CentER · Macroeconomics 1 · Problem Set 1 · Acemoglu 2.11 & 2.12

Solow Phase Lab

Graphical companion to the two exercises. Move the parameters, press Play, and hover any chart to read values.

Exercise 2.11Solow with land in fixed supply

\[ F=L^{\beta}K^{\alpha}Z^{1-\alpha-\beta}\quad\Longrightarrow\quad \dot k = s\,(Z/L)^{1-\alpha-\beta}k^{\alpha}-(n+\delta)\,k. \]

Parameters for all 2.11 figures

(a) No population growth

With \(L\) fixed, \(B=(Z/L)^{1-\alpha-\beta}\) is a constant and \(g(k)\equiv\dot k/k=sBk^{\alpha-1}-\delta\).

Fig. 1Existence, uniqueness and stability of \(k^*\)
Click inside the left or middle panel to set the initial capital stock \(k(0)\).
Solow diagram
saving \(sBk^{\alpha}\)depreciation \(\delta k\)\(k(0)\)
Growth rate \(g(k)=\dot k/k\)
\(g(k)\)\(-\delta\)\(k(0)\)
Time paths from several \(k(0)\)
path from \(k(0)\)other \(k(0)\)
What to notice
  • Middle: \(g\) falls from \(+\infty\) to \(-\delta\), so it crosses zero once, and the arrows point to that crossing from both sides.
  • Right: paths never cross \(k^*\), so each one is monotone.

(b) Population growth

With \(L=e^{nt}\), land per worker shrinks and \(g(k,t)=sBe^{-\gamma t}k^{\alpha-1}-(n+\delta)\), with \(\gamma=(1-\alpha-\beta)n\). Its zero, the temporary steady state \(k^*(t)\), now moves.

Fig. 2A steady state that keeps moving: \(k\) chases a target sliding to zero
Growth rate \(g(k,t)\) at time \(t\)
\(g(k,t)\)\(g(k,0)\)\(k^*(t)\)\(k(t)\)
\(k(t)\) against the temporary steady state
\(k(t)\)\(k^*(t)\)\(k^*\) when \(n=0\)
Factor prices (levels, log scale)
wage \(w\)land rent \(R_Z\)return to capital \(r\)
What to notice
  • Press Play: \(g\) shifts down and drags \(k^*(t)\) toward zero. At every instant \(k\) still moves toward \(k^*(t)\).
  • Middle: \(k\) rises, the falling target meets it, and \(k\) turns down. The sign of \(\dot k\) gives the direction, not where \(k\) ends up.
  • Right: the wage falls toward zero, land rent grows, and the return to capital settles at a positive constant.

(c) Malthusian population

Let population growth respond to income: \(n=n(y)\) with \(n'>0\) and \(n(\bar y)=0\). The figure uses \(n(y)=\bar n\,(1-\bar y/y)\).

Fig. 3Constant \(n\) versus Malthusian \(n(y)\): \(y\to0\) versus \(y\to\bar y>0\)
The population rule \(n(y)\)
\(n(y)\)constant \(n\)
Income per worker \(y(t)\)
constant \(n\)\(n(y)\)\(\bar y\)
Population \(L(t)\) under \(n(y)\)
\(L(t)\)\(L^*\)
What to notice
  • With constant \(n\), \(y\to0\). With \(n(y)\), \(y\to\bar y>0\): above \(\bar y\) population grows and pushes \(y\) down, below it shrinks and pushes \(y\) up.
  • Raise \(Z\) or \(s\) in the deck: \(L^*\) rises, \(y^*\) does not.

Exercise 2.12Only capitalists save

Workers consume everything; capitalists save a share \(s_K\) of capital income: \[\dot k = s_K\,f'(k)\,k-(n+\delta)k,\qquad f(k)=k^{\alpha}.\] The golden rule \(k_{gold}\) solves \(f'(k)=n+\delta\).

Parameters for Fig. 4
Fig. 4The capitalists' steady state against the golden rule
Growth rate \(g(k)=\dot k/k\)
\(s_Kf'(k)-(n+\delta)\)same with \(s_K=1\)
Saving versus required investment
saving \(s_Kf'(k)k\)capital income \(f'(k)k\)\((n+\delta)k\)
Steady-state consumption \(c=f(k)-(n+\delta)k\)
\(c(k)\)\(k^*\)\(k_{gold}\)
What to notice
  • Left: same picture as Fig. 1. The dashed curve is the economy with \(s_K=1\); its zero is \(k_{gold}\). With \(s_K\lt 1\) the zero comes earlier: \(k^*\lt k_{gold}\).
  • Middle: \((n+\delta)k\) meets capital income at \(k_{gold}\) and capitalists' saving at \(k^*\). Set \(s_K=1\) and they coincide.
  • Right: the gap costs steady-state consumption.