WWeckesser_SeparableExample_SolowGrowth

WWeckesser_SeparableExample_SolowGrowth - 38 4 APPLICATIONS...

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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
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