Question

Suppose you are a policy maker who wants to select a level of saving rate that would maximize consumption per worker. Given that output per worker is a square root of capital per worker, and 10% of capital depreciates every year.
(a) Construct a table with the following saving rates: 0.3, 0.4, 0.5 and 0.6. For each saving rate, fill the table with values for capital per worker, output per worker, depreciation per worker, consumption per worker and marginal product of capital. Do not just present at able but show all your work clearly, or you lose marks.
(b) Which saving rate would you chose and why.

Answer 1

Let y be output per worker, k be capital per worker, $\delta$ be depreciation rate, s be the savings rate, c be the consumption per worker.

y = $\tiny \sqrt{k}$

(a) At steady state, following equation holds

k/y = s/$\delta$

=> k/$\tiny \sqrt{k}$ = s/0.1 => k = 100s²

Consumption is given by

c = y - sy. At steady state, investment = depreciation.

So, c = y - $\delta$k

Now, marginal product of capital (MPK) is the derivative of the production function with respect to k

so, MPK = 1/2$\tiny \sqrt{k}$

Savings Rate	capital per worker	output per worker	depreciation	consumption per worker	MPK
	k = 100s2	y = $\tiny \sqrt{k}$	0.1*k	c = y - $\delta$k	MPK = 1/2$\tiny \sqrt{k}$
0.3	9	3	0.9	2.1	0.166667
0.4	16	4	1.6	2.4	0.125
0.5	25	5	2.5	2.5	0.1
0.6	36	6	3.6	2.4	0.083333

(b) That savings rate is chosen which maximizes consumption per worker. From the above table, it is clear that at savings rate = 0.5, consumption per worker is maximised. So, savings rate = 0.5 would be chosen.