Intermediate Macroeconomics ECN 312 – Summer 2015 Lecture 3: The Budget Constraint Introduction to Consumer Theory Consumer theory is about choice. Economists are concerned with what bundle of good people will choose to consume. This is relevant in many way. » Will society have enough food to meet the demands of the population?

Intermediate Macroeconomics ECN 312 – Summer 2015 Lecture 3: The Budget Constraint Introduction to Consumer Theory Consumer theory is about choice. Economists are concerned with what bundle of good people will choose to consume. This is relevant in many way. » Will society have enough food to meet the demands of the population? » Will people purchase an expensive luxury item like a digital smart watch? » As productivity rises, will people work more or less? » If a tax is placed on some good, how will people’s choices change? Generally, economists assume that consumers will choose the best bundle of goods they can afford. Of course, we need to be clear about what “best” and “can afford” means. What People Can Afford Our model for consumers is a simplified version of reality, where we their income as given. The budget constraint is an equation which describes what a person can afford. Suppose we live in a world with just two goods: guns (𝑥1) and butter (𝑥2). A consumer’s choice of these goods is their consumption bundle (𝑥1, 𝑥2). This is will be two numbers, such as (5, 8) or (0, 10) that represent a consumer obtaining 5 guns and 8 butters or 0 guns and 10 butters respectively. The prices of these two goods are denoted as 𝑝1 and 𝑝2, where 𝑝1 is the price of 𝑥1 (guns) and 𝑝2 is the price of 𝑥2 (butter). Taking the amount of money a person has to spend, 𝑚, as given, the budget constraint of the consumer is 𝒑𝟏𝒙𝟏 + 𝒑𝟐𝒙𝟐 ≤ 𝒎. Note that 𝑝1𝑥1 is the amount of money the consumer spends on guns, and 𝑝2𝑥2 is the amount of money the consumer spends on butter. This must be less than or equal to what the consumer has to spend. The set of affordable consumption bundles is called the budget set of the consumer. Real World Budget Constraints In the real world, budget constraints are long, and include many items like housing, cars, food, movies, schooling, music, television, games, etc. And each of those is even more detailed than we are letting on. Housing is doors, windows, carpeting, paint, furniture, etc. Cars are vehicles, insurance, gas, washes, air fresheners, etc. And so on. But models are supposed to be a simplified version of reality, which is why economists often refer to the composite good, or a good which represents “everything else”. With the composite good in play, we can keep things to a two-good scenario. Ex. Analysis of consumption of movie theater tickets: 𝑥1: movie theater tickets 𝑥2: all other goods. 𝑝1𝑥1 + 𝑝2𝑥2 ≤ 𝑚 𝑝1: price of movie theater tickets 𝑝2: price of the composite good Sometimes, to make things easier, you can think of the composite good as money itself. When that is the case, the budget constraint is simply: 𝑝1𝑥1 + 𝑥2 ≤ 𝑚 Properties of the Budget Set The budget line is the set of bundles that cost exactly 𝑚: 𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚 We can rearrange the budget line to give us: 𝑥2 = 𝑚 𝑝2 − 𝑝1 𝑝2 𝑥1 Let’s look at a defined budget line: 2𝑥1 + 5𝑥2 = 100 The shaded region is the budget set. The budget line has a slope of −𝑝1/𝑝2. The vertical intercept is 𝑚/𝑝2, which is the maximum amount of good 2 that the consumer can afford to buy. The horizontal intercept is 𝑚/𝑝1, which is the maximum amount of good 1 that the consumer can afford to buy. The Slope of the Budget Line The slope of the budget line has a nice economic interpretation. It tells us the rate at which we can “substitute” good 1 for good 2 and maintain the same level of expenditure. In other words, it is the rate at which the market will exchange these two goods. Looking back at our budget line 2𝑥1 + 5𝑥2 = 100, the slope of the line is -0.4. This means that in order to get 1 unit of 𝑥1, you have to give up 0.4 units of 𝑥2. Thus, the slope of the budget line represents the opportunity cost of consuming more 𝑥1 in terms of 𝑥2. This is the true economic cost of 𝑥1. 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 QUANTITY OF X2 QUANTITY OF X1 Changes in Your Budget Below we will look at how the budget constraint changes under difference scenarios. In each case we will start with our guns and butter case: 𝑥1: guns 𝑥2: butter 𝑝1: price of guns 𝑝2: price of butter 𝑚: money to spend 𝑝1𝑥1 + 𝑝2𝑥2 ≤ 𝑚 Initially the budget line will be: 2𝑥1 + 5𝑥2 = 100 Change in Income When m increases, the budget line shifts out and is parallel to the original line. Say m increases to 120, so that: 2𝑥1 + 5𝑥2 = 120 A decrease in income would have the opposite effect. Changes in Prices Suppose 𝑝1 increases to 3, so 3𝑥1 + 5𝑥2 = 100. The higher price rotates the budget line around the vertical intercept, and the slope of course changes. Suppose 𝑝2 drops to 4, so 2𝑥1 + 4𝑥2 = 100. Again the slope changes, but now the horizontal intercept stays the same. Suppose both changes happened at the same some, so 3𝑥1 + 4𝑥2 = 100. In this case, some bundles are newly affordable while others are now unaffordable. Suppose that there is inflation, and the prices of both guns and butter go up by 10%. How does the budget constraint look? (1 + 𝜋)2𝑥1 + (1 + 𝜋)5𝑥2 = 100 or 2𝑥1 + 5𝑥2 = 100 1 + 𝜋 So this will have the same effect as a reduction in income: Of course, if income (m) goes up by the same rate, then (1+pi) just cancels out, and there is no effect at all. Taxes and Subsidies Quantity Taxes: taxes paid per unit of the good purchased. » Ex. Gasoline taxes (18.4 cents per gallon) Ad Valorem (Sales) Taxes: taxes paid as a percentage of the price of the good. » Ex. Arizona sales tax is 5.6%. Maricopa County adds 0.7% and Tempe adds 1.8% (8.1 total) Lump-Sum Tax: a fixed tax that does not depend on anything. » Ex. Everyone pays $20. Income Taxes: taxes paid as a percentage of income. Subsidy: the opposite of a tax, where the government pays the consumer. » These can come in the same variety as taxes. Let’s try and set up budget constraints for each of these kinds of taxes, and then think about how a subsidy would be different. Quantity Taxes Suppose we put a quantity tax (𝑡) on good 1 (𝑥1). [Notice that I am moving away from the example of guns and butter to the general case] Original constraint: 𝑝1𝑥1 + 𝑝2𝑥2 ≤ 𝑚 New constraint: (𝑝1 + 𝑡)𝑥1 + 𝑝2𝑥2 ≤ 𝑚 A subsidy? New constraint: (𝑝1 − 𝑡)𝑥1 + 𝑝2𝑥2 ≤ 𝑚 Ad Valorem Taxes Suppose we put an Ad Valorem tax (𝜏) on good 2. Original constraint: 𝑝1𝑥1 + 𝑝2𝑥2 ≤ 𝑚 New constraint: 𝑝1𝑥1 + 𝑝2𝑥2 + 𝜏𝑝2𝑥2 ≤ 𝑚 or 𝑝1𝑥1 + (1 + 𝜏)𝑝2𝑥2 ≤ 𝑚 A subsidy? 𝑝1𝑥1 + (1 − 𝜏)𝑝2𝑥2 ≤ 𝑚 Lump-Sum Tax Suppose we put a Lump-Sum Tax (𝑇) on this consumer. Original constraint: 𝑝1𝑥1 + 𝑝2𝑥2 ≤ 𝑚 New constraint: 𝑝1𝑥1 + 𝑝2𝑥2 + 𝑇 ≤ 𝑚 or 𝑝1𝑥1 + 𝑝2𝑥2 ≤ 𝑚 − 𝑇 A subsidy? Income Tax Suppose we put an income tax (𝜏) on this consumer. Original constraint: 𝑝1𝑥1 + 𝑝2𝑥2 ≤ 𝑚 New constraint: 𝑝1𝑥1 + 𝑝2𝑥2 ≤ (1 − 𝜏)𝑚 A subsidy? Rationing In times of war, some nations ration goods to their people. For instance, during World War II the government rationed things like butter and meat, allowing consumers to buy only a certain amount of that good and no more. How would this look graphically? Say we are back to guns and butter, but we can only buy up to 30 guns. Or only 10 butter: Rationing with Taxes Taxes can be applied only after a certain amount of a good has been purchased. For example, second homes are often taxed at a much higher rate that first homes are. How would this constraint look? Note that this is a rather difficult budget line to write out mathematically: 𝑝1𝑥1 + 𝑝2𝑥2 ≤ 𝑚 if 𝑥1 ≤ 𝑥̅1 𝑝1𝑥1 + 𝑝2𝑥2 + 𝑡(𝑥1 − 𝑥̅1) ≤ 𝑚 if 𝑥1 > 𝑥̅1 So sometimes it is nicer to just keep to the graphical analysis. In-Kind Transfers, Gift Cards, and Food Stamps In-Kind Transfers are a transfer of a certain amount of some good to a person. A very common type of in-kind transfer are gift cards. What happens when you get a gift card for 𝑥1? Let’s say it is enough to buy you 5 units of 𝑥1. This leads to a “kink” in the budget constraint. This is why gift cards are somewhat perplexing. Because they can only be as good or worse than money. Varian has the same analysis for food stamps, which are a kind of gift card, but good only for certain items rather than only certain stores. But you can see that not only is it possible to afford more food, but also to afford more everything (or more of the composite good).

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