# 5. CONSUMER EQUILIBRIUM: MAXIMIZING UTILITY SUBJECT TO THE BUDGET CONSTRAINT

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It would be wonderful if we could all consume as much of everything as we wanted, but unfortunately, most of us are constrained by income and prices. We now superimpose the budget constraint onto the preference map to model the actual choice of our consumer. This is a constrained (by the resources available to pay for consumption) optimization problem that every consumer must solve: Choose the bundle of goods and services that gets us as high on our ranking as possible, while not exceeding our budget.

## 5.1. Determining the Consumer’s Equilibrium Bundle of Goods

In general, the consumer’s constrained optimization problem consists of maximizing utility, subject to the budget constraint. If, for simplicity, we assume there are only two goods, wine and bread, then the problem appears graphically as in Exhibit 11.

Exhibit 11. Consumer Equilibrium

Note: Consumer equilibrium is achieved at point a, where the highest indifference curve is attained while not violating the budget constraint.

The consumer desires to reach the indifference curve that is farthest from the origin, while not violating the budget constraint. In this case, that pursuit ends at point a, where she is purchasing Wa ounces of wine along with Ba slices of bread per month. It is important to note that this equilibrium point represents the tangency between the highest indifference curve and the budget constraint. At a tangency point, the two curves have the same slope, meaning that the MRSBW must be equal to the price ratio, PB/PW. Recall that the marginal rate of substitution is the rate at which the consumer is just willing to sacrifice wine for bread. Additionally, the price ratio is the rate at which she must sacrifice wine for another slice of bread. So, at equilibrium, the consumer is just willing to pay the opportunity cost that she must pay.

In contrast, consider another affordable bundle represented by point b. Certainly, the consumer is able to purchase that bundle because it lies on her budget constraint. However, the MRSBW at that point is greater than the price ratio, meaning that she is willing to give up wine to obtain bread at a rate greater than she must. Hence, she will be better off moving downward along the budget constraint until she reaches the tangent point at a. In effect, she is willing to pay a higher price than she must for each additional unit of bread until she reaches Ba. For all of the units that she consumes up to Ba, we could say that the consumer is receiving consumer surplus, a concept we visited above when discussing the demand curve. Importantly, she would not purchase slices of bread beyond Baat these prices because at a point like c, the marginal rate of substitution is less than the price ratio—meaning that the price for that additional unit is above her willingness to pay. Even though she could afford bundle c, it would not be the best use of her income.

EXAMPLE 5

# Consumer Equilibrium

Currently, a consumer is buying both sorbet and gelato each week. His MRSGS [marginal rate of substitution of gelato (G) for sorbet (S)] equals 0.75. The price of gelato is €1 per scoop, and the price of sorbet is €1.25 per scoop.

1. Determine whether the consumer is currently optimizing his budget over these two desserts. Justify your answer.

2. Explain whether the consumer should buy more sorbet or more gelato, given that he is not currently optimizing his budget.

## Solution to 1:

In this example, the condition for consumer equilibrium is MRSGS = PG/PS. Because PG/PS = 0.8 and MRSGS = 0.75, the consumer is clearly not allocating his budget in a way that maximizes his utility, subject to his budget constraint.

## Solution to 2:

The MRSGS is the rate at which the consumer is willing to give up sorbet to gain a small additional amount of gelato, which is 0.75 scoops of sorbet to gain one scoop of gelato. The price ratio, PG/PS (0.8), is the rate at which he must give up sorbet to gain an additional small amount of gelato. In this case, the consumer would be better off spending a little less on gelato and a little more on sorbet.

## 5.2. Consumer Response to Changes in Income: Normal and Inferior Goods

The consumer’s behavior is constrained by his income and the prices he must pay for the goods he consumes. Consequently, if one or more of those parameters changes, the consumer is likely to change his consumption behavior. We first consider an increase in income. Recall from Exhibit 7, Panel C, that an increase in income simply shifts the budget constraint outward from the origin, parallel to itself. Exhibit 12 indicates such a shift and shows how the consumer would respond, in this case, by buying more of both bread and wine.

Exhibit 12. The Effect of an Increase in Income on a Normal Good

Note: The effect of an increase in income when both goods are normal is to increase the consumption of both.

As we discovered, there is no restriction that the purchase of every good must respond to an increase in income with an increase in quantity. There, we defined normal goods as those with a positive response to an increase in income and inferior goods as those with a negative response to an increase in income. Suppose that bread is an inferior good for a particular consumer, whereas wine is a normal good. Exhibit 13 shows this consumer’s purchase behavior when income increases. As income rises, the consumer purchases less bread but more wine.

Exhibit 13. The Effect of an Increase in Income on an Inferior Good

Note: The effect of an increase in income on the purchase of bread, an inferior good in this example, reduces the consumption of that good.

## 5.3. How the Consumer Responds to Changes in Price

We now hold income and the price of one good (wine) constant but decrease the price of the other good (bread). Recall that a decrease in the price of bread pivots the budget constraint outward along the horizontal axis but leaves the vertical intercept unchanged—as in Exhibit 14, where we examine two responses to the decrease in the price of bread.

Exhibit 14. Elastic and Inelastic Responses to a Decrease in Price

In both cases, when the price of bread falls, the consumer buys more bread. But in the first case, he is quite responsive to the price change, responding with an elastic demand for bread. In the second case, the consumer is still responsive but much less so than the first consumer; this consumer’s response to the price change is inelastic.