# Income Elasticity of Demand vs. shift in demand curve

I've just read an econ textbook and want to verify a statement/definition about income elasticity of demand.

Changes in income shift the demand curve, and we can measure the responsiveness of demand to income changes by calculating income elasticity of demand as follows: Question: Where does this change in quantity demanded come from? Is it from a movement along demand curve, or from a shift of demand curve?

Denoting income by $I$, price by $p$, the demand function by $D(I,p)$ and income elasticity of demand by $\eta$, the definition of point elasticity of income is

$$\eta = \frac{\text{d}D(I,p)}{\text{d}I}\frac{I}{D(I,p)}.$$

So the change in demand comes from a change in income.

Without specifying your coordinate system the terms "movement along the curve" and "shift of the curve" are meaningless.

Consider the function $f(a,x) = a/x$. Suppose this function determines the value of a variable $y$, that is $y = f(a,x)$. Given $a$ we can plot $f(a,x)$ in the $(x,y)$ coordinate system. If $x$ changes there is movement along the curve in this coordinate system. If $a$ changes the curve shifts in this coordinate system. However treating $a$ as a variable and given $x$ we could also plot $f(a,x)$ in the $(a,y)$ coordinate system. Then change in $a$ would mean movement along the curve and change in $x$ would shift the curve.

As quantity demanded usually depends on both income and price, you face a similar situation, where $q = D(I,p)$. Income elasticity examines a change in income, but without specifying if your curve is ploted in $(p,q)$ or $(I,q)$ you cannot classify the change in $I$ as either "movement along the curve" or "shift of the curve".

• Isn't the normal assumption that demand curves are plotted in $(p,q)$? – Adam Bailey Jan 26 '18 at 9:39
• @AdamBailey Usually it is, but we are talking specifically about income elasticity here. Anyway the answer also covers that case. – Giskard Jan 26 '18 at 10:25
• @denesp, thanks. The book refers to demand shift when describing income elasticity, but uses "quantity demanded" in the formula. I think, "quantity demanded" and "change in demand" should not be used interchangably? – london Jan 27 '18 at 22:04
• @london Unfortunately I don't get your meaning. "quantity demanded" and "change in demand" should definitely not be used interchangably, as one describes change, whereas the other does not. – Giskard Jan 27 '18 at 22:39
• @denesp, this is what I dont get too. Pages 72-73 of: macmillanihe.com/page/detail/Economics-for-Business/… – london Jan 28 '18 at 10:35

The change would be the slope of the demand curve in relation to the income variable. For example, if your demand curve is given as follows: $$Q_x = -P_x +.5I$$

Then the elasticity would be the derivative of the demand function in relation to the income variable, times the given point at which you're calculating the elasticity: $$\frac{\partial Q}{\partial I} \frac{I}{Q} = .5 \frac{I}{Q}$$

In essence what you're doing is calculating how much would the demand curve move if your income were to change in relation to a given income and quantity of demand.