Technical Appendix. The Steady State

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1 Technical Appendix A The Steady State We denote constant, steady-state levels of variables by dropping the time subscript and assume = fe, f = f, τ = τ, L = L,andZ = Z =1. Under these assumption, the steady state of the model is symmetric: Q = Q = TOL =1and the levels of all other endogenous variables are equal across countries. Solving for Given the solution for the average export productivity, we can obtain the cutoff level z from = νz,whereν {/ [ (θ 1)]} 1/(). Wecansolvefor as follows. The Euler equation for share holdings yields: ṽ = µ β (1 δ) d D + N d. 1 β (1 δ) Combining this equation with the free entry condition ṽ = w implies: d D + N d = w. (A.1) The steady-state zero profit export cutoff equation is: d = wf θ 1 (θ 1). (A.2) Also, steady-state profits from selling at home and abroad are d D = ( ρ D ) 1 θ C/θ and d = ( ρ ) 1 θ C/θ wf, respectively. These two equations imply: µ ρ ³ d D = d + wf. (A.3) ρ D Optimal pricing yields ρ D =[θ/ (θ 1)] z D 1 w and ρ =[θ/ (θ 1)] τ z 1 w. Hence, ρ / ρ D = τ z D /, and substituting this into (A.3), we have: µ τ zd ³ d D = d + wf, TA-1

2 or, taing (A.2) into account, µ τ µ zd θ 1 d D = wf (θ 1) + wf. (A.4) The steady-state share of exporting firms in the total number of domestic firms is: N =(z min ) ( ) (θ 1). (A.5) Substituting equations (A.2), (A.4), and (A.5) into (A.1), using z D = {/ [ (θ 1)]} 1 zmin, and rearranging yields: ( ) 1 θ (τz min ) 2+( ) (z min ) (θ 1) This equation can be rewritten as: (θ 1) θ 1 (θ 1) = f. ξ 1 ( ) 1 θ + ξ 2 ( ) = ξ 3, (A.6) where 2 ξ 1 (τz min ) > 0, (θ 1) ξ 2 (z min ) θ 1 (θ 1) (θ 1) > 0, ξ 3 > 0. f The left-hand side of equation (A.6) is a hyperbola. This guarantees existence and uniqueness of > 0, the exact value of which we obtain numerically. Solving for ρ The law of motion for the total number of domestic firms implies: N E = δ 1 δ. (A.7) Steady-state aggregate accounting yields C = wl+ dd + N d N E w. Using (A.1) and TA-2

3 (A.7), this can be rewritten as: C w = L + N 1 β D. (A.8) Equation (A.2) and the expression for average export profits, d =( ρ ) 1 θ C/θ wf,imply: C w = ρ θ (θ 1) f. (A.9) The price index equation ρ 1 θ D + N ρ 1 θ ρ = µ ρ or, using ρ / ρ D = τ z D / and equation (A.5), ρ D =1yields: + N, ρ = µ τ zd + µ zmin. (θ 1) (A.10) Together, equations (A.8), (A.9), and (A.10) yield the following equation for ρ : ρ 1 θ = θ (θ 1) f K 1 1 β L 1. where K is the right-hand side of equation (A.10). SpecialCase:AllFirmsExport In this case, equation (A.9) no longer holds since the zero cutoff profit condition (A.2) no longer applies. Using d D =( ρ D ) 1 θ C/θ and d =( ρ ) 1 θ C/θ wf, equation (A.1) can be written as: ρ 1 θ C ³ τ +1 wf = θ w, which implies: C w = ρ θ τ +1 ½ f + ¾. (A.11) Equation (A.11) now replaces equation (A.9) when solving for ρ. This yields the following expression for ρ : ρ 1 θ = θ τ +1 ½ f + ¾ K 1 1 β L 1. TA-3

4 Solving for the Remaining Variables The solutions for other endogenous variables are straightforward = K 1 ρ ; ρ D = τ z D ρ using ρ / ρ D = τ z D / ; w = ρ θτ using ρ =[θ/ (θ 1)] τ z 1 w; h i 1 β C = w L + using (A.8); N E = δ 1 δ ; (1 δ)β N = (z min ) ( ) h d D = 1 θ ( ρ D) 1 θ C; d = 1 θ ( ρ ) 1 θ C wf ; () i ṽ = w (using the free entry condition); using (A.5); 1+r =1/β (using the Euler equation for bond holdings). Symmetry of the steady state ensures z =, ρ = ρ, ND =,NE = N E, N = N, d D = d D, d = d, ṽ =ṽ, in addition to C = C, w = w, andr = r. B Labor Maret Clearing Recall that a firm with productivity z produces Z t z units of output per unit of labor employed. Consider separately the labor used to produce goods for the domestic and export marets: let l D,t (z) and l,t (z) represent the amount of labor hired to produce goods for each maret. These only represent labor used in production; in addition, each new entrant hires,t /Z t units of labor to cover the entry cost, and each exporter hires f,t /Z t units of labor to cover the fixed export cost in every period. The profits earned from domestic sales for a firm with productivity z are then given by: d D,t (z) =ρ D,t (z)z t zl D,t (z) l D,t (z) = 1 θ 1 l D,t (z), using ρ D,t (z) = θ Z t z from optimal pricing. This relationship holds for a firm with average productivity z D, and also for averages across all domestic firms. This implies that the average TA-4

5 amount of production labor hired to cover domestic sales is (θ 1) d D,t /. The total amount of such labor hired at home is thus,t (θ 1) d D,t /. The profits earned from export sales for an exporting firm with productivity z are given by: d,t (z) =Q t ρ,t (z) Z tzl,t (z) τ t l,t (z)+ f,t = 1 Z t θ 1 w f,t tl,t (z), Z t using ρ,t (z) =Q 1 θ w t τ t t Z t z from optimal pricing. Note that only Z tzl,t (z)/τ t export units are sold, although Z t zl,t (z) are produced (the remaining fraction having melted away in an iceberg fashion while crossing the border). Again, this relationship holds for a firm with average export productivity,t, and also for averages across all exporters. The average amount of production labor hired to cover export sales is thus (θ 1) d,t / +(θ 1) f,t /Z t. Multiplying by N,t yields the total amount of such labor for the home economy. The total amount of production labor hired in the home economy is then θ 1,t d D,t (z)+ θ 1 N,t d,t + θ 1 Z t N,t f,t. Adding the total amount of labor hired by new entrants, N E,t,t /Z t, and that hired by exporters to cover the fixed costs, N,t f,t /Z t, yields the aggregate labor demand for the home economy: L t = θ 1,t dd,t + θ 1 N,t d,t + θ Z t N,t f,t + 1 Z t N E,t,t. Equating L t to labor supply (L) yields the equilibrium condition for home s labor maret. The derivation for foreign is analogous. Balanced Trade Implies Labor Maret Clearing We now demonstrate that balanced trade under financial autary implies labor maret clearing. ³ Using the home price index equation 1=,t ( ρ D,t ) 1 θ + N,t ρ,t 1 θ, the balanced trade ³ condition Q t N,t ( ρ,t ) 1 θ Ct = N,t ρ,t 1 θ Ct can be written: Q t N,t ( ρ,t ) 1 θ C t = h 1,t ( ρ D,t ) 1 θi C t. TA-5

6 This condition can be re-written as C t = θn,t µ f,t d,t + + θ,t dd,t, Z t since d D,t =( ρ D,t ) 1 θ C t /θ and d,t = Q t ( ρ,t ) 1 θ C t /θ f,t /Z t. Combining this with aggregate accounting (C t = L +,t dd,t + N,t d,t N E,t,t /Z t ) yields the labor maret clearing condition for the home economy: L = θ 1,t dd,t + θ 1 N,t d,t + θ Z t N,t f,t + 1 Z t N E,t,t. The proof for the foreign economy follows the same steps. TA-6