Chiang_Ch4 - Chiang& Wainwright Chiang& Wainwright...

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Unformatted text preview: Chiang & Wainwright Chiang & Wainwright Mathematical Economics Mathematical Economics Chapter 4 Linear Models and Matrix Algebra 1 Chiang_Ch4.ppt Stephen Cooke U. Idaho Ch. 4 Linear Models & Matrix Algebra Ch. 4 Linear Models & Matrix Algebra Matrix algebra can be used: a. to express the system of equations in a compact notation; b. to find out whether solution to a system of equations exist; and c. to obtain the solution if it exists. Need to invert the A matrix to find the solution for x* 2 d A adjA x A adjA A d A x d Ax = = = =-- * 1 1 * det Chiang_Ch4.ppt Stephen Cooke U. Idaho One Commodity Market Model One Commodity Market Model (2x2 matrix) (2x2 matrix) Economic Model (p. 32) 1) Q d =Q s 2) Q d = a – bP (a,b >0) 3) Q s = -c + dP (c,d >0) Find P* and Q* Scalar Algebra Endog. :: Constants 4) 1Q + bP = a 5) 1Q – dP = -c 3 d A x d Ax c a P Q d b 1 * 1 1- = = - = - d b bc ad Q d b c a P +- = + + = * * Matrix Algebra Chiang_Ch4.ppt Stephen Cooke U. Idaho One Commodity Market Model One Commodity Market Model (2x2 matrix) (2x2 matrix) 4 d A x c a d b P Q d Ax c a P Q d b 1 * 1 * * 1 1 1 1-- = - - = = - = - Matrix algebra Chiang_Ch4.ppt Stephen Cooke U. Idaho General form of 3x3 linear matrix General form of 3x3 linear matrix parameters endog. vars exog. vars. & constants Scalar algebra form parameters & endogenous variables exog. vars & const. a 11 x + a 12 y + a 13 z = d 1 a 21 x + a 22 y + a 23 z = d 2 a 31 x + a 32 y + a 33 z = d 3 5 = 3 2 1 33 3 2 3 1 2 3 2 2 2 1 1 3 1 2 1 1 d d d z y x a a a a a a a a a Matrix algebra form Chiang_Ch4.ppt Stephen Cooke U. Idaho 1. Three Equation National Income Model 1. Three Equation National Income Model (3x3 matrix) (3x3 matrix) Let ( Exercise 3.5-1, p. 47) Y = C + I 0 + G C = a + b(Y-T) (a > 0, 0<b<1) T = d + tY (d > 0, 0<t<1) Endogenous variables? Exogenous variables? Constants? Parameters? Why restrictions on the parameters? 6 Chiang_Ch4.ppt Stephen Cooke U. Idaho 2. Three Equation National Income Model 2. Three Equation National Income Model Exercise 3.5-2, p.47 Exercise 3.5-2, p.47 Endogenous: Y, C, T: Income (GNP), Consumption, and Taxes Exogenous: I and G : autonomous Investment & Government spending Constants a & d: autonomous consumption and taxes Parameter t is the marginal propensity to tax gross income 0 < t < 1 Parameter b is the marginal propensity to consume private goods and services from gross income 0 < b < 1 7 bt b G I bd a Y +- + +- = 1 ) 8 * Chiang_Ch4.ppt Stephen Cooke U. Idaho 3. Three Equation National Income Model 3. Three Equation National Income Model Exercise 3.5-1, p. 47 (substitution method)Exercise 3....
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This note was uploaded on 11/09/2011 for the course ECON 101 taught by Professor Richards during the Spring '11 term at Cambrian College.

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Chiang_Ch4 - Chiang& Wainwright Chiang& Wainwright...

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