The demand for a product is given by q=200-2 p^2 Find the elasticity of demand when the price is

The demand for a product is given by q=200-2 p^2 Find the...

Key Concepts

Price Elasticity of Demand

Price elasticity of demand measures how responsive the quantity demanded of a product is to a change in its price. It is a crucial concept in economics because it helps determine how changes in price affect total revenue. The calculation generally involves the derivative of the demand function multiplied by the ratio of price to quantity, which standardizes the measure across different products and markets.

Differentiation in Demand Analysis

Differentiation is used in demand analysis to determine the rate at which quantity demanded changes with respect to price. By finding the derivative of the demand function, one can capture the marginal effect of price on demand, which is key when calculating elasticity. This mathematical tool transforms the static demand equation into an analysis of dynamic responses to price changes.

Elastic vs Inelastic Demand

Elasticity categorizes demand based on its sensitivity to price changes. When the absolute value of the elasticity is greater than one, the demand is considered elastic, meaning consumers are relatively sensitive to price changes. Conversely, if the absolute value is less than one, the demand is inelastic, indicating that quantity demanded is relatively unresponsive to price fluctuations. This classification is essential for businesses and policymakers when making pricing decisions.

Transcript

00:01 So for this question, we have some product that has a demand function of q equals 200 minus 2p squared.

00:08 And we're being asked to find what the elasticity of this product is at $5.

00:12 So now here in red, i have written the formula for elasticity, and this is the exact formula as opposed to the non -exact approximation formula where you have delta q over q over delta p over p.

00:27 The most important thing to remember when we're using this exact formula is that the numerator and denominator is kind of switched here, where you have p over q times the derivatives of q in respect to p is of dq over dp.

00:38 So you just need to remember that those are switched in this, that one has p on top and another has p on the bottom.

00:44 But from here is just pretty easy to calculate.

00:47 So dq over dp is what we should start by finding.

00:52 So all we need to do is take the derivative of that q function we had in respect to p.

00:56 So we'll start with 200.

00:58 Derivative of 200 is 0 because the number, derivatives numbers are always 0.

01:02 And now we need to take the derivative of negative 2p squared.

01:06 By this, we do power rule, so that should just be negative 4p because we bring the 2 to the front and 2 times 2 is 4, and then that 2 in the exponent becomes a 1.

01:18 So this should be our derivative.

01:20 And now we're going to switch to green, we just need to plug all of this in.

01:25 So we have e equals the absolute value of p over q times dq over dp is negative for p.

01:37 So before we move on, we actually need to plug in the p and the q value for this.

01:42 So let's find that q value real quick...

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