<p>Even though it could seem a more complicate way, actually <strong>using multivariate calculus is the simplest way</strong>, as simple as univariate calculus.</p>
<p>In fact, a partial derivative is calculated in the same way as the derivative of a function of only <strong>one</strong> variable.</p>
<p>You can write your equation of the demand <span class="math-container">$Q$</span> of barley as:</p>
<p><span class="math-container">$Q(PB, PW, M)=1,900−4PB+0.1M+2PW \qquad (1)$</span></p>
<p>where we see that <span class="math-container">$Q$</span> is a function of three variables <span class="math-container">$PB, PW, M$</span>.</p>
<p>The elasticity <span class="math-container">$\epsilon$</span> of the demand of barley with respect to its own price is, by definition (taking the absolute value of the derivative):</p>
<p><span class="math-container">$$\epsilon =|\frac {\partial Q}{\partial PB }\frac {P}{Q}|.\qquad (2)$$</span></p>
<p>The partial derivative of <span class="math-container">$Q$</span> with respect to <span class="math-container">$PB$</span> is, by definition, the derivative (of a function of one variable, <span class="math-container">$PB$</span> only) of <span class="math-container">$(1)$</span> with respect to <span class="math-container">$PB$</span>, taking fixed <span class="math-container">$PW$</span> and <span class="math-container">$M$</span>.</p>
<p>So, we have</p>
<p><span class="math-container">$$\frac {\partial Q}{\partial PB }= -4 \qquad (3)$$</span></p>
<p>and taking the absolute we have, according to <span class="math-container">$(2)$</span>:</p>
<p><span class="math-container">$$\epsilon = 4\frac {P}{Q}.\qquad (4)$$</span></p>

CC BY-SA 4.0
2023-02-23T20:51:04.747Z
Where is the multivariate calculus here?
Giskard
CC BY-SA 4.0
2023-02-23T21:02:46.450Z
There is multivariate calculus here because we have a partial derivative, which is a derivative of a function of several variables, which belongs, by definition, to multivariate calculus. In univariate calculus we don't have the concept of partial derivative.
BakerStreet
CC BY-SA 4.0
2023-02-24T09:16:34.317Z
You are welcome!
BakerStreet
CC BY-SA 4.0
2023-02-25T19:08:43.373Z
You are welcome!
BakerStreet
CC BY-SA 4.0
2023-02-23T15:51:55.403Z
2023-02-23T15:51:55.403Z
BakerStreet
<p>There is no such thing as &quot;the&quot; elasticity of demand. Elasticities are always specified with respect to some variable the demand depends on. In your example there is an own-price elasticity, a cross-price elasticity and an income elasticity of demand. Each one is calculated using the respective partial derivative. If &quot;the elasticity&quot; is mentioned without specifying the variable, then this usually means the own-price elasticity, which is <span class="math-container">$-4P/Q$</span> here.</p>

CC BY-SA 4.0
2023-02-24T09:04:16.773Z
I wrote the same thing as a comment, but the text in the body of the question gets the terms right, so I deleted it.
Giskard
CC BY-SA 4.0
2023-02-26T10:49:38.920Z
@Giskard, that's true, I overlooked this.
VARulle
CC BY-SA 4.0
2023-02-24T04:11:53.233Z
2023-02-24T04:11:53.233Z
VARulle
<p>Own price elasticity of demand is just the elasticity of the demand w.r.t. the own price, here PB. The other variables are treated as constant parameters, ceteris paribus.</p>

CC BY-SA 4.0
2023-02-23T15:34:00.437Z
2023-02-23T15:34:00.437Z
Giskard
<p>How should one go about determining the own price elasticity of demand of the following:
Assume that the market demand for barley is given by: Q=1,900−4PB+0.1M+2PW
, where Q
is the quantity of barley demanded, PB
is the price of barley, M
is income (say per capita income of consumers) and PW
is the price of wheat. The prices of wheat and barley are each 200 (say £s per tonne) and M
is 1,000</p>
<p>Does one need to use multivariable calculus to differentiate Q with regards to PB,PW and M, or is there a simpler way?</p>
<p>Thanks</p>

<microeconomics><supply-and-demand><elasticity>
How to determine elasticity of demand when equation has more than one variable
John M Keynes
# How to determine elasticity of demand when equation has more than one variable

(Asked by: username_0 on 2023-02-23)

How should one go about determining the own price elasticity of demand of the following:
Assume that the market demand for barley is given by: Q=1,900−4PB+0.1M+2PW
, where Q
is the quantity of barley demanded, PB
is the price of barley, M
is income (say per capita income of consumers) and PW
is the price of wheat. The prices of wheat and barley are each 200 (say £s per tonne) and M
is 1,000

Does one need to use multivariable calculus to differentiate Q with regards to PB,PW and M, or is there a simpler way?

Thanks

**Tags:** microeconomics, supply-and-demand, elasticity


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## 3 answers

### Answer by username_1 on 2023-02-23. Score: 2

Even though it could seem a more complicate way, actually **using multivariate calculus is the simplest way**, as simple as univariate calculus.

In fact, a partial derivative is calculated in the same way as the derivative of a function of only **one** variable.

You can write your equation of the demand $Q$ of barley as:

$Q(PB, PW, M)=1,900−4PB+0.1M+2PW \qquad (1)$

where we see that $Q$ is a function of three variables $PB, PW, M$.

The elasticity $\epsilon$ of the demand of barley with respect to its own price is, by definition (taking the absolute value of the derivative):

$$\epsilon =|\frac {\partial Q}{\partial PB }\frac {P}{Q}|.\qquad (2)$$

The partial derivative of $Q$ with respect to $PB$ is, by definition, the derivative (of a function of one variable, $PB$ only) of $(1)$ with respect to $PB$, taking fixed $PW$ and $M$.

So, we have

$$\frac {\partial Q}{\partial PB }= -4 \qquad (3)$$

and taking the absolute we have, according to $(2)$:

$$\epsilon = 4\frac {P}{Q}.\qquad (4)$$

> **Comments:**
> **username_2**: Where is the multivariate calculus here?  
> **username_1**: There is multivariate calculus here because we have a partial derivative, which is a derivative of a function of several variables, which belongs, by definition, to multivariate calculus. In univariate calculus we don't have the concept of partial derivative.  
> **username_1**: You are welcome!  
> **username_1**: You are welcome!  

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### Answer by username_3 on 2023-02-24. Score: 0

There is no such thing as "the" elasticity of demand. Elasticities are always specified with respect to some variable the demand depends on. In your example there is an own-price elasticity, a cross-price elasticity and an income elasticity of demand. Each one is calculated using the respective partial derivative. If "the elasticity" is mentioned without specifying the variable, then this usually means the own-price elasticity, which is $-4P/Q$ here.

> **Comments:**
> **username_2**: I wrote the same thing as a comment, but the text in the body of the question gets the terms right, so I deleted it.  
> **username_3**: @username_2, that's true, I overlooked this.  

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### Answer by username_2 on 2023-02-23. Score: 0

Own price elasticity of demand is just the elasticity of the demand w.r.t. the own price, here PB. The other variables are treated as constant parameters, ceteris paribus.


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