# Solow model - max steady state output per worker vs max steady state consumption per worker

$$Y=K^{1/3}G^{2/9}(AL)^{4/9}\\\frac{\dot{A}}{A}=\frac{\dot{L}}{L}=0.01\\\dot{G}=tY-0.03G\\\dot{K}=0.2(1-t)Y-0.03K$$

So far, we have a system that shows a basic Solow model with public and private capital and a tax rate to fund the public capital (t). The 0.2 corresponds to the saving rate by consumers in the economy.

It can be shown that steady state output per worker is:

$$\left(\frac{Y}{L}\right)^* = A*(4(1-t))^{3/4}(20t)^{1/2}$$

And steady state consumption per worker is:

$$\left(\frac{C}{L}\right)^* = \frac{(0.8*4^{3/4}*20^{1/2})A*(1-t)^{7/4}t^{1/2}}{L}$$

You can show that the t that maximises SS output per worker is 0.4, whilst to max SS consumption per worker, the t is 2/9.

I don't understand why these tax rates are different. My question is why are they different?

If the gov't switch from t = 2/9 to t = 0.4 to go from maximising consumption per worker to maximising output per worker, we've presumably increased output per worker, at the expense of consumption per worker. Where does this surplus output go if it doesn't increase C nor flow abroad (as we have a closed econ)? Does it go into gov't coffers? Does the gov't count as 'workers?' Or do even those who have physical capital count as 'workers'?

I hope my question is clear!

I've charted all of this on a graphical calculator, and interestingly, consumption per worker doesn't evolve over time (the A and L differ by a factor, so A/L is a constant). I still don't understand why the additional capital (which leads to higher output per worker) from raising t to 0.4 from 2/9 decreases consumption. What have we achieved here - what else is increasing from this increase in output if we decrease consumption per worker? It seems to be just an increase in output for its own sake...) Oh, and x-axis is the tax rate, t is time in the graph!

Note to self: check steady state of capital per worker tomorrow.