Harrod's Growth model is given as under:$\begin{array}{ll} \mathrm{S}_{\mathrm{t}}=\alpha \mathrm{Y}_{\mathrm{t}} & 0

Question

Harrod's Growth model is given as under:

$\begin{array}{ll} \mathrm{S}_{\mathrm{t}}=\alpha \mathrm{Y}_{\mathrm{t}} & 0<\alpha<1 \\ \mathrm{I}_{\mathrm{t}}=\beta\left[\mathrm{Y}_{\mathrm{t}}-\mathrm{Y}_{\mathrm{t}-1}\right] & \beta>0 \\ \mathrm{~S}_{\mathrm{t}}=\mathrm{I}_{\mathrm{t}} & \end{array}$

where S _t= Savings, Y _t = Income, l _t = Investment, t = time

In this model for economic growth, the condition for economic growth is

Prepp Team · Accepted Answer

Harrod's Growth Model Explained

Harrod's growth model is a key model in economics that attempts to explain the dynamics of economic growth. It focuses on the relationship between savings, investment, and the rate of output growth needed for sustained economic expansion. The model uses several equations to describe the economy at a specific point in time, $t$.

The given equations are:

Savings: $ S_t = \alpha Y_t $, where $ 0 < \alpha < 1 $. Here, $ \alpha $ is the savings rate, indicating that savings are a fixed proportion of income $ Y_t $.
Investment: $ I_t = \beta (Y_t - Y_{t-1}) $, where $ \beta > 0 $. This equation suggests that investment is proportional to the change in income from the previous period ($ Y_{t-1} $) to the current period ($ Y_t $). $ \beta $ represents the accelerator coefficient or capital-output ratio in some interpretations.
Equilibrium Condition: $ S_t = I_t $. This states that for the economy to be in equilibrium (or on a steady growth path in this context), planned savings must equal planned investment.

We want to find the condition under which this model predicts economic growth, which means income $Y_t$ increases over time.

Setting up the Equilibrium in Harrod's Model

To find the condition for economic growth, we start by equating the savings and investment equations, as required by the equilibrium condition $ S_t = I_t $:

$ \alpha Y_t = \beta (Y_t - Y_{t-1}) $

Deriving the Income Path Equation

Now, we need to rearrange this equation to understand how income $ Y_t $ relates to income in the previous period $ Y_{t-1} $. This will show us the path of income over time.

First, distribute $ \beta $ on the right side:

$ \alpha Y_t = \beta Y_t - \beta Y_{t-1} $

Next, move the $ Y_{t-1} $ term to the left side and the $ \alpha Y_t $ term to the right side to group terms involving $ Y_t $ and $ Y_{t-1} $:

$ \beta Y_{t-1} = \beta Y_t - \alpha Y_t $

Factor out $ Y_t $ from the terms on the right side:

$ \beta Y_{t-1} = (\beta - \alpha) Y_t $

Finally, express $ Y_t $ in terms of $ Y_{t-1} $:

$ Y_t = \frac{\beta}{\beta - \alpha} Y_{t-1} $

This equation is a first-order linear difference equation. It shows that income in the current period is a constant multiple of income in the previous period. If we start with some initial income $ Y_0 $, income in subsequent periods will be:

$ Y_1 = \frac{\beta}{\beta - \alpha} Y_0 $
$ Y_2 = \frac{\beta}{\beta - \alpha} Y_1 = \left(\frac{\beta}{\beta - \alpha}\right)^2 Y_0 $
In general, $ Y_t = Y_0 \left(\frac{\beta}{\beta - \alpha}\right)^t $

Condition for Economic Growth

Economic growth occurs if income $ Y_t $ increases over time. Looking at the solution $ Y_t = Y_0 \left(\frac{\beta}{\beta - \alpha}\right)^t $, assuming initial income $ Y_0 > 0 $, income will grow if the base of the exponential term is greater than 1. That is, the factor by which income is multiplied each period must be greater than 1.

The condition for economic growth is therefore:

$ \frac{\beta}{\beta - \alpha} > 1 $

For this inequality to hold and lead to sustained growth, the term $ \frac{\beta}{\beta - \alpha} $ must be positive. Since we are given $ \beta > 0 $, this implies that the denominator $ \beta - \alpha $ must also be positive ($ \beta - \alpha > 0 $), which means $ \beta > \alpha $. If $ \beta - \alpha < 0 $, the fraction would be negative, leading to oscillations or decline in income, not continuous growth.

Analyzing the Given Options

Let's examine each option in light of the condition $ \frac{\beta}{\beta - \alpha} > 1 $ derived from Harrod's growth model:

Option 1: $ \frac{\beta}{\beta-\alpha} > 1 $. This is exactly the condition we derived for income $ Y_t $ to grow over time based on the given equations in Harrod's model.
Option 2: $ \frac{\beta}{\beta-\alpha} < 0 $. Since $ \beta > 0 $, this inequality would require $ \beta - \alpha < 0 $. In this case, the growth factor $ \frac{\beta}{\beta - \alpha} $ is negative. A negative growth factor would cause income to change sign each period or decline towards zero, which does not represent economic growth.
Option 3: $ \frac{\beta}{\alpha} > 0 $. We are given $ \alpha > 0 $ and $ \beta > 0 $. The ratio of two positive numbers is always positive. So, this condition $ \frac{\beta}{\alpha} > 0 $ is always true under the given constraints. While necessary for meaningful parameters, it doesn't specify the relationship required for income to grow period after period in this model. The growth depends on $ \frac{\beta}{\beta-\alpha} $.
Option 4: $ \frac{\beta}{\alpha-\beta} > 0 $. This can be rewritten as $ \frac{\beta}{-(\beta-\alpha)} > 0 $. Since $ \beta > 0 $, this implies $ -(\beta-\alpha) > 0 $, which means $ \beta - \alpha < 0 $. This is the same situation as Option 2, where the growth factor $ \frac{\beta}{\beta - \alpha} $ would be negative, not leading to economic growth.

Conclusion on Harrod Growth Condition

Based on the derivation from the fundamental equations of Harrod's growth model, the condition for economic growth, where income increases over time, is $ \frac{\beta}{\beta - \alpha} > 1 $.

Revision Table: Key Concepts in Harrod's Model

Concept	Equation	Meaning
Savings Function	$ S_t = \alpha Y_t $	Savings are proportional to income, $ \alpha $ is the savings rate.
Investment Function	$ I_t = \beta (Y_t - Y_{t-1}) $	Investment is proportional to the change in income, $ \beta $ is the accelerator coefficient/capital-output ratio.
Equilibrium Condition	$ S_t = I_t $	Planned savings equal planned investment.
Income Path	$ Y_t = \frac{\beta}{\beta - \alpha} Y_{t-1} $	Shows how income changes from one period to the next.
Condition for Growth	$ \frac{\beta}{\beta - \alpha} > 1 $	The factor by which income grows each period must be greater than 1.

Additional Information: Harrod-Domar Model Insights

The framework used here is closely related to the Harrod-Domar model, which was one of the earliest attempts to provide a dynamic analysis of economic growth. In a slightly different formulation, the Harrod-Domar model defines three growth rates:

Actual Growth Rate ($G$): Determined by the savings rate ($s$) and the capital-output ratio ($\nu$). $ G = s / \nu $. In our context, $s$ is like $ \alpha $. The definition of $ \nu $ varies slightly; here $I_t = \beta \Delta Y_t$, suggesting $ \beta $ is related to $ \nu $. Our income growth factor is $ \frac{Y_t}{Y_{t-1}} = 1 + g $, where $g$ is the growth rate. We found $1+g = \frac{\beta}{\beta - \alpha}$, so $g = \frac{\beta}{\beta - \alpha} - 1 = \frac{\beta - (\beta - \alpha)}{\beta - \alpha} = \frac{\alpha}{\beta - \alpha}$. So the actual growth rate is $ \frac{\alpha}{\beta - \alpha} $. Growth occurs if $ g > 0 $, which means $ \frac{\alpha}{\beta - \alpha} > 0 $. Since $ \alpha > 0 $, this requires $ \beta - \alpha > 0 $, or $ \beta > \alpha $.
Warranted Growth Rate ($G_w$): The growth rate at which producers are satisfied with their investment, meaning demand is sufficient to absorb the increased output from that investment. It is given by $ G_w = s / \nu $, where $s$ is the savings rate and $ \nu $ is the desired capital-output ratio. In our context, this might be interpreted as the growth rate required for $ \frac{\beta}{\beta - \alpha} = 1 + G_w $ to hold, matching planned saving/investment decisions. The condition for warranted growth path stability requires actual growth to equal warranted growth ($ G = G_w $).
Natural Growth Rate ($G_n$): The maximum possible growth rate determined by the growth of the labor force and technological progress. $ G_n = n + k $, where $n$ is labor force growth and $k$ is technological progress rate.

Harrod highlighted the difficulty of achieving a steady state where $ G = G_w = G_n $ due to the independent nature of savings ($\alpha$), investment decisions ($\beta$), and population/technology growth. Deviations from this path can lead to instability, often referred to as Harrod's "knife-edge" problem.

The condition $ \frac{\beta}{\beta - \alpha} > 1 $ derived here is specifically the requirement for income to show positive growth over time in the given structure, corresponding to $ Y_t $ increasing relative to $ Y_{t-1} $.

Prepp Team · Answer

$\frac{\beta}{\beta-\alpha}<0$

Prepp Team · Answer

$\frac{\beta}{\alpha}>0$

Prepp Team · Answer

$\frac{\beta}{\alpha-\beta}>0$

(a) Marginalist Revolution	(i) Samuelson
(b) Multiplier-Accelerator model	(ii) J. R. Hicks
(c) IS-LM curves	(iii) Jevous
(d) Real Business Cycle	(iv) Robert J. Borro