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Consider Solow growth model $Y=F(K,AL)$ where $A$ measures knowledge progress, $K$ is capital and $L$ is label. I have suppressed temporal dependence. I am going to assume $L$, $A$ grow exponentially at rate $g,n$ respectively. Assume $\frac{d}{dt}K=sY-\delta K$ where $\delta$ is decay rate of capital and $s$ is saving rate. Assume saving is instantaneously used for capital purchasing as investment.

Let $k^\star$ be the current equilibrium. Leveling up consumption indicates $(1-s)Y$ increases which indicates saving rate should drop. However, dropping saving rate should drop overall $\frac{Y}{AL}$ effective production per capita. Thus overall $Y$ drops below previous equilibrium growth path. If I set $a_K$ as elasticity of product to capital, then elasticity of $Y$ to saving should be $\frac{a_K}{1-a_K}$. In general most countries are with $a_K=\frac{1}{3}$. In particular elasticity of $Y$ to saving is positive. In particular, $Y$ should grows and decays as saving grows or decays. Thus decreasing saving should decrease long term growth of $Y$.

Originally, I thought $Y$ should measure GDP which should have a component measuring consumption. If consumption increases, I should expect temporal increment of GDP or increment of GDP over long term.(That should be the reason why one wants to increase consumption during recession.) Or is that statement is a statement for out of equilibrium statement? Am I wrong on that part or wrong with understanding of Solow model?

Reference. Romer, Macroeconomics Chapter 1. Sec 4.

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