The demand for a product is given by p=90-10 q . Find the elasticity of demand when p=50 . If th

The demand for a product is given by p=90-10 q . Find the...

Key Concepts

Demand Function

A demand function represents the relationship between the price of a product and the quantity of the product that consumers are willing to purchase. It encapsulates the inverse relationship that generally exists between price and quantity, forming the basis for understanding market behavior.

Price Elasticity of Demand

Price elasticity of demand measures how sensitive the quantity demanded of a product is to changes in its price. It is calculated by taking the percentage change in quantity demanded divided by the percentage change in price, indicating whether demand is elastic (sensitive to price changes) or inelastic (insensitive to price changes).

Differentiation in Elasticity Calculation

When computing elasticity, differentiation is used to find the rate of change of the quantity demanded with respect to price. This technique provides an instantaneous measure of responsiveness at a specific point on the demand curve, which is crucial for accurately assessing elasticity.

Percentage Change Analysis

Understanding percentage change analysis involves calculating small percentage variations in economic variables such as price and quantity. This analysis is essential for applying elasticity, as it allows one to interpret how a percentage change in price will lead to a corresponding percentage change in the quantity demanded.

Transcript

00:01 This question is asking us about a product that has a price function of 90 minus 10 q.

00:06 And it's asking us two things.

00:08 First of all, to find the elasticity when the price is 50.

00:11 And also it's asking us how q changes if price increases by 2%.

00:16 So now we're going to take a look at the first question first.

00:21 So we need to find the elasticity when the price is $50.

00:24 So over here written in right on the left, i have the elasticity equation, but it's going to ask us about a couple of things.

00:31 First of all we need to find dq over dp and this is a little bit problematic right now because that means we need to take the derivative of q in respect to p but the way this equation is written right now is an equation of p in terms of q so we're going to rewrite this and we're going to solve for q so let's subtract 90 from both sides so p minus 90 equals negative 10 q and then we are going to multiply or divide both sides by negative 10 so that we will be 9 minus 1 tenth p equals q.

01:08 So this because it's the equation we need, and now we can actually take that derivative.

01:13 So let's do that real quick.

01:14 We find dq over dp.

01:19 What's that going to equal? so the derivative of that nine right there, that's just zero because the derivative of a constant number is always zero.

01:26 And then the derivative of minus one tenth p is just negative one.

01:32 10th, the p goes away.

01:34 So this is our derivative, and then there's one more thing we actually defined before we start plugging this in.

01:39 And i want to do a dotted line down the middle, so we separate the two parts of this problem.

01:42 We need to figure out what q is, because that red elasticity question has a q in it.

01:48 So we need to figure out what that is.

01:51 So let's plug in the $50 into that q function we got.

01:56 So because q equals 9 minus 1 tenth p, that's going to be 9 minus 1 tenth and price is 50.

02:06 That's 9 minus 5.

02:09 And then that's going to be 4.

02:12 So our demand here seems to be 4.

02:16 So let's plug this all into the elasticity function now.

02:19 So elasticity is the absolute value of, i'm just going to copy this down...

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