WWeckesser_SeparableExample_SolowGrowth

# WWeckesser_SeparableExample_SolowGrowth - 38 4 APPLICATIONS...

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

38 4. APPLICATIONS OF DIFFERENTIAL EQUATIONS 4.1. Solow’s 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 = λ L, (4.4) where λ > 0 is the per capita growth rate. This is a first order equation for L which we can solve to find L = L 0 e λ 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 K ( t ). Thus this is a

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern