Online Appendix to Optimal Gradualism

Online Appendix to “Optimal Gradualism” A.1 Proofs for Section 1 Proof of Proposition 1:. Suppose that kx,t > 0. We verify this is the case at the end of the proof. Firms in island x must be indifferent between producing with workers or producing using the new technology. This implies wx,t = (1 + τx,t)/Ax,t. We now show that wages across non-disrupted islands are equalized. Assumption 1 ensures that this holds at time 0. Each period, a flow α ⋅ (1 − ℓt) of workers joins these islands. Directed search implies that this flow of workers is allocated in a way that preserves the equality of wages. In particular, this is the case so long as new workers are allocated in proportion to island populations at the time of their arrival. This shows that, under Assumption 1, directed search is not strictly necessary. Random search as in Alvarez and Shimer (2011) suffices. The expression pinning down the common wage wt comes from the fact that we have normalized the price of the final good to 1, which implies that island wages lie along the isocost curve 1 = cf ({wx,t}x∈D , wt). Note that this equation implies that wt is implicitly a function of the vector of after tax productivities {(1 + τx,t)/Ax,t}. To derive the expression for output and new technology utilization we use Shepard’s lemma, which implies yx,t = yt ⋅ cfx . Adding across non-disrupted islands, yields ℓt = yt ⋅ ∑ cfx = yt ⋅ cfw ({wx,t }x∈D , wt ) , x∉D which after rearrangement yields the expression for output. On the other hand, for disrupted islands we have ℓx,t + kx,t = yt ⋅ cfx ({wx,t }x∈D , wt ) . Substituting the expression for output and rearranging yields the expression for kx,t in the A1 proposition. To conclude, we show that kx,t > 0 as claimed in the proposition (and initially in the proof). Note that this is needed to ensure that capital is actually used and that firms are therefore indifferent between hiring workers or capital. Assumption 2 implies cfx ({wx,t }x∈D , wt ) cfw ({wx,t }x∈D , wt ) > cfx ({w̄}x∈D , w̄) cfw ({w̄}x∈D , w̄) = ℓx,0 ℓx,t ≥ . ℓ0 ℓt Rearranging this inequality yields kx,t > 0. Proof of Proposition 2:. The first part follows from the definition of a complete market. The second and third parts follow from an envelope logic. Workers in non-disrupted island can set αx = 0 and achieve the exact consumption path of workers in disrupted islands, since their wages are higher (or equal) at all points in time. They can then consume the excess savings generated with this policy, which implies that they can always achieve a strictly higher utility than disrupted households. Because they are making optimal decisions, we must have U0 > Ux,0 , as wanted. A.2 Proofs for Section 2 This section provides proofs for Lemma 1 and Propositions 3, 4, and A2. Proof of Lemma 1. We first derive the expression for the change in tax revenue in (1). We prove this in a slightly more general case. For simplicity, we ignore time subscripts. Suppose there are different types of labor indexed by ℓj , different (untaxed) intermediate goods yi , and different types of (taxed) capital km . Tax revenue is given by T = ∑ τm ⋅ m km , Am which implies dT = ∑ τm ⋅ m dkm km +∑ ⋅ dτm . Am m Am Each producer f operates some CRS production function. Thus, we have 0= max f y f ,yif ,km ,ℓfj y f − ∑ yif i f km ⋅ pi − ∑ ⋅ (1 + τm ) − ∑ ℓfj ⋅ wj . m Am j A2 The envelope theorem then implies 0 = − ∑ yif ⋅ dpi − ∑ m i f km ⋅ dτm − ∑ ℓfj ⋅ dwj . Am j Adding this across all producers, we obtain ∑ m km ⋅ dτm = − ∑ ℓj ⋅ dwj . Am j Plugging into the expression for dT yields a general version of the formula in the lemma: dT = ∑ τm ⋅ m dkm − ∑ ℓj ⋅ dwj . Am j We now turn to the change in welfare. The welfare function is given by ∞ W0 = ∑ ℓx,0 ⋅ W (max Ux ({wx,t + Tt , wt + Tt }∞ t=0 ; α)) + ℓ0 ⋅ W (U({wt + Tt }t=0 )) . α x∈D Recall that Ux ({wx,t +Tt , wt +Tt }∞ t=0 ; α) gives the maximum utility that a displaced household can achieve. The envelope theorem implies that, so long as cx,t > 0, an infinitesimal change in wx,t + Tt changes Ux ({wx,t + Tt , wt + Tt }∞ t=0 ; α) by −ρt ⋅ u′ (cx,t ) ⋅ (dwx,t + dTt ). dUx ({wx,t + Tt , wt + Tt }∞ t=0 ; α) = Px,t ⋅ e This is because optimizing households weakly prefer consuming the additional income to any other use. After all, households could always reduce their consumption but they choose not to do that. The right hand side gives the expected marginal utility of consuming this extra income. Likewise, so long as cx,tn ,t > 0, an infinitesimal change in wt + Tt changes Ux ({wx,t + Tt , wt + Tt }∞ t=0 ; α) by dUx ({wx,t + Tt , wt + Tt }∞ t=0 ; α) = ∫ t 0 e−ρt ⋅ u′ (cx,tn ,t ) ⋅ (dwt + dTt ) ⋅ ⋅αx ⋅ e−αx tn ⋅ dtn =(1 − Px,t ) ⋅ e−ρt ⋅ E[u′ (cx,tn ,t )∣tn ≤ t] ⋅ (dwt + dTt ), where this expression integrates over all potential histories in which the household is at the non-disrupted island at time t and can consume this extra income. A3 Finally, so long as ct > 0, an infinitesimal change in wt + Tt changes U({wt + Tt }∞ t=0 ) by −ρt dU({wt + Tt }∞ ⋅ u′ (ct ) ⋅ (dwt + dTt ). t=0 ) = e Combining these results yields the formula in equation (2). Note that the lemma also applies when there is endogenous reallocation effort. This is because changes in reallocation effort have a second order effect on Ux,0 (households are optimizing with respect to αx ). Proof of Proposition 3. The proof follows Costinot and Werning (2022). Using the definition of the χ’s, we can write the change in welfare following a variation as ∞ (A1) dW0 = ∫ 0 [ ∑ ℓx,t ⋅ χx,t ⋅ dwx,t + χ̄t ⋅ dTt ] dt. x∈X Consider a change in taxes that changes kx′ ,t by dkx′ ,t but keeps the utilization of all other types of capital unchanged. At a social optimum, this variation cannot affect welfare. Thus: ∑ ℓx,t ⋅ χx,t ⋅ dwx,t + χ̄t ⋅ dTt = 0. x∈X Using the fact that dTt = − ∑x∈X ℓx,t ⋅ dwx,t + τx′ ,t ⋅ dkx′ ,t Ax′ ,t , we obtain ∑ ℓx,t ⋅ (χx,t − χ̄t ) ⋅ dwx,t + χ̄t ⋅ τx′ ,t ⋅ x∈X dkx′ ,t = 0, Ax′ ,t which can be re-written as τx′ ,t ⋅ Dividing by that ∂ ln wx,t ∂ ln kx′ ,t dkx′ ,t Ax′ ,t dkx′ ,t χx,t = ∑ ℓx,t ⋅ ( − 1) ⋅ (−dwx,t ). Ax′ ,t x∈X χ̄t and rearranging terms yields the formula in (3). The derivation clarifies refers to a partial derivative (i.e., the percent change in wages across islands resulting from a change in kx′ ,t holding all other kx,t as well as the allocation of workers across islands constant). Proof of Proposition 4. We use Lemma 1, which continues to be valid when reallocation effort is endogenous. Suppose we are at an optimum. Consider a reform that changes kx′ ,t by dkx′ ,t but leaves all other kx,s unchanged. This reform changes αx by dαx and, because the reform kept kx,s fixed for all other x, s, it also changes wages and tax revenue at all points in time and islands. A4 Define the direct effect of the reform as the effect on welfare through wages and tax revenue holding all αx constant. This then triggers an indirect effect via changes in αx which affect wages and revenue at all other time periods. For an outcome ax,s , denote by dk ax,s the direct effect of the reform—i.e., the change induced by kx′ ,t —and by dα ax,s the indirect effect—i.e., the change induced by changes in αx . Let’s first consider the indirect effects and the determination of dαx . The first-order ′ condition for αx is κx (αx ) = Ux,α . Totally differentiating this equation we get (A2) ∂Ux,α ⋅ dαx′′ + Ux,α,d,t ⋅ (dk wx,t + dk Tt ) ⋅ dt + Ux,α,n,t ⋅ (dk wt + dk Tt ) ⋅ dt. x′′ ∈D ∂αx′′ ′′ κx (αx ) ⋅ dαx = ∑ In this equation, ∂Ux,α ∂αx′′ gives the effect of changing dαx′′ on Ux,α via changes in wages and tax revenue over time (this object also has to be computed holding kx,s constant for all x, s). The equation also shows that the direct effects of the reform also alter Ux,α , but these effects are “of the order of” dt, since the direct effect only changes wages and tax revenue at a point in time t. Equation (A2) is a system of equations that can be solved as (A3) ′′ dαx = ∑ εx′′ ,x ⋅ (Ux,α,d,t ⋅ (dk wx,t + dk Tt ) + Ux,α,n,t ⋅ (dk wt + dk Tt )) ⋅ dt x∈D The εx′′ ,x tell us how changes in the direct incentives to reallocate in island x affect reallocation rates from island x′′ . Let ℓx,0 ⋅ µx denote the welfare gains from increasing the reallocation rate out of island x. It captures all the indirect changes in wages and tax revenue generated by changes in reallocation rates (holding technology utilization constant at all points in time and islands). We can compute the welfare gains from the reform that changes kx′ ,t by dkx,t and leaves all other kx,s unchanged as dW0 = ∑ ℓx,t ⋅ χx,t ⋅ (dk wx,t + dk Tt ) ⋅ dt + ∑ ℓx′′ ,0 ⋅ µx′′ ⋅ dαx′′ . x′′ ∈D x∈X Note that there is a dt multiplying the welfare effects of the direct effect of the reform via wages and tax revenue, since this only accrue in an instant of time. This implies that both the direct and indirect effects are “of the order of” dt. A5 Using the formula for dαx′′ in equation (A3) we get dW0 = ∑ ℓx,t ⋅ χx,t ⋅ (dk wx,t + dk Tt ) ⋅ dt x∈X + ∑ ∑ ℓx′′ ,0 ⋅ µx′′ ⋅ εx′′ ,x ⋅ (Ux,α,d,t ⋅ (dk wx,t + dk Tt ) + Ux,α,n,t ⋅ (dk wt + dk Tt )) ⋅ dt, x∈D x′′ ∈D which can be rewritten as dW0 = ∑ ℓx,t ⋅ χend x,t ⋅ (dk wx,t + dk Tt ) ⋅ dt. x∈X At an optimum, we must have that this variation cannot increase welfare. Following the same steps as in the proof of Proposition 3, we get dk ln wx,t ℓx,t ⋅ wx,t χend ℓt ⋅ wt χend dk ln wt x,t ⋅( − 1) ⋅ (− )+ ⋅ ( t − 1) ⋅ (− ) mx′ ,t χ̄t dk ln kx′ ,t mx′ ,t χ̄t dk ln kx′ ,t x∈D τx′ ,t = ∑ This derivation also clarifies that dk ln wx,t dk ln kx′ ,t and dk ln wt dk ln kx′ ,t are partial derivatives: they corre- spond to the change in wages given a change in capital holding reallocation rates constant. To complete the proof of the proposition, we compute the change in welfare driven by a change in dαx holding kx,s constant at all periods. Equation (A1) implies that these welfare gains are given by µx ⋅ ℓx,0 ⋅ dαx = ∫ ∞⎢ ⎡ ⎤ ⎥ ⎢ ∑ ℓx′′ ,s ⋅ χx′′ ,s ⋅ dα wx′′ ,s + χ̄s ⋅ dα Ts )⎥ds. ⎢ ⎥ ⎢ x′′ ∈X ⎥ ⎣ ⎦ 0 The variation we are considering keeps all quantities constant (except for αx ). This implies dα Ts = ∑ dτx,s ⋅ x∈D kx,s = − ∑ ℓx′′ ,s ⋅ dα wx′′ ,s , Ax,s x′′ ∈X which we can use to write the change in welfare due to changes in reallocation rates as µx ⋅ ℓx,0 ⋅ dαx = ∫ ∞⎢ 0 ⎡ ⎤ ⎥ ⎢ ∑ ℓx′′ ,s ⋅ (χx′′ ,s − χ̄s ) ⋅ dα wx′′ ,s ⎥ds. ⎢ ⎥ ⎢ x′′ ∈X ⎥ ⎣ ⎦ The formula in (5) follows from the fact that dα wx′′ ,s = ∂wx′′ ,s ⋅ (−s ⋅ e−αx s ) ⋅ ℓx,0 ⋅ dαx . ∂ℓx,s A6 Note that these are partial derivatives since we are interested on the effect of αx′ on wages and tax revenues holding all other αx and kx,s constant. A.2.1 Deriving Formulas for Ux,α , Ux,α,d,t , Ux,α,n,t . Hand-to-mouth: In this case, we have ∞ Ux = ∫ 0 e−ρt ⋅ [Px,t ⋅ u(wx,t + Tt ) + (1 − Px,t ) ⋅ u(wt + Tt )] ⋅ dt. Differentiating this with respect to α, and then with respect to wages at time t, we obtain: ∞ Ux,α = ∫ 0 e−ρt ⋅ (t ⋅ Px,t ) ⋅ [u(wt + Tt ) − u(wx,t + Tt )] ⋅ dt, Ux,α,d,t = − (t ⋅ Px,t ) ⋅ λx,d,t , Ux,α,n,t =(t ⋅ Px,t ) ⋅ λx,n,t . No borrowing and no transition risk: Let cx,t = Px,t ⋅ wx,t + (1 − Px,t ) ⋅ wt + Tt In this case, we have ∞ Ux = ∫ 0 e−ρt ⋅ u (cx,t ) ⋅ dt. Differentiating this with respect to α, and then with respect to wages at time t, we obtain: ∞ Ux,α = ∫ 0 e−ρt ⋅ (t ⋅ Px,t ) ⋅ (wt − wx,t ) ⋅ u′ (cx,t ) dt, Ux,α,d,t = − (t ⋅ Px,t ) ⋅ λx,d,t + e−ρt ⋅ (t ⋅ Px,t ) ⋅ (wt − wx,t ) ⋅ u′′ (cx,t ) ⋅ Px,t , Ux,α,n,t =(t ⋅ Px,t ) ⋅ λx,n,t + e−ρt ⋅ (t ⋅ Px,t ) ⋅ (wt − wx,t ) ⋅ u′′ (cx,t ) ⋅ (1 − Px,t ). Borrowing with transition risk: In this case there are no simple analytical expressions for Ux,α , Ux,α,d,t , Ux,α,n,t , nor a simple way of computing these objects numerically. For this reason, we do not analyze this scenario with endogenous reallocation effort. A7 Ex-post complete markets: Assume that u(c) = c1−γ /(1 − γ) and let ∞ hx,0 = ax,0 + ∫ 0 e−rt ⋅ [Px,t ⋅ wx,t + (1 − Px,t ) ⋅ wt + Tt ] ⋅ dt denote the effective wealth of households in disrupted islands at time 0. We can solve analytically for Ux as −γ 1 Ux = [r − (r − ρ)] ⋅ h1−γ x,0 /(1 − γ). γ Differentiating this with respect to α, and then with respect to wages at time t, we obtain: −γ ∞ 1 ⋅ e−rt ⋅ (t ⋅ Px,t ) ⋅ (wt − wx,t ) ⋅ dt, Ux,α = [r − (r − ρ)] ⋅ h−γ x,0 ∫ γ 0 −γ Px,t 1 −rt Ux,α,d,t = − [r − (r − ρ)] ⋅ h−γ ⋅ (t ⋅ Px,t ) − γ ⋅ ⋅ Ux,α , x,0 ⋅ e γ hx,0 −γ 1 − Px,t 1 −rt ⋅ (t ⋅ Px,t ) − γ ⋅ ⋅ Ux,α . Ux,α,n,t = [r − (r − ρ)] ⋅ h−γ x,0 ⋅ e γ hx,0 A.2.2 Optimal Assistance Programs We now study optimal policy design for a government that can tax technology and set temporary assistance programs. As in the main text, we let λx,d,t = e−ρt ⋅ u′ (cx,t − ψ(nx,t )) denote the marginal utility of consumption at time t for households that have not reallocated, λx,n,t = E[e−ρt ⋅ u′ (cx,tn ,t − ψ(nx,t ))∣tn ≤ t] denotes the average marginal utility of consumption among households that reallocated by time t, and λt = e−ρt ⋅ u′ (ct − ψ(nt )) denotes the marginal utility of consumption of non-disrupted households at time t. We also define χx,t and χ̄t as in Proposition 3. Without loss of generality we normalize initial effort to 1 at all islands, so that work effort is given by ε wx,t ⋅ (1 − Rt ) ℓ nx,t = ( ) . w̄ Moreover, the counterfactual work effort of non-participants in non-disrupted islands is n∗t = ( wt εℓ ) . w̄ A8 Define T̃t = Tt + Rt ⋅ wt ⋅ n∗t . Households’ income at time t becomes (1 − Rt ) ⋅ wx,t ⋅ nx,t + T̃t . Moreover, we can write the condition for a balanced government budget as T̃t = ∑ τx,t ⋅ x∈D kx,t + Rt ⋅ ∑ ℓx,t ⋅ wx,t ⋅ nx,t Ax,t x∈X This shows that our formulation for temporary assistance programs is equivalent to having a time varying linear income tax Rt . We first extend Lemma 1 to this environment. Lemma A1 (Joint variations Lemma) Consider a variation in taxes on technology and the replacement rate that induces a change in wages dwt , dwx,t , technology dkx,t , tax revenue dT̃t , and reallocation effort dαx . This variation changes tax revenue and social welfare by dT̃t = ∑ τx,t ⋅ (A4) x dkx,t − (1 − Rt ) ⋅ ( ∑ ℓx,t ⋅ nx,t ⋅ dwx,t ) + mℓ,t ⋅ dRt Ax,t x∈X dRt ) 1 − Rt x∈X ⎡ ⎤ ∞⎢ ⎥ ⎢ dW0 = ∫ ⎢ ∑ ℓx,t ⋅ χx,t ⋅ nx,t ⋅ [(1 − Rt ) ⋅ dwx,t − wx,t ⋅ dRt ] + χ̄t ⋅ dT̃t ⎥⎥ ⋅ dt. 0 ⎢ x∈X ⎥ ⎣ ⎦ + εℓ ⋅ Rt ⋅ ( ∑ ℓx,t ⋅ nx,t ⋅ dwx,t − mℓ,t ⋅ (A5) Proof. The expression for the change in welfare follows from the envelope theorem and the definition of χx,t and χ̄t . Turning to the change in revenue, we have dkx,t kx,t ⋅ τx,t + ∑ ⋅ dτx,t + Rt ⋅ ∑ ℓx,t ⋅ nx,t ⋅ dwx,t + mℓ,t ⋅ dRt + Rt ⋅ ∑ ℓx,t ⋅ wx,t ⋅ dnx,t . x∈D Ax,t x∈D Ax,t x∈X x∈X dT̃t = ∑ Following the same steps from the proof of Lemma 1, we can show that ∑x∈D kx,t Ax,t ⋅dτx,t = − ∑x∈X ℓx,t ⋅ nx,t ⋅ dwx,t . Plugging this expression into the formula for dT̃t we get dkx,t ⋅ τx,t − (1 − Rt ) ⋅ ∑ ℓx,t ⋅ nx,t ⋅ dwx,t + mℓ,t ⋅ dRt + Rt ⋅ ∑ ℓx,t ⋅ wx,t ⋅ dnx,t . x∈D Ax,t x∈X x∈X dT̃t = ∑ dRt ). Equation (A5) Equation (A4) follows from the fact that dnx,t = nx,t ⋅ εℓ ⋅ (d ln wx,t − 1−R t A9 follows from the proof for Lemma 1 and the envelope condition for optimal effort. We now use Lemma A1 to prove Proposition 5. Proof of Proposition 5. The proof follows Costinot and Werning (2022). Consider a variation that changes kx′ ,t by dkx′ ,t but keeps the utilization of all other types of capital and the replacement rate of assistance programs unchanged. At a social optimum, this variation cannot affect welfare. Thus: ∑ ℓx,t ⋅ χx,t ⋅ nx,t ⋅ (1 − Rt ) ⋅ dwx,t + χ̄t ⋅ dT̃t = 0. x∈X For this specific variation, we have dT̃t = τx′ ,t ⋅ dkx′ ,t − (1 − Rt ) ⋅ ( ∑ ℓx,t ⋅ nx,t ⋅ dwx,t ) + εℓ ⋅ Rt ⋅ ∑ ℓx,t ⋅ nx,t ⋅ dwx,t , Ax′ ,t x∈X x∈X which implies the necessary condition for an optimum ∑ ℓx,t ⋅ nx,t ⋅ x∈X + τx′ ,t ⋅ χx,t ⋅ (1 − Rt ) ⋅ dwx,t χ̄t dkx′ ,t − (1 − Rt ) ⋅ ( ∑ ℓx,t ⋅ nx,t ⋅ dwx,t ) + εℓ ⋅ Rt ⋅ ∑ ℓx,t ⋅ nx,t ⋅ dwx,t = 0. Ax′ ,t x∈X x∈X Regrouping terms, we can write this as τx′ ,t ⋅ χx,t dkx′ ,t = ∑ ℓx,t ⋅ nx,t ⋅ ( − 1) ⋅ (1 − Rt ) ⋅ (−dwx,t ) − εℓ ⋅ Rt ⋅ ∑ ℓx,t ⋅ nx,t ⋅ dwx,t Ax′ ,t x∈X χ̄t x∈X Dividing by dkx′ ,t and using the fact that mx,t = kx,t /Ax,t yields (6). We now turn to the formula for optimal replacement rates. Consider a variation that changes dRt but keeps technology utilization constant at all times and islands. The change in revenue from this variation is dT̃t = − (1 − Rt ) ⋅ ( ∑ ℓx,t ⋅ nx,t ⋅ dwx,t ) + mℓ,t ⋅ dRt + εℓ ⋅ Rt ⋅ ( ∑ ℓx,t ⋅ nx,t ⋅ dwx,t − mℓ,t ⋅ x∈X x∈X dRt ) 1 − Rt The welfare change from this variation is given by equation (A5). At a social optimum, this variation cannot affect welfare, which yields the necessary condition for an optimum A10 we obtain the necessary condition for an optimum ∑ ℓx,t ⋅ nx,t ⋅ x∈X χx,t ⋅ [(1 − Rt ) ⋅ dwx,t − wx,t ⋅ dRt ] χ̄t − (1 − Rt ) ⋅ ( ∑ ℓx,t ⋅ nx,t ⋅ dwx,t ) + mℓ,t ⋅ dRt + εℓ ⋅ Rt ⋅ ( ∑ ℓx,t ⋅ nx,t ⋅ dwx,t − mℓ,t ⋅ x∈X x∈X dRt )=0 1 − Rt Regrouping terms, we can write this as εℓ ⋅ Rt ⋅ (mℓ,t ⋅ χx,t dRt − ∑ ℓx,t ⋅ nx,t ⋅ dwx,t ) = ∑ ℓx,t ⋅ nx,t ⋅ ( − 1) ⋅ [(1 − Rt ) ⋅ dwx,t − wx,t ⋅ dRt ] , 1 − Rt x∈X χ̄t x∈X which dividing through by mℓ,t ⋅ dRt yields εℓ ⋅ Rt ⋅ ( ℓx,t ⋅ nx,t ⋅ wx,t ∂ ln wx,t ℓx,t ⋅ nx,t ⋅ wx,t χx,t ∂ ln wx,t 1 −∑ )= ∑ ⋅( − 1) ⋅ [(1 − Rt ) ⋅ − 1] , 1 − Rt x∈X mℓ,t ∂Rt mℓ,t χ̄t ∂Rt x∈X which can be rearranged as in the formula in the Proposition. When implementing the formula in equation (7), we compute the general equilibrium derivatives ∂ ln wt ∂Rt and ∂ ln wx,t ∂Rt as (1/wx,t )⋅dwx,t /dRt and (1/wt )⋅dwt /dRt , where these objects are given by the solution to the system of equations dwx,t =εℓ ⋅ ∑ fyx ,yx′ ⋅ ℓx′ ,t ⋅ nx′ ,t ( x′ ∈D dwt =εℓ ⋅ ∑ fy,yx′ ⋅ ℓx′ ,t ⋅ nx′ ,t ( x′ ∈D A.3 dwx′ ,t dRt dwt dRt − ) + εℓ ⋅ fyx ,y ⋅ ℓt ⋅ nt ⋅ ( − ) wx′ ,t 1 − Rt wt 1 − Rt dwx′ ,t dRt dwt dRt − ) + εℓ ⋅ fy,y ⋅ ℓt ⋅ nt ⋅ ( − ) wx′ ,t 1 − Rt wt 1 − Rt Proofs for Section 2.5 Proof of Proposition 6. Suppose the government sets no taxes nor assistance programs. Consider a new path for technology that changes wages by dwx,t and dwt . The resulting change in welfare is ∞ dW0 = ∫ 0 ( ∑ ℓx,t ⋅ χx,t ⋅ dwx,t + ℓt ⋅ χt ⋅ dwt ) ⋅ dt. x∈D Note that changes in reallocation effort are second order. This is because in this case we are varying technology and letting kx,t adjust, which implies that wages are entirely pinned down by technology and independent of reallocation rates. A11 Differentiating the ideal-price index condition cf ({wx,t }x∈D , wt ) = 1, we obtain st ⋅ d ln wt + ∑ sx,t ⋅ d ln wx,t = 0. x∈D Substituting in the formula for the change in welfare and rearranging terms yields ∞ dW0 = ∫ 0 ( ∑ (ℓx,t ⋅ χx,t ⋅ wx,t − ℓt ⋅ χt ⋅ wt ⋅ x∈D Finally, from wx,t = 1/Ax,(1−Γ)⋅t , we get d ln wx,t = Ȧx,t Ax,t sx,t ) ⋅ d ln wx,t ) ⋅ dt. st ⋅ t ⋅ dΓ, which gives the formula in equation (8). To prove the second part of the Proposition, we show that the government can always change taxes in response to improvements in technology in a way that leaves the economy and welfare unchanged and generates additional revenue. Suppose that τx,t is the optimal path for taxes when the productivity of new technology is Ax,t . Suppose that productivity shifts up to Ãx,t ≥ Ax,t . Taxes can then be adjusted up to a level τ̃x,t defined as Ãx,t Ax,t = , 1 + τ̃x,t 1 + τx,t and which satisfies τ̃x,t ≥ τx,t while at the same time keeping transfers Tt unchanged. This choice of taxes and transfers ensures that prices, utilities, reallocation rates, and the equilibrium remain unchanged. To conclude the proof we need to show that this policy is feasible, or Tt ≤ ∑ τ̃x,t ⋅ x∈D kx,t Ãx,t . This follows from the fact that Tt = ∑ τx,t ⋅ x∈D τx,t 1 + τ˜x,t kx,t kx,t kx,t =∑ ⋅ ⋅ τ˜x,t ⋅ ≤ ∑ τ˜x,t ⋅ , Ax,t x∈D 1 + τx,t τ˜x,t Ãx,t x∈D Ãx,t as wanted. A.4 Theoretical Extensions This section provides theoretical extensions of our baseline model. A12 A.4.1 Inequality Between and Within Islands This section extends our results to an economy with ex-ante differences in labor productivity within and between islands. We also discuss a rationale for ignoring pre-existing income differences across islands when deciding how to compensate the losers of globalization and technological progress. We consider a model where workers differ in absolute advantage. Households are endowed with ξ > 0 units of labor. We refer to ξ as the type of the household. The distribution of ξ in island x has cdf Φx (ξ), and the distribution of ξ in non-disrupted islands has cdf Φ(ξ). This definition implies ∫ξ ξ ⋅ dΦx (ξ) = ℓx,0 and ∫ξ ξ ⋅ dΦ(ξ) = ℓ0 . We make the following assumptions: ˆ The utility function is of the form u(c) = 1 1−γ 1−γ c for some γ > 0. ˆ A household of type ξ receives labor income ξ ⋅ wx,t in island x and a proportional lump-sum rebate of ξ ⋅Tt . It also faces a cost of reallocation ξ 1−γ ⋅κx (α) and is endowed with initial assets ξ ⋅ a0,x . ˆ The budget restrictions faced by households imply that a consumption plan ξ ⋅ c is feasible for a household of type ξ from island x if and only if ξ ′ ⋅ c is feasible for a household of type ξ ′ from island x. These assumptions imply that all households in island x choose paths for consumption and savings that are proportional to each other. In particular, let {cx,tn ,t , cx,t , αx } denote the optimal consumption plan for a household with ξ = 1 from island x, and let {cξx,tn ,t , cξx,t , αxξ } denote the optimal consumption plan for a household of type ξ from island x. The assumptions imply cξx,tn ,t = ξ ⋅ cx,tn ,t , cξx,t = ξ ⋅ cx,t , and αxξ = αx . In addition, households’ utility ξ is Ux,0 = Ux,0 ⋅ ξ 1−γ , and their marginal utilities of consumption are λξx,d,t = λx,d,t ⋅ ξ −γ and λξx,n,t = λx,n,t ⋅ ξ −γ . The same applies to households from non-disrupted islands. Consider a welfare function of the form 1 1 ξ W0 = ∑ ∫ W (((1 − γ) ⋅ Ux,0 ) 1−γ ; ξ) ⋅ dΦx (ξ) + ∫ W (((1 − γ) ⋅ U0ξ ) 1−γ ; ξ) ⋅ dΦ(ξ). x∈D ξ ξ This welfare function accounts for heterogeneity in ξ. We do not require the welfare function to be symmetric, and in particular, we let W depend on ξ to capture societal preferences for redistribution across households with different types. We also wrote the welfare function A13 1 1 ξ in terms of consumption equivalent terms ((1 − γ) ⋅ Ux,0 ) 1−γ and ((1 − γ) ⋅ U0ξ ) 1−γ , but this is done for tractability only. Proposition A1 The results in Propositions 3, 4 and 5 apply to this general economy with Pareto weights re-defined as ξ ⋅ dΦx (ξ) ℓx,0 ξ ⋅ dΦ(ξ) ⋅ . ℓ0 gx = ∫ W ′ (ξ ⋅ ((1 − γ) ⋅ Ux,0 ) 1−γ ; ξ) ⋅ ((1 − γ) ⋅ Ux,0 ) 1−γ ⋅ γ 1 ξ g = ∫ W ′ (ξ ⋅ ((1 − γ) ⋅ U0 ) 1−γ ; ξ) ⋅ ((1 − γ) ⋅ U0 ) 1−γ γ 1 ξ ξ Proof:. Using the fact that Ux,0 = Ux,0 ⋅ ξ 1−γ and U0ξ = U0 ⋅ ξ 1−γ , we can rewrite the welfare function as 1 1 W0 = ∑ ∫ W (ξ ⋅ ((1 − γ) ⋅ Ux,0 ) 1−γ ; ξ) ⋅ dΦx (ξ) + ∫ W (ξ ⋅ ((1 − γ) ⋅ U0 ) 1−γ ; ξ) ⋅ dΦ(ξ). x∈D ξ ξ Changes in welfare are then given by dW0 = ℓx,0 ⋅ gx ⋅ dUx,0 + ℓ0 ⋅ g ⋅ dU0 , which coincides with the change in welfare in the main text. As explained in the main text (see footnote 10) all the results in the paper hold for arbitrary Pareto weights, gx , g, and so in particular, they also hold after redefining gx and g. The proposition illustrates how ex-ante inequality affects optimal policy. Suppose that W ′ (c; n) = c−η for η ≥ γ, so that the welfare function is scale free, symmetric, and concave in individual utilities. Then: −η γ 1 ξ ⋅ dΦx (ξ) ) ⋅ (((1 − γ) ⋅ Ux,0 ) 1−γ ) ⋅ ((1 − γ) ⋅ Ux,0 ) 1−γ ℓx,0 ξ −η γ 1 ξ ⋅ dΦ(ξ) g = (∫ ξ −η ⋅ ) ⋅ (((1 − γ) ⋅ U0 ) 1−γ ) ⋅ ((1 − γ) ⋅ U0 ) 1−γ . ℓ0 ξ gx = (∫ ξ −η ⋅ x (ξ) Ex-ante inequality across households only matters via the terms ∫ξ ξ −η ⋅ ξ⋅dΦ and ∫ξ ξ −η ⋅ ℓx,0 ξ⋅dΦ(ξ) ℓ0 . These terms are larger for islands x with households that have fewer units of labor on average, introducing a motive for taxing kx,t more aggressively due to its tagging value. On the other hand, within island inequality does not affect optimal taxes conditional on these tagging terms. Optimal taxes are also zero in the long run, since distorting kx,t loses its tagging value as people reallocate away from island x. A14 The proposition also identifies conditions under which ex-ante inequalities do not interact with the problem of protecting losers. Suppose that W satisfies W ′ (ξ ⋅ c; ξ) = W ′ (c). This captures a situation where the public considers it fair for households of type ξ to enjoy a higher consumption, proportional to their higher labor endowment. In this case gx =W ′ (((1 − γ) ⋅ Ux,0 ) 1−γ ) ⋅ ((1 − γ) ⋅ Ux,0 ) 1−γ γ 1 g =W ′ (((1 − γ) ⋅ U0 ) 1−γ ) ⋅ ((1 − γ) ⋅ U0 ) 1−γ 1 γ and inequality of labor endowments between and within islands is irrelevant for the problem of compensating winners and losers. This offers a rationale for ignoring ex-ante inequalities across (and within) islands when selecting optimal taxes on technologies or trade motivated exclusively by compensating the losers. In particular, we can ignore ex-ante inequalities across islands if they are considered fair. A.4.2 Retraining Subsidies and Active Labor Market Policies As our second extension, we consider the problem of taxing technologies when the government has access to other tools that allow it to select the socially optimal level of reallocation. These tools might include retraining subsidies or active labor market policies. Proposition A2 Suppose the planner has other policy tools that implement the optimal social level of reallocation. A necessary condition for an optimal tax sequence is that: (A6) τx′ ,t = ∑ x∈X ℓx,t ⋅ wx,t χx,t ∂ ln wx,t ⋅( − 1) ⋅ (− ), mx′ ,t χ̄t ∂ ln kx′ ,t where the multipliers on the right-hand side are now evaluated along an equilibrium with the socially optimal level of αx . Proof:. Optimal reallocation effort maximizes social welfare. The envelope theorem implies that the effect of any reform on welfare is equal to the direct effect holding αx constant, which leads to the same optimality condition as in Proposition 3. A15 A.5 A.5.1 Calibration and Details of Numerical Algorithms Calibration Details, China Shock Calibrating π: Industry prices are initially given by Pi = 1. Following the disruption, we get a price index Pi,tf = ci (Wt , exp(−π)), for some cost function ci with ci (1, 1) = 1. Assuming that π is small, we can log-linearize this equation around (1, 1) as ln Pi,tf ≈ share domestic productioni,tf ⋅ ln Wt − share Chinese productioni,tf ⋅ ln π. This implies ln Pi,tf =≈ share domestic productioni,tf ⋅ (ln Wt + ln π) − ln π. Let s = max{share Chinese productioni,tf } and suppose that s is small, as is the case in the data. Then ln Pi,tf =≈ ln share domestic productioni,tf ⋅ ln π − ln π. This shows that the regression in Bai and Stumpner (2019) across industries identifies ln π. Pre-existing trade: In the applications of our framework to the China Shock and Colombia’s trade liberalization, we have to deal with the fact that there was some pre-existing trade. For the China Shock, we handle pre-existing trade by assuming that there is a mass νp(i) of islands associated with industry i that were already replaced by Chinese imports and hosted no workers by 1991. We normalize the cost of Chinese imports in these islands to 1, which implies that the cost function associated with (11) becomes 1 1−σ cfu ({wx }, w) = (νp + ν ⋅ w1−σ + ∑ νx ⋅ wx1−σ ) , x∈D where νp = ∑i νp(i) . The normalization w̄ = 1 in status quo then requires νp + ν + ∑x∈D νx = 1. In our calibration, we set νp = 2.5%—the share of imports in GDP before the China Shock. We assume that the China Shock is driven by advances in the productivity of Chi- A16 nese imports at other islands, and not by cost reductions of established products. These assumptions imply that the status-quo level of imports in industry i is mi,t0 = νp(i) ; yi,t0 while imports in industry i at time t after the China Shock are given by mi,t 1 cfx = νp(i) + ⋅ (ℓt ⋅ f − ℓx,t ) , yi,t yt ⋅ Ax,t cw where x is defined as the island associated with industry i (i.e., the one for which i(x) = i). In this expression, the first term accounts for imports at islands with pre-existing trade and the second term accounting for imports in new islands. The change in normalized import shares at time t is then equal to (A7) Change in normalized import sharei(x),t = 1 1 cfx ⋅ ⋅ (ℓt ⋅ f − ℓx,t ) for x ∈ D. Ωi yt ⋅ Ax,t cw Equation (A7) provides a system of equation across industries that we use to calibrate νi(x) and si(x) in a first step to match the change in normalized imports by 2007 (recall that Ax,tf = exp(π) at this point), and then to calibrate a path for Ax,t in a second step, as described in the main text. For Colombia’s trade liberalization, we assume that a mass νp(i) of segments were already produced internationally and hosted no workers by 1989. In addition, we assume this segments were not protected by 1989, and experienced no subsequent decline in tariffs after the 1990 trade liberalization. Under these assumptions, we have that the status-quo level of imports in industry i is mi,t0 = νp(i) ; yi,t0 while imports in industry i at time t after the liberalization are mi,t 1 + τx,t cfx = νp(i) + ⋅ (ℓt ⋅ f − ℓx,t ) , yi,t yt ⋅ Ax,t cw where x is defined as the island associated with industry i (i.e., the one for which i(x) = i). A17 The change in normalized import shares at time t is then equal to (A8) Change in normalized import sharei(x),t = 1 1 + τx,t cfx ⋅ ⋅ (ℓt ⋅ f − ℓx,t ) for x ∈ D. Ωi yt ⋅ Ax,t cw Equation (A8) provides a system of equation across industries that we use to calibrate νi(x) and si(x) to match the increase in normalized import shares between 1989 and 2002. A.5.2 Implementing the Formulas in Equations (3) and (4): This subsection describes the numerical procedure used to compute optimal taxes. Exogenous reallocation effort. We compute optimal taxes with exogenous reallocation effort as follows: (0) 1. Start with τx,t = 0 (laissez-faire). (n) 2. Compute equilibrium objects for τx,t , identified with the superscript (n) below. 3. Use equation (3) to update optimal taxes as (n) (n) (n) ℓx,t ⋅ wx,t ⎛ χx,t ⎞ ⎛ ∂ ln wx,t ⎞ ℓt ⋅ wt(n) ⎛ χt(n) ⎞ ⎛ ∂ ln wt(n) ⎞ =∑ ⋅ (n) − 1 ⋅ − ⋅ (n) − 1 ⋅ − + , (n) (n) ⎝ χ̄t ⎠ ⎠ ⎠ ⎝ ∂ ln kx(n) ⎠ ⎝ ∂ ln kx(n) mx′ ,t mx′ ,t ⎝ χ̄t x∈D ′ ,t ′ ,t (n+1) τx′ ,t 4. Repeat steps 2–3 until convergence. Endogenous reallocation effort. We compute optimal taxes with exogenous reallocation effort as follows: (0) (0) 1. Start with τx,t = 0 (laissez-faire) and the observed rate of reallocation αx . (n) (0) 2. Compute equilibrium objects for τx,t and αx , identified with the superscript (n) below. 3. Use equation (3) to update optimal taxes as (n+1) τx′ ,t (n) (n) ℓx,t ⋅ wx,t ⎛ χx,t ⎞ ⎛ ∂ ln wx,t ⎞ ℓt ⋅ wt(n) ⎛ χtend,(n) ⎞ ⎛ ∂ ln wt(n) ⎞ =∑ ⋅ end,(n) − 1 ⋅ − + ⋅ end,(n) − 1 ⋅ − , (n) ⎠ ⎝ ∂ ln kx(n) ⎠ mx(n) ⎝ χ̄t ⎠ ⎝ ∂ ln kx(n) ⎠ mx′ ,t ⎝ χ̄t x∈D ′ ,t ′ ,t ′ ,t end,(n) A18 4. Update the reallocation rate using (n+1) αx (n) where ∆αx (n) (n) = αx + ∆αx , is given by ∞ (n) ∆αx = ∑ εx,x′′ ⋅ ∫ x′′ 0 (n) (n) (Ux′′ ,α,d,t ⋅ (∆(n) wx′′ ,t + ∆(n) Tt ) + Ux′′ ,α,n,t ⋅ (∆(N ) wt + ∆(n) Tt )) ⋅ dt, and ∆(n) wx′′ ,t , ∆(n) wt , ∆(n) Tt is the change in wages and tax revenue generated by the update in taxes from iteration n to n + 1. 5. Repeat steps 2–4 until convergence. This procedure only requires us to specify values for εx,x′′ and solve for the optimal tax and the equilibrium path without having to specify the κ function. This comes at the cost of assuming that the elasticities εx,x′′ remain roughly unchanged for the variations in taxes considered. It also ignores the effect of changes in household utility on the multipliers gx , which is second order due to the envelope theorem, but could be non-negligible for large changes in reallocation effort αx . As an alternative, we experimented with the following procedure, which requires parametrizing the κx function: (0) (0) 1. Start with τx,t = 0 (laissez-faire) and the observed rate of reallocation αx . (n) (0) 2. Compute equilibrium objects for τx,t and αx , identified with the superscript (n) below. (n) 3. Compute εx,x′′ by solving the system of equations in (A2). 4. Use equation (3) to update optimal taxes as (n+1) τx′ ,t (n) (n) ℓx,t ⋅ wx,t ⎛ χx,t ⎞ ⎛ ∂ ln wx,t ⎞ ℓt ⋅ wt(n) ⎛ χtend,(n) ⎞ ⎛ ∂ ln wt(n) ⎞ =∑ ⋅ − 1 ⋅ − + ⋅ − 1 ⋅ − , (n) end,(n) ⎠ ⎝ ∂ ln kx(n) ⎠ mx(n) ⎝ χ̄tend,(n) ⎠ ⎝ ∂ ln kx(n) ⎠ mx′ ,t ⎝ χ̄t x∈D ′ ,t ′ ,t ′ ,t end,(n) (n) using the values of εx,x′′ to compute the χ’s. 5. Update the reallocation rate using (n+1) κ′ (αx A19 (n) ) = Ux,α . 6. Repeat steps 2–5 until convergence. A.5.3 Details for the Savings Problem with no Risk Sharing: As explained in the text, households problem can be summarized by the following system of HJB equations ∂vx (a, t) ⋅ (ra + wx,t − c) + αx ⋅ (v(a, t) − vx (a, t)), c ∂a ∂v(a, t) ρv(a, t) − v̇(a, t) = max u(c) + ⋅ (ra + wt − c). c ∂a ρvx (a, t) − v̇x (a, t) = max u(c) + Here, vx (a, t) is the value function of households in disrupted islands at time t with assets a when they exert reallocation effort αx (kept as an implicit argument to simplify notation), and v(a, t) is the value function of households in non-disrupted islands with assets a. ∞ ∞ Let hx,t = ∫t e−(s−t)r ⋅ wx,s ds and ht = at + ∫t e−(s−t)r ⋅ ws ds. We can rewrite these HJB equations using z = a + h—effective wealth—as the state variable: ∂vx (z, t) ⋅ (rz − c) + αx ⋅ (v(z + ht − hx,t ) − vx (z, t)), c ∂z ∂v(z) ρv(z) = max u(c) + ⋅ (rz − c). c ∂z ρvx (z, t) − v̇x (z, t) = max u(c) + Note that the HJB equation for v(z) is now stationary, since interest rates are constant. For u(c) = c1−γ /(1 − γ), we can solve analytically for v(z) as −γ z 1−γ 1 . v(z) = [r − (r − ρ)] ⋅ γ 1−γ Moreover, policy functions in the non-disrupted island are given by 1 ct = [r − (r − ρ)] ⋅ zt , γ 1 żt = (r − ρ) ⋅ zt . γ This implies λx,n,t = −γ t 1 1 1 ⋅ ∫ e−ρt ⋅ αx ⋅ e−αx tn ⋅ ([r − (r − ρ)] ⋅ (zx,tn + htn − hx,tn ) ⋅ e γ (r−ρ)(t−tn ) ) ⋅ dtn , 1 − Px,t 0 γ where zx,t denotes the effective wealth held by households in disrupted islands at time t. A20 This expression uses the fact that 1 1 cx,tn ,t = [r − (r − ρ)] ⋅ (zx,tn + htn − hx,tn ) ⋅ e γ (r−ρ)(t−tn ) . γ To characterize zx,t we solve the HJB equation for vx (z, t) numerically using the finitedifferences method described in Achdou et al. (2021). This method characterizes the common path of consumption cx,t and assets zx,t for households in disrupted islands starting from zx,0 = ax,0 + hx,0 . From this method, we also obtain λx,d,t = e−ρt ⋅ c−γ x,t . Figure A1 plots typical path for consumption cx,t and assets zx,t starting from zx,0 = hx,0 = 1 in an economy where ht − hx,t is positive and rises from 0.3 to 0.5 over time. For this examples, we consider a baseline scenario with αx = 5%, r = ρ = 5%, and γ = 2 and report variants. Figure A1: Consumption and wealth path in disrupted islands. The figure reports examples of the optimal path for effective wealth and consumption in disrupted islands when households can borrow but face uncertainty regarding when they will reallocate. These paths are obtained numerically using the finite-differences method described in Achdou et al. (2021). A21