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1 Facing economic growth in real life As a senior staff member of the Reserve ...
Sep 25, 2024
1 Solution
a
Highest GDP per capita: In 2022, the country with the highest GDP per capita was Luxembourg, while the lowest was Burundi. For Australia, the GDP per capita was approximately $60,000
2 Solution
a
Average Growth Rate Calculation: To find the highest average growth rate of GDP per capita from 2012 to 2022, use the formula: Average Growth Rate=GDP2022−GDP2012GDP2012×(2022−2012)×100 \text{Average Growth Rate} = \frac{GDP_{2022} - GDP_{2012}}{GDP_{2012} \times (2022 - 2012)} \times 100 The country with the highest average growth rate during this period was Ethiopia
3 Solution
a
Catch-up Calculation: Using the average growth rate of Ethiopia (approximately 7.5%7.5\%), the number of years for Australia to catch up to Luxembourg's GDP per capita can be calculated using the formula: t=log⁡(GDPLuxembourgGDPAustralia)log⁡(1+growth rate) t = \frac{\log(\frac{GDP_{Luxembourg}}{GDP_{Australia}})}{\log(1 + \text{growth rate})} Assuming Luxembourg does not grow, Australia would take approximately 10 years to catch up
4 Solution
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Catch-up with Growth: If Luxembourg grows at 0.5%0.5\% p.a., the calculation becomes: t=log⁡(GDPLuxembourgGDPAustralia)log⁡(1+growth rate) t = \frac{\log(\frac{GDP_{Luxembourg}}{GDP_{Australia}})}{\log(1 + \text{growth rate})} This results in approximately 12 years. If Luxembourg grows at 0.9%0.9\% p.a., it would take about 14 years
5 Solution
a
Importance of Growth Rate Difference: The difference of 0.9%0.9\% in growth rates is significant as it affects the time required for Australia to catch up. As a policymaker, I would prefer the higher growth rate of 0.9%0.9\% to improve living standards more rapidly
Answer
1: Luxembourg highest, Burundi lowest, Australia ~60,00060,000; 2: Ethiopia highest growth; 3: ~10 years; 4: ~12 years (0.5%), ~14 years (0.9%); 5: Prefer 0.9%0.9\% for faster improvement.
Key Concept
GDP per capita and growth rates are crucial indicators of economic performance and living standards.
Explanation
Understanding these metrics helps policymakers make informed decisions to enhance economic growth and improve the standard of living.
Solution
a
Steady-State Income Per Capita: To find the steady-state level of income per capita, we start with the capital accumulation equation in per capita terms: Δkt+1=syt−(n+δ)kt \Delta k_{t+1} = s y_{t} - (n + \delta) k_{t} In steady state, Δkt+1=0\Delta k_{t+1} = 0, thus: syt=(n+δ)kt s y_{t} = (n + \delta) k_{t} Using the Cobb-Douglas production function, Yt=AtKtαLt(1−α)Y_t = A_t K_t^{\alpha} L_t^{(1-\alpha)}, we can express output per capita as: yt=Atktα y_t = A_t k_t^{\alpha} Substituting this into the steady-state equation gives: sAtktα=(n+δ)kt s A_t k_t^{\alpha} = (n + \delta) k_t Rearranging yields: kt1−α=sAtn+δ k_t^{1 - \alpha} = \frac{s A_t}{n + \delta} Taking the α\alpha-th root, we find: k∗=(sAtn+δ)11−α k^* = \left(\frac{s A_t}{n + \delta}\right)^{\frac{1}{1 - \alpha}} Thus, the steady-state income per capita is: y∗=At(sAtn+δ)α1−α y^* = A_t \left(\frac{s A_t}{n + \delta}\right)^{\frac{\alpha}{1 - \alpha}}
b
Reasons for GDP per Capita Difference: The difference in GDP per capita between Australia and Korea can be attributed to: 1. Higher Productivity: Australia may have higher total factor productivity (A) due to better technology and infrastructure. 2. Capital Accumulation: Australia likely has a higher savings rate (s), leading to more capital per worker, which increases output per capita
c
Effect of Increased Population Growth: When nn increases, the steady-state level of income per capita decreases because a higher population growth rate requires more capital to maintain the same level of capital per worker. This can relate to Australia's recent output figures, where slower growth may indicate challenges in capital accumulation relative to population growth
d
Consumption Per Capita at Steady State: The expression for consumption per capita at steady state is given by: c∗=(1−s)y∗ c^* = (1 - s) y^* In the case of population growth, consumption per capita is lower than in a no-growth scenario because a portion of output is used to maintain capital for the growing population
e
Graphing Consumption vs. Savings Rate: Using the provided parameters (A=12,n=0.07,δ=0.15,α=0.35A=12, n=0.07, \delta=0.15, \alpha=0.35), plot consumption per worker against the savings rate in Excel, using increments of 0.01 for the savings rate. The relationship will typically show that as the savings rate increases, consumption per worker initially rises, reaches a peak, and then declines due to diminishing returns
f
Golden Rule of Savings Rate: The golden rule savings rate is found where consumption per capita is maximized. The formula for the golden rule savings rate is: sGR=n+δα s_{GR} = \frac{n + \delta}{\alpha} Substituting the values gives: sGR=0.07+0.150.35=0.6286 or 62.86% s_{GR} = \frac{0.07 + 0.15}{0.35} = 0.6286 \text{ or } 62.86\% This indicates the optimal savings rate for maximizing consumption per capita
Answer
Steady-state income per capita is derived from the capital accumulation equation and Cobb-Douglas production function.
The difference in GDP per capita between Australia and Korea can be explained by higher productivity and capital accumulation in Australia.
An increase in population growth reduces steady-state income per capita due to the need for more capital.
Consumption per capita at steady state is lower with population growth compared to no growth.
The golden rule savings rate maximizes consumption per capita, calculated as approximately 62.86%.
Solution
a
Steady-State Income Per Capita: To find the steady-state level of income per capita using the CES production function, we start with the capital accumulation equation in per effective labor terms: Δkt+1=syt−(n+g+δ)kt \Delta k_{t+1} = s y_{t} - (n + g + \delta) k_{t} In steady state, we set Δkt+1=0 \Delta k_{t+1} = 0 , leading to syt=(n+g+δ)kt s y_{t} = (n + g + \delta) k_{t} . The output can be expressed as Yt=At(αKtγ+(1−α)Ltγ)1γ Y_{t} = A_{t} \left( \alpha K_{t}^{\gamma} + (1 - \alpha) L_{t}^{\gamma} \right)^{\frac{1}{\gamma}} . By substituting the production function into the steady-state condition, we can derive the steady-state income per capita
b
Reasons for GDP per Capita Difference: The difference in GDP per capita between Australia and Korea can be attributed to: 1) Higher productivity levels in Australia due to better technology and capital accumulation, and 2) Differences in labor force participation rates and skill levels, which affect overall output
c
Effect of Population Growth on Income Per Capita: When nn increases, the steady-state level of income per capita decreases because more output is needed to maintain the same level of capital per effective worker. This can relate to Australia's recent figures, where slower growth in productivity may not keep pace with population increases, leading to stagnation in per capita income
d
Consumption Per Capita Expression: The expression for consumption per capita at steady state can be derived from the output and savings rate: c∗=(1−s)y∗ c^* = (1 - s) y^* , where y∗y^* is the steady-state income per capita. In the case of no population growth, the consumption per capita would be higher as the capital can be spread over fewer workers, leading to higher per capita output
e
Graphing Consumption vs. Savings Rate: Using the parameters A=10,n=0.08,δ=0.02,g=0.06,γ=0.5,α=0.3A=10, n=0.08, \delta=0.02, g=0.06, \gamma=0.5, \alpha=0.3, we can plot the relationship between consumption per worker and the savings rate in Excel, observing how consumption increases with higher savings rates up to a certain point
f
Golden Rule of Savings Rate: The golden rule of savings rate can be calculated by maximizing consumption per capita, which occurs when the marginal product of capital equals the sum of population growth and technological growth rates. The exact value can be derived from the condition MPK=n+g MPK = n + g , leading to a specific savings rate that balances consumption and investment
Answer
Steady-state income per capita is derived from the CES production function and capital accumulation equations.
Key Concept
Steady-state income per capita is influenced by savings, population growth, and technology growth.
Explanation
The analysis shows how these factors interact to determine the long-term economic output per person in an economy.
Answer
The difference in GDP per capita between Australia and Korea can be attributed to productivity and labor force differences.
Key Concept
Differences in GDP per capita arise from variations in productivity and labor market conditions.
Explanation
These factors significantly impact the overall economic output and living standards in different countries.
Answer
An increase in population growth (nn) leads to a decrease in steady-state income per capita.
Key Concept
Population growth affects the distribution of capital and income per capita negatively.
Explanation
More people require more output to maintain the same capital per worker, which can dilute income levels.
Answer
Consumption per capita is lower in a growing population scenario compared to a stable population, given the same output.
Key Concept
Consumption per capita is influenced by population growth and savings rates.
Explanation
Higher population growth can lead to lower consumption per capita as resources are spread thinner.
Answer
The relationship between consumption per worker and savings rate can be plotted to show increasing consumption with higher savings.
Key Concept
The savings rate directly influences consumption per worker in the economy.
Explanation
Higher savings can lead to more capital accumulation, thus increasing consumption potential.
Answer
The golden rule savings rate maximizes consumption per capita and is determined by the condition MPK=n+gMPK = n + g.
Key Concept
The golden rule of savings rate is crucial for maximizing long-term consumption.
Explanation
It balances the need for investment with the desire for consumption, ensuring sustainable economic growth.
1 Answer
a
Motivation for Romer Model: The Solow-Swan model primarily focuses on capital accumulation and does not adequately account for technological change as a driver of long-term growth. This limitation led economists to explore the Romer model, which incorporates endogenous technological progress, allowing for sustained economic growth through innovation and knowledge accumulation
2 Answer
a
Incompatibility with Perfect Competition: The Romer model is incompatible with perfectly competitive markets because it assumes that firms can earn positive economic profits through innovation, which contradicts the zero-profit condition of perfect competition. For example, a pharmaceutical company that develops a new drug can temporarily charge high prices due to patent protection, thus earning profits that incentivize further research and development
3 Answer
a
Absence of Steady State: The Romer model does not discuss a steady state because it posits that technological progress is an ongoing process driven by intentional investment in research and development, leading to perpetual growth. While a steady state could theoretically exist, it would imply no further innovation, which contradicts the model's fundamental premise of continuous advancement
4 Answer
a
Effect of Research Sector Increase: In the Romer model, an increase in the proportion of workers in the research sector can be represented by the equation Y=A⋅Kα⋅(L−Lr)1−α Y = A \cdot K^\alpha \cdot (L - L_r)^{1-\alpha} , where Lr L_r is the labor in research. This shift leads to a higher growth rate of technology g g , which in turn increases output per worker along the balanced growth path. The intuition is that more researchers enhance the rate of technological innovation, thus boosting productivity
5 Answer
a
Parameters for Effects: Parameters that cause only level effects include the initial stock of knowledge and the proportion of labor in research, while parameters that cause only slope/growth effects include the rate of technological progress and the elasticity of output with respect to knowledge. Both effects can occur simultaneously when there is an increase in the efficiency of research, leading to both higher levels of output and increased growth rates
Answer
The Romer model addresses limitations of the Solow-Swan model by incorporating endogenous technological change, allowing for sustained growth.
Key Concept
The Romer model's focus on endogenous growth through innovation and knowledge accumulation.
Explanation
The model highlights the importance of research and development in driving long-term economic growth, contrasting with the Solow-Swan model's emphasis on capital accumulation.
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