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1 Renshw: Mths for Econoics nswers to dditionl exercises Exercise.. Given: nd B 5 Find: () + B + B 7 8 (b) (c) (d) (e) B B B + B T B (where 8 B 6 B 6 8 B + B T denotes the trnspose of ) T 8 B 5 (f) (g) B T T B 8 5 B B 87. Given: nd B [ b b b ] Find B nd B Oxford University Press, 5. ll rights reserved.

2 B b b b b b b b b b Renshw: Mths for Econoics nswers to dditionl exercises B b + b + b (Note, sclr). Given: [ ] nd b B b b Find (if they exist) B, T B, B, B T, T B T, nd T T B. In ech cse, explin why the trix product does (or does not) exist. B b + b + b B does not exist becuse nuber of eleents in row of T () does not equl nuber of eleents in colun of B (). (Or, equivlently, becuse nuber of coluns in T () does not equl nuber of rows in B () T B b b b b b b b b b T B does not exist becuse nuber of eleents in row of B T () does not equl nuber of eleents in colun of (). (Or, equivlently, becuse nuber of coluns in B T () does not equl nuber of rows in () b b b T B (Note, se s B) T T b b b b b b T B b + b + b (Note, se s B). Given: nd b B b b b b b Find (if they exist) B, T B, B, B T, T B T, nd T T B. Oxford University Press, 5. ll rights reserved.

3 Renshw: Mths for Econoics nswers to dditionl exercises B does not exist becuse nuber of eleents in row of () does not equl nuber of eleents in colun of B () b b T B b b b b b + b + b b + b + b b b b b b b Exercise.. Find the inverses (if they exist) of: B C D The inverses re: B C - does not exist, becuse ech eleent of row equls ties the corresponding eleent of row. (Eqully, ech eleent of colun equls ties the corresponding eleent of colun.) - D See section 9. of the book.. Using your nswers fro question bove, solve where possible the following sets of siultneous equtions. If you find tht no solution is possible, explin why. () x + y 5 x + y s explined in section 9. of the book, we cn write this pir of siultneous equtions in trix for s v b where x v y nd 5 b Oxford University Press, 5. ll rights reserved.

4 Renshw: Mths for Econoics nswers to dditionl exercises Therefore v - b. Fro () bove, we hve v b... tht is, x.8, y , so (You cn check tht these vlues re correct by substituting the bck into the given siultneous equtions.) (b) y 7 x + y 6 Following the se steps s in () bove, nd using B, fro question (), we find x 5 6, y.75. (Don t forget to check by substitution). (c) x + y x+ y In this cse the relevnt trix in question () is C, which hs no inverse. Therefore the siultneous equtions in this cse do not hve solution. (This is becuse they re not independent of one nother for exple, the second eqution is ties the first.) (d) x + y x + y - Following the se steps s in () nd (b) bove, nd using D, fro question (), we find x, y. (Don t forget to check by substitution).. Find the deterinnt, ll inors nd cofctors, nd the inverse of ech of the following trices: Oxford University Press, 5. ll rights reserved.

5 Renshw: Mths for Econoics 5 nswers to dditionl exercises () Deterinnt Mtrix of inors: Mtrix of cofctors ( signed inors): Inverse trix: (b) Deterinnt: Mtrix of inors: Mtrix of cofctors (signed inors) Inverse trix: (c) 5 Oxford University Press, 5. ll rights reserved.

6 Renshw: Mths for Econoics 6 nswers to dditionl exercises Deterinnt: Minors Cofctors ( signed inors): Inverse trix: (d) 5 Deterinnt: Minors: Cofctors: inverse trix: Exercise.. Use Crer's rule to find x in the eqution systes below. () y z x + y + z 5 x + y Oxford University Press, 5. ll rights reserved.

7 Renshw: Mths for Econoics 7 nswers to dditionl exercises s explined in section 9. of the book, we cn write this set of siultneous equtions in trix for s v b where 5 x v y nd z b Crer s rule tells us tht we cn solve for ech vrible s follows. (i) To solve for x, we for new trix,, obtined by replcing the first colun of with the vector b. Thus The solution vlue of the vrible x is then given by tht is, the rtio of the deterinnts of the two trices. In this exple, the two deterinnts re nd, so x 7 (ii) To solve for y, we for new trix,, obtined by replcing the second colun of with the vector b. Thus 5 The solution vlue of the vrible y is then given by tht is, the rtio of the deterinnts of the two trices. In this exple, the two deterinnts re nd 7, so y 7 9 (iii) To solve for z, we for new trix,, obtined by replcing the third colun of with the vector b. Thus 5 Oxford University Press, 5. ll rights reserved.

8 Renshw: Mths for Econoics 8 nswers to dditionl exercises The solution vlue of the vrible y is then given by tht is, the rtio of the deterinnts of the two trices. In this exple, the two deterinnts re nd, so z 7. So our solutions re: x 7, y 9, z 7 (You cn check tht these vlues re correct by substituting the bck into the given siultneous equtions.) Oxford University Press, 5. ll rights reserved.

9 (b) x z x + y + z Renshw: Mths for Econoics 9 nswers to dditionl exercises 5x + y Using the ethod of () bove: (i) Solution for x. We hve 5.5 nd So x (ii) Solution for y. We hve 5.5 nd So 5.5 y.75 (iii) Solution for z. We hve 5.5 nd 5 So z 5 So our solutions re: x, y.75, z 5 (You cn check tht these vlues re correct by substituting the bck into the given siultneous equtions.) Oxford University Press, 5. ll rights reserved.

10 Renshw: Mths for Econoics nswers to dditionl exercises. Solve the following syste of liner equtions using () trix inversion, nd (b) Crer's rule..5.5 x y z () Solution by trix inversion. We cn write this set of siultneous equtions in trix for s v b where.5.5 x v y nd z b The solution is v - b. We find - s So v b So our solutions re: x, y, z (You cn check tht these vlues re correct by substituting the bck into the given siultneous equtions.) Oxford University Press, 5. ll rights reserved.

11 Renshw: Mths for Econoics nswers to dditionl exercises (b) Solution by Crer s rule. Using the ethod of question bove: (i) Solution for x. We hve nd 8 So x 8 (ii) Solution for y. We hve.5.5 nd.5 So y (iii) Solution for z. We hve nd.5 So z So our solutions re: x, y, z, s before.. In the econoy of the Kingdo of Mononi, households spend their incoes on doesticlly produced goods, C d, nd iported goods, M. This spending is observed to follow the reltionship C d + M Y () where is preter nd Y households incoe. Iports re observed to be relted to household consuption by the reltionship M C d (where is preter) () Oxford University Press, 5. ll rights reserved.

12 Renshw: Mths for Econoics nswers to dditionl exercises The equilibriu condition for this econoy is tht ggregte output, Y, ust equl the dend for output for doestic consuption, C d, nd doestic investent, I, plus dend for exports by foreigners, X. Therefore the equilibriu condition is Y Cd + I + X () (where I nd X re ssued to be exogenous). Note tht household incoe is necessrily equl to output becuse households ern their incoes (wges, slries nd profits) by producing output. () Show tht, fter suitble rerrngeent, the set of siultneous equtions to bove cn be written s x b, where is trix contining the preters of the odel ( nd ) x is vector contining the endogenous vribles (Y, C d nd M) nd b is vector contining the exogenous vribles (I nd X). We cn present the equtions of this odel in tble for s: Y -C d I + X (fro eqution ) -C d +M (fro eqution ) Y + -C d (fro equtions & ) This helps us to see tht the equtions cn be expressed in trix for. For exple, the coefficients of Y in colun re, nd. This + gives us the first colun of the coefficient trix. Siilrly the coefficients of C d in the second colun re,, nd. This gives us the second colun of the coefficient trix. fter soe fiddling round, we find tht the equtions cn be expressed in trix for s follows: Y I + X C d M + or in trix nottion, x b. The key point here is tht we hve now the endogenous vribles Y, C d nd M on the left hnd side (the vector x). These endogenous vribles re deterined by the vlues of the exogenous vribles I nd X (the vector b), together with the vlues of the preters (the trix.) Oxford University Press, 5. ll rights reserved.

13 Renshw: Mths for Econoics nswers to dditionl exercises (b) By clculting -, find the equilibriu vlues of the endogenous vribles in ters of the vlues of the preters nd exogenous vribles. Given x b, we wnt to find x b. To find -, we follow the procedure explined in section 9. of the book. The trix of trnsposed cofctors, D', is D' + ( ) The deterinnt of is ( ) ( ) So + D' + ( ) + + Since x b, we find the solution vlues for the vector of endogenous vribles, x, s x b, tht is: Y I + X C + d + M ( ) + + Multiplying out, we get + d + + Y ( I + X ) C ( I + X ) M ( I + X ) (c) Using your nswer to (b), find the equilibriu vlues of the endogenous vribles when.75,.5, nd I + X. Check your nswer by solving the odel by siultneous eqution ethods Fro (b) bove, Y ( I + X ) () 5.75 d +.5 C ( I + X) () 5 Oxford University Press, 5. ll rights reserved.

14 + 8 Renshw: Mths for Econoics nswers to dditionl exercises M ( I + X ) () 75 (d) Use your nswer to (b) to clculte the effect on Y, C, nd M of sll increse in ech of the exogenous vribles, I or X. Our trix solution in (b) gives us the levels of Y, C d, nd M, s functions of the levels of I + X. However, we cn lso clculte the effects of sll chnges, siply by replcing I +X with di + dx, nd so on. Then we get dy di + dx dc + d + dm ( ) d + + dy ( di + dx ) dc ( di + dx ) dm ( di + dx ) (If you distrust the vlidity of this, try clculting Y, C d, nd M when I I, X X. Then clculte Y, C d, nd M when I I + di, X X + dx (where of course di nd dx denote sll chnges). Then subtrct the first solution vlues fro the second solution vlues, to rrive t the chnges in Y, C d, nd M. You will find tht you hve the se result s bove.) (e) Wht is the reltionship between exports nd iports in this odel? Exports re n exogenous vrible tht is, they re deterined by fctors lying outside of this odel. There is wek cusl link running fro exports to iports, through the reduced for eqution M ( I + X ), + but it is only the su of investent plus exports tht ppers on the right hnd side, so chnge in exports need hve no effect on iports if there is copensting chnge in investent t the se tie. Oxford University Press, 5. ll rights reserved.