2. Let k^* be the steady state level of capital per worker. Find an expression f

Question

2. Let k^* be the steady state level of capital per worker. Find an expression for k ? that only depend on the parameters of the model. How does k^* relate to n, z, d and s?

3. Suppose ? = 1/2, n = 0.03; d = 0.07; s = 0.2, z = 1. Find numerical solutions for the steady state level of capital per worker, consumption per worker, output per worker and savings per worker.

4. Remember that at the golden rule steady state we have that: MPK = n + d Show that the golden rule level of consumption is attained when the saving rate s is equal to ?.

Explanation / Answer

F = zKN1 –

Dividing both sides by B:

(F/N) = z. (K/N)

Or,

y = z. (k) where

y: Output per worker and k: Capital per worker

(1)

Capital depreciates at the rate ‘d’ & so, future Capital = k.(1 – d)

Population (workers) grows at the rate ’n’, so future Workers = N. (1 + n)

Therefore,

Future capital per worker = [k. (1 – d) / N. (1 + n)]

(2)

Steady state capital per worker, k* = (s. z / d)[1 / (1 – )]

As we can see,

- k* is independent of ‘n’

- as ‘s’ and/or ‘z’ increases (decreases), k* increases (decreases).

- as ‘d’ increases (decreases), k* decreases (increases).

(3) = ½ = 0.5, n = 0.03; d = 0.07; s = 0.2, z = 1

(i) In steady state, k* = (s. z / d)[1 / (1 – )]

= (0.2 x 1 / 0.07)[1 / (1 – 0.5)] = (2.86)2 = 8.18 [Steady state capital per worker]

(ii) y = F(k*) = z. (k*) = 1. (8.18)0.50 = 2.86 [Steady state output per worker]

(iii) Steady state savings per worker = s. F(k*) = 0.2 x 2.86 = 0.57

(iv) Steady state Consumption per worker = (1 – s). F(k*) = 0.8 x 2.86 = 2.29

NOTE: Out of 4 multi-part questions, the first 3 are answered in full.

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