经济增长(economic growth)

发表于

economic growth theory

Exogenous Growth Theory

  • The Solow Growth Model (1956)
  • The Augmented Solow Model (Mankiw, Romer and Weil, 1992)

Endogenous Growth Theory

  • The AK Model
    • The ‘Basic’ AK Model
    • An AK Model of Learning-by-Doing (Arrow 1962; Romer 1986)
    • An AK Model with Human Capital (Lucas 1988)

  • R&D Models (Romer 1990)

Solow Growth Model(长期经济增长索洛模型)

The supply and Demand for goods

The supply for goods(供给侧)

  • 总生产函数 $Y = AF(L, K, R)$
  • R: 自然资源, A: 经济中的技术水平;

  • 先不考虑技术变革, 自然资源不变

  • Assume production function $Y = F(K, L)$, and has constant return to scale $\lambda Y = F(\lambda K, \lambda L)$

  • Y: 产量, K: 物质资本量, L: 劳动量

  • set $\lambda = 1/L$ so $Y/L = F(K/L, 1)$

  • production per worker(人均资本存量) $y = Y/L$, $k = K/L$, then $$y = f(k)$$

Diminishing Returns to Capital(资本收益递减)：随着投入量的增加，每一单位额外投入得到的收益减少(资本边际产出下降)

diminishing_returns_to_captical.png

The demand for Goods(需求侧)

  • (这儿有个隐式的假设，供给和需求达到均衡，即供给=需求，故Y=Z)

  • The demand for Goods and the consumption Function $y = c + i$

  • y: 可支配收入(税后), c: 用于消费部分, i: 储蓄部分被用于投资的部分

储蓄率取决于银行的回报，银行的回报取决于银行投资率和利润率，假设高投资率

  • use S denote saving rate, so $c = (1 - s)y$(可支配收入储蓄部分以外的全部用于消费)

Keep in mind that various goverment policies can potentially influence a nation’s saving rate, so one of our goal is to find what saving rate is desirable/ For now, however, we just take the saving rate as given.

  • then $y = (1-s)y + i$, $i = sy$

  • Growth in the Capital stock(资本存量) and the Steady State(稳态)$$i = sf(k)$$

growth_in_the capital_stock.png

  • 供给等于需求，故人均产出等于消费部分加上投资部分

Solow Growth Model

索洛模型的目标是找出国民最优储蓄率

  • assume depreciation rate(资本折旧率) is $\delta$, so the change in capital stock is equal to investment minus depreciation $\Delta k = i - \delta k$

  • let supply = demand, $$\Delta k = sf(k) - \delta k$

growth_depreciationRate.png

  • 提高s(储蓄率)可以获得暂时的经济增长，直到达到新稳态点

How saving Affects Growth:

growth_HowSavingAffectsGrowth.png

  • The Solow model shows that:
    If the saving rate is low, the economy will have a small capital stock and a low level of output. If the saving rate is high, the economy will have a large capital stock and a high level of output, but it will NOT maintain a high rate of growth forever.

一个国家只靠资本积累无法获得持续经济增长(存在资本折旧率)

The Golden Rule Level of Capital

  • 目标是福利水平最优，即不能只考虑GDP最大；尽可能使用于当前消费的部分收入最大化

  • Comparing Steady States
    $y = c + i$, $c = y - i$
    $c^{} = f(k^{}) - \delta k^{*}$
    then $MPK = \delta$: $S_{gold}$, 黄金储蓄率

growth_goldenRuleLevelOfCapital.png

  • 当前福利水平尽可能最高的点在边际产出和折旧率相等点；将投资回报率调控到投资回报曲线与折旧曲线相交点

Population Growth

  • The Steady State with Population Growth
    $\Delta k = i - (\delta + n)k$, where n is population growth rate.

  • let supply = demand, $\Delta k = sf(k) - (\delta + n)k$

  • 人口增长，资本存量不变，相当于人均资本摊薄，等价于资本折旧率提高

growth_populationGrowth.png

  • The Effects of Population Growth
    $y^{} = f(k^{})$
    then $MPK = \delta + n$: $S_{gold}$

growth_populationGrowth_effect.png

Technological Progress

  • The efficiency of labour and the S-S(steady state) with Technological Progress $Y = F(K, L * E)$

  • Let $k = K/(L E)$ (capital per effective worker), $y = Y/(L E) = f(k)$ (output per effective worker), so $\Delta k = sf(k) - (\delta + n + g)k$

技术进步相当于有效劳动力增加，而资本不变，不变资本被摊薄；技术进步导致资本不够用

  • The Effects of Technological Progress
    $c^{} = f(k^{}) - (\delta + n + g) k^{*}$
    then $MPK - \delta = n + g$: $S_{gold}$

  • output per worker $y/L = y * E$

增加人均产出只能靠技术进步

Policies and Growth

What’s the Optimal Rate of Saving?

  • Changing the Saving-Rate
  • Allocating the Economy’s Investment
  • Fertility Programmes
  • Encouraging Technological Progress

Cross-Country Implications of the Solow Model

Conditional Convergence

  • assume all countries have access to the global stock of technology
  • Long-run, all will grow at g.

  • Short-run, growth-rates will depend inversely on the distance between the capital stock and steady-state level, which is determined by the savings-rate, depreciation-rate, and so on

  • In terms of cross-country comparisons, there is no tendency to convergence. Countries which save more are predicted to have permanently faster growth, and income levels will therefore diverge over time

Long-run:

  • The model predicts that long-run growth in standards of living (y) is equal to the rate of technological progress, which is exogenously given.
  • It follows that government is unable to influence growth permanently

Short-Run:

  • In the short-run, however, (where effective capital per head, k is not in its steady-state), growth is affected by a number of variables. Transitional growth (and the steady-state level of output per head) is
    • increasing in the rate of saving,
    • decreasing in the rate of population growth and rate of depreciation.
  • Government has greater scope to affect growth in the short-run, by, for example, influencing fertility-rates and encouraging more household saving