$1$c$)$ Using $(1),(8),$ and $(12$a$),$ the equilibrium variables $(c_t,l_t,w_t)$ are determined by the conditions

$c_{tt}=2l_t^{\frac{1}{2}}$

$w_t=l_t^{-\frac{1}{2}}=c_t{l}_t$

which can be solved to obtain the solution as

$l_t^{**}=2^{-\frac{11}{2}}=\frac{1}{\sqrt[\phantom{2}]{2}},c_t^{**}=2^{\frac{3}{4}}=2*2^{-\frac{1}{4}},$ and $,w_t^{**}=2^{\frac{11}{4}}.$

The fact that $c_t^{**}-=c^{**}$ can be used in $(12$c$)$ to obtain $R_{t+1}^{**}-={}^{-1}.$ Note that all these variables are independent of monetary policy. This result is referred to as classical dichotomy. It implies that monetary policy may be able to affect monetary variables, but is unable to affect the real part of equilibrium.

$F(l)=2l^{\frac{}{2}}$

$(1$a$)$ The firm's decision problem is

$ma{x}_l\{p_{t}2l^{\frac{1}{2}}-w_t^{n}l|||0\}$

The first order conditions describing interior solutions are

$p_t{l}_t^{-\frac{1}{2}}=w_t^{n}Longleftrightarrow{l}_t^{-\frac{1}{2}}=w_{t}Longleftrightarrow{l}_t=w_t^{-2}.$

Given variables are the price $p_t$ and the nominal wage $w_t^n.$

After eliminating the Lagrangean variable, the optimality conditions read

$c_t{l}_t=w_t$

$c_t{m}_t^{-4}=p_t^m$

$\frac{c_t}{c_{t+1}}=\frac{q_{t+1}}{q_t}=R_{t++1^{-}}^{-1}$Pp

Plesse use full working and explain how the equilibrium variables were determined.