Economics Dynamics Problems 52

# Economics Dynamics Problems 52 - To verify this deFne the...

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36 Economic Dynamics with initial conditions k (0) = K 0 L 0 = k 0 We cannot solve equation (2.11) because the production function is not explicitly deFned. Suppose we assume that the production function F ( K , L ) conforms to a Cobb–Douglas, i.e., we assume Y = aK α L 1 α 0 <α< 1 Y L = a ± K L ² α or y = f ( k ) = ak α (2.12) In this instance the capital/labour ratio grows according to ˙ k = sak α ( n + δ ) k (2.13) The Solow growth model with a Cobb–Douglas production function therefore conforms to the following differential equation ˙ k + ( n + δ ) k = sak α This is a Bernoulli equation, 2 and can accordingly be solved by performing a transformation that results in a linear differential equation that is readily solvable. Given such a solution, then a solution can be found for the original variable.
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Unformatted text preview: To verify this, deFne the following transformation: v = k 1 − α . . . dv dt = (1 − α ) k − α dk dt or ˙ k = k α (1 − α ) dv dt Using these results we can derive the following k − α ˙ k + ( n + δ ) kk − α = sa k − α ˙ k + ( n + δ ) k 1 − α = sa ± k − α k α 1 − α ² dv dt + ( n + δ ) v = sa i.e. dv dt + (1 − α )( n + δ ) v = (1 − α ) sa which is a linear differential equation in v with solution v ( t ) = as n + δ + ± v − as n + δ ² e − (1 − α )( n + δ ) t 2 A Bernoulli equation takes the general form dy dt + f ( t ) x = h ( t ) x α See Giordano and Weir (1991, pp. 95–6)....
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