ECON 500 Fall Exam #2 Answer Key.

Transcription

1 ECO 500 Fall 004. Eam # Answer Key. ) While standing in line at your favourite movie theatre, you hear someone behind you say: I like ocorn, but I m not buying any because it isn t worth the high rice. Letting = (buckets of ocorn) and = (dollars sent on everything else), grahically illustrate the solution to this individual s utility maimization roblem. (0 oints) X I 0 0 ) Suose D( ) = 49 5 and S( ) =. Determine the equilibrium rice and equilibrium quantity in this market. (0 oints) D ( ) = S( ) 49 5 = 49 = 7 = 7 = 49. Therefore, q = D(49) = S(49) = 4.

2 I I 3) Mario otimally consumes (,, I) = =. + + a) Is this commodity an ordinary good or a Giffen good for Mario? Elain. (5 oints) This commodity is an ordinary good for Mario. In general, a good is ordinary if the level of consumtion is inversely related to the rice of the good. For the given functional form, as increases I (,, I) = 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? Elain. (5 oints) This commodity is a normal good for Mario. In general, a good is normal if the level of consumtion is ositively related to the income of the consumer. For the given functional form, as I increases I (,, I) = also increases (since the numerator of the + fraction becomes larger, while the denominator remains constant). 4) Between 00 and 004 the equilibrium rice for lemons increased from $.04 to $.7 and the equilibrium quantity increased by,000 units. Clearly elain whether or not this observed change could be a result of: a) an increase in Suly with no change in Demand. (8 oints) Demand Initial Suly ew Suly q This observed change could not be a result of an increase in Suly with no change in Demand. As illustrated above, an increase in Suly with no change in Demand would have to lead to a decrease in rice.

3 b) a simultaneous increase in Demand and decrease in Suly. (8 oints) ew Suly Initial Suly ew Demand Initial Demand q This observed change could be a result of a simultaneous increase in Demand and decrease in Suly, as illustrated in the grah above. 5) J.R. and Diana like to consume =(hotdogs) and =(slices of izza). Diana s utility is given by the function U ( X ) = + 5 ; J.R. has monotonic references (you know nothing else about his references). Consider the following three consumtion bundles: A = ( A, A ) = (3, ) ; B = ( B, B ) = (4,3) ; and C = ( C, C ) = (,4 ). a) Can you determine which of these three bundles Diana likes the most? If so, identify her most referred bundle and elain why she finds it to be the most desirable of the three; if not, elain why it is not ossible to identify which bundle is her most referred. (6 oints) It is ossible to determine which of these bundles Diana likes the most. Her references are described by the utility function U ( X ) = + 5. As a result, her utility for each bundle is: U ( A) = =, U ( B) = = 3, and U ( C) = + 0 =. From here: B f C f A. Therefore, B = (4,3) is her most referred bundle of these three. b) Can you determine which of these three bundles J.R. likes the most? If so, identify his most referred bundle and elain why he finds it to be the most desirable of the three; if not, elain why it is not ossible to identify which bundle is his most referred. (6 oints) It is not ossible to determine which of these bundles J.R. likes the most. All we know is that his references are monotonic. From here it follows that B f A (so that A = (3, ) cannot be most referred). However, we cannot comare the relative desirability of B = (4,3) and C = (,4).

4 45 6) Suose demand is given by D ( ) =. a) Grahically illustrate Consumers' Surlus in at a rice of = 3. (4 oints) 3 q 5 Consumers' Surlus in at a rice of = 3 is illustrated by the area shaded in lime in the above grah. b) Grahically identify the change in Consumers' Surlus resulting from a decrease in rice from = 3 to =. (4 oints) 3 q 5 45 The change in Consumers' Surlus resulting from a decrease in rice from = 3 to = is illustrated by the area shaded in sky blue in the above grah.

5 c) oting that the demand curve becomes flatter as we move to lower rice, higher quantity airs, lace a numerical bound on the change in Consumers' Surlus resulting from a decrease in rice from = 3 to =. (6 oints) oting that the demand curve becomes flatter as we move to lower rice, higher quantity airs : the change in Consumers Surlus can be bounded as: ( )(5) = 0 < CS < 50 = ()(5) + (40)() d) It can be shown that the change in Consumers' Surlus as a result of a decrease in rice from = 3 to = is equal to CS = 30. Does this value fall within the bound that you identified in art (c) above? ( oints) Yes, since 0 < 30 < 50. 7) Consider a consumer with U ( X ) =. For this individual, and MU =. a) Secify MRS, as a function of and. ( oints) MU = MU MRS =, =. MU b) Based uon your answer to art (a), are the references of this consumer conve? Clearly elain. ( oints) Yes, the references of this consumer are conve, since will become smaller as is decreased and is increased. MRS =,

6 c) Determine the utility maimizing consumtion bundle for this consumer (that is, the otimal levels of and, each as a function of rices and income). (6 oints) The utility maimizing consumtion bundle is the unique air of and for which the following two conditions hold simultaneously: MRS, = = and + = I. The first condition can be stated as =. Substituting into the second condition and solving for, I we realize =. Thus, returning to the first condition, we have I =. d) Determine the level of utility (as a function of rices and income) that this individual realizes as a result of consuming this otimal bundle. (3 oints) When consuming the bundle identified in art (c), the level of utility of this consumer can be eressed as a function of rices and income as I I I V (,, I) = U (, ) = =. e) How will the level of hainess of this consumer change if income and both rices simultaneously double (so that I new = I,, new =,, and = )? Clearly elain. (3 oints), new, The level of hainess of the consumer will not change as a result of this change in rices and income. From the function determined in art (d), we have that the level of hainess after this change is equal to I I V (,,,,I ) = = so 4,,,, that V,,I ) = V (,, I ). (,,,,

7 8) At rices and income of (,, I), a consumer maimizes utility by choosing X illustrated below. For each of the following scenarios, identify the entire area in (, ) -sace where the new otimal bundle could be. Clearly elain your answer. a) good one is a comlement to good two and good two is an ordinary good ; the rice of commodity two increases, with income and the rice of commodity one held fied. (5 oints) X As a result of this change in the rice of commodity two, consumtion of each good must decrease (since commodity two is ordinary and commodity one is a comlement to commodity two). Therefore, the new consumtion bundle must be in the area shaded in gold in the above grah.

8 b) good one is normal ; the rice of commodity one increases, with income and the rice of commodity two held fied. (5 oints) X As a result of this change in the rice of commodity one, consumtion of commodity one must decrease (since if commodity one is normal it must also be ordinary). However, consumtion of commodity two may either increase or decrease. Thus, the new consumtion bundle could be anywhere in the area shaded in teal in the above grah. EXTRA CREDIT: (4 oints) oel and his brother Liam enjoy going to the ub and consuming =(cigarettes) and =(alcohol). oel s utility is given by U ( X ) = [,856 ( 40 ) ( 6 ) ], so that his marginal utility for cigarettes is MU ( X ) = ( 40 ) and his marginal utility for alcohol is MU ( X ) = ( 6 ). Suose a ack of twenty cigarettes costs 4 (so that the er unit rice of cigarettes is =. 0 ) and that each drink costs 3 (that is, each unit of alcohol costs = 3 ). If oel goes to the ub with 70, how many units of and how many units of will he consume? Clearly elain. First observe that MU ( X ) ( 40 ) ( X ) ( 6 ) = is negative for 40 and MU = is negative for > 6. Over these ranges, increased consumtion leads to a lower level of utility, indicating that oel is actually worse off. From here X = (40,6) is "oel's overall best bundle," in that U ( X ) > U ( X ) for all X X. As a result, if X = (40,6) is affordable, then oel will maimize his utility by urchasing X = (40,6). At the given rices, X = (40,6) is affordable (since 40( 5 ) + 6( 3) = 56 < 70). Therefore, oel will otimally consume 40 cigarettes and 6 drinks. >