# In Solow growth model, increasing consumption drops production which should measure GDP?

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.