Online Appendix to The Allocation of Talent and U.S. Economic Growth

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1 Onlne Appendx to The Allocaton of Talent and U.S. Economc Growth Not for publcaton) Chang-Ta Hseh, Erk Hurst, Charles I. Jones, Peter J. Klenow February 22, 23 A Dervatons and Proofs The propostons n the paper summarze the key results from the model. Ths appendx shows how to derve the results. Proof of Proposton. Occupatonal Choce As gven n equaton 5), the ndvdual s utlty from choosng a partcular occupaton,uτ,w,ǫ ), s proportonal to w g ǫ ) β η, where w g w hg s φ s ) η β /τ g. The soluton to the ndvdual s problem, then, nvolves pckng the occupaton wth the largest value of w g ǫ. To keep the notaton smple, we wll suppress the g subscrpt n what follows. Wthout loss of generalty, consder the probablty that the ndvdual chooses occupaton, and denote ths byp. Then p = Pr[ w ǫ > w s ǫ s ] s = Pr[ǫ s < w ǫ / w s ] s = F ǫ,α 2 ǫ,...,α N ǫ)dǫ, 3) wheref ) s the dervatve of the cdf wth respect to ts frst argument andα w / w. Recall that [ N Fǫ,...,ǫ N ) = exp s= T s ǫ θ s ] ρ. Takng the dervatve wth respect to ǫ and evaluatng at the approprate arguments gves { ] } F ǫ,α 2 ǫ,...,α N ǫ) = θ ρ)t T ρ ǫ θ ρ) ρ θ exp [ Tǫ 3)

2 2 HSIEH, HURST, JONES, AND KLENOW where T s T sα θ s. Evaluatng the ntegral n 3) then gves p = F ǫ,α 2 ǫ,...,α N ǫ)dǫ = T { T T ] } ρ θ ρ)ǫ θ ρ) ρ θ exp [ Tǫ dǫ = T dfǫ) T = T T T = s T sα θ s = T w θ s T s w s θ A smlar expresson apples for any occupaton, so we have where w T /θ w. p = wθ s wθ s Proof of Proposton 2. Average Qualty of Workers Total effcency unts of labor suppled to occupaton by groupgare H g = q g p g E[h ǫ Person chooses]. Recall thathe,s) = h s φ e η. Usng the results from the ndvdual s optmzaton problem, t s straghtforward to show that where h η η h s φ ) η. Therefore, h ǫ = h w τ w) ) η η ǫ η +τ h, w τ w H g = q g p ) ) η [ η g h E +τ h ǫ η ] Person chooses. 32)

3 THE ALLOCATION OF TALENT 3 To calculate ths last condtonal expectaton, we use the extreme value magc of the Fréchet dstrbuton. Lety w ǫ denote the key occupatonal choce term. Then y max {y } = max{ w ǫ } = w ǫ. Sncey s the thng we are maxmzng, t nherts the extreme value dstrbuton: Pr[y < z] = Pr[y < z] = Pr[ǫ < z/ w ] ) z z = F,..., w w N [ ] ρ = exp T s w s θ z θ s = exp{ [ Tz θ ] ρ }. 33) That s, the extreme value also has a Fréchet dstrbuton, where T s T s w θ s. Straghtforward algebra then reveals that the dstrbuton ofǫ, the ablty of people n ther chosen occupaton, s also Fréchet: Gx) Pr[ǫ < x] = exp{ [T x θ ] ρ } 34) where T N s= T s w s / w ) θ. Ths result s useful later n the paper n that t mples that the wage dstrbuton across people wthn an occupaton wll also be Fréchet, wth a parameter that depends on θ ρ). Therefore, to recover an estmate of θ from the wage dstrbuton, we wll need to adjust for the correlaton parameterρ. Fnally, one can then calculate the statstc we needed above back n equaton 32): the expected value of the chosen occupaton s ablty rased to some power. In partcular, letdenote the occupaton that the ndvdual chooses, and let λ be some postve exponent. Then, E[ǫ λ ] = ǫ λ dgǫ) = θ ρ)t ρ) ǫ θ ρ) +λ e [T ǫ θ ] ρ dǫ 35)

4 4 HSIEH, HURST, JONES, AND KLENOW Recall that the Gamma functon sγα) x α e x dx. Usng the change-of-varable x = [T ǫ θ ] ρ, one can show that E[ǫ λ ] = T λ/θ = T λ/θ Γ Applyng ths result to our model, we have [ E ǫ η ] Person chooses = T θ = x θ ρ)e λ x dx η Γ ) T θ p g λ ). 36) θ ρ) η Γ θ ρ) ) η θ ρ) η ). 37) Substtutng ths expresson nto 32) and rearrangng leads to the last result of the proposton. Proof of Proposton 3. Occupatonal Wage Gaps The proof of ths proposton s straghtforward gven the results of Proposton. Note that η η η/ η). B Data Appendx Table B.: Sample Statstcs By Census Year

5 THE ALLOCATION OF TALENT 5 Table B.2: Occupaton Categores for our Base Occupatonal Specfcaton

6 6 HSIEH, HURST, JONES, AND KLENOW Table B.3: Examples of Occupatons wthn Our Base Occupatonal Categores

7 THE ALLOCATION OF TALENT 7 Table B.4: Occupaton Categores for our Broad Occupaton Classfcaton