\(\frac{\dot c(t)}{c(t)} = \frac{f'(k(t)) - \rho - \theta g}{\theta}\)
\(\dot k (t) = f(k(t)) - c(t) - (n + g)k(t)\)
Capital-accumulation is not affected since the government is not spending the tax. Since the real after-tax rate of return on capital is now \((1 - \tau) f'(k(t))\), the maximization equations for households gives this growth rate of consumption:
\(\frac{\dot c(t)}{c(t)} = \frac{(1 - \tau)f'(k(t)) - \rho - \theta g}{\theta}\)
Thus, the after-tax rate of return must equal \(\rho + \theta g\), meaning f'(k) must be higher and k lower for consumption to remain. The \(\dot c = 0\) line shifts left.
Since households were aware ahead of time of the tax (although did not yet change their behavior) the change is smoothed by intertemporal optimization.
d) Again, c does not jump at the time of the tax. Since the new point is too high to maintain the capital stock at k*, it will fall while consumption rises since the economy is to the left of the \(\dot c = 0\) line. When it falls on the saddle-path, it will begin falling and reach the new point of k*.
e) The time paths for consumption per unit of effective labor are as follows: