Assignment 4
1. Consider two economies A and B with the following production functions:
YA(t) = KA(t)α K(̇)A (t) = SAYA(t)
And
YB(t) = KB(t)α K(̇)B (t) = SBYB(t)
Where α>1. Economies have the same initial value of K but,sA>sB . Show that Y1/Y2 is continually rising.
2. Consider the Knowledge accumulation model where:
Y(t) = A(t)(1 − aL )L(t)
And
A(̇)(t) = B[aLL(t)]YA(t)θ
Where θ < 1.
Population growth rate is constant at n.
Describe how each of the following affect thė(g) = 0 line and the growth rate of A at the moment of change and in the long run (graph Ln A overtime to show the effect).
a) An increase inn.
b) An increase in aL.
c) An increase in ϴ .
3. Suppose that output is given by the equation Y(t) = K(t)α [A(t)L(t)]1−α , that L is constant and equal to 1; thatK(̇)(t) = SY(t) and that knowledge accumulation occurs as aside effect of goods production:A(̇)(t) = BY(t).
a. Find expression for gA(t) and gK(t) in terms of A(t), K(t) and parameters of the model.
b. Sketch the ġA = 0 and ġk = 0 lines in (gA, gK) space.
c. Does the economy converge to a balanced growth path? If so, what are the growth rates of K, A and Y on the balanced growth path?
d. How does an increase in s affect long-run growth rate?
4. Consider the model with education where G(E) = e ∅E and Y(t) = K(t)α [A(t)H(t)]1−α and H(t) = L(t)G(E) and A(̇)(t) = gA(t) and L(̇)(t) = nL(t) and constant depreciation rate of δ . However, assume that E rather than being constant is increasing steadily: E(̇)(t) = mwhere m>0. With this change what is the long-run growth rate of output per worker?
5. Suppose that output is given by the equation Y(t) = K(t)α [(1 − aH )H(t)]β , H(̇)(t) = BaHH(t) andK(̇)(t) = SY(t).
Assume 0<α<1 , 0<β<1 and α+β>1.
a. What is the growth rate of H?
b. Does the economy converge to a balanced growth path? If so, what are the growth rates of K and Y on the balanced growth path?