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This is an economics question, but if I can get the correct answer to this formula I can answer the question. Any help is appreciated, thanks! If the demand $Q_x^d$ for a product given the price $P_x$ is $$\ln Q_x^d = 10 – 5 \ln P_x$$ then product $x$ is:
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The price elasticity $E$ of a product which has price $P$ given a demand $Q$ is defined as the percentage change in demand for each percentage change in price, i.e. $$E = \frac{d \ln Q}{d \ln P}\qquad (1)$$ or alternatively $$E = \frac{P}{Q} \frac{dQ}{dP}\qquad (2)$$ Since in your case you have $$\ln Q = 10 - 5\ln P$$ You should be able to differentiate $\ln Q$ with respect to $\ln P$ (equation (1)) to get the result you require. Alternatively, you could express $Q$ in terms of $P$ by exponentiating both sides of the formula, giving $$Q = e^{10 - 5\ln P} = \frac{e^{10}}{P^5}$$ and apply equation (2) to find $E$, which may be easier if you're not that experienced with differentiation. Once you have calculated $E$, you can decide whether the good in question is elastic or inelastic by noting that elastic goods have $|E|>1$ and inelastic goods have $|E|<1$. |
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