Question

4. Consider an economy in which the amount of investment is equal to the amount of saving
(i.e., the economy is closed to international flows of capital).
Any output that is not saved is consumed (all equivalently, C=Y-S; C=Y-I; c=y-s; c=y-i).
The production function is ?? = ?????? and labor force grows at rate n while productivity (A) is
constant.
The “golden rule level of saving – investment ” is the “optimal” saving – investment rate
(????) that maximises consumption per worker (?? is the fraction of income that is invested –
saved).
Prove that marginal product of capital (MPK) equals (?? + ??) at the golden rule level of saving
– investment (?? is depreciation rate).
Show the golden rule level of saving – investment on a graph.
On the same graph, show the case in which this economy is over saving – investing. (Hint: In
this case, higher saving - investment does increase GDP per worker, but not consumption
per worker.)

Answer 1

In the economy, the amount of investment is equal to the amount of saving
(i.e., the economy is closed to international flows of capital).

Any output that is not saved is consumed. i.e.

Y = C - I = C - S

And,

y = c - i = c - s

Here small alphabes means per worker measurement. For example, Y = GDP and y = GDP per worker.

The per worker production function is given as

{We are denoting alpha as 'a'}

Here labor force grows at rate n while productivity (A) is constant.

At Golden Rule level of saving-investment, the optimal saving-investment rate is ?G. The consumption per worker is maximized here.

? = depriciation rate

Now, the change in capital accumulation is defined as

∆k = ?.y - (? + n).k {? = savings rate}

Now, at steady state

∆k = 0

or, ?.y = (? + n).k..........(1)

Now, the consumption per worker is defined as

c = y - i = y - s

Now, savings per worker is s = ?.y

Hence,

c = y - ?.y

​​​​​​Now, from equation (1) we put ?.y = (? + n).k

Hence,

c = y - (? + n).k

Now, at golden rule level of saving-investment, the consumption is maximized. Hence, the derivative of 'c' with respect to 'k' will be zero. Hence,

dc/dk = 0

or, dy/dk - (? + n) = 0........(3)

Now, dy/dk is the marginal productivity of capital or MPK. Hence, dy/dk = MPK

Hence, from (3) we get

MPK - (? + n) = 0

or, MPK = ? + n [Proved]

Hence, this is proved that, at Golden Rule Saving-Investment, MPK = ? + n.

The following diagram shows the situation of Golden Rule level of consumption and savings.

The Golden Rule level of savings rate is ?G, consumption per worker is cG and GDP per worker is yG.

Now, when the economy is over saving or the savings rate increases from ?G to ?1, the savings curve shifts from ?G.y to ?1.y.

Then from the above diagram we can see that,

✓ GDP per worker increases from yG to y1.

✓ Consumption per worker decreases from cG to c1.

This is the case when the economy is over saving.

Hope the proof and the diagram is clear to you my friend.