Economics Dynamics Problems 51

Economics Dynamics Problems 51 - I = ˙ K δ K S = sY ˙ K...

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Continuous dynamic systems 35 Y = F ( K , L ), which is twice differentiable and homogeneous of degree one (i.e. constant returns to scale). Let k = K / L denote the capital/labour ratio and y = Y / L the output/labour ratio. Then Y L = F ( K , L ) L = F ± K L , 1 ² = F ( k , 1) = f ( k ) i.e. y = f ( k ) with f (0) = 0 , f ± ( k ) > 0 , f ±± ( k ) < 0 , k > 0 We make two further assumptions: 1. The labour force grows at a constant rate n , and is independent of any economic variables in the system. Hence ˙ L = nL L (0) = L 0 2. Savings is undertaken as a constant fraction of output ( S = sY ) and savings equal investment, which is simply the change in the capital stock plus replacement investment, hence
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Unformatted text preview: I = ˙ K + δ K S = sY ˙ K + δ K = sY K (0) = K Now differentiate the variable k with respect to time, i.e., derive dk / dt , dk dt = ˙ k = L dK dt − K dL dt L 2 ˙ k = ± 1 L ² dK dt − ± K L ²± 1 L ² dL dt = ± K L ²± 1 K ² dK dt − ± K L ²± 1 L ² dL dt = k ± ˙ K K − ˙ L L ² But ˙ K K = sY − δ K K = sY L ± L K ² − δ = sf ( k ) k − δ and ˙ L L = nL L = n Hence ˙ k = sf ( k ) − δ k − nk = sf ( k ) − ( n + δ ) k (2.11)...
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