction in unemployment benefit and / a fall in the worker’s bargaining power as index of labour market reforms and a fall in the price of the product produced in the frictional sector as an index of trade reforms. See Chaudhuri (2006) in this context. 3

3

our theoretical results may predict that trade reforms are superior to labour market reforms for a small open economy having both frictional as well as normal unemployment.

Our model differs from the existing works. Firstly, we extend the DMP (2000) model in twosector general equilibrium framework by adding efficiency wage theory; secondly, we consider both normal and frictional unemployment in the economy; thirdly, unlike DMP (2000) we endogenise the job destruction rate by linking it to the worker’s efficiency in the frictional 5

sector; fourthly, we compare trade reform policy with the labour market reform policies in a efficiency induced match theoretic general equilibrium model.

2. The Model: We consider a small open economy consisting of two sectors: sector1 and sector 2. Sector 1 is 6

low-skill, non-frictional, informal sector which produces and exports commodity1. The other sector 2 is high-skill, frictional and formal sector which is import-competing sector and produces Commodity2. The prices of the two commodities are exogenous as the country is small. The price of commodity 1 is chosen as numeraire. The two sectors use both labour and capital in 7

production. The production functions are of fixed-coefficient type. Capital is perfectly mobile

5

We find a strong negative relation between wage rate and separation rate in the works of Leonard (1987), Anderson and Meyer (1994), Galizzy and Lang (1998) and Christensen et al. (2005), Capelli and Chauvin (1991). See also Zahaviera (2010). 6

Grinols (1991), Chandra and Khan (1993) and Gupta (1997) have assumed in their models that the products of the urban informal sector are export goods. This is also empirically justified in developing countries like India where many hand-made products are exported to the foreign countries. 7

In the DMP (2000) model it is assumed that capital is hired per unit of labour after when labour is matched with the firm. This implies fixed capital-labour ratio which in turn means complementarity between labour and capital. 4

between the two sectors. So, we have a unique rate of return on capital. However, labour is assumed to be immobile factor due to the skill differences.

8

Both workers and the firms search each other only in sector 2 and workers are employed only when they are matched with the firms. In this sector there is match frictions and this leads to frictional unemployment of high-skill labour. But in sector 1, low-skilled workers are matched with the firms instantaneously. In this sector all workers do not get jobs but all vacancies are filled and so there is also unemployment of low-skill labour in this sector. This unemployment is non-frictional by nature and we may call it normal unemployment. Workers in sector 1 are paid efficiency wages and in the other sector 2 workers are unionized and the wage rate in this sector is determined by the Nash-bargaining solution. We assume job-destruction only in the frictional sector and it is exogenous.

Following DMP (2000) we consider the matching function as m  m  u, v  , where m stands for job-matching in the labour market, u is the frictional unemployment rate and v is the vacancy rate and m1 , m2  0, m11 , m22  0, m12  m21  0. Total flow of matches m  au and total flow of jobs is m  vq . The job arrival rate is

m m  a and the job offer rate is  q . Matching function u v

is assumed to possess CRS property and we may write a 

m m v v  .   q   where   is u v u u

the labour market tightness and q   e ) ˆ ˆ ˆ T ,   P  b 2

(17)

Taking total differentials of (8) and using (17) we get uˆ uˆ uˆ 2  0; 2 , ,  0; 2 . , ,  0 Pˆ ˆ bˆ 2

(18)

From (17) and (18) it is evident that if P falls both W , fall and then f falls and u rises. From 2 2 2 (9) we find X falls, given L , a . Equation (10) shows that X rises when X falls, given 2 2 L2 2 1

a

, a , K . This implies that u must fall, given a , h, L (see Equation (4)). Now, total K1 K 2 L1 1 1

unemployment reduces if     . However, if b /  falls u may move in any direction. 2 L1 K1 Thus, its effects on u ,U are ambiguous. These results lead to the following proposition: 1 Proposition 3: In the presence of both efficiency wage and bargained wage if job destruction is inversely related to the worker’s efficiency in the frictional sector then trade reforms may reduce aggregate unemployment where as labour market reforms create ambiguous effects on total unemployment.

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4. Conclusions: In small open economy frictional unemployment is not so much acute because this type of unemployment is basically related to the high-skill workers which are scarce in such economy. Usually normal unemployment is observed here. After globalization workers have also acquired skill in the small open economy. Thus, it is reasonable to think the co-existence of both normal as well as frictional unemployment in this economy. In our model we find that if job-destruction rate is exogenous, frictional wage cannot be determined within the firm. Nash-bargaining game plays crucial role in this context. However, if job-destruction is inversely related to the worker’s efficiency, frictional wage may or may not be determined within the firm depending on the relative bargaining strength of the worker. The development economists are very much interested regarding the effects of economic reforms on small open economy. So far as the trade reforms are concerned the traditional argument is that trade reforms raises unemployment in the import-competing sector and reduces unemployment in the export sector. Thus, the net effect is ambiguous. However, our efficiency wage induced search theoretic model does not agree with this view. We consider different versions of our matching model. For instance, (i) when non-frictional wage is determined by the efficiency of the worker, frictional wage is determined by the workers bargaining power and job-destruction is exogenous; (ii) Both wage rates are determined by the worker’s efficiency and job-destruction is inversely associated with worker’s efficiency; and (iii) Wage rates are determined like the first version of the model but job-destruction is endogenous. In all the three cases we find that trade reforms may reduce total unemployment if labour share in efficiency adjusted term exceeds capital share in the frictionless sector. However, in the case of labour market reforms we do not get uniform results. If unemployment benefit and/ worker’s bargaining power falls aggregate unemployment may fall when jobdestruction is exogenous and low-skill wage is determined by worker’s efficiency and high-skill wage is determined by worker’s bargaining power. If both wage rates are related to the worker’s efficiency and job-destruction is related to the worker’s efficiency then a reduction in unemployment benefit and / worker’s bargaining power does not have any effect on sectoral as well as total unemployment. On the other hand if wage rate in the frictionless sector is

16

determined by worker; efficiency and frictional wage is determined by worker’ bargains power and job-destruction is endogenous then such policies produces ambiguous effects both on sectoral as well as total unemployment.

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APPENDICES

Appendix A: Derivation of Nash-wage equation when  is exogenous:

The Nash–bargaining problem is

Max   W  U    J  V 1    (A.1)

w 2 The first order condition for maximization is

  J V 

  W  U   1   W  U   J V   0 w w 2 2

Using value equations

19

(A.2)

and zero-profit condition V  0 into (A.2) one gets

  1  e f ,W   2   J  V  1   W U   0      r   f    Here, for interior solution to exist, e

f ,W 2

(A.3)

 1 . So From (A.3) we get efficiency wage for the

frictional sector as

 C  C w b  r    2 q   1    1   

(A.4)

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The two value equations of unemployment ( U ) and employment ( W ) for the frictional sector are: rU  b   q  W  U  and rW  W - W  U  , where b is the unemployment benefit in 2 the frictional sector. 18

Let  

C = frictional cost of labour. Using this into (A.4) we may write Equation ((7) 1    q  

in the main text.

Appendix B: Effects of changes in P , b,  on W , , u , u , E when e  1 and  is f ,W 2 2 1 2 2 exogenous:

Total differentials of Equations (6) and (7) are W  r   C a   aL 2  L2  ˆ  Pˆ  2 1  e  Wˆ   e    f , W 2 q ,  2 f P q  P      2  2  2  ( a =a =0 due to fixed coefficient technology) K1 K 2

(B.1)

W       2 1  e   Wˆ2   r    eq,   q    ˆ  bbˆ    r     q    e ˆ f , W    f  2  

(B.2)

Solving (A.1) and (A.2) we get   ˆ   r    eq,   q    P2   1  Wˆ    2  r    C e aL2 bbˆ   r     q  e ˆ       q,  P q    2



ˆ 



     





W   1  aL 2 ˆ   bb    r     q    e ˆ  Pˆ  2 1  e  2 f  f ,W   P  2  2 

(B.3)

(B.4)

where W   aL 2    2 1  e f ,W  P f  2 2



  r    eq, 

  C         q     0 q       



19

(B.5)

Note that Equation (6) and Equation (7) are the conventional labour demand and labour supply curves. The slopes of the labour demand curve and the labour supply curve are given as

dW 2 d

dW 2 d

 DD L

 SS

 r    C eq,

W 2

W    q   2 1  e f ,W  f  2

 

0

(B.6)

   q    W q,   2 0 W    2 1  e f ,W  f  2

  r    e

L

(B.7)

Stability of the frictional labour market equilibrium implies that dW 2 d

 SS

L

dW 2 d

    C      q     0    0.   r    e   q,   q        DD L

(B.8)

Using (A.3) and (A.4) we may write

Wˆ  2     r    e   q     0  ˆ q ,   P  2

(B.3.1)

  Wˆ a 2   1 r   C e L2 b   0 q,  P    q   bˆ  2 

(B.3.2)

  Wˆ a 2   1 r   C e L2   r     q    e ˆ   0 q,  P     q   ˆ  2 

(B.3.3)

 1 W    . 2 1  e 0 f ,W   f  Pˆ 2  2

(B.4.1)

ˆ

ˆ bˆ



 1 aL2 W2  . b 1  e f ,W   0  P f  2 2

(B.4.2)

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W   ˆ 1  aL2     r     q    e  2 1  e 0  f  f ,W  ˆ   P2  2 

(B.4.3)

a X a X Let E  L1 1 , E  L2 2 and E  E  E 1 2 1 2 h f

E Eˆ   Eˆ  1    Eˆ , where   1 1 2 E

(B.5)

Now,





Eˆ  Xˆ a , h are constant 1 1 L1 Eˆ  Xˆ  fˆ a is constant 2 2 L2  Eˆ   Xˆ  1    Xˆ  1    fˆ 1 2





(B.5.1)

From (10) we get



Xˆ   K 2 Xˆ . Substituting this into (A.5.1) we get 1 2  K1 Eˆ 

Xˆ





2     1    fˆ K1 K1



(B.5.2)

Now, E

a

X

a

X

  1  L1 1  L1 1    ( E  L) E

hE

hL

L1

where    share of labour in sector 1 in efficiency term L1

If we assume that    , then automatically  > . This implies that L1 K1 K1 Xˆ ˆ E  2     1    fˆ  0 K1  K1      





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Appendix C: Effects of changes in P , b,  on W , , u , u when e  1 and     f  2 2 1 2 f ,W 2 Total differentials of (6),(7) are given as

  Ca W   C  aL 2 ˆ 2 L 2   ˆ  Pˆ  1  e   e . W  r   f e      q,   f ,W  f ,W q    P 2  2 q  P    f  2 2  2  2 

W      TM   fe  2 1  e  Wˆ  TM   r    e   q    ˆ   f , W f , W 2    f   2 2 

(C.1)

(C.2)

   =bTbˆ  T be  M   r     q     e  e   ˆ    T,  T,

where T 

1   

1   M 

,M 

1  eL  1  eF 

In simplified form we may write

A Wˆ  B ˆ  Pˆ 1 2 1 2

(C.1.1)

A Wˆ  B ˆ  bTbˆ  Dˆ 2 2 2

(C.2.1)

Where W   C  aL 2 20 A   2 1  e   e 0   1  f  f ,W  f ,W q    P  2 2   2

20

dW 2 d

W B  2 1 0 A 0 1 A 1 DD L 22

  Ca L2   0 B   r    f   e 1  q, q   P   2

W    A   2 1  e  TM   fe 0  2  f  f ,W  f ,W   2 2 

 B  TM   r    e   q     0   2 

   D  T be  M   r     q     e  e   T ,  T ,     

Solving (B.1.1) and (B.2.1) we get

Wˆ B Wˆ B bT Wˆ BD  2   2  0, 2   1  0, 2   1  0( D  0 as e  e assumed) T,     Pˆ ˆ bˆ 2

ˆ

A ˆ A bT ˆ A1D   2  0,  1  0,  0    Pˆ ˆ bˆ 2

(C.4)

(C.5)

Taking total differentials of (8) and after simplification we get   q     ˆ uˆ  e .e .Wˆ  1  e   2   f    q     , f f ,W 2  q,     2 

Using (B.4) and (B.5) from (B.6) we get

23

(C.6)

  Wˆ uˆ  q   2  e  2 .e ˆ ˆ  , f f , W   f    q    P 2  P2 2         

   1  e q,      

   ˆ  ˆ  P  2

   0  

(C.6.1)

   Wˆ  uˆ  q     ˆ   2  e  2   1  e .e , ,  0 q,    bˆ     f    q     , f f ,W2  bˆ   bˆ               

(C.6.2)

   Wˆ  uˆ  q      ˆ   2  2 e   .e  1 e , ,  0 q,    ˆ     f    q     , f f ,W2  ˆ   ˆ                

(C.6.3)

24

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