Suppose the demand function is of the form q=C p^-k, where C...

Suppose the demand function is of the form $q=C p^{-k},$ where $C$ and $k$ are positive constants. (a) Find the elasticity $E$ (b) If $0< k <1,$ what does your answer from part (a) say about how prices should be set to maximize the revenue? (c) If $k>1,$ what does your answer from part (a) say about how prices should be set to maximize the revenue? (d) If $k=1,$ what does your answer from part (a) tell you about setting prices to maximize revenue? (e) Based on your answers above, is a demand function of the form $q=C p^{-k}$ realistic? Explain your answer.

Suppose the demand function is of the form $q=C p^{-k},$ where $C$ and $k$ are positive constants.
(a) Find the elasticity $E$
(b) If $0< k <1,$ what does your answer from part (a) say about how prices should be set to maximize the revenue?
(c) If $k>1,$ what does your answer from part (a) say about how prices should be set to maximize the revenue?
(d) If $k=1,$ what does your answer from part (a) tell you about setting prices to maximize revenue?
(e) Based on your answers above, is a demand function of the form $q=C p^{-k}$ realistic? Explain your answer.

Key Concepts

Price Elasticity of Demand

This concept measures how sensitive the quantity demanded is to a change in price, and is typically defined as the percentage change in quantity demanded divided by the percentage change in price. In the context of the given demand function q = C p^(-k), the elasticity turns out to be a constant value equal to k, meaning that for every proportional change in price, there is a fixed proportional change in quantity demanded regardless of the price level.

Revenue Maximization

Revenue maximization concerns choosing a price that maximizes the total revenue, which is the product of price and quantity demanded. When the elasticity is less than one (inelastic demand), increasing price increases revenue because the percentage decrease in quantity is less than the percentage increase in price. Conversely, when elasticity is greater than one (elastic demand), an increase in price hurts revenue because the percentage drop in quantity exceeds the percentage increase in price. If elasticity is exactly one (unit elastic demand), changes in price do not affect revenue since the percentage change in quantity exactly offsets the change in price.

Constant Elasticity Demand Model

A demand function of the form q = C p^(-k) is known as a constant elasticity demand model because the elasticity remains the same at every price level. This is a simplifying assumption often used in theoretical analysis since it provides tractable results, but it also leads to peculiar outcomes such as revenue being strictly increasing or decreasing with price when k is not equal to 1, meaning there is no unique optimal price within a realistic range.

Realistic Demand Modeling

The realism of a demand function is judged by how well it captures actual market behavior. A constant elasticity demand function implies that the percentage change in quantity demanded is the same regardless of price, which is rarely observed in real markets where consumers' sensitivity typically changes with price levels. Furthermore, the lack of a clear revenue-maximizing price when demand is either strictly inelastic or elastic (except at the unit elastic point) challenges the practical applicability of such models for setting prices.

Transcript

00:01 All right, we have q is equal to c p to the negative k.

00:03 One thing to note is that c and k are both greater than zero.

00:08 We're going to go to the book, which we're doing this question eight.

00:12 Part a is to find the elasticity.

00:16 So for part a, we know that e is equal to negative p over q times dqdp.

00:25 So first step is finding dqdp.

00:29 Dq dp is equal to.

00:33 So we know c and k are both constants.

00:36 So we are going to have c times we're going to bring down that negative k.

00:44 It's a constant times p to the negative k minus 1.

00:50 Because you have to subtract one for the exponent.

00:53 That means we have an e equal to negative p over q, which is which is c p to the negative k times our derivative which is times c negative k p to the negative k minus 1.

01:19 E is equal to p c k p to the negative k minus 1 over c p to the negative k okay, so some initial cancelizations, the c's, right? that's right off the bat.

01:39 So if we rewrite this as k times p times p to the negative k minus 1 over p to the negative k.

01:49 We have p to the first power.

01:52 To combine exponents with the same basis, you add these exponents.

01:56 If we add exponents, you get k, negative k minus 1 plus 1, is k p to the negative k over p to the negative k these cancel we are left with e equal to k so now we have found part a which is just asking us to find our e so we have e equal to k if we go back to the book we can now move on to part b part b wants to know what happens if our k is between zero and one so b, we have 0 is less than k is less than 1.

02:46 What happens here? well, e is equal to k.

02:49 So if e is equal to k, and we put these conditions on k, we're also putting those conditions on e.

02:57 With those conditions on e, this implies that we're saying 0 is less than e is less than 1.

03:05 When e is less than 1, we have inelastic demand.

03:14 That's going to be the answer to part b...

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