University of California, Davis Date: August 25, PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE ANSWER KEY

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1 Unversty of alforna, Davs Date: ugust 5, Department of Economcs me: 5 hours Mcroeconomc heory Readng me: mnutes PREIMINRY EXMINION FOR HE Ph.D. DEGREE Please answer four parts (out of fve) NWER KEY QUEION. HE MRGIN UIIY OF WEH et u : be a dfferentable, strctly concave, locally nonsatated utlty functon. We restrct ourselves n what follows to a prce-wealth doman P of vectors ( pw, ) for whch the UMX[p, w] problem has a soluton... refly argue that the soluton to the UMX[p, w] problem, denoted ( pw, ) s unque and defned by the frst order equaltes. NWER. Unqueness follows from strct quasconcavty. he FO equaltes characterze the soluton because the doman of u s an open set... Verbally defne the margnal utlty of wealth, to be denoted ( p, w). tate and prove ts relaton to the agrange multpler. NWER. he margnal utlty of wealth s the partal dervatve of the ndrect utlty functon wth respect to wealth, and t concdes wth the agrangan multpler of the UMX[p, w] problem, as can be proved by a standard applcaton of the Envelope heorem..3. We now partton the set {,, } of goods nto the three subsets: {,, }, { +,, }, and { +,, }. vector = (,, ) s smlarly parttoned nto the three subvectors (,..., ), (,..., ) and (,..., ), = ( ; ; 3 ). he prce 3 vector p s smlarly parttoned nto p ( p,..., p ), p ( p,..., p ) and 3 p p p (,..., ), p = (p ; p ; p 3 ). We assume that the utlty functon u s separable n the form 3 3 u ( ) u( ) u( ) u( ),

2 (for J =,, 3, u J s dfferentable, strctly concave and locally nonsatated) and adopt the followng nterpretaton: our household, comprsed of a husband and a wfe, consumes three types of goods: s a basket of goods consumed n Vacaton Island, s a basket of goods consumed n Vacaton Island, and 3 s the basket of all other goods consumed by the household. Our household has solved the UMX[p, w] problem at some pont n the past, but they have forgotten some detals..3(a). efore engagng n ther two-sland vacaton, the husband says: all we have to remember are our planned ependtures n Island, w, and n Island, w. hen when we are n Island, we observe ts prces p and solve ma u ( ) subect to p < w, wth soluton denoted ( p, w ), and when we are n Island, we observe ts prces p and solve ma u ( ) subect to p < w, wth soluton denoted ( p, w ). hen, the husband clams, our solutons wll concde wth what we had orgnally planned,. e., J ( p J, w J ) J ( p, w), J =,. Is the husband rght? rgue your answer. NWER. Yes, he s rght, as long as he correctly remembers w as p ( p, w) and w as p ( p, w). onsder Island. uppose as contradcton hypothess that Ma ( ) p u subect to the constrant p 3 (, ( p, w), ( p, w)) ( p, w) solves w. hen u ( ) u ( ( p, w)), because ( pw, ) clearly satsfes w,. e., u (, ( pw, ), ( pw, )) > upw ( (, )), whle satsfes the constrant p w, contradctng the fact that ( pw, ) s the unque soluton to UMX[p, w]..3(b). he wfe nterects: what about ust rememberng the margnal utlty of wealth? hen when we are n Island we unconstranedly solve ma ( ( ) ), u p wth soluton ˆ (, ) p, and when we are n Island we unconstranedly solve ma ( ( ) ), u p wth soluton J J J ˆ ( p, ), and, she clams, we shall also get ˆ ( p, ) ( p, w), J =,.

3 3 Is the wfe rght? rgue your answer. Verbally nterpret the mamzaton problems proposed by the wfe. NWER. Yes, she s rght too, as long as she correctly remembers as ( p, w). onsder Island : when she solves the system of equatons n unknowns u ( ) p,,...,, (.) for = ( p, w) gven, she must come up wth the soluton ( p, w), because ( pw, ) does satsfy (.) for the same p,,...,,. e., u ( ) u ( ( p, w)). Note that ( p, w) s mpossble gven the strct concavty of u : wrtng ( p, w), f, then the strct concavty of u mples and u ( ) u ( ( p, w)) u ( ) u ( ( p, w)) [ ] u ( ) u ( ( p, w)), whch yelds the contradcton > when u ( ) u ( ( p, w)). In words, the wfe s approach equates the margnal utlty of the ependture spent n good to the margnal utlty of wealth..4. et there be a large number of slands. Do the arguments of husband and wfe n.3 carry over? ompare the two approaches. NWER. Yes, they do carry over, as long as the utlty functon s separable sland by sland. ut wth one mllon slands the husband s approach requres keepng track of the one mllon numbers w, whereas the wfe s approach only requres to keep track of..5. heck the husband s and wfe s approaches for Island n the followng eample., =. u : : u ( ) ln,,, NWER. he Walrasan demands n the obb-douglas case are w ( p, w), =,,. (.) p u he agrange multpler can be obtaned from the frst-order equalty whch by (.) yelds p, or w p p,. e., p,

4 4 ( pw, ) w. (.3) (lternatvely, we can compute the ndrect utlty functon v ( pw, ). w p w w ) w p v( p, w) ln w, wth p Husband. olves ma ln ln subect to p p w, wth soluton w p p w If he correctly recalls that (,, ),,. p w w (, ) (, ) ( ), then w p p w p p w p p w p p w( ) w p p w p w Hence, he gets the Walrasan (,, ) (, ),,. p p demands for Island rght. Wfe. olves ma( ln ln [ p p]), wth FO p,. e., p =,. If she correctly recalls that ( pw, ) (from (.3)), then she w computes ˆ ( p, p, ) w ( p, w). Hence, she also gets the Walrasan demands for Island p rght.

5 5 QUEION. MONOPOY REGUION et there be goods, and wrte (,..., ). onsder a consumer wth utlty functon, u: : u(,...,, ) u ( ) where, for =,, -, u s strctly concave wth u () =, strctly ncreasng and twce dfferentable, and satsfes the condton lm u '( ). In what follows, we set p = and restrct our attenton to strctly postve prces p >, =,, -... how that the Walrasan demand for good, =,, -, can be wrtten ˆ ( p ), and nverted, wth nverse denoted pˆ ( ). NWER. From the FO we obtan u ( ) = p, (.) wth u ( ) decreasng, and hence nvertble. frm supples goods,, - to our consumer. Its cost functon s, f,,..., (,..., ), F c otherwse where F > and c >, =,, -. Defne: : : ( ) u( ) ( ), ˆ p c F. : : ( ) [ ( ) ].. Interpret and n words. NWER. s the socal surplus functon, equal to ONUMER VUION - O, whch can also be epressed as ONUMER URPU + PROFI, or as ONUMER URPU + PRODUER URPU FIXED O. We consder the followng optmzaton problems, and postulate that any soluton to any of them s strctly postve.

6 6 Problem U (unconstraned) Ma ( ). Problem [ ] (constraned) Ma ( ) subect to ( ). Problem M[ ] (constraned) Ma ( ) subect to ( ). Problem MU (unconstraned) Ma ( )..3. Wrte the frst-order condtons of Problem U, epress them n the format of the erner equaton (. e., MRKUP = DEGREE OF MONOPOY), and verbally nterpret. What can be sad about the markups, and about the level of profts at a soluton to Problem U? NWER. Ma ( ) = u c F, Ma [ ( ) ] wth FO: u'( ) c, =,, -,. e., usng (.) above, pˆ( ) c. Hence, prces must equal margnal costs, and p ˆ( ) c p ˆ( ),. e., the markups are zero. It follows that at the soluton to Problem U, profts are gven by [ pˆ ( ) ] c F F, and the frm s sufferng losses by the amount of the fed cost F..4. For each of the Problems U,, M and MU, verbally nterpret the problem and the agent who may face t. NWER. Problem U, the unconstraned mamzaton of socal surplus, can be nterpreted as the problem of a socal planner nterested n economc effcency. Problem MU, the unconstraned mamzaton of profts, can be nterpreted as the problem of a proft mamzng, pure monopoly. Problems [ ] and M[ ] can be nterpreted as those of a regulated monopoly. Problem [ ], Ma ( ) subect to ( ), can be vewed as the second-best problem of a socal planner who wshes to mamze socal surplus but must guarantee a mnmal level of

7 7 profts to the frm. For nstance, f, then Problem [ ] mamzes socal surplus subect to the condton that the frm must at least break even, a condton whch, as ust seen, s not satsfed at the soluton to Problem U. s wll be seen below, Problem M s formally smlar to problem. Now the regulator mposes a mnmum level of socal surplus (perhaps va mnmum quanttes, or mamum prces), and lets the frm mamze profts subect to ths constrant..5. Wrte the frst-order condtons of Problem MU, epress them n the format of the erner equaton. How are the markups for two dfferent goods related to each other? NWER. Ma ( ) = ˆ p c F, Ma [ ( ) ] wth FO: pˆ'( ) pˆ c, =,, -, whch can be wrtten n the erner-equaton form pˆ c pˆ'( ), =,, -, pˆ pˆ ( p ) where ( ) s the elastcty of the (drect) demand for good. he monopolst s markups on the varous goods are nversely related to ther (absolute value of the drect) elastcty of demand..6. For each of the Problems and M, wrte ts frst-order condtons, and epress them n the format of the erner equaton. ompare across the four problems, and nterpret. NWER. Problem. he agrangan s wth FO: u'( ) c [ pˆ'( ) pˆ c ], =,, -, [ u ( ) ] [ [ ˆ ( ) ] ] c F p c F, or, usng (.), pˆ c [ pˆ'( ) pˆ c ], =,, -,. e., [ pˆ c ][ ] pˆ'( ), =,, -, whch can be wrtten n the erner-equaton form pˆ [ ˆ c p'( ) ], =,, -. pˆ pˆ [ ][ ( p )]

8 8 gan, the markups on the goods are nversely related to ther (absolute value of the drect) elastcty of demand. When the constrant s not bndng, = and we are back to.3. t the other etreme, as becomes large, we approach.5. Problem M. he agrangan s now [ pˆ ( ) c ] F [ [ u ( ) c ]] wth FO: pˆ'( ) pˆ c [ pˆ c ], =,, -, where (.) has been used,. e., pˆ c, =,, -. pˆ [ ] [ ( p )] Now, when the constrant s not bndng, = and we are back to.5. t the other etreme, as becomes large, we approach ssume now that the functons u are quadratc, of the form: u a b u a b :[, / ): ( ) (/) ( ). Prove that Problems and M are dual n the followng sense: () If solves Problem [ ], and s the value of Problem [ ], then () If then solves Problem M[ ]. solves Problem M[ ] and s the value of Problem M[ ], solves Problem [ ]. NWER. Note that the functons and are strctly concave, whch n partcular mples that ther upper level sets are conve sets. Hence, problems [ ] and M[ ] are concave problems, problem [ ] has at most one soluton, and so does problem M[ ]. (). et solve Problem [ ], wth value,. e., ( ), and ( ). y the necessty of the Kuhn-ucker condtons, there s a nonnegatve multpler such that ( ) ( ). (.) Frst note that, because ( ), satsfes the constrant n Problem M[ ].

9 9 If =, then ( ),. e., s the unque mamzer of, whch mples that the set { : ( ) } s the sngleton { }. ut ths s the constrant set of Problem M[ ]. Hence, solves Problem M[ ]. If, on the contrary, >, then, defnng / >, (.) mples that ( ) ( ), whch s the Kuhn-ucker condton of Problem M[ ]. he suffcency of these condtons n the concave case guarantees that () s proved n a parallel manner. solves Problem M[ ]..8. onsder the specal case where = and u (, ) a(/)( ). ompute and graph the functons and n the same fgure, where you are also asked to llustrate the solutons to problems U and MU, as well as the dualty theorem of the prevous secton. NWER. ( ) a(/)( ) F c, wth ( ) a c. ( ) [ a ] F c ( ) a c., wth Problem U. oluton at a c. a c Problem MU. oluton at. Dualty heorem. et be gven, as llustrated n the Fgure. onsder Problem [ ]. he constrant set s { : ( ) } = [, ], and s mamzed at, wth value ( ) =. onsder now the dual problem M[ ].he constrant set s now { : ( ) } = [, ], and s mamzed at, wth value ( ) =.

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14 Mcro Prelm. ugust. nswer keys for Questons 4 and 5 QUEION 4. (a) he von Neumann-Morgenstern normalzed utlty functons are as follows: ob onnor Donna Emly (f ) w (f (, )) w 3 (f (, )) w 3 4 For the etensve game there are two choces, one wth 3 players (Donna, Emly, Player ) as shown n the frst fgure below and the other wth 4 players, as shown n the second fgure. In some of the questons t s mplctly assumed that the representaton s n terms of 4 players. 4 Donna ob (/6) NURE onnor (5/6) Player Player Donna Emly Emly 3/ Player Donna Emly Emly Pl 6/4 3/4 Pl 3/4 7/4 7/4 Pl Pl 3/ 3/ 6/ Pl 3/ Pl

15 Donna ob (/6) NURE onnor (5/6) 5 ob onnor Donna ob 6/4 3/4 ob 3/4 7/4 Emly 7/4 ob Emly Emly onnor 3/ 3/ 6/ onnor 3/ 3/ ob onnor Donna Emly onnor (b) ob has 4 = 6 strateges (one of them s f chosen as Player, I choose (,,,) where the choces apply to ob s nodes top frst and then left to rght), onnor has 4 = 6 strateges (one of them s f chosen as Player, I choose (,,,) where the choces apply to ob s nodes top frst and then left to rght), Donna has 4 strateges (one of them s aganst ob I play, aganst onnor I play ) and Emly has 3 = 8 strateges (one of them s (,,) where the choces apply to Emly s nformaton sets from top to bottom). (c) onsder the strategy profle (,,, ),(,,, ),(, ),(,, ). hen the payoffs are,,, (d) t every subgame-perfect equlbrum ob and onnor choose the payoff-mamzng acton at each of the sngleton nodes. hus the game can be smplfed to:

16 Donna ob (/6) NURE onnor (5/6) 6 ob onnor Donna Emly Emly Emly 3/ ob onnor Donna Emly () 6/4 () 3/4 () 7/4 3/ () 6/ () 3/ () 7/4 One weak sequental equlbrum of ths game s:,,(, ),(,, ) wth belefs for Emly 5,,,,, (from top to bottom) 6 6 nother weak sequental equlbrum s,,(, ),(,, ) wth belefs for Emly (from top,,,,, to bottom) (e) hen we would have a perfect-nformaton game (the nformaton sets of Emly gven n the fgure above would be elmnated; hence Emly s strategy now has to specfy a choce at each of s nodes, so that she has 6 = 64 strateges). oth Donna and Emly then would know that aganst ob would be followed by, whle aganst onnor would be followed by. hus the backward nducton soluton would be (,,, ),(,,, ),(, ),(,,,,, ) (Emly s strategy s read as follows: top-left, top-rght, mddle left, mddle-rght, bottom left, bottom-rght). (f) he game of part (a) can be vewed as a stuaton of one-sded ncomplete nformaton among three players: Donna, Player and Emly, where Emly s uncertan about the payoffs of Player. he game s one of perfect nformaton, as follows:

17 Donna Pl. Emly Pl. 7 8 Emly Emly 6 7 Pl. 3 Pl all G the above game wth, 4, 3, 4, 5 4, 6 4, 7 4, 8 4. (hese are the payoffs of ob n the game of part a.) all G the above game wth,, 3, 4, 5, 6, 7, 8. (hese are the payoffs of onnor n the game of part a.) he stuaton of ncomplete nformaton can then be represented usng two states, as follows: Donna: Game G Game G Player : Emly: 6 5 6

18 QUEION 5. 8 (a) (a.) Whenever p G R > ( + r) X > p R (so that there does not est a w R such that pw ( rx ) but there ests a w R such that pw G ( rx ) and t). (a.) Each bank would offer any type G entrepreneur a loan of $X wth repayment X ( r) w. No bank would offer a loan to type entrepreneurs. he epected utlty p G X( r) of a type G entrepreneur would be UG pgr pgr X( r) > and the pg epected utlty of a type entrepreneur would be U =. (b) bank s epected proft from a loan s G condton yelds w p ( ) p X( r), that s, G w p ( ) p X( r). he zero proft w X( r). o p ( ) p ( ) U G = p R X r G and U p ( ) p = p R X( r). In order for ths to G p ( ) p G be an equlbrum t must be that U (whch mples UG ), that s, R X( r) : the fracton of hgh-rsk borrowers should not be too hgh. R( p p ) G (c) U G (c, w) = p ( Rw) ( ) c. U (c, w) = p ( R w) ( ) c G (d) Frst of all, note that nobody would offer a collateral c > c. Furthermore, the zero-proft X c ( r) condton requres w wˆ. he types wll not apply f and only f p p Rwˆ ( ) c. he G types wll apply f and only f G p ˆ ˆ ( R w) pg ( R w) p ˆ G Rw( ) c. hus we need c (e) (e.) (b) w = 5/7 = 7.49, U G = /7 =8.57, U = /7 =.857. (c) U G (c, w) = w c; U (c, w) = ( w) c w c. (d) w5 c, 5 c X 5 ( R ) [the RH of ( ) would gve c 5, thus the bndng constrant becomes c X ]. (e.) he lowest value of c s 5. In the separatng equlbrum w and thus U G = 45 (+ )5 = 49.5 and U =. o, relatve to poolng, sgnalng helps the G types and hurts the types. It s also more effcent: under poolng the epected average utlty s (.857) ( )8.57 whle wth sgnalng t equals () ( ) , close to the average epected utlty wthout asymmetrc nformaton, whch s ( )( 5) ( ) G