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a. the production function is constant return to scale, so divide both side of the production function by L*E, we get Y/(L*E)=F[K/(L*E), (1-u)], since u is an exdogenous variable, the new production function can be rewrite as y=f(k), remember the larger u, the lower y at a certain level of k, and y=Y/(L*E), k=K/(L*E)
b. the break-even investment should be: I=s*Y=[g(u)+n+d]*K, where K is capital stock rather than in per effective worker term, and d is the depreciation rate
c. dk=sy-[g(u)+n+d]*k
the saving curve and break-even line are very much like curves in the classic slow model. however the slope of the break-even line is now equal to [g(u)+n+d]
d. at the steady state, neither are output per effective worker nor capital per effective worker changed. Y/L=y*E, take logs and dreveatives with respect to time of both sides of the equation, we can get, d(Y/L)/(Y/L)=dy/y+dE/E, and we know that at steady state, dy=0.
So, the growth rate of output per worker=dE/E=g(u), from which we conclude that saving rate has nothing to do with the steaty-state growth rate of Y/L and the more fration of labor force in universities, the faster the steady-state Y/E grows.
e. the increasing u shifts the saving curve downward, and makes the break-even line more steepy.
the immediate effect is reducing the amount of investment that cannot cover the depreciation of k at the previous steady-state level. Gradually, the capital per effective worker falls until the new break-even level of k achieved. The result is that new steady-state k is less than the previous level. k is capital per effective worker
f. it`s not unambiguous.
in the short run, output per worker may decrease as a result of a fall in u
in the long run, however, because the growth rate of output per worker is faster with more fraction of labor force working in universities, the level of output per worker will outperform the one with slower growh rate eventually.
hence, it depends on whether we concern more about the standard of living today or more about the life in the future.
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