Exercise 2.3 Suppose that the supply and demand sets for a particular market are I S= (q,p)vert 3p-q=5 , D= (q,p)Vert 3p+q^2+2q=9 Sketch S and D and determine the equilibrium set E=Scap D . Comment briefly on the interpretation of the results.

To find the equilibrium set $E=S\cap D$, we need to find the intersection of the supply set $S$ and the demand set $D$. <br /><br />The supply set $S$ is given by the equation $3p-q=5$. We can rearrange this equation to solve for $q$: $q=3p-5$. <br /><br />The demand set $D$ is given by the equation $3p+q^{2}+2q=9$. We can rearrange this equation to solve for $q$: $q^{2}+2q-3p+9=0$. <br /><br />To find the intersection, we need to find the values of $q$ that satisfy both equations. We can do this by substituting the expression for $q$ from the supply set into the demand set and solving for $p$. <br /><br />Substituting $q=3p-5$ into the demand set equation, we get: <br /><br />$3p+(3p-5)^{2}+2(3p-5)=9$<br /><br />Simplifying this equation, we get: <br /><br />$3p+9p^{2}-30p+25+6p-10=9$<br /><br />$9p^{2}-9p+15=0$<br /><br />Dividing by 3, we get: <br /><br />$3p^{2}-3p+5=0$<br /><br />This quadratic equation has no real solutions, as the discriminant $b^{2}-4ac$ is negative. Therefore, there are no values of $p$ that satisfy both the supply and demand equations. <br /><br />Since there are no values of $p$ that satisfy both equations, the equilibrium set $E$ is empty. This means that there is no equilibrium point where the supply and demand sets intersect. <br /><br />In the interpretation of the results, we can say that there is no equilibrium point in this market, as there are no values of $p$ that satisfy both the supply and demand equations. This could indicate that the market is not in equilibrium, or that there are other factors affecting the supply and demand sets that are not accounted for in the given equations.