<p><em>(I do not agree that this is a duplicate of the answer proposed by other members here).</em>  </p>

<p>Total Revenue is</p>

<p>$$TR = Q(P)\cdot P$$</p>

<p>The derivative with respect to price is </p>

<p>$$\frac{\partial TR}{\partial P} = Q'(P)\cdot P + Q(P)$$</p>

<p>For this to be positive (and so total revenue to increase with price), we require </p>

<p>$$\frac{\partial TR}{\partial P} &gt; 0 \implies  Q'(P)\cdot P + Q(P) &gt;0 $$</p>

<p>$$\implies \frac{Q'(P)\cdot P}{Q(P)} &gt; -1$$</p>

<p>The left-hand-side is the price elasticity of demand. Now, <em>greater</em> than <em>minus</em> unity means <em>smaller</em> than unity <em>in absolute terms</em>. And it is in that sense that demand at the region is "price inelastic" (the minus sign indicating direction of movement).</p>

CC BY-SA 3.0
2017-06-15T18:03:00.110Z
2017-06-15T20:12:44.267Z
2017-06-15T20:12:44.267Z
Alecos Papadopoulos
<p>I was recently reading up on the Price Elasticity of Demand and the section of Inelastic demand mentions that if a firm has inelastic demand for its product and wishes to increase its total revenue then it should raise the price of the product. The author has done one calculation to somehow convince the reader, but I think that anyone can conjure numbers just for the sake of argument. I am looking for a mathematical argument that will be rigorous enough to convince me. Please note that I get the intuition behind why this might happen but my textbook does not provide any mathematical reasoning as to why this must happen. Any insights will be much appreciated.</p>

CC BY-SA 3.0
2016-01-14T17:53:37.960Z
Consider following these steps http://economics.stackexchange.com/a/1728/1601 without the costs, as you are only interested in revenue.
Giskard
CC BY-SA 3.0
2016-01-14T19:44:35.337Z
What do you understand "inelastic demand" to mean, in mathematical terms?
410 gone
CC BY-SA 3.0
2016-01-15T04:55:52.160Z
By inelastic demand, I mean 0<e<1.
Student
CC BY-SA 3.0
2016-01-16T15:33:17.883Z
Right, and what do you know the definition of price elasticity?
410 gone
<microeconomics><elasticity>
Mathematical Proof as to why does the total revenue increase when the prices are increased for an inelastic product?
Student
# Mathematical Proof as to why does the total revenue increase when the prices are increased for an inelastic product?

(Asked by: username_0 on 2016-01-14)

I was recently reading up on the Price Elasticity of Demand and the section of Inelastic demand mentions that if a firm has inelastic demand for its product and wishes to increase its total revenue then it should raise the price of the product. The author has done one calculation to somehow convince the reader, but I think that anyone can conjure numbers just for the sake of argument. I am looking for a mathematical argument that will be rigorous enough to convince me. Please note that I get the intuition behind why this might happen but my textbook does not provide any mathematical reasoning as to why this must happen. Any insights will be much appreciated.

**Tags:** microeconomics, elasticity
> **Comments:**
> **username_1**: Consider following these steps http://economics.stackexchange.com/a/1728/1601 without the costs, as you are only interested in revenue.  
> **username_2**: What do you understand "inelastic demand" to mean, in mathematical terms?  
> **username_0**: By inelastic demand, I mean 0<e<1.  
> **username_2**: Right, and what do you know the definition of price elasticity?  

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## 1 answers

### Answer by username_3 on 2017-06-15. Score: 1 | Accepted answer

*(I do not agree that this is a duplicate of the answer proposed by other members here).*

Total Revenue is

$$TR = Q(P)\cdot P$$

The derivative with respect to price is

$$\frac{\partial TR}{\partial P} = Q'(P)\cdot P + Q(P)$$

For this to be positive (and so total revenue to increase with price), we require

$$\frac{\partial TR}{\partial P} > 0 \implies Q'(P)\cdot P + Q(P) >0 $$

$$\implies \frac{Q'(P)\cdot P}{Q(P)} > -1$$

The left-hand-side is the price elasticity of demand. Now, *greater* than *minus* unity means *smaller* than unity *in absolute terms*. And it is in that sense that demand at the region is "price inelastic" (the minus sign indicating direction of movement).


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