School of Economic Sciences - Washington State University

Proof. From Chaney, aggregate state i exports to country j are given by (5),( ) γX˜θjij = µL i Y j × (w i f ij ) −a .w i τ ijReplace the untreated fixed cost to export f ij with the effective fixed export cost η ij ,X ij = µL i Y j(θjw i τ ij) γ× (w i η ij ) −awhich is legitimate due to the assumptions. Substitute in for η ij using (2.2), η ij = f ij. PositiveMijbfirst derivative shows state exports are increasing in missions:dX ijdM ij= ab ˜X ij M ab−1ij> 0for a, b > 0 and M ij ≥ 1. Since there is no upper bound on the Pareto distribution, ˜Xij > 0; thereis always a mass of firms with productivity greater than the threshold ˆφ ij .Lemma 2. State profit as a function of trade missions,Π s (M i,1 , M i,2 , ..., M i,J ) = σ − 1σγJ∑j=1˜X ij × M abij ,is increasing in M ij .Proof. By definition,Π i =J∑j=1L i∫ ∞ˆ φijπ ij (φ)dH(φ).σ w i τ ijσ−1 φPlug in for ˆφ ij and π ij = p ij (φ)q ij (φ) − c ij (φ) using the equilibrium values of p ij (φ) =( ) −σ−1 ( )and, conditional on exporting at all, q ij (φ) = σ σ γγ wi τ 1−σ ijµφ σ−1 σ−1γYj. Given thedistribution is Pareto, there is an analytic solution,Π i =J∑j=1γ−(σ−1)σ − 1σγ µL iY j(θjw i τ ij) γ(w i η ij ) −( γσ−1 −1) .θ j27