# Calculating the substitution effect with the derivative

Substitution Effect (SE) for a price increase of $$P_x$$ to $$P_x'$$ can be written as:

$$h(P_x', P_y, U) - h(P_x, P_y, U) = ∆h$$, where $$h$$ is Hicksian demand. Correct?

The Slutsky equations decomposes a change in Marshallian Demand into the SE and Income Effect (IE), where the SE is written as: $$\frac{\partial h}{\partial p_{x}}$$

Question:

For small $$∆P_x$$ is this correct: $$∆h = h(P_x', P_y, U) - h(P_x, P_y, U) \approx \frac{\partial h}{\partial p_{x}}∆P_x$$

I.e. Unless our Hicksian demand was linear, it would be inappropriate to actually calculate the SE for large changes in P using:

$$\frac{\partial h}{\partial p_{x}}∆P_x$$

This would only give us a decent linear approximation of the value of the SE for small changes in $$P_x$$. Am i correct, or am i getting my calculus confused? Thanks!

You wrote:

Unless our Hicksian demand was linear, it would be inappropriate to actually calculate the SE for large changes in P using:

$$\frac{\partial h}{\partial p_{x}}∆P_x$$

You are right, a differential of a function of one variable $$f(x)$$ is a good approximation of the increase of the function only for small changes of $$x$$.

To summarize the reason why this approximation is not good for large changes of $$x$$, I synthesize below, in the picture, the geometric meaning of the differential of a function $$y=f(x)$$ of one single variable.

As I think you already know, the derivative of a function $$f: \mathbb {R} \rightarrow \mathbb {R}$$ in a point $$x_0$$ represents geometrically the slope of the tangent line to the function at $$x_0$$ (the trigonometric tangent of the angle $$\alpha$$ in the picture above).

The differential $$dy$$ of $$f(x)$$ in $$x_0$$ is, by definition:

$$dy= f'(x_0) \Delta x \;\;\;\;\;\;\;(1)$$

where $$f'(x_0)$$ is the derivative of $$f$$ at $$x_0$$.

When $$x$$ increase from $$x_0$$ to $$x_0+\Delta x$$, the function increases from $$f(x_0)$$ to $$f(x_0+\Delta x)$$, and its increase is $$\Delta Y$$, which in the picture is equal to the segment $$\bar {BS}$$.

The differential $$dy$$ in the picture is equal (according to $$(1)$$) to the segment $$TS$$, that is the increase of the ordinate of the tangent line following the increase $$\Delta x$$.

The segment $$\bar {BT}= \bar {BS} -\bar {TS}$$ represents the 'error' we make when approximating the increase of the function $$\Delta Y$$ with the differential $$dy$$.

As can be seen in the picture, this error becomes larger and larger as $$\Delta x$$ becomes larger, and smaller as $$\Delta x$$ becomes smaller, and tends to $$0$$ as $$\Delta x \rightarrow 0$$.

Of course this is a geometric, informal argument, but what we said can be showed formally and rigorously.

$$\;$$

OBSERVATION. Just an observation to avoid a possible confusion.

In your text we have a partial derivative. But the argument above is completely correct, even if we have a partial derivative.

$$\frac{\partial h}{\partial p_{x}}∆P_x$$ is not the differential of the function of several variables $$h(P_x, P_y, U)$$, of course. But $$\frac{\partial h}{\partial p_{x}}∆P_x$$ is nevertheless a differential, the differential of the function of one variable $$h(P_x, \bar P_y, \bar U)$$, where $$\bar P_y$$ and $$\bar U$$ are fixed value.

A partial derivative is, actually, a derivative of a function of one variable, by definition, because it is the derivative of the function which is obtained keeping fixed all the variables, except the variable with respect to we are taking the derivative.

In our case, the function is $$h(P_x, \bar P_y, \bar U)$$. As $$\frac{\partial h}{\partial p_{x}}∆P_x$$ is the differential of the function of one variable $$h(P_x, \bar P_y, \bar U)$$, what we said above about the differential of a function of one variable applies.

• This seems like an exceptional answer, I really appreciate you taking the time to clarify the matter for me. I just had one question about your note at the end which I will add in a separate comment: Commented Feb 10, 2023 at 13:23
• 1) You are correct, that if I was calculating/graphing the Hicksian demand curve for $x$, then I should technically have $h(P_x, \bar P_y, \bar U)$, i.e. clearly show $p_x$ and $U$ are fixed. Should I do that, is my notation technically wrong? Commented Feb 10, 2023 at 13:29
• Question 1). I hope I understand you exactly, you want to know if you should indicate that a variable is fixed when you take a partial derivative. No, it is implicit, for the very definition of partial derivative. And, in general, we mustn't indicate in notation a fixed variable every time we use it, if it has been said and it is clear from the context. To indicate it in notation every time will be heavy unnecessary notation. So, I think that your notation is correct. As for question 2) I'll read it and I answer you later, in few time. Thank you for your kindness. Commented Feb 10, 2023 at 13:49
• Amazing, I appreciate it! Commented Feb 10, 2023 at 14:03
• In short, at the moment don't worry, wait to study functions of several variables. These should be studied on a good book of mathematical analysis, or of more advanced calculus. Commented Feb 10, 2023 at 17:15