# Lesson 5: Prices in The Budget Line, The Law of Demand

In the last lesson, we discussed how income factors into the budget line. Now, we will consider price in the budget line.

Just as we defined income effect in the previous lesson, we define price effect as the effect of changes in relative price on consumption. The Law of Demand, then, tells us that as relative price of a good decreases, consumption increases, assuming that income is held constant.

However, there is another facet to the Law of Demand – it holds only for compensated price changes. A price change is compensated if wealth is changed along with price to make the previously chosen bundle on the budget line, so that it is ‘just barely’ available. A price change is uncompensated if there is no wealth adjustment. Why does it make sense that the Law of Demand holds for only compensated price changes? Consider this graph:

The black budget line is the original budget line, and the black point is the chosen bundle on that budget line. When we shift to the blue budget line, the relative price of food in terms of all other goods decreases. This price change is uncompensated, since the previously chosen bundle is no longer available. The Law of Demand, without compensated price changes, would say that consumption of food will increase. However, that is clearly impossible due to the budget constraints. When we include compensation, it is possible to consume more food:

Compensated price changes are rarely brought up in introductory economics classes; they are not even introduced in McCloskey until Chapter 4. I decided to say a bit about them here, because I found myself getting confused while reading Section 1.3, where compensated price changes are not mentioned. We will go into it in more detail as we progress through the book.

Often, for small uncompensated price changes, people will still use the Law of Demand. When the price change is small, it is ‘close enough’ to compensated. For example, there is an uncompensated price change in this graph, but we still may use the Law of Demand:

In this section and this lesson, if you encounter an uncompensated price change, feel free to use the Law of Demand for analysis.

Now, try answering a question: If absolute prices of all goods (including labor) doubled, what would you expect to happen to consumption?

Nothing, because all relative prices remain the same, and in fact, you face the same budget line as before. The original budget line, for say, bread and all other goods is

(1)

Then, doubling prices of bread, all other goods, and labor (i.e. income) gives

(2)

which is the same as equation (1).

The Law of Demand seems intuitive, but is there a better reason to believe it? Economists have proven that a property known as the Weak Axiom of Revealed Preference (WARP) is equivalent to the Law of Demand. This means that if WARP is true, the Law of Demand (for compensated price changes) is true. While WARP is a bit harder to understand than the Law of Demand, it may be an easier pill to swallow.

A bundle A is said to be revealed preferred to a bundle B if A is chosen when B is available. So, if you are faced with a budget line, and choose a point on the budget line, that point is revealed preferred to every other point on (and under) the budget line.

WARP says that if A is revealed preferred to B, then B cannot be revealed preferred to A. This makes sense – you would not expect someone to act as if two things are simultaneously better than one another.

Consider now the following graph of All Other Goods vs. Food:

Suppose that bundle 0 is chosen on the blue budget line. Then, there is a compensated price change resulting in the red budget line. We know that the consumer chooses either bundle 1 or bundle 2 on the red budget line. If we take WARP to be true, which bundle must be chosen?

From the blue budget line to the red budget line, the relative price of food increases. The Law of Demand, which applies since the price change is compensated, would tell us that consumption of food would decrease. Therefore, the consumer must choose bundle 1.

Now, take WARP to be primitive, as the question asks. Then, note that when faced with the blue budget line, bundle 0 and bundle 2 are both attainable. Since bundle 0 is chosen, we know that 0 is revealed preferred to 2. WARP tells us that bundle 2 cannot also be revealed preferred to bundle 0. This means that bundle 2 cannot be chosen on the red budget line, so bundle 1 must be chosen.

We get the same result (bundle 1 chosen) whether starting with the Law of Demand or WARP.

So far, we have been building up  some basic theory, but not doing too much economics. Now, let’s get into the economics: try these questions from the book.

Question: When in Italy, Lou Cain spends his monthly income of $100 on 20 kg spaghetti at$2 per kg and 12 lb hamburger at $5 per lb. When he moves to Italy, he spends his income of$110 on 22 kg spaghetti at $3 per kg and 11 lb hamburger at$4 per lb. Is Cain consistent between Italy and Britain (in the WARP sense)?

First, we draw Lou Cain’s budget line in Italy in blue. We can do this drawing the line through the point where all income is spent on spaghetti, and the point where all income is spent on hamburgers. Otherwise, you can find the equation for the line any way you like, as is done in Lesson 4. Either way, here is Lou Cain’s budget line in Italy, with the chosen bundle plotted as well:

We can similarly draw Lou Cain’s budget line in Britain in red, along with the chosen bundle:

It is hard to see what is going on near both bundles, so let’s zoom in:

Now, we can see what’s going on. In Italy, both the blue chosen bundle and the red bundle are affordable, but the blue bundle is chosen. So, the blue bundle is revealed preferred to the red bundle. In Britain, both the red chosen bundle and the blue bundle are affordable, but the red bundle is chosen. So, the red bundle is revealed preferred to the blue bundle. This is a clear contradiction of WARP.

T/F: Simultaneous increases in the first-class postage rate from 20 to 30 cents and the overnight delivery rate from $1.00 to$1.10 will leave unaltered the proportion in which the two types of mail are consumed.

The price of mail relative to all other goods has increased, so less mail will be purchased. Of the mail we purchase, a greater proportion than before will be spent on overnight delivery because the price of overnight relative to first-class has decreased.

Since these price changes are small-ish, we can ignore that the price changes are uncompensated.

During Prohibition in the USA, much to the dismay of some, alcoholic beverages continued to be produced and sold, despite the threat of confiscation, fine, or jail.

T/F: One would expect the average quality of alcoholic beverages to be lower than before Prohibition.

You’re going to get punished just the same if you get caught with cheap beer or fine wine – there is a flat cost of punishment added to any type of alcohol consumption or production. So, the higher quality/absolute price alcohol gets cheaper relative to the lower quality/absolute price alcohol. Therefore, in a similar situation as the previous question, a greater proportion of alcohol purchased is high-quality. The average quality of alcoholic beverages increases after Prohibition.

That’s all for this lesson. We’re finally done with Chapter 1! Next time we’ll get started on Chapter 2, which is all about indifference curves.

Assignment: Read all of Section 1.3. Do Exercises 1-3 and Problems 1 -6. Lots of questions this time, but putting work into understanding this section will help out later, when we cover the same material at a higher level.

Note 1: Demand curves are introduced in Section 1.3, but I choose not to include them in this lesson. They are given only a cursory treatment which may be confusing right now. We will talk about them in further detail once we reach Chapter 4. I’ve vetted the problems, and demand curves won’t be necessary to answer them.

Note 2: There are many criticisms of the theory we have detailed in the last 5 lessons. What we have done so far is not the last say on the matter. But it is the first say, and the one that is most important to learn. Certainly, you should be critiquing economic theory, but without learning it well, you run the risk of critiquing in a naive manner.