3. Industry
3. Industry
本章导读 本章从厂商层面过渡到行业层面：厂商层面常假设边际成本递增、完全竞争；而行业层面在长期、规模报酬不变（CRS）下供给可水平。随后建立只含劳动与资本两要素的行业模型，写出四个均衡条件并全微分，叠加 CRS 条件，借欧拉定理与替代弹性 \(\sigma\) 推出四条比较静态关系：\(\Delta P=s_L\Delta W+s_K\Delta R\)、\(\Delta Y=s_L\Delta L+s_K\Delta K\)、\(\Delta L-\Delta K=-\sigma(\Delta W-\Delta R)\)、\(\Delta Y=\varepsilon^D\Delta P\)。
3. Industry
Overview This chapter moves from the firm level to the industry level: at the firm level we often assume increasing marginal cost and perfect competition, while at the industry level, in the long run under constant return to scale (CRS), supply can be horizontal. We then build a two-factor (labor and capital) industry model, write four equilibrium conditions and totally differentiate them, add the CRS condition, and use Euler's theorem and the elasticity of substitution \(\sigma\) to derive four comparative-statics relations: \(\Delta P=s_L\Delta W+s_K\Delta R\), \(\Delta Y=s_L\Delta L+s_K\Delta K\), \(\Delta L-\Delta K=-\sigma(\Delta W-\Delta R)\), \(\Delta Y=\varepsilon^D\Delta P\).
3.1 Difference between firm level and industry level
在厂商层面，可合理假设边际成本递增、完全竞争，即需求完全弹性、供给向上倾斜。然而在行业层面、长期中，若假设规模报酬不变（CRS），则供给可为水平（完全弹性），需求向下倾斜。
3.1 Difference between firm level and industry level
At the firm level, it is reasonable to assume increasing marginal cost and perfect competition, which means demand is perfectly elastic and supply is upward sloping. However, at the industry level in the long run, supply can be horizontal (perfectly elastic) if we assume constant return to scale (CRS), and demand is downward sloping.
注记 3.1（CRS）/ Remark 3.1 (Constant return to scale) CRS 是一个自然假设：若有一家新厂商把现有厂商的一切完全照搬，那么它的生产应当完全相同。于是唯一的问题是它能否做到完美复制。若我们相信生产要素是生产函数中唯一要紧的东西，则生产函数必为 CRS。但若存在某些影响生产过程却未被计入要素的未观测因素，那么复制所有可观测的东西并不包含这些未观测因素，从而生产函数就不是 CRS。一言以蔽之，生产函数是否 CRS，取决于生产过程中是否存在未观测因素。CRS is a natural assumption: if there is a new firm that copies everything exactly from existing firms, its production should be exactly the same. So the only question is whether it can do the perfect copy. If we believe production factors are all that matters in the production function, then the function is constant return to scale for sure. But if there are unobserved factors that also affect the production process but are not counted as a factor, then copying everything observable does not include the unobserved factors, so the function is not CRS. In a word, whether the production function is CRS depends on whether there are unobserved factors in the production process.
3.2 Two inputs model
考虑一个生产只需两种要素——劳动与资本——的经济。定义：
- \(W\)：劳动的要素价格；
- \(R\)：资本的要素价格；
- \(P\)：产出价格；
- \(L\)：生产用劳动；
- \(K\)：生产用资本；
- \(Y\)：总产出。
该经济的四个均衡条件为
3.2 Two inputs model
Consider an economy in which production needs only two factors: labor and capital. Define:
- \(W\): factor price of labor;
- \(R\): factor price of capital;
- \(P\): output price;
- \(L\): labor used in production;
- \(K\): capital used in production;
- \(Y\): total output.
The four equilibrium conditions for this economy are
$$ \begin{aligned} 1.&\quad P=C_Y(W,R,Y)\\ 2.&\quad L=C_W(W,R,Y)\\ 3.&\quad K=C_R(W,R,Y)\\ 4.&\quad Y=D(P) \end{aligned} $$
其中 \(D\) 是行业的需求函数。对这四个均衡方程全微分：
where \(D\) is the industry's demand function. Totally differentiating these four equilibrium equations gives
$$ \begin{aligned} 1.&\quad dP=C_{YW}(W,R,Y)\,dW+C_{YR}(W,R,Y)\,dR+C_{YY}(W,R,Y)\,dY\\ 2.&\quad dL=C_{WW}(W,R,Y)\,dW+C_{WR}(W,R,Y)\,dR+C_{WY}(W,R,Y)\,dY\\ 3.&\quad dK=C_{RW}(W,R,Y)\,dW+C_{RR}(W,R,Y)\,dR+C_{RY}(W,R,Y)\,dY\\ 4.&\quad dY=\frac{dD(P)}{dP}\,dP \end{aligned} $$
注意这些微分方程是不需额外假设即成立的均衡条件。若进一步假设 CRS，则
Note these differential equations are equilibrium conditions that hold without any further assumptions. Then, if we further assume CRS, we have
$$C_{YW}(W,R,Y)=\frac{L}{Y},\qquad C_{YR}(W,R,Y)=\frac{K}{Y},\qquad C_{YY}(W,R,Y)=0\tag{3.1}$$
注记 3.2 / Remark 3.2 由 CRS 的定义 \(F(\alpha L,\alpha K)=\alpha F(L,K)\)。设 \(K=K^\star\)、\(L=L^\star\) 解 \(F(L,K)\) 的成本最小化问题，则 \(K=\alpha K^\star\)、\(L=\alpha L^\star\) 必解 \(F(\alpha L,\alpha K)\) 的成本最小化问题，这意味着最优产出与最优要素之比保持不变，即 \(\tfrac{K^\star}{Y^\star}\) 与 \(\tfrac{L^\star}{Y^\star}\) 不随产出 \(Y\) 变化。此外，CRS 也意味着边际成本不变。By the definition of CRS, \(F(\alpha L,\alpha K)=\alpha F(L,K)\). Suppose \(K=K^\star\) and \(L=L^\star\) solve the cost minimization problem for \(F(L,K)\); then \(K=\alpha K^\star\) and \(L=\alpha L^\star\) must solve the cost minimization problem for \(F(\alpha L,\alpha K)\), which implies the ratio between optimal output and optimal factor remains constant, i.e. \(\tfrac{K^\star}{Y^\star}\) and \(\tfrac{L^\star}{Y^\star}\) don't change with output \(Y\). Moreover, CRS also implies constant marginal cost.
由上面的注记，\(C_{YW}(W,R,Y)=C_{WY}(W,R,Y)=\tfrac{\partial L^\star}{\partial Y}=\tfrac{L}{Y}\)，对 \(C_{YR}(W,R,Y)\) 同理。现把 CRS 条件 (3.1) 叠加到这组均衡方程上。
第一个方程（用 CRS 条件）：
From the remark above, \(C_{YW}(W,R,Y)=C_{WY}(W,R,Y)=\tfrac{\partial L^\star}{\partial Y}=\tfrac{L}{Y}\), and the same argument holds for \(C_{YR}(W,R,Y)\). Now add the CRS condition (3.1) to the set of equilibrium equations.
First equation (use CRS conditions):
$$ \begin{aligned} dP&=C_{YW}(W,R,Y)\,dW+C_{YR}(W,R,Y)\,dR+C_{YY}(W,R,Y)\,dY\\ &=\frac{L}{Y}dW+\frac{K}{Y}dR\\ \Rightarrow\ \frac{dP}{P}&=\frac{WL}{PY}\frac{dW}{W}+\frac{RK}{PY}\frac{dR}{R}\\ \Rightarrow\ \Delta P&=s_L\Delta W+s_K\Delta R \end{aligned} $$
其中 \(s_L=\tfrac{WL}{PY}\) 为劳动的收入份额，\(s_K=\tfrac{RK}{PY}\) 为资本的收入份额。
第二、三个方程合并（用 CRS 条件）：最终得到 \(s_L\Delta L+s_K\Delta K=(s_L+s_K)\Delta Y\)。详细合并见下方推导。
where \(s_L=\tfrac{WL}{PY}\) is labor's share of revenue and \(s_K=\tfrac{RK}{PY}\) is capital's.
Second and third equations combined (use CRS conditions): the end result is \(s_L\Delta L+s_K\Delta K=(s_L+s_K)\Delta Y\). The detailed combination is below.
推导 / Derivation：second + third equation → \(\Delta Y=s_L\Delta L+s_K\Delta K\)
先把第二、第三个方程各自化为弹性形式（用 \(C_{WY}=\tfrac{L}{Y}\)、\(C_{RY}=\tfrac{K}{Y}\)、\(L=C_W\)、\(K=C_R\)）：
First put the second and third equations into elasticity form (using \(C_{WY}=\tfrac{L}{Y}\), \(C_{RY}=\tfrac{K}{Y}\), \(L=C_W\), \(K=C_R\)):
$$ \begin{aligned} dL&=C_{WW}(W,R,Y)\,dW+C_{WR}(W,R,Y)\,dR+C_{WY}(W,R,Y)\,dY\\ \Rightarrow\ \frac{dL}{L}&=\frac{WC_{WW}(W,R,Y)}{L=C_W}\frac{dW}{W}+\frac{RC_{WR}(W,R,Y)}{L=C_W}\frac{dR}{R}+\frac{L}{YL}dY\\ \Rightarrow\ \Delta L&=\frac{WC_{WW}(W,R,Y)}{C_W}\Delta W+\frac{RC_{WR}(W,R,Y)}{C_W}\Delta R+\Delta Y \end{aligned} $$
$$ \begin{aligned} dK&=C_{RW}(W,R,Y)\,dW+C_{RR}(W,R,Y)\,dR+C_{RY}(W,R,Y)\,dY\\ \Rightarrow\ \frac{dK}{K}&=\frac{WC_{RW}(W,R,Y)}{K=C_R}\frac{dW}{W}+\frac{RC_{RR}(W,R,Y)}{K=C_R}\frac{dR}{R}+\frac{K}{YK}dY\\ \Rightarrow\ \Delta K&=\frac{WC_{RW}(W,R,Y)}{C_R}\Delta W+\frac{RC_{RR}(W,R,Y)}{C_R}\Delta R+\Delta Y \end{aligned} $$
再作 \(\tfrac{WL}{PY}\times\)第二个方程 \(+\tfrac{RK}{PY}\times\)第三个方程：
Then take \(\tfrac{WL}{PY}\times\) second equation \(+\tfrac{RK}{PY}\times\) third equation:
$$ \begin{aligned} \frac{WL}{PY}\Delta L+\frac{RK}{PY}\Delta K&=\left(\frac{W^2LC_{WW}(W,R,Y)}{PYL}+\frac{WRKC_{RW}(W,R,Y)}{PYK}\right)\Delta W\\ &\quad+\left(\frac{WLRC_{WR}(W,R,Y)}{PYL}+\frac{R^2KC_{RR}(W,R,Y)}{PYK}\right)\Delta R+(s_L+s_K)\Delta Y\\ \Rightarrow\ s_L\Delta L+s_K\Delta K&=\left(\frac{WC_{WW}(W,R,Y)+RC_{WR}(W,R,Y)}{PY}\right)W\Delta W\\ &\quad+\left(\frac{WC_{RW}(W,R,Y)+RC_{RR}(W,R,Y)}{PY}\right)R\Delta R+(s_L+s_K)\Delta Y\\ \Rightarrow\ s_L\Delta L+s_K\Delta K&=(s_L+s_K)\Delta Y \end{aligned} $$
最后一个等号成立（脚注 3.1）是因为 \(L=C_W(W,R,Y)\) 与 \(K=C_R(W,R,Y)\) 关于 \(W\) 与 \(R\) 是零次齐次的，于是由欧拉定理 \(WC_{WW}+RC_{WR}=0\)、\(WC_{RW}+RC_{RR}=0\)（脚注 3.1：对 \(C_W(tW,tR,Y)\) 与 \(C_K(tW,tR,Y)\) 关于 \(t\) 在 \(t=1\) 处求导，应都为零）。于是方程变为
The last equality holds (footnote 3.1) because \(L=C_W(W,R,Y)\) and \(K=C_R(W,R,Y)\) are homogeneous of degree 0 in \(W\) and \(R\), so by Euler's theorem \(WC_{WW}+RC_{WR}=0\) and \(WC_{RW}+RC_{RR}=0\) (footnote 3.1: simply take the derivative of \(C_W(tW,tR,Y)\) and \(C_K(tW,tR,Y)\) w.r.t. \(t\) evaluated at \(t=1\), which should both be zero). Hence the equation becomes
$$s_L\Delta L+s_K\Delta K=(s_L+s_K)\Delta Y$$
由 CRS 可推得 \(s_L+s_K=1\)，故 \(\Delta Y=s_L\Delta L+s_K\Delta K\)。\(\blacksquare\)
Since \(s_L+s_K=1\) is implied by CRS, \(\Delta Y=s_L\Delta L+s_K\Delta K\). \(\blacksquare\)
注记 3.3 / Remark 3.3 注意 \(s_L\) 与 \(s_K\) 定义为收入份额，一般并不必然加总为一。但如前所述，CRS 意味着生产过程中没有未观测因素，从而总要素成本等于总收入——这正是欧拉定理的本质。因此 \(s_L\)、\(s_K\) 也可解释为成本份额，它们必然加总为一。Note \(s_L\) and \(s_K\) are defined as the share of revenue, not necessarily adding up to one in general. But as discussed, CRS implies there are no factors unobserved in the production process, which means total factor cost equals total revenue — exactly the essence of Euler's theorem. Therefore \(s_L\) and \(s_K\) can also be explained as the share of cost, which definitely add up to one.
引入一个新方程（不用 CRS 条件）：
Introducing a new equation (does not use CRS conditions):
$$\Delta L-\Delta K=-\sigma(\Delta W-\Delta R)$$
其中 \(\sigma\) 为替代弹性，定义为
where \(\sigma\) is the elasticity of substitution, defined as
$$-\sigma=\frac{d\log\left(\frac{L}{K}\right)}{d\log\left(\frac{W}{R}\right)}=\frac{d\log(L)-d\log(K)}{d\log(W)-d\log(R)}=\frac{\Delta L-\Delta K}{\Delta W-\Delta R}$$
最后一个方程（不用 CRS 条件）：
The last equation (does not use CRS conditions):
$$ \begin{aligned} dY&=\frac{dD(P)}{dP}dP\\ \frac{dY}{Y}&=\frac{\frac{dD(P)}{dP}P}{Y}\frac{dP}{P}\\ \Delta Y&=\varepsilon^D\Delta P \end{aligned} $$
小结（Summary）：
- \(\Delta P=s_L\Delta W+s_K\Delta R\)（用 CRS）；
- \(\Delta Y=s_L\Delta L+s_K\Delta K\)（用了 CRS，但也可不用 CRS 推得，只是会失去 \(s_L+s_K=1\) 这一条件）；
- \(\Delta L-\Delta K=-\sigma(\Delta W-\Delta R)\)（不用 CRS）；
- \(\Delta Y=\varepsilon^D\Delta P\)（不用 CRS）。
Summary:
- \(\Delta P=s_L\Delta W+s_K\Delta R\) (use CRS);
- \(\Delta Y=s_L\Delta L+s_K\Delta K\) (use CRS but can also be derived without CRS, though we lose the condition \(s_L+s_K=1\));
- \(\Delta L-\Delta K=-\sigma(\Delta W-\Delta R)\) (does not use CRS);
- \(\Delta Y=\varepsilon^D\Delta P\) (does not use CRS).