Math Problem Statement
Suppose that the production function f(k) = 2k and A =1 so that the growth of the labor-augmenting technology (A/A) = 0. What is the growth rate of (k.)/(k)?
Solution
To find the growth rate of , let's break down the equation given in the image and use the information provided.
Given Information
The equation for capital accumulation per effective worker is:
where:
- ,
- ,
- ,
- ,
- .
However, you've specified a modified production function:
with and a zero growth rate for labor-augmenting technology .
This simplifies the growth rate analysis because we’re dealing with and not .
Steps to Solve for
Given the production function , we can substitute this into the capital accumulation equation without considering the adjustments for effective labor (since is constant and does not grow). Therefore, the equation for becomes:
Substitute :
Expanding:
Now, divide both sides by to find :
Substitute the Values
Using the given values:
- ,
- ,
- .
Calculate each term:
- ,
- ,
- .
Final Answer
The growth rate of is 0.28 or 28%.
Would you like additional details on this calculation or further analysis?
Related Questions:
- How does the growth rate change if the depreciation rate changes?
- What impact does the savings rate have on ?
- How would the solution differ if were growing at a positive rate?
- What does this growth rate imply for long-term capital accumulation?
- How would labor growth affect the growth rate ?
Tip
Understanding how each parameter affects the growth rate can help in economic modeling, especially in long-term growth predictions.
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Math Problem Analysis
Mathematical Concepts
Economic Growth Models
Differential Equations
Capital Accumulation
Steady-State Analysis
Formulas
k̂(t) = s f(k̂) - (n + δ + g) k̂
f(k) = 2k
Growth rate: (k̇ / k) = 2s - n - δ
Theorems
Solow Growth Model
Capital Accumulation Equation
Suitable Grade Level
Undergraduate (Economics or Mathematics)
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