WWeckesser_SeparableExample_SolowGrowth

WWeckesser_SeparableExample_SolowGrowth - 38 4....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 38 4. APPLICATIONS OF DIFFERENTIAL EQUATIONS 4.1. Solows Economic Growth Model (Draft version 1 .) We consider a model from macroeconomics. Let K be the capital, 2 L the labor, and Q the production output of an economy. We are interested in a dynamic problem, so K ( t ), L ( t ) and Q ( t ) are all functions of time, but we will suppress the t argument. In elementary economics, one learns that a common assumption is that Q can be expressed as function of K and L : Q = f ( K, L ) . (4.1) We assume that f has, using economics terminology, constant returns to scale . Mathematically, this means that multiplying K and L by the same amount results in Q being multiplied by the same amount. That is, for any constant b , f ( bK, bL ) = bf ( K, L ) . (4.2) For example, the Cobb-Douglas function f ( K, L ) = K 1 / 3 L 2 / 3 satisfies this assump- tion. We make two more assumptions. We assume that a constant proportion of Q is invested in capital. This means that the rate of change of K is proportional to Q : dK dt = sQ, (4.3) where s > 0 is the proportionality constant. We also assume that the labor force is growing according to the equation dL dt = L, (4.4) where > 0 is the per capita growth rate. This is a first order equation for L which we can solve to find L = L e t . If possible, we would like to combine (4.1), (4.3), and (4.4) into a single equation that we may easily analyze. A natural first attempt is to substitute (4.1) into (4.3) to obtain dK dt = sf ( K, L ) (4.5) Since L ( t ) is a known function, the only unknown function is...
View Full Document

This note was uploaded on 12/06/2010 for the course ECON 3020 taught by Professor Williamson during the Spring '10 term at FSU.

Page1 / 4

WWeckesser_SeparableExample_SolowGrowth - 38 4....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online