Chapter 2 / Preferences, Constraints and Consumer Choice

2.3 Budget Constraint, Optimal Choice and Demand

Now that we have discussed what the consumer wants, let us move on to describing what the consumer can do and analyze their optimal choice.

Affordable Bundles and Budget Constraint

We call affordable a bundle that the individual can afford to purchase. Denoting the individual’s income by $M$ and the prices of the two goods by $P_X$ and $P_Y$, a bundle $(X,Y)$ is therefore affordable if $ P_XX + P_YY \leq M$.

In the case of indivisible goods such as restaurant pizzas and movie tickets, the consumer might not spend their entire income. For example, it is impossible to spend 35 euros on pizzas and cinema if, as in the example just seen, their prices are respectively 6 and 8.

In the case of divisible goods such as general consumption and housing, it becomes meaningful (and important, as we will see) to ask which bundles require an expenditure exactly equal to the individual’s income. These are the bundles that satisfy the consumer’s budget constraint, i.e., the equation $P_XX + P_YY = M$. Rewriting the budget constraint in the form

$\begin{gathered} Y = \frac{M}{P_Y} - \frac{P_X}{P_Y}X \end{gathered}$

we note that it defines a line (called the budget line) with intercept $M/P_Y$ and slope $-P_X/P_Y$. Affordable bundles are those located on the budget line (in dark green in the graph below) or below it (light green area).

Optimal Choice and Demand Functions

We are now ready to precisely define and calculate the consumer’s optimal choice, that is, the preferred bundle among those they can afford. Graphically, the optimal choice is the bundle that satisfies the following two properties: it belongs to the set of affordable bundles, and the set of bundles preferred to it does not overlap with the set of affordable bundles.

The figure allows us to make two important observations. First, the optimal bundle is not only affordable, but it lies on the budget line: the consumer spends their entire income. Second, moving along the budget line, the sign of the difference between $MRS_{XY}$ and $P_X/P_Y$ guides us toward the optimal choice. Starting from a bundle where the indifference curve is steeper than the budget constraint, that is, where $MRS_{XY} > P_X/P_Y$, the consumer has an incentive to reduce spending on $Y$ and increase spending on $X$ by an equal amount. Conversely, if the indifference curve is less steep than the budget constraint, that is, if $MRS_{XY} < P_X/P_Y$, the consumer has an incentive to reduce spending on $X$ and increase spending on $Y$ by an equal amount.

From these observations, we obtain the consumer’s optimal choice rule:

(i) If at the bundle $(M/P_X,0)$ where the income is entirely spent on $X$, that is, the horizontal intercept of the budget constraint, we have $MRS_{XY} > P_X/P_Y$, then the optimal choice is the bundle $(M/P_X,0)$. The consumer has an incentive to reduce $Y$, but cannot do so, since $Y$ is already zero.

(ii) If at the bundle $(0, M/P_Y)$ where the income is entirely spent on $Y$, that is, the vertical intercept of the budget constraint, we have $MRS_{XY} < P_X/P_Y$, then the optimal choice is the bundle $(0, M/P_Y)$. The consumer has In cases (i) and (ii) we speak of a corner solution because the optimal bundle lies on one of the two axes. an incentive to reduce $X$, but cannot do so, since $X$ is already zero.

(iii) If neither of the two previous situations applies, then we can calculate the optimal choice by solving a system of two equations (in two unknowns, i.e., the quantities of the two goods). The first equation is the budget constraint. The second requires that $MRS_{XY}$ is neither greater nor less than $P_X/P_Y$, that is, that there is tangency between the indifference curve and the budget constraint:

$\begin{aligned} \text{Budget Constraint: }\ \ \ & P_XX + P_YY = M\\ \text{Tangency Condition: }\ \ \ & MRS_{XY} = P_X/P_Y \end{aligned}$

Having calculated the optimal choice, we have also calculated the individual demand function for each good. An individual demand function is the relationship between the The quantity demanded of a good is the quantity of that good within the optimal bundle. price of a good and the quantity of that good that a single consumer demands at that price, holding income and the price of the other good constant. Describing this relationship is useful if we are interested (as we will be in the rest of these notes) in discussing the market Another advantage is simplification: it highlights the relationship between the quantity demanded of a good and its price through a concept easily accessible even to those who (unlike us 🤓) do not possess the full toolkit (utility, budget constraint, etc.) necessary. for a single good at a time, and therefore, the issue to focus on is how the consumption of that good changes when the price of that good increases or decreases.

Figure 2.12 illustrates the optimal choice for two particular utility functions and three configurations of income and prices. The following figure shows how to compute the optimal choice, and from it the demand function for $X$, more generally. Recall from the previous section that the marginal rate of substitution derived from the utility function $U=(X+\sigma)^\alpha(Y+\sigma)^\beta$ is

$\begin{gathered} MRS_{XY} = \dfrac{\alpha}{\beta} \times \dfrac{Y+\sigma}{X+\sigma} \end{gathered}$

The figure allows us to make some observations:

A change in the price of the good ($P_X$) causes a movement along the demand curve for $X$, and the law of demand holds: as $P_X$ increases, the quantity demanded of $X$ decreases.
The quantity demanded of $X$ An inferior good, on the other hand, is one whose quantity demanded decreases as income increases. In these notes, we consider only normal goods. increases as income increases: $M$ appears in the numerator in the formula for $X$. A change in income therefore causes a shift of the demand curve for $X$, to the right or left depending on whether income increases or decreases. Economists define a good as normal if its quantity demanded increases as income increases.
With Cobb-Douglas preferences ($\sigma=0$) we are always in case (iii) and we have $$ X=\frac{[\alpha/(\alpha+\beta)]M}{P_X} \qquad Y=\frac{[\beta/(\alpha+\beta)]M}{P_Y} $$ In other words, whatever the prices of the goods, the consumer always spends the same fraction $\alpha/(\alpha+\beta)$ of income on $X$ and the remaining fraction, $\beta/(\alpha+\beta)$, on $Y$.
In particular, in the Cobb-Douglas case the quantity demanded of $X$ does not depend on the price of the other good: when $\sigma=0$, indeed, $P_Y$ disappears from the formula for $X$. In other words, changes in $P_Y$ do not shift the demand curve for $X$. Economists call two goods with this characteristic independent.
When $\sigma>0$, instead, an increase in $P_Y$ changes not only the quantity demanded of $Y$, making it decrease (unless we are in case (i), where it is already zero), but also that of $X$, making it increase (unless we are in case (i) or, if $P_Y$ increases slightly, case (ii)). Two goods are called complementary if the quantity demanded of one decreases when the price of the other increases. We do not consider complementary goods in these notes. In other words, a change in $P_Y$ causes a shift of the demand curve for $X$ to the right or left depending on whether $P_Y$ increases or decreases. Economists call two goods with this characteristic substitutes.
When the two goods are equally important ($\alpha=\beta$) and the substitutability parameter $\sigma$ is very high (tends to infinity), the two goods are homogeneous. In this situation, case (iii) can occur only if $P_X$ and $P_Y$ are almost equal, i.e. we are almost always in case (i) or (ii). Unless $X$ and $Y$ have almost the same price, the consumer only buys the cheaper of the two.

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