Chapter 2: Budget Constraint transcript

Welcome to chapter 2 budget. The author of this textbook Hal Varian is well known for being very tur and he begins chapter 2 with this one sentence which I've given here. Economists assume consumers choose the best bundle they can afford. This one sentence is going to be the sentence that themes our analysis for the next four chapters. And what we will consider today is the last word of this sentence. What does it mean for uh what does it mean the best bundle that they can afford?

As we start, we want to become more familiar with the notation that Vyian starts out with in chapter 2 and we'll use for the remainder of the text. He uses lowercase X to denote goods. So x lowercase 1 or x subscript 1 is consumer good one. X sub 2 is units of consumption good two and p are the corresponding prices for each of these consumption goods. M is the amount of income that the consumer has available to spend.

Now a typical consumer will choose among thousands of goods. Varian is not telling us that consumers choose only two goods. But if we restrict our analysis to the two good case, it will make the math much simpler. In fact, we'll be able to illustrate choice with charts and diagrams, which will help with the intuition considerably. So, we'll make this simplifying assumption. We'll combine these bits of notation from above into this final budget equation.

So if the consumer chooses X1 units of good one and pays price P1 for each unit consumed, this consumer has spent P1 * X1 on consumption of good one. And likewise P2X2 on good two. So the total consumption spending is the sum of those two expressions that must be less than or equal to the budget. And what we're assuming here is that consumers are not allowed to go into debt. The upper limit of their consumption is what they can what they can afford.

Now, rather than to stay with aneseptic textbook notation, I'd like you to consider going to lunch at VTEX. So, VTEX barbecue. And if we want to have a stereotypical Waco lunch, we have to drink Dr. pepper, which as you know was invented in Waco. So, we'll drink Dr. Pepper and we'll eat small gut packs, which is the signature lunch dish at VTEX. Uh, let's assume that I have $10 to spend on lunch at VTEX, and that cans of Dr.

Pepper cost $1. Gut packs actually do cost a little more than $5 each, but just to make the math easy, let's assume they cost $5 each. So the first bit of math is the stereotypical antiseptic expression of a budget equation. P1X1 plus P2X2= M. And then the line below it, we substitute in the parameters that we have here. The the price of Dr. Pepper is $1 a can. Gut packs costs $5 each. And I have $10 to spend. Now, you'll notice that the budget line down below represents all combinations of Dr.

Pepper and VTEX that cost exactly. Now, since this is a continuous line, what Vyian is assuming is that we can consume partial cans of Dr. Pepper and partial gut packs. This, of course, is a heroic assumption, but it'll make our analysis much easier. Now, the budget set is defined as all combinations of goods that the consumer can afford. You'll notice that uh the budget line on top represented combinations that cost exactly $10 and the shaded gray region underneath the budget line are combinations that cost less than $10. the budget line uh the slope of the budget line is going to be very important for our analysis as we go forward and uh as we develop what the slope of the budget line is.

One way we could do it is graphically and we could think about what the horizontal and the vertical intercept should be in this particular case. So if I have m dollars to spend and good one costs P1 each, M divided by P1 would be the total amount of commodity one that the consumer can afford. So if I were going to VTEX, I could afford $10 divided by a price of one or 10 cans of Dr. Pepper. The vertical intercept is derived in much the same way.

I have $10 to spend. And if I spent that on only gut packs at $5 each, I could consume at most $2, excuse me, two gut packs. So, uh, with the equation of the line, I could solve for rise over run in the normal way. A mathematical way of solving for the budget line is I could take my typical budget equation which I have below the chart as P1X1 plus P2X2= M. And I could rearrange the budget equation to solve for the stereotypical junior high equation for the line.

My vertical variable is equal to the intercept plus the slope. So when I rearrange the budget equation, I've isolated X2, which is my vertical variable is equal to M over P2. That's the vertical intercept. That's how many gut packs I could afford if my lunch consisted only of gut packs with nothing to drink. And then minus P1 over P2 is the slope of the budget line. Okay, so that's the rate at which I can trade Dr. Pepper for gut packs when I go to VTEX.

What would happen to the budget equation if instead of $10, I had $15? Well, I could afford more of everything. And the way we illustrate that graphically is a horizontal shift outward or a parallel shift outward in the budget line. Right? So, the slope of the budget line remains the same whether I had $10 or $15 to spend on lunch. But if I have $15, I can buy more of everything. If we illustrate a price cut. So in this case, if uh VTEX runs a fantastic sale on gut packs and the price dropped to 250, but the price of Dr.

Pepper remained un unchanged, what the way we would implement that in our graph, the horizontal intercept remains the same, right? I can still afford 10 and no more cans of Dr. Pepper, but I can afford twice as many gut packs as I did before if the price fell in half. So, the way this is illustrated, you'll notice it's a pivoting of the line, right? And indeed, my budget set has become larger. I can afford combinations of Dr.

Pepper and Gut Packs that I couldn't afford before. uh but remember again the horizontal intercept remains unchanged. Taxes should be thought of as a change in price. So for instance if VTEX raised the price of Dr. pepper by 10%. Uh, I claim that for the the purpose of analyzing consumer choice, it doesn't matter whether VTEX raises the price of Dr. Pepper by 10% or the city of Waco taxes Dr. Pepper at 10%. It doesn't matter to the consumer choice problem whether these additional expenses go to VTEX or to the city.

All the consumer cares about is that Dr. Pepper now costs 10% more than it did before. Now, Varian gives us three different types of taxes. The first one is what he calls a value or an adorum tax. And in this case, as the language would suggest, it's a tax on the value. So, in other words, uh the tax here is at a rate of lowercase t. So in the example that that I started out with, if the city of Waco imposed a 10% tax on Dr.

Pepper, T would be 0.1. So if Vex is charging a price of P1 for Dr. Pepper and the city of Waco imposes a 10% tax on Dr. Pepper, the after tax price that the consumer sees is 1.1* P1. Okay? and we would get a pivoting inward of the budget set that that would be typical of higher prices or higher taxes. The second type of tax that varian goes through is a quantity tax. So a quantity tax is where we don't impose a percentage but rather a fixed amount on the tax of the good.

So if there was a 25 cent tax on a can of Dr. pepper regardless of what it costs. That would be an example of a quantity tax. Now a value or ad valorum tax is proportional to the price. So for instance, a typical ad valorum or value tax that you're very familiar with would be the sales tax. When you buy anything in the city of Waco, you have to pay an 8.25% sales tax. If you bought an item that costs $1 per unit, you pay uh $8.25.

If you purchased an item that costs $100 per unit, you're paying $8.25. So the tax per unit sold depends upon the underlying price of the good, right? So, as the unit becomes more expensive per unit, uh the the tax collected goes up. It's it's a proportion of the price. A quantity tax, that's not the case. The quantity tax that you may be most familiar with is when you buy fuel, whether that be gasoline or diesel fuel, the tax is levied on a per gallon basis.

So, uh, last I checked, the price or the the fuel tax is about 40. So, that's between taxes at the federal level and taxes imposed by the state of Texas. So, when gasoline was near $4 about a year ago, $4 a gallon, the government collected 40 cents a gallon. When gasoline fell to about $2 a gallon, the tax remains 40. So the tax doesn't change. It's a tax on a per gallon basis. The final type of tax that Varian introduces us to is a lumpsum tax.

And that's where the government takes a certain dollar amount from you regardless of how rich you are, how poor you are, what you spend on good one or good two. Therefore, it's called a lumpsum tax. Uh we don't see these. In fact, I can only think of one instance in which a lumpsum tax was in imposed. Uh the British government in the late 1980s attempted to impose a lumpsum tax. And in fact, I think that was widely credited with what caused Margaret Thatcher to no longer be the prime minister of Great Britain.

But be that as it may, a lump sum tax is a tax that's a fixed dollar amount. There's no way that the consumer can change taxes paid by changing their consumption or their income or any of that sort. Subsidies are mathematically negative taxes. So each category of tax that we just talked about, a parallel category of subsidies exist. We can think of value or ad valorum subsidies where the government subsidizes a certain purchase or a certain percentage of the purchase price regardless of what the price is.

You could think of a 10% subsidy on something. A quantity subsidy is where the government imposes a per unit subsidy regardless of price. One example I could think of right off the top of my head was uh the electric vehicle subsidy. Up until very recently, the federal government would give a $7,500 subsidy to anyone who purchased a new electric car. It didn't matter whether you got kind of a uh inexpensive economy electric car like a Nissan Leaf or a Chevy Bolt or if you bought a Lucid Air or a Tesla S, right?

You got $7,500. Therefore, it's a quantity subsidy. And finally, we could think of lumpsum subsidies where the government gives you cash to spend any way that you want. In this slide, we think about a subsidy with threshold. So in other words, the consumer has to do something to qualify for the subsidy. And in in this particular example that I made up, suppose that the city of Waco gives tourists $50 in cash to spend any way they want if they spend at least $250 at the Magnolia silos.

So I've drawn a picture of what this budget line would look like. So if our two goods are on the horizontal axis dollar, the vertical axis uh dollars spent anywhere other than Magnolia. The slope of the budget line represents the rate at which a consumer trades dollars spent at Magnolia for dollars spent not at Magnolia. So the tourist shows up with $500 in sp, right? So each dollar that they spend in Magnolia takes $1 away from the amount that they have to spend on other goods until they to the point 250.

And when the consumer spends the $250th at Magnolia now they qualify to get this $50 cash grant from the city of Waco. So in the act of spending the 250 at Magnolia, the consumer's available cash to spend on other things jumps up to $300. Therefore, we have this discontinuous jump in the budget line and then the budget line continues as before with a slope of -1. Finally, we can think of rationing. So rationing would be uh a real world example that I can think of of rationing was in World War II, the federal government rationed gasoline.

So consumers were not permitted to buy as much gasoline as they could afford, but rather um the amount that any given consumer could purchase was restricted to save gasoline for the war effort. So, trying to keep with uh the Waco theme of stories, what if the city of Waco rationed Dr. Pepper at lunchtime to four cans per meal. And what that does is it basically cuts off the right tail of the budget set, right? The budget set becomes truncated.

So, it's a very simple idea. Now if you look at the canvas page for this course you'll notice that with each chapter we consider from variance intermediate micro text I've posted uh what I call practice quizzes. So you have the practice quizzes and then you have my answer key that I've submitted with it. And these practice quiz questions are are questions from Ted Bergstrom's intermediate micro workbook. and they're excellent questions, great preparation for taking the exam.

I'm not going to grade them, but I'm I'm uh giving them to you so you become familiar with the type of multiplechoice questions that that you'll see in this course. And at the end of each lecture, I'd like to go over one or maybe two questions. So today, I'm going to go over question 2.5. And we're going to consider Martha who's taking this semester an economics course and a sociology course. And we're given two different combinations of readings that she could do in the same amount of time.

So she could read 40 pages of economics and 30 pages of sociology in the same amount of time that she could read 20 pages of economics but 90 pages of sociology. So there's a trade-off. Martha has this tradeoff where you'll notice in the first bundle she's reading 40 pages of economics. In the second bundle she's reading 20. So by cutting back 20 pages in the uh the number of pages of economics she reads she's able to read 60 more pages of sociology.

So this implies then that Martha can read three pages of sociology for each page she reads in economics. Now uh we're given four different choices of budget lines and asked which one of these equations would describe Martha's choices between readings in economics and sociology. Now you remember from junior high math that if you have two points, two points define a line, right? So we can we could derive a line from these two points.

And what we also know is that uh each each one of these points must uh be described by the equation of the budget line. So uh one way of solving this problem would be simply to take each budget equation. So we have four candidates and apply them to both bundles. So the f the first one uh the first possible budget equation is I added the number of pages of economics and sociology. So for the first one 40 econ and 30 sociology that sums to 70 and the budget equation candidate A does describe that but 20 + 90 is not equal 70.

So the first candidate is a bad candidate. And you'll notice when we go through all these options, the only uh candidate equation that describes both points is option E, which is 3 E plus 150. So that's the only one that contains both points. So that's the only reasonable answer this problem. Another way about thinking, another way of thinking about this problem is that the uh that if I look at the typical form of a budget equation, P1 X1 + P2X2= M three is the price of reading a page of economics.

So, what this correct budget equation tells me is that it takes Martha three times as much time to read a page of economics as it takes her to read a page of sociology, which captures the intuition of uh this substitution that we talked about in in the last slide of substituting 20 pages of economics for 60 pages of sociology. So that that's more of an economic reasoning way of solving the problem. Okay, that does it for chapter 2 and uh we'll see you in chapter three.