Chapter 5: The Solow Growth Model - Hikaru Saijo University ✓ Solved

Chapter 5: The Solow Growth Model Hikaru Saijo University

By what proportion does per capita output change in the long run in response to the following changes? 1 Saving rate decreases by 10%. 2 The productivity level falls by 20%. 3 The capital stock increases by 50% as a result of foreign investment.

Paper For Above Instructions

The Solow Growth Model is a fundamental framework in economic theory that analyzes long-term economic growth by examining the interaction of capital accumulation, labor or population growth, and increases in productivity, typically understood through technology advancements.

The key element of the Solow model is its assumption that economies can eventually reach a steady state where capital per worker and output per worker become constant. This steady state reflects the underlying balance between investment (savings) and depreciation of capital. In the steady state, economic growth is driven by technological progress rather than capital accumulation, making it essential to study how changes in specific parameters, such as the savings rate, productivity level, and capital stock, influence long-term output.

This paper will investigate how per capita output alters in the long run due to (1) a decrease in the savings rate by 10%, (2) a 20% decline in productivity levels, and (3) a 50% increase in the capital stock due to foreign investment.

1. Effect of a 10% Decrease in the Savings Rate

A reduction in the savings rate directly impacts how much of the output is invested to replace depreciated capital. According to the Solow model, if we denote the savings rate as s, a decrease of 10% would mean moving to a new savings rate of 0.9s. The formula for steady-state capital per effective worker is given by:

$K_{<i></i>}^{\infty }=\frac{s}{d}$ where K is the steady-state capital, s is the savings rate, and d is the depreciation rate.

With a lower savings rate, the steady state will now yield a lower capital stock per worker. The new steady state can be calculated with the new savings rate:

Let’s assume the original savings rate is s = 0.2 and depreciation rate d = 0.1:

K = (0.18 / 0.1) = 1.8 is the new steady-state capital per worker, compared to K = (0.2 / 0.1) = 2 originally.

From the Solow equation, output per person (or per capita) is directly correlated with capital per person, so the output will also be affected by the change in capital per person. Any reduction in capital per worker leads to a fall in per capita output theoretically by the same proportion. Therefore, in the long run, a 10% decrease in savings rate would result in a decrease of capital and output per person, hence a proportion decrease in per capita output in the model.

2. Effect of a 20% Decrease in Productivity Level

In the Solow model, productivity level (often denoted as A) significantly influences the output of an economy. A decline in productivity means that each unit of capital and labor generates less output. If productivity decreases by 20%, it can be represented as:

A = 0.8A0.

The steady-state level of output per effective worker (Y) can be modeled as:

Y = AK^αL^β where α + β = 1 in the context of the Cobb-Douglas production function.

With a lower productivity parameter, equilibrium output drops proportionally, thereby reducing the output per capita. Assuming the previous state of output was Y₀, the new output Y would equal:

Y = 0.8Y₀.

This calculation indicates that a 20% decline in productivity results in an equivalent decrease in output per person over the long run.

3. Effect of a 50% Increase in Capital Stock due to Foreign Investment

An increase in the capital stock by 50% enhances the capacity of the economy to produce goods and services. This adjustment is important since it shifts the capital-per-worker ratio higher, assuming labor remains constant. This increment can be introduced into the existing model’s formula. Let’s denote K as the original capital stock, then a 50% increase leads to:

K = 1.5K0.

In terms of output, the relationship notably implies an increase as follows:

Y = A(1.5K)^αL^β = 1.5^α * AK^αL^β.

The output will experience a boost due to the exponential nature of capital's contribution, given that alpha (α) typically ranges between 0 and 1.

Consequently, this increase in capital stock will positively influence the steady-state equilibrium, pushing total output per capita up considerably depending on the value of α.

Conclusion

In summary, the adjustments in parameters of the Solow growth model greatly influence per capita output in the long run. A 10% drop in the savings rate or a 20% reduction in productivity significantly decreases per capita output, while a 50% increase in capital stock enhances it appreciably. Each of these examples illustrates the model's utility in understanding economic responses to policy or external changes, emphasizing the role of investment and productivity as major determinants of economic growth. Further empirical studies are essential to highlight the intricacies of these relationships in real-world economics.

References

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