Y_{in} = Omega_{in}L_{in}^{(1-phi)}x_{in}^{phi}, (A3)
x_{in} = exp{int_0^1 gamma_{j}log x_{in,j}dj}
x_{in,j} = (Z_{inj}^{(epsilon)/(1+epsilon)} + sum_{k
eq n} M_{inkj}^{(epsilon)/(1+epsilon)})^{(1+epsilon)/(epsilon)}
For brevity, we drop the firm identifier i, industry identifier g, and the home country identifier n. Firms minimize total cost given by
W^{**}L + int_0^1 V_{j}^{**}Z_{j}dj + int_{J_{0,i}} (sum_k e_{k}U_{kj}M_{kj})dj
subject to (A3)-(A5). Let lambda, psi, zeta denote the Lagrangian multiplier of equations (A3)-(A5), respectively. The first-order conditions are respectively:
where J_{0,i} is the optimal number of input variety for firm i
W^{**} = (1-phi)lambda (Y)/(L),
psi = lambda phi (Y)/(x),
zeta = psi gamma_{j}(x)/(x_{j}), j in [0,1]
V_{j}^{**} = zeta ((x_{j})/(Z_{j}))^{(1)/(1+epsilon)}, j in [0,1]
(U_{kj})/(e_{k}) = zeta ((x_{j})/(M_{kj}))^{(1)/(1+epsilon)}, j in J_{0}, k
eq n
Give me step-by-step proof to get these results
(A3)
X_{in} = exp{Y_{j} log X_{in,j}dj}
(A4)
1
(A5)
kAn
For brevity, we drop the firm identifier i, industry identifier g, and the home country identifier n. Firms minimize total cost given by
W*L + VZdj - >e_{k}U_{kj}M_{kj}dj K J_{0,i}
(A20)
subject to (A3)-(A5). Let X_{i} denote the Lagrangian multiplier of equations (A3)-(A5), respectively. The first-order conditions are respectively: where J_{0,i} is the optimal number of input variety for firm i
W* = (1-)XY/L, psi = XqY/X, zeta = 3bYjX/Xj, V = S(Xj/Z)e Ukj/ek = S(Xj/Mkj)e
j in [0,1] j in [0,1] j in J_{0}, k
eq n
Give me step-by-step proof to get these results
