(Solved): Solow model with effective labour and government spending Consider a variant of the Solow model, wh ...

Solow model with effective labour and government spending Consider a variant of the Solow model, where labour is measured in efficiency units and government spending, $G_t$, is included on the assumption that government spending makes private capital $\left(K_t\right)$ and labour $\left(N_t\right)$ more productive. The technological progress variable, $A_t$, in this model is called labour augmenting because it affects aggregate output by increasing the effectiveness of labour. The production function can be written in the following form: $Y_t=K_t^{?}{\left(A_t{N}_t\right)}^{?}{G}_t^{?}$ where $?+?+?=1$. Assume that the government runs a balanced budget, i.e. $G_t=?Y_t$ where $?$ is a tax rate on output. Consumers save a constant fraction $s$ of disposable income. The rate of population growth is equal to $n$, and the depreciation rate equals $d$. (a) Derive the intensive form production function in terms of effective labour. [Hint: To derive intensive form production function, you divide both sides of the production function by $A_t{N}_t$. Use the relation $1??=?+?$ to substitute $?+?$ by $1??$. In effective labour terms, balanced budget can be written as $g_t=?y_t$. From your function, how does the tax rate affect output? Q2.b (10 points) The capital stock, $K_t$, evolves according to: $K_{t+1}=I_t+(1?d)K_t$ (b) Using the above law of motion, what is the steady-state condition for this economy? [Hint: Start by dividing both sides of the equation with $A_{t+1}{N}_{t+1}$. Note that the goods market is in equilibrium and that saving is $s(1??)Y_t$. Also, assume that $A_{t+1}{N}_{t+1}=(1+n+z)A_t{N}_t$, where $z$ denotes a measure of labour productivity growth.] Q2.c (7 points) (c) Derive the steady-state level of capital per effective labour, $k^{?}$, in terms of $s,n,d,z$, and $?$. [Hint: Again, you need to use the relation $1??=?+?$ to substitute $?+?$ by $1??$ when finding $k^{?}$.] Q2.d (10 points) (d) Find the Golden Rule level of capital per effective worker, $k_G$. What is the saving rate that can help this economy to achieve $k_G$ in the steady state? [Hint: Again, use the relation $1??=?+?$ to substitute $?+?$ by $1??$.]