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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?
Short answer:
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.
Long answer:
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".
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.
The most important consideration of "ceteris Paribus" is missing from the question. The demand curve is a representation of relationship between price and quantity demanded of a good remaining everything else constant. This is also the law of demand. In other words, movement along the demand curve can only occur when the price of a good is interfered while everything else remain constant. On the other hand, shift in demand curve will only happen when there is any change in, either, consumers' income, consumers' taste/preference, prices of related goods and or substitute goods, while the price of the concerned good remains constant. Although, Income elasticity of demand have change in quantity demanded with respect to change in the income, which can confuse learners, however, the essence here is that at this point the price of the concerned good is not interfered, so, the income elasticity of demand can not cause movement along the demand curve, rather, it causes shift in the demand curve.
