Assume that: the aggregate production function takes the form Yt = F(Kt , Nt) =
ID: 1195416 • Letter: A
Question
Assume that: the aggregate production function takes the form Yt = F(Kt , Nt) = Kt Nt 1 with 0<<1;
capital depreciates at a rate ;
there is no government spending or taxes;
no technological change;
the labor force is constant, i.e. Nt = N;
aggregate saving is a constant fraction s (the saving rate) of aggregate output.
Denote with kt the amount of capital per worker (Kt /N) and with yt output per worker (Yt /N).
a) What is the marginal product of capital (Yt /Kt)? Is it constant or decreasing in Kt?
b) Write the production function in per capita (per worker) terms, i.e. yt as a function of kt.
c) Obtain the equation that determines the change in capital per worker (kt+1 – kt) and, from that, the equation that determines the growth rate of capital per worker (kt+1 – kt) / kt. Then show the graphical representation of each, indicating the steady state level k*.
d) How is the growth rate of output per worker (yt+1 – yt) / yt related to the growth rate of capital per worker (kt+1 – kt) / kt ?
e) Solve for the steady state level of capital per worker k* and output per worker y*, as a function of the parameters of the model (, , s). Then compute the value of k* and y* if, for example, = 1/2, = 1%, and s = 0.10.
f) Assume the economy is at the steady state when, in period T, a permanent increase in the saving rate s occurs (i.e. the new saving rate is s > s). Show analytically (with your results from e)) and graphically (with the graphs from c)) how this change affects the long run levels of capital and output per worker (the new steady state) and the growth rate during the transition.
g) If for example the new saving rate is s = 0.50, what will be the new steady state levels k** and y** towards which the economy will converge? Compute also the growth rate of k when the change in s occurs (at time T), starting from the steady state k*, and immediately after (at time T+1 and T+2): what is the response of the growth rate to the change?
Explanation / Answer
a) dYt/dKt = aKta-1Nt1-a
It is decreasing in Kt as a<1. As Kt increases Yt will decrease.
b) yt = (kt)a
c) Change in capital = sf(kt) - dk
s= savings rate
d = depreciation
Growth rate of capital= change in capital/kt = s(kt)a-1 - d
e) s(MPK) = d
0.1(0.5)(kt)-0.5 = 0.01
kt* = (5)1/2
yt* = 5