<p>I'm not quite sure what <span class="math-container">$Z$</span> stands for in this model (looks like some kind of multiplier over the standard utility) but I can generally guess the rest. <span class="math-container">$C$</span> stands for consumption, <span class="math-container">$L$</span> for real money holding, <span class="math-container">$N$</span> for hours worked, <span class="math-container">$B$</span> is one-period bond, <span class="math-container">$M$</span> is nominal money, <span class="math-container">$W$</span> is wage, <span class="math-container">$D$</span> is dividend, and <span class="math-container">$T$</span> is tax.</p>
<p><span class="math-container">$U(C,L)$</span> denotes the utility derived from consumption and real money holding while <span class="math-container">$V(N)$</span> is disutility from working. Rewriting the problem as follows,</p>
<p><span class="math-container">$$\underset{C_t,N_t, L_t, B_t}{\max} E_0\sum^{\infty}_{t=0}\beta^t\mathcal U(C_t,L_t,N_t;Z_t) \\ \text{subject to} \\ 
P_tC_t+M_t+B_t\leq B_{t-1}(1+i_{t-1})+M_{t-1}+W_tN_t+D_t-P_tT_t
$$</span></p>
<p>You can now set up the Lagrangian
<span class="math-container">$$\mathcal L= E_0\sum^{\infty}_{t=0}\beta^t\mathcal U(C_t,L_t,N_t;Z_t) + \lambda_t [B_{t-1}(1+i_{t-1})+M_{t-1}+W_tN_t+D_t-P_tT_t-P_tC_t-M_t-B_t]$$</span>
Taking the partial derivative with respect to the choice variables,</p>
<p><span class="math-container">$${\partial \mathcal L\over \partial C_t}=\beta^tZU_{C,t}-\lambda_tP_t \tag{1}
$$</span>
<span class="math-container">$${\partial \mathcal L\over \partial N_t}= -\beta^tZV_{N,t}+W_t \tag{2}$$</span>
<span class="math-container">$${\partial \mathcal L\over \partial L_t}=\beta^tZU_{L,t}-\lambda_t P_t+\lambda_{t-1}P_{t-1} \tag{3}$$</span>
<span class="math-container">$${\partial \mathcal L\over \partial B_t}= \lambda_{t+1}(1+i_t)-\lambda_t \tag{4}$$</span></p>
<p>Solving with equality (I'll leave the steps for you to complete the exercise), you can derive the Euler equation from (1) &amp; (4). The intratemporal labor supply can be derived from (1) &amp; (2). Lastly, the money demand schedule can be derived from (1), (3) and (4).</p>

CC BY-SA 4.0
2021-04-23T22:43:36.757Z
2021-04-25T08:13:58.753Z
2021-04-25T08:13:58.753Z
Rei
<p>I am trying to solve the household's optimization problem in the New Keynesian model framework, where utility is given by</p>
<p><span class="math-container">$$
E_0\sum_{t=0}^\infty \beta^t \mathcal{U}(C_t,L_t,N_t;Z_t)
$$</span>
and period utility is defined as <span class="math-container">$\mathcal{U}(C,L,N;Z)=(U(C,L)-V(N))Z$</span>.</p>
<p><span class="math-container">$U(\cdot)$</span> is increasing and concave, <span class="math-container">$V(\cdot)$</span> is increasing and convex and <span class="math-container">$h(L/C)\equiv U_l/U_c$</span> is a continuous and decreasing function that satisfies <span class="math-container">$h(\bar{\varkappa})=0$</span> for some <span class="math-container">$0&lt;\bar{\varkappa}&lt;\infty$</span>.</p>
<p><span class="math-container">$L_t\equiv M_t/P_t$</span> and it is assumed that <span class="math-container">$C_t,N_t,L_t\geq 0$</span> for all t.</p>
<p>The household's budget constraint sequence is given by
<span class="math-container">$$
P_t C_t+B_t+M_t=B_{t-1}(1+i_{t-1})+M_{t-1}+W_t N_t+D_t-P_t T_t 
$$</span>
for <span class="math-container">$t=0,1,2,...$</span> and we rule out Ponzi schemes by imposing
<span class="math-container">$$
\lim_{T\to\infty}\Lambda_{0,T}\mathcal{A}_T\geq0
$$</span>
with <span class="math-container">$\mathcal{A}_T\equiv [B_{t-1}(1+i_{t-1})+M_{t-1}]/P_t$</span> as the household's real financial wage at the beginning of period t. <span class="math-container">$W_t$</span> are nominal wages and <span class="math-container">$D_t$</span> are dividends paid by firms.</p>
<p>I am supposed to arrive at the optimality conditions below</p>
<ol>
<li><p>Euler equation: <span class="math-container">$U_{c,t}=\beta(1+i_t)(P_t/P_{t+1})U_{c,t+1}$</span></p>
</li>
<li><p>Intratemporal labor supply: <span class="math-container">$W_t/P_t=V_{n,t}/U_{c,t}$</span></p>
</li>
<li><p>Money demand schedule: <span class="math-container">$h(L_t/C_t)=i_t/(1+i_t)$</span></p>
</li>
</ol>
<p>In combination with the transversality condition: <span class="math-container">$\lim_{T\to\infty}\Lambda_{0,T}\mathcal{A}_T=0$</span></p>
<p>How do I set up the Lagrangian?</p>

<optimization><new-keynesian-economics><intertemporal-choice>
Deriving optimality conditions in the New Keynesian model framework with an undefined consumption function
Spotdoddy
# Deriving optimality conditions in the New Keynesian model framework with an undefined consumption function

(Asked by: username_0 on 2021-04-16)

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?

**Tags:** optimization, new-keynesian-economics, intertemporal-choice


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## 1 answers

### Answer by username_1 on 2021-04-23. Score: 2 | Accepted answer

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).


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