Teacher Quality Policy When Supply Matters: Online Appendix

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1 Teaher Qualiy Poliy When Supply Maers: Online Appendix Jesse Rohsein July 24, 24 A Searh model Eah eaher draws a single ouside job offer eah year. If she aeps he offer, she exis eahing forever. The ouside offer arrives afer he eaher learns her previous year s performane and is paid on ha basis). Ouside offers are indexed by he oninuaion value ha hey provide, ω. I assume ha he ouside offer reeived prior o year >, ω, has a ensored Pareo disribuion: if ω V ζ λ ) F ω ) = V ζ λ ω if V ζ λ < ω < HV ) if HV ω. Here, V is he oninuaion value if he eaher remains in eahing in year under he baseline, single salary onra whih is onsan aross eahers), λ is he annual exi hazard under his onra, and H is he maximum ouside wage, expressed as a fraion of he inside oninuaion value. Imporanly, he disribuion of ω is independen of he eaher s abiliy as a eaher, τ i. Thus, as he eaher learns abou τ i she does no simulaneously learn abou her fuure ouside opions and vie versa). Under he ouside disribuion ), he probabiliy ha a eaher who would obain oninuaion value V [ V λ ], HV in eahing will insead exi is ζ ) V ζ ln λv) λ V ) = Pr {ω > V } = λ V, wih ln V = ζ. The model in he main ex is developed in erms of he negaive of he elasiiy of he exi hazard ln λ wih respe o he inside wage under he baseline onra, ζ ln w = ln λ ln V ln w = ζ ln V ln w. The laer fraion varies wih. I hus solve ln V The use of a ensored disribuion ensures ha V is finie for any ζ. I has no effe on he resuls so long as he ensoring poin is high enough ha offers a ha poin are always aeped. I se H = 2, saisfying his rierion for he onras under onsideraion here.

2 reursively for his elasiiy whih depends on ζ s, s >, bu no on ζ iself ) and use i o define he elasiiy parameer in ) as ζ ln V. ζ ln w The disribuion of he iniial non-eahing offer, ω, is similar o ha of offers laer in he areer, hough here he shape parameer is ompued as ) ζ ln V. η ln w B Solving he model Equaion 4) in he main ex does no have a losed-form soluion, bu for any speified onra i an be solved reursively. Under he learning model developed above, he disribuion of period- performane measure given θ is ) y θ N ˆτ, h)σ + τ 2 σɛ 2 + σ 2 ɛ. 2) This is a univariae disribuion ha an easily be ompued for any speified value of ˆτ. Given ˆτ and y, ompuaion of ˆτ is rivial. The reursive soluion hus has hree seps. Firs, I ompue wt C y,..., y T ), he final period wage under onra C as a funion of he performane signals o dae. Seond, I ompue he value of remaining in eahing in period T, V T θ T ; C), as a funion of θ T, by inegraing wt C over he ondiional disribuion of y T given by 2). Third, for eah < T, given esimaes of V + θ ; C) as a funion of θ, I ompue w C y,..., y ) as a funion of y and θ, hen inegrae eah over he disribuion of y and herefore of θ ) given θ o obain V θ ; C). The sae spae θ is of dimension +, reaing a dimensionaliy problem for areers of reasonable lengh. Noe, however, ha eah of he onras onsidered above redues he sae spae for ompuaion of w C from he - dimensional disribuion {y,..., y } o a one- or wo-dimensional disribuion: {y, y } for he bonus onra and {ȳ } for he enure and alernaive firing onras. Meanwhile, he eaher s assessmen of her own abiliy a he end of period an be summarized eiher by he single variable ˆτ i, or by he pair {µ, ȳ }. I an hus fous on sae spaes of only wo dimensions, θ = {ˆτ, y } for he bonus onra or ˇθ = {µ, ȳ } for he enure and firing onras. I approximae he join disribuions of hese wo-dimensional sae variables and y wih grids of 49 3 poins spaed o have equal probabiliy mass. Having ompued V θ, C) for eah, θ, and C, I simulae he impa of poliies by drawing poenial eahers from he {µ, τ} disribuion, hen drawing performane measures {y,..., y T } for eah. For eah areer, I ompue θ and V a eah year, and use hese o ompue he effes of onra C on he probabiliy of enering he profession and, ondiional on enering, on surviving o year. Noe ha I need no model he disribuion of {µ, τ} in he populaion of poenial eahers under my onsan elasiiy assumpions, hanges in he reurns o eahing indue proporional hanges in he amoun 2

3 of labor supplied o eahing by eah ype ha do no depend on he number of people of ha ype in he populaion. C Marke learing Alernaive onras may yield greaer or lesser enry or persisene in aggregae. For example, adding performane bonuses wihou reduing base pay will yield more enry from high-µ eahers and greaer persisene of high-ˆτ eahers, wihou offseing reduions from eahers wih low µ or ˆτ. Under eah alernaive onra, I ompue he seady-sae size of he eaher workfore, assuming ha he onra has been in plae for a leas T years and ha he same number of enering eahers have been hired in eah year. I onsider wo senarios for labor demand. Firs, I assume ha he eduaion sysem will require he same number of eahers under he alernaive onras as are required under he baseline onra; where my ompuaion yields a larger or smaller workfore han in baseline, I assume ha he base salary is adjused upward or downward o yield he appropriae number of eahers. The α B and α F parameers lised for he fixed quaniy senaro in Table are he adjusmens required given he oher parameers lised here; hese are found via a numerial searh algorihm. My seond senario assumes insead ha he sysem s oal budge is fixed, so ha inreases in average eaher salaries mus be offse by reduions in he number of eahers and herefore by inreases in lass size). The fixed budge senario rows in Table show he α parameers ha balane he disri s budge, given suiable hanges in lass size. D Opimal firing hresholds In Seion IV.B of he main paper, I onsider he opimal hoie of hresholds i.e., uoff values of ȳ ) for firing eahers a eah year. I ompue hese hresholds as he soluion o he disri s dynami opimizaion problem, assuming ha he disri pays a os of firing a eaher ha is proporional o he expeed number of years remaining in he eaher s areer and ha he disri is myopi abou poenial labor supply responses. Speifially, le x represen he number of years ha a eaher wih years of experiene an be expeed o remain in eahing given an annual exi probabiliy of λ and erain reiremen afer year T. I an be shown ha x = λ λ λ ) T ). Le W ȳ ; fire) represen he value of reaining a eaher i.e., offering her employmen for he nex year) afer year < T if her average performane o dae is ȳ and he firing os is fire ; le W fire ) represen he value of hiring a new eaher from he baseline abiliy disribuion; and le Z fire ) 3

4 W fire ) fire x represen he value of firing a eaher afer year. Then he oninuaion value of reaining a eaher afer year = T is: W ȳ ; fire) = λ W fire ) + λ ) φ ȳ + r + ) + δw fire )), 3) σ 2 τ σ 2 ɛ +στ where φ = and hus E [τ ȳ ] = φ ȳ. δ is he disoun rae.) The oninuaion value of reaining a eaher afer year < T is: W ȳ ; fire) = λ W fire ) + λ ) φ ȳ + r + ) 4) +δe [ max { W + ȳ+ ; fire), Z + fire )} ȳ ] ), Thus, he value of hiring a new eaher mus be W fire ) = + r) + δe [ max { W y ; fire), Z fire )}]. 5) Given a hoie of fire and a hypohesized value for W, one an use 3), 4), and 5) reursively o solve for he implied value of W. The fixed poin for his is he value W fire ). Moreover, he firing hresholds a year are he values of ȳ ha equae W ȳ ; fire) wih Z fire ), and hese an be used o ompue he share of enering eahers who will be fired a some poin in heir areers. The esimaes in Figure 7 in he main paper are obained by hoosing a range of values for fire ; using a numerial searh algorihm o find he fixed poin W given fire ; ompuing he firing hresholds implied by hese values and he resuling firing shares; and hen solving he labor supply model given hese hresholds for he marke-learing wages and average produiviy levels. E Appendix Figures 4

5 Figure A: Empirial one-year ariion hazards from he 999/ Shools and Saffing Survey/Teaher Follow-Up Survey.3 All exis from lassroom Exlude family leave, adminisraion Ariion rae Years of experiene Noes: Solid line shows fraion of eahers a eah experiene level in he Shools and Saffing Survey who are no eahing a he ime of he one-year Teaher Follow-Up Survey. Dashed line odes as non-exis a) individuals aring for family members a he ime of he follow-up who say hey plan o reurn o eahing wihin a year and b) individuals who oninue o work for sae & loal governmens in non-eahing jobs in elemenary and seondary eduaion e.g., as prinipals). Verial line orresponds o he assumed reiremen dae T = 3) used in simulaions here. Horizonal lines orrespond o he assumed annual ariion hazards used in he main solid line; λ =.8) simulaions and in he alernaive simulaion in Table 3, Row 3 doed line; λ =.6 for experiene 5 and λ =.3 hereafer). Sample sizes average 22 eahers per year of experiene below 3. 5

6 Figure A2: Join effes of enure onras and budge inreases Base parameers Effe on avg. oupu Fraion denied enure Baseline budge 5% budge inrease Effe on avg. oupu Pessimisi parameers Fraion denied enure Effe on avg. oupu Opimisi parameers Fraion denied enure Noes: Panels show hanges in average oupu, relaive o he single salary onra under he baseline budge and saled in suden-level sandard deviaions, assoiaed wih alernaive enure denial raes and/or budge alloaions. Parameers are as indiaed in Table ; base wages are assumed se o fix he oal disri budge. Marked poins indiae he onra parameers 2% denied enure, wih deisions afer he seond year) used for Table 3, Row. Dashed line models a 5% budge inrease. 6

7 Figure A3: Probabiliy of ever being fired over a 3-year areer under differen deision rules, by rue abiliy Fraion ever fired Firing deision based on: Performane over firs 2 years enure) Average performane o dae Poserior mean abiliy "Opimal" deision rule Perenile of rue abiliy Noes: See Seion IV.C for desripion of he deision rules. Eah rule is se so ha, given he urren abiliy disribuion, he unondiional probabiliy of being fired before he end of a 3 year areer, equals 2%. Figure shows probabiliies ondiional on abiliy, τ. 7

8 Figure A4: Cumulaive firing probabiliy by rue abiliy deile and experiene under differen deision rules Lowes deile Seond deile Cumulaive fraion fired.5.5 Third and fourh deiles Deiles five o en Years of experiene Graphs by abiliy group Firing deision based on: Firs 2 years enure) Average performane o dae Poserior mean abiliy "Opimal" deision rule Noes: See Seion IV.C for desripion of he deision rules. Eah rule is se so ha, given he urren abiliy disribuion, he unondiional probabiliy of being fired before he end of a 3 year areer, equals 2%. Figure shows he probabiliy ha a eaher will be fired on or before year under eah deision rule, given abiliy in he indiaed group and assuming no volunary exi. 8