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Can you answer the math problems in this document?

The document includes several economics-based problems centered on analyzing productivity, factor accumulation, and growth rates using production functions and comparative analysis. Here is a breakdown of the problems and suggested solutions:

### Problem Summaries and Solution Approach

1. **Drawing Comparative Diagrams**:
   - **Problem**: Sketch diagrams for cases where Country 1 has different levels of productivity and factors of production relative to Country 2.
   - **Solution**: You would need to represent output per worker and productivity levels graphically, showing hypothetical axes where output per worker is on the vertical axis and factors of production on the horizontal axis. Two cases are required:
     - Case (a): Higher productivity but lower factors of production for Country 1.
     - Case (b): Higher output per worker and factors of production but lower productivity for Country 1.

2. **Calculating Productivity for Sylvania and Freedonia**:
   - **Problem**: Given the production function \( y = A k^{0.5} h^{0.5} \), calculate productivity \( A \), and compare outputs under productivity-only and factor-only differences.
   - **Solution**: Plug in values for output per worker (\( y \)), physical capital per worker (\( k \)), and human capital per worker (\( h \)) to solve for \( A \). Then, use these productivity levels to compare outputs under hypothetical scenarios where differences arise solely from productivity or from factor accumulation.

3. **Productivity Growth Comparison for Countries A and B**:
   - **Problem**: Compare productivity growth between 2000 and 2010 for two countries using the provided production function.
   - **Solution**: Calculate productivity \( A \) for each year by rearranging the production function, then analyze the growth rates by comparing the calculated productivity values for each country across the two years.

4. **Factor Accumulation and Productivity Relative to the United States**:
   - **Problem**: For three countries, calculate their factor accumulation and productivity relative to the United States.
   - **Solution**: Using the ratios of physical and human capital per worker relative to the U.S., evaluate factor accumulation and productivity. Determine which factor (factor accumulation or productivity) plays the largest role in each country's income relative to the U.S.

5. **Growth Rates of Productivity and Factor Accumulation**:
   - **Problem**: Calculate productivity and factor accumulation growth rates for three countries.
   - **Solution**: Using the given growth rates for output, physical capital, and human capital, decompose the output growth into contributions from factor accumulation and productivity growth. Compare the results to identify which factor primarily drives growth in each country.

Would you like a detailed step-by-step calculation for each problem, or help with specific questions? 

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### Additional Questions to Explore
1. How does productivity influence output per worker when factors of production are equal across countries?
2. How would changes in physical capital per worker affect productivity, assuming other variables are constant?
3. Can variations in human capital alone significantly impact relative income levels between countries?
4. How can growth in productivity be sustained in an economy with stagnating factor accumulation?
5. What policies could a country implement to improve its productivity relative to others?

### Tip:
For productivity-focused calculations, always isolate productivity \( A \) in the production function to simplify comparisons across different scenarios.

Learn to solve economic growth problems involving productivity, factor accumulation, and growth rate comparisons using production functions. This resource breaks down complex issues like calculating productivity for fictional countries, comparing growth rates, and understanding output variations due to different factors.

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