Output as a combination of capital and labour - capital accumulation

生产是资本和劳动力的组合 - 资本积累

     Why did 19th political economists think of capital as “fixed”?  The reasoning seems to be that all tools required for production are themselves products of labour, and so when aggregating the entire production chain, the only variable input is the amount of labour hours employed.  However, production takes place on land, and also uses natural resources (water, air, etc), but these forms of capital are “exogenous” and so treated as “fixed”.

     With modern production processes, physical capital and labour input are substitutable; it’s possible to increase output by employing more capital for a given amount of labour input (but subject to diminishing returns).  Solow proposed that the amount of capital added over time (dK/dt) depends on the rate of savings and the rate of capital depreciation,  dK/dt = s*Y(t) - 𝞭*K(t) where:

• “s” is the share of output reinvested as capital 

• “𝞭” is the rate at which capital stock depreciates

     After some mathematical manipulation, he showed that:  K̂/Y = s/(n + g + 𝞭) where K̂ is the “steady state” level of capital that the economy maintains (dK/dt = 0). [Note 1] The amount of capital relative to output (K̂/Y), also known as the economy’s “capital intensity”:

• rises in its savings rate (s), 

• falls in the growth of labour (n), growth labour productivity (g) and capital depreciation rate (𝞭). 

     This result seems intuitive.  The higher the savings rate of an economy, the more capital it employs.  However, less capital is employed when effective labour grows at a high rate, or when capital depreciates quickly.  The result also seems to match the observation that the fast growing East Asian economies have high rates of savings, more capital stock (e.g., factories, physical infrastructure), and higher per capita output.  

     The lesson seems to be that high savings rate enables faster capital accumulation [Note 1], which leads to faster “catch up” growth. [Note 2]  

****

Note 1:  This is equivalent to K̂(n + g + 𝞭) = sY.  The LHS represents capital added, the RHS represents savings.  Suppose for some reason the savings rate, s rises relative to (n + g + 𝞭), then sY > K̂(n + g + 𝞭), in which the economy is no longer at steady state, and capital accumulates.

Note 2:  “Catch up” growth is the notion that less developed economies with low amounts of capital stock will enjoy higher growth relative to developed economies.  As per capita output converges with those of developed economies, growth slows down.

---

     19世纪的政治经济学家，为什么认为资本是“固定的”？其推理看似是，生产所需的所工具本身都是劳动力的产品；因此在汇总整个生产链时，唯一的可变输入是所使用的劳动时间。然而，生产是在土地上发生的，也使用自然资源（水、空气等），但是这些形式的资本都是“外生的”，因此被视为“固定的”。

     有了现代化的生产过程，物质资本和劳动输入都可以互相替代；在给定量的劳动力输入下，通过使用更多资本，可以增加产生（但是受收益递减的）。索洛提出，随着时间的推移增加资本量（dK/dt）取决于储蓄率和资本折旧率，dK/dt = s*Y(t) - 𝞭*K(t) 其中：

• “s”是为资本再投资的产出份额

• “𝞭”是资本存量折旧的速率

     经过一些数学运算，他表明：K̂/Y = s/(n + g + 𝞭) 其中 K̂是经济维持的“稳态”资本水平 (dK/dt = 0)。[注1] 资本量相对于产出的量（K̂/Y），也被称经济体的“资本强度”：

• 在储蓄率中上升；

• 在（i）劳动力增长（n）、（i i）劳动生产率增长（g）、（i i i）资本折旧率（𝞭）各个中下降。

     这个结果，看起来很直观。一个经济体的储蓄率越高，其使用的资本物越多。然而，当有效劳动力高速增长，或资本快速折旧时，资本的使用就会减少。这结果看似也符合观察结果，即快速增长的东亚经济体，储蓄率高、资本存量（如：工厂，有型基础设施）更多、人均产生更高。

     教训看似是，高储蓄率可以加快资本积累[注1]，从而导致更快的“追赶”增长。[注2]

****

注1：这相当于K̂(n + g + 𝞭) = sY。左边代表资本增加，右边代表储蓄。假设由于某些原因，储蓄率 s 相对于（n + g + 𝞭）上升，则 sY > K̂(n + g + 𝞭) ，此时经济不再处于稳定状态，因此资本积累。

注2：“追赶”增长是指资本存量较低的欠发达经济体将比发达经济体享有更高的增长。随着人均产出与发达经济体趋同，增长率放缓。