Question

Suppose a country’s has the following production function: y_t = k_t^1/4, where k and y denote capital and output respectively. The capital depreciation rate is fixed at ? = 0.1 each period, i.e., k_t+1 = k_t ? 0.1k_t + sk_t^1/4,

where s > 0 represents the saving rate.

(a) Calculate the steady state of capital and output:

k^? =?y^? =?

(b) Is this steady states stable?

(c) Suppose there are only two usages of output, con- sumption and saving(investment), i.e., y_t = c_t + sy_t. The golden rate of saving rate is defined as the saving rate that maximizes the steady state consumption. Using results in (a) to calculate the golden saving rate: s_g =?

Answer 1

a)

$y_t = k_t^{1/4}\\ k_{t+1} = k_t - 0.1k_t +sk_t^{1/4}\\ at\:\:steady\:\:state\:,k_{t+1}=k_t=\bar{k}\\ Hence, 0.1\bar{k} = s\bar{k}^{1/4}\Rightarrow \bar{k} = (10s)^{4/3}.\\ Thus, \bar{y} = \bar{k}^{1/4} =((10s)^{4/3})^{1/4} = (10s)^{1/3}$

b) Clearly, this steady state is stable.

c) We have,

ct = (1-s)It. At steady state 2(1- s)(10s)/ For golden rate of savings rate, we put derivative of steady state consumption with respect to s as zero