ECO 0: ELEENTS OF ICROECONOICS The Theor of Consumer ehviour SALE QUESTIONS AND ANSWERS ) Dorcs level 00 student of Ketsco, clls her prent to send her mone to u ilo which cost $ ech nd ooks which cost $8 ech. The sent her 0 pirs of shoes nd 0 pirs of new jens trousers which she cn freel sell on the mrket for $0 nd $0 respectivel.. Write down the eqution of Dorcs s udget line nd interpret the result.. Crefull sketch the udget line c. If Dorcs discovers tht sustituting two () tins of ilo for three () will not chnge her level of stisfction, wht must e the lowest price of ook efore she will sustitute ilo for ooks given tht the price of ilo remins t $ rice of ilo, m $ rice of red, $8 Income from shoes (s) = 0 pirs of shoes multiplied $0=$00 Income from Jens (j)=0 pirs of Jens multiplied $0 = $800 Totl income () =00+800=$00 () -------------------udget line m Dorcs udget = ilo +8 ooks =00 8 00 00 8 00 ----------()---------udget Eqution (ilo in terms of red) Alterntivel, 8 00 8 00 8 8 8 50 --------()----------udget Eqution (red in terms of ilo) Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge
Interprettion: From (), if Dorcs us ook the numers of ilo she cn u will rece two (). Alterntivel, if Dorcs spends ll her income on ilo she will u 00. From (), if Dorcs us tins of ilo the numer of ooks she cn u will rece one (). Alterntivel, if Dorcs spends ll her income on ooks she us 50 ) From eqution () 00 If 0, 00 If 0 0 00 00 50 00 =00-0 50 From eqution () 50 If 0, 50 If 0 0 50 Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge
50 00 50 =50-/ 0 00 c) At equilirium, ut RS $ RS cross multipl 8 $.666 RS The consumer will e in equilirium when the price of ook is $.6667. Impliction: n price of ooks ove $.666, Dorcs will sustitute ilo for ooks until she reches equilirium. ) A. Suppose consumer hs $0 ville to e divided etween commodities A nd nd the unit price of is fied t $. Wht will e his demnd eqution for A if his utilit function is U =?. Estlish the lgeric reltionship etween rginl Revenue (R) nd elsticit of demnd. Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge
Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge ). U -------------- ojective function =0 Implicit constrint function setting the lgrngin function, ) ( L Differentiting with respect to, nd 0 L ------------() 0 L ------------() 0 L ----------() From () & () ------------(5) -----------(6) 6 5 (7) ut =0 0 5 Demnd function for commodit A
Alterntivel: U U () necessr condition ()Sufficient condition U d d U...() U.() ut () nd () into () ut (5) into () (5) -------(6) -----(7) 0, =0 Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge 5
). R 5 Demnd function of A TR Q R dtr d Q dq dq dq dq R Q Fctorizing R Q d dq Q d dq d dq ut dq E d Q E d Q dq E d Q dq R E E (ecuse we consider solute figure) The elsticit coefficient is inversel relted to price level nd mrginl revenue. The higher the price level, the lower will e the elsticit nd the higher will e the mrginl revenue (R) nd vice vers Summr If demnd is inelstic, n price leds to n increse in totl revenue nd decrese in price leds to fll in totl revenue If the demnd is elstic Ed n increse in price will result in decrese of the totl revenue while decrese in price will result in n increse in the totl revenue If the demnd hs unitr elsticit totl revenue is not ffected chnges in price, since if Ed= then R O Ed Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge 6
). use the lw of diminishing mrginl utilit to unquote the phrse we will onl know the worth of wter when the well is dr. rovide proof tht the slope of the norml consumer preference curve is equivlent to the rtio of mrginl utilit c. Wh does the mrginl utilit diminishes? d. Stte three () ioms of the crdinlist pproch nd hence the equi-mrginl principle. d). This is n etension of the wter-dimond prdo. The fct is tht, e to the reltive undnce of wter, it is reltivel cheper thn dimond, thus ssocited with greter mrginl utilit (U). However when wter ecomes reltivel scrce, it will now ecome reltivel epensive hence, the consumer cn onl mimize his/her utilit consuming less of wter.. U f, () ut d d d d d d d () 0 () d d d 0------------------------() U (5) U (6) ut (5) nd (6) into () Ud Ud 0 (7) d U d U U U U d U d d d Divide through d nd U d d U U Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge 7
RS d d U U c) rginl utilit diminishes ecuse of the successivel less importnt uses of dditionl quntities of commodit. d) Rtionlit of the consumer Disposle income of the consumer is given Tstes nd preferences re given There is perfect competition in the goods mrket, tht is prices of goods re given rginl utilit is positive Utilit is mesurle/quntifile The equi-mrginl principle sttes tht the utilit derived from spending n dditionl unit of mone must e the sme for ll commodities purchsed. Tht is, U U p U z z ). Use the demnd schele in the tle elow to find the consumer surplus if quntit of si is purchsed. rice Qt dded rginl utilit(u) Totl utilit(tu) 00 00 00 80 80 80 65 65 55 55 00 50 5 50 50 5 6 5 95 N. U nd TU were not given Solution How much do we hve to p for si units? 6 5=$70 Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge 8
How much would we e willing to p for these si units Totl utilit 95 Thus the consumer surplus $ 95 $70 $5 Also, Totl utilit for five units is $50 ut we would hve to p 5 50 Therefore consumer surplus=50-50=$00. ) Assume tht there re onl commodities nd tht the totl utilit function is multiplictive of the form The Us re U U dq dq At equilirium, U q U q q q q q U q q q q We m derive the demnd for commodit sustituting q p in the udget line q q q q q q Similrl, q Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge 9
Thus, the demnd for nd re ll inversel relted to its own price p nd p respectivel nd positivel relted to income 5) Suppose consumer s preference etween food nd clothing cn e represented the utilit function U clothing. where mesures the numer of units of food nd the numer of units of ) show tht consumer with this utilit function elieves tht more is etter for ech good. ) show tht U of food nd clothing is diminishing c) show tht the indifference curve ehiit diminishing mrginl rte of sustitution for this utilit function d) is the curve conve? Differentite U d Solution U d ) emining the utilit function we cn see tht U increses whenever or increses. This mens tht the consumer likes more of ech good. ore is etter simple mens tht the mrginl utilit is positive. ) First emine U d. As increses (holding constnt) U flls, therefore the consumer s U of food is diminishing. Now emine U. As increses d U flls. Therefore, the U of clothing is diminishing. Thus, U diminishing. Alterntivel Tke first Derivtive of U stisfies the ssumption tht more is etter nd those mrginl utilities re U Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge 0
Tke second derivtive of U or first derivtive of U U 0 Also, U U 0 Diminishing Diminishing Thus, the U of clothing is diminishing nd U of food is diminishing. U U c) Simplif () RS RS / Since RS /, / RS RS / (Use Quotient rule) 0 Therefore the indifference curve ehiit diminishing mrginl rte of sustitution for this utilit function. d) Tke second derivtive of RS (Use Quotient Rule) RS RS From eqution one () RS=/ Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge
0 Hence conve to the origin (Ke, for minimum point the second derivtive should e greter thn zero - rememer qudrtic curves) 6) Show whether is good or d or neuter. U ) Solution ) U, increses. ) U decreses. U c. U, d) U is d nd is good since s increses U decreses nd s increses U is good nd is d since s increses U increses nd s increses U c) U,, oth nd re goods d) U is neuter nd is good since hs no effect on U 7) Suppose consumer preferences etween two goods tht cn e represented the utilit function U=AU A where A nd re positive constnts. For this utilit function U A nd U Show tht the RS is incresing Solution Since oth U nd U re positive, indifference curves will e negtivel sloped. This mens tht s increses long n indifference curve must decrese. We know tht A RS U / U A / / incresing nd decresing,. As we move long the indifference curve RS will increse. So we hve n incresing RS of for. / / 8) Consider the proction function whose eqution is given the formulq K L. ) Wht is the eqution of the Isoqunt corresponding to Q 0 Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge
) For the sme proction function, wht is the eqution of the Isoqunt corresponding to n ritrr level of output Q? Solution ) The Q 0 Isoqunt shows ll of the comintions of lour nd cpitl tht llow the firm to proce 0 units of output. To find the eqution of the 0 unit Isoqunt, we solve this eqution for K in terms of L K 00.This is the eqution of the 0 unit Isoqunt. L Q ) K L Eric purchses food (mesured ) nd clothing (mesured ) nd hs the utilit function U,. His mrginl utilities re U nd U. He hs monthl income of 800. The rice of food is 0, nd the price of clothing is 0.Find his optiml consumption undle. Solution I 0 0 800 () Since the optimum is interior the tngenc requires U U 0 0 Or simpl () Sustituting 0 800 0 0 nd 0 Thus, Eric s optiml sket involves the purchse of 0 units of food nd 0 units of clothing ech month. Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge
9) Given U where = is n positive constnt, =mrgrine utter. Show tht (the) utter nd mrgrine re sustitutes Solution U U m It follows tht RS U U RS U U RS =RS Therefore RS is constnt (not diminishing). Since the RS is lso, the consumer is lws willing to give up unit of utter to get nother unit of mrgrine. ut RS is the slope of the indifference curves the slope of the indifference curves will e constnt nd equl to - 0)rove tht the slope of consumer s udget line is. () () () Eqution () is stright line with - slope nd intercept ) Which of the following sttements contrvenes the lws of preference? Eplin I. if I choose either of them, I know I would e filled with sorrow. II. The re not the sme t ll, I cn t mke choice. III. I don t cre, just mke the choice for me Solution I. violtes the iom of comprison. It indictes inconsistenc. II. violtes the iom of comprison III. Epresses indifference ut does not violte n lws Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge
) The tle elow shows kwesi ensh s Totl Utilit (TU) schele for nn nd pple. Suppose tht the price of nn is GH nd tht of pple is GH 0.50; Kwesi ensh s income is GH 6 per time period nd ll is spent on nn nd pple. Quntit 5 6 7 8 TU of nn 6 0 5 60 66 70 7 TU of Apple 0 8 5 5 56 60. Stte ioms of the Crdinlist pproch nd hence the equi-mrginl principle. Find the optimum of Kwesi ensh c. Derive Kwesi ensh s demnd curve for nn if the price of nn flls from GH to GH 0.50 nd the price elsticit of demnd over the price rnges. d. If ensh consumed units of nn nd units of pple, will he e t n optimum? Eplin our nswer. Solution U Q TU TU A U U A U 5 6 7 8 6 0 5 60 66 70 7 0 8 5 5 56 60 6 0 8 6 0 9 8 7 6 5 6 () 0 8 6 0 8 6 () 0 8 A A U (p=0.50) 8 0 6 () 8. i. Rtionlit of the consumer ii. Disposle income of the consumer is given iii. Tstes nd preferences re given iv. There is perfect competition in the goods mrket. tht is, prices of goods re given v. mrginl utilit is positive Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge 5
vi. Utilit is mesurle U A A U U Z Z equi mrg inl U U. A U U A A A A 6 0. 50A 6 0.50 6 6 6 6 0.50 0. 50A when price flls The optimum of the consumer occurs t the point where he consumes units of nns nd 6 units of pples c) rice of nns 0.5 Demnd curve for nns Q p Ed Q Ed Ed 0.50 0 6 Quntit of nns p Q.50.5 9.5 0.50 0.50 6 Thus, the Ed over the given price rnge is unitr. d) No, ecuse the consumption of those quntities does not full ehust his disposle income. Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge 6
) Suppose tht the price of red is GH 0 nd tht of utter is GH while income is GH 00.. Distinguish etween udget line nd udget spce.. Wht is the eqution of the udget line nd wht re the coordintes c. Wht is the grdient of the udget line? d. If the price of red reces one-qurter, wht will e the new coordintes of the udget line? Will the grdient of the udget line chnge or remin the sme nd wh? e. Descrie wht hppens to the coordintes nd grdient of the udget line if income increses 50% ) udget spce/set udget line ) 00=0 red + utter if no red is purchsed red =0 00= utter 00 utter= 50 Coordintes 0,50 If No utter is purchsed, utter =0 00 red 0 0 Coordintes 0,0 c) 00=0 red + utter utter=00-0 red 00 0 utter= red utter = 50-5 red The grdient is -5 utter u u 0 50 5 Or 5 red r r 0 0 0 d) 00=5 red + utter Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge 7
If red =0 00 utter 50 If utter =0 red 00 5 red.,50.,0 0 New coordintes utter = 00-5 red utter 00 5 red 5 The grdient will now de, Thus the grdient will chnge e to the chnge in price of red. e) 00=0red+ utter utter = 00-0 red utter = 5-5 red,5 0 5,0 The grdient remins sun chnged ut intercept chnges indicting odil shift of the udget line. ) ention the fctors tht rtionl consumer must consider in deciding how much of good or goods to u. The income of the consumer The price of the commodit The vlue of the commodit of the consumer 6) Wht informtion is depicted single indifference curve? It represents ll the possile comintions of goods which the consumer in question finds to the equll stisfing Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge 8
7). Define n inferior good nd norml good. Cn good e oth? Wh? An inferior good is defined s one of which n indivil purchses less when his or her income rises nd more when his or her income flls. A norml good is defined s one of which n indivil purchses more when his or her income rises nd less when his or her income flls. es, the sme good cn e oth most goods re norml up to some level of income eond which the ecome inferior. 8). Given the dt elow, compute the RS of movies for concert nd vice vers. Comintion A C D ovies per month Concerts per month 7 5 Comintion ovies per month Concerts per months RS mc RS cm A 7 - - 5 C D Compiled : enedict Afful Jr. 006 skiuk@hoo.com ge 9