# Price elasticity of demand coefficient

I am working on the following question: AS price increases along a straight line demand curve, will the price elasticity of demand coefficient increase, decrease or remain unchanged?

The answer states that the price elasticity of demand coefficient will decrease, or remain unchanged in two different cases. If the straight line demand curve is downward sloping, the price elasticity of demand coefficient decreases when price increases. If the straight line demand curve is vertical, the price elasticity of demand curve remains unchanged when price increases.

In my point of view, when the price increases, the good takes up a larger proportion of the consumers’ expenditure, so the price elasticity of demand coefficient increases, but not decreases.

What’s wrong with my interpretation?

• A simple geometric interpretation: the elasticity is the relative variation of Y with X. For a decreasing straigth line, as X increase of a fixed amount, Y decreases always of the same amount. But in relative terms if it is going smaller, this fixed decrement will always be more important. Do you see that in absolute terms the elasticity will increase, the variation in relative terms becomes more and more important. Jun 26 at 21:02

The price elasticity of demand is $$-\frac{dQ}{dP}\frac{P}{Q}.$$ (Note that here and elsewhere in my answer I have defined the elasticity so that is positive when demand is downward sloping.) For a linear demand curve, the slope ($$dQ/dP$$) is constant and so the size of the elasticity is proportional to $$P/Q$$. Since the demand curve is downward sloping, when $$P$$ is higher $$Q$$ is lower and so when $$P$$ is higher $$P/Q$$ is also higher. Hence the elasticity is higher when $$P$$ is higher.
$$\varepsilon_p=\varepsilon^h_p+\varepsilon_Ib$$
where $$\varepsilon_p$$ is the standard price elasticity of demand for the good, $$\varepsilon^h_p$$ is the elasticity of the Hicksian (compensated) demand, $$\varepsilon_I$$ is the income elasticity and where $$b$$ is the budget share.
For a normal good, the income effect is positive ($$\varepsilon_I>0$$) and so the larger the budget share, the larger the price elasticity of demand (all else being equal). Thus your intuition is correct for the case of a good with positive income effects. For a good with no income effects ($$\varepsilon_I=0$$) the second term in the equation disappears (only the substitution effect is relevant: $$\varepsilon_p=\varepsilon^h_p$$).