Consider an economy described by the production function: Y = F (K, L ) = K^(2/3
ID: 1195130 • Letter: C
Question
Consider an economy described by the production function: Y = F (K, L ) = K^(2/3)L^(1/3).
a) Find the per worker production function.
b) Assuming population growth (n) and technological change (g), find the steadystate capital stock per worker as a function of the savings and depreciation rate, population growth and technological change.
c) Now assume the depreciation rate is 3% a year, population growth is 2% a year and technological change is 1% and the savings rate is 24%. Find the steady-state level of capital per worker and the corresponding levels of output per worker.
d) Now using the Marginal Product of Capital (per worker) find the level of capital that maximizes consumption per worker in the steady state.
e) What savings rate is necessary for the economy to reach this consumption maximizing steady state? (Hint use the answer from the first part of question b). How does this compare to the current savings rate (24%)?
Explanation / Answer
a) Y/L = (K/L)2/3
b) Let Y/L = y and K/L =k
Steady state is where:
s(MPK) = n+g+d where d= depreciation rate
c) 0.24(2/3)k-1/3 = 0.03+0.02+0.01
k= (8/3)3
k=19
y= 7
d) To maximize consumption per worker, maximize
y - sy
In steady state sy = (n+g+d)k
Maximize k2/3 -(n+g+d)k
Differentiate w.r.t k
MPK =2/3k-1/3 = n+g+d
2/(3*0.06) = k1/3
k = 1371