Question one ??1 = 15 − ?1 + 2?2 + ?3 ; ??1 = −7 + ?1...
Question one
??1 = 15 − ?1 + 2?2 + ?3 ; ??1 = −7 + ?1
??2 = 9 + ?1 − ?2 − ?3 ; ??2 = −4 + 4?2
??3 = 8 + 2?1 − ?2 − 4?3 ; ??3 = −5 + 2?3
a) State the equilibrium condition for this three-commodity model.
b) Write the equilibrium system in a compact matrix of form Ax=b.
c) Now, by using the Cramer’s rule, determine the equilibrium price for this three-commodity model.
d) Next, find the equilibrium quantity for this three-commodity model.
e) Explain the relationship between
(i) Good 1 and Good 2.
(ii) Good 2 and Good 3.
Homework Answers
2. Consider the following model of Supply and Demand. where P is the price of the...
2. Consider the following model of Supply and Demand. where P is the price of the good, Qd is quantity demanded and Q5 is quantity supplied. (i) What condition should b satisfy in order for the first equation to be a reasonable demand function? (ii) What condition should b and d satisfy in order for this system to have a unique equilibrium? (ii) Assuming a unique equilibrium exists express the system in matrix form and use matrix algebra to find...
Pcoer IS approximately ES0 2. Consider the following model of Supply and Demand. where P is...
Pcoer IS approximately ES0 2. Consider the following model of Supply and Demand. where P is the price of the good, Q is quantity demanded and QS is quantity supplied. (i) what condition should δ satisfy in order for the second equation to be a reasonable supply function. (ii) What condition should B and 6 satisfy in order for this system to have a unique equilibrium. Ģi Assuming a unique equilibrium exists express the system in matrix form and use...
2. Consider the following model of Supply and Demand. where P is the price of the...
2. Consider the following model of Supply and Demand. where P is the price of the good, Q is quantity demanded and Qs is quantity supplied. G) What condition should o satisfy in order for the second equation to be a reasonable supply function. (ii) What condition should ß and satisfy in order for this system to have a unique equilibrium. uming a unique equilibrium exists express the system in matrix form and use matrix algebra to find the equilibrium...
5. (22 pts) Consider the three-commodity market model given by: Qi = 16 – 2p1 +...
5. (22 pts) Consider the three-commodity market model given by: Qi = 16 – 2p1 + 2p2 + P3 and Qi = 2p1 – 7 Q2 = 8+2p1 – P2 – P3 and Q2 = 4p2 – 4 Q% = 4+421 – P2 – 4p3 and Qs = 2p3 – 3 where Q4, Q and pi denote quantity demanded, quantity supplied and price of good i = 1, 2, 3, respectively. (a) What is the relationship among the three goods?...
experiment and records the following observations: 2. An engineer conducts an 3 2 1 X 9.3...
experiment and records the following observations: 2. An engineer conducts an 3 2 1 X 9.3 1.2 3.6 y model of the form, y = ax, find the optimal least squares Assuming that the recorded data obeys estimates of "a and b". Also, using the above estimated values of "a and b," find a predicted value of y for x 5. a [Note: You need not check the Hessian Matrix for sufficiency condition]. Hint: First convert the given model to...
experiment and records the following observations: 2. An engineer conducts an 3 2 1 X 9.3...
experiment and records the following observations: 2. An engineer conducts an 3 2 1 X 9.3 1.2 3.6 y model of the form, y = ax, find the optimal least squares Assuming that the recorded data obeys estimates of "a and b". Also, using the above estimated values of "a and b," find a predicted value of y for x 5. a [Note: You need not check the Hessian Matrix for sufficiency condition]. Hint: First convert the given model to...
3 (b) Write the following systems of linear equations as matrix equation and then as an...
3 (b) Write the following systems of linear equations as matrix
equation and then as an augmented matrix:
(4marks)
(d) Use Cramer’s rule to solve the system of 2 linear equations
in 3(b). (7marks)
We were unable to transcribe this imageWe were unable to transcribe this image
7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by Gaus...
7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by Gaussian elimination and express the general solution in vector form. (b) (5 points) Write down the corresponding homogenous system Ax-0 explicitly and determine all non-trivial solutions from (a) without resolving the system
7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by...
1 1 -2 Given the LTI system -Ax Bu where A3 3 2and B0 a) Check the controllability using i) the c...
1 1 -2 Given the LTI system -Ax Bu where A3 3 2and B0 a) Check the controllability using i) the controllability matrix, and ii) the Hautus-Rosenbrock test. b) Identify the controllable and uncontrollable subspaces, and convert the system to a Kalman con- 0 trollable canonical form c) Suppose that we start from the initial state z(0) (1,1, 1)T. Is there a control u(t) that drives the state to (1(3,-1,1)7 at some time t? Is there a control u(t) that...
eclass.srv.ualberta.ca 2 of 2 1. Consider the matrix 3-2 1 4-1 2 3 5 7 8...
eclass.srv.ualberta.ca 2 of 2 1. Consider the matrix 3-2 1 4-1 2 3 5 7 8 (a) Find a basis B for the null space of A. Hint: you need to verify that the vectors you propose 20 actually form a basis for the null space. (Recall: (1) the null space of A consists of all x e R with Ax = 0, and (2) the matrix equation Ax = 0 is equivalent to a certain system of linear equations.)...
2. Consider the following model of Supply and Demand. where P is the price of the good, Qd is quantity demanded and Q5 is quantity supplied. (i) What condition should b satisfy in order for the first equation to be a reasonable demand function? (ii) What condition should b and d satisfy in order for this system to have a unique equilibrium? (ii) Assuming a unique equilibrium exists express the system in matrix form and use matrix algebra to find...
Pcoer IS approximately ES0 2. Consider the following model of Supply and Demand. where P is the price of the good, Q is quantity demanded and QS is quantity supplied. (i) what condition should δ satisfy in order for the second equation to be a reasonable supply function. (ii) What condition should B and 6 satisfy in order for this system to have a unique equilibrium. Ģi Assuming a unique equilibrium exists express the system in matrix form and use...
2. Consider the following model of Supply and Demand. where P is the price of the good, Q is quantity demanded and Qs is quantity supplied. G) What condition should o satisfy in order for the second equation to be a reasonable supply function. (ii) What condition should ß and satisfy in order for this system to have a unique equilibrium. uming a unique equilibrium exists express the system in matrix form and use matrix algebra to find the equilibrium...
5. (22 pts) Consider the three-commodity market model given by: Qi = 16 – 2p1 + 2p2 + P3 and Qi = 2p1 – 7 Q2 = 8+2p1 – P2 – P3 and Q2 = 4p2 – 4 Q% = 4+421 – P2 – 4p3 and Qs = 2p3 – 3 where Q4, Q and pi denote quantity demanded, quantity supplied and price of good i = 1, 2, 3, respectively. (a) What is the relationship among the three goods?...
experiment and records the following observations: 2. An engineer conducts an 3 2 1 X 9.3 1.2 3.6 y model of the form, y = ax, find the optimal least squares Assuming that the recorded data obeys estimates of "a and b". Also, using the above estimated values of "a and b," find a predicted value of y for x 5. a [Note: You need not check the Hessian Matrix for sufficiency condition]. Hint: First convert the given model to...
experiment and records the following observations: 2. An engineer conducts an 3 2 1 X 9.3 1.2 3.6 y model of the form, y = ax, find the optimal least squares Assuming that the recorded data obeys estimates of "a and b". Also, using the above estimated values of "a and b," find a predicted value of y for x 5. a [Note: You need not check the Hessian Matrix for sufficiency condition]. Hint: First convert the given model to...
3 (b) Write the following systems of linear equations as matrix
equation and then as an augmented matrix:
(4marks)
(d) Use Cramer’s rule to solve the system of 2 linear equations
in 3(b). (7marks)
We were unable to transcribe this imageWe were unable to transcribe this image
7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by Gaussian elimination and express the general solution in vector form. (b) (5 points) Write down the corresponding homogenous system Ax-0 explicitly and determine all non-trivial solutions from (a) without resolving the system
7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by...
1 1 -2 Given the LTI system -Ax Bu where A3 3 2and B0 a) Check the controllability using i) the controllability matrix, and ii) the Hautus-Rosenbrock test. b) Identify the controllable and uncontrollable subspaces, and convert the system to a Kalman con- 0 trollable canonical form c) Suppose that we start from the initial state z(0) (1,1, 1)T. Is there a control u(t) that drives the state to (1(3,-1,1)7 at some time t? Is there a control u(t) that...
eclass.srv.ualberta.ca 2 of 2 1. Consider the matrix 3-2 1 4-1 2 3 5 7 8 (a) Find a basis B for the null space of A. Hint: you need to verify that the vectors you propose 20 actually form a basis for the null space. (Recall: (1) the null space of A consists of all x e R with Ax = 0, and (2) the matrix equation Ax = 0 is equivalent to a certain system of linear equations.)...
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