Question 2: Harrod-neutral technical progress
Now consider a variation of the Swan-Solow model that has a deterministic trend growth in technology. The technology is embodied directly into labor. So often economists call this a model of labor-augmenting technical progress. Specifically, let xt denote the level of labor-augmenting technology at time t, and assume that
xt+1 = (1 + g)xt,
where g > 0 is the growth rate. So now, the only variation in the model is the production function:
Yt = zF(Kt, xtNt).
where x is now embodied in the labor input.
1. Sketch the trend of the process {xt}t=0 to infinity := {x0, x1, x2, ...} where x0 is known. Sketch also the trend of the process {ln(xt)}t=0 to infinity.
2. Show that along the long run path, per-worker capital, k*, and per-worker income, y*, are no longer constant but they grow as a function of time.
3. However, show that now, if we define kt~ = Kt/xtNt = kt/xt, interpreted as "capital per effective units of labor", the model has a well defined steady-state path (k*~, k*~, k*~, ...) that is constant (i.e. does not grow with time).
4. Sketch in an appropriate diagram, the determination of the steady state in terms of capital per effective units of labor. What happens to this steady state if g is increased permanently?
5. Sketch the time path of log-income ln(Y) and log-consumption ln(C), before, during, and after the increase in g.