1 p 2 p q b b1 b2 supply for good 1 q q 0 equilibrium

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Unformatted text preview: uilibrium By conditions) we can reduce the system of six equations to that of only two: conditions) ( a0 −b0 ) + ( a1 −b1 ) P + ( a2 −b2 ) P2 = 0 1 1 (α0 − β0 ) + (α1 − β1 ) P + (α2 − β2 ) P2 = 0 We have a system of two equations with six parameters: is there a problem? 18 Reducing the Number of Parameters We can reduce the number of parameters by introducing new parameters: ci ≡ ai − bi , i = 0,1,2 γ i ≡ αi − βi , i = 0,1,2 Our system then becomes c1 P + c2 P2 = −c0 1 γ1 P + γ 2 P2 = −γ 0 1 We solve this system again by eliminating the variables: We Find Find P from the first equation: P * = c2γ 0 −c0γ 2 1 1 c1γ 2 − c2γ1 Substitute P Substitute 2 * into the second equation to find P2 = c0γ 1 − c1γ 0 c1γ 2 − c2γ 1 What should be the restrictions on the parameters for What these prices to make sense? these 19 General Market Equilibrium In case we have n commodities, we are talking about a model of In general market equilibrium. We will then have n equilibrium conditions (one for each good): for Qdi − Qsi = Qdi ( P , P2 ,..., Pn ) − Qsi ( P , P2 ,..., Pn ) = 0, i = 1..n 1 1 Pi = Pi ( a1 , a2 ,..., am ) , i = 1..n, m ≠ n * * * Equilibrium prices P will in the end be functions of the model’s Equilibrium i parameters ai The number of the model’s parameters may not be equal to the number The may of the model’s equations representing equilibrium conditions. of 20 Consistency and Independence It may happen that a particular equation system (1) may not even have a It solution—consistency problem (2) may have more than one solution— solution—consistency independence problem Consistency problem: the two equations below the are contradicting each other are x y 8 + = y x y 9 + = 9 y =9−x y = 8− x 8 The two lines are parallel to each The other, so no intersection point exists: exists: our system is inconsistent our inconsistent x 21 Consistency and Independence Independence problem: there may be fewer (independent) equations there compared to the number of parameters. compared There are an infinite number There 2 12 x +y = of (x,y) that satisfy these two equations. 4 x +2 y =24 The intersection of a straight y line with itself makes an infinite number of points. infinite 12 y = 12 − 2 x 2 y = 24 − 4 x 6 x These two equations are functionally These dependent, so the system of these two dependent so equations has an infinite number of solutions. solutions. 22...
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