Substituting total derivative d for partial derivative \partial

Question

In economic models it seems to be commonplace to substitute a total derivative derived from one equation, say $\frac{d k}{d \tau}$, for the partial derivative derived from another equation, say $\frac{\partial k}{\partial \tau}$.

Why is it allowed to perform this substitution? After all, $d$ and $\partial$ are different concepts, are they not?

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Here is a (simplified) example from a model on tax competition by Köthenbürger (2002):

A firm maximizes profits from the production and sale of a single good. One of the resulting first-order conditions is $f_k(k) = r(\tau) + \tau$, where $f_k$ is the derivative of the production function with respect to the input capital $k$, $r$ is the interest rate, and $\tau$ is the tax rate of a tax on capital. Differentiating with respect to $k$ and $\tau$ on both sides, we can rearrange to arrive at

$$ \frac{d k}{d \tau} = \frac{\frac{\partial r}{\partial \tau} + 1}{f_{kk}} $$

The model also contains the maximization problem of a government which yields the first-order condition

$$ \frac{d u}{d \tau} = u_c \left( - f_{kk} \frac{\partial k}{\partial \tau} k + \dots \right) + \dots \overset{!}{=} 0 $$

The analysis of the model continues with a substitution of $\frac{d k}{d \tau}$ from the first equation for $\frac{\partial k}{\partial \tau}$ in the second equation.

Answer 1

I think the reason why people do this is that, in a typical economic model, there are usually a lot of variables that are intercorrelated in one way or another. It is difficult to isolate just two variables where one is dependent on the other, and only on this single independent variable but not anything else. Therefore, most such comparative static relationship should be indeed partial derivatives. Whenever we substitute the total derivative for the partial, what we assume is that, at this moment, only the relationship between these two variables, ceteris paribus, is of our interest. Therefore the $d$ notation effectively indicates that we are holding everything else constant. In this regard, the total and the partial are the same indeed.

Answer 2

In this particular case, the substitution actually makes sense, if you put on your economist's hat.

The first expression $\frac{dk}{d \tau}$ comes from differentiating firm's optimal condition, marginal product = marginal return of capital.

In the government's problem, $\frac{\partial k}{\partial \tau}$ is the rate of change of capital conditioned on all other variables controlled by the government being held constant. But presumably this is a general equilibrium model. Also, those other variables are exogenous to the firm. So in equilibrium, this $\frac{\partial k}{\partial \tau}$ must be the same as the firm's $\frac{dk}{d \tau}$.

For the same reason, one assumes government's capital is the same as firm's capital, as is always done in decentralized equilibrium models.