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Harrod Domar Growth

"Things change over a period of time"

The Harrod-Domar growth model is a macroeconomic model that explains economic growth through the relationship between capital, savings, and output. It posits that a country's growth rate is directly proportional to its savings rate and inversely proportional to its capital-output ratio. The model is an early attempt to explain the dynamics of growth in capitalist economies.

The Core Harrod-Domar Equation

The model's fundamental equation is derived from several key assumptions, including that all savings are invested and that there's a constant capital-output ratio.

Basic Identities:

  • National Income: \(Y_t = C_t + S_t\) (Income is either consumed or saved)2
  • Macroeconomic Balance: \(S_t = I_t\) (Savings equal investment)
  • Capital Accumulation: \(K_{t+1} = I_t + (1 - \delta)K_t\), where \(\delta\) is the depreciation rate. New investment + Capital remaining after depreciation (exhaustion)
  • Savings: If the MPC for an individual is \(s\).

Capital-Output Ratio (\(\theta\)):

The capital-output ratio (\(\theta\)) is the amount of capital required to produce one unit of output.

\(\theta = 1\) teaspoon tea leaves per cup, if tea was your output.

$$ \theta = \dfrac{K_t}{Y_t} \implies K_t = \theta Y_t $$ By substituting these into the capital accumulation equation and simplifying, we derive the fundamental growth equation: $$ \dfrac{Y_{t+1} - Y_t}{Y_t} = \dfrac{s}{\theta} - \delta $$ This can be expressed as: $$ g = \dfrac{s}{\theta} - \delta $$ Where: - \(g\) is the economic growth rate. - \(s\) is the savings rate. - \(\theta\) is the capital-output ratio. - \(\delta\) is the depreciation rate.

A more simplified version of this, often used in introductory contexts, is: $$ g = \dfrac{s}{c} $$ This highlights the core tenet of the model: the growth rate (\(g\)) is directly proportional to the savings rate (\(s\)) and inversely proportional to the capital-output ratio (\(c\)).

Intuition

  • Investment \(\implies\) Growth
  • Savings fund Investment
  • Not all investment is equally productive. A high \(c\) implies inefficient investment. Thus it penalizes \(s\). Even if you have more savings but they are being directed to inefficient projects, growth will be much lower.

Harrod's Three Growth Rates

Roy Harrod extended the model by proposing three distinct types of growth rates, which he used to explain the potential instability of a capitalist economy.

1. Actual Growth Rate (\(g_a\))

This is the observed or ex-post growth rate of an economy. $$ g_a = \dfrac{s}{c} $$

This rate reflects the current state of demand and supply in the economy. - \(g_{a} = \dfrac{s}{c}=\dfrac{S/{Y}}{{\Delta K}/{\Delta Y}}\) - Since \(\Delta K = I\), \(g_{a}= \dfrac{S}{Y} \cdot \dfrac{\Delta Y}{I}\) - Since \(S = I\), \(\boxed{g_{a}=\dfrac{\Delta Y}{Y}}\)

2. Warranted Growth Rate (\(g_w\))

This is the "full-capacity" growth rate that would keep entrepreneurs satisfied that they are operating at the optimal level.

\[ g_w = \dfrac{s}{c_r} \]

where, - \(c_{r}:\)required capital-output ratio (the capital-output ratio that would be needed to maintain steady growth, \(g\))

The model's famous "knife-edge" instability arises from any deviation of the actual growth rate from the warranted growth rate. - If \(g_a > g_w\), demand is higher than supply, leading to inflation.1 - If \(g_a < g_w\), supply is higher than demand, leading to unemployment and recession.

For steady state - Resources are optimally utilized - What is available is produced \(a_{o}\)

Think of this as

  • Actual growth rate as the growth of national income (real)
  • Represents growth of demand or income
  • Warranted (Ideal) growth rate is one that would keep investors and entrepreneurs satisfied with their business decisions.
  • Represents growth of supply or output

3. Natural Growth Rate (\(g_n\))

This is the maximum possible growth rate allowed by the growth of the labor force and technological progress. It is the rate required to maintain full employment over the long run. $$ g_n = \text{Growth of labor force} + \text{Growth of labor productivity} $$ For a stable, full-employment economy, all three growth rates must be equal (\(g_a = g_w = g_n\)). The Harrod-Domar model suggests that this alignment is highly improbable.

  • \(G_{w} \gt G_{n}\) \(\implies\) Underutilization
  • \(G_{w} \lt G_{n}\) \(\implies\) Unemployment

  1. The actual growth of income and demand is faster than the growth of productive capacity. ↩

  2. In a simplified, closed economy with no government, total output (Y) is the sum of consumption (C) and investment (I). \(Y = C+I\). At the same time total income (Y) is either spent on consumption (C) or is saved (S). \(Y = C + S\) \(\implies\) \(C + I = Y = C + S \implies I = S\) ↩