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