By a state wherein the values of endogenous variables

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Unformatted text preview: he level given by the solution to the system? What implications does it bear for the consumers and producers? 2.What if the price is below P*? The equilibrium conditions must represent the reason why, if satisfied, the values of the endogenous variables will tend to keep satisfying these 8 conditions. Finding Equilibrium by Elimination of Variables Mathematically, we just found the equilibrium by solving the system of 3 equations in 3 variables: Q d = a −bP Q c s = − +dP Q d =Qs However, the equilibrium condition allows us to reduce the number of variables to just two: Q = a −bP Q c = − +dP We can even reduce this system to one equation only: a −bP = − +dP c The equilibrium prices and quantities are now found in a straightforward way: a +c * b( a + c ) ad − bc * P= ; Q = a − bP = a − = b+d b+d b+d * 9 Interpretation of the Result Mathematical results should not only conform to the system of the underlying equations, they should also be economically meaningful. What are the meaningful requirements we should impose on the equilibrium values of price and quantity? a+ c P* = b+ d Since b>0 and d>0 by assumptions on the consumers’ behavior, b+d>0. Since a>0 and c>0 by assumptions on the producers’ behavior, a+c>0. Hence, our equilibrium price will be always positive, which makes sense. Q * ad − bc = b+ d The equilibrium quantity should be positive as well, but is it? For that to be satisfied, we must make sure that ad − bc > 0 10 Equilibrium from the Set Perspective Demand line can be thought of as a set: Same is true for the supply line: D = { ( P, Q ) | Q = a − bP} S = { ( P, Q ) | Q = −c + dP} The equilibrium prices and quantities may then be thought of as an intersection of the two sets that contains only one element: (P , Q ) = D S * * A few warning words about finding an equilibrium: 1)An equilibrium may not be unique 2)An equilibrium may not always exist 11 Supply and Demand: a Quadratic Model Consider the following supply-demand system: Q Q d= s Q P2 d =4 − Q 1 s =4 P − Supply is still a linear function, but quantity demanded depends on the price in a non-linear, quadratic way. Similarly to the linear case, we can reduce this system of three equations in three variables to just one equation: Qd = Qs = 4 − P 2 = 4 P −1 ⇓ P 2 + 4 P −5 = 0 12 The Set Perspective As in linear case, our demand and supply can be represented in terms of the sets of ordered pairs: 2 { D = ( P, Q ) | Q = 4 − P } S = {( P, Q ) | Q = 4 P −1} Our equilibrium is given by the intersection of the supply and demand sets: ( P , Q ) = D S * * Qs = 4 P − 1 Q*=3 P*=1 Qd = 4 − P 2 13 Higher­Degree Polynomial Equations How do we go about solving this cubic equation? x −x −4 x +4 =0 3 2 One way to do this is to factor the cubic function: One factor x 3 −x 2 −4 x +4 =( x −1)( x +2 )( x −2 ) =0 Clearly, the roots are: Clearly, x1 = 1, x2 = −2, x3 = 2 How do we search for cubic roots in the general ca...
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