Country A has the production function: Y = F (K, L) = K 0.5 L 0.5 a. Express the
ID: 2495339 • Letter: C
Question
Country A has the production function: Y = F (K, L) = K0.5L0.5
a. Express the production function in per worker terms. Thus, y = f (k) where y = Y/L and k=K/L
b. Suppose that both countries start off with a capital stock per worker of 2 (i.e. k = 2). What are the levels of income per worker and consumption per worker? Illustrate this on a graph
c. Assume that neither country experiences population growth or technological progress. Assume further that country A saves 10 percent of its output and the stock of capital depreciate by 5 percent each year. Using your answer from part (a) and the steady-state condition that investment equals depreciation
i. Find the steady-state level of capital per worker for each country.
ii. Find the steady-state levels of income (output) per worker and consumption per worker for each country
iii. Show on a graph the steady-state capital stock per worker, output per worker and consumption per worker for each country
d. State the condition for the Golden Rule and find the Golden Rule level of capital.
Explanation / Answer
a)
The per-worker production function is: f(k) = k1/2 since,
y = F (K, L) / L = ( K1/2 L1/2) / L = ( K/L)1/2 (L/L)1/2 = k1/2 11/2 = k1/2
b)Suppose at time zero (t=0) both countries have 2 units of capital per worker
time kA yA cA kB yB cB
0 2.00 1.414 1.2730 2.00 1.414 1.1314
1 2.04 1.428 1.2850 2.18 1.477 1.1819
2 2.08 1.442 1.2980 2.37 1.539 1.2313
3 2.12 1.456 1.3104 2.56 1.599 1.2800
4 2.16 1.469 1.3227 2.75 1.658 1.3267
It will take four years for consumption in country B to exceed that of country A
C)
We assume that :
d=0.05
Savings rate in country A = sA = 0.10
Savings rate in country B = sB = 0.20
At BGP: s f(k*) = * k* (see explanation in I above before equation (3)) so we can use this
equation , the relevant savings rate and our particular functional form for f(k) to calculate k* for
each country.
Country A:
sA f (k)= * k can be written as : 0.10 k1/2 = 0.05 k
We solve for k:
0.10 / 0.05 = k k-1/2
2 = k1/2
4 = k
Then at the BGP :
Capital per worker : kA* = 4
Income per worker: yA* = f(kA*) = 41/2 = 2
Consumption per worker: cA* = (1-sA) yA* = 0.90 @ 2 = 1.8
Country B:
sB f (k)= * k can be written as : 0.20 k1/2 = 0.05 k
We solve for k:
0.20 / 0.05 = k k-1/2
4 = k1/2
16 = k
Then at the BGP :
Capital per worker : kB* = 16
Income per worker: yB* = f(kB*) = 161/2 = 4
Consumption per worker: cB* = (1-sB) yB* = 0.80 @ 4 = 3.2