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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: \(\dfrac{CF}{L}\)

Production Function

  • \(Y = F(K,L)\) assuming constant returns to scale1
  • 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 \(\dfrac{\text{capital}}{worker}\)
  • \(\dot{K}\) in capital stock or growth in cs
  • \(\boxed{\dot{K} = sy - \delta k}\)
    • \(\dot{k}\gt {0} \implies sy \lt \delta k\)
    • \(\dot{k}\lt {0} \implies sy \lt \delta k\)
    • \(\dot{k} = {0} \implies sy = \delta k\)
  • In a closed economy
    • \(Y = C + I + G\) \(\implies\) \(Y - C -G=I\)
    • \(\underbrace{Y-C-I}_{\text{Private}} + \underbrace{T-G}_{\text{Public}} = I\)
    • \(S = I\) \(\implies\) \(sy = I\)2
  • If \(\dot{K}\) is net investment, \(sY\) is gross savings and \(\delta k\) is depreciation. Here, we are writing gross savings in terms of gross investments or vice versa. We are saving the number of machines adding in the economy on \(sY\) and \(\delta k\) is the number of machines needed to be added to keep the capital unchanged.
  • \(k = \dfrac{K}{L}\) \(\implies\) \(\ln k = \ln K - \ln L\)
  • Differentiating wrt time
\[ \begin{align*} & \dfrac{1}{K} \dfrac{dk}{dt} = \dfrac{1}{K} \dfrac{dK}{dt} - \dfrac{1}{L} \dfrac{dL}{dt} \\ \implies & \dfrac{\dot{k}}{k}=\dfrac{\dot{K}}{K}-\underbrace{\dfrac{\dot{L}}{L}}_{\Delta\text{ population}} \\ \implies & \dfrac{\dot{k}}{k} = \dfrac{sy-\delta k}{k} - n = \dfrac{sy/L}{k/L} - (n+\delta) \\ \\ \implies & \dot{k} = sy - (n+\delta)k \\ \implies & \dot{k} = \underbrace{sk^\alpha}_{\frac{\text{cap avl}}{worker}} - \underbrace{(n+\delta)k}_{\text{Required} \frac{\text{cap}}{\text{worker}}} \end{align*} \]
  • \(sk^\alpha\) should always \(\uparrow\) to keep cap stock positive
  • \(sy\) is telling us the number of machines added per worker.
  • \((n+\delta)k\) tells us the number of machines needed per worker.
    • \(\dot{k} = 0\) \(\to\) ideal/steady state \(\sim\) cap stock remains constant
    • if \(\dot{k} \gt 0\), \(sy \gt (n+\delta)k\)
    • if \(\dot{k} \lt 0\), \(sy \lt (n+\delta)k\)
  • If the number of machines added per worker is more than the number of machines needed to keep the capital per worker unchanged. Then the capital per worker will increase.
  • 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.

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

Technologia

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

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:

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

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.

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.

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 residual3 - Same LF, \(L\)[^4] - 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 \(\tilde{k}\uparrow\), \(s\tilde{k}^\alpha \downarrow\)

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.

  1. When input \(\uparrow\) , how much does the change? If it both show commensurate percentage change, then we say that it is constant returns to scale. ↩

  2. Here, \(s = \dfrac{S}{Y}\) ↩

  3. Solow's residual indicates technology ↩