Exercise 2.3 Suppose that the supply and demand sets for a particular market are 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. I

To find the equilibrium set $E=S\cap D$, we need to find the intersection of the supply set $S$ and the demand set $D$. This means we need to find the values of $q$ and $p$ that satisfy both equations.<br /><br />Let's start by solving the system of equations:<br /><br />1. $3p - q = 5$<br />2. $3p + q^2 + 2q = 9$<br /><br />We can solve this system of equations using substitution or elimination method. Let's substitution method:<br /><br />From equation 1, we can express $p$ in terms of $q$:<br /><br />$p = \frac{q + 5}{3}$<br /><br />Now, substitute this expression for $p$ into equation 2:<br /><br />$3\left(\frac{q + 5}{3}\right) + q^2 + 2q = 9$<br /><br />Simplify the equation:<br /><br />$q + 5 + q^2 + 2q = 9$<br /><br />$q^2 + 3q + 5 = 9$<br /><br />$q^2 + 3q - 4 = 0$<br /><br />Now, we can solve this quadratic equation using factoring or the quadratic formula. Let's use the quadratic formula:<br /><br />$q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$<br /><br />In this case, $a = 1$, $b = 3$, and $c = -4$. Plugging these values into the quadratic formula, we get:<br /><br />$q = \frac{-3 \pm \sqrt{3^2 - 4(1)(-4)}}{2(1)}$<br /><br />$q = \frac{-3 \pm \sqrt{9 + 16}}{2}$<br /><br />$q = \frac{-3 \pm \sqrt{25}}{2}$<br /><br />$q = \frac{-3 \pm 5}{2}$<br /><br />This gives us two possible solutions for $q$:<br /><br />$q_1 = \frac{-3 + 5}{2} = 1$<br /><br />$q_2 = \frac{-3 - 5}{2} = -4$<br /><br />Now, we can substitute these values of $q$ back into the expression for $p$ to find the corresponding prices:<br /><br />For $q = 1$:<br /><br />$p = \frac{1 + 5}{3} = 2$<br /><br />For $q = -4$:<br /><br />$p = \frac{-4 + 5}{3} = \frac{1}{3}$<br /><br />So, the equilibrium set $E$ consists of two points: $(1, 2)$ and $(-4, \frac{1}{3})$.<br /><br />The interpretation of these results is that there are two possible equilibrium points in this market. The first point, $(1, 2)$, represents a higher quantity and price, while the second point, $(-4, \frac{1}{3})$, represents a lower quantity and price. This suggests that there may be two different market conditions that can lead to equilibrium, depending on factors such as demand and supply.