Question

Consider the Solow growth model that we developed in class. Output at time t is given
by the production function $Y_{t}=AK_{_{_{t}}}^{^{1/3}}L^{2/3}$ $Yt=AK_{_{_{t}}}^{^{1/3}}L^{2/3}$where A is total factor productivity, Kt
is total capital at time t and L is the labour force. Total factor productivity A and labour force
L are constant over time. There is no government or foreign trade and $Y_{t}=C_{t}+I_{t}$ where
Ct is consumption and It is investment at time t. Every agent saves s share of his income and consumes the rest. Therefore, $(1+s)Y_{t}=C_{t}$ and $sY_{t}=S_{t}$. Each period, savings equal investment:$I_{t}=S_{t}$ . Capital evolves according the transition equation $K_{1+t}=(1-d)K_{t}+I_{t}$ ,where d is the depreciation rate.
a) Combine the production function and the transition equation for capital to express $K_{t+1}$ as a function of $K_{t}$ and the parameters of the model.
b) Express the transition equation in per worker terms, letting kt =
Kt/L denote capital per worker.
Suppose that A = 5, L = 1, s = 0.2, d = 0.1. Furthermore kt = 8.
c) Let yt = Yt/L denote output per worker. Express output per worker in terms of capital per worker using the above production function. Calculate output per worker at time t.
d) Calculate how much capital (per worker) depreciates at time t. Calculate investment (per worker) at time t. Calculate the level of capital per worker in t+1 . Did capital per worker increase?

Answer 1