4. Growth

Jun He June 01, 2026
中文English 增长Growth 资本积累Capital Accumulation 稳态Steady State 无套利No-Arbitrage 全要素生产率Total Factor Productivity

4. Growth

Note

本章导读 人均产出增长不能只靠人口增长，须考虑其他增长动力。本章聚焦资本这一逐渐折旧、提供商品与服务流的特殊耐用品：先给出资本的记号与按租金率加总的方法，写出离散与连续时间下资本品的四个方程（使用市场出清、投资市场出清、资本运动方程、无套利），并求解稳态均衡；再把技术进步作为全要素生产率（TFP）增长引入生产函数 \(Y=F(K,L,T)\)，推出 TFP 增长的四条分解关系。

4. Growth

Note

Overview Per-capita output growth cannot rely on population growth alone; we must consider other growth drivers. This chapter focuses on capital, a special durable good that depreciates gradually and provides a flow of goods and services: we set up the notation for capital and a way to aggregate it by rental rates, write the four equations for capital goods in discrete and continuous time (use-market clearing, investment-market clearing, law of motion of capital, no-arbitrage), and solve the steady-state equilibrium; then we introduce technological improvement as total factor productivity (TFP) growth into the production function \(Y=F(K,L,T)\), deriving four decomposition relations for TFP growth.

4.1 Durable goods investment and capital accumulation

总产出增长可来自人口增长。但若我们关心人们生活水平的变化，即人均产出增长，就需考虑人口增长之外的增长动力。本节主要聚焦资本——一种特殊的耐用品（脚注 4.1：注意耐用品不同于可储存品。例如一瓶酒可放很久，但一旦开瓶喝掉就用尽了，这使它更像可储存品而非耐用品；而一件艺术品我们把它当作展示并持续从中获得效用，这才是一种耐用资本品），它提供商品与服务的流量并逐渐折旧。

4.1.1 Set-up

  • \(P_t\)：时刻 \(t\) 资本品（所有权）的价格；
  • \(K_t\)：时刻 \(t\) 的资本存量；
  • \(\delta\)：折旧率，假设随时间不变；
  • \(r\)：利率，假设随时间不变；
  • \(R_t\)：时刻 \(t\) 仅使用一期的资本品的租金价格；
  • \(I_t\)：时刻 \(t\) 的投资。

4.1.2 How do we aggregate different types of capitals

计算加总资本时常需对不同类型资本加总，这要求为每种资本品找到合适权重。一个自然想法是

4.1 Durable goods investment and capital accumulation

Total output growth could result from population growth. However, if we care about the change in people's living standard, i.e. per-capita output growth, we need growth drivers other than population growth. This subsection focuses mainly on capital, a special durable good (footnote 4.1: note a durable good differs from a storable good. For example, a bottle of wine can last a long time, but when opened and drunk it is used up, which makes it more like a storable good than a durable one. On the other hand, a piece of art we treat it as something to display and always extract utility from displaying it — that is a kind of durable capital good) that provides a flow of goods and services and depreciates gradually.

4.1.1 Set-up

  • \(P_t\): price of capital good at time \(t\) for the ownership;
  • \(K_t\): capital stock at time \(t\);
  • \(\delta\): depreciation rate, assumed constant over time;
  • \(r\): interest rate, assumed constant over time;
  • \(R_t\): rental price of capital good for only one period use at time \(t\);
  • \(I_t\): investment at time \(t\).

4.1.2 How do we aggregate different types of capitals

When calculating aggregate capital, we often need to aggregate over types of capital, which requires a proper weight for each type. One natural idea is

$$K^t_{\text{aggregate}}=\sum_{i=1}^N R^t_i\cdot K^t_i.$$

注意这里用租金率而非价格作为资本权重，因为租金率代表彼此可比的边际产品价值。

4.1.3 Four equations for capital goods: discrete time

Note we use rental rates instead of prices as the weights of capital, because rental rates represent the value of marginal products that are comparable with each other.

4.1.3 Four equations for capital goods: discrete time

1.（使用市场的市场出清条件）租赁市场上对资本的需求应等于资本存量：

  1. (market clearing condition in the use market) Demand for capital in the rental market should equal capital stock:

$$K_t=D_t(R_t)$$

2.（投资市场的市场出清条件）投资应等于新资本品的供给：

  1. (market clearing condition in the investment market) Investment should equal the supply of new capital goods:

$$I_t=S_t(P_t)$$

3.（资本运动方程）

  1. (law of motion of capital)

$$K_t=(1-\delta)K_{t-1}+I_t$$

4.（无套利条件）

  1. (no-arbitrage condition)

$$ \begin{aligned} P_t&=\sum_{n=0}^\infty\frac{R_{t+n}(1-\delta)^n}{(1+r)^n}\\ \text{recursive version}&=R_t+\frac{P_{t+1}(1-\delta)}{1+r} \end{aligned} $$

4.1.4 Four equations for capital goods: continuous time

1.（使用市场出清）\(K(t)=D_t(R(t))\) 2.（投资市场出清）\(I(t)=S_t(P(t))\) 3.（资本运动方程）\(\dot K(t)=I(t)-\delta K(t)\)，其中 \(\dot K(t)\) 为对时间的导数 \(\tfrac{dK(t)}{dt}\)。 4.（无套利条件）

4.1.4 Four equations for capital goods: continuous time

  1. (use-market clearing) \(K(t)=D_t(R(t))\)
  2. (investment-market clearing) \(I(t)=S_t(P(t))\)
  3. (law of motion of capital) \(\dot K(t)=I(t)-\delta K(t)\), where \(\dot K(t)\) is the derivative w.r.t. time, \(\tfrac{dK(t)}{dt}\).
  4. (no-arbitrage condition)

$$P_t=\int_0^{+\infty}e^{-(r+\delta)\tau}R(t+\tau)\,d\tau$$

为简便，后文用离散设定讨论稳态。

4.1.5 Steady state conditions

稳态中所有变量保持不变：\(P_t=\bar P\)、\(R_t=\bar R\)、\(I_t=\bar I\)、\(K_t=\bar K\)。若进一步假设需求与供给不随时间变化，即 \(D_t(R_t)=D(R_t)\)、\(S_t(P_t)=S(P_t)\)，则稳态条件为

1.（使用市场出清）租赁市场需求等于资本存量 \(\bar K=D(\bar R)\)； 2.（投资市场出清）投资等于新资本品供给 \(\bar I=S(\bar P)\)； 3.（资本运动方程）\(\bar I=\delta\bar K\)； 4.（无套利条件）

For simplicity, later on we use the discrete set-up to discuss the steady state.

4.1.5 Steady state conditions

In steady state, all variables stay constant: \(P_t=\bar P\), \(R_t=\bar R\), \(I_t=\bar I\), \(K_t=\bar K\). If we further assume unchanging demand and supply, i.e. \(D_t(R_t)=D(R_t)\) and \(S_t(P_t)=S(P_t)\), then the steady-state conditions are

  1. (use-market clearing) demand equals capital stock \(\bar K=D(\bar R)\);
  2. (investment-market clearing) investment equals supply of new capital \(\bar I=S(\bar P)\);
  3. (law of motion of capital) \(\bar I=\delta\bar K\);
  4. (no-arbitrage condition)

$$ \begin{aligned} \bar P&=\sum_{n=0}^\infty\frac{\bar R(1-\delta)^n}{(1+r)^n}\\ &=\bar R\frac{1+r}{r+\delta} \end{aligned} $$

用稳态条件，可通过令需求等于供给在租赁市场上建立均衡：

Using the steady-state conditions, we can establish an equilibrium in the rental market by setting demand equal to supply:

$$ \begin{aligned} D(\bar R)&=\bar K&&\text{by market clearing in the use market}\\ &=\frac{\bar I}{\delta}&&\text{by law of motion of capital}\\ &=\frac{1}{\delta}S(\bar P)&&\text{by market clearing in the investment market}\\ &=\frac{1}{\delta}S\left(\bar R\frac{1+r}{r+\delta}\right)&&\text{by no-arbitrage condition} \end{aligned} $$

可把 \(S_{ss}=\tfrac{1}{\delta}S\!\left(\bar R\tfrac{1+r}{r+\delta}\right)\) 视为稳态租赁市场供给曲线（向上倾斜），\(D_{ss}=D(\bar R)\) 视为稳态租赁市场需求曲线（向下倾斜）。由于稳态下用尽了全部均衡条件，系统可被唯一钉死，这与两条曲线只有一个交点相吻合。类似地，也可把全部方程转入投资市场来钉死系统，结果完全相同。

4.2 Total factor productivity growth: technological improvement

前面的讨论中生产函数只有两种要素：劳动与资本，这意味着真实人均收入增长只能由资本深化产生。然而我们也可把技术进步纳入模型，把这种变化定义为全要素生产率（TFP）增长。考虑生产函数 \(Y=F(K,L,T)\)，全微分：

We can regard \(S_{ss}=\tfrac{1}{\delta}S\!\left(\bar R\tfrac{1+r}{r+\delta}\right)\) as the steady-state rental-market supply curve (upward sloping) and \(D_{ss}=D(\bar R)\) as the steady-state rental-market demand curve (downward sloping). Since we have used all the equilibrium conditions at steady state, the system can be uniquely pinned down, which coincides with these two curves having only one intersection. Similarly, we can transfer all equations into the investment market to pin down the system, giving exactly the same result.

4.2 Total factor productivity growth: technological improvement

The previous discussion uses a production function with only two factors: labor and capital, which assumes the real per-capita income growth can only be generated by capital deepening. However, we can also include technological improvement into our model and define its change as total factor productivity (TFP) growth. Consider the production function \(Y=F(K,L,T)\) and totally differentiate it:

$$ \begin{aligned} dY&=F_K\,dK+F_L\,dL+F_T\,dT\\ \because F_K=\frac{R}{P}\text{ and }F_L=\frac{W}{P}\Rightarrow\ \frac{dY}{Y}&=\frac{KR}{PY}\frac{dK}{K}+\frac{LW}{PY}\frac{dL}{L}+\frac{F_T}{Y}dT\\ \Rightarrow\ \Delta Y&=s_K\Delta K+s_L\Delta L+\frac{F_T}{Y}dT\\ \Rightarrow\ \Delta TFP=\frac{F_T}{Y}dT&=\Delta Y-(s_L\Delta L+s_K\Delta K) \end{aligned} $$

于是可正式定义 \(\Delta TFP\)：

So we can formally define \(\Delta TFP\):

$$\Delta TFP=\Delta Y-(s_L\Delta L+s_K\Delta K).\tag{4.1}$$

注意这里 \(s_L\)、\(s_K\) 是收入份额，一般未必加总为一。若假设关于 \(K\) 与 \(L\) 是 CRS，则由欧拉定理 \(PY=WL+KR\)，可得 \(\Delta TFP\) 的另一表达：

Note here \(s_L\) and \(s_K\) are shares of revenue, which may not add up to one. If we assume CRS in \(K\) and \(L\), then by Euler's theorem \(PY=WL+KR\), we can have another expression of \(\Delta TFP\):

$$\Delta TFP=(s_L\Delta W+s_K\Delta R)-\Delta P$$

此式可通过对 \(PY=WL+KR\)（CRS 下成立）全微分证明：

This expression can be shown by totally differentiating \(PY=WL+KR\), which is true under CRS:

$$ \begin{aligned} YdP+PdY&=LdW+WdL+KdR+RdK\\ \Rightarrow\ \frac{dP}{P}+\frac{dY}{Y}&=\frac{WL}{PY}\frac{dW}{W}+\frac{WL}{PY}\frac{dL}{L}+\frac{KR}{PY}\frac{dR}{R}+\frac{KR}{PY}\frac{dK}{K}\\ \Rightarrow\ \Delta P+\Delta Y&=s_L\Delta W+s_L\Delta L+s_K\Delta R+s_K\Delta K \end{aligned} $$

由 (4.1) \(\Rightarrow\Delta P=s_L\Delta W+s_K\Delta R-\Delta TFP\)，故

By (4.1) \(\Rightarrow\Delta P=s_L\Delta W+s_K\Delta R-\Delta TFP\), hence

$$\Delta TFP=(s_L\Delta W+s_K\Delta R)-\Delta P\tag{4.2}$$

整理 (4.1)：

Rearrange equation (4.1):

$$ \begin{aligned} \Delta Y&=(s_L\Delta L+s_K\Delta K)+\Delta TFP\\ \Rightarrow\ \Delta\frac{Y}{L}=\Delta Y-\Delta L&=\left(\underbrace{(s_L-1)}_{=-s_K:\text{CRS}}\Delta L+s_K\Delta K\right)+\Delta TFP\\ \Rightarrow\ \Delta\frac{Y}{L}&=s_K\Delta\frac{K}{L}+\Delta TFP \end{aligned} $$

这表明真实人均收入增长确实可分解为资本深化与技术增长。整理 (4.2)：

which shows that real per-capita income growth can indeed be decomposed into capital deepening and technological growth. Rearrange equation (4.2):

$$ \begin{aligned} \Delta TFP&=(s_L\Delta W+s_K\Delta R)-\Delta P\\ \Rightarrow\ \Delta TFP&=s_L(\Delta W-\Delta P)+s_K(\Delta R-\Delta P)\quad\because\text{ CRS}\\ \Rightarrow\ \Delta TFP&=s_L\Delta\frac{W}{P}+s_K\Delta\frac{R}{P} \end{aligned} $$

这表明技术增长的好处可分为真实工资增长与真实租金率增长。

小结（Summary）：

  1. \(\Delta TFP=\Delta Y-(s_L\Delta L+s_K\Delta K)\)（\(\Delta TFP\) 的定义，不用 CRS）；
  2. \(\Delta TFP=(s_L\Delta W+s_K\Delta R)-\Delta P\)（用 CRS）；
  3. \(\Delta\frac{Y}{L}=s_K\Delta\frac{K}{L}+\Delta TFP\)（用 CRS）；
  4. \(\Delta TFP=s_L\Delta\frac{W}{P}+s_K\Delta\frac{R}{P}\)（用 CRS）。

which shows the benefit of technological growth can split into real wage growth and real rental rate growth.

Summary:

  1. \(\Delta TFP=\Delta Y-(s_L\Delta L+s_K\Delta K)\) (definition of \(\Delta TFP\), does not use CRS);
  2. \(\Delta TFP=(s_L\Delta W+s_K\Delta R)-\Delta P\) (use CRS);
  3. \(\Delta\frac{Y}{L}=s_K\Delta\frac{K}{L}+\Delta TFP\) (use CRS);
  4. \(\Delta TFP=s_L\Delta\frac{W}{P}+s_K\Delta\frac{R}{P}\) (use CRS).