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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

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3 Answers 3

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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 $(2)$:

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

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  • $\begingroup$ Where is the multivariate calculus here? $\endgroup$
    – Giskard
    Commented Feb 23, 2023 at 20:51
  • $\begingroup$ 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. $\endgroup$ Commented Feb 23, 2023 at 21:02
  • $\begingroup$ You are welcome! $\endgroup$ Commented Feb 24, 2023 at 9:16
  • $\begingroup$ You are welcome! $\endgroup$ Commented Feb 25, 2023 at 19:08
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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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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.

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    $\begingroup$ 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. $\endgroup$
    – Giskard
    Commented Feb 24, 2023 at 9:04
  • $\begingroup$ @Giskard, that's true, I overlooked this. $\endgroup$
    – VARulle
    Commented Feb 26, 2023 at 10:49

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