The Minimum Wage and a Non-Competitive Market for Qualifications

Transcription

1 The Minimum Wage and a Non-Competitive Market for ualifications Gauthier Lanot Department of Economics Umeå School of Business and Economics Umeå University 9 87 Umeå Seden gauthier.lanot@umu.se Panos Sousounis Keele Management School Keele University Keele ST5 5BG UK p.sousounis@keele.ac.uk Abstract: In this paper e consider an euilibrium model of demand and supply for several ualifications first in a competitive setting and then in a noncompetitive setting. We propose a tractable analytical frameork i.e. hen orkers choose beteen ualifications according to a multinomial logit model of choice and hen a CES production function describes the substitutions possibilities beteen the different types of labour. While in the competitive case the effects of the minimum age are those e expect in the imperfectly competitive case e find that a minimum age can create unemployment and e find that the elfare of the population as a function of the minimum age is not unimodal. We sho furthermore that alloing one ualification to be exempted from the minimum age does not mean that its relative demand is unaffected by changes to the minimum age. Keyords: Minimum age age differentials segmented labour markets monopsony. JEL-Codes: J3 J38J42. Acknoledgements: We ish to thank seminar participants in Keele Bilgi and Umeå for their helpful comments. We are grateful to Thomas Aronsson for his insights and comments on a previous draft.

2 Introduction The effects of a minimum age on employment and ages is often studied ithin the context a single labour market involving homogenous orkers facing a variety of market conditions hich invalidate the competitive predictions: the most famous example arises in the context of monopsony (see Manning 23 for an analysis of its many forms). In such conditions a minimum age ill redistribute income to the least paid. The key policy uestion concerns the cost of the minimum age in terms of market efficiency. When the labour market is dominated by a single monopsony firm a minimum age that is less than the competitive age ill improve efficiency: it ill reduce unemployment and it ill increase the age level. This is the textbook analysis hich argues for the beneficial role of the minimum age in the presence of market poer on the labour market. By extension one may believe that any market here the demand side has market poer over the supply labour ill benefit from the introduction of a minimum age. In this paper e study the effect of a minimum age in the context of a segmented labour market. The labour market is segmented since orkers must choose among several ualifications hich have distinct marginal productivities. We assume that orkers decide on their preferred ualification by comparing the ages that each ualification commands. The total output in the economy depends on the particular mix of ualifications and ualifications can be substitutes or complements. We introduce some imperfections on this segmented labour market by alloing the aggregate production sector to determine the relative ages. The objective of the paper is to describe a simple analytical frameork ithin hich e can analyse the effect of the minimum age on the distribution of ualifications among the employed orkers and in the population and on the distribution of the euilibrium ages across ualifications. We do not account here for the important role of information asymetry search frictions or strategic interactions on the labour market. We sho nonetheless that the case for the minimum age may not be as straightforard as expected despite in our apparently elementary frameork. Grossman (983) proposes a related analysis of the firm demand for labour for to distinct skill groups here the firm takes the age of unskilled orkers as given and determines the age of skilled orkers. Unskilled orks is assumed to be subjected to the minimum age and the effort of skilled orkers responds both to their real age and to their age relative to the minimum age. The model is therefore a model of efficiency ages on the market for skilled ork. The analysis suggests that as the minimum age increases the age the firm pays for skill ork increases because of the firm substitutes aay from unskilled ork and because the firm has an incentive to maintain the relative age of skilled orkers. Our modeling maintains the assumption that the proportion of orkers supplied to the distinct ualification groups depends on the ages of each ualification relative to the competitive ualification. --

3 In a related 2-sector frameork Lee & Saez (22) analyse the contribution of the minimum age in an optimal taxation schedule in a model of competitive labour markets. They assume that orkers in the lo paid sector are heterogenous in their productivity. An increase in the minimum age is then elfare improving because it is able increases the average productivity of the lo paid in ork hile the least productive orkers become unemployed first. They sho that the optimal tax schedule takes advantage of this effect of the minimum age. Because our focus is not on optimal taxation this is not the transmission mechanism e follo here. Although the structure of the model is simple the role of the different components of the model give a rich enough structure so that the effect of the minimum age on the distribution of ages beteen ualifications depend on the model parameter values in a non-trivial ay and possibly unexpected fashion given the model simple structure. We sho that the kind of market poer e consider does not create unemployment but misallocation of orkers to ualifications relative to the competitive outcome. A large enough minimum age hoever can create unemployment and ill not be able to correct completely for the efficiency loss due to the presence of market poer. The intuition is clear: the minimum age distorts relative ages hich in turn distort the choices beteen ualifications and the demand for employment. Furthermore e sho that the effect of the minimum age on the population elfare is not necessarily monotone: elfare as a function of the minimum age may have multiple maxima. This in turns imply that the optimal minimum age may involve an optimal level of unemployment. Finally e sho that the demand for orkers ith a ualification that is exempted from the minimum depends on the minimum age. Our model can capture some of the ell documented heterogeneity observed beteen the lo paid sectors in the UK for example. The reports of the UK Lo Pay Commission (Lo Pay Commission 23) describe the bite of the MW in several lo pay occupations and sho that the bite varies beteen occupations and that this pattern persists over time. In their revie of the US evidence both Flinn (2) and Neumark & Wascher (28) similarly report that the bite of the minimum age differs beteen lo paid occupations. The paper is organised as follos e first describe the analytical frameork in a competitive environment ithout a minimum age and then discuss the effect of the introduction of a minimum age. In a third section e extend the analysis to a model here the production sector acts in a monopsonistic manner on the market for ualifications and then sho the effect of the The 23 reports that 4 to 6% of men employed are covered by the UK national MW for omen this proportion is 6 to 9%. Moreover 9% of unualified orkers are paid at or belo the NMW. For all orkers 22 and over the bite of the minimum age (the ratio of the minimum age to the median age) is greater than 45% in 999 and rising sloly to about 52% over the period to 22. For the lo paid sectors and over the same period the bite ranges from 6% to 8% hereas for unualified orkers the bite ranges beteen 8 and 9% over the period 27/

4 minimum age in this context. Finally e discuss the effect of exemptions from the minimum age in this frameork and e conclude. An Analytically Convenient Model of the Markets for ualifications We consider a large population of individuals ho choose beteen + different ualifications. We normalise the population size to. We think of these individuals as prospective ne entrants on the labour market. Their aim is to decide on the ualification hich is best suited to their tastes given the age that each ualification commands. We then model the demand side in the aggregate to determine endogenously the euilibrium ages hich each ualification ill command given all the parameters of the model. This captures in a reduced form manner the many interactions and frictions on the demand side hich determine the overall techniue. For example it may reflect the technological constraints of the production of ualification or it may reflect the difficulty of retraining once a ualification has been acuired. These are important further issues hich e do not consider here. We assume that the choices beteen ualifications are such that it is the distribution of relative ages hich determines the distribution of ualification choices. In particular e select a Multinomial Logit model of choice hich simplifies the analysis (for a more general discussion of the logit model and its use in economic theory see Anderson De Palma & Thisse 992). We assume that individual i s utility for ualification is I = + ln + η =... () i i here <+ is a ualification specific shifter (including for example the costs of obtaining the ualification evaluated in utility terms) and e set = i.e. the th ualification is set to be the reference ualification. ith measures the marginal utility to changes in the ualification specific age and η i is a unobserved individual and ualification specific random component hich follos a Type extreme value distribution. The proportion of the population hich chooses ualification can be evaluated simply as: π exp + ln exp( + ln ) S = = p exp( p ln p) + exp p+ ln p= p=. (2) + The model of choice beteen ualifications is related to a Roy model hich it becomes if =+. In that extreme case orkers choose the ualification hich yields the largest age ith probability. In our context this ould lead all orkers to make the same choice. Teulings (23) analyses a more general frameork here orkers ith distinct skills make different choices and are -3-

5 assigned to a large number of alternative jobs according to a Roy model of choice in a competitive general euilibrium context. Our analysis here is limited to a discrete number of labour markets and the assignment problem the orkers solve in our case is simpler. Workers are heterogeneous and decide hich ualification to acuire; furthermore e specify only one aggregate production technology here distinct ualifications have distinct marginal productivities. The individual unobserved component do not determine individual productivity and therefore all orkers ith a given ualification command the same age. On the production side e assume that the aggregate technology hich determines the demand for labour inputs ith alternative ualification is characterised by an increasing and concave function ψ ( ) of a labour aggregate F ( ): ( ( )) Y= ψ F N... N here N measures the uantity of input labour (for a particular generation) ith ualification. The labour aggregate is of the CES form: ρ ρ F( N... N) = N + αn ρ (3) = ith ρ such that ρ= σ here σ ith σ measures the elasticity of substitution beteen labour types. Assuming that either aggregate profits are maximised or costs are minimised given the technology the first order conditions for the demand of labour of type relative to the demand for the th ualification are N = = α N ρ D π D α π σ ( ) σ (4) here π =... stands for the demand for a proportion of orkers ith D ualification are the ages relative to the th ualification. This particular functional form implies that the proportions on the demand side are of the multinomial logit type: π D p= ( σ α σ ( )) exp ln ln = + exp ln ln ( σ α p σ ( p) ) for all =... (5) and π D = + exp ln ln p= ( σ α p σ ( p) ). (6) -4-

6 The euilibrium beteen the demand and the supply of the various ualification ill determine the euilibrium relative ages =... such that: π S ( ) π ( ) = for all =.... D In hat follos e make the additional assumption that + σ> so that the euilibrium relative ages are such that ( ) + ln = σ lnα σ ln for all =... hich implies that ln ( ) σ lnα = + σ for all =.... (7) We assume that ualification commands such a high age that ln is negative for all i.e. for all. Furthermore e ill assume throughout that the list of ln is sorted in ascending order e have the log of the euilibrium ages relative to ualification ln ln... ln. (8) 2 Since + σ is positive this amounts to a restriction on the sign and the relative σ α. Note that the distribution of the sizes of the differences ( ) uantities σ ln( α) ln determines the distribution of relative euilibrium ages and therefore determines the distribution of euilibrium ages. Unsurprisingly the model predicts that the characteristics of the distribution of ages across ualifications depend on both the demand and supply parameters. µ lnα ν = σ lnα so that e can express the Let = ( ) + and ( ) euilibrium probabilities and relative ages as σ exp µ + σ π= for all =... (9) σ + exp µ p + σ p= and ν ln = for all =... () + σ ith µ = ν=. Condition (8) implies that ν ν2... ν but does not imply a similar order beteen the uantities µ (these uantities are each increasing ith ν but this does not determine their ordering). Hoever assuming that the number of orkers ith lo paid ualifications i.e. such that are a small proportion of the orkforce ith the th -5-

7 ualification in the competitive euilibrium that is such that =... implies that µ =... since π < π σ lnπ ln = µ for all =.... () π + σ Assuming µ =... it is then easy to conclude that for all =... : ln lnπ lnπ lnπ = = (2) lnσ + σ lnσ + σ ln ln lnπ ln = = σ (3) ln + σ ln + σ The competitive uantities decrease ith the (technical) elasticity of substitution σ. In an economy here the competitive ages and uantities are small relative to the th ualification a technology change hich increases the substitutability beteen ualifications (as measured by a higher value of σ) all other things eual leads to smaller competitive relative ages and relative uantities for ualifications =. Furthermore the relative age bill for any ualification = must decrease since lnπ + = lnπ. lnσ + σ Considering the supply side a larger value of all other things eual leads to a larger sensitivity of ualification choice to relative ages larger competitive relative ages hile the competitive relative uantities are smaller. Overall the effect on the relative age bill depends on σ since: lnπ σ = ln ln + σ if σ< the effect on the age bill is positive and negative otherise. We note further that the data of distribution of the competitive relative ages... π... π ould not determine uniuely all the and uantities ( ) parameters of the model. There are several values for the parameters α... α... σ hich ould lead to the same values for the relative ( ) competitive ages and uantities ( ) π π. In the competitive euilibrium the population is employed in its entirety: there is no unemployment caused by a mismatch beteen the supply and the demand for ualifications. Any possible diseuilibrium is resolved by the price mechanism. The euilibrium e describe here leaves the euilibrium age of the th ualification unspecified ithout any conseuence for the distribution of the population across ualifications. -6-

8 The Effect of a Minimum Wage in Competitive Markets When a minimum age is introduced several effects come into play. First because the minimum involves an increase in the marginal cost of production the scale of production may vary this is the usual scale effect. Second the minimum age ill affect some ualifications and not others. This ill generate a redistribution of the demand and the supply of ualifications because the relative reards and costs are modified. The ne euilibrium set of ages must accounts for both effects. In the presence of a minimum age the calculus of the euilibrium ages and proportions is modified significantly since euating the proportions demanded and supplied of each ualification does not account for the possibility of some unemployment created by a mismatch beteen the demand side and the supply side. Instead the market euilibrium arises hen the uantity of labour supplied ith each ualification matches the uantity demanded. For any ualification here the minimum age bites the supply side is determined relative to the age of the th ualification hile the demand side is determined by the same relative age in euilibrium and the scale of production overall. In our simple model here the demand and supplies depend only on the ages relative to the age paid for the th ualification all the analysis on the euilibrium for ualifications to can be carried out in terms of the minimum age relative to the age for the th ualification say ω here is the age of ualification hen the minimum age is in force. The set of euilibrium relative ages in the labour markets here the minimum age is imposed ( = ) is such that ɶ π D ( ) π ( ) S ɶ and such that ɶ D ( ) = π ( ) for all =. (4) S π (5) ɶ ɶ for all ualification here the minimum age does not bite i.e. such that > ω. Whenever ω > for any of the r least paid occupations =... r r the relative supply of orkers at the relative minimum age ω ill be larger than the demand at that age. This is a conseuence of our earlier assumption relative to the competitive age these ualifications are more expensive hence the demand for their services fall as described by (4) hereas the relative supply increases as described by (2). -7-

9 Workers ith ualifications hich are not covered by the minimum age are paid at the competitive relative ages > ω such that relative demand and supply meet as described by (7) for any > r. In our model the introduction of the minimum age therefore affects directly the pay of the loest paid orkers (if they are employed) creates unemployment among these orkers and leave the relative pay conditions for all orkers uncovered by the minimum age relatively unaffected (i.e. their pay differential relative to the th ualification remains at the competitive level the euilibrium uantities ill change hoever). Introducing more labour markets and alloing for the endogenous distribution of the orkers beteen sectors as e do here does not change the textbook analysis of the effect of the minimum age on a competitive labour market: the minimum age creates some unemployment (e provide a complete analysis in the online appendix). Monopsony Poer and the Labour Markets for ualifications In this section e describe the euilibrium outcomes hen the demand side (the aggregate production sector) recognizes and incorporates into its decision problem the effects prices have on the distribution of orkers across ualifications. Hence folloing the textbook s intuition e ould expect the minimum age to be a relevant policy tool to improve elfare. In general it is possible for the demand side to decide on leaving some orkers unemployed (as ould be the case in the textbook exposition of the monopsony model) not because of any mismatch beteen demand and supply but simply because of the exercise of market poer. Hoever in the current model this is not a feasible euilibrium outcome. In euilibrium any non-productive ualification i.e. such that α = ill be rearded ith a age relative to the th ualification eual to and therefore ill neither be supplied nor demanded. The analysis follos the textbook treatment of a monopsony although in the current context the production side is involved in the management of the supply of labour across several sectors (our ualifications). An implicit assumption of the model is that orkers cannot move beteen ualifications i.e. a orker ith ualification enters the market for that ualification only and is not able to move to another close substitute ualification or even retrain. This is clearly a real phenomenon but beyond the scope of the current paper. The model assumes that the aggregate production side does not determine the age paid for orkers ith the th ualification indeed monopsonistic profits ould be maximised.r.t. ith i.e. the production side ould understand that the supply of orkers ith the th ualification is infinitely inelastic and therefore if the production side as to exploit its market poer it ould pay ages as close to as possible for all ualifications (or even ould reuire the orkers to pay for the right to ork). This is a conseuence of our early modelling assumptions hich state that the supply for each ualification -8-

10 depends only on the relative ages. To depart from such an extreme form of market poer e assume instead that the production side determines its scale competitively i.e. it determines the level of employment N ɶ for the orkers ith the th ualification or euivalently it determines total employment N ɶ so as to maximise profit taking the age of the highest paid ualification as given. Hence e consider an intermediate form of market poer here the production sector controls relative reards and determines the employment of the highest paid orker competitively but does not determine absolute reards (hich ould drive ages don as close to as possible given our assumptions on ualification choice). Instead of taking prices as given on the lo paid ualifications the production understands the relationship beteen the ages relative to the age of the th ualification and the distribution of the population beteen ualifications that is for a given vector of relative ages. < the firms expects the ɶ population to be distributed beteen ualification according to the probabilities: π S p= ( + ln ) exp = + exp + ln ( p p) for =.... If = the proportions of orkers ith the different ualifications are fixed and independent of the relative ages hile if + the ualification ith the largest age is supplied by the hole population and all the other ualifications are not supplied. Assuming < <+ e can inverse these expression to find a relationship S S S beteen the relative ages and the ratios π π π e find: lnπ = + ln for =... S or euivalently e consider the inverse supply curves: ith φ S φπ = for =... (6) exp for all >. For a given employment plan N= ( N N... N) and alloing the monopsony ɶ to account for the relative supplies the total age bill becomes: ( ) = + φ ( π ) N N N p p p p p= ɶ p= ith γ +. γ The production side maximises its joint profits -9-

11 γ pψ( NF( π π2... π) ) N φp( π p) + (7) p= ith respect to π... π and N. Define NF( π π2... π) L = to measure the aggregate labour input and let ( π π2... π) + ( p) p ɶ p= measure the relative costs of C F φ π ( π π2... π) γ employing π... πin each ualification = relative to the employment of orkers ith the th ualification. We observe that π π2... π ( L) L. Cɶ ( π π2 π) max pψ.... ( L) L. ɶ( ππ 2 π) pψ. min C... π π2... π therefore the optimal relative uantities π... π are obtained from the Cɶ π π2... π. minimisation of the relative costs ( ) ( 2... ) Cɶ π π π is a strictly uasi-convex function 2 of the uantities π... π. Therefore the first order conditions are sufficient to describe optimality and they imply that the optimal uantities π ɶ... p = must satisfy α π ( ɶ ) φ π γ = γ + φ π + ( ɶ ) ( ɶ ) ρ ( ɶ ) p p p p p= p= α π ρ =... (8) In turn this last expression implies that for any to distinct ualifications and r e must have at the optimum: α σ γ ρ σ π φ π αφ + ɶ ɶ r = = π α ɶ r r πɶ r αrφ φ r and e conclude that πɶ for =... must take the form: 2 This is a direct conseuence of our functional assumptions: F( ) function and φ ( π ) = π π... π is a strictly concave 2 γ + is a strictly convex function of the relative proportions. The relative C π π... π is not a convex function in general hoever (consider the ratio 2 costs function ( ) γ ( ) ( ) + x + x ρ ρ ith ρ<< γ x for example ith ρ=.5 and.5 see Schaible (976). γ= for [ ] x ) --

12 σ σ α + φ πɶ =Λ ɶ (9) for some positive constant Λ ɶ independent of. We can then sho that Proposition. i) The aggregate monopsony employs less of each ualification = than it ould under perfect competition on these labour markets that is Λ ɶ is strictly less than. ii) The aggregate monopsony pays relatively less and preserves the competitive age differentials beteen ualifications =. Hence e can rite πɶ =Λ ɶ π. Since Λ ɶ is strictly less than e conclude that the monopsonistic production sector sets πɶ =... such that πɶ < π and therefore πɶ > π. We interpret Λ ɶ as a measure of the market poer of the monopsony over the labour markets for the ualifications. Geometrically the monopsony reduces all relative uantity by a factor Λ ɶ along the ray going through the competitive allocation π =... At the optimum the production side pays ages for all ualification hich are smaller relative to the age of the th ualification than they ere in the competitive setting: ln ɶ = ln ln π ln ln ln Λ+ ɶ = Λ+ ɶ < i.e. the relative ages are uniformly lnλ ɶ less than the competitive relative ages and e verify that the th ualification remains the highest paid ualification since all the relative ages are less than. Hence the ranking of the monopsony s relative ages is identical to the ranking of the competitive relative ages and the monopsony preserves the competitive relative age differentials. Geometrically the monopsony reduces all relative ages by a factor =... Λ ɶ < along the ray going through the competitive relative ages Given our assumptions the exercise of market poer here does not create any unemployment: the relative uantities are on the relative supply curves at the monopsony relative ages. The cost of market poer here is in term of misallocation of labour across ualification compared to the competitive distribution. As a result a larger proportion of the population chooses to obtain the th ualification given the age structure. Assume that e ish to compare competitive euilibria such that the relative ages and proportions are kept constant in the competitive euilibrium hile changing some of the characteristics of the technology or of the preferences. --

13 That is e imagine changing ρ or γ hile maintaining constant. In such circumstance p= p p of ρ or γ. Given this assumption e can conclude: and π =.. A= π is kept unchanged by a variation γ ρ Λɶ Λɶ lnλɶ = γ ρ Λɶ ρ ρλɶ. A constant This uantity is negative for all values of ρ. Comparing to labour markets ith the same competitive euilibrium distribution of orkers across ualifications and identical relative ages the more substitutable the ualifications (the larger ρ) the larger the ability of the monopsony to reduce the relative ages and relative employment. Conversely the smaller ρ i.e. the larger the complementarity beteen the ualifications the smaller the monopsony poer (the closer Λ ɶ is to ). A similar calculation this time dealing ith γ gives: γ ρ γ γ Λɶ Λɶ +AΛ ɶ + lnλɶ = γ ρ Λɶ γ ρλɶ A constant hich e cannot sign in general. Assume that γ from above or + in the limit Λ ɶ and the elasticity is negative for all positive values of A. In this case the supply side is very sensitive to changes to the relative ages. Reducing this sensitivity i.e. reducing hich increases γ reduces Λ ɶ and therefore increases the production sector s market poer. The production side demands and employs a (strictly) larger proportion of orkers ith the th ualification hen it acts as a monopsony. Furthermore since the relative ages and the probabilities for each ualification are smaller than the corresponding competitive relative ages and probabilities the monopsony production sector generates a relative age distribution for ualifications hich is more compressed than the one generated in the competitive setting. Finally e describe the tax policy hich corrects for the actions of the monopsonistic production sector in our model. We assume that the government can tax/subsidise all relative ages at the margin at a rate θ θ> and redistribute the costs/subsidies through a lump sum tax. This means that the production sector perceives the relative supply for each ualification to be: S θφπ = for =... (2) for some positive value of θ. In effect the minimisation problem ith a tax/subsidy θ has the same general form as the original minimisation problem if e adjust the relative supply functions by a factor θ. The monopsony sets the relative uantities for each ualification such that: -2-

14 γ ρ =Λ ɶ =Λɶ θ π πɶ π here the uantities ith primes denote the optimal uantities set by the monopsony in the presence of the tax/subsidy θ. The last expression arises directly from our earlier analysis by replacing φ ith θφ. The value of the subsidy hich leads the monopsony to employ the competitive relative uantities is such that: γ ρ θ Λ ɶ =. Adjusting A to ( σ ) θ σ + A A and applying the argument hich proves proposition e find that θ ( γ ( γ )) = + A - <. Hence at the margin the corrective instrument is a subsidy that is the government pays the orker a share θ of the age and the monopsony sector pays the other fraction θ of the age. This subsidy is financed by a lump sum tax on profits exactly eual θ A. While this solution is to the subsidy value at the competitive level: ( ) simple enough it is not the one that is used to correct for the effect of market poer on the labour market. The next section describes the effect of the minimum age on all the different labour markets as its relative size increases. Monopsony and the Effect of the Minimum Wage In this section e analyse the effect of the minimum age on the actions of the monopsony. We maintain the assumption that the age of the highest paid ualification is determined outside of the model all other relative prices being given. The structure of our model and in particular of the supply curves for each ualification simplifies the analysis substantially. Given the age for the th ualification the introduction of a minimum age introduces technical difficulties in particular the profit function is no longer necessarily continuously differentiable: the minimum age creates a (ell knon) discontinuity of the marginal cost for each ualification =. In the textbook analysis at the optimum for the monopsony the introduction of the minimum age decreases the marginal cost and therefore implies that the employment that maximises profits is larger. In our context the intuition concerning the marginal cost for each ualification is the same: the introduction of the minimum age decreases the marginal cost for the nely affected ualification. Hoever it ill increase the marginal cost for all other ualifications already covered by the minimum age. Hence the textbook analysis does not extend straightforardly (even given our functional form assumptions). We keep the age of the th ualification fixed at. In our context it is natural to consider ω the minimum age relative to for ω. The relative supplies are increasing in the relevant relative age that is for each -3-

15 ualification the value π ( ωφ) that the relative costs for each ualification S ( ) is such πɶ < π ɶ < ω. Finally note π π = φπ + = (2) are strictly increasing and convex functions of the relative supply π. For a given value of and of the relative minimum age ω the monopsony production sector maximises total profits over Nand π π2... π : ( ( )) ( ) pψ NF π π2... π N π max ωφ + π. (22) = The problem can euivalently be set up as a to stage optimisation problem hereby the production sector minimises aggregate units costs in terms of relative employment of each ualification and then maximise profit ith L = NF π π2... π and respect to overall aggregate labour input. Define ( ) rerite the previous expression as ( ).. C( ωπ ; π2... π) pψ L L ith C( ωπ ; π2... π) and e note that L π π2... π + max π ωφ( π) = F ( π π2... π) ( L) L. C( ωπ π2 π) max pψ. ;... ( L) L. ( ωπ π2 π) max pψ. min C ;.... L π π2... π Hence the effect of the minimum age on the profit of the monopsony aggregate production sector is determined by the effect of the minimum age Cωπ ; π2... π. on the minimum of the costs per unit of aggregate labour ( ) For any given value of ω ( ; 2... ) function of ( π π2... π) 3. Therefore ( ; 2... ) some uniue value πω ( ) ( π( ω) π2 ( ω)... π( ω) ) Cωπ π π is a strictly uasi-convex Cωπ π π is minimised at. Because of the presence ɶ of the minimum age the first order conditions for the ualification ill depend on hether or not a particular group of ualifications is affected by the minimum age or not. Furthermore for some ualification the optimal 3 Again this is a direct conseuence of our functional assumptions: F( ) + max concave function and ( ) given the relative minimum age. π π... π is a strictly 2 π ωφ π is a convex function of the relative proportions = -4-

16 uantities are located on a kink of the relative cost function. We can summarise the situation as follos: π ω is such that i) The optimal solution for ualification ( ) ω π( ω) < φ and: ρ α( π( ω) ) ω =. ρ α ( π ( ω) ) π ( ω) max + ωφ + ( π ( ω) ) k k k k k k= k= ii) The optimal solution for ualification is at a kink π ( ω) relative cost is decreasing to the right of π ( ω) π( ω) ) and : (23) ω = (the φ and increasing the left of ω < πk ( ω) max + ωφk( πk ( ω) ) k= ρ (24) α( π ( ω) ) φ( π ( ω) ) < γ. ρ + αk( πk ( ω )) πk ( ω) max ωφ + k( πk ( ω) ) k= k= π ω is such that iii) The optimal solution for ualification ( ) ω π( ω) > (the relative age on the market for the th ualification φ is larger than the minimum age) and: ρ α( π ( ω) ) φ( π( ω) ) = γ. ρ α ( π ( ω) ) π ( ω) max + ωφ + ( π ( ω) ) k k k k k k= k= σ We assume throughout that the uantities αφ are increasing ith = that is the competitive relative ages are increasing ith. We can sho the folloing properties of the solution of this minimisation problem given ω. First e sho that the minimum age affects each ualification in turn according to their competitive ranking. (25) -5-

17 Proposition 2. Assume that the minimum age ω does not cover ualification s hen the aggregate monopsony maximises profits: then the minimum age does not bite on all ualifications r ith r s. In particular this implies that the smallest value of the relative minimum age hich bites (i.e. such that its affects the age of a single ualification) is eual to the smallest value of the age paid by the monopsony. We turn to the employment effects of the minimum age and e sho next that hen the competitive differentials are large enough for some values of the minimum age it is possible that all the ualifications covered by the minimum age experience unemployment. Proposition 3. Assume that the minimum age ω is such that the first p p< ualifications are covered by the minimum age. A sufficient condition for orkers ith ualifications to p to experience unemployment is that the competitive relative age differentials beteen the ualifications covered by the minimum age and the ualifications not covered by the minimum age are larger than σ γ +. By implication proposition 3 suggests that ualifications ith small enough competitive relative age differentials ill respond to the minimum age σ similarly as far as unemployment is concerned. The bound γ + on the relative age differentials is decreasing ith either parameters or σ. Consider to independent labour markets ith identical competitive ages but different substitution elasticities. The proposition implies that the labour market ith the larger substitution elasticity is more likely to exhibit unemployment in the covered ualifications than the other. Moreover there exists a large enough (relative) minimum age such that all ualifications are covered and if the relative minimum age is strictly larger than this value all lo paid ualifications ill experience unemployment. Proposition 4. For a large enough relative minimum age ω ω all ualifications are covered by the minimum age. In that case the optimal relative uantities set by the monopsony are such that: σ σ π ω = ω α and ω= = max. ( ) =... Furthermore if ω> ω all ualifications ith > experience unemployment. Finally e study ho the relative proportions for each ualification vary ith the minimum age. Our purpose here is to sho hat properties on the relative demands that the model structures imposes and to suggest that despite -6-

18 the strong assumption on the model structure the relative employment responses to an increase in the minimum age are not straightforard : they depend in a complicated ay on all parameters in the model. Indeed the optimal relative uantities that the monopsony sector determines for any ualification can be on the demand curve hich ill imply the existence of unemployment for that ualification or on the supply curve. To proceed e define the folloing uantities: ( ) ( ) ( ( )) C ω + πk ω max ωφ k π k ω k= + k k k= ρ ( ω) α π ( ω) F S S S ω + ( ω) ( ) r s.t. ω πr < φ r ωπ C s s.t. ω πs = φ s ω < = φ ; r r ith S ( ω) = if r {... } : πr ( ω) ρ ωπs απ s s ω γ ρ C( ω) F( ω) ( ω) and e sho: ith S ω ( ω ) = if s {... } : πs t s.t. ω πt > φ t φπ C γ ω = = φ ; s ω S+ ω = if t {... } : π t> = φ t t t ith ( ) ( ω) Proposition 5. Outside of any non-differentiability of the solution ith respect to ω the elasticity of the relative demands π p ith respect to ω is: i) for any ualification r covered by the minimum age and such that ω πr< φ (the monopsony relative uantity for occupation r is on the r demand curve ) : γ ( S ( ω) + Sω( ω) )( γ ρ) S+ ( ω) e e σ ω r ω= σ ith e ω σ ; γ ( S ( ω) )( γ ρ) S+ ( ω) σ ii) for any ualification s covered by the minimum age such that ω π s= φ (the monopsony relative uantity for occupation s is on the s relative supply curve ): eωω e s ω= ; -7-

19 ω iii) for all ualifications t not covered by the minimum age i.e. πt> φ (the t monopsony relative uantity for occupation t is on the relative supply curve ): Sω( ω) e+ ω et ω= and et ω. γ ( S ( ω) )( γ ρ) S+ ( ω) σ Furthermore e + ω and e ω are such that: ( γ ρ) e ( ρ) = + e. + ω ω The key finding here is the euality ( γ ρ) e ( ρ) = + e hich suggests + / ω / ω that the elasticities of the relative uantities ith respect to the minimum age across the ualifications that are covered and those that are not obey a strict relationship. This relationship is clearly the conseuence of our functional assumption concerning the ualification choice probabilities and the CES technology. Finally these imply that the elasticity of the labour aggregate ith respect to the minimum age is a mixture of the three elasticities e ω e + ω : ρ π r αs πs π s π t r s s t e ω e ω S ( ω) e S ( ). ω ω L = + + ( ω) + + π F k π + + k + π k k= k= k= In general hoever e cannot determine the sign of this expression. The proof of the proposition suggests that there exist regimes here e L ω is proportional to e / ω and e / ω can be negative or positive. Hence the effect of the minimum age on the total employment as measured by L = NF( π π2... π) is not necessarily monotonous. Hence even though our functional form assumptions are restrictive it is interesting to observe that the model is consistent ith a variety of ualitative responses to an increase in the relative minimum age. Furthermore Proposition 5 is useful to calculate (numerically) the solution of the optimisation problem for increasing value of the relative minimum age. An Example We illustrate the effect of the minimum age on the economy and in particular on orkers elfare. To do so e complete the model by specifying ψ ( ) as a simple poer function: ψ x = x α. ( ) In keeping ith our assumptions e assume that the market for orkers ith the th ualification is competitive hence that demand for labour is such that the aggregate production sector adjusts the marginal productivity to the marginal costs. Since the relative uantities for all other sectors are obtained -8-

20 from the minimisation of the relative costs and given the logit specification for the supply of orkers to each market hich depends only on the relative ages the supply of orkers ith th ualification is determined from the minimisation of relative costs. In other ords the aggregate labour input L is knon. The competitive assumption on the market for the th ualification means that the euilibrium age is such that pψ L =. C ωπ ; π... π. ( ) ( 2 ) In all our calculations e assume that α=.7 and e normalise p to. To evaluate the elfare of the population of orkers e use the elfare measure derived from the logit model evaluated at the relative ages π ω. In the absence of unemployment our model suggest that the ( ( )) population s elfare is (see Anderson De Palma & Thisse 992) B( b) = E max { I I... I } ɶ = ln exp( p ln p) + + γ (26) p= = ln + ln exp( p ln p) γ p= here γ is Euler s constant 4. The last expression clarifies the role of the relative ages for ualifications = from the role of the age of the th ualification. This measure needs to be adapted hen unemployment takes place in one or more markets. We assume that unemployed orkers receive a level of benefit b and value unemployment ith ualification as b I = + ln b+ η i i Note that this implies that I b ln i I i= and this difference does not b depend on the unobservable term η i. This is an additional assumption hich e adopt to simplify the elfare calculations. It implies that unemployed orkers ho have made a particular ualification choice suffer only from the opportunity cost of the age loss relative to the unemployment benefit they receive. We consider first the possibility of unemployment in the market for orkers ith ualification =. We assume that all orkers in that market experience unemployment ith the same probability p u. The population elfare B( b pu) must be modified to account for the orkers ho are affected by ɶ the likelihood of unemployment i.e. if they choose the = and the others: 4 We ignore the effect on elfare of other sources of income. -9-

21 b B( b pu) = E ( ) E E I I I + pu I I I + pu I I I < < ɶ = E I E I p ( )( ( ) ( )) I I + I I uπ m m b (27) < ɶ = B( b) puπ( ) ( m( ) m( b) ) ɶ ɶ I = max I I2 I3... I and m( ) + ln =... and here { } ( ) m b + ln b. The first expression distinguishes beteen individuals ho chose = i.e. such that I < I and the others ho are such that I I. Among the former group a proportion p u is unemployed and the elfare in b that case is I. The second expression uses the property that the difference b I i I i does not depend on the unobservable component η i and therefore b pue ( I I) π( ) ( m( ) m( b) ) I < I =. ɶ Keeping the age distribution constant and increasing the unemployment rate for orkers ith the first ualification the change in indirect utility relative to full employment is: B( b pu) B( b) = puπ( ) ( m( ) m( b) ) (28) ɶ ɶ ɶ hich is negative for level of benefits smaller than. More generally if the distribution of unemployment across ualifications to + is p= ( pu pu 2... p u + ) the population elfare becomes: ɶ + B( b p) = B( b) puπ( ) ( m( ) m( b) ). ( 29) ɶ ɶ ɶ = ɶ We set the benefit level to half of the (absolute) minimum age. A strictly positive value for the benefit allos us to consider the trade-off beteen a higher minimum age and a higher level of unemployment. If the benefit level is eual to zero the costs of unemployment are infinite. This ould reduce the usefulness of the minimum age as a tool against market poer. The benefit level is financed ith a proportional income tax τ on all income. For a distribution of unemployment in the population the tax τ is defined as the ratio of total expenditure on unemployment benefit (the proportion of unemployed orkers multiplied by the benefit level) divided by the total income (age and benefit) in the population. The tax has no conseuence on the choices of orkers since orkers determine the ualification choice on the basis of relative ages only. It has an effect on elfare hoever as e can see clearly in euation (26) and (27) here the effect of the proportional income ln τ. Because the tax is raised on all sources of tax on elfare amounts to ( ) income it does not have any effect on the elfare costs of unemployment either (given our specification) since they depend on the ratio of the ualification age to the benefit level. This ratio does not vary ith a proportional tax on all income. We consider =4 lo paid ualifications and e set the model parameters as follos: -2-

22 ( 2 ) α α... α = ( ) ( 2 )... = ( ) =.75 and σ=.75. Hence the supply side is responsive to relative age changes hile the technology exhibits substantial complementarities beteen labour inputs. The competitive age differentials ith the competitive ualification are substantial:. = ( ). These differentials ɶ are large enough for unemployment to arise for some ualifications for some values of the relative minimum age. The value of the competitive age for the th ualification is ɶ =.927 and the euilibrium distribution across ualification gives a large eight to the competitive sector π = ( ). As e expect the relatively large ɶ substitution elasticity gives substantial market poer to the production sector i.e. Λ= ɶ.636 and the relative ages for the lo paid ualifications are reduced to 77.2% of their competitive values. These particular values allo us to illustrate the (possibly) surprising non monotonic effect of the minimum age on elfare. The top right hand pane in -2-

23 Figure shos the proportion demanded and supplied ith each ualification. Each ualification is represented by a single curve for small values of the relative minimum age in this case the demand matches the supply. As the size of the relative minimum age increases the demand for the ualification is less than the supply i.e. there is unemployment in the market for that ualification. The curve for the ualification splits into to curves the top one represents the proportion supplied hile the one belo represents the proportion demanded of each ualification. When the supply is larger than the demand the minimum age bites hereas hen the supply and the demand match the relative ages are determined by the relative supply curves. The top right hand pane describes the age of the th ualification hile middle left hand pane shos the relative ages as a function of the relative minimum age. Since the first ualification contributes little to the production a larger minimum age does affect the demand for the other ualification much. As the minimum age increases the age of the th ualification increases steadily. Intuitively this maintains the size of the age differentials and ensures that the supply of orkers ith the th ualification is large enough. As e noticed earlier the increase of contributes positively to the population overall elfare. The variance beteen the relative ages of the lo paid ualification shrinks as the minimum age increases and for relatively large minimum ages all relative ages are eual. This has a further positive effect on elfare as shon in the middle right hand pane. The minimum age creates unemployment and unemployment has an increasing elfare cost hoever this is illustrated for each ualification in the bottom left hand pane. For each ualification e dra the proportions supplied and demanded. As the minimum age increases the demand eventually falls belo the supply and at that point the schedule for each ualification splits into to curves. The loest right hand pane shos ho elfare varies overall. The total elfare schedule is multimodal. Consider the first local maximum. This arises because the orkers ith the 2 nd ualification first mostly gain from the increase of the minimum age and beyond a certain level as unemployment increases in this market the elfare gains are overtaken by the elfare losses that unemployed orkers face. The second more pronounced maximum arises for similar reasons hen the market for the 4 th ualification is affected by the minimum age. Total elfare is maximised hen the relative minimum age reaches 33.24% of the th ualification age. The aggregate profits (of the aggregate monopsony) on the other hand fall monotonously ith the minimum age (not shon) hile the reuired tax rate increases to capture more than 65% of all income hen the minimum age is eual to the competitive age for the th ualification. This pattern is robust to changing the substitution elasticity to σ=.25. The competitive relative ages are then. = ( ) and the ɶ competitive age for the th ualification is =.35. The orkers are distributed among the ualifications such that -22-

24 π = ( ). When the production sector behaves as a ɶ monopsony on the lo paid ualifications it sets Λ ɶ to Λ= ɶ.776 and reduces the relative share of ualifications to 4 to 77.6% of their competitive values and employs 36.5% of the orkers in the market for the th ualification. Furthermore all relative ages are about 86.5% of the competitive relative ages and th ualification orkers are paid ɶ =.366. The relative minimum age that maximises overall elfare is 38.69% of the th ualification age. The second (local) maxima arises hen the relative minimum age is 56.8% of the th ualification age but yields a loer overall elfare value. -23-

25 Figure The effect of the minimum age on elfare and costs -24-

26 Exemptions from the Minimum Wage We complete our analysis by commenting on the effect of an exemption from the minimum age for the relative demand of a particular ualification. Exemptions of this kind exist for example in the UK apprentices ere exempt of a minimum age until 22 and thereafter the rate that applies to them is about 5% of the regular rate. Clearly our model does not capture all the relevant considerations that ere part of the argument for exemption and against the full rate. The UK rate as set at a distinct level so as to compensate firms (usually small firms) for the costs of training their apprentices. Nor does the model account for the long run age gains that ualified apprentices ill capture. Assume ualification only is exempt from the coverage of the minimum age. The aggregate production sector maximises its profits: ( ( )) ( ) γ pψ NF π π2... π N φπ π max ωφ π + + = 2 ith respect to Nand π π2... π given value of and of the relative minimum age ω. Folloing the analysis described in the previous section in order to determine the optimal distribution of employment among ualifications the production sector reuires to minimise the ne relative costs function C ( ωπ ; π2... π) ith respect to π π2... π here γ + φπ + max π ωφ( π ) = C ( ωπ ; π2... π). F( π π2... π) Denote πω ( ) the vector of relative uantities hich minimises C ( ωπ ) ɶ ɶ. The first order conditions for the optimal uantities for ualifications =2 are almost identical to the one e describe in euations (23) (24) and (25) the expressions involving the costs must recognize the exemption from the minimum age for the st ualification. Because of the exemption the optimal π ω satisfies: uantity for the first ualification ( ) ρ α( π( ω) ) γφ( π( ω )) = ( 3) ρ γ + αk( πk ( ω )) φπ π ( ω) max ωφ + + ( π( ω) ) k= = 2 π ω = denotes the optimal uantities for a given relative here ( ) minimum age ω. The exemption of one ualification from the minimum is not a marginal change to the optimisation problem e consider earlier on. It is clear that the relative costs to the monopsony production sector are smaller ith the exemption than ithout hoever it is unclear hat the effect of any exemption on the allocation across ualifications should be in general. Our approach here is to sho the effect of a change of the relative minimum age hen one single ualification is exempted. -25-