ECON 500 – Fall 2004. Exam #2 – Answer Key. 1) While standing in line at your favourite movie theatre, you hear someone behind you say: “I like popcorn, but I’m not buying any because it isn’t worth the high price.” Letting =1x (buckets of popcorn) and =2x (dollars spent on everything else), graphically illustrate the solution to this individual’s utility maximization problem. (10 points) 2) Suppose ppD 549)( −= and ppS 2)( =. Determine the equilibrium price and equilibrium quantity in this market. (10 points) )()( pSpD = ⇔ pp 2549 =− ⇔ p749 = ⇔ 7=p ⇔ 49*=p. Therefore, 14)49()49(*=== SDq. 2x 1x *X 0 I 03) Mario optimally consumes 211221212211),,(ppIpppppIpIppx+=+=. a) Is this commodity an ordinary good or a Giffen good for Mario? Explain. (5 points) This commodity is an ordinary good for Mario. In general, a good is “ordinary” if the level of consumption is inversely related to the price of the good. For the given functional form, as 1p increases 21212211),,(pppIpIppx+= decreases (since the numerator of the fraction remains constant, while the denominator becomes larger). b) Is this commodity a normal good or an inferior good for Mario? Explain. (5 points) This commodity is a normal good for Mario. In general, a good is “normal” if the level of consumption is positively related to the income of the consumer. For the given functional form, as I increases 21212211),,(pppIpIppx+= also increases (since the numerator of the fraction becomes larger, while the denominator remains constant). 4) Between 2002 and 2004 the equilibrium price for lemons increased from $1.04 to $1.27 and the equilibrium quantity increased by 2,000 units. Clearly explain whether or not this observed change could be a result of: a) an increase in Supply with no change in Demand. (8 points) This observed change could not be a result of an increase in Supply with no change in Demand. As illustrated above, an increase in Supply with no change in Demand would have to lead to a decrease in price. p q Demand Initial Supply New Supplyb) a simultaneous increase in Demand and decrease in Supply. (8 points) This observed change could be a result of a simultaneous increase in Demand and decrease in Supply, as illustrated in the graph above. 5) J.R. and Diana like to consume 1x =(hotdogs) and 2x =(slices of pizza). Diana’s utility is given by the function 2152)( xxXU+=; J.R. has “monotonic preferences” (you know nothing else about his preferences). Consider the following three consumption bundles: )1,3(),(21==AAxxA ; )3,4(),(21==BBxxB ; and )4,1(),(21==CCxxC . a) Can you determine which of these three bundles Diana likes the most? If so, identify her most preferred bundle and explain why she finds it to be the most desirable of the three; if not, explain why it is not possible to identify which bundle is her most preferred. (6 points) It is possible to determine which of these bundles Diana likes the most. Her preferences are described by the utility function 2152)( xxXU+= . As a result, her utility for each bundle is: 1156)(=+=AU, 23158)( =+=BU , and 22202)(=+=CU . From here: ACB ff. Therefore, )3,4(=B is her most preferred bundle of these three. b) Can you determine which of these three bundles J.R. likes the most? If so, identify his most preferred bundle and explain why he finds it to be the most desirable of the three; if not, explain why it is not possible to identify which bundle is his most preferred. (6 points) It is not possible to determine which of these bundles J.R. likes the most. All we know is that his preferences are “monotonic.” From here it follows that ABf (so that )1,3(=A cannot be most preferred). However, we cannot compare the relative desirability of )3,4(=B and )4,1(=C. p q Initial Demand New Supply Initial Supply New Demand6) Suppose demand is given by 245)(ppD = . a) Graphically illustrate Consumers' Surplus in at a price of 3=p . (4 points) Consumers' Surplus in at a price of 3=p is illustrated by the area shaded in “lime” in the above graph. b) Graphically identify the change in Consumers' Surplus resulting from a decrease in price from 3=p to 1=p . (4 points) The change in Consumers' Surplus resulting from a decrease in price from 3=p to 1=p is illustrated by the area shaded in “sky blue” in the above graph. 3 5 p q 3 p q 1 45 5c) Noting that the “demand curve becomes flatter” as we move to “lower price, higher quantity pairs,” place a numerical bound on the change in Consumers' Surplus resulting from a decrease in price from 3=p to 1=p . (6 points) Noting that the “demand curve becomes flatter” as we move to “lower price, higher quantity pairs”: the change in Consumers’ Surplus can be bounded as: )2)(40()5)(2(5010)5)(2(21+=<∆<= CS d) It can be shown that the change in Consumers' Surplus as a result of a decrease in price from 3=p to 1=p is equal to 30=∆CS. Does this value fall within the bound that you identified in part (c) above? (2 points) Yes, since 503010<< . 7) Consider a consumer with 21)( xxXU = . For this individual, 12121xxMU= and 21221xxMU=. a) Specify 2,1MRS as a function of 1x and 2x . (2 points) 12212,1xxMUMUMRS== . b) Based upon your answer to part (a), are the preferences of this consumer “convex”? Clearly explain. (2 points) Yes, the preferences of this consumer are “convex,” since 122,1xxMRS = will become smaller as 2x is decreased and 1x is increased.c) Determine the utility maximizing consumption bundle for this consumer (that is, the optimal levels of 1x and 2x , each as a function of prices and income). (6 points) The utility maximizing consumption bundle is the unique pair of 1x and 2x for which the following two conditions hold simultaneously: 21122,1ppxxMRS == and Ixpxp=+2211. The first condition can be stated as 2211xpxp = . Substituting into the second condition and solving for 2x , we realize 2*22 pIx= . Thus, returning to the first condition, we have 1*12 pIx= . d) Determine the level of utility (as a function of prices and income) that this individual realizes as a result of consuming this optimal bundle. (3 points) When consuming the bundle identified in part (c), the level of utility of this consumer can be expressed as