3. Consider the following form for the production function of an economy \[ Q = F(K, L, t) \] where \( Q, K, L \) and \( t \) denote output, capital, labour and time respectively. If \( K \) and \( L \) depend only on \( t \), find an expression for \( dQ/dt \) in terms of the partial derivatives of the production function and the time derivatives of \( K \) and \( L \). Now suppose that \( K \) and \( L \) have constant proportionate rates of growth \( m \) and \( n \) respectively. Find the rate of growth of output when the production function takes the following form: \[ Q = e^{rt} K^\alpha L^\beta \quad (r, \alpha, \beta > 0). \]
3. Consider the following form for the production function of an economy \[ Q = F(K, L, t) \] where \( Q, K, L \) and \( t \) denote output, capital, labour and time respectively. If \( K \) and \( L \) depend only on \( t \), find an expression for \( dQ/dt \) in terms of the partial derivatives of the production function and the time derivatives of \( K \) and \( L \). Now suppose that \( K \) and \( L \) have constant proportionate rates of growth \( m \) and \( n \) respectively. Find the rate of growth of output when the production function takes the following form: \[ Q = e^{rt} K^\alpha L^\beta \quad (r, \alpha, \beta > 0). \]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![3. Consider the following form for the production function of an economy
\[ Q = F(K, L, t) \]
where \( Q, K, L \) and \( t \) denote output, capital, labour and time respectively.
If \( K \) and \( L \) depend only on \( t \), find an expression for \( dQ/dt \) in terms of the partial derivatives of the production function and the time derivatives of \( K \) and \( L \).
Now suppose that \( K \) and \( L \) have constant proportionate rates of growth \( m \) and \( n \) respectively. Find the rate of growth of output when the production function takes the following form:
\[ Q = e^{rt} K^\alpha L^\beta \quad (r, \alpha, \beta > 0). \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffba66248-862a-43fb-91b6-16cc301cecb0%2F1fdb728d-c4bf-4d55-9ceb-c5f0844b3ea8%2F36gwgu_processed.png&w=3840&q=75)
Transcribed Image Text:3. Consider the following form for the production function of an economy
\[ Q = F(K, L, t) \]
where \( Q, K, L \) and \( t \) denote output, capital, labour and time respectively.
If \( K \) and \( L \) depend only on \( t \), find an expression for \( dQ/dt \) in terms of the partial derivatives of the production function and the time derivatives of \( K \) and \( L \).
Now suppose that \( K \) and \( L \) have constant proportionate rates of growth \( m \) and \( n \) respectively. Find the rate of growth of output when the production function takes the following form:
\[ Q = e^{rt} K^\alpha L^\beta \quad (r, \alpha, \beta > 0). \]
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