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.

To find the equilibrium set, 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 />$3p - q = 5$<br />$3p + q^2 + 2q = 9$<br /><br />We can solve this system of equations using substitution or elimination method. Let's use substitution method:<br /><br />From the first equation, we can express p in terms of q:<br /><br />$p = \frac{q + 5}{3}$<br /><br />Now, substitute this expression for p into the second equation:<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 />Combine like terms:<br /><br />$q^2 + 3q + 5 = 9$<br /><br />Subtract 9 from both sides:<br /><br />$q^2 + 3q - 4 = 0$<br /><br />Now, we can solve this quadratic equation using factoring, completing the square, 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 />Simplify the expression inside the square root:<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 />$q_2 = \frac{-3 - 5}{2} = -4$<br /><br />Now, let's substitute these values of q back into the expression for p:<br /><br />For $q_1 = 1$:<br /><br />$p_1 = \frac{1 + 5}{3} = 2$<br /><br />For $q_2 = -4$:<br /><br />$p_2 = \frac{-4 + 5}{3} = \frac{1}{3}$<br /><br />So, the equilibrium set E is:<br /><br />$E = \{(1, 2), (-4, \frac{1}{3})\}$<br /><br />The interpretation of these results is that there are two equilibrium points in this market. The first equilibrium point is when the quantity demanded is 1 and the price is 2. The second equilibrium point is when the quantity demanded is -4 and the price is $\frac{1}{3}$. These points represent the prices and quantities at which the supply and demand sets intersect.