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Melitz, J.M. “The impact of trade on intra-industry reallocations and aggregate industry productivity,”Econometrica 71(6), 1695-1725 (2003)
・Dixit-Stiglitzモデル・企業間の限界生産性の差異・氷解型輸送費
地域経済モデルに適用 → 原理的にはMori-Turriniと同様の立地行動を再現可能
技術格差 → 市場範囲差
2
国際貿易：n ＋1 国, 各2国間輸送費 ＝ 同一
効用関数：
U =[∫
ω∈Ωq(ω)
σ−1σ dω
] σσ−1
P =[∫
ω∈Ωp(ω)1−σdω
] 11−σ
価格指数：
3
各バラエティ需要：
q(ω) = Q
[p(ω)P
]−σ
≡ U
各バラエティ需要：
r(ω) ≡ p(ω)q(ω) = R
[p(ω)P
]1−σ
R ≡ PQ =∫
ω∈Ωr(ω)dω地域/国総支出：
4
生産技術：
l = f + q/ϕ
企業特有の生産性(ランダムに決定)
労働投入量：
利潤最大化価格：
p(ϕ) =σ
ϕ(σ − 1)w =
σ
ϕ(σ − 1)(∵ w ≡ 1)
r(ϕ) = R [Pϕ(1− 1/σ)]σ−1
5
p(ϕ)p(ϕ′)
=ϕ′
ϕ
q(ϕ)q(ϕ′)
=(
ϕ
ϕ′
)σ
r(ϕ)r(ϕ′)
=(
ϕ
ϕ′
)σ−1
6
利潤：π(ϕ) = pq − w(f + q/ϕ)
= (p− 1/ϕ)q − f
=[p− 1− 1/σ
(1− 1/σ)ϕ
]q − f
=pq
σ− f
=r(ϕ)σ
− f
営業利潤
=R
σ[Pϕ(1− 1/σ)]σ−1 − f
r(ϕ) = R [Pϕ(1− 1/σ)]σ−1
7
“平均”技術水準：
ϕ ≡[∫ ∞
0ϕσ−1µ(ϕ)dϕ
] 1σ−1
(実現した)技術水準分布 : µ(ϕ)
※ 企業数(バラエティ規模)に独立
0 ϕ
8
集計変数：
P =[∫ ∞
0p(ϕ)1−σMµ(ϕ)dϕ
] 11−σ
= M1
1−σ p(ϕ)
R ≡ PQ =∫ ∞
0r(ϕ)Mµ(ϕ)dϕ = Mr(ϕ)
Q = Mσ
σ−1 q(ϕ)
Π ≡∫ ∞
0π(ϕ)Mµ(ϕ)dϕ = Mπ(ϕ)
r ≡ R
M= r(ϕ)
π ≡ ΠM
= π(ϕ)
平均収益：
平均利潤：
9
企業の参入／退出：
1）(埋没)参入費用： fe > 0
2) 技術水準の決定 ← ランダム
0 ϕ
g(ϕ)
3) 退出 ← ランダム：確率 δ
：潜在的技術水準分布
10
v(ϕ) = max
0,∞∑
0
(1− δ)tπ(ϕ)
= max 0, φ(ϕ)/δ
企業価値関数：
∞∑
0
(1− δ)t = 1/δ
生産性閾値：
ϕ∗ ≡ infϕ : v(ϕ) > 0
i.e., 営業利潤 ＞ 0 ⇒ 操業
11
均衡生産性分布：
µ(ϕ) =
g(ϕ)1−G(ϕ∗)
, ϕ ≥ ϕ∗
0, otherwise
0 ϕ∗ ϕ
pin ≡ 1−G(ϕ∗)
12
平均利潤と操業可能技術水準：
r ≡ r(ϕ) =[ϕ(ϕ∗)
ϕ∗
]σ−1
r(ϕ∗)
π ≡ π(ϕ)
=[ϕ(ϕ∗)
ϕ∗
]σ−1 r(ϕ∗)σ
− f
=
[ϕ(ϕ∗)
ϕ∗
]σ−1
− 1
︸ ︷︷ ︸≡k(ϕ∗)
f ∵ π(ϕ∗) = 0⇔ r(ϕ∗)σ
= fπ
(π, ϕ∗)
ϕ(ϕ∗) =[
11−G(ϕ∗)
∫ ∞
ϕ∗ϕσ−1g(ϕ)g(ϕ)dϕ
] 1σ−1
← (p.5)
13
ve ≡ pinv − fe =1−G(ϕ∗)
δπ − fe
参入の純期待価値：
v =∞∑
t=0
(1− δ)tπ = π/δ
pin自由参入条件（期待利潤＝ゼロ)：
π =δfe
1−G(ϕ∗)ve = 0⇒
14
1704 MARC J. MELITZ
(Zero Cutoff Profit) (Free Entry)
FIGURE 1.-Determination of the equilibrium cutoff (p* and average profit I-T.
uniqueness of the equilibrium qp* and 7, which is graphically represented in Figure 1.15
In a stationary equilibrium, the aggregate variables must also remain con- stant over time. This requires a mass Me of new entrants in every period, such that the mass of successful entrants, pinMe, exactly replaces the mass 8M of incumbents who are hit with the bad shock and exit: pinMe = 8M. The equi- librium distribution of productivity ,u(p) is not affected by this simultaneous entry and exit since the successful entrants and failing incumbents have the same distribution of productivity levels. The labor used by these new entrants for investment purposes must, of course, be reflected in the accounting for aggregate labor L, and affects the aggregate labor available for production: L = L p + Le where L p and Le represent, respectively, the aggregate labor used for production and investment (by new entrants). Aggregate payments to pro- duction workers Lp must match the difference between aggregate revenue and profit: Lp = R - H (this is also the labor market clearing condition for produc- tion workers). The market clearing condition for investment workers requires Le = Mefe. Using the aggregate stability condition, pinMe = 8M, and the free entry condition, 7T = 8fe/[l - G(p*)], Le can be written:
Le Mefe =-fe=MiTH. pin
Thus, aggregate revenue R = Lp + HI = Lp + Le must also equal the total pay- ments to labor L and is therefore exogenously fixed by this index of country
15The ZCP curve need not be decreasing everywhere as represented in the graph. However, it will monotonically decrease from infinity to zero for Sp* E (0, +oo) as shown in the graph if g(Xp) belongs to one of several common families of distributions: lognormal, exponential, gamma, Weibul, or truncations on (0, +oo) of the normal, logistic, extreme value, or Laplace distributions. (A sufficient condition is that g(Gp)$p/[1 - G(p)] be increasing to infinity on (0, +oo).)
自由参入条件操業停止利潤
均衡最低技術水準と平均(営業)利潤
15
参入企業数：
pinMe
Me
操業開始企業数：
退出企業数：
定常均衡 =
Le = Mefe参入のための労働投入
生産のための労働投入Lp = R−Π
L
Le = Mefe =δM
pinfe = Mπ = Π
R = Lp + Π = Lp + Le = L
総労働者数：
δM自由参入条件 (p.13)
(w ≡ 1)
π = pq − w(f + q/ϕ)
16
M =R
r=
L
σ(π + f)
r
σ= π + f
定常均衡企業数：
R = L
(p.6)
(p.15)
P = M1
1−σ1
(1− 1/σ)ϕ
定常均衡生計費：
17
国際貿易
輸出費用／輸出先国(埋没費用)： fex > 0
国内調達価格： pd(ϕ) =1
(1− 1/σ)ϕ
国外調達価格： px(ϕ) =τ
(1− 1/σ)ϕ= τpd(ϕ)
国内収益：
国外収益：
rd(ϕ) = R[P (1− 1/σ)ϕ]σ−1 (p.4)
rx(ϕ) = τ1−σrd(ϕ)
18
r(ϕ) =
rd(ϕ)
rd(ϕ) + nrx(ϕ) = (1 + nτ1−σ)rd(ϕ)(輸出無し)
(輸出有り)
fex =∞∑
t=0
(1− δ)tfx = fx/δ
輸出埋没費用のフロー化：
∴ fx = δfex
πd(ϕ) =rd(ϕ)
σ− f
πx(ϕ) =rx(ϕ)
σ− fx : πx(ϕ) > 0⇒ 輸出
輸出利潤／国
19
ϕ∗x = infϕ : ϕ ≥ ϕ∗, πx(ϕ) > 0
輸出企業の最低技術水準：
国際貿易下の利潤：
π(ϕ) = πd(ϕ) + max0, nπx(ϕ)
πd(ϕ∗) = 0⇒ rd(ϕ∗)σ
= f
πx(ϕ∗x) = 0⇒ rx(ϕ∗x)σ
=τ1−σrd(ϕ∗x)
σ= fx
20
πx(ϕ∗) < 0⇒ τ1−σrd(ϕ∗)σ
< fx
⇒ τ1−σf < fx
⇒ τσ−1fx > f
定常均衡における輸出＋非輸出企業の共存：
0 < ϕ∗ < ϕ∗x
21
定常均衡生産性分布：
µ(ϕ) =g(ϕ)
1−G(ϕ∗), ∀ϕ ≥ ϕ∗
px =1−G(ϕ∗
x)1−G(ϕ∗)
輸出企業シェア：
各国における購入可能バラエティ数：
Mt = M + nMx
22
1) 生産性↑ → 市場範囲↑ 企業の生産性差異 → 労働者生産性差異に変更 → MTモデルと同様の立地行動2) 輸出固定費 → 非氷解型輸送費 (e.g., MTモデル) (対称 n 国の設定は一般的立地空間に直接適用できない)
地域格差／技術水準による住み分けモデルへの応用
