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Exercises · Q23

Q.Consider the demand curve D(p)=10−3pD(p) = 10 - 3p. What is the elasticity at price 53\frac{5}{3}?

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Price elasticity of demand measures how responsive quantity demanded is to a price change. For the linear demand D(p)=10−3pD(p) = 10 - 3p, the elasticity at p=53p = \frac{5}{3} is -1 (unit elastic).

The concept of price elasticity of demand is about proportional change, not absolute change. A steep demand curve doesn't automatically mean inelastic demand — elasticity depends on where you are on the curve. The formula is:

Price elasticity of demand:

Ed=dQdp⋅pQE_d = \frac{dQ}{dp} \cdot \frac{p}{Q}

Here, dQdp\frac{dQ}{dp} is the slope of the demand function (how much quantity changes when price changes by one unit), and pQ\frac{p}{Q} scales it by the current price-quantity point. The product tells you the percentage change in quantity for a 1% change in price.

For the given demand curve D(p)=10−3pD(p) = 10 - 3p, the slope is constant: dQdp=−3\frac{dQ}{dp} = -3. That negative sign reflects the law of demand — price up, quantity down.

Now, at p=53p = \frac{5}{3}, we first find the quantity demanded:

Q=10−3×53=10−5=5Q = 10 - 3 \times \frac{5}{3} = 10 - 5 = 5

So at this point, p=53p = \frac{5}{3} and Q=5Q = 5.

Plug into the elasticity formula:

Ed=(−3)×535=(−3)×53×15=(−3)×13=−1E_d = (-3) \times \frac{\frac{5}{3}}{5} = (-3) \times \frac{5}{3} \times \frac{1}{5} = (-3) \times \frac{1}{3} = -1 …

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