Harrod Domar Growth
\(Y_{t} = C_{t} + S_{t}\)
- Consumption spent on Goods and services, \(C_{t}\)
- Savings, \(S_{t}\) = \(I_{t}\) (always, assume macroeconomic balance)
- \(I\uparrow \implies \Delta Y\uparrow\), investment results in the change in national income
\(K_{t+1} = I_{t} + (1-\delta)K_{t}\)
- Investment + depreciation of capital
\(K_{t+1} = I_{t} + (1-\delta)K_{t}\)
- Capital output ratio, \(\theta\)
\[
\begin{align*}
\dfrac{K_{t}}{Y_{t}} & = \theta \\
K_{t} & = \theta Y_{t} \\
K_{t+1} & = \theta Y_{t+1}
\end{align*}
\]
So, \(\theta Y_{t+1} = I_{t} + (1-\delta) K_{t}\) sairam
\[
\begin{gather*}
\quad & \theta Y_{t+1} &=& S_{t} + (1-\delta)K_{t} \\
\implies &\theta Y_{t+1} &=& sY_t + (1-\delta)K_t \\
\implies &\theta Y_{t+1} &=& s\theta Y_{t} + (1-\delta)\theta Y_{t} \\
\implies &\theta Y_{t+1} - \theta Y_t &=& s\theta Y_{t} + (1-\delta)\theta Y_{t} - \theta Y_t \\
\implies &\theta(Y_{t+1} - Y_t) &=& (s\theta + \theta - \delta\theta - \theta)Y_t \\
\implies &\theta(Y_{t+1} - Y_t) &=& (s\theta - \delta\theta)Y_t \\
\implies &\dfrac{Y_{t+1} - Y_t}{Y_t} &=& \dfrac{s\theta - \delta\theta}{\theta} \\
\implies& \underbrace{\dfrac{Y_{t+1} - Y_t}{Y_t}}_{\text{Growth}} &=& s - \delta \\
\implies& g = s - \delta \\
\implies & \boxed{g + \delta = s}
\end{gather*}
\]
The boxed is the H.D. equation. - \(\theta\): How much additional units of capital to spend on additional unit of input. - With \(\uparrow\) in savings, g \(\uparrow\) - Sustained long run growth - Steady state equilibrium \(\implies\) no unemployment / inflation
Harrod Proposed 3 types of growth rates¶
Here, - \(g_{a} \to\) actual growth rate - \(g_{w} \to\) warranted growth rate
- Actual Growth Rate
- Demand or income
- \(g_{a} = \dfrac{s}{c}\left[\dfrac{s/{y}}{{\Delta k}/{\Delta y}}\right]\)
- Since \(\Delta K = I\), \(g_{a}= \dfrac{S}{Y} \cdot \dfrac{\Delta Y}{I}\)
- Since \(S = I\), \(\boxed{g_{a}=\dfrac{\Delta Y}{Y}}\)
- Reflects the demand existing in the economy,, but is prone to the same fluctuations
- Warranted Growth Rate (because changes are inevitable)
- Supply or Output
- For steady state
- Resources are optimally utilized
- What is available is produced \(a_{o}\)
- Capital is utilized to its fullest (\(C_{r} \to\) required capital)
- \(C_{r}\) is such that \(g\) is maintained the capital is utilized to its fullest.
- \(g_{a} = g_{w}\)
- No scope for under- or over-utilization
-
Flows¶
- Output \(\to\) Supply
- Income \(\to\) Demand
- \(g_{a} \gt g_{w}\) and \(c \lt c_{r}\)
- Income > Output
- Supply < demand
- \(\implies\) inflation
- Deficiency in capital \(\to\) output \(\downarrow \implies\) Price inflation (\(\uparrow\))
- \(g_{a} \lt g_{w}\) and \(c \gt c_{r}\)
- Income < Output; supply > demand
- \(\implies\) unemployment
- Overproduction \(\to\) Unemployment \(\to\) Recession \(\to\) Marginal efficiency of capital \(\to\) Output less \(\to\) Unemployment to cut production costs.
- Domar says "Investment is a double edged sword. Investment with multiplier changes demand"
- \(\Delta Y_{A} = \Delta I \left(\dfrac{1}{1-C}\right)\)
- \(\implies\) \(\Delta Y = \dfrac{\Delta I}{s}\)
- \(\implies\) \(\Delta Y_{t} = \Delta I_{t}\) (accelerator principle)
- \(\Delta Y_{S} = s_{\sigma}\)
- \(\sigma = \dfrac{1}{c_{r}}\)
- Natural Growth Rate (includes labor also)
- Capital and Labor
- \(\dfrac{\Delta Y}{Y} = l + q\)
- \(l:\) Labor Force
- \(q:\) Productivity of Labor Force
- \(G_{w} \gt G_{u}\) \(\implies\) Underutilization
- \(G_{w} \lt G_{n}\) \(\implies\) Unemployment
- Due to low access to technology
- Low savings \(\implies\) Low investment \(\implies\) Lower capital
Endogeneity¶
- Endogeneity \(\to\) Bidirectional causation
- Endogeneity in savings
- savings always generate income/growth
- growth also promotes savings
- Countries
- Poor: \(s\downarrow\)
- Middle: \(s\uparrow\)
- High income: \(s\downarrow\)
- Population
- \(I\) (low income): BR \(\uparrow\) DR \(\downarrow\)
- \(II\)(Mid income): BR \(\uparrow\) DR \(\downarrow\)
- \(III\) (High income): BR \(\downarrow\) DR \(\downarrow\)
- Population and growth impact each other