A monopolistic competition model of social security, unemployment and product innovation

Transcription

1 Centrum voor de Economsche Stude van Innovate en Technologe Centre for the Economc Study of Innovaton and Technology Unversty of Antwerp A monopolstc competton model of socal securty, unemployment and product nnovaton W Meeusen G Rayp CESIT Dscusson paper No 2000/09 October

2 A monopolstc competton model of socal securty, unemployment and product nnovaton W Meeusen (Unversty of Antwerp) G Rayp (Unversty of Ghent) 1 Introducton In ths paper we present a one-country Grossman-Helpman-type of expandng product varety general equlbrum model n whch the equaton expressng contnuous clearng of the labour market s replaced by an equaton that mposes contnuous equlbrum on the socal securty budget (cf Grossman and Helpman, 1991) We wll assume that, although the process of product nnovaton s solely under the control of prvate frms, the accumulaton of knowledge has partly a publc character and thereby creates postve spllovers for other frms The model wll therefore exhbt sustanable (endogenous) growth, even f, for reasons of smplcty and transparency, physcal and human captal accumulaton s dscarded At the same tme however the economy, even n the long-run, wll not necessarly operate at full employment levels snce the labour market does not automatcally clear: the wage rate s the result of a barganng process between unons and employers The government leves a unform tax rate on labour and entrepreneural ncome and fnances n ths way the payment of allowances to the unemployed The basc research queston s how the preferences of the government wth respect to dstrbutonal ssues affect the equlbrum rate of nnovaton n the economy and the equlbrum growth rate, and, n a more generalsed context, nternatonal compettveness We present the model and analyse the propertes of ts soluton n sectons 2 and 3 We examne the long-run propertes n secton 4 In secton 5 we draw conclusons, and relate the present exercse to the sustanablty ssue of natonal socal securty systems n the face of ncreasng globalsaton n nternatonal trade 2 The model We start from the famlar Dxt-Stgltz specfcaton of the utlty of a representatve consumer under condtons of monopolstc competton (Dxt and Stgltz, 1977): σ ( 1) ( σ 1) σ n σ u = = 1c (1) c represents the consumpton of a good by the representatve consumer n s the number of dfferentated goods n the economy The demand sde of the economy s charactersed wth a unform elastcty of substtuton between goods, equal to σ > 1 Ths utlty s maxmsed under the followng budget constrant: 2

3 n =1 p c e e s total expendture by the representatve consumer A fundamental result from dualty theory s that a prce-ndex P s assocated wth u of the followng form: 1 n 1 σ [ ] ( 1 σ p ) P = (2) From the frst-order condtons follow the demand functon of an ndvdual consumer for a good of type : c σ p e = (3) P P We obtan the total demand for good by aggregatng over the ndvdual demands of all consumers: x = C = Nc = p P σ Ne P = p P σ E P (4) N s the number of ndvduals n the economy The demand for a good, consequently, s negatvely dependent on ts relatve prce and postvely lnked to total demand The choce of a numerary ensures complete determnaton of that expresson For that we follow Grossman and Helpman (1991) and assume that total nomnal spendng E s equal to 1, whch s another way of sayng that the government follows a strct polcy of zero-growth nomnal money supply It holds n ths case that n E = = p x = 1 (5) 1 Followng Sørensen (1994) and others, we keep the supply sde of the economy as smple as possble The n goods are produced usng one and the same constant returns to scale producton functon wth only one nput, labour (l ) Each good s produced by a sngle frm that enjoys ts (relatve) monopoly as a result of proprety rghts obtaned as nnovator As an expresson of the producton of goods we have therefore: x = al ( = 1,, n) (6) a s labour productvty and s assumed to be unform over all producton unts Profts per frm can now be wrtten as π = p x w l 3

4 If we assume that entrepreneurs take no account of the effect that a prce change n the ndvdual varety wll have on the prce ndex P, then the frst-order condton for maxmsng proft s the followng: p σ w = σ 1 a (7) w s the sectoral money wage rate It s possble to consderably smplfy the model f we now frst consder wage formaton Wth respect to the determnaton of the wage rate, we assume that all workers n the manufacturng sector are members of a trade unon operatng n ther sector, exercsng monopsony power over the labour supply, n the sense that t s the trade unon that s solely responsble for wage barganng Note that ths does not mean that there s no ntersectoral labour moblty It however does mean that the trade unon actve n the sector only cares about the well-beng of ts own members The trade unon organsatons are not drectly concerned wth matters of government fnance and therefore do not consder the equlbrum of the socal securty system as an external lmtaton They behave n a utltaran manner, whch mples that they endeavour to maxmse the global utlty of ther members: l u + ( M l ) u, e u where l are the employed unon members, and M s the total member number The subscrpts e and u refer respectvely to the utlty of workng and unemployed members (cf Oswald, 1985) After revertng to ndrect utlty functons, we can descrbe the objectve of the trade unon organsatons thus : Max w ( 1 τ ) O U = l w ( 1 τ ) P + ( M l ) b P (8) b, the level of the unemployment beneft (unform across sectors), s determned by the government and s therefore exogenous as far as the unon s concerned Note that, from a more generalsed pont of vew, b can be symbolc, not only for unemployment benefts, but also for other personal allowances, lke state pensons, health care allowances etc τ s the unform tax rate The soluton for the optmal gross sectoral wage rate, f the unons do not care about the effect of ther clams on the overall prce level, has agan a mark-up form: σ b w = = w ( σ 1) ( 1 τ ) (9) The net money wage w (1 - τ) s therefore unform across sectors and s a fxed mark-up above the socal securty beneft b It s nvarant wth respect to the level of socal 4

5 securty contrbutons A rse n taxes, n other words, s fully recovered on the gross wage and employment adjusts tself accordngly (see eg Holmlund et al, 1989, p 27) As a consequence of (9), we may now smplfy the notaton and drop the subscrpts wth respect to the varables p, π, l and x : p = p π l = l x = x = π for all = 1,, n It now also holds that npx = 1, (10) P n /(1 ) = p, (11) px 1 π = px wl = = σ nσ (12) If v(t) denotes the value of the clam to the, supposedly nfnte, stream of profts that accrues to the representatve frm at tme t, we get v( t) = t ( R( υ) R( t) ) 1 e n( υ) σ dυ (13) R(υ) represents the cumulatve dscount factor Separate from ther productve actvty, entrepreneurs are also nvolved n R&D and nnovaton They fnance product development costs by ssung equty If there are no knowledge spllovers, and f A s the (constant) labour ntensty of R&D actvty, then the rate at whch new products ( varants n the termnology of Grossman and Helpman) are created s the followng: l d n = R dt, A where l R s the economy-wde amount of labour devoted to nnovatve actvty If, on the contrary, part of the knowledge created n the nnovatve frm cannot be approprated and splls over to the rest of the economy, then the above equaton should be amended If we assume that the knowledge stock K R that becomes avalable for the other frms reduces n a proportonate way the labour requrements necessary for desgnng new products, we get the followng expresson: 5

6 lr d n = dt A/ K R We agan follow Grossman and Helpman by assumng that K R can be proxed by the exstng number of varants n Ths fnally yelds for the rate of nnovaton at each moment of tme, under condtons of knowledge spllovers from R&D: n lr g & = (14) n A Free entry to nnovatve actvty then means that an unlmted amount of addtonal workers wll be hred for ths purpose as long as the margnal returns of the newly hred researcher (nv/a)dt exceed the margnal cost wdt Ths s of course not compatble wth a stuaton of general equlbrum as t means that labour demand would be unbounded In the opposte case product development would come to a standstll We must therefore conclude that the free entry condton takes the followng form: wa v n wth equalty holdng whenever n& > 0 (15) The tax rate τ and the unemployment beneft b s set by the government n terms of ts own socal preferences, and under the constrant that the socal securty budget balances at each moment of tme: ( L nl ga) b = τ ( nl + ga) w (16) L s the actve populaton The balanced budget requrement of course means that n actual fact the government dsposes of only one free polcy parameter Because b s an nstrument expressed n money terms, the optmal value of whch may therefore be assumed to vary over tme, and because τ s a proporton, t seems to be more natural to concentrate on the latter as the polcy varable, and to treat b as beng mplctly determned by (16), gven τ Fnally, we need an equaton explctly expressng the fact that frms are valued at ther fundamental value and that, as a result of t, there s no arbtrage on the captal market Ths equaton s obtaned by dfferentatng equaton (13) Ths yelds v& = rv π (17) r s of course the rate of nterest For that matter, when drectly solvng (13), we also have on the equlbrum growth path that v = π r + g (18) 6

7 3 Soluton of the model n terms of the rate of nnovaton and the tax rate In order to solve for the equlbrum values of the model we can concentrate on the producton functon (6), the two mark-up equatons (7) and (9), the monetary growth rule (10), the free entry-equaton (15) taken n the sense of the equalty, the budget constrant (16), and the no-arbtrage condton (17) The analytcal soluton s relatvely easy f we ntroduce the new varable 1, nv beng defned as the nverse of the aggregate equty value of all frms Ths wll also allow us to present the soluton graphcally On the one hand, we can make use of the free-entry condton to express n terms of b: = 1 σ 1 1 τ = Aw σ Ab (19) Combnng the equatons for the socal securty budget constrant wth the mark-up equatons and the monetary growth rule, we obtan, after some manpulatons, an expresson for b n terms of g and τ : 1 2 σ (1 τ )( σ + τ 1) b σ = ( σ 1)(1 τ ) L ga( σ + τ 1) (20) Substtuton of (20) n (19) yelds an expresson for n terms of g and τ : 1 σ ( σ 1)(1 τ ) = L g A f1( g; τ ) A σ 1 σ + τ 1 (21) From the no-arbtrage equaton we learn, from the other hand, that v& = rv π 1 = rv σn v& = r v σ It follows that & = g r + σ, (22) 7

8 and therefore also that = σ ( r + g) f2 ( g) when & = 0, (23) e, when the rate of declne n the share prce of the representatve frm s exactly equal to the rate of nnovaton (that s, the rate of the ntroducton of new products) Accordng to (22), above the & = 0 lne, the rate of nnovaton n the (g,)-graph would, at the same levels of, be too low n comparson to the rate of decrease of the value of the typcal share (remember that share values of exstng frms drop as new products, and thus new frms, make ther entry; see (12) and (18)) The aggregate equty of the economy would therefore fall, meanng that would rse The economy would then run nto contradctons, and expectatons could never be fulflled Take for nstance the above case where ncreases g would then be drven to zero, meanng that nnovaton stops Ths means however that would contnue to ncrease ndefntely, whch, at a then constant level of n, n ts turn means that the equty value of the exstng frms (v) would vansh Ths s n conflct wth the fact that profts would by defnton reman postve (equaton (12)) Ths sgnfes that expectatons would be contnuously contradcted It s straghtforward to show that n the opposte case (decreasng ) expectatons would necessarly be contradcted as well Snce n equlbrum we have to assume that consumers contnue to maxmse utlty, entrepreneurs contnue to maxmse profts, and the socal securty budget has to reman balanced, the economy should fnd tself on the pont of ntersecton E of the f 1 and f 2 curves (see Fgure 1) E corresponds to the famlar saddlepont-soluton where economc agents form ther expectatons ratonally, jumpng at each moment of tme to the equlbrum poston Solvng (21) and (23) for the equlbrum rate of nnovaton yelds: σ 1 1 τ L g = r σ ( σ + τ 1) A (24) g It holds obvously that < 0 Socal protecton clearly, and not unexpectedly, reduces τ the nnovatveness of the economy: the taxes that have to be leved n order to fnance the socal securty budget ncrease the wage cost, whch, n turn, nctes nnovators to abandon the development of new products earler (see the free-entry condton) When τ = 0, expresson (24) reduces to the form obtaned by Grossman and Helpman (1991, p 61) In a way the model analysed n ths paper can therefore be seen as some form of generalsaton of Grossman and Helpman s If τ = 0, and thus also b = 0, the unemployed receve no compensaton But can there be unemployment n that case? The answer s no The unon wll ndeed maxmse ts objectve when a wage s set that 8

9 ensures full employment n the manufacturng sector Ths follows, n ths partcular case, from the unambguously negatve sgn of the slope of the O U functon for values of w larger than the market-clearng wage, and from the fact that f the wage clam s decreased beyond the level of full employment O U would of course fall proportonately (see Fgure 2) Let τ* be the tax rate that s chosen by the government n functon of ts own socal welfare crteron, that probably wll relate as well to the average ncome of ts ctzens, as to the varance of that ncome Wll the rate of nnovaton g(τ*) that corresponds to ths choce concde wth the equlbrum rate of growth of the economy? In order to answer ths queston we have to take a closer look at the long-run propertes of the model Ths s to what we turn n the followng secton As regards the other parameters appearng n (24), the followng conclusons can be drawn (see also Grossman and Helpman, 1991, p 63): the rate of nnovaton wll be hgher, frstly, f resources are used more effectvely n R&D (a lower A), secondly, at a lower dscount rate snce ths means a lower cost of captal, and thrdly, at lower levels of σ snce that mples a greater taste for varety 9

10 Fgure 1: Equlbrum soluton of the expandng product varety model wth socal protecton S S S S E =0 g* a/a x/x S ( τ 2 ) ~ S ( τ) S ( ) τ 1 S ( τ ) g 10

11 Fgure 2: Wage barganng by the unon n the absence of socal protecton O u w* w 11

12 4 Long-run propertes We shall now assume that, next to (endogenous) product nnovaton, there s also process nnovaton, whch takes the form of a constant, but exogenous, rate of ncrease of labour productvty n the manufacturng sector ( a& / a = constant ) We now have the followng relatons between dfferent rates of change: w & n& v& = (thefree entry condton (15)), w n v v & π π& v& n& = r = = v v π v n w& = 0, w p & w& a& a& = = p w a a (the no - arbtrage condton + (the mark - up equaton for prce (7)) eq (12)), (25) (26) (27) (28) Note that, because of the zero-growth monetary rule, the steady rse n labour productvty s reflected n a negatve rate of change of the prce-level and n the steady growth of real wages Together wth the monetary zero-growth condton (10) ths mples that the followng relaton must hold n the long-run: n & a& x& g = (29) n a x It s obvous that a & / a cannot be anythng else than the long-run rate of growth of the economy From the producton pont of vew, real ncome can ndeed be defned as GDP R GDP = p N & 1 vn v = nx + = nx + ng p p Balanced growth means that the manufacturng sector and the R&D sector of the economy must grow at the same rate From (25)-(29) t drectly follows that the rate of growth of the economy (and of both ts sectors) must therefore be equal to a & / a : rate of growth of GDP R gr( GDP R ) x& = gr( nx) = g + x v v& = gr( ng) = p v p& a& + g = p a If, lke n Fgure 1, a & / a, the rate of growth of the economy, would exceed the rate of 1 We use n ths partcular context the commonly used p varable as the GDP-deflator, rather than P, snce the latter varable not only contans a component expressng average prce changes, but also one expressng changes n qualty 12

13 nnovaton g = g*, then that would mean that part of the growth s attrbutable to the growth of the number of new varants, and the other part s absorbed by the expanson of the producton of exstng products It also would mean that the growth of value creaton n the R&D sector of the economy s partly attrbutable to the rate of nnovaton tself, and partly to the ncrease n valuaton n real terms of the equty that s used to fnance ths nnovaton For the rate of growth to concde wth the rate of nnovaton n Fgure 1, the tax rate would have to decrease from the optmal value τ* to the value τ ~ Rather than lookng at the growth rate of the economy, as tradtonally measured, we can also consder the growth rate of utlty, makng use of (1), (3), (12) and the above growth rate relatons: u & u 1 a& = g + σ 1 a (30) It s therefore clear that, when a & / a = 0, the economy may stll grow n terms of experenced utlty even f n ths case the economy s not fuelled by steady growth of labour productvty n the manufacturng sector, e, when there s no overall growth n real terms, and declne of producton for exstng varetes (30) also means that real growth measured n utlty terms, as opposed to a measurement n tradtonal terms, may ncrease as nnovatvty ncreases, eg by lowerng the socal securty tax level In ths partcular sense ths model, lke Grossman and Helpman s orgnal model, dsplays truly endogenous growth 5 Concluson We analysed a model that can be consdered as a generalsaton of the Grossman- Helpman (1991) one country expandng product varety model of endogenous growth wth knowledge spllovers The proposed model does no longer requre the labour market to clear and allows the government to pursue a polcy of socal protecton We analysed, n partcular, the way n whch the fnancng burden of the socal securty system weghs upon the nnovatvty of the prvate sector, and therefore, mplctly, on the nternatonal compettveness of the country The prce that has to be pad for socal protecton, ceters parbus, s, not unexpectedly, a lower rate of nnovaton The model also shows however that the government may have other possbltes to compensate for ths adverse effect or to mtgate ts mpact, the productvty level of R&D actvty and a judcous choce of the tax rate beng the most obvous ones It s evdent therefore that the model presented, f further generalsed n a 2-country framework, would be well suted to serve as a useful tool to assess the sustanablty of dfferent natonal levels of socal protecton n the face of the ncreasng globalsaton of world markets A frst attempt n ths drecton based on a smulaton approach can be found n Meeusen and Rayp (2000) 13

14 References Dxt, A and JE Stgltz (1977), Monopolstc competton and optmum product dversty, Amercan Economc Revew, 67 : Grossman, G and E Helpman (1991), Innovaton and Growth n the Global Economy, Cambrdge, Mass, MIT-Press Holmlund, B, K Løfgren and L Engstrøm (1989), Trade Unons, Employment and Unemployment Duraton, Oxford, Clarendon Press Meeusen, W and G Rayp (2000), Socal Protecton Competton n the EMU, paper presented at the Second Annual Conference of the European Trade Study Group, Glasgow, September 2000 Oswald A (1985), The Economc Theory of Trade Unons : An Introductory Survey, Scandnavan Journal of Economcs, 87(2) : Sørensen, JR (1994), Market Integraton and Imperfect Competton n Labor and Product Markets, Open Economes Revew, 5 :