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**Unformatted text preview: **Economics 3010
Fall 2009 Professor Daniel Benjamin
Cornell University Problem Set 1 Solutions 1. (Opportunity cost: Common mistakes) (a) Ezra and Cornell are both students at the same upstate New York Ivy League university. Ezra wins a ticket from a radio station to see a jazz band perform at an outdoor concert. Cornell has paid $18 for a ticket to the same concert. On the evening of the concert, there is a tremendous thunderstorm. Ezra decides that because he won’t enjoy the concert while sitting in the rain, he won’t go. Cornell, however, reasons that he doesn’t want the $18 he’s already paid to go to waste, so he’ll go to the concert, even though he would not go if his ticket had been free. Cornell has committed what economists call the “sunk cost fallacy” because he is allowing his behavior to be influenced by costs he incurred in the past (costs which are already “sunk”). (i) Using the concept of opportunity cost, explain why Cornell’s cost‐benefit analysis is mistaken. Opportunity cost is what you give up when you make a choice. When Cornell is deciding whether or not to go to the concert, he should realize that he loses $18 regardless of what he decides to do. Therefore, this money should not factor into his decision, since it’s gone in all situations. The actual opportunity cost of the concert is a movie at home (or whatever his next best option would be). Because Cornell would not go to the concert if it were free, we know that this opportunity cost is greater than the benefit of live music. (ii) Some Americans who continue to support the war in Iraq argue that we need to keep fighting to justify all the money that has already been spent and the lives that have already been lost. Explain why this argument is an example of the sunk cost fallacy. Justifying war based on past losses is like Cornell justifying the concert based on the fact that he already paid $18. Stay in or leave Iraq, lives have already been lost. The relevant information here is whether or not the future benefits of war exceed its opportunity costs. [ Common mistake: The question asks to analyze an argument. Many students gave answers saying that staying in war is a mistake. This is not complete. Staying in war based on the argument in the question is a mistake. Whether or not we should stay in the war (based on other arguments) is beyond the scope of the question. When analyzing the validity of an argument, it is important to judge it based on the logic used, not on personal opinions. ] (b) Jane has just moved to a new city and needs to buy a clock radio and a television set. She is about to buy a clock radio at her local electronics store for $20 when a friend sends her a text message saying that the exact same radio is on sale at Wal‐Mart for $10. Even though Wal‐Mart is a 15‐minute drive away, Jane decides to drive there to buy the clock radio since she can get it half off. A week later, she is at her local electronics store shopping for a TV set. The one she likes costs $1010. Before she purchases it, she checks with her friend (who seems to spend a lot of free time at Wal‐Mart), who tells her it sells for $1000 at Wal‐Mart. This time, Jane decides it’s not worth the trip for a 1% savings. Explain why Jane’s behavior is inconsistent with a proper understanding of opportunity cost. [ Note for interestedstudents: The facts that people often commit the sunk cost fallacy (part a) and think in proportions (part b) themselves are relevant for understanding real‐world decision‐making. For that reason, these are ongoing topics of research within the field of behavioral economics (the intersection of economics with psychology). ] In the first situation, Jane is willing to drive 15 minutes to save $10. If her behavior correctly weighs benefits and costs, then it implies that the opportunity cost of the trip is worth less than $10. In the second situation, Jane is not willing to drive to Wal‐Mart in order to save $10. This is inconsistent and suggests that she is making a mistake in one case or the other. [ Note: Rather than thinking about the problem in terms of the opportunity cost, which is measured here in terms of the number of dollars, Jane has instead thought about money saved in terms of proportions. She figures that because $10 is such a small percentage of $1000, it’s not worth making the trip. This is a decision‐making error that people commonly make in their everyday lives. ] 2. (Supply and Demand: Log‐linear supply and demand functions) [ You can find solutions to some parts of this question in chs. 15.5, 15.8, 16.4, and the appendix to ch.15. Feel free to use what you learn from those parts of the textbook in writing up your answer, but try the problems first with the textbook closed, and make sure you write up solutions in your own words. ] (a) First suppose the demand function in a market is “linear”: Qd = D(P, a) = max{a – b P, 0} where a > 0, b > 0, and P > 0. (i) Explain why it makes sense to write the demand function this way (with the max) rather than just as D(P, a) = a – b P. Take any a>0, b>0 and fix them. If we had Qd = D(P, a ) = a ‐ bP , Qd would become negative for P large enough. Of course, this does not make sense: we want consumers to have “zero” demand for such P’s. A simple way to achieve that is to use the Max function in the way specified in the problem, so that Qd = 0 for any such P’s. Note that a and b were fixed at an arbitrary value, so the argument goes through for any a>0 and b>0. (ii) Calculate the price elasticity of demand when a – b P < 0 (the part of the demand curve where Qd = 0). Calculate the price elasticity of demand when a – b P > 0 (the part of the demand curve where Qd > 0). Show that the price elasticity of demand in this latter case depends on which point of the demand curve we’re interested in (i.e., it depends on the level of P). ∂D As discussed in class, the elasticity of demand is defined as ∂P Q , which is equivalent to ∂D P after ∂P Q
P rearranging terms. We’ll consider the two cases a – b P < 0 and a – b P > 0 one by one. 1. Case 1 (a ‐ bP < 0): Since Qd = max{a – b P, 0}, we have Qd = 0. Recall that the elasticity of demand is defined at a point, i.e. a given price/quantity pair. So take any price P=P1 such that a ‐ b P1 < 0 and fix the price/quantity at this value. We can now evaluate the elasticity of demand at this point. In a “neighborhood” of P=P1 (i.e. for P’s close enough to P1), we’ll still have a – b P < 0 and so Qd = 0. Restated in terms of derivatives, that means that ∂D
(P1 ) = 0 , and so the price elasticity of demand is equal to 0. Since the price/quantity ∂P
pair was chosen arbitrarily, we’ve shown that the price elasticity of demand is equal to 0 for every price for which a ‐ bP < 0. 2. Case 2 (a ‐ bP > 0): In this case, we have D = a – b P and so the elasticity of demand is ∂D
= −b , and Q = a ‐ bP . So ∂P P
∂D P
−bP = ‐ b = . Equivalently, the elasticity of ∂P Q
a - bP (a - bP) demand could be expressed in terms of Q as Q-a by noting that the numerator (– bP) is Q equal to Q – a and the denominator (a ‐ bP) is equal to Q. It is important to note that in contract with Case 1, the elasticity of demand here is different for every price/quantity pair. (b) Now suppose the demand and supply functions in a market are given by: Qd = D(P, a) = exp{a} × P^{‐b} (1) (2) Qs = S(P, c) = exp{c} × P^{d} where a, b, c, d, P > 0, and exp is the exponential function. (i) Draw a graph of the supply and demand curves (with P on the y‐axis and Qd, Qs on the x‐
axis). P S D
Q (ii) Explain why it is not necessary to use the max formulation (that we used in part (a) of this question) with this specification of the demand and supply functions. Because f(x) = exp{x} > 0 for any x on the real line and g(y) = P^y > 0 for any y on the real line (the latter inequality also requires P >0), Qd = exp{a} × P^{‐b} > 0 and Qs = exp{c} × P^{d}> 0, and so using the Max formulation would be redundant. (iii) Calculate the price elasticity of demand and the price elasticity of supply. Do these depend on the levels of price and quantity? The elasticity of demand is equal to ∂Qd P
QP
P
= (−b exp{a}P −b−1 )
= (−b d ) = −b . By symmetry ∂P Qd
Qd
PQ of the functional forms for demand and supply, the elasticity of supply is ∂Qs P
= d . As we can see, ∂P Qs neither of these elasticities depends on the levels of price and quantity. (iv) Parts (ii) and (iii) explain why economists often find it convenient to use equations (1) and (2) as approximations to the demand and supply functions, rather than linear functions (which are the approximations usually used in math classes). Now, define new variables qd = ln(Qd), qs = ln(Qs), and p = ln(P), where ln is the natural logarithm. Show that equations (1) and (2) can be rewritten as linear in the logarithms of the original variables: (3) qd = a – b p qs = c + d p (4) It is for this reason that equations (1) and (2) are called a log‐linear specification of the demand and supply functions. By taking the natural logarithm of equations (1) and (2), we get respectively (and as required): • ln Qd = ln({exp(a)}P −b ) = ln{exp(a)} − b ln P = a − bp and • ln Qs = ln({exp(c)}P d ) = ln{exp(c)} + d ln P = c + dp (v) Use equations (3) and (4) to solve for the equilibrium q* and p*. To solve for the equilibrium q* and p*, we just need to set qd = qs. Using equations (3) and (4), this is equivalent to setting a ‐ bp = c + dp, which gives us p* = (a ‐ c )/(b + d) after re‐arranging terms. We can then substitute back for p* in either using (3) or (4) to get q* = (ad + bc)/(b + d). (vi) Notice that the equilibrium (log) quantity, q*(a, c), and (log) price, p*(a, c), can be considered as functions of a and c (both of which are exogenous variables). Calculate each of the following comparative statics: ∂q*/∂a, ∂p*/∂a, ∂q*/∂c, and ∂p*/∂c. Illustrate these comparative statics on two supply and demand diagrams, one showing the effects of a change in a and one showing the effects of a change in c. (The axes can be either P and Q, or their logarithms, p and q; it’s up to you.) From our earlier work, we have that p*(a, c)= (a ‐ c )/(b + d) and q*(a, c) = (ad + bc)/(b+d). Taking partial derivatives, we then get the following comparative statics: ∂p*
1
∂q*
d
=
=
> 0, > 0 ∂a b + d
∂a b + d
∂p*
1
∂q*
b
=−
=
< 0, > 0 ∂c
b+d
∂c b + d Note: When you draw a graph, please clearly label all the lines and axis. When you describe any shift of the line, please clearly identify which direction you try to show (increase or decrease of demand/supply). Just drawing two parallel demand curves without any remarks or just saying “change of demand” is not sufficient. In the following graphs, we graph (a) the effect of an increase of a on p* and q* and (b) the effect of an increase of c on p* and q*. Note that given the form of equations (3) and (4), in the (a) only the demand curve will shift and in (b) only the supply curve will shift. However, both p* and q* will change in equilibrium. (a) Effect of an increase of a on p* and q* (b) Effect of an increase of c on p* and q* p p
S
S
S’ p1 p* p*
p1 D’
D D q* q1 q q* The demand curve shifts from D to D’, and the new
equilibrium pair is (p1, q1). As expected from the
comparative statics formulas derived above, both the
equilibrium price and quantity increase following an
increase of a. q1 q The supply curve shifts from S to S’, and the new
equilibrium pair is (p1, q1). As expected from the
comparative statics formulas derived above, the
equilibrium price decreases and the equilibrium
quantity increases following an increase of c. Many people shifted the supply curve in the wrong direction, probably because they had the intuition that an increase in supply should lead to an upward shift of the supply curve. Here’s a useful way to think about shifts in the supply curve to avoid falling into this trap in the future (the same logic can be used to trace out shifts in the demand curve). Consider the equation qs = d p + c. We’re interested in how an increase in c will affect the supply curve. So fix the price at any value p^. Now letting c increase from c2 to c3>c2, we see that dp^ + c3> dp^ + c2. Graphically, that means that for any given p^, a larger quantity will be supplied for c3. So if you draw a line L parallel to the x‐axis that goes through S and S’ (this is equivalent to fixing the price), the intersection of S’ with L should be to the right of the intersection of S with L. (vii) Total revenue for sellers in this market is R = P Q. Hence in equilibrium, ln(R) = p*+ q*. Prove that ∂(p*+ q*)/∂a > 0. Prove that ∂(p*+ q*)/∂c > 0 if and only if demand is elastic. What do these results imply about how total revenue changes when supply or demand shifts? Use these results to predict how government policies that reduce the supply of heroin will affect the revenues of heroin producers. Also use these results to predict how government policies that reduce the demand for heroin will affect the revenues of heroin producers. Since we have p*(a, c) = (a ‐ c )/(b + d) and q*(a, c) = (ad + bc)/(b+d ), p* + q* = (1 + d )a + (b − 1)c
. We b+d can then find the partial derivatives that the problem asked us to find: (1) ∂ ( p * + q*) (1 + d )
∂ ( p * + q*)
. Since b>0 and d> 0, > 0 =
(b + d )
∂a
∂a (2) ∂ ( p * + q*) (b − 1)
∂ ( p * + q*)
= Since b>0 and d>0, > 0 if and only if b>1. Since we ∂c
(b + d )
∂c found earlier that the elasticity of demand is –b, we have that ∂ ( p * + q*)
> 0 if and only if ∂c the elasticity of demand is <‐1. This is precisely what having an elastic demand means. Technical note if you’re curious why it’s okay to work with p*+q* rather than P*Q*: We have p*+q*= ln (P*) + ln (Q*) = ln (P*Q*), where P*Q* is equal to total revenue at the equilibrium. Since the logarithmic function is an increasing function, p*+q* is a monotonic transformation of P*Q*, and so doing comparative statics with p*+q* rather than P*Q* will give ∂ ( p * + q*) will have the same sign as ∂a
∂ ( PQ)
∂ ( p * + q*)
∂ ( PQ)
and will have the same sign as . ∂a
∂c
∂c us the right directions for all the changes. In other words, (1) Effect of an increase in a (2) Effect of an increase in c
with elastic demand (3) Effect of an increase in c
with inelastic demand p p p S
S’ S S p
p’ p’ S’ p D
p’ p
D
q q’ D D’
q An increase in a leads to an
unambiguous increase in both price
and quantity. This was expected
since ∂(p*+q*)/∂a > 0 q q’ q With elastic demand, an increase in
c leads to an increase in p*+q*.
Graphically, it is clear that the
decrease in p is more than offset by
the increase in q. Again, this is
consistent since ∂(p*+q*)/∂c > 0 if
–d < –1. q q’ q With inelastic demand, an increase
in c leads to a decrease in p*+q*.
Graphically, it is clear that the
increase in p is more than offset by
the decrease in q. Again, this is
consistent since ∂(p*+q*)/∂c < 0 if
–d > –1. Since demand for heroin is inelastic (given that it’s an addictive substance), we expect government policies that reduce the supply for heroin (i.e. decrease c in the context of our model) to increase the supplier’s revenue. On the other hand, we expect government policies that reduce the demand for heroin (i.e. decrease a in the context of our model) to decrease the supplier’s revenue. 3. (Supply and Demand: Creating a market for live kidneys) [ The figures and numbers for this question are drawn directly from a recent research paper: Becker, Gary S., and Julio Jorge Elias (2006), “Introducing Incentives in the Market for Live and Cadaveric Organ Donations,” University of Chicago mimeo. If you are interested, feel free to read more details: http://home.uchicago.edu/~gbecker/MarketforLiveandCadavericOrganDonations_Becker_Elias.pdf ] Ill individuals often die while waiting years for a kidney transplant because fewer kidneys are donated than are needed. Of the kidneys that are donated, most are from cadavers, and relatively few are from live donors, even though donating a kidney has become much safer in recent years. For the U.S. in the years 1990‐2005, Figure 1 shows the total number of patients on the waiting list, the number of transplants performed, and the number donated from live donors. The gap between the size of the waiting list and the number of transplants performed is increasing every year. Figure 4 shows the annual excess demand for kidneys, the gap between the quantity demanded for kidneys and the quantity supplied of kidneys. (The “annual excess demand” equals the annual number who join the waiting list, as opposed to the number who are on the waiting list. That is the appropriate notion of annual excess demand if we imagined that the market was in equilibrium last year, and this year is the first year where there is excess demand. This is the thought experiment we are interested in because we want to use data from when the market is in disequilibrium to predict what would happen if the market were in equilibrium.) (a) It is currently illegal in the U.S. for a person to receive payment for donating a kidney. The vast majority of live donors are family members of individuals on the waiting list. Some people would say that there is no market for kidneys in the U.S. An economist would say that there is a market, but the market has excess demand because there is a price ceiling of $0. Explain the economist’s point of view. By not allowing individuals to be paid for donating a kidney, the government has created a price ceiling of $0 for kidneys. At such a price, the quantity of kidneys demanded is obviously very high. This demand includes people with life threatening illnesses, as well as those with minor problems. On the other hand, the quantity of kidneys supplied at a price of $0 is low; only family members and loved ones with a personal interest in the sick individual’s survival are in general willing to give up a kidney for free (their compensation is in a way “non‐monetary”, as in the case of a parent enjoying the fact that her son/daughter stays alive). Now let’s think about what will happen if a government were to eliminate the price ceiling. On the one hand, the quantity supplied would increase. People who were ready to give up a kidney for free will still be willing to do so, but new people will also enter the market as suppliers (e.g. non‐family members who wouldn’t be willing to donate a kidney for free to a stranger, but would for a high enough price). As the price rises, quantity demanded would decrease, as individuals without life threatening illnesses “drop out” of the market. Eventually, price would adjust so that quantity supplied equals quantity demanded. (b) Explain why the total quantity of kidneys demanded in 2005 is the number of kidney transplants in 2005 plus the excess demand in 2005. What is the total quantity demanded in 2005? The total quantity demanded is equal to the number of transactions that occur at a given price plus the excess demand at that price. In this example, the total number of transactions is the number of kidney transplants (from figure 1). The excess demand is given by the gap between supply and demand (from figure 4). Plugging in numbers, we have Total Quantity Demanded = 15,000 (total number of transplants) + 8,000 (the gap between supply and demand) = 23,000. Many students said that the waitlist represents excess demand. This logic is reasonable, but not the relevant notion of excess demand for predicting what will happen a few years out. When the market first opens, the waitlist will be excess demand. However, after a few years, the waitlist clears, and the only relevant figure is new excess demand. (c) Figure 5 shows a supply and demand diagram for the live kidney market. The current situation is illustrated by the demand curve DD (the curve that starts at “D” and ends at “D”) and the supply curve SS (the curve that starts at “S,” takes a 90‐degree turn, and ends at “S”). The average cost of performing a kidney transplant operation is about $160,000. As long as someone (the patient or insurance company) pays this cost, and as long as a family member is able to donate a live kidney, the operation will be done. Imagine that all family members who can make a donation do so (even without being paid for it). How does that explain why the supply curve is perfectly elastic up to a point and then becomes perfectly inelastic? The perfectly elastic portion of the supply curve represents the supply of kidneys provided by the family members. If there are Q0 relatives, then these people are willing to supply the kidneys for free (only the cost for the operation needs to be paid). The demanders can therefore receive any number of kidneys between 0 and Q0 for only the cost of an operation. Once all the relatives are used up, no more kidneys are supplied. Thus, supply becomes perfectly vertical at Q0. (d) We want to predict what would happen if it became legal to receive payment in exchange for a kidney donation. The main costs faced by a potential live kidney donor are: • Risk of death. The risk of dying as a result of a kidney transplant is less than 0.1%. A variety of other research studies suggest that individuals who earn $35,000 per year often make decisions about what kind of job to take (in terms of a tradeoff between higher salary and higher risk of death) consistent with valuing their own life at around $5 million. This suggests that many people would be willing to accept an increase of 0.1% in their own chance of death if they were paid at least $5,000. • Recovery from surgery takes about 4 weeks. For an individual who earns $35,000 per year, what the opportunity cost in terms of income from losing a month of work is about $35,000 * 4 / 52 ≈ $2,700. • Small reduction in quality of life from only having one kidney. Suppose a potential live donor would be willing to accept this reduction in quality of life in exchange for $7,500. Why does this calculation suggest that a large supply of live kidneys would be available if each donor were paid at least $15,200? Why is the new supply curve S*S* drawn as perfectly elastic at a price of $175,200? Explain why the predicted quantity demanded, Q1, is larger than the number of transplants actually performed today, Q0, but smaller than the quantity demanded today, Q0’. If people were given $15,200 (=$5000+ $2700 + $7500) for a kidney, then all individuals of the type described in the setup of the question would be willing to donate. If we assume there are a large number of these people, then we have a large supply of kidneys at a price of $15,200. We can therefore provide a large number of operations at a price of $175,200, which includes the cost for the operation ($160,000) plus the kidney cost ($15,200). Because there are a large number of operations that can be provided at a cost of $175,200, we assume the supply curve is perfectly elastic. It is important to note that it won’t be perfectly elastic over all quantities. Eventually, we’d run out of people willing to supply kidneys at $15,200. So if we go far enough to the right, the supply curve will start sloping up. The key assumption is that this will happen past the intersection point of the supply and demand curves, and so will have no effect on equilibrium values for price and quantity. The quantity of transplants demanded at a price of $175,200 (Q1) is smaller than the quantity demanded today at a price of $160,000 (Q0’). This quantity (Q1), however, is larger than the number of transplants performed today (Q0) because there are more than Q0 people that are willing to pay $175,200 for a kidney. Since the supply is perfectly elastic at price $175,200, the number of kidneys operations performed in equilibrium will be precisely the quantity demanded, Q1. It is important to note that the conclusion reached in the last paragraph depends heavily on the assumptions we made in the question. We can see from the diagram that if supply was perfectly elastic as a price that is much higher than $175,200, then the equilibrium quantity could have been less than Q0. Try redrawing the diagram with SS*higher up to check that this indeed true. (e) What is the predicted price for kidneys? If the price of kidneys is added to the cost of the surgery, what is the predicted price of the transplant operation for ill individuals? What is the percent change in price? If the elasticity of demand is ‐1, what is the predicted percent decline in quantity demanded? As we showed in the last paragraph, given the assumptions of the problem the predicted cost for kidneys is $15,200 and the predicted price for the full operation is $175,200. The percent change in price is given by (new price‐ old price)/old price i.e. (175,200‐ 160,000)/160,000 = 0.095, or 9.5%. We know that the elasticity of demand is the percent change in quantity demanded divided by the percent change in price, so we have: ‐1 = % change in quantity demanded/9.5%. We conclude that the percent change in quantity demanded is ‐9.5%. (f) Combining your answers from parts (b) and (e), what is the predicted the number of kidneys demanded at the new equilibrium? How many more kidney transplant operations are predicted to occur if it were legal to give monetary compensation to kidney donors? In part (b), we found that the total quantity demanded was 23,000 and in part (e), we showed that raising the price by 9.5% makes the quantity demanded fall by 9.5%. Thus, the predicted number of kidneys in the new equilibrium (let’s call it x) is given by: ‐9.5% = (x – 23,000)/23,000 and so x = 20,815. We can now find the increase in the number of kidney operations by subtracting the previous number of transplants from x, which gives us $5,815. [Note for interested students: Clearly this analysis is very rough, since it’s based on a variety of estimates that could be inaccurate. This should make you wonder whether the results are robust to alternative assumptions! When results are based on estimates that may be inaccurate, economists often perform a sensitivity analysis (The last paragraph in part (d) takes a crack at that). That is, they re‐do the analysis with different assumptions to see how much the results change. We will not ask you to do this, but the authors of the original research paper do perform such an analysis. This entire analysis is an example of positive economics. When the authors take the next step of recommending that the government allow a free market in live kidneys, they are now doing normative economics. ] ...

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