Income and Consumption: Finding Your New Optimal Bundle

The Effect of an Income Change on the Budget Constraint

Let's begin by examining how an income change affects the consumer's budget constraint. Recall the budget line equation:

where $P_X$ and $P_Y$ are the prices of goods X and Y, and $I$ is income. The slope of this line is $-P_X/P_Y$, and the vertical and horizontal intercepts are $I/P_Y$ and $I/P_X$, respectively.

When income $I$ changes, but prices $P_X$ and $P_Y$ do not, the slope of the budget line remains constant. This means the new budget line will be parallel to the original one.

• An increase in income shifts the budget line outward, expanding the set of affordable consumption bundles.
• A decrease in income shifts the budget line inward, contracting the set of affordable bundles.

An outward shift allows the consumer to reach a higher indifference curve, resulting in a new optimal bundle with greater utility. Conversely, an inward shift forces the consumer to a lower indifference curve.

The following video provides a straightforward numerical example of this process, picking up exactly where our last lesson on the equi-marginal principle left off.

Normal and Inferior Goods

As income changes, the quantity demanded for a good will also change. The direction of this change allows us to classify goods into two main categories.

1. Normal Goods: A good is considered normal if the quantity demanded increases as consumer income rises, and decreases as income falls. For these goods, the relationship between income and quantity demanded is positive. Most goods, like restaurant meals, new clothing, and vacations, fall into this category.

2. Inferior Goods: A good is considered inferior if the quantity demanded decreases as consumer income rises, and increases as income falls. The relationship is negative. Examples might include instant noodles or bus travel; as people become wealthier, they may substitute away from these goods towards more expensive alternatives like dining out or buying a car.

Given your background in statistics and quantitative methods, we can define this formally using calculus. If $x(P_X,P_Y,I)$ is the demand function for good X, then:

• Good X is normal if $\frac{\partial x}{\partial I}>0$.
• Good X is inferior if $\frac{\partial x}{\partial I}<0$.

The following reading provides a detailed explanation of these concepts and introduces graphical tools for visualizing them.

Deriving the Engel Curve

While the Income-Consumption Curve shows the path of optimal bundles in a two-good space, the Engel Curve provides a more direct relationship between income and the quantity demanded of a single good. It is derived by solving the consumer's utility maximization problem for one good as a function of income, holding prices constant.

The derivation involves the same steps we've been practicing:

1. Set up the utility maximization condition: $MRS=P_X/P_Y$ or, equivalently, $M{U}_X/P_X=M{U}_Y/P_Y$.
2. Solve this condition to express the quantity of one good in terms of the other (e.g., solve for Y in terms of X).
3. Substitute this expression into the budget constraint, $P_{X}X+P_{Y}Y=I$.
4. Solve for the quantity of the remaining good (e.g., X) as a function of income $I$. This equation, $X=f(I)$, defines the Engel curve for good X.

The video below demonstrates this algebraic derivation, which should align well with your preference for formal problem-solving.

Practice Problem

Let's apply these concepts. In our previous lesson, we considered a consumer with the utility function $U(X,Y)=X^{1/3}{Y}^{2/3}$. Their income was $I=\$600$, with prices $P_X=\$10$ and $P_Y=\$20$. We found the optimal bundle to be $X^{\ast }=20,Y^{\ast }=20$.

Now, suppose this consumer's income increases to $900.

1. Calculate the new optimal consumption bundle ($X^{\ast \ast },Y^{\ast \ast }$).
2. Based on your result, are goods X and Y normal or inferior?
3. Derive the equation for the Engel curve for good X.

Solution:

1. Find the new optimal bundle:

   • From our previous work, we know that for this utility function, the optimality condition $M{U}_X/P_X=M{U}_Y/P_Y$ simplifies to the relationship $Y=X$. This relationship depends on the utility function and prices, but not on income.
   • Now, we substitute this relationship into the new budget constraint: $$P_{X}X+P_{Y}Y=900$$ $$10X+20(X)=900$$
   • Solving for X: $$30X=900⟹X^{\ast \ast }=30$$
   • Since $Y=X$, the new optimal quantity of Y is: $$Y^{\ast \ast }=30$$
   • The new optimal bundle is (30, 30).

2. Classify the goods:

   • When income increased from $600 to $900, the quantity demanded of good X increased from 20 to 30. Therefore, good X is a normal good.
   • Similarly, the quantity demanded of good Y increased from 20 to 30. Therefore, good Y is also a normal good.

3. Derive the Engel curve for good X:

   • We follow the same procedure but use the general income variable $I$. We substitute the optimality condition $Y=X$ into the general budget constraint: $$10X+20Y=I$$ $$10X+20(X)=I$$ $$30X=I$$
   • This equation, $I=30X$, is the Engel curve for good X. It can also be written as $X=I/30$. It shows a positive, linear relationship between income and the quantity of X demanded, confirming that X is a normal good for all levels of income.

Conclusion

In this lesson, we have extended our analysis of consumer choice to see how the optimal consumption bundle responds to changes in income. We have established a clear analytical and graphical framework for this "income effect."

Key Takeaways:

• A change in income causes a parallel shift in the budget line, leaving its slope (the relative price ratio) unchanged.
• An increase in income allows the consumer to reach a higher indifference curve and a new optimal bundle.
• Normal goods are those for which demand increases as income rises ($\partial x/\partial I>0$).
• Inferior goods are those for which demand decreases as income rises ($\partial x/\partial I<0$).
• The Income-Consumption Curve (ICC) traces the path of optimal bundles as income changes.
• The Engel Curve isolates the relationship between income and the quantity demanded for a single good.

In our next lesson, we will investigate how consumption changes in response to a price change. We will find that a price change creates both a substitution effect and an income effect. Our work today in understanding how a pure income change affects behavior is the essential first step to mastering that more complex analysis.