<p>(Without using differentiation) When <span class="math-container">$w \leq w'$</span> it follows that <span class="math-container">$pf(x) − w · x \geq pf(x) − w' · x$</span> and so <span class="math-container">$\pi(p,w) \geq \pi(p,w')$</span>.</p>
<p>EDIT 1. The last inequality (first left as an exercise) can be justified as follow: <span class="math-container">$w \leq w'$</span> implies that
<span class="math-container">$$pf(x) − w · x \geq pf(x) − w' · x$$</span>
for any <span class="math-container">$x \geq 0$</span> and admissible. The inequality is in particular true for <span class="math-container">$x=x^*(p,w')$</span> and so
<span class="math-container">$$ pf(x^*(p,w')) − w · x^*(p,w') \geq pf(x^*(p,w')) − w' · x^*(p,w').$$</span>
However, <span class="math-container">$x^*(p,w')$</span> does not maximize profits for input prices equal to <span class="math-container">$w$</span> and so
<span class="math-container">$$ pf(x^*(p,w)) − w · x^*(p,w) \geq pf(x^*(p,w')) − w' · x^*(p,w')$$</span>
or equivalently
<span class="math-container">$\pi(p,w) \geq \pi(p,w')$</span>.</p>
<p>EDIT 2. If the output price <span class="math-container">$p$</span> is endogenous and adjusts to aggregate output supply and demand, as it seems to be the case in your question, the issue has been treated by:<br />
Heiner, R. A. (1982): “Theory of the Firm in “Short-Run” Industry Equilibrium,” American Economic Review, 72, 555-62.<br />
Braulke, M. (1984): “The Firm in Short-Run Industry Equilibrium: Comment,” American Economic Review, 74, 750-753.</p>

CC BY-SA 4.0
2020-10-30T20:54:16.343Z
Note that $\pi(p,w)={\color{red}{\max_{x\ge0}}pf(x)-w\cdot x}=pf(x^*(p,w))-w\cdot x^*(p,w)$, where $x^*(p,w)$ is the maximizer given $p$ and $w$. So it not necessary that $x^*(p,w)=x^*(p,w')$ when $w\le w'$.
Herr K.
CC BY-SA 4.0
2020-11-01T22:04:54.127Z
@Herr K. Yes of course, this is not necessary (and I do not assume it). I do not understand your comment very well...
Bertrand
CC BY-SA 4.0
2020-11-01T22:18:44.113Z
Perhaps you can try to provide a more explicit explanation of the first sentence in your answer. Specifically, how did you derive the last two inequalities from knowing $w\le w'$?
Herr K.
CC BY-SA 4.0
2020-11-01T23:14:55.917Z
Thanks for the update. But I think there is still some inconsistencies. By definition, $x^*(p,w')$ maximizes profit when output price is $p$ and input prices are $w'$. If you agree with this interpretation of notation, then how is it possible to have $$ pf(x^*(p,w')) − w · x^*(p,w') \color{red}{\overset{??}{\geq}} pf(x^*(p,w')) − w' · x^*(p,w').$$
Herr K.
CC BY-SA 4.0
2020-11-02T06:35:45.277Z
@Herr K. This is due to $w \leq w' \implies w · x^*(p,w') \leq w' · x^*(p,w')$, I think this point was clear in the edit.
Bertrand
CC BY-SA 4.0
2020-11-02T07:12:41.740Z
Thanks again. And yes, you're right! I must have misread something in your answer earlier. Sorry.
Herr K.
CC BY-SA 4.0
2020-11-04T12:13:05.777Z
This is clearly more elegant solution than that I have posted. Upvoted!
Dayne
CC BY-SA 4.0
2020-10-30T20:17:25.747Z
2020-11-02T06:36:28.817Z
2020-11-02T06:36:28.817Z
Bertrand
<p>From FOC, we know that:</p>
<p><span class="math-container">\begin{align}
\nabla_x\pi(\mathbf{x},\mathbf{w})=p\nabla f(\mathbf{x})-\mathbf{w}=\mathbf{0} \tag{1}
\end{align}</span></p>
<p>This will be true at equilibrium, i.e. for any given <span class="math-container">$\mathbf{w}$</span>, the input vector <span class="math-container">$\mathbf{x}$</span> will adjust so that the above holds.</p>
<p>Now consider <span class="math-container">$d\pi(\mathbf{x},\mathbf{w})/d w_i$</span> (and using <span class="math-container">$(1)$</span>):</p>
<p><span class="math-container">\begin{align}
\frac{d\pi(\mathbf{x},\mathbf{w})}{d w_i} &amp;=\nabla_x\pi\cdot\nabla_{w_i} \mathbf{x} \, + \nabla_w\pi\cdot\nabla_{w_i} \mathbf{w} \\
&amp;=0 +\nabla_w\pi\cdot\nabla_{w_i} \mathbf{w} \tag{*}\\
&amp;=\nabla_w(pf(\mathbf{x})-\mathbf{w}\cdot \mathbf{x)} \cdot \nabla_{w_i} \mathbf{w} \\
&amp;= \Big(p\nabla_wf(\mathbf{x})-\nabla_w(\mathbf{w}\cdot \mathbf{x)}\Big)\cdot \nabla_{w_i} \mathbf{w} \\
&amp;= \Big(p \frac{\partial\mathbf{x}}{\partial\mathbf{w}}\nabla f(\mathbf{x})-\frac{\partial\mathbf{x}}{\partial\mathbf{w}}\mathbf{w} - \mathbf{x}\Big)\cdot \nabla_{w_i} \mathbf{w}\\
&amp;= \mathbf{J}(p\nabla f(\mathbf{x})-\mathbf{w})\cdot \nabla_{w_i} \mathbf{w}  -\mathbf{x} \cdot \nabla_{w_i} \mathbf{w} \\
&amp;= \mathbf{J} \nabla_x\pi(\mathbf{x},\mathbf{w})\cdot \nabla_{w_i} \mathbf{w}-\mathbf{x} \cdot \nabla_{w_i} \mathbf{w} \tag{**}\\
&amp;= -\mathbf{x} \cdot \nabla_{w_i} \mathbf{w} \\
&amp;= -x_i 
\end{align}</span></p>
<p>where, in steps <span class="math-container">$(*), (**)$</span>, FOC is used and <span class="math-container">$\mathbf{J}$</span> is the Jacobian matrix.</p>
<p>Since, <span class="math-container">$x_i \geq0$</span>, we have:</p>
<p><span class="math-container">$$\frac{d\pi(\mathbf{x},\mathbf{w})}{d w_i} \leq0$$</span></p>

CC BY-SA 4.0
2020-10-29T13:10:35.347Z
This is correct, but perhaps you would want to consider adding a line to show clearly why $\nabla_w(pf(\mathbf{x})-\mathbf{w}\cdot \mathbf{x)} = -\mathbf{x}$
Alecos Papadopoulos
CC BY-SA 4.0
2020-10-29T13:44:03.227Z
Ok. Thanks for the advice. Will edit accordingly.
Dayne
CC BY-SA 4.0
2020-10-29T02:15:02.113Z
2020-10-29T14:45:36.093Z
2020-10-29T14:45:36.093Z
Dayne
<p>Prove that the profits of the firm weakly decreases with input prices. More formally, suppose that
the firm has a production function f, so that its profit function is</p>
<p>π(p, w) = max(x≥0) <span class="math-container">$pf(x) − w · x$</span>,</p>
<p>where p denotes the output price and w denotes the input price vector. Then show that if <span class="math-container">$w$</span> and <span class="math-container">$w'$</span> are two input price vectors such that <span class="math-container">$w'_j$</span> = <span class="math-container">$w_j$</span> for all <span class="math-container">$j \neq i$</span> and <span class="math-container">$w'_i$</span> &gt; <span class="math-container">$w_i$</span>
, then <span class="math-container">$π(p, w') ≤ π(p, w)$</span></p>
<p>I know that as input prices decrease, the toal costs for the firm decrease as well. Due to the decrease in TC, the firm then produces more output at the same cost. However, due to the increase in supply, the price of the output falls to a new equilibrium, causing profits to fall as well unless the increase in q is proportionate to the decrease in price that would prevent profits from dropping.</p>
<p>However, I am unsure on how to start the proof using <span class="math-container">$\pi (p, w)$</span>. Since it is <span class="math-container">$w$</span> that is changing, would I hold price of output fixed at <span class="math-container">$p$</span>?</p>
<p>Then I would assume that <span class="math-container">$x'$</span> is profit maximizing at:</p>
<p><span class="math-container">$$pf(x) − w'· x ≤ pf(x') − w'· x'$$</span></p>
<p>and <span class="math-container">$x$</span> is profit maximizing at:</p>
<p><span class="math-container">$$pf(x) − w · x \geq pf(x') − w· x'$$</span></p>
<p>Then I would multiply the second equation by -1, getting</p>
<p><span class="math-container">$$-pf(x) + w · x \leq -pf(x') + w· x'$$</span></p>
<p>I would then add that to the first equation getting:</p>
<p><span class="math-container">$$ (− w'· x) + (w \cdot x) ≤ (− w'· x') + (w \cdot x')$$</span></p>
<p>which simplifies to:</p>
<p><span class="math-container">$$ (w - w') (x - x') \leq 0 $$</span></p>
<p>Since all components of <span class="math-container">$w'− w$</span> are 0 except the ith</p>
<p><span class="math-container">$$ (w_i - w_i') (x_i - x_i') \leq 0 $$</span></p>
<p>We are left with:</p>
<p><span class="math-container">$$(x_i - x_i') \leq 0 $$</span>
<span class="math-container">$$ x_i \leq x_i' $$</span></p>
<p>Leaving us with the fact that factor input demands at <span class="math-container">$x' \geq x$</span> therefore we assume that the demand for output is greater, causing prices to fall.</p>
<p>However, I have no idea how to prove that <span class="math-container">$π(p, w') ≤ π(p, w)$</span> if <span class="math-container">$w'_j$</span> = <span class="math-container">$w_j$</span> for all <span class="math-container">$j \neq i$</span> and <span class="math-container">$w'_i$</span> &gt; <span class="math-container">$w_i$</span>.</p>
<p>I understand that the idea is based on the fact that <span class="math-container">$w'_j$</span> = <span class="math-container">$w_j$</span> and <span class="math-container">$w'_i$</span> &gt; <span class="math-container">$w_i$</span>, <span class="math-container">$w' &gt; w $</span>, that means that the <span class="math-container">$w' \cdot x$</span> in the profit function is greater than <span class="math-container">$w \cdot x$</span>. Since the cost is greater in <span class="math-container">$\pi (p, w')$</span>, that means that <span class="math-container">$π(p, w') ≤ π(p, w)$</span>. However, I am confused as to how I can approach this proposition using the given profit function.</p>

CC BY-SA 4.0
2020-10-28T10:18:27.383Z
If you don't want this question closed - at the least show how you tried to solve it.
KarmaPeasant
CC BY-SA 4.0
2020-10-28T10:52:30.730Z
Hi, Welcome to Economics SE! we have a policy regarding homework questions where we require that you show some work before an answer is provided. More on this topic here:https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions. I am closing this q because currently it very clearly violates the policy but if you edit it to make it in line with our policy it will be reopened.
1muflon1
CC BY-SA 4.0
2020-10-28T19:23:15.973Z
@1muflon1 I have edited the question to show how far I got into it! I am still confused however on the latter part of the question
DH00325
CC BY-SA 4.0
2020-10-28T19:23:56.357Z
@DH00325 I reopened your Q.
1muflon1
CC BY-SA 4.0
2020-10-29T01:07:29.840Z
Hint: differentiate $\pi (\mathbf{x,w})$ w.r.t $w_j$ and use the FOC for profit maximization.
Dayne
CC BY-SA 4.0
2020-10-30T21:05:21.507Z
It is not true that "Due to the decrease in TC, the firm then produces more output at the same cost", the contrary can happen if the input is "inferior".
Bertrand
<microeconomics><proof>
Prove that the profits of the firm weakly decreases with input prices
DH00325
# Prove that the profits of the firm weakly decreases with input prices

(Asked by: username_0 on 2020-10-28)

Prove that the profits of the firm weakly decreases with input prices. More formally, suppose that
the firm has a production function f, so that its profit function is

π(p, w) = max(x≥0) $pf(x) − w · x$,

where p denotes the output price and w denotes the input price vector. Then show that if $w$ and $w'$ are two input price vectors such that $w'\_j$ = $w\_j$ for all $j \neq i$ and $w'\_i$ > $w\_i$
, then $π(p, w') ≤ π(p, w)$

I know that as input prices decrease, the toal costs for the firm decrease as well. Due to the decrease in TC, the firm then produces more output at the same cost. However, due to the increase in supply, the price of the output falls to a new equilibrium, causing profits to fall as well unless the increase in q is proportionate to the decrease in price that would prevent profits from dropping.

However, I am unsure on how to start the proof using $\pi (p, w)$. Since it is $w$ that is changing, would I hold price of output fixed at $p$?

Then I would assume that $x'$ is profit maximizing at:

$$pf(x) − w'· x ≤ pf(x') − w'· x'$$

and $x$ is profit maximizing at:

$$pf(x) − w · x \geq pf(x') − w· x'$$

Then I would multiply the second equation by -1, getting

$$-pf(x) + w · x \leq -pf(x') + w· x'$$

I would then add that to the first equation getting:

$$ (− w'· x) + (w \cdot x) ≤ (− w'· x') + (w \cdot x')$$

which simplifies to:

$$ (w - w') (x - x') \leq 0 $$

Since all components of $w'− w$ are 0 except the ith

$$ (w\_i - w\_i') (x\_i - x\_i') \leq 0 $$

We are left with:

$$(x\_i - x\_i') \leq 0 $$
$$ x\_i \leq x\_i' $$

Leaving us with the fact that factor input demands at $x' \geq x$ therefore we assume that the demand for output is greater, causing prices to fall.

However, I have no idea how to prove that $π(p, w') ≤ π(p, w)$ if $w'\_j$ = $w\_j$ for all $j \neq i$ and $w'\_i$ > $w\_i$.

I understand that the idea is based on the fact that $w'\_j$ = $w\_j$ and $w'\_i$ > $w\_i$, $w' > w $, that means that the $w' \cdot x$ in the profit function is greater than $w \cdot x$. Since the cost is greater in $\pi (p, w')$, that means that $π(p, w') ≤ π(p, w)$. However, I am confused as to how I can approach this proposition using the given profit function.

**Tags:** microeconomics, proof

> **Comments:**
> **username_1**: If you don't want this question closed - at the least show how you tried to solve it.  
> **username_2**: Hi, Welcome to Economics SE! we have a policy regarding homework questions where we require that you show some work before an answer is provided. More on this topic here:https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions. I am closing this q because currently it very clearly violates the policy but if you edit it to make it in line with our policy it will be reopened.  
> **username_0**: @username_2 I have edited the question to show how far I got into it! I am still confused however on the latter part of the question  
> **username_2**: @username_0 I reopened your Q.  
> **username_3**: Hint: differentiate $\pi (\mathbf{x,w})$ w.r.t $w_j$ and use the FOC for profit maximization.  
> **username_4**: It is not true that "Due to the decrease in TC, the firm then produces more output at the same cost", the contrary can happen if the input is "inferior".  

---

## 2 answers

### Answer by username_3 on 2020-10-29. Score: 4 | Accepted answer

From FOC, we know that:

\begin{align}
\nabla\_x\pi(\mathbf{x},\mathbf{w})=p\nabla f(\mathbf{x})-\mathbf{w}=\mathbf{0} \tag{1}
\end{align}

This will be true at equilibrium, i.e. for any given $\mathbf{w}$, the input vector $\mathbf{x}$ will adjust so that the above holds.

Now consider $d\pi(\mathbf{x},\mathbf{w})/d w\_i$ (and using $(1)$):

\begin{align}
\frac{d\pi(\mathbf{x},\mathbf{w})}{d w\_i} &=\nabla\_x\pi\cdot\nabla\_{w\_i} \mathbf{x} \, + \nabla\_w\pi\cdot\nabla\_{w\_i} \mathbf{w} \\
&=0 +\nabla\_w\pi\cdot\nabla\_{w\_i} \mathbf{w} \tag{\*}\\
&=\nabla\_w(pf(\mathbf{x})-\mathbf{w}\cdot \mathbf{x)} \cdot \nabla\_{w\_i} \mathbf{w} \\
&= \Big(p\nabla\_wf(\mathbf{x})-\nabla\_w(\mathbf{w}\cdot \mathbf{x)}\Big)\cdot \nabla\_{w\_i} \mathbf{w} \\
&= \Big(p \frac{\partial\mathbf{x}}{\partial\mathbf{w}}\nabla f(\mathbf{x})-\frac{\partial\mathbf{x}}{\partial\mathbf{w}}\mathbf{w} - \mathbf{x}\Big)\cdot \nabla\_{w\_i} \mathbf{w}\\
&= \mathbf{J}(p\nabla f(\mathbf{x})-\mathbf{w})\cdot \nabla\_{w\_i} \mathbf{w} -\mathbf{x} \cdot \nabla\_{w\_i} \mathbf{w} \\
&= \mathbf{J} \nabla\_x\pi(\mathbf{x},\mathbf{w})\cdot \nabla\_{w\_i} \mathbf{w}-\mathbf{x} \cdot \nabla\_{w\_i} \mathbf{w} \tag{\*\*}\\
&= -\mathbf{x} \cdot \nabla\_{w\_i} \mathbf{w} \\
&= -x\_i
\end{align}

where, in steps $(\*), (\*\*)$, FOC is used and $\mathbf{J}$ is the Jacobian matrix.

Since, $x\_i \geq0$, we have:

$$\frac{d\pi(\mathbf{x},\mathbf{w})}{d w\_i} \leq0$$

> **Comments:**
> **username_6**: This is correct, but perhaps you would want to consider adding a line to show clearly why $\nabla_w(pf(\mathbf{x})-\mathbf{w}\cdot \mathbf{x)} = -\mathbf{x}$  
> **username_3**: Ok. Thanks for the advice. Will edit accordingly.  

---

### Answer by username_4 on 2020-10-30. Score: 2

(Without using differentiation) When $w \leq w'$ it follows that $pf(x) − w · x \geq pf(x) − w' · x$ and so $\pi(p,w) \geq \pi(p,w')$.

EDIT 1. The last inequality (first left as an exercise) can be justified as follow: $w \leq w'$ implies that
$$pf(x) − w · x \geq pf(x) − w' · x$$
for any $x \geq 0$ and admissible. The inequality is in particular true for $x=x^\*(p,w')$ and so
$$ pf(x^\*(p,w')) − w · x^\*(p,w') \geq pf(x^\*(p,w')) − w' · x^\*(p,w').$$
However, $x^\*(p,w')$ does not maximize profits for input prices equal to $w$ and so
$$ pf(x^\*(p,w)) − w · x^\*(p,w) \geq pf(x^\*(p,w')) − w' · x^\*(p,w')$$
or equivalently
$\pi(p,w) \geq \pi(p,w')$.

EDIT 2. If the output price $p$ is endogenous and adjusts to aggregate output supply and demand, as it seems to be the case in your question, the issue has been treated by:  
Heiner, R. A. (1982): “Theory of the Firm in “Short-Run” Industry Equilibrium,” American Economic Review, 72, 555-62.  
Braulke, M. (1984): “The Firm in Short-Run Industry Equilibrium: Comment,” American Economic Review, 74, 750-753.

> **Comments:**
> **username_5**: Note that $\pi(p,w)={\color{red}{\max_{x\ge0}}pf(x)-w\cdot x}=pf(x^*(p,w))-w\cdot x^*(p,w)$, where $x^*(p,w)$ is the maximizer given $p$ and $w$. So it not necessary that $x^*(p,w)=x^*(p,w')$ when $w\le w'$.  
> **username_4**: @username_5. Yes of course, this is not necessary (and I do not assume it). I do not understand your comment very well...  
> **username_5**: Perhaps you can try to provide a more explicit explanation of the first sentence in your answer. Specifically, how did you derive the last two inequalities from knowing $w\le w'$?  
> **username_5**: Thanks for the update. But I think there is still some inconsistencies. By definition, $x^*(p,w')$ maximizes profit when output price is $p$ and input prices are $w'$. If you agree with this interpretation of notation, then how is it possible to have $$ pf(x^*(p,w')) − w · x^*(p,w') \color{red}{\overset{??}{\geq}} pf(x^*(p,w')) − w' · x^*(p,w').$$  
> **username_4**: @username_5. This is due to $w \leq w' \implies w · x^*(p,w') \leq w' · x^*(p,w')$, I think this point was clear in the edit.  
> **username_5**: Thanks again. And yes, you're right! I must have misread something in your answer earlier. Sorry.  
> **username_3**: This is clearly more elegant solution than that I have posted. Upvoted!  

---

