Mathematics Textbooks for Science and Engineering Shapoor Vali Principles of Mathematical Economics II Solutions Manual, Supplementary Materials and Supplementary Exercises Mathematics Textbooks for Science and Engineering Volume More information about this series at http://www.springer.com/series/10785 Shapoor Vali Principles of Mathematical Economics II Solutions Manual, Supplementary Materials and Supplementary Exercises Shapoor Vali Department of Economics Fordham University New York, NY USA Mathematics Textbooks for Science and Engineering ISBN 978-94-6239-087-4 ISBN 978-94-6239-088-1 DOI 10.2991/978-94-6239-088-1 (eBook) Library of Congress Control Number: 2013951796 Published by Atlantis Press, Paris, France www.atlantis-press.com © Atlantis Press and the authors 2015 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher Printed on acid-free paper Series Information Textbooks in the series ‘Mathematics Textbooks for Science and Engineering’ will be aimed at the broad mathematics, science and engineering undergraduate and graduate levels, covering all areas of applied and applicable mathematics, interpreted in the broadest sense Series editor Charles K Chui Stanford University, Stanford, CA, USA Atlantis Press 8, square des Bouleaux 75019 Paris, France For more information on this series and our other book series, please visit our website www.atlantis-press.com Editorial Recent years have witnessed an extraordinarily rapid advance in the direction of information technology within the scientific, engineering, and other disciplines, in which mathematics play a crucial role To meet such urgent demands, effective mathematical models as well as innovative mathematical theory, methods, and algorithms must be developed for data information understanding and visualization The revolution of the data information explosion as mentioned above demands early mathematical training with emphasis on data manipulation at the college level and beyond The Atlantis book series, “Mathematics Textbooks for Science and Engineering (MTSE),” is founded to meet the needs of such mathematics textbooks that can be used for both classroom teaching and self-study For the benefit of students and readers from the interdisciplinary areas of mathematics, computer science, physical and biological sciences, various engineering specialties, and social sciences, contributing authors are requested to keep in mind that the writings for the MTSE book series should be elementary and relatively easy to read, with sufficient examples and exercises We welcome submission of such book manuscripts from all who agree with us on this point of view This fourth volume consists of the solution manual and supplementary materials of the previous volume: “Principles of Mathematical Economics,” by the same author It is divided into 13 chapters, covering such topics as: Market equilibrium model, Rates of change and the derivative, Optimal level of output and long run price, Nonlinear models, Economics Dynamics, and Mathematics of interest rates and finance It is an important companion of the author’s previous volume Charles K Chui vii Preface It was part of my original plan in writing my book, Principles of Mathematical Economics, Volume III of this MTSE book series, to provide answers to some of the problems in the exercise sections of the book However, when the text grew to about 500 pages and the number of problems to over 600, I decided that simply providing numerical answers, without going through the steps of formulating and solving the problems, would not be very helpful to students or general readers My own experience from teaching quantitative courses convinced me that providing mere answers was even harmful: in many cases chasing “the answer” leads some students to set problems up incorrectly, but such that its solution is the same as the given answer This, on some occasions, as I am sure many of my academic colleagues have experienced, leads to debate over the “proper” grade for a homework or exam problem, and periodic sermon, undoubtedly boring, about the impossibility of consistently reaching correct conclusions through wrong reasoning I believe that the old saying “the correct formulation of a problem is 50 % of the solution” is an expression of the accumulated wisdom of scientific inquiries For the above reasons, I decided to prepare this manual which gives a full-fledged formulation and solution to each of the problems in the text In some cases I go beyond setting up and solving a problem and include additional materials relevant to the subject of the chapter This book naturally accompanies Principles of Mathematical Economics, but it can also be used independently from the text The book can be treated as a standalone collection of solved problems in different areas of mathematical economics and as additional sets of exercises, over 500, that can be used to sharpen students’ skill and depth of understandings of many economic topics Therefore, students can benefit from this manual even if the course they take in quantitative economics uses a different textbook The manual is organized as follows: exercises from each chapter of the text are listed, followed by their solutions Where a problem can be solved using different methods, sketches of alternative methods are also provided If the solution references an equation in the text, the equation number is used But if in the process of solving a problem a new equation is derived its number is tagged by “SM” ix x Preface (for “Solution Manual”) to distinguish it from the text equation For example, the first equation derived in Chap is labeled (4.1 SM) If a modified version of a text formula is introduced, it is tagged “MOD” to indicate the modification Finally, each set of solutions to chapter exercises is followed by an additional set of unsolved problems under the heading “Supplementary Exercises” In the process of preparing this manual, I discovered a number of typographical errors in the text These errors are all mine and escaped me in the process of proofreading I have corrected the errors if they appear in the Exercise sections of the chapters, which are repeated here in the manual I hope there are no errors in this manual, but given the sheer volume of numbers and mathematical expressions, it is still likely that you encounter some errors In case you do, I would appreciate it if you would let me know by sending an email to vali@fordham.edu As it always the case, writing a book of this nature requires help from many individuals In particular, I would like to thank the publishers Dr Keith Jones and Dr Zeger Karssen, as well as the book series editor, Professor Charles Chui, of Atlantis Press, for their support in the publication of this book I am grateful to Dr Michael Malenbaum, Ellen Fishbein, and Behrang Vali for their help in editing and proofreading parts of the manuscript Some of my students also helped me by checking some of the solutions I thank them all, especially Tyler Shegerian Above all, I am again indebted to my wife Firoozeh for her support and encouragement Spring 2014 Shapoor Vali Contents Household Expenditure Variables, A Short Taxonomy 17 Sets and Functions 21 Market Equilibrium Model 33 Rates of Change and the Derivative 57 Optimal Level of Output and Long Run Price 81 Nonlinear Models 123 Additional Topics in Perfect and Imperfect Competition 151 Logarithmic and Exponential Functions 177 10 Production Function, Least-Cost Combination of Resources, and Profit Maximizing Level of Output 205 11 Economics Dynamics 227 12 Mathematics of Interest Rates and Finance 239 13 Matrices and Their Applications 263 xi 276 13 Matrices and Their Applications with solution EN = Coe−1 EX ⎡ ⎤ ⎡ Y ⎢C ⎥ ⎢−0.80 ⎢ ⎥=⎢ ⎣ T ⎦ ⎣−0.10 I −0.15 −1 0 ⎡ ⎤ ⎡ 7.69231 Y ⎢C ⎥ ⎢5.53846 ⎢ ⎥=⎢ ⎣ T ⎦ ⎣0.76923 1.15385 I 0.8 ⎤−1 ⎡ ⎤ 60 −1 ⎢40⎥ 0⎥ ⎥ ⎢ ⎥ ⎦ ⎣10⎦ 38 −6.15385 −5.23077 0.38462 −0.92308 7.69231 6.53846 0.76923 1.15385 ⎤ ⎤⎡ ⎤ ⎡ 1000 60 7.69231 ⎥ ⎢ ⎥ ⎢ 5.53846⎥ ⎥ ⎢40⎥ = ⎢ 752 ⎥ 0.76923⎦ ⎣10⎦ ⎣ 110 ⎦ 188 38 2.15385 Note that the absolute value of the determinant of inverse of the coefficient matrix is the model’s autonomous expenditure multiplier, that is m is equal to abs(|Coe−1 |) = 7.69 Alternatively, we can treat Y D as a separate variable and write the model as ⎧ ⎪ ⎪Y = C + I + 110 + 40 − 90 ⎪ ⎪ ⎪ ⎪ ⎨Y D = Y − T → C = 40 + 0.80Y D → ⎪ ⎪ ⎪ T = 10 + 0.10Y → ⎪ ⎪ ⎪ ⎩ I = 38 + 0.15Y → → Y − C − I = 60 −Y + Y D + T = −0.80Y D + C = 40 −0.10Y + T = 10 −0.15Y + I = 38 This model in matrix form is ⎤⎡ ⎤ ⎡ ⎤ ⎡ Y 60 −1 −1 ⎥ ⎢Y D ⎥ ⎢ ⎥ ⎢ −1 1 ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ −0.80 0 ⎥ ⎥ ⎢ C ⎥ = ⎢40⎥ ⎢ ⎦ ⎣−0.10 0 ⎣ T ⎦ ⎣10⎦ 38 −0.15 0 I with solution EN = Coe−1 EX ⎡ ⎤ ⎡ 7.69231 6.15385 Y ⎢Y D ⎥ ⎢6.92308 6.53846 ⎢ ⎥ ⎢ ⎢ C ⎥ = ⎢5.53846 5.23077 ⎢ ⎥ ⎢ ⎣ T ⎦ ⎣0.76923 0.61538 1.15385 0.92308 I #20 The model in Example is C = 60 + 0.75Y D 7.69231 6.92308 6.53846 0.76923 1.15385 −6.15385 −6.53846 −5.23077 0.38461 −0.92308 ⎤⎡ ⎤ ⎡ ⎤ 1000 60 7.69231 ⎢ ⎥ ⎢ ⎥ 6.92308⎥ ⎥ ⎢ ⎥ ⎢ 890 ⎥ ⎢ ⎥ ⎢ ⎥ 5.53846⎥ ⎢40⎥ = ⎢ 752 ⎥ ⎥ 0.76923⎦ ⎣10⎦ ⎣ 110 ⎦ 188 38 2.15385 13 Matrices and Their Applications 277 I = 30 + 0.1Y − 7r0 T = 15 + 0.1Y Originally, the government expenditure G is 125 The country’s parliament decides to cut the budget by 30 % to 87.5 and change the marginal tax rate such that the government has a balanced budget The question is how they arrived at the marginal tax rate of 14.15 % If the budget is set at 87.5 and the country must have a balanced budget, then the volume of tax must be 87.5 too In that case the consumption function would change to C = 60 + 0.75(Y − T ) = 60 + 0.75(Y − 87.5) = 0.75Y − 5.625 After plugging for r0 in the investment function, we have I = 30 + 0.1Y − ∗ = 0.1Y − Substituting for C and I in the national income equation Y = C + I + G , we have Y = 0.75Y − 5.625 + 0.1Y − + 87.5 Y − 0.85Y = 76.875 −→ Y = −→ Y = 0.85Y + 76.875 78.875 = 512.5 0.15 With the equilibrium value of Y determined, we can use the tax function expressed as T = 15 + tY and find t, the marginal tax rate 87.5 = 15 + tY → 87.5 − 15 = 515.5t, leading to t = 72.5 = 0.1415 or 14.15 % 512.5 #21 After substituting 820.8, −37.5, and 3323.3 for G , N X , and M0 in the equations we can write the system as ⎧ Y = C + I + 820.8 − 37.5 → Y − C − I = 783.3 ⎪ ⎪ ⎪ ⎪ ⎪ C = −228.78 + 0.832(Y − T ) → 0.832Y − C − 0.832T = 228.78 ⎪ ⎪ ⎪ ⎨ I = −41.951 + 0.255(Y − T ) − 11.511r → ⎪0.255Y − I − 11.551r − 0.255T = 41.951 ⎪ ⎪ ⎪ ⎪ ⎪ r = −0.178 + 0.010Y − 0.012 ∗ 3323.3 → 0.010Y − r = 40.0576 ⎪ ⎪ ⎩ T = 135 + 0.15Y → −0.15Y + T = 135 278 13 Matrices and Their Applications (a) This system in matrix form is ⎡ −1 ⎢ 0.832 −1 ⎢ Coe EN = E X → ⎢ ⎢ 0.255 ⎣ 0.010 −0.15 ⎤ ⎤⎡ ⎤ ⎡ 783.3 Y −1 0 ⎥ ⎢ ⎥ ⎢ 0 −0.832⎥ ⎥ ⎢C ⎥ ⎢ 228.78 ⎥ ⎢ ⎥ ⎢ ⎥ −1 −11.551 −0.255⎥ ⎢ I ⎥ = ⎢ 41.951 ⎥ ⎥ −1 ⎦ ⎣ r ⎦ ⎣40.0576⎦ 135 T 0 (b) The system’s solution is obtained by EN = Coe−1 EX, where the inverse of the matrix of coefficients is ⎡ 5.22029651 ⎢3.69179369 ⎢ Coe−1 = ⎢ ⎢0.52850282 ⎣0.05220297 0.78304448 −5.22029651 −4.69179369 −0.52850282 −0.05220297 −0.78304448 −5.22029650 60.2996450 −3.69179369 42.6439090 −1.52850282 17.6557361 −0.05220297 −0.3970035 −0.78304448 9.0449468 ⎤ −5.67446231 −4.84497975⎥ ⎥ −0.82948256⎥ ⎥ −0.05674462⎦ 0.14883065 and the vector of equilibrium values of the endogenous variables is ⎤ ⎡ ⎤ ⎡ 4325.169 Y ⎢C ⎥ ⎢2717.659⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ EN = ⎢ ⎢ I ⎥ = ⎢ 824.209 ⎥ ⎣ r ⎦ ⎣ 3.194 ⎦ 783.775 T (c) The absolute value of the determinant of Coe−1 , which is 5.22, is the model’s autonomous spending multiplier, m The tax multiplier, m , is −bm, where b = 0.832 Hence m = −0.832 ∗ 5.22 = −4.34 The interest rate multiplier, m , is −km, where k = −11.511 Hence m = −11.511 ∗ 5.22 = −60.1 #22 Expanding the money supply to $3,500 billion changes E X and leads to a new vector of equilibrium values for the endogenous variables E N ⎡ ⎤ 783.300 ⎢228.780⎥ ⎢ ⎥ ⎥ EX = ⎢ ⎢ 41.951 ⎥ ⎣ 42.178 ⎦ 135.000 −→ ⎤ ⎡ ⎤ ⎡ 4453.028 Y ⎢C ⎥ ⎢2808.082⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ EN = ⎢ ⎢ I ⎥ = ⎢ 861.647 ⎥ ⎣ r ⎦ ⎣ 2.352 ⎦ 803.954 T It is clear that expansion of the money supply is the source of decline in interest rate form 3.194 to 2.352 % The rest of the endogenous variables show increases compare to exercise # 21, indicating that declining interest rate had a stimulating effect on the economy 13 Matrices and Their Applications 279 #23 An additional $40 billion of government expenditure along with the expansion of the money supply, changes E X to a new vector, leading to a new vector of values for the endogenous variables, ⎡ ⎤ 823.300 ⎢228.780⎥ ⎢ ⎥ ⎥ EX = ⎢ ⎢ 41.951 ⎥ ⎣ 42.178 ⎦ 135.000 −→ ⎤ ⎡ ⎤ ⎡ 4661.840 Y ⎢C ⎥ ⎢2955.753⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ EN = ⎢ ⎢ I ⎥ = ⎢ 882.787 ⎥ ⎣ r ⎦ ⎣ 4.440 ⎦ 834.276 T The change in the interest rate from 2.352 to 4.449 %, after an about % increase in the government expenditure, is indicative of the crowding out #24 Change in the tax function changes both the matrix of coefficients Coe and the vector of exogenous values E X These changes lead to a new matrix formulation of the original system in exercise #21, ⎡ ⎢ 0.832 ⎢ Coe EN = EX → ⎢ ⎢ 0.255 ⎣ 0.010 −0.13 The inverse of Coe is ⎡ 5.8885879 ⎢4.2623954 ⎢ Coe−1 = ⎢ ⎢0.6261924 ⎣0.0588858 0.7655164 −1 −1 0 ⎤ ⎤⎡ ⎤ ⎡ −1 0 783.3 Y ⎥ ⎢ ⎥ ⎢ 0 −0.832⎥ ⎥ ⎢C ⎥ ⎢ 228.78 ⎥ ⎢ I ⎥ = ⎢ 41.951 ⎥ −1 −11.551 −0.255⎥ ⎥ ⎥⎢ ⎥ ⎢ −1 ⎦ ⎣ r ⎦ ⎣40.0576⎦ 100 T 0 −5.8885879 −5.2623954 −0.6261924 −0.0588858 −0.7655164 −5.8885879 −4.2623954 −1.6261924 −0.0588858 −0.7655164 68.019079 49.234930 18.784148 −0.319809 8.842480 ⎤ −6.40089507 −5.46522388⎥ ⎥ −0.93567118⎥ ⎥ −0.06400895⎦ 0.16788364 and the vector of equilibrium values of the endogenous variables is ⎤ ⎡ ⎤ ⎡ 5102.899 Y ⎢C ⎥ ⎢3381.703⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ EN = ⎢ ⎢ I ⎥ = ⎢ 937.897 ⎥ ⎣ r ⎦ ⎣ 10.917 ⎦ 763.376 T The impact of reducing the marginal tax rate by 13.3 % and the autonomous taxes by 26 % is evident by comparing this result with that of exercises #21, #22, and #23, presented in Table 13.1, especially a dramatic rise in the interest rate 280 13 Matrices and Their Applications Table 13.1 Results of problem # 24 EN Original #21 #21 After tax change Y C I r T 4325.169 2718.659 824.209 3.194 783.775 5102.899 3381.703 937.897 10.971 763.376 #22 After tax and M0 change #23 After tax, M0 and G change 5247.127 3486.100 977.727 10.293 782.126 5482.670 3656.596 1002.774 12.649 812.747 #25 (a) We create the matrix of technical coefficients by using Eq (13.16) The matrix of inter-sectoral transaction, T , the sum vector of size 4, S, and the vector of final demand, FD, are ⎤ ⎡ ⎡ ⎤ ⎤ ⎡ 69980 82550 1109 246867 14595 ⎢ 897154 ⎥ ⎢1⎥ ⎢ 9418 143422 653526 204375 ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ T =⎢ ⎣100120 322107 1834966 868271 ⎦ S = ⎣1⎦ FD = ⎣1845577⎦ 850000 65865 213120 861584 4863250 We calculate the vector of sectoral outputs as, ⎡ ⎤ 415101 ⎢1907895⎥ ⎥ Q = T S + FD = ⎢ ⎣4971041⎦ 6853819 We form D Q, a diagonal by matrix with the sectoral outputs on the main diagonal, zero elsewhere, ⎡ ⎤ 415101 0 ⎢ ⎥ 1907895 0 ⎥ DQ = ⎢ ⎣ ⎦ 4971041 0 0 6853819 Then the matrix of coefficients A is calculated as ⎡ 0.19886726 0.0005812689 ⎢ 0.02268845 0.0751729000 A = T D Q −1 = ⎢ ⎣0.24119431 0.1688284733 0.15867223 0.1117042605 0.04966103 0.13146663 0.36913113 0.17332064 ⎤ 0.00212947 0.02981914⎥ ⎥ 0.12668426⎦ 0.70956791 13 Matrices and Their Applications 281 Table 13.2 Results of part(d) problem # 25 Sector Agriculture Const and mining Ag C and M Manu Ser GDP % Change 417833 1908286 4972722 6856465 14155306 0.053 415975.9 1939503.9 4983914.4 6874136.7 14213531 0.46 Manufacturing Services 422193.8 1926320.6 5081937.9 6930960.5 14361413 1.51 416854.3 1914706.2 4994721.3 6959328.4 14285610 0.97 (b) After creating I − A and a new vector of final demand FD , by changing the first element of FD from 69980 to 1.03 ∗ 69,980 = 72079.4, we find the new sectoral gross outputs, Q , as ⎡ ⎤ 417833 ⎢1908286⎥ ⎥ Q = (I − A)−1 FD = ⎢ ⎣4972722⎦ 6856465 (c) The sum of sectoral outputs is the total output, GDP, of the economy The original volume of output in part (a) is $14,147,856, about 14.15 trillion dollars The total output after % increase in agricultural sector is $14,155,306, about 14.16 trillion dollars A % increase in the U.S agricultural activities contribute 0.053 % to the overall national output (d) Every sector’s final demand are increased by % and then the sectoral outputs and the total national output, GDP are estimated The result is organized in the following table Under each sector the result of % increase in the final demand of that sector is listed The “% Change” in the last row is the percentage change in the GDP from the original model in part (a) As the table indicates, the biggest impact of increase in final demand belongs to the Manufacturing sector, followed by Services, Construction and Mining, and Agriculture (Table 13.2) Following is a short R snippet to problem #25 # We enter elements of inter-sectoral transaction matrix T by row T = matrix(c(82550,1109,246867,14595,9418,143422, 53526,204375, 100120,322107,1834966,868271,65865,213120 861584,4863250),4,b=T) # We enter element of column vector of final demand FD FD = matrix(c(69980, 897154, 1845577, 850000)) #Next we make the column sum vector S S = matrix(c(1),4) 282 13 Matrices and Their Applications #We now calculate the vector of sectoral outputs Q as Q = T%*%S +FD # Then print Q Q [,1] [1,] 415101 [2,] 1907895 [3,] 4971041 [4,] 6853819 # We add the outputs of sectors to get the original GDP GDP= t(S)%*%Q #Where t(S) is the transpose of the sum vector S GDP [,1] [1,] 14147856 # To form the diagonal matrix DQ, we first make a by # null matrix DQ = matrix(c(0),4,4) #Note here that ‘‘c(0),4,4’’ tells R that we want a by # matrix all #Next we replace the diagonal elements of DQ with elements of Q for (i in 1:4) { DQ[i,i] = Q[i] } #which is equivalent to DQ[1,1]=Q[1], DQ[2,2]=Q[2],DQ[3,3]=Q[3], # and DQ[4,4]= Q[4] #We can now calculate A A = T%*% solve(DQ) # ‘‘solve(DQ)’’ finds the inverse of DQ # Next we create a by unit matrix I Here we use ‘‘diag(n)’’ # command that makes a unit matrix of size n I = diag(4) IA = I-A # We now make (I-A) inverse IAinv = solve(IA) # To make sure that we have correctly created the matrix of # technical coefficients A, we form (I-A)ˆ(-1)FD, which must give # us the original vector of sectoral outputs Q 13 Matrices and Their Applications 283 IAinv%*%FD [,1] [1,] 415101 [2,] 1907895 [3,] 4971041 [4,] 6853819 # # # # # It does Next we go around a loop, increase the final demand for each sector by 3\,% and call this new vector FDn We use this vector and find the new vector of sectoral outputs Qn, the new total national output GDPn and calculate the rate of growth RG with respect to the original GDP for (i in 1:4) { + FDn=FD + FDn[i]=FD[i]*1.03 + Qn = IAinv%*%FDn + GDPn = t(S)%*%Qn + RG = ((GDPn-GDP)/GDP)*100 + print(Qn) + print(GDPn) + print(RG) } # print(Qn), print(GDPn) and print(RG) prints new sectoral # outputs, new GDP, and rates of growth #26 After increasing the final demand vector by % (a) The new sectoral outputs are $427,554, $1,965,132, $5,120,172 and $7,059,434 (b) The national output is $14,572,292, which is exactly % more than the original value of the GDP in problem # 25 This result is a reaffirmation of the linear homogeneity of the Input-Output models Chapter 13 Supplementary Exercises Given matrices A, B, and C, A= (a) (b) (c) (d) Find Find Find Find −10 −2 20 −10 ⎡ ⎤ −3 ⎢2 ⎥ ⎥ B=⎢ ⎣ −10⎦ −1 A , B , and C AB and (AB)C B A and show that it is equal to (AB) A A and A A Are they the same? C= −3 12 284 13 Matrices and Their Applications Find the inverse of the following matrices by method of adjoint ⎡ A= −3 ⎤ B = ⎣2 ⎦ −2 Find the determinant of the following matrices If the matrix is invertible, find its inverse by using elementary row operations ⎡ ⎤ A = ⎣2 4⎦ ⎡ ⎤ B = ⎣2 −3⎦ Find the inverse of the following matrices by using elementary row operations ⎡ ⎤ A = ⎣ 3⎦ ⎡ ⎢0 B=⎢ ⎣0 −2 −1 0 ⎤ 2⎥ ⎥ 6⎦ −4 Solve the following system of equations with unknowns x1 , x2 , and x3 2x1 + 5x2 − 4x3 = 20 5x1 − 2x2 + 4x3 = 15 3x2 − 2x3 = Write the following system in matrix form and solve it x1 − 2x2 + 2x3 + 2x4 = 30 2x1 − 4x3 + 5x4 = 10 x2 − 2x3 = x1 + 2x3 + 3x4 = 20 Solve the following system of equations with unknowns by hand x1 + x3 = 25 x2 + x3 = 18 x1 + x2 = 23 x3 + x4 = 15 Write the system in the previous problem in matrix form and solve it 13 Matrices and Their Applications 285 Courses Students 82 89 65 75 85 95 92 68 79 72 80 97 60 64 70 The above table shows grades of students in different courses (a) Write a matrix operation that determines the grade average for each course (b) Write a matrix operation that determines the grade average for each student (c) Write a matrix operation that gives the grade average for all students in all courses 10 The demand and supply functions of two related goods are given by Q d1 = 40 − 6P1 + 2P2 Q s1 = −40 + 8P1 Q d2 = 250 + 3P1 − 4P2 Q s2 = −60 + 3P2 (a) Use the equilibrium condition and reduce the system to a by matrix system and solve for equilibrium prices Then determine the equilibrium quantities (b) Write the system in a full by matrix form and solve for the equilibrium prices and quantities 11 The demand and supply function of three related goods are given by Q d1 = 120 − 4P1 + 4P3 Q s1 = −40 + 2P1 Q d2 = 140 + 2P1 − 2P2 + 4P3 Q s2 = −60 + 3P2 Q d3 = 150 − 3P1 + 2P2 − 2P3 Q s3 = −90 + 3P3 286 13 Matrices and Their Applications Write the system in a full by matrix form and find the equilibrium prices and quantities 12 An investor has $20,000 to invest He has two investment opportunities; one relatively low risk with % return, and another relatively high risk with % return Use matrix formulation and determine the amount he should invest at each rate in order to generate $850 annual interest 13 Assume an economy with sectors: Agriculture, Manufacturing, and Services To produce $1 of agricultural output requires $0.10 worth of agricultural products, $0.30 worth of manufacturing products, and $0.20 of services To produce $1 of manufacturing output requires $0.35 of manufacturing, $0.05 of agricultural, and $0.30 of services worth of input To produce $1 of services requires $0.15 worth of services and $0.25 of manufacturing products The annual final demands for outputs of the sectors are $176, $320 and $650 billion, respectively (a) Determine the value of sectoral and national outputs (b) What impact an increase of % in final demand for manufacturing products has on the sectoral and national outputs (c) What impact an increase of % in final demands for all sectors has on the sectoral and national outputs 14 A vertically integrated steel company owns and operates coal mines, iron ore mines, and a rail road system Production of a $1 worth of steel requires inputs of $0.10 of steel, $0.05 of coal, $0.02 of iron ore, and $0.08 of rail road (transportation) Production of a $1 worth of coal requires $0.15 of steel, $0.10 of coal, and $0.03 of rail road Production of $1 worth of iron ore requires $0.10 of steel, $0.12 of coal, and $0.02 of rail road The input requirements for $1 worth of rail road are $0.35 of steel and $0.05 of coal (a) Determine the output of steel, coal, iron ore, and rail road companies needed to satisfy a final demand of $550 million of steel, $300 million of coal, $150 million of iron ore, and $120 million worth of rail road service (b) What is the value of coal and iron ore used in production of steel or sold in the market 15 Consider an economy with four sectors, agriculture (A), construction and mining (C), manufacturing (M), and services (S) The following table provides the input requirements for $1 worth of output for each sector and the final demand (FD, in million dollars) for each sector (a) Determine the value of sectoral and national outputs (b) Assume there is a 2.5 % increase in final demand Calculate the new sectoral and national outputs 13 Matrices and Their Applications 287 Input-Output Table Output Input A C M S A C M S FD 0.12 0.00 0.08 0.05 0.02 0.06 0.25 0.12 0.05 0.20 0.20 0.18 0.03 0.15 0.10 0.10 1100 2500 3700 4000 16 An electronic company produces different electronic devices in California and in Zhengzhou, Henan province, in China The following matrix gives the labor hour per unit of each device in California and China Assume the labor costs, on the average, is $20 per hour in California and $5 per hour in China Labor Hour per Device Location Dev Dev Dev California China 0.10 0.30 0.08 0.20 0.05 0.10 (a) Write a matrix expression to calculate the labor cost of producing each device in California and China (b) Write a matrix expression to calculate the average labor cost of producing these devices in California and China 17 The original Input-Output table prepared by the U.S Bureau of Labor Statistics, consisted of 42 industries that comprised the U.S economy in its entirety and showed the exchange of goods and services in the U.S for 1947 (this table appeared as Table 2.1 on pages 16–19 of Wassliy Leontiefs book Input-Output Economics.) The following table is an aggregated 3-sector version of the original table (figures in millions of 1947 dollars) (a) Use the table and calculate the matrix of inter-sectoral coefficients (b) Calculate the sectoral gross outputs if the new vector of final demand, as row vector, for year 1948 is given as FD = 42100 68500 156700 (c) What is the percentage change in the GDP? 288 13 Matrices and Their Applications Table of Inter-sectoral Transactions US 1947 Agriculture Manufacturing Services Agriculture Manufacturing Services Final Demand 34690 5280 10450 4920 61280 25950 5620 22990 42030 39240 60020 130650 18 The following table is an aggregated version of US Department of Commerce, Bureau of Economic Analysis (BEA) 15-sector input-out table estimated for 2012 (See www.bea.gov/industry/io_annual.htm) In this table Mining, Utilities, and Construction industries are aggregated into one sector labeled MUC Similarly, different service sectors are aggregated into one called Ser Other sectors are Agriculture (Ag), Manufacturing (Manu) and Government (Gov) (figures in millions of 2012 dollars) Table of Inter-sectoral Transactions US 2012 Ag MUC Manu Ser Gov Ag MUC Manu Ser Gov Final Demand 86810 8285 80416 69030 37 1502 97899 328561 302979 384 270144 648351 1903440 883549 5597 934112 298770 934112 5132521 76513 2929 114869 370598 667024 9537 74554 989442 2021814 10670276 2619157 (a) Use the table and calculate the matrix of inter-sectoral coefficients (b) Calculate the sectoral gross outputs if the new final demand vector for year 2013 is given as FD = 74664 998445 2031426 10809234 2867920 (c) What is the percentage change in the GDP? 19 In the previous exercise ignore an eliminate Government and aggregate MUC in the Manufacturing sector This creates a 3-sector model similar to 1947 table given in exercise #17 (a) Use the table and calculate the matrix of inter-sectoral coefficients (b) Calculate the sectoral gross outputs if the new final demand vector for year 2013 is given as FD = 74664 3029871 10809234 13 Matrices and Their Applications 289 (c) What is the percentage change in the GDP? (d) What is the impact of ignoring government as a sector on the GDP? 20 Compare the results of exercise #19 and #17 (a) What sector of the US economy has grown the most over the last 65 years? (b) What is the size of the US economy in 2012 compared to 1947? 21 Consider a close economy with the following consumption and investment functions: C = 250 + 0.67Y I = 150 + 0.07Y Assume the government expenditure G is 80 (a) Write the model in matrix form and determine the equilibrium values of national income, consumption, and investment (b) From the matrix of coefficients, determine the model’s multiplier (c) Assume the government expenditure changes to 90 Determine the new equilibrium values of Y, C and I 22 Consider the following national income determination model for a close economy: C = 220 + 0.84Y D I = 130 + 0.09Y D − 20.5r0 T = 85 + 0.12Y where r0 is the interest rate Assume the government expenditure G is 100 and r0 = 4.2 (a) Write the model in matrix form and determine the equilibrium values of national income, consumption, investment, and tax (b) From the matrix of coefficients, determine the model’s government expenditure multiplier and tax multiplier (c) Assume the government expenditure changes to 90 Determine the new equilibrium values of Y, C, I and T 23 Assume the central bank of the economy described in the last problem expands the money supply and manages to reduce the interest rate to 3.2 % Determine the new equilibrium values of the endogenous variables 24 The following is an extended version of the model in exercise #22 In this model r is treated as an endogenous variable with its value determined by income and the money supply, 290 13 Matrices and Their Applications C = 220 + 0.84Y D I = 130 + 0.09Y D − 10.5r r = −5 + 0.016Y − 0.02M0 T = 85 + 0.12Y (a) Treat Y D as a separate variable and write the model in matrix form (the coefficient matrix is now a by matrix) Determine the equilibrium values of the endogenous variables assuming G = 80 and M0 = 800 (b) Calculate the model’s multipliers 25 What would be the equilibrium values of the exogenous variables in problem #24 if the Federal Reserve decides to cut the money supply by 20 %? 26 Write problem #6 of Chap Supplementary Exercises in matrix form and find the household’s optimal bundle 27 Assume a household allocates its $4,500 monthly income to food, housing, transportation, health care, and all other items According to Table 1.2 in the text, allocation of a typical household’s budget to housing is almost equal to the sum of its allocations to food, transportation, and health care Allocation to all other items is twice as allocation to transportation plus $500 Allocation to transportation is equal to sum of allocations to food and health care Finally, allocation to all other items is also equal to one half of allocation to food, plus the allocations to transportation and health care If the unit price of food, housing, transportation, health care, and other items are $20, $50, $30, $15 and $35 respectively, determine the household optimal bundle 28 In problem #27, assume that household’s income decreases by % What is the household’s new optimal bundle? 29 In problem #27, assume that due to inflation prices increase by % What is the new optimal bundle? 30 Does comparing the results of problems #28 and #29 lead you to believe that inflation is similar to the loss of income? ... 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