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In Solow's "A Contribution to the Theory of Economic Growth" paper, how do we go from this equation

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to

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such that

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Could you please give me a hint about that derivation?

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1 Answer 1

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We have that: $$ \dot K = s K^\alpha L_0^b e^{nbt} $$ Rewriting the differential equation gives: $$ K^{-\alpha} \frac{dK}{dt} = s L_0^b e^{nbt} $$ Integrate both sides with respect to $t$ from $0$ to $T$ gives: $$ \frac{1}{b} [K^b]^T_0 = s L_0^b \frac{1}{nb}[e^{nbt}]^T_0 $$ So: $$ K^b_T = K^b_0 - \frac{s}{n} L_0^b + \frac{s}{n} L_0^b e^{nbT} $$ Equivalently: $$ K_T = \left[K^b_0 - \frac{s}{n} L_0^b + \frac{s}{n} L_0^b e^{nbT}\right]^{\frac{1}{b}} $$

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