Question $4(24$ points$)$

Suppose that we are in the Neoclassical model. This problem will give specific functional forms for the equations underlying the model. Begin with the supply side. Suppose that labor demand and supply are given by:

$N_t=a_1{w}_t-a_2{}_t$

$N_t=-b_1{w}_t+b_2{A}_t+b_3{K}_t$

where $(11)$ is the labor supply curve and $(12)$ is labor demand. $a_1>0,a_2>0,$ and $b_1>0,b_2>0$ and $b_3>0$ are parameters.

$($a$)$ Use $(11)$ and $(12)$ to solve for expressions for $N_t$ and $w_t$ as a function of parameters and exogenous variables.

$($b$)$ Suppose that $a_1=1,a_2=0\mathrm{.}4,b_1=2,b_2=0\mathrm{.}5,$ and $b_3=0\mathrm{.}3.$ Suppose further that ${}_t=3,A_t=1,$ and $K_t=20.$ Solve for numerical values of $N_t$ and $w_t$ using your answer from the previous part.

$($c$)$ Suppose that the production function is $Y_t=A_t{K}_t^a{N}_t^{1-}.$ Suppose that $=\frac{1}{3}.$ Use your answer from the previous parts, along with the given values of exogenous variables and parameters, to solve for $Y_t.$

Now let us turn to the demand side. Suppose that the consumption and investment demand functions are:

$C_t=c_1(Y_t-G_t)+c_2(Y_{t+1}-G_{t+1})-c_3{r}_t$

$I_t=-d_1{r}_t+d_2{A}_{t+1}-d_3{K}_t$

The aggregate resource constraint is $Y_t=C_t+I_t+G_t.$

$($d$)$ Use the aggregate resource constraint, plus $(13)$ and $(14),$ to derive an expression for $r_t$ as a function of $Y_t$ and other variables $($for the purposes of this exercise, treat $Y_{t+1}$ as exogenous$).$ In other words, derive an expression for the IS curve.

$($e$)$ Suppose that $c_1=0\mathrm{.}5,c_2=0\mathrm{.}4,c_3=1,d_1=20,d_2=4,$ and $d_3=0\mathrm{.}1.$ Suppose further that $A_{t+1}=1,Y_{t+1}=1\mathrm{.}2,G_t=0\mathrm{.}2,$ and $G_{t+1}=0\mathrm{.}2.$ Given your answer for the value of $Y_t^{\text{'}}$ above, your expression for the IS curve, and these parameter values, solve for numeric values of $r_t,C_t,$ and $I_t.$

$($f$)$ Suppose that $A_t$ increases from $1$ to $1\mathrm{.}2.$ Solve for new numeric values of $Y_t,N_t,w_t,r_t,C_t,$ and $I_t.$ Comment briefly on why $r_t$ and $Y_t$ change in the direction they do$.$