2
$\begingroup$

What are the rules of total differentials? I. e., given an arbitrary expression of which to take the total differential, what rules can be applied to arrive at the desired result, and how does one go about doing so?

$\endgroup$

closed as off-topic by Giskard, Art, jmbejara, Maarten Punt, Adam Bailey Sep 8 at 21:11

  • This question does not appear to be about economics, within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    $\begingroup$ This question should be in math.stackexchange, or it is likely already there. You could revise the question and make it more economics-specific such as “total differentials and their applications in economics”. The answer provided does indeed connect this but this question as it stands is about math $\endgroup$ – Brennan Aug 25 at 2:45
  • 5
    $\begingroup$ I'm voting to close this question as off-topic because it is about math, not economics. $\endgroup$ – Giskard Aug 25 at 10:53
  • $\begingroup$ Fair enough. I posted it here on the assumption that economists (students in particular, like me) would be more likely to search for / find this information here on this site. Feel free to edit, amend, close and / or move as you see fit. $\endgroup$ – chsk Aug 26 at 14:07
2
$\begingroup$

Recall that the total differential $df$ of a function $f : \mathbb{R}^n \to \mathbb{R}$, $x = (x_1, \dots, x_n) \mapsto f(x)$, is defined as

$$df = \sum_{i = 1}^n \frac{ \partial f }{ \partial x_i } dx_i = \sum_{i = 1}^n f_{x_i} dx_i = \nabla f \cdot \left(dx_1, \dots, dx_n\right)^\top$$

Let $X$, $Y$, $Z$ be variables, and let $a$, $b$, $c$, $k$ and $n$ be constants. Then the following rules are valid:

$$\begin{alignat}{3} dk & = 0 && \qquad \text{(constant rule)} \\ d(cX^n) & = cnX^{n-1} dX && \qquad\text{(power rule)} \\ d(aX + bY) & = a\cdot dX + b \cdot dY && \qquad\text{(sum rule)} \\ d(XY) & = Y \cdot dX + X \cdot dY && \qquad\text{(product rule)} \\ d(XYZ) & = YZ \cdot dX + XZ \cdot dY + XY \cdot dZ && \qquad\text{(product rule for three factors)} \\ d\left( \frac{X}{Y} \right) & = \frac{ Y \cdot dX - X \cdot dY }{ Y^2 } && \qquad\text{(quotient rule)} \\ d(X^{-1}) & = -\frac{ dX }{ X^2 } && \qquad\text{(inverse rule)} \\ d(X \circ Y) & = X'(Y) \cdot dY && \qquad\text{(chain rule)} \end{alignat}$$

When evaluating around $(X, Y, \dots) = (X_0, Y_0, \dots)$ (e. g. around the steady state of a model), replace $X$ by $X_0$, $Y$ by $Y_0$ etc. in the above rules. For $X = X_0$ or $Y = Y_0$ respectively:

$$\begin{align} d(cX^n) \biggr\rvert_{X = X_0} & = cnX_0^{n-1} dX \\ d(X^{-1}) \biggr\rvert_{X = X_0} & = -\frac{ dX }{ X_0^2 } \\ d(X \circ Y) \biggr\rvert_{Y = Y_0} & = X'(Y_0) \cdot dY \end{align}$$

For $(X, Y) = (X_0, Y_0)$:

$$\begin{align} d(aX + bY) \biggr\rvert_{(X, Y) = (X_0, Y_0)} & = a\cdot dX + b \cdot dY \\ d(XY) \biggr\rvert_{(X, Y) = (X_0, Y_0)} & = Y_0 \cdot dX + X_0 \cdot dY \\ d\left( \frac{X}{Y} \right) \Biggr\rvert_{(X, Y) = (X_0, Y_0)} & = \frac{ Y_0 \cdot dX - X_0 \cdot dY }{ Y_0^2 } \end{align}$$

For ($X, Y, Z) = (X_0, Y_0, Z_0)$:

$$\begin{align} d(XYZ) \biggr\rvert_{(X, Y, Z) = (X_0, Y_0, Z_0)} & = Y_0Z_0 \cdot dX + X_0Z_0 \cdot dY + X_0Y_0 \cdot dZ \end{align}$$

As an example, consider the IS equation

$$Y = C((1 - t) \cdot Y) + I(i, E) + G$$

of an IS/LM model of a closed economy with a flat tax rate $t$. Take the total differential around the current state of the economy, using the rules above and setting $Y^v = (1 - t) Y$:

$$\begin{align} dY & = d\left[ C((1 - t) Y) + I(i, E) + G \right] \\ & = d\left[ C((1 - t) Y) \right] + d\left[ I(i, E) \right] + dG \\ & = C_{Y^v} d\left[ (1 - t) Y \right] + I_i \, di + I_E \, dE + dG \\ & = C_{Y^v} \left[ Y_0 d(1 - t) + (1 - t_0) \, dY \right] + I_i \, di + I_E \, dE + dG \\ & = C_{Y^v} \left[ Y_0 (-dt) + (1 - t_0) \, dY \right] + I_i \, di + I_E \, dE + dG \\ & = C_{Y^v} \left[ (1 - t_0) \, dY - Y_0 \, dt \right] + I_i \, di + I_E \, dE + dG \end{align}$$

References:

  • Chiang, Alpha C. and Kevin Wainwright, Fundamental Methods of Mathematical Economics, 4th ed., McGraw-Hill 2005, chapter 8.
$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.