# Our development of economic theory has two main parts, consumers and producers. We will start with the consumers.

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1 Lecture 1: Budget Constraints c 2008 Je rey A. Miron Outline 1. Introduction 2. Two Goods are Often Enough 3. Properties of the Budget Set 4. How the Budget Line Changes 5. The Numeraire 6. Taxes, Subsidies, Rationing 1 Introduction Our development of economic theory has two main parts, consumers and producers. We will start with the consumers. We want to develop a model of consumer behavior that makes speci c predictions about what goods consumers buy, how much, when, what factors determine this, and so on. This will allow us to analyze how consumers react in various situations (in response to di erent kinds of information or goods, government policies, rm advertising). We start with the most basic component of the model: the budget constraint. This should mainly be review, with little new or hard material. I include it for completeness, to make sure we re all using the same ideas and notations. Plus, we will add a few wrinkles relative to what you have seen. The basic framework is the following: at any point in time, a consumer can choose from a particular set of goods. For now, we will deal with exactly two goods, but this is mainly for exposition. 1

2 The consumer will end up, after the choice is made, with a particular consumption bundle. Call this (x 1 ; x 2 ) It is always important to be clear about the units of measurement. There are many instances where the answer to an economics question is entirely about getting the units right. Also, it always helps to know exactly what we are measuring. In particular, we may be measuring physical units versus monetary units (like dollars), real versus nominal, etc. So, in this case, we measure goods 1 and 2 in terms of physical units: loaves of bread; minutes of cell phone use; gallons of water, number of cd s; rounds of golf; etc. Say each good has some price at which the consumer can purchase the good. Denote these by (p 1 ; p 2 ) These are the nominal prices, i.e., the sticker prices measured in, say, dollars. (This could be measured in some other units, not necessarily just currency. Ignore real versus nominal for the time being.) Then the consumer s budget constraint is p 1 x 1 + p 2 x 2 m where m is the amount of money the consumer has to spend. measured in dollars. Again, think of m as Thus, p 1 x 1 is expenditure on good 1 and p 2 x 2 is expenditure on good 2. The budget constraint simply says that total spending by the consumer must be less than or equal to the amount of money the consumer has to spend. Combinations of (x 1 ; x 2 ) that satisfy this budget constraint are sometimes referred to as a ordable consumption bundles. The set of a ordable consumption bundles at a given set of prices and income is the budget set. This is the rst component of our model. As presented so far it is totally straightforward. 2

3 2 Two Goods are Often Enough Much of the analysis we will consider assumes there are in fact only two goods. This is obviously convenient because it allows for a graphical presentation, which we will consider in a moment. It also avoids the need for a calculus tool known as Lagrange multipliers. I ll provide an example of those once or twice, but they are not required for the course. We could, of course, consider an n good model, p 1 x 1 + p 2 x 2 + : : : + p n x n m Almost nothing other than ease of presentation would change. The two good assumption is reasonable because most of the main insights although not all are independent of whether the number of goods is two, three, fty, or whatever. The assumption is also often used when we want to focus on one good in particular, as we can de ne the other good as everything else. In this case, we would think of good 1 as something speci c and good 2 as a composite good. It then makes sense to think of good 2 as being measured in dollars, i.e., the amount of money the consumer spends on all other goods besides good 1. So, in that case the price of good 2 is simply 1: a dollar costs a dollar. The budget constraint becomes p 1 x 1 + x 2 m In words, the amount of money spent on good 1 plus the amount spent on all other goods is less than or equal to the total money a consumer has to spend. This is a simple variation on the basic setup; we will use it several times during the course. 3

4 3 Properties of the Budget Set Now we want to examine the properties of the budget set. We will consider this both using algebra and graphs. The set of consumption bundles that exactly exhausts the consumer s possible expenditure is the set of (x 1 ; x 2 ) that satis es p 1 x 1 + p 1 x 2 = m The only di erence is that we now have only equality in our budget constraint. This is the equation of a line relating x 1 and x 2, assuming we take p 1 ; p 2 and m as given. The standard way to present this the way you have already seen is in a graph of x 2 versus x 1. 4

5 Graph: Standard Budget Line (x 2 = 20 2x 1 ) x x1 Note what is on the axes (that is the rst thing you should check do when confronted with any graph, including noting the scale of the graph). Algebraically, the equation for the budget line is x 2 = (m=p 2 ) (p 1 =p 2 )x 1 So, this is just the equation for a straight line with intercept m=p 2 and slope p 1 =p 2. 5

6 Of course, the only parts of the line that are relevant to our analysis are those in the rst quadrant, since those are the points where the amounts of the two goods are positive or zero. There are some settings where you might want to think about negative quantities e.g., short-selling stocks but ignore these for now. What are the intercept points? be If he spent everything on good 1, the consumer s expenditure on good 1 would m=p 1 Why? Because p 1 x 1 + p 2 0 = m implies this. So, the point where the budget line intersects the x 1 axis is m=p 1. The same reasoning shows that the intersection point for the other axis has to be m=p 2. The slope of the budget line is p 1 =p 2 In economics terms, the slope is the rate at which this consumer can trade one good for another in the marketplace. That is, if you reduce consumption of x 1 by one unit, you save p 1 dollars; with these extra p 1 dollars, you can now a ord an extra p 1 =p 2 units of x 2. Thus, we see that the extra x 2 you can a ord is p 1 =p 2 times the change in x 1. Now, all points inside the line and the axes are also a ordable. So those points, plus the line, constitute the budget set. Whether the consumer is always on the budget line, or inside it, is something we will discuss. For the most part, the natural assumption is that the consumer spends everything; note, however, that this is an assumption that might not hold in some circumstances. For example, some consumers might be irrational, or some consumers might get satiated. Satiation per se does not necessarily mean you are inside the budget line; 6

7 if you have access to free disposal, you do not care whether you have extra. But in many instances, disposing of goods that you do not want can be costly, so we might want to model that explicitly in certain instances. A di erent terminology for describing the slope of the budget line is to say that it measures the opportunity cost of consuming good 1. This is because p 1 =p 2 is the amount of x 2 you give up for each unit of x 1 ; it is the lost opportunity, i.e., what it costs economically to have a unit of x 1. If p 1 is big, then the cost of consuming x 1 is large since you could have purchased a lot of x 2, holding the price of good 2 constant. If p 2 is large, then the cost of consuming x 1 is small, since you would not have gotten much x 2 by reducing your consumption of x 1. This is not saying anything di erent; it is simply a choice of language or description that can be helpful in some instances. 4 How the Budget Line Changes Next we want to examine how the budget line and budget set change when prices or income change. 4.1 A Change in Income It is obvious from the equation that an increase in income causes a parallel shift of the budget line, i.e., it a ects the intercept but not the slope. This make sense intuitively: a higher level of income does not a ect the rate at which you can trade the two goods in the marketplace; that is determined by prices, which are beyond the consumer s control. Remember that the intercept of the budget line is m=p 2 so an increase in m shifts the line up. 7

8 Graph: A Shift in the Budget Line Due to an Increase in Income (x 2 = 20 2x 1 ) >(x 2 = 30 2x 1 ) x x1 Of course, it makes sense that the budget set is bigger if you have more income. 4.2 One Price Changes Price changes (for one good at a time) mean that we rotate the budget line, holding one of the two intercepts xed. Say we increase p 1. This does not a ect the vertical (x 2 ) intercept, assuming p 2 is held constant. This makes sense, since if you are spending all your money on good 2, the price of good 1 is irrelevant. But at any other level of x 2, the amount of x 1 you can buy is now lower; thus, the line rotates inward. Equivalently, if you spent everything on good 1, you would be able to buy fewer units, so the horizontal intercept has to shift to the left. We now have a steeper slope ( p 1 =p 2 is greater in absolute value). 8

9 Graph: An Increase in p 1 Rotates the Budget Line (x 2 = 20 2x 1 ) >(x 2 = 20 4x 1 ) x x1 We see that the opportunity cost of good 1 is now higher: to get an extra unit of x 1, we now have to give up more x 2. Of course, an increase in p 2 rotates relative to the other intersection. 4.3 Both Prices Change by the Same Amount Say we start with p 1 x 1 + p 2 x 2 = m and then move to tp 1 x 1 + tp 2 x 2 = m Then this is the same as 9

10 p 1 x 1 + p 2 x 2 = m=t So, an increase in both prices by t percent is equivalent to a decrease in income by that same percent; thus, the budget line shifts inward. Note that this is not a general in ation: we normally think of in ation as raising all prices, including wage rates, by the same percentage; so m would also increase in proportion to t. Of course, the example considered here might be the relevant one for a person whose nominal income is xed, whether because of not getting salary increases or because a government transfer payment is not indexed to in ation. But it is a di erent situation than general in ation. 4.4 Both Prices and Income Change by the Same Percentage tp 1 x 1 + tp 2 x2 = tm Nothing changes: just divide t out of the budget line. This is the case of general in ation. So, that s the basic budget set / budget constraint. This is nothing di erent from EC10, except that we have used an equation. We will start to make it a bit more interesting in a minute. 5 The Numeraire The next item is a small piece of terminology that coincides with an economic insight. The budget line is de ned by two prices and one income. One price is redundant. The original budget line, 10

11 is the same as, p 1 x 1 + p 2 x 2 = m x 1 + x 2 (p 2 =p 1 ) = m=p 1 This is the same line as before, with the same set of points. This process is called choosing one good as the numeraire. It is a useful tool sometimes. We will see examples later The numeraire also explicitly represents the fact that only relative prices matter; we ll see that in more detail in a few lectures. 6 Taxes, Subsidies, Rationing One issue we frequently want to consider is the e ect of government policies on economic outcomes. Let s take a brief look at certain policy toolds as a way of illustrating the budget set. 6.1 Taxes A quantity tax imposes a certain amount of tax per unit of the good. Examples in the US include: 15 cents in tax per gallon of gas; some kinds of alcohol taxes; certain airline, taxes and phone service taxes, cable TV taxes. The tax is set to be so many \$ per ticket, etc. The tax changes the price from p 1 to p 1 + t. This makes budget line steeper. (p 1 + t)x 1 + p 2 x 2 = m or x 2 = (p 1 + t) p 2 x 1 + m p 2 11

12 A value, or ad valorem tax, is a tax on the value, not the quantity. It usually is seen as a percent, e.g., = :05. Examples: sales taxes, value-added taxes. A value tax changes the price to (1 + )p 1. p 1 goes to seller, and p 1 goes to the government. The total cost to consumer is therefore the sum. This makes the budget line steeper again, although not in exactly the same way. p 1 (1 + )x 1 + p 2 x 2 = m A subsidy is the opposite of a tax; we have a minus sign where we originally had a plus sign. A lump sum tax is a xed amount that is independent of behavior. Thus, the budget line shifts in or out, depending on whether we are dealing with a tax or subsidy, but the relative prices are una ected. p 1 x 1 + p 2 x 2 = m T Note the di erence between the rst two and the lump-sum. The rst two change the relative price of the two goods. For the quantity tax, the degree to which it a ects the relative price depends on the level of prices. For the ad valorem tax, it a ects the relative price, but the e ect is the same no matter what the level of prices. Lump-sum taxes do not a ect relative prices at all. This result about lump-sum taxes turns out to be important in public nance, speci cally, in the design of optimal tax systems. Lump sum taxes are thought of as non-distorting because they do not change relative prices. Of course, you might think this statement is incomplete, since if income is earned and we tax income, it would seem that we are distorting an economic decision - the choice of how much to work. Leave this aside for now; we will clarify it in later lectures. For the moment, we have just a hint of this deeper result that the kind of taxes we might prefer should depend on whether the tax a ects a relative price. 6.2 Rationing Another kind of intervention is rationing constraints. Lots of examples exist: 12

13 1. During wars, governments have frequently rationed consumer purchases of meat, sugar, tires, gasoline, and so on. 2. Planned economies like the former Soviet Union have often rationed purchases of consumer durables like toasters or dishwashers. 3. The Brazilian government rationed purchases of consumer goods from Manaus when the tari rates there were low. 4. Countries with government-provided medical insurance often ration the provision of certain procedures that are expensive and in high demand, like fancy scans or elective surgeries. Say good 1 is rationed: x 1 x 1 Then the budget set is as shown in the picture. 13

14 Graph: Budget Line with Rationing x 2 = 20 2x 1 > x 1 = 10 x 2 =2 10 x2 =2 if 12 x x 1 = if x 2 < 12 x x1 Things can get more interesting due to the complicated features of certain policies or options in the marketplace. Say a tax only applies after a certain amount of expenditure, e.g., luxury taxes that apply only if the price of a good is above a certain threshold. Or, the prices that apply change depending on the amount consumed, e.g., 400 free cell phone minutes per month and then \$1.00 per minute thereafter. Or, the rst airline ticket is full fare and the second is 50% o. To give a speci c example, say the consumer faces p 1 up to x 1 but p 1 + t to the right of x 1. 14

15 Graph: Budget Line with Rationing and Taxes 20 2x1 if 0 x x 2 = x 1 if 3 < x 1 x x1 There are many other examples; see the Food Stamp Program discussion in the textbook. 15

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