1 Growth & Development¶

Growth	Development
One-dimensional	Multi-dimensional
e.g. Continuous increase in the output over a period of time	More comprehensive
Quantitative	Qualitative
- Some indicators
- GDP $\to$ Floor variable/ indicator (dynamic or continuous)
- indicates demand
- $c + i + g$ (India is a consumption driven economy, but may change)
- includes Goods and Services
- CPI
- Not services, not all goods
- GNP / GNI = GDP + NFIA
- GDP $\gt$ GNI, because in developing countries NFIFA is usually negative

Extensive	Intensive
Standard of Living (Economies of scale)	Given more importance to the 'per-capita' aspect
Commodity (goods etc)	Factor markets (skill sets)

Why is the process of growth important
- Policy Making (e.g. Recession, monetary policy)
- Capital formation

1.1.1 Importance of economic growth¶

Choices are more when the country is developed. Don't need to depend only on one sector
Social conflicts in developing country. A zero sum game. Need to remove for one in order to provide to another
Poverty & inequality paradox: > As income increases, inequality rises, for many emerging economies, growth led people out of poverty but it increased inequality.
Spill-over effects of growth

2 Traditional Growth Theories¶

2.1 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, $\theta Y_{t+1} = I_{t} + (1-\delta) K_{t}$

$$ \begin{align} &\quad\theta Y_{t+1} = S_{t} + (1-\delta)K_{t} \ \implies&\quad \theta Y_{t+1} = \theta Y_{t} = S_{t} + (1-\delta)K_{t} - \theta Y_{t} \ \implies&\quad \theta(Y_{t+1}-Y_{t}) = sY_{t} + (1-\delta)K_{t} - \theta Y_{t} \ \implies&\quad \theta(Y_{t+1}-Y_{t}) - (1-\delta)K_{t} = sY_{t} - \theta Y_{t} \ \implies&\quad \theta(Y_{t+1}-Y_{t}) = sY_{t} - \theta Y_{t} + (1-\delta) \theta Y_{t} \ \implies&\quad \theta(Y_{t+1}-Y_{t}) = [(s-\theta) + (1-\delta)\theta]Y_{t}\ \implies&\quad \theta(Y_{t+1}-Y_{t}) = [s- \theta + \theta - \theta\delta] Y_{t} \ \implies&\quad \theta(Y_{t+1}-Y_{t}) = [ s - \theta\delta] Y_{t} \

\implies&\quad \underbrace{\dfrac{Y_{t+1} -Y_{t}}{Y_{t}}}_{\text{Growth}} = \dfrac{S- \delta\theta}{\theta} \implies&\quad g = \dfrac{s}{\theta} - \delta\ \implies&\quad \boxed{g + \delta = \dfrac{s}{\theta}} \ \end{align}

$$

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

2.1.1 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

2.1.2 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

2.2 Neo-classical or Solow's Growth¶

Technology is exogenous
Economic growth
assumed entire population participates in the labor force (LF)
Production function (PF): $Y = F(L,K)$ (total)
- Intensive form of PF: $\dfrac{PF}{L}$ (cause we assumed entire population is in LF)
Capital formation (CF): $K_{t+1} = I_{t} + (1-\delta)K_{t}$
- Intensive CF: $\dfrac{CF}{L}$

2.2.1 Production Function¶

$Y = F(K,L)$ assuming constant returns to scale²
So, $Y = F(K,L) = K^\alpha L^{1-\alpha};\quad 0 \lt \alpha \lt 1$
Intensive form: $\dfrac{Y}{L} = \dfrac{K^\alpha L^{1-\alpha}}{L}$
- $y = \left(\dfrac{K}{L}\right)^\alpha \cdot (1)^{1-\alpha}$
- $y$ (output/labor) = $k^\alpha$ (capital/labor)
- FOC: $\dfrac{\partial{y}}{\partial{k}}=\alpha k^{\alpha-1}$ (increasing function)
- SOC: $\dfrac{\partial^2{y}}{\partial{k^2}} = (\alpha)(\alpha-1)k^{(\alpha-2)}$ (decreasing function)
Concave function: meaning it increases at a decreasing rate similar to DMU, so $\uparrow$ in $k$ $\implies$ smalleincrease in output. Diminishing output of capital (Diminishing returns to capital)
With more capital per worker ($k$), firms produce more output per worker. However, there is diminishing returns to $\dfrac{\text{capital}}{worker}$
GRAPH
- At $k^*$, $sy = (n+\delta)k$. It is called steady state and stable equilibrium. At $k^0$, $sy \gt (n + \delta)k$. Here, the capital per worker will increase till steady state is reached. On the other hand, $k'$ will start falling until the steady state is reached.

2.2.2 Comparative Statics¶

Savings
Population Growth

"If these change, what will happen to the equilibrium?" is the key question.

GRAPH: steady state when change in savings

2.2.3 Technologia¶

We looked at $Y = F(K,L)$.

2.2.4 Three types of Augmenting¶

But to generate sustained economic growth, in per capital income, we must assume technological progress ($A$). Three kinds of production functions considering technology are:

Labor Augmenting / Harrod Neutral
- $Y = F(K, \mathbf{A}L) = \boxed{(AK)^\alpha L^{1-\alpha}}$
Capital Augmenting / Solow Neutral
- $Y = F(\mathbf{A}K, L) = \boxed{(AK)^\alpha L ^{1-\alpha}}$
Hicks Neutral
- $Y = A \cdot F(K,L) = A \cdot K ^\alpha \cdot L^{1-\alpha}$
- $\dfrac{\dot{A}}{A} = g$

2.2.4.1 Labor Augmenting¶

We are assuming that technological progress is exogenously given. Therefore, growth in technology will be $\dfrac{\dot{A}}{A} = g$.

\[ \begin{align} Y & = K^\alpha (AL)^{1-\alpha} \\ y & = \dfrac{K^\alpha(AL)^{1-\alpha}}{L} \\ & =\dfrac{K^\alpha}{\cancel L} A^{1-\alpha}\cdot L^{\cancel{1}-\alpha} \\ & = A^{1-\alpha} \left(\dfrac{K}{L}\right)^\alpha \\ y & = A^{1-\alpha} k^\alpha \\ \ln y & = \alpha \ln k \cdot (1-\alpha) \ln A \end{align} \]

Differentiating wrt time, $t$ we get

$$ \begin{align} \dfrac{1}{y} \dfrac{dy}{dt} & = \dfrac{\alpha}{k} \cdot \dfrac{dk}{dt} + \dfrac{1-\alpha}{A} \dfrac{dA}{dt} \ \dot{y} & =\dfrac{\alpha}{k}\dot{k} \dfrac{(1-\alpha)\dot{A}}{A} \ y & = \alpha g_{k} + (1-\alpha)g \ \dot{k} & = sy - \delta k \end{align} $$ Dividing both sides by $k$, we get

\[ \begin{align} \dfrac{\dot{k}}{k} & = \dfrac{sy}{k} - \dfrac{\delta \cancel k}{\cancel k} \\ \text{Growth rate in } k & = \dfrac{sy}{k} - \delta \end{align} \]

$s$ & $\delta$ $\to$ constant
So, if the growth needs to increase/grow at a constant rate, $\dfrac{y}{k}$ also should grow at the same rate.

Here, $s$ & $\delta$ are exogenously given. So, if we want the growth of capital to be constant, $\dfrac{Y}{K}$ also needs to grow at constant rate as $s$ & $\delta$ are not changing.

2.2.4.2 Balanced Growth Rate¶

In intensive form, if capital growth per labor must grow/increase at a constant rate, then even output / capital per labor will also grow at the same rate $\dfrac{Y}{K}$. Thus, $\dfrac{y}{k}$ is also constant. $\implies$ Balanced Growth Rate.

A situation in which capital/output/consumption & population are growing at the constant rate is called Balanced Growth Rate.

\[ \begin{align} g_{y} & = \alpha g_{k} + (1-\alpha)g \\ \because g_{k} & = g_{y} \\ g_{y} & = \alpha g_{y} + (1-\alpha) g \\ \implies(1-\alpha)g_{y} & = (1-\alpha)g \\ & \boxed{g_{y} = g = g_{k}} \end{align} \]

Therefore, along the balanced growth path, output per worker and capital per worker are growing at a constant rate i.e., at the rate of technological progress.

In the model, without technology, we said $\dfrac{\dot{k}}{k} = g_{k}$, and $g = \dfrac{\dot{y}}{y}$ are equal. But here, we are saying that $\dfrac{\dot{y}}{y} = g$. Meaning, technology is the source of sustained growth.

2.2.4.3 Harrod Neutral¶

$$ Y = F(K,L) = K^\alpha (AL)^{1-\alpha} $$ Dividing by $AL$,

\[ \begin{align} \dfrac{Y}{AL} & = \dfrac{K^\alpha(AL)^{1-\alpha}}{AL} \\ \dfrac{Y}{AL} & = K^\alpha AL^{1-\alpha - 1} \\ \dfrac{Y}{AL} & = \left(\dfrac{K}{AL}\right)^\alpha \end{align} \]

With a new notation, $\tilde{\boxed{\cdot}} = \dfrac{\boxed{\cdot}}{AL}$ ($\boxed{\cdot}$ per augmented labor) we can write: $$ \dot{\tilde{y}} = \tilde{k}^\alpha $$ Applying $\ln$ and differentiation,

\[ \dfrac{\dot{\tilde{k}}}{k} = \dfrac{\dot{K}}{k} - n - g \]

Substituting $\dot{K} = sy- \delta K$ we get

\[ \dfrac{\dot{\tilde{k}}}{k} = \dfrac{sy}{K} - (n + g + \delta) \]

Dividing by augmented labor, $$ = \dfrac{sy/AL}{K/AL} - (n + g + \delta) $$ $$ \implies\dfrac{\dot{\tilde{k}}}{\tilde{k}} = s \dfrac{\tilde{y}}{\tilde{k}} - (n + g + \alpha) $$ Multiplying $\tilde{k}$ on both sides,

$$ \boxed{\dot{\tilde{k}}= s\tilde{y} - (n + g + \delta)\tilde{k}} $$ is nothing but the Solow's intensive growth model, but taking into account the labor augmented with technology.

GRAPH

At steady state, $\dot{\tilde{k}} = 0$. So, $s\tilde{k}^\alpha(s\tilde{y}) = (n + g + \delta)\tilde{k}$

$$ \implies \dfrac{s}{(n+ g + \delta)} = \tilde{k}^{(1-\alpha)} $$ At $\tilde{k}$, it is a steady state and at $\tilde{k}_{0}$, the investment being undertaken exceeds the amount needed to keep the capital technology ratio (CTR) constant. So the CTR will increase till the steady state is reached.

2.2.4.4 Convergence¶

Poor countries converge with rich countries
Catching up with growth of rich countries

Assumptions - two identical countries Initial conditions - Total factor productivity / Solow's residual¹ - Same LF, $L$ - Same savings rate, $s$

Poor countries remain poor cause the capital to labor is low in comparison with the rich (high $\dfrac{K}{L}$, high $\dfrac{Y}{L}$) - GRAPH - There is an inverse relationship between CLY and CLYGR - Because of low technology, the poor and rich countries can converge. - The reason for this low technology could be: Intervention: Labor union and trade barriers.

GRAPH - When $\tilde{k}\uparrow$, $s\tilde{k}^\alpha \downarrow$

2.2.4.5 Convergence & Solow's Growth Model¶

We know that $\dfrac{\dot{\tilde{k}}}{k} = s\tilde{k}^{\alpha-1} - (n+g+\alpha)\tilde{k}$. - As $\tilde{k}\uparrow$, $s\tilde{k}^\alpha \downarrow$ because of diminishing returns to capital. Among countries that have the same steady state, the countries converge, i.e., poor countries should grow faster on average than rich countries. - For instance, industrialized countries have similar initial conditions so convergence holds true. The differences among the income levels around the world largely reflect the differences in steady state, because all countries don't have similar initial conditions.

Unconditional Convergence: Whatever be the parameter, all countries converge at a steady state. It's not proved.

Conditional Convergence: Similar parameters would result in convergence. For example, industrialized countries among the poorer countries would converge.

3 Modern Growth Theories¶

They give importance to human capital also

Mankiw, Romer & Weil (1992) is an extension to the Neo-classical or Solow's Growth model (but considering human capital)
Learning by Doing (Kenneth Arrow): There is spillover of technology from one sector to another
Romer Endogenous Growth Model emphasizes on the Role of RND on economic growth
AK Model
Lucas Model

3.1 MRW¶

3.2 Learning by Doing¶

Higher the knowledge $\implies$ Higher capital stock
The production function depending on the technology accumulation will have the form $$ Y_{t} = K^\alpha (A_{t}L_{t})^{1-\alpha} $$ where,
$Y_{t}$ output
$K$ capital
$A_{t}$ stock of knowledge
$L_{t}$ Labor force

The simplest case of LBD is found in those situations where learning occurs as a consequence of production of new capital, i.e., increase in the stock of knowledge $= f(\text{increase in stock of capital})$.²

As capital is exogenous , which grows through savings, since the knowledge is a function of capital, knowledge also becomes endogenous.

\[ \underbrace{A_{t}}_{\text{Stock of knowledge}} = \underbrace{BK^\beta}_{\text{Stock of capital}} \]

\[ Y_{t} = K^\alpha(B\cdot K^\beta L_{t})^{1-\alpha} \]

Full economy, $Y_{t} = \underbrace{K^\alpha}_{\text{Direct Contribution}} B^{1-\alpha} \underbrace{K^{\beta(1-\alpha)}}_{\text{Indirect impact}} L_{t}^{(1-\alpha)}$ has,

Direct contribution: infrastructure, machinery
Indirect impact: helps in generating stock of knowledge

This is the additional benefit of capital, through its role in generating knowledge.

For a single firm,

\[ \boxed{ Y_{t} = A(K) F(\underbrace{K_{i}, L_{i}}_{\text{Inputs}}) } \]

It captures how an individual firm output depends not just on its own inputs ($K_{i},L_{i}$) but also on the economy wide capital stock , $A(K)$ which enhances productivity through LBD, and thus has a spillover effect.

3.3 Romer Endogenous Growth Model¶

Learning by investment²
Inspired from Learning by Doing

\[ Y = A(R) \times f(R_{i},K_{i},L_{i}) \]

where, - $A(R)$ is economy wide, technology developed with $R$ - $R$ is the economy wide RND - $R_{i}$ is the RND of individual firm

There are three components: 1. Externalities: $+$ve spillover effects 2. Increasing returns to production of output 3. Decreasing returns to production of new knowledge

The new knowledge after a certain point, will face DMR because it now doesn't increase the output as much as it did initially. But even though new knowledge may have DR, output will still $\uparrow$.³

Due to weak patents, though they have DR to new knowledge. The knowledge has spillover to benefit the entire economy.¹

3.4 AK Model¶

Paul Romer and Robert Barro
Assumes linearity

$$ Y = AK $$ is the production function where, - $A$ technology - $K$ capital, which can be - Physical - Human

There are no diminishing returns, marginal productivity of capital cannot be negative.

\[ \begin{align} \dot{k} &= \underbrace{sY}_{\text{Grows > }\delta K} - \delta K \\ \dfrac{\dot{k}}{K} & = \dfrac{sY}{K} - \delta \\ \dfrac{\dot{k}}{K} & = \dfrac{sAK}{K} - \delta \\ \dfrac{\dot{k}}{K} & = sA - \delta \\ \end{align} \]

\[ \begin{align} Y & = AK \\ \ln Y & = \ln A + \ln K \end{align} \]

$\dfrac{\dot{Y}}{Y} = \dfrac{\dot{K}}{K}$ ( assuming $A$ is constant) = $sA - \delta$

Output is an $\uparrow$ function of investment.

Ideas (increasing returns to scale and monopolistic competition and due to parents)
- non-rivalrous
- public goods
- non-excludable
For ideas, perfect competition not fit, because they incur losses. Because, they need to set $MC = P$ and $MC = 0$, so they need to price higher and these is no efficient pricing.

3.5 Lucas Model¶

based on Uzawa and Baker
Theory of human capital
More investment in education - $\uparrow$ in human capital (productivity of Human Resources $\implies$ leads to growth)

Two effects: - Internal - External

\[ Y_{i} = A(K_{i}) (H_{i}) (H^e) \]

where, - $K_{i}$ physical capital @ firm - $H_{i}$ human cap @ firm - $H^e$ economy level

At firm, CRS but as a $g$ #doubt included, $IRS$

\[ \dot{H} =\dfrac{dH}{dt} = (1 - \mu)H \]

where, - $\mu$ number of hours worked

$$ \dfrac{\dot{H}}{H} = 1 - \mu $$ where, - LHS = Growth in HC - $1-\mu$ is an increasing function

5 Theories of Underdevelopment¶

6 Sectorial Aspects of Development¶

Since the knowledge has no barrier to contain it, everyone can use that knowledge and anchor on it to create better output, that benefits the economy. Piracy example. ↩↩
Knowledge keeps growing and it is sustained by investment. ↩↩↩
In research sector predominantly. But in Goods sector, output will be much higher than the amount of knowledge used. ↩