c)
The production function is Cobb–Douglas, \(f(k)= k^\alpha\), and capital’s share, α, rises.
The break-even investment line, \((n + g + \delta)k\), is
unaffected by the rise in \(\alpha\).
The effect of a change in \(\alpha\) on the actual investment
curve, i.e. on \(sf(k)\), can be determined by looking at the
derivative \(\frac{df(k)}{d\alpha}\).
\(\frac{df(k)}{d\alpha} = ln(k)k^{\alpha}\)
Since \(k>0\) and \(\alpha >0\) therefore \(k^\alpha > 0\).
\(ln(k) < 0\) when \(k<1\) and \(ln(k) > 0\) when \(k>1\)
Hence:
\(\frac{df(k)}{d\alpha} > 0\) when \(k > 1\) and \(\frac{df(k)}{d\alpha} < 0\) when \(k < 1\)
Graphically we can represent the effect of an increase in \(\alpha\) on the actual investment curve as follows: