Problem Set 4 Solutions - Economics 3010 Fall 2010...

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Unformatted text preview: Economics 3010 Fall 2010 Professor Daniel Benjamin Cornell University Problem Set 4 Solutions 1. (Slutsky equation: The law of demand) Let’s return to Steven Landburg’s claim (1993, pp.6‐7) from the last problem set about the effect of a better birth control technique on the number of unwanted pregnancies: Will the invention of a better birth control technique reduce the number of unwanted pregnancies? Not necessarily – the invention reduces the ‘price’ of sexual intercourse (unwanted pregnancies being a component of that price) and thereby induces people to engage in more of it. The percentage of sexual encounters that lead to pregnancy goes down, the number of sexual encounters goes up, and the number of unwanted pregnancies can go either down or up. Illustrate in a budget constraint / indifference curve diagram an individual’s tradeoff between number of sexual encounters (measured in units of quality, where higher risk of unwanted pregnancy represents a lower quality encounter) and all other uses of time. Show what happens to the budget constraint when the price of sexual encounters falls. What will happen to the individual’s consumption of sexual encounters if the number of sexual encounters is a normal good? Can we make a confident prediction if it is an inferior good? Which assumption seems more reasonable to you? What data could you look at to test whether the number of sexual encounters is normal or inferior (for a typical individual)? Figure A illustrates how the budget line shifts out when the price of sexual encounters falls: since we take “sexual encounters” to be on the x‐axis, the budget line (BL0) moves to the right (BL1). When the price of “sexual encounters” falls, an individual’s consumption of “sexual encounters” definitely increases if sex is a normal good. But we can’t make a confident prediction if “sexual encounters” is an inferior good. The logic is laid out in the next paragraphs and figures. The price decrease of “sexual encounters” can be broken down into two effects: an income effect and a substitution effect. The substitution effect tells us what happens to consumption of “sexual encounters” when the price changes, adjusting income so that the original bundle (the dark green point in Figures B/C below) is affordable. In Figures B/C, the original bundle is the dark green point, the original budget line is the black line, and the budget line obtained after the price fall and adjustment of income is the green line. Note that the substitution effect always has the opposite sign of the price change. So in this example, since price falls, the substitution effect says that consumption of “sexual encounters” goes up. The income effect can be deduced by shifting out the green line to the budget line we get if we don’t adjust income after the price change (the red line in Figures B/C). Unlike the substitution effect, the sign of the income effect depends on whether the good is normal or inferior. If “sexual encounters” is a normal good, then the sign of the income effect is the same as the sign corresponding to the income change needed to go from the green line to the red line. If “sexual encounters” is an inferior good, then the sign of the income effect is the opposite of the sign of this income change. Thus, for people for whom sex is a normal good, both the income and substitution effects are positive when price falls, which means the drop in price will result in an unambiguous increase in sexual encounters (see Figure B). But for people for whom sex as an inferior good, the total change in the consumption of “sexual encounters” is ambiguous, since the substitution effect and income effect have opposite signs. The substitution effect leads to higher consumption, while the income effect leads to less. Figure C graphs the case when the net change is an increase in the consumption of “sexual encounters”. Try drawing a diagram for the other case, when the income effect is so large that its magnitude is larger than that of the substitution effect, so that the net change following a fall in price is a decrease in consumption. Figure B Result when sex is a normal good Other uses of time IC2 IC1 IC0 SE Sexual intercourse IE Figure C Result when sex is an inferior good Other uses of time IC2 IC1 SE IC0 IE Sexual intercourse Note: We can also read out the effect of a price change on consumption of “sexual encounters” by using the Slutsky equation, x,p = sx,p – B.S. x,m: - sx,p < 0 holds whether “sexual encounters” is an inferior good or a normal good. - x,m > 0 if “sexual encounters” is a normal good. Using the Slutksy equation, we then see that x,p < 0 (a decrease in price will lead to an increase in consumption – this is the law of demand). - x,m < 0 if “sexual encounters” is an inferior good. Using the Slutksy equation, we then see that the sign of x,p is ambiguous. Assuming income is “total time available”, the assumption that sex is a normal good seems more reasonable: if people had more time, they would probably have more sex. To test whether sex is a normal or an inferior good, we can look at survey data to see whether individuals with fewer time commitments (i.e. with more flexibility for how they can spend their time) have more sex. For example, we could look at individuals after their senior year of college, comparing their behavior in the month before they start a full‐time job and the month after they’ve started working full‐time. Using the same approach, we could look at retirement‐age individuals, comparing their behavior in the year before they retire and the year after they retire. Side note for people interested in econometrics / empirical work: Testing hypotheses in the way outlined in the paragraph above is called implementing a “regression discontinuity design”. Note that for this “identification strategy” to make sense, people on both sides of the “discontinuity” (getting a full time job and retiring) need to be more or less identical (e.g. in terms of demographics). Since we don’t expect 22 year olds to be all that different from 23 year olds, this approach to testing is probably valid in the context of the problem. 2. (Buying and Selling: An advantage of owning your house) (a) First we’ll consider the case of someone who rents her home. Write down her budget constraint for housing services versus all other goods. (This is the standard budget constraint you are used to.) Show in a diagram what happens when the price of housing services goes up. Use the principle of revealed preference to argue that if the price of housing services goes up (i.e., her rent goes up), then she is worse off, assuming that she was consuming a positive amount of housing services before the price change. Recall: The principle of revealed preferences says that people are choosing the best bundle they can afford. And so to infer people’s preferences, we just need to observe their choices. Formally, if (x1, x2) is chosen when prices are (p1, p2) and p1x1 + p2x2 p1y1 + p2y2 for some bundle (y1, y2), then (x1, x2) (y1, y2), since people choose the best bundle they can afford (See Chapter 7.2). x2 x1: housing services and x2: All other goods On the diagram on the left, the original budget line is B1, and the budget line after the increase in the price of housing services is B2. Suppose that her choice before the price increase was (x1*, x2*). Using the principle of revealed preferences, we can say that (x1*, x2*) is (x1*, x2*) preferred to all other affordable bundles in (I1) Budget Set 1. But as rent goes up, her budget (I2) line shifts inward to (B2). So the bundles that are (B2) (B1) affordable at the new price were all already * * x1 x1 ' x1 affordable at the old price (i.e. Budget Set 2 is strictly contained in Budget Set 1). We can conclude that (x1*, x2*) is preferred to all affordable bundles in Budget Set 2. (b) Now suppose that she owns her home instead of renting it, and her endowment is her currently optimal mix of housing services and all other goods. Write down her budget constraint, and illustrate it in a diagram. Draw an indifference curve that is tangent to her budget constraint at her endowment. x2 Her endowment (ω1, ω2) is her currently optimal mix of housing services and all other goods. The budget constraint is p1ω1 + p2ω2 p1x1 + p2x2 (We can think of ω1 as the amount of square footage of housing.) 2 (B1) 1 x1 (c) In the diagram, illustrate how the budget line rotates when the price of housing services goes up. How will she change her consumption bundle? Is she better off or worse off? Now illustrate how the budget line rotates when the price of housing services goes down. How will she change her consumption bundle? Is she better off or worse off? As the housing price changes, the budget line rotates around the endowment point, since the consumer “owns” the house and all other goods. The reason is that the consumer can still afford her endowment (she owns it!) regardless of the change in price. Regardless of the change in price of housing services, we can see from the two diagrams below that she becomes better off! (see part (d) to understand the intuition behind this result). When the price goes up, she will consume less housing services. When it goes down, she will consume more housing services. House price goes up House price goes down x2 x2 (I2) (I1) (I1) 2 2 (B2) 1 (B1) (B1) x1 1 (I2) (B2) x1 (d) How is it possible that the consumer is better off, regardless of whether the price of housing services goes up or down?! As argued above, when the consumer owns an endowment of housing services and other goods, her new budget constraint (regardless of the direction of the price change) will include the original bundle. But this change also allows her to reallocate her budget between housing and consumption (so she now effectively has more choice). Mathematically, while consuming her endowment was the optimal course of action at the original prices, i.e. MRS(endowment) = ‐p1_original/p2_original, this equality will not hold at the new prices. This is why she chooses to consume a bundle other than her endowment. Optional Problem. (Consumer’s surplus: The case of Cobb‐Douglas utility) [ You can find solutions to some parts of this question in the appendix to ch. 14, and other parts closely follow an example in ch. 14.8. Feel free to use what you learn from those parts of the textbook in writing up your answer, but try the problem first with the textbook closed, and make sure you write up solutions in your own words. ] (a) Consider a household’s choice between charitable giving (Good 1) and all other goods (Good 2). Suppose we observe that expenditure on charitable giving across households is a constant fraction α of household income and does not depend on the prices of other goods. Hence a household’s demand function is x1*(p1, p2, m) = αm/p1, where m is the household’s income, and p1 is the “price” of charitable giving (the number of dollars you have to contribute to the charity per dollar that gets transferred to poor people). Suppose the price p1 falls from p to q<p (perhaps because the charity becomes more efficient at converting contributions to charitable transfers or because the government increases the tax deduction on charitable giving). Explain why the increase in consumer’s surplus is ∫t[p,q] (am/ t) dt. Illustrate on a demand curve. Calculate the increase in consumer’s surplus in terms of p, q, a, and m. Note that given the definition of consumer’s surplus, at price p CS is the area bounded by the demand curve, the y‐axis and the horizontal line price = p. At price q, CS is then the area bounded by the demand curve, the y‐axis and the horizontal line price = q. Subtracting the first area from the second area gives us the yellow area in the figure above. This area can be represented as an integral, where the integrand is the demand function x(t) = am/t. △CS = ∫t[p,q] (am/ t) dt = am ln (p) ‐ am ln (q) = am ln (p/q) (b) Consumer’s surplus has the advantage that it can be calculated directly from the observed demand function, but it is only an approximate measure of welfare (except in the case of quasi‐linear utility, when it is an exact measure). However, you showed in the last problem set that Cobb‐Douglas preferences generate this demand function. So in this case, we can infer from the demand function that the underlying utility function is Cobb‐Douglas. We can use that fact to calculate exact measures of the change in welfare: equivalent variation and compensating variation. Assume that m = $100,000, a = 0.02, p = $1.50, and q = $1.30. What is the change in consumer’s surplus, equivalent variation, and compensating variation? Show that the change in consumer’s surplus is in between the equivalent variation and the compensating variation. From part (a), we know that the change in consumer’s surplus is given by am ln (p/q). Plugging in the numbers given in the problem, we get △CS = (0.02)*(100,000)*ln(1.5/1.3) = 286.202 To find the compensating variation (CV), we set the original utility (am/p)^a)*((1‐a)m/p2)^(1‐a)) equal to the “CV” utility (i.e. utility obtained when prices are (q, p2) and wealth is m +CV, (a(m+CV)/q)^a)*((1‐ a)(m+CV)/p2)^(1‐a))) and solve for CV: ((am/p)^a)*((1‐a)m/p2)^(1‐a) = (a(m+CV)/q)^a)*((1‐a)(m+CV)/p2)^(1‐a) (am)*((1/p)^a)*(1/p2)^(1‐a) = a(m+CV)*((1/q)^a)*(1/p2)^(1‐a) (am)/(p^a) = a(m+CV)/(q^a) m(q/p)^a = m + CV CV = m((q/p)^a) ‐ m So CV = 100000((1.3/1.5)^0.02) – 100000 = ‐285.7925 Note: If q is lower than p, then CV<0. This makes sense since when price falls, people move to a higher utility level and so if we want to bring them back to their old utility level, we need to take money away from them rather than compensate them. To find the equivalent variation (EV), we take the utility level obtained at the new prices (q, p2) with income m, ((am/q)^a)*((1‐a)m/p2)^(1‐a), set it equal to the “EV” utility evaluated at the old prices (p, p2) and wealth m‐EV, ((a(m‐EV)/p)^a)*((1‐a)(m‐EV)/p2)^(1‐a)), and solve for EV: ((am/q)^a)*((1‐a)m/p2)^(1‐a) = ((a(m‐EV)/p)^a)*((1‐a)(m‐EV)/p2)^(1‐a) am*((1/q)^a)*((1/p2)^(1‐a)) = a(m‐EV)*((1/p)^a)*((1/p2)^*(1‐a)) m(p/q)^a = m – EV EV = m‐m(p/q)^a So EV =100000 ‐100000((1.5/1.3)^0.02) =‐ 286.611 Note: If q is lower than p, then EV<0. This makes sense since to achieve the same utility level at the new (lower) price and the old (higher) price, the consumer needs to have more money at the new (lower) price. We conclude from our calculations that the change in consumer’s surplus is in between the equivalent variation and the compensating variation. In fact, |CV|< |CS| < |EV| (285.79< 286.202< 286.611). ...
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