Tecnical Details of: US-Europe Differences in Tecnology-Driven Growt Quantifying te Role of Education Dirk Krueger Krisna B. Kumar July 2003 Abstract In tis companion document to our paper, US-Europe Differences in Tecnology- Driven Growt: Quantifying te Role of Education we present a complete analytical analysis of te Balanced Growt Pat of our model. Analytic Caracterization of te Balanced Growt Pat Given te motivation of our paper, we are particularly interested in te comparative statics wit respect to te education policy parameter, s, parameters intended to capture product and labor market frictions, C, f, and te speed of tecnological innovation λ. Our strategy is to firstsolvetedifferent stages of te adopting firm s problem for a scedule tat relates te optimal productivity tresold Ē, beyond wic te firm cooses te adoption option, to a given measure of ouseolds available for work in te ig-tec sector. Tis can be viewed as a uman capital demand scedule. We ten derive a similar scedule from te ouseolds education problem te productivity tresold Ē, wit its implied wage differential, tat is necessary to induce a given measure of ouseolds to work in te ig-tec sector. A ig Ē is indicative of te ig wages ouseolds need to expect before tey are willing to supply a given. For tis reason te ouseold condition, denoted by ĒHH, can be viewed as a uman capital supply scedule. Finally we combine tese two scedules to solve for te balanced growt pat education allocation and productivity tresold. All oter BGP equilibrium values follow.. Firms We first discuss te solution of te optimal tecnology adoption problem, conditional on an acceptable E draw and ten analyze te determination of te optimal tresold Ē.
.. Te Tecnology Adoption Decision Recall tat H a = n a E is te effective labor used for production. Te adoption problem is max x λ xhθ a w a H a 2 (x )2, wit associated first order condition ( (H a ) θ = x if x<λ(interior growt) λ if x = λ (maximal growt). Te labor market equilibrium implies n a = and terefore H a = E. Motivated by te above first-order condition, it is convenient to define, for an arbitrary, te tresold E b beyond wic x = λ is optimal (i.e. te speed of tecnology adoption is maximal): ( ) ) (λ ) be min (max θ,,. In general, tere exist cutoffs η g and η 3 g, suc tat, for,η g only interior growt occurs, for η g,η 3 g eiter interior (wen E < be ) or maximal growt (wen E be ) is possible, and for η 3 g, only maximal growt occurs. Tese cutoffs aredefined by: η g (λ ) θ < (λ ) θ η 3 g. () Figure depicts te maximal-growt cutoff function b E ( ). Te b E function and te cutoffs depend only on te model parameters. It can be seen tat E b (0) = ; tat is, as 0, only interior growt is possible. A sufficient condition foronlyinteriorgrowttooccur,forall is (λ ) θ. Wen, if (λ ) θ, E b ( )=, and only maximal growt occurs; if < (λ ) θ <,tene b is interior and growt could eiter be interior (wenever te draw E < E b ) or maximal (wenever E E b ). 2
Ê 0 3 Figure : Cutoff b E for maximal growt Using te adopting firm s first order conditions we obtain, wen E growt) < be (interior x = +(H a ) θ w a = θx (H a ) θ = θ (H a ) θ + θ (H a ) 2θ µ π i (E, ) = ( θ)(h a ) θ + 2 θ (H a ) 2θ. (2) Wen E b E (maximal growt) we obtain, x = λ w a = θλ (H a ) θ π m (E, ) = ( θ) λ (H a ) θ 2 (λ )2. (3) In all tese expressions, H a = E, is a random variable. Te wage rate in te adoption sector, w a, as well as te actual wage for a worker, w a E,is decreasing in te availability of adoption labor, ; te wage is increasing in te productivity draw, E. Te profit function, π(e, ), is an increasing function of E in bot segments it is subscripted wit i for te interior region and m for te maximal region wit a kink at E b. 2 Profits are also increasing in, wic is relevant for te discussion of firm 2 One can sow, by evaluating bot π s at E = E b ( ), tat π(e, ) is continuous in E,andtatte second part (wen growt is maximal) is steeper tan te first (wen growt is interior). 3
beavior below. For low, te firm relies on a ig enoug productivity draw to recoup te cost of adoption and to earn profits; for ig, te firm can afford to proceed wit production for a lower productivity draw. Te optimality condition for te non-adopting firm in te main text, and te nonadoption labor market clearing condition implies tat w n = θβ θ...2 Te Dynamic Problem Te ig-tec firms decision x(e, ) = A0 A is a purely static one; given te complete spillover assumed, te adoption decision does not affect te tecnology tat tis firm as access to in te next period. However, its second stage problem of deciding wic implementation route to follow to increase productivity is dynamic across sub-periods witin a model period and is specified by v(e,n a )= ½ max ( C)π(E,n a ), fn a E + adopt, not adopt Z v(e 0,n a)df (E 0 ) ¾ Profits π are an increasing function of E ; te dismiss option is decreasing in E. We terefore anticipate a tresold Ē [,] suc tat for all E < Ē te firm fires and searces for a better workforce-tecnology matc and for all E Ē it proceeds wit production using te current workforce. In tis sub-section, we caracterize te firm s BGP condition Ē( ). Based on te productivity draw required for maximal growt, E b ( ), one of two cases arises. In te first case, depicted in Figure 2, te tresold productivity Ē is less tat te productivity needed for maximal growt. Terefore, some or all of te E draws tat te firm decides to proceed wit will not result in te maximal growt rate and tus te expected aggregate growt rate E (x) is less tan λ. Two sub-cases are possible. In case a, if <η g, te measure of generally-educated agents is low enoug to cause E b =, wic implies tat every draw of E will result in non-maximal growt. For greater tan η g, but less tan a yet-to-be determined valued η 2 g, some draws of E will result in interior growt and some will result in maximal growt for tat firm (case b). 3 (4) 3 Case b is listed only for completeness. It can be sown tat tis case does not arise on a BGP. Tis is te rationale for using λ, rater tan an ever-increasing tecnological gap relative to te frontier, as te constraint on an individual firm s adoption problem. It can be sown tat tis is also true during te transition to a more rapid rate of tecnological cange, λ 0,provided(λ 0 ) θ >.Detailed transitional analysis is available from te autors on request. 4
Case a: < Ē < Ê ( ) = (Only x < λ possible; Ex < λ ; 0 <= < ) Case b: < Ē < Ê ( ) < (x < λ or x = λ possible; Ex < λ; Є [, 2 ]) Firm s Value Firm s Value Proceed Proceed Fire Fire E E Ê = Ê Ē Ē Figure 2: Firm s Bellman Equation wen E < b E Case 2a: < Ê ( ) < Ē < (x < λ or x = λ possible, but Ex = λ) Case 2b: = Ê ( ) < Ē < (Only x = λ possible; Ex = λ ; 3 <= <=) Firm s Value Firm s Value Proceed Proceed Fire Fire E E Ê = Ê Ē Ē Figure 3: Firm s Bellman Equation wen E b E In te second case, depicted in Figure 3, Ē is greater tan or equal to te productivity needed for maximal growt. Every draw of E for wic te proceed option is cosen by an individual firm will result in maximal growt; terefore, te expected aggregate growt rate E (x) equals λ. Again, tere are two sub-cases. If isgreatertanorequaltoη 2 g, but less tan te η 3 g,somedrawsofe will not be compatible wit maximal growt, but te firm will not coose to proceed wit tese (case 2a). In te second sub-case, for large enoug, 5
tat is, for η 3 g, only draws tat are compatible wit maximal growt will even occur (case 2b). Te expression tat determines te cutoff η 2 g is derived as a by-product of te derivation of te firm s Ē curve. For ease of exposition, te above figures ave been drawn assuming tat te tresold Ē is interior for all. We will consider te corner cases later. If te proceed option lies everywere below te fire option, Ē =, and if it lies everywere above te fire option, Ē =...3 Te Firm Condition Ē Te solution for te Bellman equation depends on weter case or case 2 is assumed. We first assume tat case applies and derive te Ē function. In te appendix (section A.), we sow tat te Ē -scedule is implicitly defined by ( Z ) Ē f F Ē Ē + E df (E ) T " Z be Z ( C) π i E, df (E )+ π m E, df (E ) F # Ē πi Ē,. (5) Ē be Teleftandsideistemarginalcostoffiring te workforce and waiting for a better draw and te rigt and side is te marginal benefit of suc a draw in te form of extra expected net profits. 4 If (5) olds wit > at Ē =, ten te cost of firing is so ig tat te firm accepts any productivity draw, and Ē( )=. On te oter and, if (5) olds wit < at Ē =, te benefit of iring is so ig relative to te cost tat te firm is willing to wait for te igest draw, and Ē( )=. Oterwise an interior productivity tresold Ē ( ) is uniquely determined, wit (5) olding as an equality. In te appendix (section A.2), we caracterize te Ē condition in detail. It is convenient to divide te above condition by trougout; we will argue tat as 0, Ē, separately. In te following discussion, wen we refer to condition (5), it is tis modified condition we refer to. For a given set of cost parameters, f, C, and labor availability,, te left and side of (5) is an increasing function of Ē; waiting for a iger E s entails more rounds of firings and tus iger firing costs. Te rigt and side is decreasing in Ē. Wile a iger Ē implies a iger profit forteadoptingfirm, it reduces te range of E over wic actual production takes place for tose profits to be realized. Te decreasing rigt and side evidently captures te diminising returns inerent in te process of experimenting wit E. Since te left and side is increasing and te rigt and side is decreasing in Ē, an intersection in [,] occurs. However, were te intersection occurs depends on ; we turn to tis next. 4 In te range of in wic be = (case a), te π m term is absent. 6
Te left and side of (5) is independent of. As argued in te appendix, one can find asufficient condition to ensure tat te rigt and side is decreasing in. Given tese properties of (5) wit respect to Ē and, te intersection of te two sides, wic gives te tresold Ē, is decreasing in. We sow in te appendix tat te Ē scedule starts at (and potentially stays at for a neigborood of zero); for very low supply of adoption labor, te firing costs are negligible relative to te profits associated wit te best possible draw so tat te firm is willing to wait for it. It ten decreases monotonically. Te firm condition is depicted in Figure 4. 5 Ē [0, ) : Case a [, 2 ) : Case b [ 2, 3 ) : Case 2a [ 3, ] : Case 2b Ê ( ) 0 2 Ē ( ) 3 Figure 4: Firm Condition Ē At some point te decreasing function Ē crosses te E b function, depicted as a dotted line in Figure 4. Beyond tat point, case 2 applies. Te tat corresponds to tis switc-over, wic we ave labeled η 2 g, is implicitly defined by setting Ē = E b = (λ ) θ in η 2 g (5). Te condition, depicted in Figure 4, sould terefore be interpreted as a composite of te two cases. In te appendix (section A.3), we sow tat te Ē condition for case 2 is implicitly R i 5 Assowninteappendix,iff<( C)( θ) λ (E ) θ df (E ),weobtainē() >. 7
given by: 6 ( Z Ē f F Ē Ē + ) E df (E ) = " Z ( C) π m E, df (E ) F # Ē πm Ē,. (6) Ē Te caracterization of Ē for tis case parallels te one for te previous case. In te appendix (section A.4), we argue tat te Ē scedule is monotonically decreasing in. It is depicted as te η 2 g, -portion of te Ē -curve in Figure 4. We also provide a condition on f and oter parameters of te model to ensure Ē() >. Taken togeter, te decreasing Ē -curve is interpreted as follows. 7 As increases, te firing cost increases linearly since it is paid for every person ired. Profits increase, but less tan linearly, given te diminising returns to H a. Terefore te firm experiments less in adoption. Additionally, profits are ig on account of ig, and terefore te firm does not ave to set a ig productivity tresold in order to realize ig profits and meet adoption costs...4 Dependence of Ē on f For a given, an increase in te unit firing cost of labor, f, increases te marginal cost of drawing anoter E, and te adopting firm experiments less and accepts lower productivity matces for any given ;tatis,ē ( ) decreases. 8 We grap te equilibrium implication of tis sift in te firm condition once we ave caracterized te ouseold condition below...5 Dependence of Ē on C Unlike an increase in te firing cost, an increase in te entry or regulation costs decreases te benefit of redrawing instead of te cost. But te outcome is similar; te adopting firm experiments less and accepts lower productivity matces for any given. We grap te equilibrium implication of tis sift in te firm condition later. To study ow te growt gap between te US and Europe canges wit λ, we next examine te dependence of te adopting firm s condition on λ. at η 2 g. 6 We make assumptions in te appendix to guarantee tat tis condition olds as an equality. 7 It is easy to see from (5) and (6), by setting π i = π m at Ē = E b, tat te scedule Ē is continuous 8 In te appendix (section A.5), we provide analytical details on te dependence of Ē( ) on f,c, and λ. 8
..6 Dependence of Ē on λ Tere are two effects of an increase in te growt rate of te available tecnology, λ. Te minimum adoption labor supply,, needed to induce te firm to adopt at te maximum possiblerateincreases. Sincetefixed cost of adopting at te maximum rate, 2 (λ )2, increases, te wage in te adoption sector as to be low enoug to keep profits ig, wic is ensured by iger.tiseffect is captured by te rigtward sift of E b. Wen λ increases, te expected profit from a iger draw increases, unless only interior draws occur. Tis increase in te marginal benefit of a redraw causes te firm to set a iger adoption tresold; te firm can afford te firing cost for lower productivity draws. Tis effect causes Ē( ) to increase. We grap tis condition later...7 Dependence of Ē on parameters: Summary In summary, te dependence of te firm condition on and te various parameters is given by:.2 Houseolds Ē Ē ; f, C, λ ;,, + Workers take as given te adopting firm beavior, as caracterized by Ē( ), and decide on teir education, wic will result in a BGP education allocation. In order to calculate te expected utilities, denote te value of a vocationally educated agent on te BGP by v v and tat of a generally educated agent by v g. Conditional on being R kept by te firm, te expected utility of a latter agent is ( F(Ē)) Ē log (w a E ) df (E ). Te likeliood of tis outcome is F Ē. Wit probability F Ē, e gets fired, earns noting in te current period, but can expect a utility v g in te future. Terefore, v g = F " Z # Ē log (w a E ) df (E ) + F Ē vg, F Ē Ē wic implies Z v g = log (w a E ) df (E ). F Ē Ē Te condition tat pins down te tresold ability for obtaining general education is given by v g v v Z log (w a E ) df (E ) log (w n )=log(e (a F Ē )) log Ē µ sg s v. 9
Assuming a parametric form for te cost function, e (a) = a,notingtata =,and recalling te definition of our education policy variable, s = sg s v, te tresold condition can be written as Z µ log (w a E ) df (E ) log (w n )=log log (s). (7) F Ē Ē From te optimality condition of te non-adopting firm s problem, and te market clearing condition in te low-tec sector, it follows tat w n = θβ θ / θ. (8) From (2) and (3) we observe tat wages in te ig-tec sector depend on weter growt is interior maximal. In particular ( θ θ η w a E = g (E ) θ + θ 2θ (E ) 2θ wen x<λ θλ θ (E ) θ (9) wen x = λ Te equilibrium ouseold condition depends on wic of tese two cases occurs..2. Houseold Condition ĒHH In te appendix (section A.6), we derive te expression tat implicitly defines te ouseold condition ĒHH by using (8) and (9) in (7). For te case of x<λtis yields T ( Z be F Ē Ē log θ θ (E ) θ + θ (E ) 2θi Z df (E )+ log θλ ) θ (E ) θi df (E ) be log θβ θi +( 2θ)log (2 θ)log log (s). (0) Te left and side of tis expression can be viewed as te benefit of general education in te form of iger expected wages, and te rigt and side as te cost net of subsidy, wic includes te foregone wages in te non-adoption sector, in addition to te disutility of general education. IfteleftandsideisgreateratĒ =or te rigt and side greater at Ē =, a corner solution obtains. In te appendix (section A.7), we caracterize te ĒHH scedule in detail. Te rigt and side of (0) is independent of Ē, wile te left and side increases wit Ē: te benefit of general education is increasing in expected wages, wic is in turn increasing in te productivity tresold. An intersection in [,] occurs. However, as in te firm condition, were te intersection occurs depends on. Itcandirectlybeseentatterigtandsideof(0)isincreasingin ; tis results from an increase in te foregone wages in te labor-scarce non-adoption sector, as well as te increased disutility of inducing inframarginal agents to acquire general education. It can be sown tat te left and side is decreasing in : adoption wages decrease wit te labor supplied. 0
Ē Ē HH ( ) Ê ( ) 0 3 Figure 5: Te Houseold Condition ĒHH Given tese properties of (0) wit respect to Ē and, te intersection of te two sides, wic gives te tresold ĒHH ( ), is increasing in. In te appendix we sow tat wen 0, te wage premium is so ig tat even an Ē =is enoug to attract entry into general education. Te ĒHH curve starts at, (potentially) stays at for an interval, and increases after tat. Tis condition is depicted in Figure 5. As wit te Ē( ) scedule case applies only up to te intersection wit te E b ( ) curve; case 2 applies beyond tat, and is discussed below. We denote by η HH g te point were tis intersection occurs. In te appendix (section A.8), we sow tat te expression tat implicitly defines te ouseold condition ĒHH for te case tat x = λ is: ( Z ) log θλ (E ) θi df (E ) F Ē Ē log θβ θ +( θ)log (2 θ)log log (s). () Te analysis of tis condition parallels tat of te first case, and te details are given in te appendix (section A.9). Te left and side of () is strictly increasing in Ē and te rigt and side is independent of Ē ; te left and side is independent of and te rigt and side is strictly increasing wit. Tis indicates tat te intersection of te two sides, wic gives te tresold Ē necessary for any given labor supply, is increasing in. We can sow tat as, Ē. Wen is very ig, even te lowest ability agents obtain general education. Te disutility cost is so proibitive tat te Ē, and tus
te expected wage premium, as to be very ig to induce entry. Terefore, te ĒHH curve increases up to, and (potentially) stays at for an interval. Tis interval is depicted as te η HH g, portion of te ĒHH curve in Figure 5. Taken togeter, te increasing ĒHH curve is interpreted as follows. 9 As increases, a downward pressure on adoption wages arises, and te disutility of te marginal general education enrollee increases. To counter tese, te productivity levels E > Ē wit wic te firm produces and pays wages ave to be ig..2.2 Dependence of ĒHH on s From bot (0) and (), we see tat te rigt and sides decrease wit s. Given te increasing left and sides, tis means tat te point of intersection, Ē, if interior, sifts down, for a given. 0 As te subsidy for general education increases, ouseolds are willing to supply a given for lower tresolds Ē and tus lower wage premia. We will grap te equilibrium implication of tis sift in te ouseold condition induced by a cange in s below..2.3 Dependence of ĒHH on λ As in te case of te analysis of firms, () implies tat te E b tresold curve sifts to te rigt wen λ increases, and te tresolds η g and η 3 g increase. In bot (0) and (), te rigt and sides te cost of acquiring general education do not depend on λ. A mere examination of te left and side of () sows tat it is increasing in λ. In te appendix (section A.0), we sow tis is also true for (0). Given te flat rigt and sides, te tresold, Ē, decreases. Expected adoption wages are increasing in λ; terefore, suc an increase induces supply of a given even for lower tresolds Ē. Ē HH 9 As in te firm condition, it is easy to see from (0) and () by setting Ē = E b tat te scedule is continuous at η HH g. 0 Te intervals for for wic corner solutions arise cange, owever. Te corner Ē =occurs because te benefit of general education exceeds te cost for any Ē. Since te cost of general education decreases wit s, te benefit of general education exceeds te cost for a larger interval of. Likewise, te corner Ē = occurs because te cost of general education exceeds te benefit foranyē. Since te cost of general education as decreases in s, te cost of general education exceeds te benefit forasmaller interval of. Te beavior of te intervals in wic Ē =or Ē = is very similar to te one discussed above for an increase in s; te only difference is tat an increase in λ works by increasing te benefit wilete increased s works by decreasing te cost of general education. 2
.2.4 Dependence of ĒHH on parameters (summary) To summarize, te ouseold condition is given by Ē HH ĒHH ; sλ. +;,.3 Existence and Uniqueness of Balanced Growt Pat Equilibrium Given te strictly decreasing, continuous firm condition, Ē, and te strictly increasing, continuous ouseold condition,, we obtain a unique BGP equilibrium. It is Ē HH depicted in Figure 6; te intersection of te two conditions yield te equilibrium education allocation and productivity tresold Ē. Ē Ê ( ) Houseolds Intersection in case 2a (Ex = λ) Ē * Firms 0 * 3 Figure 6: Te BGP Equilibrium In te example sown in Figure 6, te intersection between Ē and Ē HH occurs in case 2a of te firm s condition were Ē Ê. Terefore all firms always coose maximal growt. As discussed in section..2, if te intersection occurs in case, te expected growt rate of te economy satisfies E (x) <λ.note tat once and Ē are determined, all oter variables, suc as te expected growt rate, expected wages, and profits on te BGP can be determined. For instance, te average growt rate in te economy is given by E (x) = ³ F ( Z ( be ) ³ Ē (Ē) + (E ) i θ ³ ³³ df (E )+λ F be ). 3
If ³Ê Ē, tis equation reduces to E (x) =λ. 2 Te expected growt rate E(x) is increasing in te equilibrium values of bot and Ē..4 Comparative Statics Tepreviousanalysisofowtefirm and ouseold conditions depend on te parameters (f,c,s,λ) now makes it straigtforward to caracterize teir influence on te BGP equilibrium..4. Increase in Firing Cost, f An increase in te firing cost parameter f (our proxy for labor market regulations) affects only te firm condition by sifting it down. As a result, te equilibrium tresold as well as te general education attainment and te expected growt rate decreases. Firms are willing to produce wit lower E -draws rater tan fire employees; tis translates into lower wages in te adoption sector and a lower incentive to obtaining general education. Figure 7 depicts tis situation. Ē Ê ( ) Houseolds: Ē HH Ē * Increase in f or C moves intersection from Ex = λ to Ex < λ. 0 * 3 Firms: Ē ( ) Figure 7: Effect of an increase in f, C 2³ be = (λ ) θ (). 4
.4.2 Increase in Entry Cost, C An increase of te entry cost parameter C (our proxy for product market regulations and bureaucratic costs) also only sifts te firm s condition downward, again decreasing te equilibrium tresold, general education attainment, and expected growt E(x). Firmsare willing to accept lower E -draws as te benefits of iger draws are diminised, wic results in lower wages in te adoption sector and a lower incentive to obtain general education..4.3 Increase in General Education Subsidy, s Te subsidy parameter s (our proxy for general education focus) only impacts te ouseold condition; an increase in s sifts te ouseold s condition down, reducing te equilibrium productivity tresold E, but increasing general education attainment. If te latter effect dominates, expected growt E(x) increases. Wen te effective cost of general education decreases, ouseolds require a lower tresold Ē and wage premium to coose general education. Wile a decrease in f or C or an increase in s will all result in iger general education attainment, te subsidy increase reduces Ē wile te oters increase it. Tis implies tat te average number of draws before successful production, given by / F Ē, declines wit s (and increases wit f,c). 3 Ē Europe Houseolds US (Ē * ) Eur Intersection in case 2a for US (Ex = λ); in case b for Europe (Ex < λ). (Ē * ) US Firms ( * ) Eur ( * ) US 3 Figure 8: Effect of an increase in s 3 Te mean number of attempts is given by P j= j F j Ē F Ē =. We ave cosen F(Ē) to abstract from discounting in our setup for sake of simplicity; te advantage of te subsidy sceme is even more readily apparent if discounting is taken into account. 5
.4.4 Dependence of BGP on λ Analyzing te dependence of te BGP equilibrium on te speed of tecnology availability, λ, is more complicated because all tree scedules E b ( ) (sifts rigt), Ē ( ) (sifts up), and ĒHH ( ) (sifts rigt) are affected. Te effect on te equilibrium productivity tresold Ē is ambiguous, wereas te equilibrium general education attainment increases unambiguously. Te iger productivity tresold set by te firm, as well as te direct increase in te maximal growt rate, increase te expected wage in te adoption sector. Terefore te incentive to acquire general education increases, and te resulting is iger. Figure 9 depicts tis situation. 4 Ē Houseolds Ē * Increase in λ leaves intersection in case b (Ex < λ) as Ê also sifts out. 0 * 3 Firms Figure 9: Effect of an increase in λ Wat appens to te US-Europe (expected) growt gap wen λ increases? Te answer to tis question is again complicated by te sifts of all tree curves. In Figure 0, we old te ouseold curves at teir old positions to acieve grapical tractability. Te accompanying box explains ow a growt gap, initially zero, can expand wit an increase in λ. Te beavior 4 In te particular example sown in Figure 9, case b obtains bot before and after te increase. However, if te sifting of te b E curve is more muted, wic will appen if θ is not too low, it is possible tat te new intersection is in case 2a, and maximal growt obtains. For ease of drawing, te entire firm condition is sown as sifting upward. In te interval,η g, wen be =, and only interior growt is possible, a cange in λ as no effect. Te argument made in section..6 is for a marginal cange in λ tat leaves te relevant case for Ē uncanged. As long as te intersection of te ouseold and firm conditions does not correspond to case a (only interior growt draws), te figure will be an accurate representation of an increase in λ. 6
of te maximal growt cutoff function, E b, is important for tis possibility. Wile te sifts of te firm and ouseold conditions increase te equilibrium general education attainment for Europe, te productivity tresold corresponding to tis increased attainment may still fall sort of te new E b needed for maximum growt. Alternately, te wages in te adopting sector ave to be low enoug to induce te firm to adopt at te iger maximum speed. In te US, were general education subsidy and attainment are iger, can exceed even te increased cutoff η g. It may terefore continue to grow at te iger potential rate. Ē Europe Houseolds US Ex = λ for bot US and Europe initially. Wen λ increases, US intersection is still above te Ê curve, implying Ex = λ, but for Europe te intersection is below te Ê line, implying Ex < λ. Houseold curves sift out as well, but are eld constant ere for simplicity; te growt gap will increase if tese do not sift too muc. Firms Figure 0: Dependence of te US-Europe growt gap on λ 7
A Appendix A. Firm Condition for Case Assume tat case is te relevant one, and equate te payoffs in(??) attetresoldtoget: Ev f Ē S ( C) ( θ) µ θ 2 Ē + θ 2θ Ē, (2) wit equality if Ē (,). Given te distribution function F (wit density f)one e, integrating te appropriate v over te tree segments, substituting for Ev from te tresold condition and simplifying we obtain te firm condition tat implicitly defines Ē, in a form suitable for comparative statics: ( Z ) " Ē Z be ( C)( θ) f F Ē Ē + E df (E ) T θ (E ) θ df (E ) F # θ Ē Ē + ( C) 2 θ " Z be 2θ Ē (E ) 2θ df (E ) F # 2θ Ē Ē + Ē ( C) A more concise version is presented in te main text as (5). Z be ( θ) λ θ (E ) θ 2 (λ )2 df (E ). (3) A.2 Caracterizing Ē for Case A.2. Dependence on Ē For a given,f,c, te LHS of (3) is an increasing function of Ē. Use Leibniz formula for te term witin curly braces to find tat te derivative is F Ē f e Ē Ē + f e Ē Ē > 0. Te LHS starts out at f wen Ē =and increases to fe(e ) >f at Ē =, independent of. In te RHS of (3) te tird term does not depend on Ē. Again use Leibniz formula for te terms witin te first set of square brackets to find te derivative as θ Ē f e θ Ē θ F Ē Ē + θ Ē f e Ē < 0. Similarly, te terms witin te second set of square brackets are also decreasing in Ē. So te RHS is decreasing in Ē. Tis is true, independent of weter case a or case b is relevant, since E b does not enter te calculation. Since te LHS is increasing and te RHS decreasing in Ē, an intersection is likely to occur. However, were te intersection occurs depends on. A.2.2 Dependence on Te LHS of (3) does not cange wit. We now argue tat te RHS of (3) is decreasing in. Wen b E = (case a), te tird term vanises and te first two terms are directly seen to decrease in. If (λ ) θ >, b E =canberuledout. Tecaseofinterior b E is more involved, as b E = (λ ) θ directly depends on. Te derivative of te first tree terms of te RHS of (3) wit respect 8
to can be sown to reduce to: " Z be ( C)( θ)2 2 θ Ē ( C) θ " Z be ( 2θ) 2 2( θ) (E ) θ df (E ) F Ē Ē θ Ē (E ) 2θ df (E ) F Ē Ē 2θ Z ( C) 2 ( θ) 2 λ θ (E ) θ 2 (λ )2 df (E ). be Since te terms witin te square brackets are all not necessarily positive, te entire derivative cannot be readily signed. Since E b decreases wit, tereby increasing te range over wic maximal profits result, tere is ambiguity regarding te dependence of te RHS of (3) on. For te above derivative to decrease in, intuitively it appears tat te concavity of profits in sould be strong enoug; tat is, θ sould be low enoug. Using (3), one can sow tat te above derivative is negative if: ( Z ) ( θ) f Ē F Ē Ē + E df (E ) > ( C) µ " Z be θ 2 θ 2θ (E ) 2θ df (E ) F # 2θ 2θ Ē Ē + Ē θ ³ ³ 2 (λ )2 F be. To find a sufficient condition we investigate a low value for te LHS and a ig one for te RHS. Given te definition of E b, we can write (λ ) 2 = ³ 2θ η be 2θ g. Using tis, evaluating te integral in te RHS at E b, and simplifying we find one ig value for te RHS is: θ 2 (λ )2 θ 2 ³ ³ 2θ η be 2θ g F be µ θ 2 θ ³ µ 2θ be 2θ F Ē θ 2 θ 2θ 2θ F Ē Ē. For case b tat we are discussing, it is true tat < Ē < E b <,and >η g = (λ ) θ. Using tesefacts,teigvalueforterhscanbemademorestringent,andtesufficient condition reduces to: ( θ) f (λ ) θ ( C) ( F Ē Ē + Z Ē ) E df (E ) + θ (λ )2 2θ # µ 2 θ + θf Ē > θ 2 (λ )2. Since te LHS is increasing in Ē, setting it to its minimum value of yields te sufficient condition we seek for te RHS of (3) to decrease in : " ( θ) f (λ ) θ 2 >θ ( C) 2 2 θ # 2θ. Tis condition is satisfied for θ 0; as conjectured earlier, enoug concavity of profits in will ensure tat te sign of te derivative will be te one we seek. Wen θ is at its maximum value of 2, te above condition reduces to te simpler, but more stringent condition of f> 2. # 9
A.2.3 TeNatureofĒ( ) Te above analysis indicates tat te LHS is strictly increasing and te RHS is strictly decreasing in Ē; te LHS is invariant in and te RHS is strictly decreasing wit. Tese observations indicate tat te intersection of te two sides, wic gives te tresold Ē, is decreasing in. Consider condition (3), multiplied by so tat we can analyze te case of 0; E b, and maximal growt is not possible for any draw in tis case. Bot sides of tis modified condition tend to zero as 0, but te ratio of LHS to RHS tends to zero using L Hospital s rule. Since te RHS, te marginal benefit term of waiting for a draw, dominates te LHS, te marginal cost over all possible Ē, we get Ē (0) =. Indeed Ē =, for in a neigborood of 0 for wic te RHS of (3) exceeds fe(e ). Consider te following expressions for te two sides of (3) wen : µ (λ ) θ ( C) Z 2 θ LHS Ē =; = = f; LHS Ē = ; = = fe (E ) > 0, RHS Ē =; = Z (λ ) θ =( C)( θ) (E ) θ df (E ) + (E ) 2θ df (E ) +( C) Z (λ ) θ RHS Ē = ; = =0. ( θ) λ (E ) θ 2 (λ )2 df (E ), If f < RHS Ē =; =, Ē >. Since te RHS is decreasing in, Ē > at = guarantees te same for all. Te condition says tat te firing cost f cannot be ig enoug to cause te firm to make do wit any E. We will assume tis for now and update it wen we discuss te firm condition for case 2. Terefore Ē starts at (and potentially stays at for a neigborood of zero) and decreases monotonically to a value less tan, if te above assumption is satisfied. Attepointatwicte decreasing function Ē crosses E b, te assumption tat we are in case of te firm s Bellman equation ceases to be valid because Ē E b. Define η 2 g implicitly by setting Ē = E b = (λ ) θ η in 2 g (3). Te following condition determines η 2 g : Ã Ã!! (λ ) θ (λ ) Z (λ ) θ Z θ η f F + 2 g ( C) E η 2 g η 2 df (E ) = g η 2 ( θ) λ η 2 θ (λ ) θ g (E ) θ (λ )2 df (E ) g 2 η 2 g (4) Ã!! Ã! ( C)( θ) (λ ) θ η 2 θ Ã (λ ) θ θ F ( C) θ Ã!! Ã! 2 (λ ) θ η 2 g g η 2 g η 2 2θ Ã (λ ) 2θ θ F. η 2 g g η 2 g We assume tat te parameters of te model are suc tat η 2 g < ; tat is, wen η 2 g =, te LHS of (4) is lower tan te RHS. A.3 Firm Condition for Case 2 Given tat case 2 is te relevant one, equating te payoffs in(??) attetresoldyields: Ev f Ē =( C) ( θ) λ θ Ē (λ )2. 2 20
We ave assumed tat Ē > as discussed above; if we are in tis case, Ē <,and terefore we can write te tresold condition as an equality. Integrate v over te two relevant segments, substitute for Ev from te tresold condition, ten simplify to obtain te firm s condition for tis case as: f ( F Ē Ē + Z Ē ) E df (E ) = " Z ( C)( θ) λ θ A more concise version is presented in te main text as (6). (E ) θ df (E ) F # θ Ē Ē. Ē (5) A.4 Caracterizing Ē for Case 2 A.4. Dependence on Ē Te LHS of (5) is uncanged from case ; it starts out at f wen Ē =and increases to fe(e ) >f at Ē =, independent of. InteRHS,usingLeibniz formulafortetermwitinsquarebrackets we find te derivative as θ E f e θ θ Ē θ F Ē Ē + E f e Ē < 0. As in case, te RHS is decreasing in Ē. A.4.2 Dependence on Since te LHS of (5) is increasing and te RHS decreasing in Ē, an intersection is likely to occur. However, were te intersection occurs depends on. Te LHS does not cange wit. In case 2, unlike case, detailed derivatives are not needed to argue tat te RHS is decreasing in ; mere examination of (5) reveals tis. Since Ē > E b, E b does not even figure in te condition for case 2, and te ambiguity referred to in case does not arise ere. A.4.3 TeNatureofĒ Te above analysis indicates tat te intersection of te two sides, wic gives te tresold Ē, is decreasing in. Since bot sides of te expression in (3) and (5) coincide at = η 2 g, te Ē scedule is continuous at η 2 g. To caracterize te relationsip in te interval η 2 g,, we compute: LHS Ē =; = = f; LHS Ē = ; = = fe(e ) > 0, RHS Ē =; = " Z # = ( C)( θ) λ (E ) θ df (E ) ; RHS Ē = ; = =0. An intersection clearly exists. To ensure, Ē >, we need f<rhs Ē =; =. Since we will be in case 2, rater tan case, for tis corner to be relevant we update te earlier assumption on f wit te following assumption: " Z # f<( C)( θ) λ (E ) θ df (E ). Terefore, Ē starts at te case value for η 2 g and decreases monotonically to a value larger tan, if te above assumption is satisfied. Te firm condition depicted in Figure 4 is a composite of te two cases discussed tus far. 2
A.5 Dependence of Ē on f,c,λ In bot (5) and (6), an increase in f sifts te left and side upward. Given tat te rigt and sides of tese conditions are decreasing in Ē and independent of f, te points of intersection sift leftward; tat is, Ē( ) decreases. In te neigborood of =0, it still is te case tat Ē (0) = ; te firm scedule for Ē( ) <sifts down wit an increase in f. For a given, wen te cost of entry, C, increases te rigt and sides of bot (5) and (6) sift downward. Given tat te left and sides are increasing in Ē and independent of C, te points of intersection sift leftward; tat is, Ē ( ) decreases. In te neigborood of =0, it will still be tecasetatē (0) = ; te firm scedule for Ē( ) <sifts down wen tere is an increase in C. Wecanseefrom()tatteE b tresold curve sifts rigt wen λ increases, and te tresolds η g and η 3 g increase. In bot (5) and (6), te left and sides te marginal cost of firingandredrawing donotdepend on λ. Below, we provide sufficient conditions to ensure tat te rigt and sides are increasing in λ for cases oter tan case a, were only interior growt rates are realized. Given te nature of te two sides discussed above, te points of intersection sift rigtward; tat is, Ē ( ) increases. Tus Ē sifts up for cases oter tan case a. A mere examination of te RHS of (5) sows tat it is increasing in λ. To see if tis is true for te RHS of (3), we take te derivative wit respect to λ. Tis derivative can be simplified to: ( C) Z ( θ) i θ (E ) θ (λ ) df (E ) be Tis expression is not automatically positive; te fact tat profits are positive wen growt equals λ does not imply tat its derivative is positive. We need to make distributional assumptions to unambiguously sign te derivative. Anticipating te calibration, we assume a uniform distribution for productivity draws: f e (E )=. Carrying out te integration in te above derivative, and using te definition of be we can sow tat te expression is positive if: ( θ) θ () θ > (λ ) 2θ (λ ) + θ. ( + θ) ( + θ) For tis case (b), b E, implies tat θ () θ in te LHS cannot be smaller tan (λ ), and te RHS is largest at =. Make tese substitutions to get a stringent sufficient condition for te derivative to be positive as: (λ ) θ >. Consistent wit te sufficient condition in section (A.2), a low θ and a low are more likely to satisfy te above condition and cause te RHS of (3) to increase in λ. Given te caracterization of Ē discussed earlier, te firm condition increases (sifts upward) wen λ increases. 22
A.6 Houseold Condition for Case Substituting te relevant expressions for te wages in (7), we find: T ( Z be F Ē log log θ θ (E ) θ + θ 2θ (E ) 2θi Z df (E )+ θλ ) θ (E ) θi df (E ) log Ē be θβ θ / i θ log log (s). (6) Factoring out 2θ from te two integrals wen > 0 we will discuss te =0case separately on te left and side, transposing terms, and noting tat 2θ >0, we obtain te expression tat implicitly defines te ouseold condition ĒHH as: T ( Z be F Ē Ē log θ θ (E ) θ + θ (E ) 2θi Z df (E )+ log θλ ) θ (E ) θi df (E ) be log θβ θi +( 2θ)log (2 θ)log log (s). A.7 Caracterizing ĒHH A.7. Dependence on Ē for Case Te RHS of (0) is independent of Ē. Differentiate te LHS wit respect to Ē to get: ef ( Z Ē be 2 F Ē + F Ē Ē n log log θ θ (E ) θ + θ (E ) 2θi Z df (E )+ θ θ Ē θ + θ Ē 2θ i ef Ē o. be log θλ ) θ (E ) θi df (E ) Since wages are iger in te maximum growt region (at E = E b, interior and maximal growt wages are te same, and for iger E, maximal growt wages are iger), te second term is larger tan te first. Terefore te derivative is: > > ef Ē F Ē ( ef ( Ē F Ē F Ē Z F Ē log Ē log θ θ (E ) θ + θ (E ) 2θi df (E ) log θ θ θ i ) 2θ Ē + θ Ē θ θ Ē θ + θ Ē 2θ i F Ē log θ θ Ē θ + θ Ē 2θ i ) =0. Terefore te LHS of (0) is increasing in Ē. Taken togeter wit te flat RHS, tis means tat an intersection is likely to occur. However, were te intersection occurs depends on. A.7.2 Dependence on It can directly be seen tat te RHS of (0) is increasing in. To study te dependence of te LHS of (0) on, differentiate te terms witin curly braces and simplify to obtain te derivative as: Z be 2 θ θ (E ) θ Z Ē θ θ (E ) θ + θ (E ) df (E ) θ η 2θ g df (E ) < 0. be Tat is, te LHS of (0) is decreasing in. 23
A.7.3 TeNatureofĒHH Te above analysis indicates tat te LHS of (0) is strictly increasing and te RHS is independent of Ē; te LHS is strictly decreasing wit and te RHS is strictly increasing wit. Tese results indicate tat te intersection of te two sides, wic gives te tresold Ē necessary for any given labor supply, is increasing in. Firstexaminetecaseof 0. Working wit (6), it can be seen tat bot LHS Ē =; =0 and LHS Ē = ; =0 diverge to. Examination of te RHS of (0) reveals tat RHS =0 log θβ θ log (s), afinite quantity. Since te benefit of general education exceeds te cost for any Ē, te corner case of Ē =results. It is likely for a neigborood of =0tat LHS > RHS and Ē =. Wen is very low, relative wages are so ig tat even an Ē =is enoug to attract entry into general education. Te situation is reversed for. We ave tat te is LHS of (0) well-defined at bot endpoints of Ē: LHS Ē =; = ( Z be = log θ (E ) θ + θ (E ) 2θi Z ) df (E )+ log θλ (E ) θi df (E ) be LHS Ē = ; = = log θ () θ + θ () 2θi. Wen Ē =, given case requires E b Ē, we need to set E b = ; L Hospital s rule is needed to evaluate te LHS. However, RHS =. Since te cost exceeds te benefit for any Ē, te corner case of Ē = results. It is likely for a neigborood of =tat LHS < RHS and Ē =. Wen is very ig, even te lowest ability agents ave to obtain general education, and te disutility is so proibitive tat te Ē, and tus te expected wage premium of generally educated agents, as to be very ig to induce entry. Tis analysis indicates tat te ĒHH curve starts at, (potentially) stays at for an interval, increases up to, and (potentially) stays at for an interval. At some = η HH g curve will intersect wit b E. Define η HH g µ F µ (λ ) θ η HH g te ĒHH implicitly, by setting Ē = E b = (λ ) θ in (0): η HH g Z log θλ η HH θ g (E ) θi df (E ) (λ ) θ η HH g = log θβ θ +( 2θ)log η HH g Case 2, wic is discussed below, becomes relevant beyond η HH g. (2 θ)log η HH g log (s). A.8 Houseold Condition for Case 2 Substituting te relevant expressions for te wages in (7), factoring out θ from te integral on te left and side, and transposing terms we obtain te expression tat implicitly defines te ouseold condition ĒHH in tis case as: F Ē ( Z Ē log ) θλ (E ) θi df (E ) log θβ θ +( θ)log (2 θ)log log (s). 24
A.9 Caracterizing ĒHH for Case 2 A.9. Dependence on Ē Te RHS of () is independent of Ē. Te derivative of te LHS of () wit respect to Ē is: ef ( Z Ē = θλ (E ) θi df (E ) log θλ i ) θ Ē F Ē F Ē > ef ( Ē F Ē F Ē log Ē log θλ Ē θ i F Ē log θλ Ē θ i ) =0. A.9.2 Dependence on Te LHS of () is independent of. Mere examination reveals tat te RHS is increasing in. A.9.3 TeNatureofĒHH Te above analysis indicates tat te LHS of () is strictly increasing and te RHS is independent of Ē; te LHS is independent of and te RHS is strictly increasing wit. Tese results indicate tat te intersection of te two sides, wic gives te tresold Ē necessary for any given labor supply, is increasing in. Since bot sides of te expression in (0) and () coincide at = η HH g, te ĒHH scedule is continuous at η HH g. To caracterize te relationsip in te interval η HH g,, we compute for (): LHS ( Z ) Ē = ĒHH = ĒHH log θλ (E ) θi df (E ) > log θλ () θi F LHS Ē = = log θλ () θi f e () e f () Ē HH =log θλ () θi, were L Hospital s rule as been used for te second evaluation. Te LHS is independent of and is an increasing function in Ē, wit finite endpoints. However, te RHS = diverges to. Terefore, wen =, te cost of general education exceeds te benefit for any Ē, and Ē = results; tis is likely to be true for a neigborood of =. A.0 Dependence of ĒHH on λ Take te derivative of te terms in curly braces in te LHS of (0), noting E b = (λ ) θ to λ. Tis derivative is R be λ df (E ) > 0., wit respect 25