# Total derivative evaluates to zero: problem while doing comparative statics

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 $$\eta$$.

– 1muflon1
Jan 26, 2022 at 0:45
• Updated the post. Jan 26, 2022 at 1:05

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.

• 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 Jan 26, 2022 at 12:26
• @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
Jan 27, 2022 at 6:27
• 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? Jan 27, 2022 at 7:16
• 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
Jan 27, 2022 at 12:50
• 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. Jan 27, 2022 at 15:01