<p>Consider the minimisation problem
<span class="math-container">$$
OF(N,u,\eta,c) = \min_\phi C_1 S(\phi; N,u,\eta,c)
$$</span>
Here, I'm assuming that the minimisation is with respect to <span class="math-container">$\phi$</span>. (don't know if this is correct as it is not specified in your model).</p>
<p>Let <span class="math-container">$\phi^\ast(N,u,\eta,c)$</span> be the (assumed unique) optimal solution. And assume that it is identified by the suitable first order condition.</p>
<p>Then we can use the enveloppe theorem to conclude that:
<span class="math-container">$$
\frac{d OF(N,u,\eta,c)}{d N} = C_1 \frac{d S(\phi^\ast; N, u, \eta, c)}{d N} = C_1 \phi^\ast F(\phi^\ast(u-\eta) + \eta - c).
$$</span></p>
<p>I'm mainly guesssing here as it is not very clear from you question which variable is endogenously set by the equilibrium condition and what variable you are minimising over.</p>

CC BY-SA 4.0
2022-01-26T12:26:00.853Z
Thank you for your response! For additional information: There is a pool of N players, and a proportion NF(.) players turn up who have to "serve" S customers. S, N, u, $\eta$, c are parameters and common knowledge (distribution function F(.) is common knowledge too). $\phi$ is endogenously determined by the implicit equation above which also defines the equilibrium. By Intermediate Value Theorem, we know a unique solution to $\phi$ exists. Now, even though the OF is Min, I don't have a decision variable (yet) and just want to analyze how OF behaves wrt different parameters. Hope it helps
gradstudent
CC BY-SA 4.0
2022-01-27T06:27:11.470Z
@gradstudent If you minimize something, e.g. $\min C_1 S$, you need to have a decision variable: something that is allowed to change in order to get a minimum. Otherwise this is not really a minmisation problem.
tdm
CC BY-SA 4.0
2022-01-27T07:16:29.620Z
fair enough! Actually the expression C1S is the outcome of a simplification of a larger objective function which I didn't outline here. Suppose I were to only analyze how this function C1S behaves wrt the parameters. What am I missing/doing wrong according to you?
gradstudent
CC BY-SA 4.0
2022-01-27T12:50:44.640Z
Then you can't use the enveloppe theorem to simplify things. Also, either you treat $S$ as a parameter or as a function, you can't do both.
tdm
CC BY-SA 4.0
2022-01-27T15:01:34.023Z
I see. S is definitely a parameter, but my idea was to substitute its expression from the equilibrium equation containing $\phi$. I thought that is the correct way of doing things, and then I proceeded to use total derivative formula, which is how I ended up getting 0.
gradstudent
CC BY-SA 4.0
2022-01-26T09:02:41.350Z
2022-01-26T09:02:41.350Z
tdm
<p>I am playing with a toy model version for my research and I have an implicit equation defining the equilibrium:
<span class="math-container">$$\phi = \frac {S}{NF(\phi(u-\eta)+\eta - c)}$$</span></p>
<p>where <span class="math-container">$S,N,u,\eta,c$</span> are parameters and <span class="math-container">$F(.)$</span> is the cdf function. I have a cost minimizing objective function defined as <span class="math-container">$Min C_1S$</span>, where <span class="math-container">$C_1$</span> is a separate constant cost coefficient.</p>
<p>I need to perform comparative statics analysis of the objective function wrt different parameters, so I substituted the value of <span class="math-container">$S$</span> from the implicit equation above and then proceeded to compute the total derivative of the objective function wrt the different parameters. My expressions evaluate to zero, so I am clearly doing something wrong and I can't understand what.</p>
<p>Here's my attempt:
Say the objective function is OF. Then I need <span class="math-container">$\frac{dOF}{dN}$</span>. According to total derivative formula, we should have: <span class="math-container">$\frac{dOF}{dN}=\frac{\partial OF}{\partial \phi}\frac{d\phi}{dN}+\frac{\partial OF}{\partial N}$</span>. Now I wasn't sure of <span class="math-container">$\frac{d\phi}{dN}$</span>, so I tried two approaches.</p>
<p>Firstly, I tried <span class="math-container">$\frac{\partial\phi}{\partial N}$</span> because <span class="math-container">$\phi$</span> is affected by all these parameters <span class="math-container">$N, S,$</span> etc. I used the implicit function theorem to arrive at <span class="math-container">$\frac{-\phi F(\phi(u-\eta)+\eta -c)}{NF(\phi(u-\eta)+\eta -c) + \phi N(u-\eta)f(\phi(u-\eta)+\eta -c)}$</span>.</p>
<p>Multiplying this with <span class="math-container">$\frac{\partial OF}{\partial \phi}$</span> I finally get <span class="math-container">$-C_1\phi F(\phi(u-\eta)+\eta -c)$</span>. But <span class="math-container">$\frac{\partial OF}{\partial N}$</span> evaluates to <span class="math-container">$C_1\phi F(\phi(u-\eta)+\eta -c)$</span>, so the two terms cancel each other out.</p>
<p>Then I thought, perhaps I need to calculate <span class="math-container">$\frac{d\phi}{dN}$</span> since this is the equilibrium and we don't need the ceteris paribus condition. From the implicit function, I took derivative of the entire expression wrt N. This is what I get:</p>
<p><span class="math-container">$\frac{\partial \phi}{\partial N}NF(\phi(u-\eta)+\eta -c)+\phi F(\phi(u-\eta)+\eta -c)+\phi Nf(\phi(u-\eta)+\eta -c)\frac{\partial \phi}{\partial N}(u-\eta)$</span>, here <span class="math-container">$f()$</span> is the pdf.</p>
<p>I substitute the value of <span class="math-container">$\frac{\partial \phi}{\partial N}$</span> from the implicit function theorem and <span class="math-container">$\frac{d\phi}{dN}$</span> evaluates to zero. This also happens when I follow this procedure for other parameters, say <span class="math-container">$\eta$</span>.</p>

CC BY-SA 4.0
2022-01-26T00:45:04.813Z
Can you show us your attempt please?
1muflon1
CC BY-SA 4.0
2022-01-26T01:05:47.610Z
Updated the post.
gradstudent
<microeconomics><comparative-statics><product-differentiation><equilibrium>
Total derivative evaluates to zero: problem while doing comparative statics
gradstudent
# Total derivative evaluates to zero: problem while doing comparative statics

(Asked by: username_0 on 2022-01-26)

I am playing with a toy model version for my research and I have an implicit equation defining the equilibrium:
$$\phi = \frac {S}{NF(\phi(u-\eta)+\eta - c)}$$

where $S,N,u,\eta,c$ are parameters and $F(.)$ is the cdf function. I have a cost minimizing objective function defined as $Min C\_1S$, where $C\_1$ is a separate constant cost coefficient.

I need to perform comparative statics analysis of the objective function wrt different parameters, so I substituted the value of $S$ from the implicit equation above and then proceeded to compute the total derivative of the objective function wrt the different parameters. My expressions evaluate to zero, so I am clearly doing something wrong and I can't understand what.

Here's my attempt:
Say the objective function is OF. Then I need $\frac{dOF}{dN}$. According to total derivative formula, we should have: $\frac{dOF}{dN}=\frac{\partial OF}{\partial \phi}\frac{d\phi}{dN}+\frac{\partial OF}{\partial N}$. Now I wasn't sure of $\frac{d\phi}{dN}$, so I tried two approaches.

Firstly, I tried $\frac{\partial\phi}{\partial N}$ because $\phi$ is affected by all these parameters $N, S,$ etc. I used the implicit function theorem to arrive at $\frac{-\phi F(\phi(u-\eta)+\eta -c)}{NF(\phi(u-\eta)+\eta -c) + \phi N(u-\eta)f(\phi(u-\eta)+\eta -c)}$.

Multiplying this with $\frac{\partial OF}{\partial \phi}$ I finally get $-C\_1\phi F(\phi(u-\eta)+\eta -c)$. But $\frac{\partial OF}{\partial N}$ evaluates to $C\_1\phi F(\phi(u-\eta)+\eta -c)$, so the two terms cancel each other out.

Then I thought, perhaps I need to calculate $\frac{d\phi}{dN}$ since this is the equilibrium and we don't need the ceteris paribus condition. From the implicit function, I took derivative of the entire expression wrt N. This is what I get:

$\frac{\partial \phi}{\partial N}NF(\phi(u-\eta)+\eta -c)+\phi F(\phi(u-\eta)+\eta -c)+\phi Nf(\phi(u-\eta)+\eta -c)\frac{\partial \phi}{\partial N}(u-\eta)$, here $f()$ is the pdf.

I substitute the value of $\frac{\partial \phi}{\partial N}$ from the implicit function theorem and $\frac{d\phi}{dN}$ evaluates to zero. This also happens when I follow this procedure for other parameters, say $\eta$.

**Tags:** microeconomics, comparative-statics, product-differentiation, equilibrium
> **Comments:**
> **username_1**: Can you show us your attempt please?  
> **username_0**: Updated the post.  

---

## 1 answers

### Answer by username_2 on 2022-01-26. Score: 4

Consider the minimisation problem
$$
OF(N,u,\eta,c) = \min\_\phi C\_1 S(\phi; N,u,\eta,c)
$$
Here, I'm assuming that the minimisation is with respect to $\phi$. (don't know if this is correct as it is not specified in your model).

Let $\phi^\ast(N,u,\eta,c)$ be the (assumed unique) optimal solution. And assume that it is identified by the suitable first order condition.

Then we can use the enveloppe theorem to conclude that:
$$
\frac{d OF(N,u,\eta,c)}{d N} = C\_1 \frac{d S(\phi^\ast; N, u, \eta, c)}{d N} = C\_1 \phi^\ast F(\phi^\ast(u-\eta) + \eta - c).
$$

I'm mainly guesssing here as it is not very clear from you question which variable is endogenously set by the equilibrium condition and what variable you are minimising over.

> **Comments:**
> **username_0**: Thank you for your response! For additional information: There is a pool of N players, and a proportion NF(.) players turn up who have to "serve" S customers. S, N, u, $\eta$, c are parameters and common knowledge (distribution function F(.) is common knowledge too). $\phi$ is endogenously determined by the implicit equation above which also defines the equilibrium. By Intermediate Value Theorem, we know a unique solution to $\phi$ exists. Now, even though the OF is Min, I don't have a decision variable (yet) and just want to analyze how OF behaves wrt different parameters. Hope it helps  
> **username_2**: @gradstudent If you minimize something, e.g. $\min C_1 S$, you need to have a decision variable: something that is allowed to change in order to get a minimum. Otherwise this is not really a minmisation problem.  
> **username_0**: fair enough! Actually the expression C1S is the outcome of a simplification of a larger objective function which I didn't outline here. Suppose I were to only analyze how this function C1S behaves wrt the parameters. What am I missing/doing wrong according to you?  
> **username_2**: Then you can't use the enveloppe theorem to simplify things. Also, either you treat $S$ as a parameter or as a function, you can't do both.  
> **username_0**: I see. S is definitely a parameter, but my idea was to substitute its expression from the equilibrium equation containing $\phi$. I thought that is the correct way of doing things, and then I proceeded to use total derivative formula, which is how I ended up getting 0.  

---

