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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.

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    $\begingroup$ Consider following these steps economics.stackexchange.com/a/1728/1601 without the costs, as you are only interested in revenue. $\endgroup$ – Giskard Jan 14 '16 at 17:53
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    $\begingroup$ What do you understand "inelastic demand" to mean, in mathematical terms? $\endgroup$ – EnergyNumbers Jan 14 '16 at 19:44
  • $\begingroup$ By inelastic demand, I mean 0<e<1. $\endgroup$ – model_checker Jan 15 '16 at 4:55
  • $\begingroup$ Right, and what do you know the definition of price elasticity? $\endgroup$ – EnergyNumbers Jan 16 '16 at 15:33
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(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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