Budget Constraints

This post will be on chapter 2 of the textbook which talks about budget constraints. I would like to apologize for the lack of post yesterday. This week has been very difficult for me to keep motivated and work on my studying. Today I woke up though and I just said to myself that I would go for a walk, clear my head and work. I am not trying to make up for lost time this week, I am just going to pick up where I left off and continue working.

To sum up the economic theory of the consumer is very simple. Economists assume that consumers choose the best bundle of goods they can afford. This deals with some vague terminology that we have to define more carefully in order to study it. First off is what is meant by what a consumer can afford. The next chapter in the textbook talks about what is meant by best.
So now we want to examine what we mean by what a consumer can afford. This has to do with the concept of the budget constraint. People consumer all sorts of things and this can be very difficult to study. When we look at some set of goods that a consumer can choose from we want to do some things to make it a little easier to study. The main thing that economists do is simplify what a consumer can choose from down to two goods. When we only deal with two goods we are then able to display the consumer’s preferences graphically.
While there can be some problems with the two good assumption it works fairly well. The reason for this is that you can make the assumption that one good is a particular item and the second good can represent everything else. As such with two goods you are able to represent fairly accurately what choices a consumer might have in regards to a good. One of the tricks to do this is to say that good 2 is just money. This sets the price of good 2 to 1 and it is able to represent what you could spend on everything else.
We indicate the consumer’s consumption bundle by (x1, x2). This is simply numbers that tell us how much the consumer is choosing to consume of good 1, x1, and how much the consumer is choosing to consumer of good 2, x2. By adding in the prices of each of the goods, (p1, p2) and the amount of money the consumer has to spend, income (m). We can then write the budget constraint of the consumer as:

p1x1 + p2x2 ≤ m.

p1x1 is equal to the amount of money the consumer is spending on good 1.

p2x2 is equal to the amount of money the consumer is spending on good 2.

The budget constraint of the consumer requires that the amount of money spent on the two goods to be no more than the total amount the consumer has to spend. The consumer’s affordable consumption bundles are those that don’t cost any more than m. We call this set of affordable consumption bundles at prices (p1, p2), and income m, the budget set of the consumer.
When p1x1 + p2x2 = m we call it the budget line. These are the bundles of goods that just exhaust the consumer’s income.

If we want to rearrange the formula for the line to make it look more like the standard line for an equation in formation y = mx + b we can do that as well and it will look like this.
x2 = m/p2 – (p1/p2)x1.
In this form, m/p2 represents the vertical intercept and –p1/p2 represents the slope of the budget line. The formula tells us how many units of good 2 the consumer needs to consumer in order to just satisfy the budget constraint if they are consuming x1 units of good 1.
The slope of the budget line has a nice economic interpretation. It measures the rate at which the market is willing to substitute good 1 for good 2, alternatively you can also say that it measures the opportunity cost of consuming good 1.
Changes in the price of goods as well as changes in income all change the budget line. If we start with income it is fairly easy to see how income affects the budget line. If we look back at our equation we can see that income (m) affects the intercept but not the slope of the line. This means that a change in income will result in a parallel shift in the budget line.

Changes in price act a little bit differently though. If we increase the price of good 1 (p1) while holding the price of good 2 (p2) and income (m) constant then if we look at the equation we can see that it will not shift the vertical intercept but that it will change the slope.

If we change the price of both goods it gets slightly more complicated. If we increase the price of both goods by, let’s say, double. Than the slope will not change at all but both of the intercepts will change. If we look at our original budget line of p1x1 + p2x2 = m and multiply both prices by a constant t so that it becomes tp1x1 + tp2x2 = m, we can see that we could rewrite the equation as p1x1 + p2x2 = m/t. So multiply both prices by the same amount is the same as dividing the income by the same amount. So a change in both prices will change the budget line depending on how much the change in each price is compared to each other.
When we look at the budget line we can see that it is made up of two prices and one income. We can do some mathematical tricks though and peg one of the variables to some fixed value, and adjust the other variables to describe exactly the same budget set. Thus we can take the equation
p1x1 + p2x2 = m
with a bit of movement it is the exact same as
(p1/p2)x1 + x2 = m/p2
Or
(p1/m)x1 + (p2/m)x2 = 1
These all represent the same budget line but what we are doing is, in the first equation pegging p2 to 1 and in the second equation doing the same for income (m).
When we do this we refer to the price as the numeraire price. The numeraire price is the price relative to which we are measuring the other price and income.
There are other things besides changes in income and price that can affect a consumer’s budget set. Economic policy often this by taxes or subsidies.
There are a couple of different types of taxes and how the tax is implemented affects how it interacts with a consumer’s budget set. For example if there is a quantity tax this means that the consumer has to pay a certain amount for each unit of the good they purchase. From the consumer’s viewpoint this is basically just an increase in price. So a quantity tax of t dollars per unit of good 1 simply changes the price of good 1 from p1 to p1 + t. This will cause the budget line to get steeper.
Another kind of tax is a value tax. This is a tax on the value, or price, of a good rather than the quantity purchased. This would be what sales taxes are. Value taxes are also known as ad valorem taxes and they changed the price of the good as well. In the case of an ad valorem tax of τ on the price of good 1, the price of the good will change from p1 to (1 + τ)p1. τ is the Greek letter tau.
A subsidy is just the opposite of a tax and it operates in the same way just doing everything the opposite.
Another kind of tax or subsidy that the government can use is a lump-sum tax or subsidy. While quantity and ad valorem taxes tilt the budget line one way or the other, a lump-sum tax or subsidy shifts the budget line out or in.
Taxes and subsidies are not the only affect that the government can have on a consumer’s budget line. A government can also ration goods. This means that the level of consumption for some good is fixed to be no larger than some amount. Suppose for example that good 1 were rationed so that no more than x1* could be consumed by a given consumer. Then the budget set of the consumer would look like this.

Sometimes taxes, subsidies and rationing are all combined. For example perhaps over a certain level of consumption a good is taxed but under that it isn’t. In that case then the consumer’s budget set will look like the following.

A good example of these in real life is the food stamp program in the U.S. Using the information in the textbook I will outline the example they give.
Before 1979, households who met certain eligibility requirements were allowed to purchase food stamps, which could then be used to purchase food at retail outlets. In January 1975 for example, a family of four could receive a maximum monthly allotment of $153 in food coupons by participating in the program. The price of these coupons depended on the household income.
The pre-1979 Food Stamp program was an ad valorem subsidy on food. The rate at which the food was subsidized depended on the household income. The budget line of it would look like this.

After 1979 the Food Stamp program was modified. Instead of required that household purchase food stamps, they are now just given to qualified households. Suppose that now a household receives a grant of $200 of food stamps a month. This means that regardless of how much it is spending on other goods the budget line will shift to the right by $200. The slope of the budget line will not change. That will look like this.

This has been a brief look at budget sets of consumers, that is, looking at what is affordable for people. Tomorrow will be looking at what consumers want to consumer and how we figure that out.