Specialization, Matching Intensity and Income Inequality of Sellers

Transcription

1 MPRA Mnich Personal RePEc Archive Specialization, Matching Intensity and Income Ineqality of Sellers Konstantinos Eleftherio and Michael Polemis University of Piraes 1 October 016 Online at MPRA Paper No , posted 16 October :39 UTC

2 Specialization, matching intensity and income ineqality of sellers Konstantinos Eleftherio ; y Michael L. Polemis Abstract We develop a simple model with heterogeneos agents and search frictions to stdy how increases in matching intensity between byers and sellers determine the level of income ineqality among sellers. Or ndings indicate that a redction in search frictions leads to higher ineqality and indces byers to prchase goods and services only from specialized sellers. JEL classi cation: C78; O30 Keywords: game theory; income ineqality; matching; technology; vale fnctions 1 Introdction The impact of technological change on income ineqality is well-docmented in the literatre (Acemogl, 00). Technological advances increase the demand for skilled labor leading to a widening of the wage gap between skilled and nskilled workers (Acemogl, 1998; Card and Dinardo, 00). Moreover, the redction of the power of low-skilled workers de to closer monitoring of their e ort, triggered by adoption of new technologies, (Skott and Gy, 007) and Department of Economics, University of Piraes, 80 Karaoli & Dimitrio Street, Piraes , Greece. mpolemis@nipi.gr (Polemis); kostasel@otenet.gr (Eleftherio). y Corresponding athor. Tel: ; Fax:

3 the so-called general prpose technology (Aghion et al., 00) -as expressed by skill transferability and vintage capital compatibility- can lead to higher ineqality. In the crrent paper, the e ect of technology on income ineqality is examined within a di erent context; we tilize a simple matching model to stdy how increases in matching intensity between byers and sellers 1 a ects income ineqality among sellers. Or ndings indicate that a redction in frictions de to technological advancements leads to eqilibrim otcomes where byers prchase goods and services only from specialized sellers. In this case, income ineqality is high. In contrast, when frictions are high, nding a specialized seller is more di - clt leading to matches with non-specialized sellers and hence a less polarized distribtion of sellers earnings. Or reslts are consistent with the ndings of Brynjolfsson et al. (016). In this stdy, a positive trend in income ineqality among professional sellers in ebay is identi ed. Frthermore, Brynjolfsson et al. (016) conclde that growth is experienced most likely by sellers providing specialized o erings. The rest of the paper is strctred as follows. The next Section presents the model while Section 3 discsses market eqilibria. Or main reslts are presented in Section 4. Finally, Section 5 concldes the paper. The Model.1 Environment We consider a market for a service. The market comprises of a large nmber (normalized to one) of byers and the same nmber of sellers who live in 1 These increases can be translated into redctions in search frictions enabling byers to nd the seller meeting their needs best. Decreases in frictions may be cased by developments in information technology (e.g., development of e-commerce).

4 continos time. Byers and sellers are of two types; h and w. The proportion of h-type byers and h-type sellers is and, respectively (where 0 ; 1). Sellers and byers meet each other at a Poisson arrival rate (see Mortensen, 1986). The tility that byers get from the service they receive from sellers is. Seller i has cost (distility of work) zero for i job, i = h; w, where seller i has cost c (0; ] for not i job. As soon as they match, agents bargain over the price throgh a Nash bargaining process (we assme 50=50 bargaining power). When agents of di erent type meet, then the match will form if the vale of c is small enogh and more precisely if it is below a threshold vale (reservation vale). We assme that all participants are risk netral disconting the ftre at the same rate r. Byers leave the market after trading and they are replaced by identical ones (clones). Sellers are in nite lived and stay in the market forever.. Exchange The price that sellers charge byers will be p ij = arg max p ij [ p ij V b j ] 1 [pij c] 1 (1) where i; j = h; w. The rst sbscript denotes the type of seller and the second the type of byer. Hence p ij is the price charged by i seller to j byer. V b j represents the vale of being a j type byer. contination vales. Solving (1) we get The otside options are the p ij = 1 [ + c V b j ] () h (w) type byers demand sellers for h (w) type jobs. h (w) type sellers are specialized in h (w) type jobs. 3

5 where c = 0 if i = j, and c (0; ] if i 6= j. 3 Market Eqilibria Eqilibria are patterns of trading from which no one wants to deviate. Hence or objective, is to identify candidate eqilibrim patterns of trade and nd the portions of the parameter space for which they are self-sstaining. In type 1 eqilibrim pattern all agents trade with each other. Let V tn i be the vale 3 of being a grop t individal of type i when everyone behaves according to type n trading pattern. 4 In the case of type 1 eqilibrim pattern we have that rv b1 h = [( p hh ) + (1 )( p wh ) V b1 h ] (3a) rv b1 w = [( p hw ) + (1 )( p ww ) V b1 w ] (3b) rv s1 h = [p hh + (1 )(p hw c)] (3c) rv s1 w = [(p wh c) + (1 )p ww ] (3d) By sbstitting p ij s and doing the algebra we get (r + b1 )Vh = [(=) + (1 )( c)=] (4a) (r + b1 )Vw = [( c)= + (1 )=] (4b) 3 where t = byer (b), seller (s). 4 Alternatively, Vi tn is the disconted expected vale of a grop t individal of type i nder n eqilibrim trading pattern. 4

6 rvh s1 = [ b1 ( Vh ) + rvw s1 = [ b1 ( c Vh ) + (1 ) ( c Vw b1 )] (1 ) ( Vw b1 )] (4c) (4d) In order to nd the portions of the parameter space which ensre the existence of type 1 eqilibrim, we have to consider whether an individal of any type and grop may want to deviate from the above de ned trading pattern. Firstly, we ensre that participation in the market is worthwhile. Hence, we get the following participation (rationality) constraints; Vh b1; V w b1 ; Vh s1 and Vw s1 0 (soltion of (3a)-(3d) shold satisfy the rationality constraints). Therefore, trading with agents of di erent type is worthwhile only if the vale of trading (and re-entering the market 5 ) exceeds that of keep searching for a trading partner. In other words, we get the following p wh Vh b1 ; p hw Vw b1 ; p wh + Vh s1 c Vh s1 ; p hw + Vw s1 c V s1 w Hence the relevant incentive compatible constraints are p wh V b1 h ; p hw V b1 w ; p wh c 0; p hw c 0 (5) By sbstitting the relevant vales for p ij s from () and doing some algebra, we get that the above constraints are satis ed if the following conditions hold. and 5 This holds for sellers. 1 b1 [ c Vh ] 0 (6) 5

7 1 b1 [ c Vw ] 0 (7) It can be easily shown that if (6) and (7) are satis ed then the trade with agents of the same type is worthwhile as well. By sbstitting (4a) and (4b) into (6) and (7), respectively and solving with respect to c we get the following reservation vales: where = =r and ~c h () = [1=(1 + )] (8) ~c w () = [1=(1 + )] (9) where = (1 )=r. If c exceeds the above reservation vales then type 1 trading pattern is no longer an eqilibrim since there is an incentive for deviation. If the proportion of h-type sellers is greater than 1= then ~c w > ~c h and therefore type 1 trading pattern is an eqilibrim if c ~c h. Doing the same for = 1= and < 1= and smmarizing all the above reslts we get the following proposition. Proposition 1. Type 1 trading pattern is an eqilibrim if c ~c h () for > 1=; c ~c w () for < 1=; c ~c = ~c w () = ~c h () for = 1=, given ; r. In type eqilibrim pattern h-type byers do not trade with w-type sellers (bt w-type byers trade with h-type sellers). The relevant eqations for byers and sellers will be (r + =)V b h = (=) (10a) 6

8 (r + b )Vw = [( c)= + (1 )=] (10b) rvh s = b ( Vh ) (10c) rvw s = [ b ( c Vh ) + (1 ) ( Vw b )] (10d) Given the above eqations and Proposition 1, we get Proposition. Type trading pattern is an eqilibrim if ~c w c > ~c h and > 1=, given ; r. In type 3 trading pattern w-type byers do not trade with h-type sellers (bt h-type byers trade with w-sellers). The relevant eqation for each individal is (r + b3 )Vh = [(=) + (1 )( c)=] (11a) [r + (1 )=]V b3 w = (1 )= (11b) rvh s3 = [ b3 ( Vh ) + Following the same logic, we get (1 ) ( c Vw b3 )] rvw s3 (1 ) = ( Vw b3 ) (11c) (11d) Proposition 3. Type 3 trading pattern is an eqilibrim if ~c h c > ~c w and < 1=, given ; r. In type 4 eqilibrim byers of type i do not trade with sellers of type j. The relevant eqations are (r + =)V b4 h = (=) (1a) [r + (1 )=]V b4 w = (1 )= (1b) 7

9 rvh s4 = b4 ( Vh ) (1c) rvw s4 (1 ) = ( Vw b4 ) (1d) Proposition 4. Type 4 trading pattern is an eqilibrim if c > ~c w for > 1=; c > ~c h for < 1=; c > ~c = ~c w = ~c h for = 1=, given ; r: Finally, when c = ~c h or c = ~c w there is a continm of mixed strategy eqilibria. In sch cases, we assme that all individals (weakly) prefer to stick to the trading pattern speci ed each time. 4 Matching rate and income di erentials Figre 1 graphs ~c w and ~c h against. The regions marked 1,, 3 and 4 correspond to the vales of c and for which the niqe eqilibrim trading patterns are respectively Type 1 (every meeting generates transactions), Type ( h-type eqilibrim ),Type 3 ( w-type eqilibrim ) and Type 4 (i type byers do not trade with j type sellers). Moreover ~c h () is decreasing and convex, ~c w () is increasing and convex, ~c h (1) > 0 and ~c w (1) = : By di erentiating ~c w and ~c h with respect to matching rate, we = r (r + ) 0 = r(1 ) [r + (1 )] 0 (14) The above derivatives indicate the amont by which ~c w and ~c h will decrease if the matching rate will increase by one per cent. Since both derivatives are negative, an increase in will rotate (rightward for ~c w and leftward for ~c h ) the 8

10 graph of each reservation vale arond. The new reservation crves will be ~c 0 w and ~c 0 h as illstrated in gre. [Figre 1 abot here] [Figre abot here] As we can see from gre an increase in matching rate decreases area 1 and increases area 4. In other words, an increase in increases the range of c s leading to a type 4 eqilibrim (byers of type i do not trade with sellers of type j) and decreases the range of c s leading to a type 1 eqilibrim (every meeting generates transactions). Let assme now that we are initially in type 1 eqilibrim trading pattern and = 1= (i.e., the proportion of h and w type sellers is the same). Moreover, we assme that ~c reservation cost for = 1= and r < c ~c = 4r (r+=) 4r+ r (r+=) decreases when increases by one per cent. We set ~c, where 4r 4r+ is the vale of is the vale by which reservation cost r (r+=) < c, in order to ensre that an increase in by one nit will shift s to eqilibrim trading pattern 4 (look at gre ). between expected lifetime income of sellers is 1 = V s1 h = c r Vw s1 = r In type 1 eqilibrim the absolte di erence V b1 r h + r + c r c r =) 1 = j r c r + V b1 r h 1j V b1 r w r + c r + V b1 r r w + r + V b1 r w V b1 r w r c (15) For > 1= (i.e., the proportion of h-type byers is greater than the proportion of w-type ones) the di erence is positive and j 1j c = r 9 ( 1) c. If r

11 < 1= then the di erence is negative and j 1j c = r vale which 1 can take is j that c > ~c r (r+=) 1j 4r+ absolte di erence of sellers income will be ( 1) c. the greater r (i.e., when c = ~c). Under the assmption, an increase in will lead to type 4 eqilibrim. The 4 = V s4 h Vw s4 r r V h b4 r + r V w b4 + r V b4 r h Bt for = 1=, Vw b4 = Vh b4 and hence 4 = r ( 1)( V b4 h ). By sbstitting Vh b4, we get that 4 = 4 r r V w b4 (16) j 1j 4r+ 4 + V b4 r r h =. The sbscript in indicates that in 4 is di erent from in 1 (actally 4 >, where is the matching rate in type 1 eqilibrim trading pattern). When increases then 4r+ increases as well. Hence 4 j 1j > 4r+ 4 j 1j. Since 4r+ 4 is greater than the greater vale that 1 gets, then 4 is always greater than 1. It can be easily proven that shifts from eqilibrim 1 to eqilibria, 3 or 4 (de to ), lead to the same reslt. Therefore, we conclde that an increase in leads to an increase in income di erentials. Hence, factors which can increase matching rate (sch as developments in information technology) intensify the ineqality of sellers income. 5 Conclsion Utilizing a simple model with heterogeneos byers and sellers, where informational frictions are present, we show that factors redcing their level, sch as the introdction and development of information technology, will aggravate income ineqality among sellers. This paper calls for frther research on this topic either by a theoretical or an empirical perspective. 10

12 References [1] Acemogl, D., Why do new technologies complement skills? Directed technical change and wage ineqality. Qarterly Jornal of Economics 113(4), [] Acemogl, D., 00. Technical change, ineqality, and the labor market. Jornal of Economic Literatre 40(1), 7-7. [3] Aghion, P., Howitt, P., Violante, G.L., 00. General prpose technology and wage ineqality. Jornal of Economic Growth 7(4), [4] Brynjolfsson, E., Bar-Gill, S., Coles, P., Hak, N., 016. Income mobility and distribtion in e-commerce: The case of ebay sellers. MIT Initiative on the Digital Economy Research Project (available at [5] Card, D., DiNardo, J.E., 00. Skill-biased technological change and rising wage ineqality: Some problems and pzzles. Jornal of Labor Economics 0(4), [6] Mortensen, D., Job search and labor market analysis. Handbook of Labor Economics, [7] Skott, P., Gy, F., 007. A model of power-biased technological change. Economics Letters 95(1),

13 Figres 4 4 c / θ Figre 1: Eqilibrim trading patterns Notes: denotes c ~ ( w ) denotes c ~ ( ) h

14 4 4 c / θ Figre : The impact of an increase in β on reservation vales Notes: denotes c ~ ( w ) denotes c ~ ( h ) denotes ~ c w ( ) denotes ~ c ( ) h