Show analytically that if elasticity of demand satisfies E>1, then the derivative of revenue wit

Show analytically that if elasticity of demand satisfies...

Key Concepts

Total Revenue Function

The total revenue function represents the income a firm obtains from selling a product and is mathematically expressed as the product of the price and the quantity sold. In economics, it is a fundamental concept that links pricing decisions with sales volume and helps in analyzing how changes in price affect overall revenue.

Price Elasticity of Demand

Price elasticity of demand is a measure of how sensitive the quantity demanded of a good is to a change in its price. It is calculated as the percentage change in quantity demanded divided by the percentage change in price. When the elasticity is greater than one, it indicates that consumers are relatively responsive to price changes, meaning that a small price increase leads to a relatively large decrease in quantity demanded.

Differentiation and the Product Rule

To determine the effect of a change in price on total revenue, one employs the process of differentiation using the product rule, since total revenue is the product of price and quantity. This technique allows us to decompose the derivative into two components, one reflecting the direct impact of price change and the other accounting for the change in quantity demanded as price changes.

Relationship Between Elasticity and Revenue Changes

Analytically linking price elasticity to changes in revenue involves rewriting the derivative of the revenue function in terms of elasticity. When elasticity exceeds one, the negative effect of a price increase on quantity demanded outweighs the positive effect of raising the price itself, thereby ensuring that the derivative of revenue with respect to price is negative. This demonstrates that revenue falls as price rises when the demand is elastic.

Transcript

00:01 So we're being asked to prove that dr over dp is greater than zero when the elasticity is less than one, and the dr of dp is less than zero when elasticity is greater than one.

00:09 They should be inverse.

00:12 So what we're going to do is we're going to just kind of get into that.

00:16 We're going to make sure that's true.

00:17 I have the elasticity equation written here.

00:20 And the first thing we're going to do is we're actually going to analyze that.

00:23 And we'll see why later.

00:25 But for now, let's just start analyzing it.

00:27 So we know that dq over dp is going to be negative because when price increases, demand falls, and when price decreases, demand rises.

00:37 Because lower price means more people will want to buy it.

00:40 It's just how economics works.

00:42 And p over q should be positive because price is always going to be positive.

00:49 Nothing is going to have a negative price.

00:50 And demand is always going to be positive because a negative demand means people are like selling it, i guess.

00:55 So together combined, this entire thing's negative.

00:59 And since we're taking the absolute value of it, the way we'd do that is just by multiplying everything by negative 1.

01:07 So we'd get negative p over q times dq over dp.

01:13 And right now, it doesn't really make sense why we're doing that, but we'll get back to that in a second.

01:18 So we have this equation right here where r equals p times q.

01:22 And what we want to do is we want to find something about dr over dp.

01:27 So it makes sense that the first thing we should be doing is just finding dr over dp.

01:31 Hopefully we'll shed some light on what's happening.

01:34 Now we need to remember that q is a function of p because q depends on p when p changes, q changes.

01:40 So that means that we actually have to do product rule here because p is a function of p and q is a function of p.

01:46 So product rule derivative of the first, that's derivative of p, which is one, times the second.

01:52 Just q.

01:54 And then plus the first one stays the same...

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