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Economics 352: Intermediate Microeconomics EC 352: Intermediate Microeconomics, Lecture 6 Economics 352: Intermediate Microeconomics Notes and Sample Questions Chapter 6: Demand Relationships Among Goods This chapter will use the consumer theory we have established to study demand relationships across goods. Substitutes and Complements We will start with the concepts of pairs of goods being substitutes or complements. To review, substitutes are two goods that can replace one another, such as butter and margarine. To be more precise, when the price of one rises, demand for the other increases. Examples include: butter and margarine Budweiser beer and Miller beer beef and chicken caffeine and sleep electric light and candle light old cars and new cars bicycles and cars Complements are goods that are typically used together. To be more precise, when the price of one falls, demand for the other rises. Examples include: electricity and computers computer hardware and computer software roads and automobiles left shoes and right shoes beer and salty snacks It is worth noting that when the price of salty snacks falls, demand for beer rises. This is why taverns will very often give patrons salty snacks for free. Substitutes and complements can be illustrated in the standard consumer diagram. Consider the change in the quantity of x demanded when the price of y falls. If the quantity of x demanded rises, the two goods are complements, of the quantity of x demanded falls, the two goods are substitutes: EC 352: Intermediate Microeconomics, Lecture 6 Two graphs showing the effect of a decrease in the price of good y on the quantity of good x that a consumer demands under the assumption that the two goods are complements and under the assumption that they are substitutes. Mathematically, the effect of a change in the price of y on the quantity demanded of x can be shown as: ∂x(px ,p y ,I) ∂x ∂x = − y ⋅ ∂p ∂p ∂I y y U=U This comes from the Slutsky stuff from the never-ending Chapter 5 as follows: c x ()px ,py , U = x[]px ,p y ,E()px ,py , U ∂xc ∂x ∂x ∂E = + ⋅ ∂py ∂py ∂E ∂py ∂E ∂x ∂x But = y and = so we have: ∂py ∂E ∂I ∂xc ∂x ∂x = + y ⋅ ∂py ∂py ∂I or EC 352: Intermediate Microeconomics, Lecture 6 ∂x ∂xc ∂x = − y ⋅ ∂py ∂py ∂I or the total effect of a change in the price of y on the quantity demanded of x is equal to a ∂xc ∂x substitution effect and an income effect − y ⋅ . ∂py ∂I As the book points out, the term y represents how much of good y this person consumes. As consumption of y increases, the income effect is likely to become much larger. If x is a normal good and y is large, then the income effect will be a large negative number and these two goods will likely be substitutes regardless of the substitution effect. For example, if a person spent a large portion of their income on food and the price of food rises, they will likely reduce their consumption of all other normal goods to avoid starving to death (a huge income effect) even if some other goods would otherwise be complements for food. Try Example 6.2 You should work through Example 6.2 in the textbook. It would help to know that ln x is d ln x 1 the natural log of x and = . dx x Net versus Gross – Some Terminology The book distinguishes between net substitutes or complements and gross substitutes or complements. ∂x The basic idea is that net refers only to the substitution effect while gross ∂p y U=U refers to the sum of substitution and income effects. The important difference is that the ∂x net or substitution effect is symmetric. That is, it is the same for as it is for ∂p y U=U ∂y . Put another way: ∂px U=U EC 352: Intermediate Microeconomics, Lecture 6 ∂x ∂y = ∂p ∂p y U=U x U=U With the income effect included, however, we are looking at whether things are gross substitutes or gross complements, and if one of the goods makes up a larger share of consumption, then there will be a larger income impact from a change in the price of that good and the symmetry that existed in the net (substitution effect only) relationship won’t exist in the gross (substitution plus income effect) relationship. The book presents this analysis for the case of many (more than two) goods by calling the goods involved xi and xj. I’ve called them x and y here for the sake of familiarity, but this analysis of net and gross effects is only technically correct when there are more than two goods. Composite Commodities The analysis gets tough when the number of goods increases. Economists typically avoid this difficulty by looking at one good specifically and lumping all other goods together into a composite commodity, which it may help to call “all other stuff.” For example, if you wanted to analyze demand for gasoline, you might use the graph: A graph showing a budget line and an indifference curve in the case where the two goods are gasoline and a composite good named “All other Stuff”. where the price of all other stuff is given by some sort of price index, and is usually set to paos=1, just to make the math easier. Income and the price of gasoline are adjusted so that this price of all other stuff (AOS) makes since. Also, since paos=1, the quantity of AOS EC 352: Intermediate Microeconomics, Lecture 6 consumed is also equal to expenditures on AOS, so it is sometimes said that expenditures on AOS rather than the quantity of AOS (which isn’t well defined) is the other good. Try Example 6.3 Example 6.3 has two things going for it. First, it uses a CES rather than a Cobb-Douglas utility function, so the math is a bit different. You should try to work through from the utility function to the Marshallian demand functions, just to be sure you can do it. Second, and more importantly, this example shows that if you have a model with three goods (x, y and z) and you analyze the result of a change in income on x, you will get results identical to what you get from a model with only two goods (x and h, where h is a composite of y and z). That is, making a composite good out of y and z didn’t change the results for x. This lends credibility to the idea of modeling consumer behavior using composite goods to simplify the task. Practice Problems Illustrate the following in consumer diagrams. 1. Increase in the price of x, x and y are substitutes. 2. Increase in the price of x, x and y are complements. 3. Decrease in the price of x, x and y are substitutes. 4. Decrease in the price of x, x and y are complements. 5. Increase in the price of y, x and y are substitutes. 6. Increase in the price of y, x and y are complements. 7. Decrease in the price of y, x and y are substitutes. 8. Decrease in the price of y, x and y are complements. .
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