Consider an economy described by the production function: Y=f(K,L)=K^(.3)L^(.7)
ID: 1230601 • Letter: C
Question
Consider an economy described by the production function: Y=f(K,L)=K^(.3)L^(.7)a. What is the per-worker production function?
b. Assuming no population growth or technological progress, find the steady-state
capital stock per worker, output per worker, and consumption per worker as a
function of the saving rate and the depreciation rate.
c. Assume that the depreciation rate is 10 percent per year. Make a table showing
steady- state capital per worker, output per worker, and consumption per worker for
saving rates of 0 percent, 10 percent, 20 percent, 30 percent and 40 percent. What
saving rate maximizes output per worker? What saving rate maximizes consumption
per worker?
Explanation / Answer
a) y=Y/L k=K/L K^0.3 • L^0.7 / L = (K/L)^0.3 = k^0.3 y=k^0.3 b) ?k=0 ?k=sy-sk sy=sk y=k^0.3 sy=sk^0.3 sk^0.3=sk k^0.7=s/s k=(s/s)^(1/0.7) = (s/s)^(10/7) c) s=10% c=y-i i=sy c=y-sy=y(1-s) c?MAX y-sk ? MAX if ?(y-sk)/?k=0 ?(y-sk)/?k = 0.3/k^0.7 - s 0.3/k^0.7 = s k^0.7 = 0.3/s = 0.3/0.1 = 3 k* = 3^(1/0.7) = 3^(10/7) y* = k*^0.3 = [3^(10/7)]^(0.3) = 3^(3/7) sy = sk s* = sk/y = 0.1•3^(10/7) / 3^(3/7) = 0.1•3 = 0.3 = 30% c* = y-sy = y(1-s) = 3^(3/7) • (1-0.3) = 0.7•3^(3/7)