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

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?

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