Question

Suppose there are 2 countries i,j  where both the countries are identical in every aspect excepting quality of education. Although both the countries have same levels of factors of production, however the quality of education per person in country i is better than that in country j.

1. Set up a Cobb-Douglas production function in per capita terms by incorporating q to be the quality of education. Note q should be associated with human capital per capita (h) but the level of h (may be represented by years of education) is same across the 2 countries. It is q that differs across the country. (0.75 marks)

2. Express the output per capita ratio in terms of the quality of education ratio. (0.5 marks)

3. Find the development accounting equation in the current scenario and explain how the quality difference will influence the development accounting equation. (2 marks)

4. Find the growth accounting equation in the current scenario and explain how the quality difference will influence the growth accounting equation. (2 marks)

Answer 1

Let q_j = quality of education in country j and q_i = quality of education in country i

The Cobb Douglas production function will be of the form:

Y_i(t) = A(t) K(t)^a (q_iH(t)L(t))^(1-a)

Y_j(t) = A(t) K(t)^a (q_jH(t)L(t))^(1-a)

Output per capita for country i = Y_i(t)/L(t) = y_i(t) = A(t) k(t)^a (q_iH(t))^(1-a)

Output per capita for country j = Y_j(t)/L(t) = y_j(t) = A(t) k(t)^a (q_jH(t))^(1-a)

Taking the ratio of both we get:

Output per capita ratio = (q_i / q_j)^(1-a)

The development acounting equation would be as follows:

We calculate the productivity of two countries by dividing the output per capita ratio with the ratio of the factors of production of both the countries. Since the factors of production of both the countries is the same, we can say that we are dividing by 1. Hence we get productivity as:

Productivity ratio = Output per capita ratio / (Ratio of factors of production) = (q_i / q_j)^(1-a) / 1 = (q_i / q_j)^(1-a)

Productivity ratio = (q_i / q_j)^(1-a)

The above equation is the development accounting equation.

We can see that since q_i > q_j and that (1-a) < 1; we can say that productivity ratio > 1 and that the productivity in country i is greater than the productivity in country j.

The growth accounting equation would be as follows:

Y(t) = A(t) K(t)^a (qH(t)L(t))^(1-a)

Taking the natural logarithm on both sides we get:

ln(Y(t)) = ln(A(t)) + aln(K(t)) + (1-a)ln(q) + (1-a)ln(H(t)) + (1-a)ln(L(t))

Taking the differential wrt t we get:

(1/Y(t))Y'(t) = (1/A(t))A'(t) + (a/K(t))K'(t) + ((1-a)/q)(dq/dt) + ((1-a)/H(t))H'(t) + ((1-a)/L(t))L'(t)

g_Y = g_A + a*g_Y + (1-a)(g_q + g_H + g_L)

The above equation is the growth accounting equation.

The quality difference affects as g_qi is greater than g_qj and hence the growth is more in country i as compared to country j.