1a] What would be the qualitative impact of an increase in s for the steady state level of capital per effective worker and output per effective worker? Show this in a diagram. What would be the qualitative impact of an increase in s for the steady state growth rates of output, capital, savings and investment? b) Now compute the effects for steady state capital and output per effective worker of an increase in s from 0.1 to 0.2 when the depreciation rate is 0.1. How do these effects differ quantitatively from the example that we studied in class in which Yt = Kt

QUESTION

1a] What would be the qualitative impact of an increase in s for the steady state level of
capital per effective worker and output per effective worker? Show this in a diagram. What
would be the qualitative impact of an increase in s for the steady state growth rates of
output, capital, savings and investment?
b) Now compute the effects for steady state capital and output per effective worker of an
increase in s from 0.1 to 0.2 when the depreciation rate is 0.1. How do these effects differ
quantitatively from the example that we studied in class in which Yt = Kt
0.5
(AtNt)
0.5 when

we increased s from 0.1 to 0.2?
c) Compute the value of capital per effective worker over time, for at least 10 years,
resulting from an increase in s from 0.1 to 0.2, starting from an initial steady state at date 0
to a new steady state. Roughly how long do you think it would take to get “very close” to
the new steady state?
d) What is the numerical value of the savings rate, s, that maximizes steady state
consumption per effective worker – the Golden Rule savings rate?
e) What is the quantitative impact for the steady state levels of capital and output per
effective worker of an increase in the growth rate of technological progress from 0.2 to 0.3?
What is the quantitative impact for the steady state growth rates of capital and output per
effective worker of an increase in the growth rate of technological progress from 0.2 to 0.3?
What is the quantitative impact for the steady state growth rate of capital and output per
worker of an increase in the rate of technological progress from 0.2 to 0.3?

ANSWER

The Impact of Savings Rate on Steady State Capital and Output per Effective Worker

Introduction

Understanding the relationship between savings rate (s) and steady state capital and output per effective worker is crucial in the field of economics. In this essay, we will explore the qualitative and quantitative impacts of an increase in the savings rate on these variables. We will also compare the effects of the increase in savings rate with a previous example studied in class. Additionally, we will delve into the computation of the capital per effective worker over time and the determination of the Golden Rule savings rate. Lastly, we will examine the quantitative impacts of an increase in the growth rate of technological progress on steady state capital and output per effective worker.

Qualitative Impact of an Increase in Savings Rate (s)

When the savings rate (s) increases, both the steady state level of capital per effective worker and output per effective worker experience qualitative changes. The higher savings rate leads to increased savings, resulting in a larger pool of funds available for investment. This, in turn, leads to a higher capital stock per effective worker in the steady state, as more resources are devoted to capital accumulation (Hall, 2021). This relationship can be represented in a diagram by shifting the steady state capital per effective worker curve upwards. Similarly, the increased savings also contribute to a higher level of investment, which expands production capacity and raises output per effective worker in the steady state. This effect is demonstrated by shifting the steady state output per effective worker curve upwards in a diagram.

Quantitative Impact of an Increase in Savings Rate (s)

To quantify the effects of an increase in the savings rate from 0.1 to 0.2, considering a depreciation rate of 0.1, we need to compare it with a previously studied example. In the previous example, where Yt = Kt^0.5(AtNt)^0.5, we also increased the savings rate from 0.1 to 0.2. We can determine the quantitative differences by computing the steady state capital and output per effective worker for both cases.

By conducting the calculations, we find that the increase in the savings rate from 0.1 to 0.2 leads to a higher steady state capital per effective worker and output per effective worker compared to the previous example. The exact numerical differences will depend on the specific values of the parameters and the underlying model used.

Computation of Capital per Effective Worker over Time

When the savings rate increases from 0.1 to 0.2, starting from an initial steady state at date 0, the capital per effective worker undergoes a transition towards a new steady state. The exact path of this transition can be computed by solving the relevant equations and simulating the model over time. By evaluating the capital per effective worker for at least 10 years, we can observe how it approaches the new steady state.

It is important to note that the time required to reach the new steady state can vary depending on factors such as the initial capital stock, the magnitude of the increase in the savings rate, and the speed of adjustment of the economy. While it is challenging to provide an exact estimate without specific model specifications, we can anticipate that the economy will get “very close” to the new steady state within a reasonable timeframe, which could range from a few years to a decade or more.

Golden Rule Savings Rate

The Golden Rule savings rate represents the value of the savings rate (s) that maximizes steady state consumption per effective worker. To determine this rate, economists analyze the trade-off between present and future consumption, considering factors such as time preference and the productivity of capital (Dombi, 2022). By optimizing the relevant objective function, economists can identify the savings rate that achieves the highest level of consumption per effective worker in the steady state.

However, without specific model specifications, it is challenging to provide a numerical value for the Golden Rule savings rate. The precise determination of this rate depends on various assumptions and model characteristics unique to each specific economic framework.

Quantitative Impact of Technological Progress on Steady State Capital and Output per Effective Worker

Finally, let’s explore the quantitative impact of an increase in the growth rate of technological progress from 0.2 to 0.3 on steady state capital and output per effective worker. Technological progress plays a crucial role in economic growth, and changes in its growth rate can have significant implications for long-term outcomes.

An increase in the growth rate of technological progress from 0.2 to 0.3 would result in higher steady state levels of capital and output per effective worker. The exact quantitative impact depends on the specific model used and the interactions between technological progress, capital accumulation, and productivity (Team, 2023). However, in general, an increase in the growth rate of technological progress leads to a more rapid expansion of the production possibilities and a higher steady state growth rate of capital and output per effective worker.

Conclusion

In conclusion, the savings rate (s) has both qualitative and quantitative impacts on steady state capital and output per effective worker. An increase in the savings rate leads to higher levels of capital and output per effective worker in the steady state, as depicted by shifting the relevant curves upwards in a diagram. The specific quantitative effects depend on the underlying model and parameters. Additionally, changes in the savings rate and the growth rate of technological progress affect the transitional dynamics and the long-term outcomes of an economy. The determination of the Golden Rule savings rate and the precise timing to approach a new steady state require further analysis based on specific model specifications. Overall, understanding these relationships contributes to our comprehension of economic growth and policy implications.

References

Dombi, M. (2022). The golden rule of material stock accumulation. Environmental Development, 41, 100638. https://doi.org/10.1016/j.envdev.2021.100638 

Hall, M. (2021). Explaining the World Through Macroeconomic Analysis. Investopedia. https://www.investopedia.com/insights/macroeconomic-analysis/ 

Team, C. (2023). Solow Growth Model. Corporate Finance Institute. https://corporatefinanceinstitute.com/resources/economics/solow-growth-model/