Deriving optimality conditions in the New Keynesian model framework with an undefined consumption function

Question

I am trying to solve the household's optimization problem in the New Keynesian model framework, where utility is given by

$$ E_0\sum_{t=0}^\infty \beta^t \mathcal{U}(C_t,L_t,N_t;Z_t) $$ and period utility is defined as $\mathcal{U}(C,L,N;Z)=(U(C,L)-V(N))Z$.

$U(\cdot)$ is increasing and concave, $V(\cdot)$ is increasing and convex and $h(L/C)\equiv U_l/U_c$ is a continuous and decreasing function that satisfies $h(\bar{\varkappa})=0$ for some $0<\bar{\varkappa}<\infty$.

$L_t\equiv M_t/P_t$ and it is assumed that $C_t,N_t,L_t\geq 0$ for all t.

The household's budget constraint sequence is given by $$ P_t C_t+B_t+M_t=B_{t-1}(1+i_{t-1})+M_{t-1}+W_t N_t+D_t-P_t T_t $$ for $t=0,1,2,...$ and we rule out Ponzi schemes by imposing $$ \lim_{T\to\infty}\Lambda_{0,T}\mathcal{A}_T\geq0 $$ with $\mathcal{A}_T\equiv [B_{t-1}(1+i_{t-1})+M_{t-1}]/P_t$ as the household's real financial wage at the beginning of period t. $W_t$ are nominal wages and $D_t$ are dividends paid by firms.

I am supposed to arrive at the optimality conditions below

1. Euler equation: $U_{c,t}=\beta(1+i_t)(P_t/P_{t+1})U_{c,t+1}$

2. Intratemporal labor supply: $W_t/P_t=V_{n,t}/U_{c,t}$

3. Money demand schedule: $h(L_t/C_t)=i_t/(1+i_t)$

In combination with the transversality condition: $\lim_{T\to\infty}\Lambda_{0,T}\mathcal{A}_T=0$

How do I set up the Lagrangian?

Answer 1

I'm not quite sure what $Z$ stands for in this model (looks like some kind of multiplier over the standard utility) but I can generally guess the rest. $C$ stands for consumption, $L$ for real money holding, $N$ for hours worked, $B$ is one-period bond, $M$ is nominal money, $W$ is wage, $D$ is dividend, and $T$ is tax.

$U(C,L)$ denotes the utility derived from consumption and real money holding while $V(N)$ is disutility from working. Rewriting the problem as follows,

$$\underset{C_t,N_t, L_t, B_t}{\max} E_0\sum^{\infty}_{t=0}\beta^t\mathcal U(C_t,L_t,N_t;Z_t) \\ \text{subject to} \\ P_tC_t+M_t+B_t\leq B_{t-1}(1+i_{t-1})+M_{t-1}+W_tN_t+D_t-P_tT_t $$

You can now set up the Lagrangian $$\mathcal L= E_0\sum^{\infty}_{t=0}\beta^t\mathcal U(C_t,L_t,N_t;Z_t) + \lambda_t [B_{t-1}(1+i_{t-1})+M_{t-1}+W_tN_t+D_t-P_tT_t-P_tC_t-M_t-B_t]$$ Taking the partial derivative with respect to the choice variables,

$${\partial \mathcal L\over \partial C_t}=\beta^tZU_{C,t}-\lambda_tP_t \tag{1} $$ $${\partial \mathcal L\over \partial N_t}= -\beta^tZV_{N,t}+W_t \tag{2}$$ $${\partial \mathcal L\over \partial L_t}=\beta^tZU_{L,t}-\lambda_t P_t+\lambda_{t-1}P_{t-1} \tag{3}$$ $${\partial \mathcal L\over \partial B_t}= \lambda_{t+1}(1+i_t)-\lambda_t \tag{4}$$

Solving with equality (I'll leave the steps for you to complete the exercise), you can derive the Euler equation from (1) & (4). The intratemporal labor supply can be derived from (1) & (2). Lastly, the money demand schedule can be derived from (1), (3) and (4).