<p>The subproblem gives:
<span class="math-container">$$
v(H,\theta) = \max_{h_i} \sum_{i = 1}^I \frac{(\theta_i h_i)^{1 - 1/\eta_i}}{1 - 1/\eta_i} \text{ s.t. } \sum_{i = 1}^I h_i \le H.
$$</span>
The first order condition for <span class="math-container">$h_i$</span> gives,
<span class="math-container">$$
\theta_i (\theta_i h_i^\ast)^{-1/\eta_i}  = \lambda^\ast,
$$</span>
where <span class="math-container">$\lambda$</span> is the Lagrange multiplier.
From the Envelope theorem, we get:
<span class="math-container">$$
v_H =  \lambda^\ast, \tag{1}
$$</span>
and
<span class="math-container">$$
v_{\theta_i} = h_i^\ast (\theta_i h_i^\ast)^{-1/\eta_i} = \frac{h_i^\ast}{\theta_i} \lambda^\ast. \tag{2}
$$</span>
where the latter equality uses the first order condition.</p>
<p>Subsituting out <span class="math-container">$(1)$</span> into <span class="math-container">$(2)$</span> gives:
<span class="math-container">$$
\theta_i v_{\theta_i} = h_i^\ast v_H.
$$</span></p>

CC BY-SA 4.0
2023-10-06T06:04:09.200Z
2023-10-06T06:04:09.200Z
tdm
<p>Say the agent's problem is
<span class="math-container">$$\max_{c,\{h\}, N}\{U(c, v(\boldsymbol{h} ; \boldsymbol{\theta}))+\lambda(w N-c)\}$$</span>, subject to <span class="math-container">$\sum_{i=1}^{I} h_{i}+N \leq 1, \quad N \in \mathcal{N}$</span>.</p>
<p>Assume <span class="math-container">$U(c, v(\boldsymbol{h} ; \boldsymbol{\theta}))$</span> is weak separability,  <span class="math-container">$v\left(\boldsymbol{h} ; \boldsymbol{\theta}\right)=\sum_{i=1}^{I} \frac{\left(\theta_{i} h_{i}\right)^{1-\left(1 / \eta_{i}\right)}}{1-\left(1 / \eta_{i}\right)}$</span>, and the analysis is done for a fixed <span class="math-container">$\lambda$</span>.</p>
<p>FOC: <span class="math-container">$U_{c}=\lambda$</span>; <span class="math-container">$U_{v} v_{i}=\omega \quad \text { for } i=1, \ldots, I $</span>, where <span class="math-container">$v_{i}=\partial v / \partial h_{i}$</span>, and denote  <span class="math-container">$\hat{\omega} \equiv \omega / U_{v}$</span>.</p>
<p>Weak separability allows us to consider the subproblem: <span class="math-container">$$\mathrm{v}(H ; \boldsymbol{\theta}) \equiv \max _{\left\{h_{i}\right\}} v\left(h_{1}, \ldots, h_{I} ; \boldsymbol{\theta}\right), \quad \text { subject to } \sum_{i} h_{i} \leq H$$</span> with  <span class="math-container">$\mathrm{v}_{H}=\hat{\omega}$</span>.</p>
<p>At an optimum, we have <span class="math-container">$U(c, v(\boldsymbol{h} ; \boldsymbol{\theta}))=U(c, \mathrm{v}(H ; \boldsymbol{\theta}))$</span>, and  <span class="math-container">$$U_{v} \mathrm{v}_{H}=\omega$$</span>.</p>
<p>Differentiating the last equation: <span class="math-container">$$\begin{aligned}
\frac{\partial \ln \omega}{\partial \ln \theta_{i}} &amp;=\left(\frac{U_{v v}-U_{c v}^{2} / U_{c c}}{U_{v}}\right) \mathrm{v}_{\theta_{i}} \theta_{i}+\frac{\partial \ln \mathrm{v}_{H}}{\partial \ln \theta_{i}}
\end{aligned}$$</span>.</p>
<p>My question is that how should we derive the <span class="math-container">$\mathrm{v}_{\theta_i}$</span>. If we don't consider any effect of <span class="math-container">$\theta_i$</span> on <span class="math-container">$H$</span>, <span class="math-container">$\mathrm{v}_{\theta_i} = h_i/ \theta_i$</span> following the Envelop theorem. However, I feel that then the derivation is entirely conditional on <span class="math-container">$H$</span>, while we should somehow also take into account of how <span class="math-container">$\theta_i$</span> affects <span class="math-container">$H$</span>. Or maybe we should do the analysis at a given amount of leisure <span class="math-container">$H$</span>? Either way, the paper I am reading says it should be something like <span class="math-container">$h_{i} \mathrm{v}_{H}=\mathrm{v}_{\theta_{i}} \theta_{i}$</span>  but I have no idea where the <span class="math-container">$\mathrm{v}_{H}$</span> comes from.</p>

<microeconomics><mathematical-economics><consumer-theory>
Consumer theory with subproblem
Alalalalaki
# Consumer theory with subproblem

(Asked by: username_0 on 2023-10-06)

Say the agent's problem is
$$\max\_{c,\{h\}, N}\{U(c, v(\boldsymbol{h} ; \boldsymbol{\theta}))+\lambda(w N-c)\}$$, subject to $\sum\_{i=1}^{I} h\_{i}+N \leq 1, \quad N \in \mathcal{N}$.

Assume $U(c, v(\boldsymbol{h} ; \boldsymbol{\theta}))$ is weak separability, $v\left(\boldsymbol{h} ; \boldsymbol{\theta}\right)=\sum\_{i=1}^{I} \frac{\left(\theta\_{i} h\_{i}\right)^{1-\left(1 / \eta\_{i}\right)}}{1-\left(1 / \eta\_{i}\right)}$, and the analysis is done for a fixed $\lambda$.

FOC: $U\_{c}=\lambda$; $U\_{v} v\_{i}=\omega \quad \text { for } i=1, \ldots, I $, where $v\_{i}=\partial v / \partial h\_{i}$, and denote $\hat{\omega} \equiv \omega / U\_{v}$.

Weak separability allows us to consider the subproblem: $$\mathrm{v}(H ; \boldsymbol{\theta}) \equiv \max \_{\left\{h\_{i}\right\}} v\left(h\_{1}, \ldots, h\_{I} ; \boldsymbol{\theta}\right), \quad \text { subject to } \sum\_{i} h\_{i} \leq H$$ with $\mathrm{v}\_{H}=\hat{\omega}$.

At an optimum, we have $U(c, v(\boldsymbol{h} ; \boldsymbol{\theta}))=U(c, \mathrm{v}(H ; \boldsymbol{\theta}))$, and $$U\_{v} \mathrm{v}\_{H}=\omega$$.

Differentiating the last equation: $$\begin{aligned}
\frac{\partial \ln \omega}{\partial \ln \theta\_{i}} &=\left(\frac{U\_{v v}-U\_{c v}^{2} / U\_{c c}}{U\_{v}}\right) \mathrm{v}\_{\theta\_{i}} \theta\_{i}+\frac{\partial \ln \mathrm{v}\_{H}}{\partial \ln \theta\_{i}}
\end{aligned}$$.

My question is that how should we derive the $\mathrm{v}\_{\theta\_i}$. If we don't consider any effect of $\theta\_i$ on $H$, $\mathrm{v}\_{\theta\_i} = h\_i/ \theta\_i$ following the Envelop theorem. However, I feel that then the derivation is entirely conditional on $H$, while we should somehow also take into account of how $\theta\_i$ affects $H$. Or maybe we should do the analysis at a given amount of leisure $H$? Either way, the paper I am reading says it should be something like $h\_{i} \mathrm{v}\_{H}=\mathrm{v}\_{\theta\_{i}} \theta\_{i}$ but I have no idea where the $\mathrm{v}\_{H}$ comes from.

**Tags:** microeconomics, mathematical-economics, consumer-theory

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## 1 answers

### Answer by username_1 on 2023-10-06. Score: 2 | Accepted answer

The subproblem gives:
$$
v(H,\theta) = \max\_{h\_i} \sum\_{i = 1}^I \frac{(\theta\_i h\_i)^{1 - 1/\eta\_i}}{1 - 1/\eta\_i} \text{ s.t. } \sum\_{i = 1}^I h\_i \le H.
$$
The first order condition for $h\_i$ gives,
$$
\theta\_i (\theta\_i h\_i^\ast)^{-1/\eta\_i} = \lambda^\ast,
$$
where $\lambda$ is the Lagrange multiplier.
From the Envelope theorem, we get:
$$
v\_H = \lambda^\ast, \tag{1}
$$
and
$$
v\_{\theta\_i} = h\_i^\ast (\theta\_i h\_i^\ast)^{-1/\eta\_i} = \frac{h\_i^\ast}{\theta\_i} \lambda^\ast. \tag{2}
$$
where the latter equality uses the first order condition.

Subsituting out $(1)$ into $(2)$ gives:
$$
\theta\_i v\_{\theta\_i} = h\_i^\ast v\_H.
$$


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