ECONOMICS RIPOS Par I Friday 7 June 005 9 Paper Quaniaive Mehods in Economics his exam comprises four secions. Secions A and B are on Mahemaics; Secions C and D are on Saisics. You should do he appropriae number of quesions from each secion. he number of quesions o be aemped is given a he beginning of each secion. Answers from he Mahemaics and he Saisics Secions mus be ied up in separae bundles, wih he leer of he Secion wrien on each cover shee. his wrien exam carries 80% of he marks for Paper. Secion A carries 4% of he marks, Secion B carries 6% of he marks, Secion C carries 4% of he marks and Secion D carries 6% of he marks. SAIONERY REQUIREMENS SPECIAL REQUIREMENS 0 Page bookle Lis of saisical formulae Meric graph paper New Cambridge Elemenary Saisical ables Rough Work Pad Approved calculaors allowed ags You may no sar o read he quesions prined in he subsequen pages of his quesion paper unil insruced o do so by he invigilaor
SECION A MAHEMAICS Answer four quesions A consumer has he uiliy funcion ( x x ) x ( x + ), x, x 0. u, If p, p, and income I 0, wha are he opimal consumpion levels x and x? (Use he Lagrange mehod o answer). Suppose ha a firm, producing oupu Y from labour L and capial K according o he echnology Y L / ( K ) /, wishes o minimise is coss given a producion arge Y. he wage rae is w and he renal rae of capial is r. (a) Derive he condiional facor demands by using he Lagrange mehod. (b) Derive he long run oal cos funcion. (c) Derive he long run average cos funcion and marginal cos funcion. Consider he following equaion sysem: 5x x x x x + x x + x x 0 8 6 (a) Wrie he sysem in marix noaion. Is he marix of he coefficiens non-singular? (b) Solve he sysem by using Cramer s rule.
4 Solve he following inegrals: (a) / x dx, x 0 / + dx (c) ( ax + bx + c)dx (b) 4 x( x ) dy 5 Find he derivaives of: dx ln 4 8 x (a) y ( x ) + e (b) y ( x + ) / ( x ) / 6 Find he parial derivaives of: z f ( x, y) ( x + y )( x y / ) (URN OVER
4 SECION B - MAHEMAICS Answer one quesion A consumer has a uiliy funcion given by ( x, y) xy, prices p x and p y respecively. U an income I and faces (a) Wrie down he consumer maximizaion problem by using he Lagrange mehod. Find he demand funcions for he goods x and y as funcions of prices and income. (b) Is he demand funcion of good x homogeneous of degree zero in all prices and income? Wha abou he demand of good y? Explain your answers. (c) Wha quaniies will he consumer choose in he case when p 4, p, and I 60? x y (d) Assume now ha he price of good y increases o 4. Wha are he quaniies now chosen by he consumer? (e) Calculae he income and he subsiuion effecs.
5 An economy is described by he following equaions: Y C C d I I τy + cy ( τ ) Y br G G M S m M ay dr D 0 0 d Where Y is aggregae oupu, C is consumpion, I invesmen, axes, G governmen spending, M money supply, X money demand and r he ineres rae. S D (a) Deermine he equaion of he IS curve. (b) Deermine he equaion of he LM curve. (c) Compue he equilibrium of his economy and give a graphical analysis of i. (d) Wrie down he economy in marix form and check your previous answer. (e) Suppose ha he governmen decides o adop an expansionary fiscal policy by increasing G, and, in order o avoid he crowding-ou effec increases M S o keep he ineres rae consan. By how much would income go up? How would your answer change if an income ax is used o finance he increase in G? (URN OVER
6 SECION C SAISICS Answer four quesions (a) he following se of scores is obained on a saisics es: 8 6 8 6 48 48 48 5 he examiner compues all of he descripive indices of cenral endency and variabiliy on hese daa, and hen discovers ha an error was made, and one of he 48 s is acually a 4. Which of he following indices will be changed from he original compuaion? (i) (ii) (iii) (iv) (v) Median Mode Range Sandard deviaion None of he above (b) he following daa shows he disribuion of scores in a calculus es aken by 50 sudens. Score Frequency 90-00 80-89 70-79 60-69 6 50-59 6 40-49 Compue he sample mean, sample median and sample variance of his daa.
7 he change in value (expressed as a percenage) of 0 residenial properies in Cambridge during 004 was found o be: (a) Calculae.0 5.6 -. 8.5 6. 0.6 4.4 -.9 9. 5. (i) (ii) Poin esimaes, and 90% confidence inervals, for he mean and variance of he change in value of all residenial properies in Cambridge during 004. (b) A he sar of he year, housing marke expers forecas ha annual house price inflaion in Cambridge during 004 would be 9%. Assess saisically (a he 5% level of significance) he hypohesis ha house price inflaion in Cambridge was lower han ha forecas by he expers. A researcher ineresed in he link beween child illieracy and public library provision obains he following daa for counries. Counry Library Provision (Libraries per 0000 children) Child Illieracy (% of populaion failing basic lieracy ess) 6. 8.6 0.5.0. 8. 4..0 5 5.0 8. 6.0 0.7 7.9.0 8. 6.9 9.5.6 0 5.5. 5.8.9 4.7 7.7 Using he daa given in he able above calculae a 95% confidence inerval for he populaion correlaion coefficien beween library provision and child illieracy. (URN OVER
8 4 An invesigaor has annual aggregae ime-series daa on per capia consumpion of perol Q (in gallons per year), he real price of perol P (an index number), and real per Y capia disposable income (an index number). he daa cover he 4 years from 960 o 00. A demand curve for per capia perol consumpion is specified in erms of logarihms of consumpion, prices and income as follows q γ + γ y γ p + ε 0 y p q log Q y Y p P γ 0 γ γ are consan coefficiens where ( ), log( ) and log( )., y and p and ε is a random error. he invesigaor has a srong prior belief ha γ, and reformulaes he demand curve in erms of perol expendiure as a share S of disposable income, where S PQ / Y. wo specificaions of he regression equaion are considered, one for he log share s log on he log price p log( P ), and he oher for he share, wihou any ( ) S ransformaion, on he unransformed price P : y S Specificaion (A) : Specificaion (B) : s α + β + ε p S a + bp + η (a) Show how specificaion (A) can be obained from he original demand curve by imposing he invesigaor s belief ha γ. (b) Find an expression for he price elasiciy of demand for each specificaion. y
9 5 UK oal employmen: 000 Quarer o 00 Quarer 4 oal Employmen Quarer (housands) Quarer oal Employmen (housands) 000Q 746 00Q 7765 000Q 7440 00Q 7850 000Q 757 00Q 7846 000Q4 7497 00Q4 8000 00Q 7604 00Q 8049 00Q 766 00Q 8 00Q 7670 00Q 80 00Q4 775 00Q4 85 Source: Economic rends Annual Supplemen 004 An economis decomposes he ime series daa in he able above according o he following addiive model: D + C + S + R Where D refers o he daa series, o he rend componen, C o he cyclical componen, S o he seasonal facor, and R o he random residual. Using he mehod of moving average, calculae he four seasonal facors, S, found by he economis. 6 (a) he probabiliy disribuion funcion of a random variable x is P ( x) x / k, x 0,,,,4,5 oherwise P( x) 0 (i) Find k. (ii) Calculae he mean and variance of x. (b) Exacly wo axi companies operae in Small own. he Red Company has red axis, and he Orange Company has orange axis. 85% of he axis are red and he oher 5% are orange. A axi was involved in a hi-and-run acciden a nigh. A winess idenified he axi as orange. Careful ess were done o ascerain peoples abiliy o disinguish beween red and orange axis a nigh. he ess showed ha people were able o idenify he colour correcly 80% of he ime, bu hey were wrong 0% of he ime. Wha is he probabiliy ha he axi involved in he acciden was indeed an orange cab? (URN OVER
0 SECION D SAISICS Answer one quesion (a) Describe he mehod of ordinary leas squares esimaion, making clear he condiions under which is applicaion is appropriae. (b) An economis ineresed in he relaionship beween he price of overnigh accommodaion in a seaside own and he proximiy of ha accommodaion o he beach collecs he following daa for hoels: Hoel Disance o neares beach (Km) Price of sandard room ( ) 0. 99 0.9 95. 8 4. 65 5.4 70 6. 60 7.9 65 8 5.0 59 9 5. 5 0 6. 48 6.8 6 8.0 65 Using hese daa, deermine he parameers of he following regression model: Y α + βx + ε where Y is he price of hoel accommodaion in pounds X is he disance of he hoel from he neares beach (c) Using your resuls assess he claim ha he proximiy of a hoel o a beach is a significan facor in he price charged by hoels for a room.
Monhly daa on prices and dividends are colleced for wo asses (A and B). A he A B beginning of monh he prices of asses A and B are P and P respecively. If an asse is held during monh - i pays a dividend a he beginning of monh, which is denoed by A B D and D respecively for asses A and B. he monhly percenage reurns on holding each asse from he beginning of monh - o he beginning of monh are denoed by R A and R. A B (a) Describe how he reurns R and R are calculaed. (b) Monhly reurns for asses A and B are calculaed for a period of 5 monhs. You are given he following informaion: B R A 0.57 A A ( R R ) 6. 98 B R 0.88 B B ( R R ). 847 A A ( R R ). 64 A A ( R R ) 0. 894 4 B B ( R R ) 4. 757 B B ( R R ) 79. 488 4 Calculae he sandard deviaion, he skewness, and he kurosis of reurns for boh asses, using he following formulae for he skewness and kurosis of a variable : X 4 / Skewness m / m Kurosis m m m where ( ) j j Give a shor descripion of wha he measures of skewness and kurosis represen. (c) If he reurns were normally disribued, wha values would you expec for he skewness and kurosis measures? Commen on he appropriaeness of he normaliy assumpion for he wo asses reurns. (d) es a he 5% level of significance wheher he sandard deviaion of he monhly reurn is higher for asse A han asse B. (e) Asse A has a higher average reurn han B and so is clearly a superior invesmen o asse B. Commen on his saemen in he ligh of your answer o par (d). X X END OF PAPER