My Math Forum Differentiate both sides of equation (labour demand and supply)

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December 20th, 2016, 11:42 PM   #1
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Differentiate both sides of equation (labour demand and supply)

Hi all

I am looking at the simple economic theory behind the impact of wage subsidies and I'm struggling with a simple differentiation of the labour demand and labour supply functions.

In Page 13 of the linked pdf (extracted image attached), the author differentiates Equation (2) to get Equation (3). It appears to be a simple differentiation but I'm not able to work through the steps to get (3), and I'm not sure how to differentiate both sides of an equation that has different function arguments.

Any help will be greatly appreciated!

https://ink.library.smu.edu.sg/cgi/v...t=soe_research
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 December 21st, 2016, 02:56 PM #2 Global Moderator   Joined: May 2007 Posts: 6,788 Thanks: 708 $\displaystyle s^w$ is the independent variable. Everything needs to be expressed as functions of it.
December 21st, 2016, 06:10 PM   #3
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Quote:
 Originally Posted by mathman $\displaystyle s^w$ is the independent variable. Everything needs to be expressed as functions of it.
But the right hand side is not a function of $\displaystyle s^w$?

Am I suppose to differentiate the left hand side totally like the attached image?
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 December 22nd, 2016, 05:13 PM #4 Global Moderator   Joined: May 2007 Posts: 6,788 Thanks: 708 As a math question, too much is undefined. What is the relationship (if any) between w^f, w^h, and s^w? Last edited by mathman; December 22nd, 2016 at 05:17 PM. Reason: typos
December 22nd, 2016, 06:10 PM   #5
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Quote:
 Originally Posted by mathman As a math question, too much is undefined. What is the relationship (if any) between w^f, w^h, and s^w?
Economists have their own system of notation, and it makes virtually no sense from a mathematical standpoint. In this case for example, what anyone else would write as an indices, which are quite unnecessary with only four or five variables, are written as exponents. It is, in my opinion, pure obscurantism that lets economists pretend to be profound.

I am in no mood to plow through a bunch of implied math written in weird notation. Basically, what the author is saying is:

$p = P(h) = labor\ provided,\ where\ h = wage\ taken\ home\ by\ employee;$

$u = U(w) = labor\ used,\ where\ w = wage\ paid\ by\ employer;\ and$

$h = w(1 + s),\ where\ s = subsidy\ paid\ by\ government.$

At a neoclassical equilibrium, $p = u \implies P(h) = U(w) \implies P\{w(1 + s)\} = U(w).$

So far so good, though the notation is less than clear.

Next we jump into elasticities, an admittedly useful technical concept in economics. If the author thinks he is writing for anyone except fellow economists, he might think to define:

$\eta = \left | \dfrac{dp}{dh} * \dfrac{h}{p} \right|\ and\ \epsilon = \left | \dfrac{du}{dw} * \dfrac{w}{u} \right |.$

I am good up to here because, for my manifest sins, I once long ago studied some economics.

I have no idea what the author is doing when he differentiates with respect to s, which up to now appeared to be a constant such as 0.1.

Last edited by skipjack; December 24th, 2016 at 03:23 AM.

 December 23rd, 2016, 01:50 PM #6 Global Moderator   Joined: May 2007 Posts: 6,788 Thanks: 708 Using the new notation, what is the question?
 December 23rd, 2016, 06:44 PM #7 Senior Member   Joined: May 2016 From: USA Posts: 1,310 Thanks: 551 Mathman I am not quite sure. A standard neo-classical argument would posit a demand curve U(w) for labor depending on w, the wage to be paid by the employer, and a supply curve P(h) depending on h, the wage taken home by the employee. We define s = (h - w) / w, entailing that h = (1 + s)w. So redefine P(h) as P(s, w). Absent taxes or subsidies on wages, s would equal 0 by definition, and equilibrium would occur at w = w_1 such that U(w_1) = P(0, w_1). The equilibrium would be stable if $\dfrac{dU(w_1)}{dw} \le 0\ and\ \dfrac{\delta P(0,\ w_1)}{\delta w} \ge 0.$ This is standard Marshallian theory, taught in first semester microeconomics. I think where the argument is headed is to show that if s > 0, then there $\exists\ w_2\ such\ that\ w_2 < w_1 < (1 + s)w_2,\ P(s,\ w_2) = U(w_2) > U(w_1),$ $\dfrac{dU(w_2)}{dw} \le 0\ and\ \dfrac{\delta P(s,\ w_2)}{\delta w} \ge 0.$ But I am lost with respect to taking the derivative with respect to s. If both s and w are variables, then I do not grasp why we are not dealing with partial derivatives. I completely lose the thread here. I may not even get what is trying to be demonstrated. I am not at all saying that there is any error in the text, but the notation is so unnecessarily convoluted and the presentation so diffuse that I do not have the patience to decipher it. I think the OP should send an email to the author and ask for clarification directly. Last edited by skipjack; December 24th, 2016 at 01:41 AM.
 December 24th, 2016, 01:58 PM #8 Global Moderator   Joined: May 2007 Posts: 6,788 Thanks: 708 I'm just guessing, but it looks like the analysis is for fixed s.
 December 24th, 2016, 11:09 PM #9 Global Moderator   Joined: Dec 2006 Posts: 20,835 Thanks: 2162 If $w^f$ is treated as a constant, $\dfrac{dw^h}{ds^w} = \dfrac{d}{ds^w}(w^f(1 + s^w)) = w^f$. On that basis, $\dfrac{d\log w^h}{ds^w} = \dfrac{1}{w^h}\cdot\dfrac{dw^h}{ds^w} = \dfrac{\dfrac{dw^h}{ds^w}}{w^f(1 + s^w)}= \dfrac{1}{1 + s^w}$. That supposedly equals $\dfrac{\eta}{\eta + \epsilon}$, i.e. $\dfrac{1}{1 + \epsilon/\eta}$, which would require that $\epsilon/\eta = s^w = w^h/w^f - 1$. As $\epsilon$ and $\eta$ are respectively the wage elasticity of labor supply and the wage elasticity of labor demand, and labor demand equals labor supply in labor-market equilibrium, one might suppose that $\epsilon/\eta = \dfrac{dw^f}{dw^h}$ or something like that, but I don't see how he gets $w^h/w^f - 1$. Maybe someone else will do better.

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