Question

Suppose that the production function is given by Y=K^(1/2)L^(1/2)

a. Derive the steady state levels of capital per worker and output per worker in terms of the saving rate, s, and the depreciation rate, δ.

b. Suppose δ = 0.05 and s = 0.2. Find out the steady state output per worker.

c. Suppose δ = 0.05 but s increases to 0.5. Find out the steady state output per worker and compare your result with your answer in part b. Explain the intuition behind your results.

Answer 1

The production function is given as

Now, the capital per worker is defined as

k = K/L........(1)

And, the output per worker is defined as

y = Y/L =

or, {As, k=K/L}.......(2)

Now, we wil answer the following answers with the help of the production function.

(a) The savings rate is = s

And, the depriciation rate = δ

Now, savings per worker is = S = s.y

And, Investment per worker is = i = δ.k

Now, the change in capital accumulation is

∆k = S - i

or, ∆k = s.y - δ.k

At the steady state, there is no change in capital accumulation. Hence, at steady state equilibrum,

∆k = 0

or, s.y - δ.k = 0

or, s. = δ.k

or, = s/δ

or, k* =

And, putting the value of k* in y, we get

or, y* =

Hence, steady state capital per worker is

k* =

And, steady state output per worker is

y* =

(b) If, δ = 0.05 and s = 0.2, we put these values in y*.

Hence, y* = s/δ

or, y* = 0.2/0.05

or, y* = 4

Hence, the steady state output per worker is 4.

(c) Now, suppose the steady state remains same at δ=0.05 but savings rate increases from s=0.2 to s'=0.5.

Hence, we put δ=0.05 and s'=0.5 in y* and get

y' =s/δ

or, y' = 0.5/0.05

or, y' = 10

Hence, if savings rate s increases to 0.5, then output per worker is 10.

​​​​​​The output per worker in part (b) was 4.

Hence, due to rise in savings rate, the output per worker has increased.

The reason behind increase in output due to rise in savings rate is following.

When, savings rate increases, the overall savings of the economy increases. As a result of it, the investment in capital also increases. Hence, capital accumulation of the economy increases and the economy reaches the steady state at a higher level of capital per worker.

As a result of the increase in the steady state capital per worker, the steady state output per worker also increases.

This is the intuition behind the change in output per worker.

Hope the solutions are clear to tou my friend.