02b_Computing_GE

02b_Computing_GE - 1 1.1 Computing General Equilibrium:...

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1 Computing General Equilibrium: Examples 1.1 Linear Quadratic Example Consider an economy with two agents A and B de f ned by W A ( x, y )=1 0 0 x 1 2 x 2 + y x A , ¯ y A )=( 4 , 5000) W B ( x, y )=3 0 x 1 2 x 2 + y x B , ¯ y B )=( 8 0 , 1000) The welfare functions are linear in y and quadratic in x . I have used the notation ¯ x h and ¯ y h to denote the initial endowments of goods x and y by agent h . What will the equilibrium be? We must search for prices and consumptions for both goods by both agents such that at those prices the consumption bundles are optimal for each agent, and such that supply equals demand. Mathematically, that meanswemust f nd values for the six variables p x p y x A y A x B y B such that x A + x B x A x B =84 y A + y B y A y B =6000 andsuchthat ( x A ,y A ) solves max ( x,y ) 100 x 1 2 x 2 + y p x x + p y y p x 4+ p y 5000 andsuchthat ( x B ,y B ) solves max ( x,y ) 30 x 1 2 x 2 + y p x x + p y y p x 80 + p
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the marginal utility of the last dollar spent on each good must be the same. Hence equilibrium can be reduced to the six equations 100 x A p x = 1 p y p x x A + p y y A = p x ¯ x A + p y ¯ y A = p x 4+ p y 5000 30 x B p x = 1 p y p x x B + p y y B = p x ¯ x B + p y ¯ y B = p x 80 + p y 1000 x A + x B x A x B =84 y A + y B y A y B =6000 The f rst equation says that the marginal utility per dollar of good x must be the same as the marginal utility of a dollar’s worth of good y to agent A . It is Gossen’s equation for agent A . The second equation says that A spends all his money. The third and fourth equations say the same thing about agent B .T h e f fth equation says that total consumption of good x must equal total supply. Equation 6 says the same thing about good y . The reader might well wonder how agent A knows how much to buy of each good be foreheknowsthepr ice ,andhowthepr
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This note was uploaded on 12/08/2009 for the course ECON 251 at Yale.

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02b_Computing_GE - 1 1.1 Computing General Equilibrium:...

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