### A Note on How to Take Total Differential of a Function

$$ \mathrm{To}\ \mathrm{illustrate},\mathrm{consider}\ \mathrm{the}\ \mathrm{function}\;Z=F\left(x,y\right) $$

(2.33)

To take total differential of this function, you have to differentiate *F*(*x*,*y*) partially with respect to *x* and multiply it by *dx*; again differentiate *F*(*x*,*y*) partially with respect to *y* and multiply it by *dy* and, then, add the two.

Thus, total differential of

*Z* =

*F*(

*x*,

*y*) is given by

$\begin{array}{c}dZ=\frac{\partial F\left(x,y\right)}{\partial x}dx+\frac{\partial F\left(x,y\right)}{\partial y}\mathit{dy}\text{\u2261}{F}_{x}\left(x,y\right)dx+{F}_{y}\left(x,y\right)\mathit{dy},\phantom{\rule{0.24em}{0ex}}\mathrm{where}\phantom{\rule{0.24em}{0ex}}{F}_{x}\left(x,y\right)\\ \text{\u2261}\frac{\partial F\left(x,y\right)}{\partial x}\phantom{\rule{0.24em}{0ex}}\mathrm{and}\phantom{\rule{0.24em}{0ex}}{F}_{y}\left(x,y\right)\text{\u2261}\frac{\partial F\left(x,y\right)}{\partial x}\end{array}$

(2.34)

(When you differentiate

*F*(

*x*,

*y*) partially with respect to

*x*, you treat

*y* as given. Similarly, for

*y* or any other variable).

Let us explain the above equation. Value of *Z* is determined by the values of *x* and *y*. The change in the value of *Z* is, therefore, due to changes in the values of *x* and *y*. Consider any given values of *x* and *y*, say, (*x*_{0},*y*_{0}). The value of *Z* corresponding to (*x*_{0},*y*_{0}) is denoted by Z_{0} so that Z_{0} = *F*(*x*_{0},*y*_{0}). If from (*x*_{0},*y*_{0}) the value of *x* changes by a very small amount *dx*, with *y* remaining unchanged at *y*_{0}, then per unit change in *x*, the value of *Z* changes by F_{x}(*x*_{0},*y*_{0}). Therefore, the total change in the value of *Z*, when *x* changes from (*x*_{0},*y*_{0}) by *dx* with *y* remaining unchanged at *y*_{0}, is given by F_{x}(*x*_{0},*y*_{0})*dx*. Similar interpretation holds for F_{y}(*x*_{0},*y*_{0})*dy*. (Interpret it). Therefore, the total change in the value of *Z*, when *x* and *y* change from (*x*_{0},*y*_{0}) by *dx* and *dy*, respectively, is given by dz = F_{x}(*x*_{0},*y*_{0})*dx* + F_{y}(*x*_{0},*y*_{0})*dy*. Hence, if values of *x* and *y* change from any given (*x*,*y*) by *dx* and *dy*, respectively, then the resulting change in the value of *Z* is given by the RHS of (2.34).

Let us illustrate with an example. Suppose (*x*,*y*) represent quantities of two goods *x* and *y*. They are bought and sold at given prices *P*_{x} and *P*_{y}. The value of (*x*,*y*) denoted *Z* is given by *Z* = *P*_{x}*x* + *P*_{y}*y*. Take total differential of *Z* and explain it.

Consider the budget equation of a consumer given by

*M* =

*P*_{x} *x* +

*P*_{y}*y*. Taking total differential of the budget equation, with

*P*_{x} and

*P*_{y} fixed, we get

$$ dM={P}_x dx+{P}_y dy $$

(2.35)

Let us explain the above equation. The LHS gives the change in the consumer’s income. Let us now focus on the RHS. Consider any (*x*,*y*) that satisfied the initial budget equation of the consumer *M* = *P*_{x} *x* + *P*_{y}*y*. If the consumer from this given (*x*,*y*) changes *x* and *y* by *dx* and *dy*, respectively, then the value of his purchases will change by the RHS of (2.35). If the LHS equals the RHS, the consumer’s new budget equation will be satisfied. More precisely, (2.35) yields all combinations of *dx* and *dy*, the value of each of which equals the change in the consumer’s budget.

### How to Represent an Equation in a Diagram (Roughly)

For this purpose, consider the equation of the indifference curve

To present this equation in a diagram, you have to first derive the slope of this equation. To derive the slope, take total differential to get

From (

2.37) we get

$$ dy/ dx=-\left(y/x\right) $$

(2.38)

From (2.38) we find that the slope is negative for positive values of *x* and *y*. (Note that *y* and *x* cannot be negative as they denote quantities of two different goods). This implies that the equation represents a negatively sloped schedule in the (*x*,*y*) plane.

We also find from (2.38) that the slope is not a constant. Its value depends upon the values of *x* and *y*. Therefore, the schedule representing the equation is non-linear. To have an idea as regards the curvature of the schedule, we have to examine how the absolute value of the slope behaves as we raise *x* and, therefore, lower *y* along the indifference curve. It is clear from (2.38) that mrs_{xy} (given by −*dy*/*dx* = (*y*/*x*)) falls as *x* rises and *y* falls along the indifference curve. This means that the indifference curve is strictly convex to the origin. Along such a curve, the absolute value of its slope falls as *x* rises and *y* falls along it. Check it for yourself.

We can examine how the absolute value of the slope of a curve behaves as we move along the curve mathematically also. This may be necessary, if the expression giving the value of the slope of the curve is complicated. Let us do it in the case of the given indifference curve for the purpose of illustration. Let us first write (

2.38) as

$$ -\left( dy/ dx\right)=y/x $$

(2.39)

Taking total differential of (

2.39), we get

$$ \mathrm{d}\left[-\left( dy/ dx\right)\right]=-\left(1/{x}^2\right) ydx+\left(1/x\right) dy $$

(2.40)

Equation (

2.40) gives us the change in the value of mrs

_{xy}, when from any given (

*x*,

*y*) satisfying (

2.36),

*x* and

*y* change by

*dx* and

*dy*, respectively. However, we do not want

*dx* and

*dy* to be arbitrary. We want

*dx* and

*dy* to be such that (

2.37) is satisfied so that we remain on the given indifference curve. Therefore, for any given

*dx*, the value of

*dy*, as derived from (

2.37), should equal −(

*y*/

*x*)

*dx*. Substituting it into (

2.40), we get

$$ \mathrm{d}\left[-\left( dy/ dx\right)\right]=-\left(1/{x}^2\right) ydx+\left(1/x\right)\left(-y/x\right) dx=-2\left(1/{x}^2\right) ydx<0,\mathrm{when}\; dx>0 $$

(2.41)

Equation (2.41) gives us the change in the value of mrs_{xy}, when from any given (*x*,*y*) on the given indifference curve, *x* and *y* are changed by such values that we remain on the given indifference curve. We find from (2.41) that if from any given (*x*,*y*) satisfying (2.36) *x* is increased and *y* is decreased commensurately so that we remain on the given indifference curve, mrs_{xy} falls. Therefore, the indifference curve is strictly convex to the origin.