<p>Assuming certain regularity conditions, the first order conditions for 
<span class="math-container">$$
\max_{x, \lambda} U(x) - \lambda (p \cdot x - m)
$$</span>
are
<span class="math-container">\begin{align*}
&amp;D_{x}U(x(p, m)) - \lambda p = 0 
\\
\text{and} \quad
&amp; p \cdot x(p, m) - m = 0.
\end{align*}</span>
Moreover <span class="math-container">$x(p, m)$</span> will be differentiable with respect to <span class="math-container">$m$</span> at <span class="math-container">$(p, m)$</span>, and this fact together with the second equation implies
<span class="math-container">$$
p \cdot D_{m}x(p, m) - 1 = 0.
$$</span>
So, by the first equation,
<span class="math-container">$$
D_{x}U(x(p, m)) \cdot D_{m}x(p, m)
=
\lambda p \cdot D_{m}x(p, m)
=
\lambda.
$$</span>
The derivative <span class="math-container">$D_{x}U(x(p, m))$</span> of the utility function at <span class="math-container">$x(p, m)$</span> corresponds to <span class="math-container">$\nabla U(x)$</span> in your notation.</p>

CC BY-SA 4.0
2019-11-10T16:38:28.593Z
I'm not sure I understand what you mean. Is the time spent working viewed as an entry of the consumption decision $x$, and would then overtime correspond to a restriction of $x$ that must be satisfied in addition to the normal budget constraint?
induction_is_a_laddah
CC BY-SA 4.0
2019-11-11T07:58:01.190Z
It's not mentioned there, it just asked what's workers strategy to accept or reject the overtime working if he could discuss the payment rent, and I can't get the relation between what has been proved? so you mean overtime working is considered as x and the payment will b part of the m?
Aisa.Imn
CC BY-SA 4.0
2019-11-10T13:51:24.907Z
2019-11-10T16:32:37.627Z
2019-11-10T16:32:37.627Z
induction_is_a_laddah
<p>Why inner multiplication of the gradient of utility function in derivative of demand function with respect to income is equal to shadow price?
This is the equation which is given but I don't know where it comes from, I would be grateful if sb can give me a hint
<a href="https://i.sstatic.net/EFzQq.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/EFzQq.png" alt="enter image description here"></a></p>

<utility><demand>
How can I prove $∇U(x).D_m x(p,m)= \text{shadow price}$?
Aisa.Imn
# How can I prove $∇U(x).D_m x(p,m)= \text{shadow price}$?

(Asked by: username_0 on 2019-11-10)

Why inner multiplication of the gradient of utility function in derivative of demand function with respect to income is equal to shadow price?
This is the equation which is given but I don't know where it comes from, I would be grateful if sb can give me a hint
[![enter image description here](https://i.sstatic.net/EFzQq.png)](https://i.sstatic.net/EFzQq.png)

**Tags:** utility, demand

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## 1 answers

### Answer by username_1 on 2019-11-10. Score: 3 | Accepted answer

Assuming certain regularity conditions, the first order conditions for
$$
\max\_{x, \lambda} U(x) - \lambda (p \cdot x - m)
$$
are
\begin{align\*}
&D\_{x}U(x(p, m)) - \lambda p = 0
\\
\text{and} \quad
& p \cdot x(p, m) - m = 0.
\end{align\*}
Moreover $x(p, m)$ will be differentiable with respect to $m$ at $(p, m)$, and this fact together with the second equation implies
$$
p \cdot D\_{m}x(p, m) - 1 = 0.
$$
So, by the first equation,
$$
D\_{x}U(x(p, m)) \cdot D\_{m}x(p, m)
=
\lambda p \cdot D\_{m}x(p, m)
=
\lambda.
$$
The derivative $D\_{x}U(x(p, m))$ of the utility function at $x(p, m)$ corresponds to $\nabla U(x)$ in your notation.

> **Comments:**
> **username_1**: I'm not sure I understand what you mean. Is the time spent working viewed as an entry of the consumption decision $x$, and would then overtime correspond to a restriction of $x$ that must be satisfied in addition to the normal budget constraint?  
> **username_0**: It's not mentioned there, it just asked what's workers strategy to accept or reject the overtime working if he could discuss the payment rent, and I can't get the relation between what has been proved? so you mean overtime working is considered as x and the payment will b part of the m?  

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