2. Supply Theory
2. 供给理论
本章导读 供给曲线与需求曲线共同决定均衡价格与数量。本章先用均衡位移分解说明供给弹性为何重要，再把厂商的生产函数与消费者的效用函数类比，建立利润最大化与成本最小化两个对偶问题，给出要素需求函数与成本函数；通过两个生产函数例子（Leontief 与完全替代）说明替代性；最后把要素需求的价格效应分解为替代效应与规模效应，并据此给出获得要素需求函数的三种方式及其弹性排序。
2. Supply Theory
Overview The supply curve and demand curve together pin down equilibrium price and quantity. This chapter first uses an equilibrium-shift decomposition to show why supply elasticity matters, then draws an analogy between a firm's production function and a consumer's utility function, sets up the dual problems of profit maximization and cost minimization to obtain the factor demand and cost functions; two production-function examples (Leontief and perfect substitutes) illustrate substitutability; finally, the price effect on factor demand is decomposed into a substitution effect and a scale effect, yielding three ways to obtain the factor demand function and their elasticity ordering.
2.1 Assumptions about the supply curve elasticity
需求曲线与供给曲线共同钉住均衡价格与数量。不同假设下供给曲线弹性不同，给出不同均衡。为看出供给弹性的重要性，分解均衡位移：
2.1 Assumptions about the supply curve elasticity
Demand and supply curves work together to pin down equilibrium price and quantity. Under different assumptions the supply elasticity differs, giving different equilibria. To see why supply elasticity matters, decompose the equilibrium shift:
$$ \begin{cases} \Delta Q&=\Delta D+\varepsilon^D\Delta P\\ \Delta Q&=\Delta S+\varepsilon^S\Delta P \end{cases}\tag{2.1} $$
其中 \(\Delta D\) 为需求曲线位移（每个价格处需求量的百分比变化），\(\Delta S\) 为供给曲线位移（每个价格处供给量的百分比变化）；\(\varepsilon^D\Delta P\)（沿需求曲线移动）与 \(\varepsilon^S\Delta P\)（沿供给曲线移动）分别度量价格变动引起的需求量与供给量的百分比变化。整理 (2.1)：
where \(\Delta D\) is the shift of demand curve (percentage change in demanded quantity at each price), \(\Delta S\) the shift of supply curve (percentage change in supplied quantity at each price); \(\varepsilon^D\Delta P\) (move along demand curve) and \(\varepsilon^S\Delta P\) (move along supply curve) measure the percentage change in quantity demanded and supplied due to price change. Rearranging (2.1):
$$ \begin{cases} \Delta P&=\dfrac{\Delta D-\Delta S}{\varepsilon^S-\varepsilon^D}\\[2mm] \Delta Q&=\dfrac{\varepsilon^S\Delta D-\varepsilon^D\Delta S}{\varepsilon^S-\varepsilon^D} \end{cases} $$
显然关于供给弹性的不同假设影响很大，而对供给弹性作不同假设的理由很多。例如大厂商可能难以调整其生产规模，因此往往比小生产者弹性更低。
2.2 Comparing production function with utility function
考虑生产函数
Clearly, different assumptions about supply elasticity matter a lot, and there are many reasons to make different assumptions. For example, a large firm may be hard to adjust its production scale, so it tends to be less elastic than small producers.
2.2 Comparing production function with utility function
Consider the production function
$$Y=F(L,K)$$
其中 \(F\)：输入 \(\to\) 输出，是厂商的技术，\(Y\) 为产出，\(L\) 为生产用劳动，\(K\) 为生产用资本。效用函数
where \(F\): input \(\to\) output is the firm's technology, \(Y\) is output, \(L\) labor used in production, \(K\) capital used in production. The utility function
$$U=U(X_1,X_2,\dots,X_N)$$
也可看作一种生产函数：投入是商品、产出是效用。但二者不同之处在于：产品可在市场上出售，而效用不能出售。所以我们通常假设厂商没有预算约束——擅长生产的厂商可以借钱、生产许多产品在市场出售，而家庭无法出售效用。
2.3 The profit maximization problem
设问题为
can also be regarded as a kind of production function with inputs of goods and output of utility. But they differ: products can be sold in the market, while utility cannot. So we usually assume no budget constraint on firms, because a firm good at producing can borrow money and produce many goods to sell, while a household cannot sell utility.
2.3 The profit maximization problem
Set up the problem as
$$\max_{X_1,X_2,\dots,X_N} P\cdot F(X_1,X_2,\dots,X_N)-\sum_{i=1}^N X_iW_i\tag{2.2}$$
其中 \(P\) 为产出价格，\(F\) 为把投入 \(X_1,\dots,X_N\) 转为产出的生产函数（技术），\(W_i\) 为 \(X_i\) 的价格。一阶条件
where \(P\) is the output price, \(F\) the production function (technology) transferring inputs \(X_1,\dots,X_N\) into output, and \(W_i\) the price of \(X_i\). The f.o.c. gives
$$[X_i]:\quad P\frac{\partial F}{\partial X_i}=W_i\quad\text{for }\forall i\tag{2.3}$$
左边 \(P\tfrac{\partial F}{\partial X_i}\) 可定义为 \(VMP_i\)，即 \(X_i\) 的边际产品价值（value of marginal product）；右边 \(W_i\) 为 \(X_i\) 的边际成本。
2.4 The cost minimization problem
等价地，可设成本最小化问题得到同样条件：
The LHS \(P\tfrac{\partial F}{\partial X_i}\) can be defined as \(VMP_i\), the value of marginal product of \(X_i\); the RHS \(W_i\) is the marginal cost of \(X_i\).
2.4 The cost minimization problem
Equivalently, set up the cost minimization problem to get the same condition:
$$\min_{X_1,X_2,\dots,X_N}\sum_{i=1}^N X_iW_i\quad\text{s.t.}\quad F(X_1,X_2,\dots,X_N)=\bar Y.$$
构造拉格朗日量 \(\mathcal{L}=\sum_{i=1}^N X_iW_i+\mu\left[\bar Y-F(X_1,\dots,X_N)\right]\)，一阶条件
Form the Lagrangian \(\mathcal{L}=\sum_{i=1}^N X_iW_i+\mu\left[\bar Y-F(X_1,\dots,X_N)\right]\); the f.o.c.
$$ \begin{aligned} [X_i]:&\quad W_i-\mu\frac{\partial F}{\partial X_i}=0\quad\text{for }\forall i,\\ [\mu]:&\quad F(X_1,X_2,\dots,X_N)=\bar Y, \end{aligned} $$
给出在生产计划 \(\bar Y\) 下每种要素的需求
give the demand for each factor good under the production plan \(\bar Y\):
$$X_i(W_1,W_2,\dots,W_N,\bar Y),\quad\text{for }\forall i\tag{2.4}$$
类似地，用要素需求函数 (2.4) 定义生产者的成本函数
Analogously, define the producer's cost function using the factor demand function (2.4):
$$C(W_1,W_2,\dots,W_N,\bar Y)\tag{2.5}$$
2.5 Two examples of production function
2.5 Two examples of production function
Example 2.1（Leontief / 无替代） \(F(X_1,X_2,\dots,X_N)=\min\left\{\tfrac{X_1}{a_1},\tfrac{X_2}{a_2},\dots,\tfrac{X_N}{a_N}\right\}\)。该技术的成本最小策略是让所有 \(\tfrac{X_i}{a_i}\) 相等。\(F(X_1,X_2,\dots,X_N)=\min\left\{\tfrac{X_1}{a_1},\tfrac{X_2}{a_2},\dots,\tfrac{X_N}{a_N}\right\}\). The cost-minimizing strategy is to equalize all \(\tfrac{X_i}{a_i}\).
$$ \begin{aligned} \frac{X_1}{a_1}&=\frac{X_2}{a_2}=\dots=\frac{X_N}{a_N}=\bar Y\\ \Rightarrow X_i&=a_i\bar Y\\ \Rightarrow C(W_1,W_2,\dots,W_N,\bar Y)&=\bar Y(a_1W_1+a_2W_2+\dots+a_NW_N) \end{aligned} $$
要素之间完全无替代，故成本函数对每个要素价格都是线性的。
There is no substitution among factors at all, so the cost function is linear in each factor price.
Example 2.2（完全替代） \(F(X_1,X_2,\dots,X_N)=b_1X_1+b_2X_2+\dots+b_NX_N\)。该技术的成本最小策略是找出最高效的要素作为唯一使用的要素。\(F(X_1,X_2,\dots,X_N)=b_1X_1+b_2X_2+\dots+b_NX_N\). The cost-minimizing strategy is to find the most efficient factor to be the only factor used.
$$C(W_1,W_2,\dots,W_N,\bar Y)=\min\left\{\frac{W_1}{b_1},\frac{W_2}{b_2},\dots,\frac{W_N}{b_N}\right\}$$
要素之间完全替代，故只使用最高效的那一个要素。
2.6 Scale effect and substitution effect in factor demand
规模效应来自产出 \(Y\) 的可调整性。下面讨论价格变动对要素需求的影响时，允许产出调整。假设只有要素 \(j\) 的价格变动：
There is perfect substitution among factors, so only the most efficient factor is used.
2.6 Scale effect and substitution effect in factor demand
The scale effect comes from the adjustability of output \(Y\). In the following discussion of the effect of price change on factor demand, we allow output to adjust. Assume only the price of factor \(j\) changes:
$$ \begin{aligned} \frac{dX_i(W_1,W_2,\dots,W_N,Y)}{dW_j}&=\frac{\partial X_i}{\partial W_j}+\frac{\partial X_i}{\partial Y}\frac{dY}{dW_j}\\ &=\frac{\partial^2C}{\partial W_i\partial W_j}+\frac{\partial^2C}{\partial W_i\partial Y}\frac{dY}{dW_j} \end{aligned}\tag{2.6} $$
最后一步成立是因为由包络定理 \(X_i=\tfrac{\partial C}{\partial W_i}\)。由于厂商成本最小化（利润最大化），\(P=\tfrac{\partial C}{\partial Y}\)，即边际成本等于边际收入。用此事实处理 (2.6) 第二项中的 \(\tfrac{dY}{dW_j}\)：
The last equality holds because by the envelope theorem \(X_i=\tfrac{\partial C}{\partial W_i}\). Since firms are cost-minimizing (profit-maximizing), \(P=\tfrac{\partial C}{\partial Y}\), i.e. marginal cost equals marginal revenue. Use this to handle \(\tfrac{dY}{dW_j}\) in the second term of (2.6):
$$ \begin{aligned} 0&=\frac{dP}{dW_j}\quad\because\text{ price is exogenous}\\ &=\frac{\partial^2C}{\partial Y\partial W_j}+\frac{\partial^2C}{\partial Y^2}\frac{dY}{dW_j}\\ \Rightarrow\ \frac{dY}{dW_j}&=-\frac{\partial^2C/\partial Y\partial W_j}{\partial^2C/\partial Y^2} \end{aligned}\tag{2.7} $$
把 (2.7) 代入 (2.6)：
Plug (2.7) into (2.6):
$$\frac{dX_i(W_1,W_2,\dots,W_N,Y)}{dW_j}=\underbrace{\frac{\partial^2C}{\partial W_i\partial W_j}}_{\text{substitution effect}}-\underbrace{\frac{\partial^2C}{\partial W_i\partial Y}\frac{\partial^2C/\partial Y\partial W_j}{\partial^2C/\partial Y^2}}_{\text{scale effect}}\tag{2.8}$$
在 \(i=j\) 时改写 (2.8)：
Rewrite (2.8) for the case \(i=j\):
$$\frac{dX_i(W_1,W_2,\dots,W_N,Y)}{dW_i}=\underbrace{\frac{\partial^2C}{\partial W_i^2}}_{\text{substitution effect}<0}-\underbrace{\frac{\left(\partial^2C/\partial Y\partial W_i\right)^2>0}{\partial^2C/\partial Y^2>0\ \because\text{ increasing marginal cost}}}_{\text{scale effect}>0}$$
显然规模效应总是强化替代效应。换言之，若允许产出水平调整，则某要素价格上升不仅通过使生产者转向其他要素、也通过缩减总产出，来降低对该要素的需求。
2.7 Three ways of obtaining factor demand function
1.（弹性较大） 固定产出，但允许所有要素数量变动（给定产出下成本最小化）：
Clearly the scale effect always reinforces the substitution effect. In words, if we allow output level to adjust, a price increase in one factor decreases demand for that factor not only by making the producer substitute toward other factors, but also by shrinking total output.
2.7 Three ways of obtaining factor demand function
1. (elastic) Fix output, but allow quantities of all factors to change (cost minimizing at a given output level):
$$X_i^{D_1}=\frac{\partial C(W_1,W_2,\dots,W_N,Y)}{\partial W_i}$$
\(X_i^{D_1}\) 是 \((W_1,W_2,\dots,W_N,Y)\) 的函数。
2.（弹性最小） 固定其他所有要素的数量，只允许一种要素数量变动（甚至不成本最小化）：
\(X_i^{D_1}\) is a function of \((W_1,W_2,\dots,W_N,Y)\).
2. (least elastic) Fix the quantities of all other factors, and only allow one factor quantity to change (even not cost minimizing):
$$X_i^{D_2}:\quad W_i=P\frac{\partial F(X_1,X_2,\dots,X_N)}{\partial X_i^{D_2}}$$
\(X_i^{D_2}\) 是 \((X_1,X_2,\dots,X_N,W_i)\) 的函数。
3.（弹性最大） 什么都不固定，允许产出与所有要素数量变动（利润最大化）：
\(X_i^{D_2}\) is a function of \((X_1,X_2,\dots,X_N,W_i)\).
3. (most elastic) Fix nothing, and allow output and quantities of all factors to change (profit maximizing):
$$\pi_{\text{profit}}=\max_Y PY-C(W_1,W_2,\dots,W_N,Y)$$
$$\Rightarrow X_i^{D_3}=-\frac{\partial\pi_{\text{profit}}}{\partial W_i}$$
\(X_i^{D_3}\) 是 \((W_1,W_2,\dots,W_N)\) 的函数。
如上所示，规模效应总强化替代效应，故 \(X_i^{D_3}\) 弹性最大（允许产出调整、引入规模效应）；\(X_i^{D_3}\) 也含替代效应（允许所有要素数量变动）。\(X_i^{D_1}\) 比 \(X_i^{D_3}\) 弹性小（只含替代效应），但比 \(X_i^{D_2}\) 弹性大（后者既无替代效应也无规模效应）。
图 2.1（已转述）：横轴要素数量 \(X_i\)，纵轴要素价格 \(W_i\)。三条需求曲线 \(X_i^{D_1}\)、\(X_i^{D_2}\)、\(X_i^{D_3}\) 由构造均过同一点 \((W_i^0,X_i^0)\)，但在该点处弹性各不相同——\(X_i^{D_2}\) 最陡（弹性最小），\(X_i^{D_3}\) 最平（弹性最大），\(X_i^{D_1}\) 居中。
\(X_i^{D_3}\) is a function of \((W_1,W_2,\dots,W_N)\).
As shown, the scale effect always reinforces the substitution effect, so \(X_i^{D_3}\) is the most elastic factor demand since it allows output to change and thus introduces the scale effect; \(X_i^{D_3}\) also has a substitution effect since it allows all factor quantities to change. \(X_i^{D_1}\) is less elastic than \(X_i^{D_3}\) because it only has a substitution effect, but more elastic than \(X_i^{D_2}\), which has neither a substitution nor a scale effect.
Figure 2.1 (paraphrased): horizontal axis = factor quantity \(X_i\), vertical axis = factor price \(W_i\). By construction the three demand curves \(X_i^{D_1}\), \(X_i^{D_2}\), \(X_i^{D_3}\) cross the same point \((W_i^0,X_i^0)\), but still have different elasticities across that point — \(X_i^{D_2}\) is steepest (least elastic), \(X_i^{D_3}\) flattest (most elastic), \(X_i^{D_1}\) in between.