An economy has a Cobb–Douglas production function:
Y=Kα(LE)1−αY=Kα(LE)1−α
The economy has a capital share of 0.30, a saving rate of 42 percent, a depreciation rate of 5.00 percent, a rate of population growth of 2.50 percent, and a rate of labor-augmenting technological change of 4.0 percent. It is in steady state.
. At what rates do total output and output per worker grow?
Total output growth rate:
%
Output per worker growth rate:
%
Answer
(a)
Production function is given by :
Y=Kα(LE)1−α
=> Y/(EL) = (Kα(LE)1−α)/(LE) = (K/(EL))α = kα
=> Y/(EL) = y = kα
Steady state occurs when change in k = 0 where k = (K/(EL))
In steady state k is constant and thus y = kα is also constant. So, Y/(EL) = is constant.
Formula :
% change in (AB) = % change in A + % change in B
% change in (A/B) = % change in A - % change in B
% change in Y = % change in ((Y/(EL))*E*L) = % change in (Y/(EL)) + % change in E + % change in L
It is given that % change in E = rate of labor-augmenting technological change = 4% and % change in L = % change in Population growth = 2.5%. As discussed above that Y/(EL) is constant implies % change in (Y/(EL)) = 0
Hence, % change in Y = % change in (Y/(EL)) + % change in E + % change in L = 0 + 4 + 2.5 = 6.5
Hence, % change in Y = Total output growth rate = 6.5%
(b)
% change in (Y/L) = % change in ((Y/(EL))*E) = % change in (Y/(EL)) + % change in E
It is given that % change in E = rate of labor-augmenting technological change = 4% and % change in L = % change in Population growth = 2.5%. As discussed above that Y/(EL) is constant implies % change in (Y/(EL)) = 0
Hence, % change in (Y/L) = % change in ((Y/(EL))*E) = % change in (Y/(EL)) + % change in E = 0 + 4 = 4
Hence, % change in (Y/L) = Output per worker growth rate = 4%
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