Question

Dacia and Romalia are two countries with the production function given by the following relationship: f(k) = 6 k1/2. Capital to labour ratio in Dacia is twice of that of Romalia.

Dacia has a 10% saving rate, 10% population growth rate, and 5% depreciation rate, while Romalia has a 20% saving rate, 10% population growth rate, and 20% depreciation rate.

Compute the following:

a) Steady-state capital- labour ratio for each country. Does the initial capital-labour ratio affect the results?

b) Output per worker and consumption per worker for each country.

Answer 1

Here there are 2 countries “Dacia” and “Romalia” with same production function given below.

=> f(k) = A*k^1/2, where A=6.

Let’s assume that “s” be the savings rate “n” be the population growth and “d” be the depreciation rate.

=> the equation motion of the capital stock per worker is given below.

=> ∆k = s*f(k) – (n+d)*k, => at the equilibrium “∆k”, must be “0” in the steady-state equilibrium.

=> s*f(k) = (n+d)*k, => s*A*k^1/2 = (n+d)*k, => (s*A)/(n+d) = k^1/2, => k* = [(s*A)/(n+d)]^2.

=> So, the “k*” be the steady-state equilibrium here.

=> So, “Dacia” has “10%” saving rate, “10%” population growth and “5%” depreciation rate.

=> the steady-state equilibrium of “Dacia” is given by, “k*d = [(0.1*6)/(0.1+0.05)]^2 = [(0.6)/(0.15)]^2”.

=> k*d = 4^2 = 16.

=> So, “Romalia” has “20%” saving rate, “10%” population growth and “20%” depreciation rate.

=> the steady-state equilibrium of “Romalia” is given by, “k*r = [(0.2*6)/(0.1+0.2)]^2 = [1.2/0.3]^2”.

=> k*r = 4^2 = 16.

So, the steady-state capital stock per worker in both country are given by, “k*d=k*r=4”, => both country have same steady state capital stock per worker.

So, as we have given that “K/L” in “Dacia” is twice of that of “Romalia”, => for “Dacia” “∆k” is negative, => the “K/L” will fall until “∆k=0” condition will be satisfied. Finally at the steady state “∆k=0” for both counties and the corresponding “K/L” is also same.

b).

So, the output per worker is given by, “f(k) = A*k^1/2”, => f(k) = 6*4^1/2 = 6*2 = 12. Now ,since both country have same "production function" and the same "steady state capital per worker", => both country have same “output per worker”.

Now, “Dacia” has saving rate “10%”, => the “consumption per worker”, is given by, “c*d = (1 – 0.1)*f(k)”.

=> c*d = 0.9*12 = 10.8”.

Now, “Rumalia” has saving rate “20%”, => the “consumption per worker”, is given by, “c*r = (1 – 0.2)*f(k)”.

=> c*r = 0.8*12 = 9.6”.