Question

Consider an economy at the steady state. Labor force grows at the gN and technology grows at the rate gA. Consider the effects of three shocks: (a) A higher rate of population growth; (b) An increase in the rate of technological change; (c) A decline in the rate of savings. i. Explain and illustrate how each shock would affect the steady-state of the 3. change from the initial to the new steady state? What are the rates of growth of K/AN, Y/AN, KN, YN, K, and Y at the initial and the new steady state?

Answer 1

Consider the given problem;

Labor force grows at the gn

technology grows at the rate ga

here “k = K/AN = capital stock per effective worker”,

“y = Y/AN = output stock per effective worker”

Now, “N=labor force” is growing at the rate “gn"

& the “A = technology“ is growing at the rate “ga”.

Now, under this situation the steady state level of “k” and “y” will be determined by the following condition.

=> ∆k = s*y – (ga + gn + d)*k, where “d” be the rate of depreciation. Now, in the steady state “∆k” must be zero.
=> s*y = (ga + gn + d)*k. Now, “s*y”, be a concave function of “k”, and “(ga + gn + d)*k”, be a straight line, => as population growth increases, => decrease in the steady state level of “k”, => decrease in “y” also. So, here the value of “k=K/AN” and “y=Y/AN” decreases temporary once a new steady state equilibrium will establish both of them become constant.
Now, “K/N = k*A” and “Y/N = y*A”, => as “k” decreases temporarily, => “K/N” also decreases temporarily, as “y” decreases temporarily, => “Y/N” also decreases temporarily.
Now, initially “k=K/AN” and “y=Y/AN” were constant in steady state equilibrium, => the growth rate was “0” for both of them. Now as “gn” increases, => “k” and “y” both decreases temporarily and further become constant in the new equilibrium, => the growth of “k” and “y” in the steady state equilibrium are “0”.
Now, “K/N = k*A” and “Y/N = y*A”, both are growing at the rate “ga”, since “k” and “y” are constant in the steady state equilibrium and “ga” is growing in the same rate as before, => “K/N” and “Y/N” are also growing at the same rate as before.
Now, “K = k*A*N” and “Y = y*A*N” are growing at the rate “ga+gn”, since “k” and “y” are constant. So, if “gn” increases, => growth rate of “K” and “Y” both increases.
b). An increase in technology growth:

as “ga” increases, => decrease in the steady state level of “k”, => decrease in “y” also, with the same reason as of “gn”. So, here the value of “k=K/AN” and “y=Y/AN” decreases temporary once a new steady state equilibrium will establish both of them become constant.
Now, “K/N = k*A” and “Y/N = y*A”, => as “k” decreases temporarily, => “K/N” also decreases temporarily, as “y” decreases temporarily, => “Y/N” also decreases temporarily.
Now, “k=K/AN” and “y=Y/AN” are constant in steady state equilibrium, => the growth rate of both of them is “0”. Now as “ga” increases and “K/N = k*A” and “Y/N = y*A”, both are growing at the rate “ga”. So, since “ga” increases => “K/N” and “Y/N” are also growing faster rate than before.
Now, “K = k*A*N” and “Y = y*A*N” are growing at the rate “ga+gn”, since “k” and “y” are constant. So, if growth rate of technology increases, => “ga” increases, => growth rate of “K” and “Y” both increases.
c). A decline in the rate of savings:

Now, as the savings rate decreases, => decrease in “a*y”, => given the “(ga + gn + d)*k”, the steady state “k” and “y” both decreases but become further constant in the new steady state. Now, “K/N = k*A” and “Y/N = y*A”, => as “k” decreases temporarily, => “K/N” also decreases temporarily, as “y” decreases temporarily, => “Y/N” also decreases temporarily.
Now “k=K/AN” and “y=Y/AN” are constant in steady state equilibrium, => growth rate of both of them is “0”. Now as “s” decreases, => the growth rate of “K/N = k*A” and “Y/N = y*A”, are remain same as “k” is constant and “A” is growing at the same rate as before.
Now, “K = k*A*N” and “Y = y*A*N” are growing at the rate “ga+gn”, since “k” and “y” are constant, => the growth rate of “K” and “Y” remain same as before
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