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# Microeconomics 312: Consumer Theory Economics 312

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This 147 page Bundle was uploaded by Paul Hickey on Tuesday January 5, 2016. The Bundle belongs to Economics 312 at Arizona State University taught by Brian Goegan in Summer 2015. Since its upload, it has received 215 views. For similar materials see Microeconomics in Economcs at Arizona State University.

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Intermediate Macroeconomics ECN 312 – Summer 2015 Lecture 1: Introduction and Review Syllabus The textbook is optional but helpful. My lectures will be based more on Varian than Landsburg. We will not have class on Friday, June 25 . There is also no class on Monday, May 25 . Problem sets cannot be turned in late. They are there to keep you up to speed, so if you fall behind, you don’t get the credit. I know Fernando curves his grades, but I don’t. My grading system is straightforward and easy. This class is not easy, but it can be made a lot easier by coming to class every day and staying on top of the assignments and material. This class is an applied math class. We will be using lots of math and lots of calculus. But all of it will be for a purpose. All of it is used to analyze the behavior of people. Economics Review Microeconomics can be summed up in one sentence: People choose where marginal benefit is equal to marginal cost. Marginal Benefit: the additional satisfaction or utility that a person receives from consuming an additional unit of a good or service. Marginal Cost: the additional cost a person must pay in order to obtain one additional unit of a good or service. Economics is about identifying those marginal benefits and marginal costs, and determining how they change under different conditions. Economists Do it With Models Models are a simplified version of reality. An example are grades. Grades are a simplified model of your performance in a class. They reduce your studies and efforts to a series of variables (problems set 1, 2, 3, …, exam 1, 2, 3) and then combine them through some process into a single grade (A+, A, etc.) that is supposed to correlated with learning and effort. Models are generally comprised of two kinds of variables: » Exogenous (or independent) variables: variables which are determined by factors not within the scope of the model. » Ex. Your grades on assignments. » Endogenous (or dependent) variables: variables which are determined by forces described in the model. » Ex. Your final grade is determined by the model. Models are simplified versions of reality because reality does not have exogenous variables. » This is actually a pretty controversial statement, as the current state of science does take a number of things about the universe as given. » In this class though, we are generally trying to model people’s behavior as it relates to consumption and production, or the distribution and use of resources in society. In a Principles of Economics class (micro or macro), the models are generally distilled into their simplest forms, and often only shown as graphs. The Principles of Economics There are two main principles I hope you left your principles class knowing about: » The optimization principle: people try to choose the best patterns of consumption that they can afford. » The equilibrium principle: prices adjust until the amount that people demand of something is equal to the amount that is supplied. The optimization principle is just another way of saying that people choose where marginal benefit is equal to marginal cost. » Let’s think about education, and choose the optimal amount of education. What are the marginal benefits, and what are the marginal costs? The equilibrium principle requires a little more review. Demand, Supply, and Equilibrium The economic concept of demand is built on the idea that people want to consume things. It gives them utility when they do, which means it makes them happy or improves their well-being. » Consumption doesn’t have to be material either. I often consume good views of nature, the emotional warmth of family, and the satisfaction of choosing political leaders. The model of demand combines people’s reservation price, or the maximum price at which they are willing to buy a good or service (or other thing). » This price may depend on many things, but includes compliments and substitutes. Demand for Movie Tickets Each person values movies differently, they have a different reservation price. And many people would purchase more than one movie ticket at certain prices. Price Quantity 4 5 2.25 billion 3.5 6 2 billion 3 7 1.75 billion 8 1.5 billion 2.5 2 9 1.25 billion 10 1 billion 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 1 We can model this demand by the equation: ???? ???? 3.5 − ???? 4 Note that my graph does not follow the convention of putting the independent variable on the horizontal axis. It has flipped them. So if we want to graph this demand curve, we need to find the inverse function. 7 1 ???? ???? − ???? 2 4 1 7 ???? ???? ???? = 4 2 1 7 ???? = − ???? ???? 4 2 ???? = 14 − 4???? ???? This is a simplified model of the demand for movie tickets. In one linear equation, it claims to describe the wishes and wants of billions of people, who are motivated by many things other than the price of a ticket. It also simplifies the discontinuities in demand. A ticket price of 7.38 might not attract a single additional patron as compared to a ticket price of 7.39 (let alone 7.381). But with billions of people, we presume there is someone at every point on this line, and that isn’t such a bad assumption to make. Supply of Movie Tickets Different theaters face different costs when supplying movies. Some theaters are located in big cities, where there is a high opportunity cost to using the land. Some theaters are located in rural areas, where there might be higher costs associated with shipping in projectors and posters and film reels. As the price of movie tickets rises, more and more entrepreneurs will enter the industry, and build new theaters to reap the reward of high ticket prices. Each theater will expand the supply of movie tickets. Short Run Price Quantity 12 5 0.75 billion 6 1 billion 10 7 1.25 billion 8 8 1.5 billion 9 1.75 billion 6 10 2 billion 4 2 0 0 0.5 1 1.5 2 2.5 1 1 We can model this supply by the equation: ???? ???? ???? 4 2 Again, to see how to graph this, we need the inverse function: ???? = 2 + 4???? In the long run, there might be different costs faced, because there is more opportunity to supply new theaters. In the short run, we may be limited to adding seats to current theaters or converting spaces already built to house audiences. In the long-run, supply is more elastic. %∆???? ????????= ???? %∆???? Likewise, %∆???? ???? = ???? ???? %∆???? Equilibrium for Movie Tickets Per our “equilibrium principle”, the market will settle where the quantity demanded is equal to the quantity supplied. 16 14 12 10 8 6 4 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 We can see in the charts that both supply and demand hit 1.5 billion tickets at $8 per ticket. We can also solve this algebraically. ???? = ???? ???? ???? 1 1 7 1 ???? − = − ???? 4 2 2 4 1 ???? = 4 2 ???? = 8 Now put P into either supply or demand 1 1 ????????= 4 8 − 2 = 1.5 7 1 ????????= 2 − 4(8 = 1.5 Equilibrium is always down hill The “optimization principle” is what drives the market to equilibrium. If the price were too high, more tickets would be offered than would be sold. This surplus would either be wasted, and the surplus providers would go out of business, or they would compete with other sellers by cutting their price, driving the market price down. If the price were too low, the lines would be longer than the ticket count, and there would be a shortage. Those with a lower reservation price would have an incentive to sell their tickets (if they get them) to those with the higher reservation price. This competition among consumers would raise the market price, incentivizing supplier to increase their supplies. Shifts in Supply and Demand Either curve will shift when there is a change in market conditions that does not involve the price. For demand, this means anything that would impact consumers’ reservation price. » Ex. A change in consumer income (the shift depends on normal vs. inferior good). For supply, this means anything that would impact producers’ marginal costs. Imperfect Markets Movie theaters do not operate in a perfectly competitive environment though. Theaters are often local monopolies, or operate in an oligopoly situation. The ordinary monopolist, who supplies the entire market, will reduce their output to raise the price and increase revenues. Revenue = Price x Quantity For the monopolist, price is a function of quantity, and they base their choice of output on the marginal revenue. [Draw in the marginal revenue curve and note the monopolist’s choice]. The discriminating monopolist will segment the market when possible, identifying characteristics which correlate with a consumer’s reservation price. » Movie theaters segment based on time (matinee vs regular price), age (senior discounts), vocation (student), and amenities (3D). Of course, there are also things like taxes, subsidies, price controls, and other regulations. Pareto Efficiency You may have heard about the difference between normative and positive economics. Economics was born out of the moral and political philosophy of utilitarianism as a positive way to measure well-being. However, economists have not shed their normative utilitarian roots, and are often concerns about “efficiency”. A Pareto improvement is one which makes at least one person better off without making anyone worse off. » Pareto efficient is a condition in which there are no possible Pareto improvements. While preferring Pareto efficiency is a normative judgement, it is an easy one to defend. Where things get tricky is when we note that there are alternative designs which could be Pareto improvements, when we have market conditions like monopolies, taxes, or price controls, but those designs are impossible to implement. Our chief way of identifying those possibilities is by looking at: » Consumer Surplus: the net benefit of trade to consumers. » Producer Surplus: the net benefit of trade to producers. » Deadweight Loss: regions where both producers and consumers could see a net benefit to trade, but are prohibited from making the trade for some reason. o OR wasted resources due to overproduction at an artificially low price. Intermediate Macroeconomics ECN 312 – Summer 2015 Lecture 2: Math Review Functions A function is a rule that describes a relationship between numbers. 2 » Ex. ???? = ???? or ???? = 2???? or ???? = ???? ???? or ???? = ????(???? ,1 )2 » The variable x is often called the independent variable. » The set of values from which x can be drawn is the domain. » The variable y is often called the dependent variable. » The set of all corresponding y values is the range. Example: If ???? ???? = ???? − 4???? + 2, then 3 » ???? 1 = (1) − 4 1 + 2 = 1 − 4 + 2 = −1 » ???? −2 = (−2) − 4 −2 + 2 = −8 + 8 + 2 = 2 3 » ???? ???? = (????) − 4 ???? + 2 Graphs A graph of a function depicts the behavior of a function pictorially. 25 ???? = 2???? ???? 0 = 0 20 ( ) ???? 2 = 4 15 ???? 4 = 8 ???? 6 = 12 10 ???? 8 = 14 5 ???? 10 = 20 0 0 2 4 6 8 10 2 120 ???? = ???? 100 ???? 0 = 0 ???? 2 = 4 80 ???? 4 = 16 60 ???? 6 = 36 ???? 8 = 64 40 ???? 10 = 100 20 Remember that shifts in a curve 0 represents changes to the equation, not 0 2 4 6 8 10 to our input value for x. Properties of Functions A continuous function is one that can be drawn without lifting a pencil from the paper: there are no jumps in a continuous function. A smooth function is one that has no “kinks” or corners. » Example of a non-smooth function: ???? = |????| A monotonic function is one that always increases or always decreases. » A positive monotonic function always increases as x increases. » A negative monotonic function always decreases and x increases. Both ???? = 2???? is positive monotonic, but ???? = ???? is not, because it is decreasing when x is negative. Equations and Identities An equation asks when a function is equal to some particular number. » Ex. 2???? = 8, ???? = 9, ???? ???? = 0. The solution to an equation is a value of x that satisfies the equation. » Ex. 4, 3 and -3, x* (we say that x* satisfies the equation???? ???? = 0. An identity is a relationship between variables that holds for all values of the variable. » (???? + ????)2 ≡ ???? + 2???????? + ???? 2 » 2 ???? + 1 ≡ 2???? + 2 Changes and Rates of Change The notation Δ???? is read as “the change in x.” If x changes from x* to x**, then the change in x is just Δ???? = ????∗∗− ???? , which can also be written as = ???? + Δ????. Typically, when we write Δ????, we mean a very small change in x, or a marginal change. A rate of change is the ratio of two changes. If y is a function of x, and is given by ???? = ????(????), then the rate of change of y with respect to x is denoted by: Δ???? ???? ???? + Δ???? − ????(????) Δ???? = Δ???? Example: ???? ???? = ???? + ???????? Δ???? = ???? + ???? ???? + Δ???? − ???? − ???????? = ????Δ???? = ???? Δ???? Δ???? Δ???? 2 Example: ???? ???? = ???? 2 2 2 2 2 Δ???? (???? + Δ???? ) − ???? ???? + 2????Δ???? + Δ???? ( ) − ???? = = = 2???? + Δ???? Δ???? Δ???? Δ???? If Δ???? is a marginal change, then it is close to zero, and the rate of change of y with respect to x will be approximately 2x. Slopes and Intercepts The rate of change of a function can be interpreted graphically as the slope of the function. The slope of a linear equation is the “rise” over 25 the “run”. In the case of ???? = 2????, a “run” of 1 (x 20 y = 2x increase by 1) leads to a “rise” of 2 (y increases by 15 2). This gives us the slope of 2. 10 Note that the slope of this equation is independent of any additional terms in the 5 equation which do not include x. That is, the 0 slope of ???? = 2???? + 10 is also 2. 0 2 4 6 8 10 120 For a non-linear equation, the rate of change 100 depends of the value of the independent variable, 80 but it will be equal to the slope of the tangent line 60 at that particular point. 40 20 The tangent line is one that touches the same 0 point as our function at the given x-value, and -200 2 4 6 8 10 shares the same rate of change at that point. -40 The vertical intercept of a function is the value of y when x = 0. The horizontal intercept of a function is the value of x when y = 0. Often we will be interested in these intercepts, because they will represent the limits to what is possible in our model. For example, the vertical intercept of the demand curve gives the highest willingness to pay, and marks the maximum value an individual’s consumer surplus could take on. Derivatives The derivative of a function ???? = ????(????) is defined to be: ????????(????) ???? ???? + Δ???? − ????(????) = lim ???????? Δ???? →0 Δ???? Differentiation rules: In the following formulas, u, v, and w are all functions of x, and c and m are constants. ???? (???? = 0 ???????? ???? (???? = 1 ???????? ???? ???? ???? (???? + ???? + ⋯ = (???? + (???? + ⋯ ???????? ???????? ???????? ???? ???? (???????? = ???? (????) ???????? ???????? ???? ???? ???? ???????? (???????? = ???? ???????? ???? + ???? ???????? (????) ???? (???????????? = ???????? ???? ???? + ???????? ???? (???? + ???????? ???? (????) ???????? ???????? ???????? ???????? ???? ???? ???? ???? ???? (???? − ???? (????) ( ) = ???????? 2 ???????? ???????? ???? ???? ???? (????????) = ???????? ????−1 ???????? ???? ???? ????−1 ???? ???????? (???? ) = ???????? ???????? (????) ???? (ln????) =1 ???????? ???? ???? ???? = ???? ln???? ???????? ???? (????????)= ???? ???? ???????? Differentiation Examples 2 3 4 5 ???? ???? = 4 + 2???? − 3???? − 5???? − 8???? + 9???? ???? 2 3 4 = 0 + 2 − 3 2???? − 5 3???? ) − 8 4???? )+ 9(5???? ) ???????? 3 − 2???? ???? ???? = 3 + 2???? (3 + 2???? )???? (3 − 2???? − 3 − 2???? ) ???? (3 + 2???? ) ???? = ???????? ???????? ???????? (3 + 2????)2 (3 + 2???? −2 − (3 − 2????)(2) −12 = (3 + 2????)2 = (3 + 2????)2 ???? ???? = ???? + 3 ???????????? ???? ???? = 2???? + 1 2 2 ???? = ????(???? ???? ) = 2???? + 1 ) + 3 = 4???? + 4???? + 4 ???????? = 8???? + 4 ???????? Chain Rule Given ????(????) and ????(????), for ???? = ????(???? ???? ): ???????? = ????????(???? ???? ) ????????(????) ???????? ????????(????) ???????? So ???? = 2 2???? + 1 )1 ∗ 2 = 4 2???? + 1 = 8???? + 4 ???????? Second Derivatives The second derivative of a function is the derivative of the derivative of that function. » The second derivative measures curvature. A negative second derivative has a slope which is decreasing (concave). A positive second derivative has an increasing slope (convex). ′ ′′ First derivatives are often denoted as ???? ???? while second derivatives are noted???? . ???? Partial Derivatives When there are more than one independent variables, we take the partial derivative with respect to a single independent variable. In other words, you treat the other independent variables as constants. Example: 0.3 0.7 ???? ????1,???? 2)= ???? 1 ????2 0.7 ???? −0.7 0.7 ????2 ????2 0.7 ???????? = 0.3????1 ????2 = 0.3 0.7= 0.3( ???? 1 ????1 1 0.3 0.3 ???? = 0.7????0.3????−0.3= 0.7 ????1 = 0.7( )1 ???????? 2 1 2 ????2.3 ????2 Optimization The maximum of a function is found under the following conditions: ???????? ???? ) = 0 ???????? 2 ???? ???? ???? ) ≤ 0 ???????? 2 The minimum of a function is found under the following conditions: ???????? ???? ) ???????? = 0 ???? ???? ???? ) 2 ≥ 0 ???????? ???????? ????) The condition ???????? = 0 is known as the first-order condition. ???? ???? ????) The condition ???????? 2 ≤≥ 0 is known as the second-order condition. Example: ???? ???? = 10 − ???? )2 ???????? ????) ???????? = 2 10 − ???? −???? = 2???? − 20 First Order Condition 2???? − 20 = 0 → ???? = 10 Second Order Condition ???? ???? ???? ) ????(2???? − 20) = = 2 ≥ 0 → ???????????????????????????? ????????2 ???????? So the function ???? ???? = 10 − ????)2is minimized when x = 10. 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 (???? ) and butter (???? ). 1 2 A consumer’s choice of these goods is their consumption bundle1(???? 2???? ). 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 as1???? and 2 , where 1 is the price o1 ???? (guns) and ????2is the price of2???? (butter). Taking the amount of money a person has to spend, ????, as given, the budget constraint of the consumer is ???? ???? + ???? ???? ≤ ????. ???? ???? ???? ???? Note that ????1 1is the amount of money the consumer spends on guns, and ???? 2 2s 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: ???? : movie theater tickets 1 ????2: all other goods. ???? 1 1 ???? ????2 2???? ????1: price of movie theater tickets ????2: 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: ????1 1+ ???? 2 ???? Properties of the Budget Set The budget line is the set of bundles that cost exactl1 1: ???? 2 2 ???? ???? = ???? We can rearrange the budget line to give us: ???? ???? ???? = − 1???? 2 ???? 2 ???? 2 1 Let’s look at a defined budget line: 2????1+ 5???? 2 100 20 18 16 14 12 10 8 QUANTITY OF X2 6 4 2 0 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 X1 The shaded region is the budget set. The budget line has a slope of −???? /????1. 2 The vertical intercept is ????/???? 2 which is the maximum amount of good 2 that the consumer can afford to buy. The horizontal intercept is ????/???? ,1which 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 2???? + 1???? = 102, the slope of the line is -0.4. This means that in order to get 1 unit of ???? 1 you have to give up 0.4 units of ???? .2 Thus, the slope of the budget line represents the opportunity cost of consuming more ???? in 1 terms of ???? 2 This is the true economic cost of ???? .1 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: ????1: guns ????2: butter ????1: price of guns ????2: price of butter ????: money to spend ????1 1+ ???? 2 2 ???? Initially the budget line will be: 2????1+ 5???? =2100 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: 2???? + 5???? = 120 1 2 A decrease in income would have the opposite effect. Changes in Prices Suppose ???? 1ncreases to 3, so 3???? +15???? = 120. The higher price rotates the budget line around the vertical intercept, and the slope of course changes. Suppose ???? 2rops to 4, so 2???? +14???? = 120. Again the slope changes, but now the horizontal intercept stays the same. Suppose both changes happened at the same some, so 3???? + 4???? = 100. 1 2 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 + ???? 2???? 1 1 + ???? 5???? = 120 or 100 2????1+ 5???? =2 1 + ???? 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-SumTax: 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 (????) on good 1 (1 ). [Notice that I am moving away from the example of guns and butter to the general case] Original constraint:1 1???? + ????2 2≤ ???? New constraint: ???? 1 ???? ???? +1???? ???? ≤2 2 A subsidy? New constraint: ???? 1 ???? ???? +1???? ???? ≤2 2 Ad Valorem Taxes Suppose we put an Ad Valorem tax (????) on good 2. Original constraint1 1 ???? +2 2???? ≤ ???? New constraint: ???? ???? + ???? ???? + ???????? ???? ≤ ???? or ???? ???? + (1 + ????)???? ???? ≤ ???? 1 1 2 2 2 2 1 1 2 2 A subsidy? 1 1 + (1 − ????)???? 2 2 ???? Lump-Sum Tax Suppose we put a Lump-Sum Tax (????) on this consumer. Original constraint1 1 ???? +2 2???? ≤ ???? New constraint: ???? ???? + ???? ???? + ???? ≤ ???? or ???? ???? + ???? ???? ≤ ???? − ???? 1 1 2 2 1 1 2 2 A subsidy? Income Tax Suppose we put an income tax (????) on this consumer. Original constraint1 1 ???? +2 2???? ≤ ???? New constraint: 1 1 + ????2 2≤ 1 − ???? ???? ) 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: ???? ???? + ???? ???? ≤ ???? if ???? ≤ ????̅ 1 1 2 2 1 1 ????1 1+ ???? ????2 2????(???? − ????1 ) ≤1???? if ???? > ????1 1 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 ???? 1 Let’s say it is enough to buy you 5 units of 1 . 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). Intermediate Macroeconomics ECN 312 – Summer 2015 Lecture 4: Utility and Preferences Trolley Problem Introduction Economics is born out of political and moral philosophy. Specifically, it is built on the philosophy of utilitarianism, which simply put believes “It is the greatest happiness of the greatest number that is the measure of right and wrong.” Its early proponents like Jeremy Bentham and his student John Stuart Mill proposed what they called hedonic calculus to count up units of pleasure and pain. » Benthamite Francis Edgeworth (who came up with a box) imagined that this hedonic calculus could one day be done by a “hedonometer”, which he thought might be something like a wristwatch. From them we get the economic concept of Utility: the total satisfaction a person receives from a given level of consumption or from an activity. Economists use the tool of utility to describe consumer preferences. » Preferences are a ranked list of bundles from the most favored to the least favored. Example: Dr. Goegan’s Movie Preferences 1. Jurassic Park 2. The Empire Strikes Back 3. Amadeus 4. Schindler’s List 5. Forrest Gump 6. The Dark Knight 7. Apollo 13 8. Seven 9. The Sixth Sense 10. Lawrence of Arabia Importantly: Consumer preferences are assumed to be ordinal and not cardinal. That is, only the ranking matters, and it is not relevant (or even perhaps possible) to determine whether or not one bundle makes us two times, or three times as well off as another. Utility People will choose the best option available to them. In this framework, that means people will choose the option which offers the greatest utility. Utility is represented as a function of the choice variables, usually something like “good 1” and good 2”. Examples of Utility Functions ???? ???? ,???? )= ???? ???? 1 2 1 2 ???? ???? ,???? ) = min{???????? ,???????? } 1 2 1 2 ???? ???? ,???? )= ???????? + ???????? 1 2 1 2 For each of these utility functions, rank the following bundles of goods from most preferred to least preferred assuming both “a” and “b” are equal to 1: (4,6); (10,3); (0,15). Indifference Curves An indifference curve visualizes utility at a fixed level, allowing the inputs to vary, but constraining them to a particular result. Example: Graph ???? ???? ,1 2) = ????1 2when ???? = 10 10 10 = ???? 1 2 ???? = 2 ???? 1 As utility increases, from 10 to 20 to 30, the indifference curve shifts away from the origin. Things to note about indifference curves: » They never cross. This is due to the transitive property of preferences, which states that if A is preferred to B, and B is preferred to C, then A is preferred to C. » Everything strictly above the indifference curve is strictly preferred to everything on the indifference curve. » Everything on the indifference curve is strictly preferred to everything strictly below the indifference curve. » A consumer whose preferences are described by this utility would be equally as well off if you gave them any combination of ???? a1d ???? th2t falls on the indifference curve. Now let’s graph the other two utility functions we saw: ???? ???? 1???? 2 )= min{ ????????1,???????? 2 ???? ???? 1???? 2 )= ???????? 1 ???????? 2 What kinds of goods do these indifference curves represent? Specifically, which one shows perfect substitutes and which one shows perfect compliments? Let’s also try: A Neutral Good (The Lost World: Jurassic Park): ???? ????1,????2) = ???????? 1 0???? 2 » What does increasing utility look like here? A Bad (Jurassic Park 3): ???? 1 ,2) = ???????? 1 ???????? 2 Marginal Utility Utility offers a measure of the benefit of consumption, and we will want to know what the benefit of a little bit more consumption will be. The marginal utility is the rate of change in utility with respect to a particular good. Example: What is the marginal utility o1 ???? if 1 ????2,????= ???????? 1 ???????? ?2 » ???????? = ???? 1 » One more unit of ????1will increase utility by a. Example: What is the marginal utility o1 ???? if 1 ????2,????= ???? 1 2 » ???????? =1???? 2 » One more unit of ????1will increase utility b2 ???? . Example: What is the marginal utility of ???? if ???? ???? ,????= ???? ???? ? 2 1 2 1 2 » ???????? =2???? 1 » One more unit of ???? will increase utility by ???? . 2 1 The Marginal Rate of Substitution From an economic perspective, it is helpful to this about the rate of change along an indifference curve. Like the budget constraint, it has a nice economic interpretation. Δ????2 is the rate at which the consumer is willing the exchange ???? for ???? . Δ????1 1 2 Another interpretation of the MRS is that it gives the value of additional uni1s of ???? in terms of ????2, so that we could say “a little bit more1of ???? would be like having MRS more 2f ???? ." Calculating MRS To calculate MRS, let’s start by noting the following: Δ???? = ???????? Δ???? + ???????? Δ???? 1 1 2 2 Along the indifference curve, Δ???? = 0. Therefore: Δ???? 2 = − ???????? 1 Δ???? ???????? 1 2 A quick and important thing to note is the negative sign. Generally speaking, this will be ignored for reasons which become clearer as we go. But it makes sense, if you get mor1 ???? then you have to give up some ???? to keep the same utility. 2 From Start to Finish Suppose a consumers preferences are described by the following equation: ???? ???? 1???? 2 )= ???? 1 3????223 Rank the following bundles from most preferred to least preferred: (0,18 ; 2,9 ; 9,2 ; 18,0 ) Graph the indifference curve for ???? = 16. Use {1,4,16,25,64,100} as values for ????1to help. Calculate the Marginal Rate of Substitution. Evaluate the MRS at 5,10 .) Interpret the MRS at this point. Intermediate Macroeconomics ECN 312 – Summer 2015 Lecture 5: Consumer Choice So far we have described two aspects of consumer life. First we have what people want: ???? ???? 1???? 2) Second we have what people can afford: ????1 1+ ???? 2 2 ???? With preferences that assume more is always better, we can safely presume that we will end up somewhere on the budget line, spending all of the money we have available to spend. » Obviously there are considerations that can be made, but it is best to start with a simple but “good enough” model and make it more complex and useful from there. So we will take the budget constraint as: ???? ???? + ???? ???? = ???? 1 1 2 2 The goal of the consumer is to maximize their own well-being given their budgetary constraint: max????(???? ,???? ) ????.????. ???? ???? + ???? ???? = ???? ????1,2 1 2 1 1 2 2 To solve this problem, we are going to need to call in a LagrangianMultiplier. This method sets up the Lagrangian: ???? = ???? ???? ,???? )− ???? ???? ???? + ???? ???? − ???? ) 1 2 1 1 2 2 Lagrange’s Theorem says that the optimal choice (???? ,???? ) must satisfy the three first-order 1 2 conditions: ???????? ???????? = − ????????1= 0 ????????1 ????????1 ???????? = ???????? − ???????? = 0 ????????2 ????????2 2 ???????? = ????1 1+ ???? 2 2 ???? = 0 ???????? It is important to see this method, because it will allow you to expand beyond a two good scenario, where each additional good adds one more first-order condition. But for now we will proceed with just two. ???????? ???????? Note that:???????? = ???????? a1d ???????? = ???????? 2 so: 1 2 ???????? − ???????? = 0 1 1 ???????? −2???????? = 2 Which becomes ???????? =1???????? 1 ???????? = ???????? 2 2 Since both of these must be true at the optimal choice, then it follows that this is also true at the optimal choice: ???????? 1 ????????1 = ???????? 2 ????????2 Thus ???????? 1 = ????1 ???????? 2 ????2 This equation can be used along with the third first-order condition (which is just the budget ∗ ∗ constraint) to solve fo1 ????2,???? . But this result is economically very beautiful. On the left-hand side you have the marginal rate of substitution, which gives the rate at which a consumer is willing to exchange one good for another. On the right-hand side you have the price ratio, which gives the rate at which the market is willing to exchange one good for another. The MRS gives us the benefit of consuming one more unit of good 1 in terms of good 2. It is the marginal benefit of additional consumption. The price ratio gives us the cost of consuming one more unit of good 1 in terms of how much of good 2 we have to give up. It is the marginal cost of additional consumption. An Example with Numbers Suppose a consumer’s preferences are described by the following utility function: ???? ???? 1???? 2)= ???? 1 2 The price of1???? is $2 and the price 2f ???? is $5. This consumer has an income of $100. Step 1: Set up the Lagrangian ???? = ????1 2− ???? 2???? +15???? − 200 ) Step 2: Get the First Order Conditions ???????? = ???? 2 ????2 = 0 ????????1 ???????? = ???? − ????5 = 0 ???????? 2 1 ???????? = 2???? 1 5???? −2100 = 0 ???????? Step 3: Find the Optimization Condition ????2 ????2 = → 2???? 1 5???? 2 ????1 ????5 Step 4: Plug it into the Budget Constraint and Solve 2????1+ 5???? 2 100 2????1+ 2???? 1 100 4????1= 100 ????????= ???????? ????????= ???????? Plug these back into utility and double check with nearby possibilities within our budget like (24, 10.4) and (26, 9.6). Graphically Let’s think about consumer choice graphically and see what we come up with. Here we have 3 indifference curves at utilities of 100, 250, and 300. Crossing through them is the budget constraint. The area under and including the budget constraint is what the consumer can afford. As indifference curves are shifted away from the origin, a higher utility is represented. That is, of the indifference curves, the rightmost curve is preferred to the leftmost curve. The goal of the consumer then is to pick a bundle that they can afford, and which lies on the highest indifference curve possible. That point is described by the middle indifference curve and the budget constraint. They share a single common point, and there is no other point which is both affordable, and lies on a higher indifference curve. Mathematically, the budget constraint is the tangent line for the indifference curve at the optimal bundle. So it could be found where MRS equal the price ratio. Let’s try graphs for perfect compliments (L-shaped) and perfect substitutes(\ shaped). Application: Which Tax is Preferred? Let’s look at our previous example again: ???? ???? 1???? 2)= ???? 1 2 The price of1???? is $2 and the price 2f ???? is $5. This consumer has an income of $100. But now let’s add a quantity tax on good 1 of $2. What happens to this consumer’s choice? The budget constraint is now: 1???? + 52 = 100 ????2 ???????????? = ????1 ???????? = 4 5 5???? 2 4???? 1 ????1= 12.5 ????2= 10 ???? = 125 What is Tax Revenue? 2 ∗ 12.5 = 25 Suppose instead we placed a $25 lump-sum tax on this person? The budget constraint is now: 1???? + 52 = 75 ????2 2 ???????????? = ; ???????? = ; 2????1= 5???? 2 ????1 5 ???? = 18.75;???? = 7.5 1 2 ???? = 140.625 The consumer prefers the lump-sum tax to the quantity tax. Why might it not be feasible to apply the lump sum tax to this consumer? Graphically Intermediate Macroeconomics ECN 312 – Summer 2015 Lecture 6: Demand Suppose our preferences are described by the following utility function: ???? ???? 1???? 2)= ???? 1 ???? +2???? ???? 1 2 ∗ ∗ What is this consumer’s optimal choice fo1 (2 ,???? )? Step 1: Write down the budget constraint. ????1 1+ ???? 2 2 ???? We don’t have any more information than that. But that is ok. We can proceed using the variables. Step 2: Set up the Lagrangian ???? = ????1+ ???? 2 ???? ????1 2???? ???? ???? 1 1 ???? −2 2 ] Step 3: Get the First-Order Conditions ???????? = 1 + ????2− ???????? 1 0 ????????1 ???????? = 1 + ???? − ???????? = 0 ????????2 1 2 Step 4: Find the optimality condition 1 + ????2 ???? 1 = 1 + ????1 ???? 2 Step 5: Substitute into the budget constraint. 1 + ????2 ????1 ????1− ???? 2 ???? ????1 1 = → ???? 2 ???? 2 2 ???? + 1 ???? →1 1= 2 1 + ????1 ????2 ????2 ???? − ???? + ???? ???? ????1 1+ ???? 2 2 ???? → ???? ???? +1 1[ 2 1 2 1 1] = ???? ????2 ????1 1+ ???? 1 ???? +2???? ???? 1 1 → 2???? ???? = 1 1 ???? + ???? 1 2 ???? − ???? + ???? ????1= 1 2 2???? 1 By symmetry and by solution, we will also find that: ???? − ???? 2 ???? 1 ????2= 2???? 1 What are these equations? Demand The consumer’s demandfunctions give the optimal amounts of each good as a function of the prices and income faces by the consumer. ????1= ???? 1 ,1 ,2 ) ????2= ???? 2???? 1???? 2????) Let’s take stock of this for a moment. Things to notice are that our set up claims to describe the consumer’s decision as a function of only prices and income. The nature of this function is defined by their preferences, which may capture more than just money, but those things are assumed to be static, and so their consequence is of little relevance to us. This is part of economic thinking. We want to reduce the world into a simpler model to help us understand what is happening in people’s heads. And certainly things like prices and income are major components of the consumer decision. But other things matter too, and we should keep that in mind whenever we proceed with this type of analysis. Normal and Inferior Goods When income increases, demand for a good usually increases. I like was Varian says about this: “Economists, which a singular lack of imagination, call such goods normal goods.” ???????? 1 > 0 ???????? ????−???? +???? Example: ????1= 1 2 2????1 ????????1 1 ???????? = 2???? > 0 ???????? ????1> 0 1 When income increases, but the quantity of a good demanded decreases, it is called an inferior good. ????????1 < 0 ???????? Example: Inferior goods are rather hard to generate with utility functions, but I believe that 1 2 ???? ????1,????2) = ln????1+ ????2g2ves ???? a1 an inferior good. Income Lanes How a person responds to changes in income can be charted graphically. The dotted line is the income lane (aka “income offer curve”, “income expansion path”, or “income-consumption curve”). An upward sloping line is a normal good. A downward sloping line is an inferior good [draw example]. Income Elasticity of Demand Elasticity is a helpful measure of the responsiveness of demand. Now that we have advanced math at our side, we can define it precisely: Δ????1/????1 ???? ???????? 1 ???????? = = Δ????/???? ????1???????? Example ???????? 1 1 ???? 1 ???????? = 2???? → ???? ???? ???? 2???? 1 1 1 Note that we can put in our demand f1r ???? : ???? 1 ???? ????????= ???? − ???? + ???? = 1 2 2???? 1 ???? − ???? 1 ???? 2 2????1 Try ???? ,???? ) = 1,1 ; 2,1 ; 1,2) 1 2 Luxury Goods are those defined as having an income elasticity of demand greater than 1. When the income elasticity is below 1, we call this a necessary good. Price Lanes Generally speaking, when the price increases, the quantity demanded goes down. ???????? 1 < 0 ???????? 1 ( ) ????????1 = − ???? + ????2 < 0 ????????1 2???? 1 When it doesn’t, this is called a Giffen good. There are not many examples of this if any (wine is the common one). The logic is that the price signals something, or that a lower price frees up incomes such that you want to buy more of something else, and less of the lower price good. We expect this line to be positively sloped when the price of1???? is changing. A positive slope means we are shifting consumption away from ???? and 1 towards 2 when the price of ???? goes 1 up. Compliments and Substitutes Getting an equation for demand also gives us an easy way to check for compliments and substitutes: Compliments Substitutes ???????? 1 ????????1 < 0 > 0 ???????? 2 ????????2 Example ???????? 1 1 = > 0 → ???????????????????????????????????????????? ????????2 2????1 Graphing Demand Importantly, when you graph demand, you are graphing the inverse demand curve. Example ???? − ????1+ ????2 ????1= 2???? 1 ???? 1 ????2 ????1= − + 2????1 2 2????1 ???? + ???? 1 ???? = 2 − 1 2????1 2 1 ???? + ????2 ????1+ 2 = 2???? 1 ???? + ???? 2 2????1+ 1 = ????1 ???? + ???? ????1= 2 2????1+ 1 For ???? = 20 and ???? = 1 2 Market Demand We’ve got a way of finding individual demand, but what about market demand? The market demand for a good is the aggregate demand for all consumers: ???? 1 1 ???? ???? 1???? 2???? ,1,???? ????) = ∑???? (???? ,1 ,2 ) ???? ????=1 For example, imagine the market consists of three consumers, whose demands take the form: Consumer 1 Consumer 2 Consumer 3 ???? ???? − ????2 ???? ????1= ????1= ????1= 2????1 3???? 1 ???? 1 ???? 2 First thing to note is that in each of these, ???? is specific to the consumer. But for ease, let’s say it is $100 for each of them, and t2at ???? is 1. 100 99 100 ???? = + + 2????1 3????1 ????1+ 1 Consumer 1 Consumer 2 Consumer 3 1 1 ???? = 150 − 2???? ????1= 200 − ???? 1 ????1= 100 − ???? 1 1 1 2 3 The trouble here is that these consumer’s don’t demand negative amounts of the product. So the market demand curve is not smooth. You would need a piecewise function to describe this demand. This is the caution that must be applied with “linear” demand functions. Intermediate Macroeconomics ECN 312 – Summer 2015 Lecture 7: Income and Substitution Effects Let’s start with our favorite utility function: ???? ???? 1???? 2)= ???? 1 ???? +2???? ???? 1 2 This yields the following demand functions: ???? + ???? − ???? ???? 1 ,1 ,2 = ) 2 1 2???? 1 ( ) ???? + ???? 1 ???? 2 ???? 2 ,1 ,2 = 2???? 2 Let’s evaluate this consumer’s choice at the following sets: (????1,????2,???? = 5,5,100 ;(3,5,100) Why are we consuming more of x 1 Substitution Effect: Our quantity demanded has changed because the rate of exchange between the two goods has changed. » Good 1 is relatively cheaper than good 2, meaning we will substitute towards good 1. Income Effect: Our quantity demanded has changed because our purchasing power has changed. » Bundles which include good 1 are now cheaper to purchase. The Slutsky Method In the graph above, we moved from the set (5, 5, 100) to the set (3, 5, 100), which moved us from a choice for (1 2x ) of (10, 10) to (17, 9.8). The dotted lines represent a hypothesized change, where we keep the new set of prices, but adjust our income so that the old bundle (10, 10) is still affordable. So m = 3*10 + 5*10 = 80, meaning we find the optimal choice for the set (3, 5, 80). The substitution effect is the change 1n x seen from moving from (5, 5, 100) to (3, 5, 80). The income effect is the change in 1 seen from moving from (3, 5, 80) to (3, 5, 100). The Substitution Effect First let’s define a few things: ????1= new price for good 1. ′ ′ ∗ ∗ ∗ ∗ ???? = ???? ????1 1???? ???? w2e2e ???? and ????1are the2original choices at the original prices. The substitutioneffect is defined as: Δ????1= ???? ????1,????1,???? 2 ???? (???? ,???? 1????)1 2 So in our example: ???? = 3 1′ ???? = 3 ∗ 10 + 5 ∗ 10 = 80 ???? Δ????1= ???? 315,80 − ???? 5,5,100 ) 80 + 5 − 3 100 + 5 − 5 = − = 13.67 − 10 = 3.67 2 ∗ 3 2 ∗ 5 This consumer buys 3.67 more units of x1 because it is now relatively cheaper than x2. The substitution effect is always positive for a price decrease. This follows from the law of demand (or rather it leads to it). The Income Effect The income effect is defined as: Δ???? 1 = ????1???? ,1 ,2 − ???? (???? 1???? ,1 )2 ′ So in our example: Δ???? ???? = ???? 3,5,100 − ???? 3,5,80 ) 1 1 1 100 + 5 − 3 80 + 5 − 3 =

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