How to Calculate the productivity multiplier?
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Given a Cobb Douglas $Y_t = A (K_t^alpha L_t^{1-alpha}) $ $ K_{t+1} = sY_t + (1-delta) K_t$ How do we get the multiplier on productivity to be equal to $ frac{1}{1-alpha}$ ? I understand that if productivity increases, output increases, thus we get more capital and thereby more output and so on. But I can't reach this multiplier. My Attempt: If have x increase in A, then $Y_t=(1+x)A (K_t^alpha L_t^{1-alpha}) $ That is an x increase in Y. $ K_{t+1} = s(1+x)A (K_t^alpha L_t^{1-alpha}) + (1-delta) K_t$ $Y_{t+1}=(1+x)A * K_{t+1}^alpha *L_{t+1}^{1-alpha} $ $Y_{t+1}= (1+x)A * (s(1+x)A (K_t^alpha L_t^{1-alpha}) + (1-delta) K_t)^alpha * L_{t+1}^{1-alpha} $ I think the increase here should be $x * alpha $ but I can't see it. so that for a unit increase in productivity i....