An aggregate equilibrium market for two goods is represented by the linear model presented in table...
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An aggregate equilibrium market for two goods is represented by the linear model presented in table 1 below, where Q; is the equilibrium quantity obtained by equating demand and supply in the market for good i, and prices are P, i=1,2. Aggregate income, which is exogenous, is denoted Y. The letters a, b, c, d, denote numerical constants. Table 1: Equilibrium model of two markets Market for good 1 2, = 10 + aP, + bP, + Y -2 + 3 P, Market for good 2 2, = 15 + cP, + dP, + Y Demand (la) (2a) %3D Supply (2a) 2, = -1+ 2P, (2b) (a) Use equations (2a) and (2b) to eliminate Q2, giving a single equation in terms of the P, P2, and Y. (b) Express the resultant three equation system in form AX=B where A is a coefficient matrix, Xa vector of the remaining endogenous variables, Q1, P1, and P2, (in that order), and B is a vector of constants and the exogenous variable, Y. Now let a=-2, b=1, c=1, and d=-1. (c) (i) Determine if the inverse of A exists. (ii) Use Cramer's rule to obtain solutions for endogenous variables, Q1, P1, and Pz and hence also obtain a solution for Q2. (iii) Discuss the impact of an increase in income, Y, on the solution for Q, obtained in part (ii) above. An aggregate equilibrium market for two goods is represented by the linear model presented in table 1 below, where Q; is the equilibrium quantity obtained by equating demand and supply in the market for good i, and prices are P, i=1,2. Aggregate income, which is exogenous, is denoted Y. The letters a, b, c, d, denote numerical constants. Table 1: Equilibrium model of two markets Market for good 1 2, = 10 + aP, + bP, + Y -2 + 3 P, Market for good 2 2, = 15 + cP, + dP, + Y Demand (la) (2a) %3D Supply (2a) 2, = -1+ 2P, (2b) (a) Use equations (2a) and (2b) to eliminate Q2, giving a single equation in terms of the P, P2, and Y. (b) Express the resultant three equation system in form AX=B where A is a coefficient matrix, Xa vector of the remaining endogenous variables, Q1, P1, and P2, (in that order), and B is a vector of constants and the exogenous variable, Y. Now let a=-2, b=1, c=1, and d=-1. (c) (i) Determine if the inverse of A exists. (ii) Use Cramer's rule to obtain solutions for endogenous variables, Q1, P1, and Pz and hence also obtain a solution for Q2. (iii) Discuss the impact of an increase in income, Y, on the solution for Q, obtained in part (ii) above.
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