https://wiki.math.ucr.edu/index.php?title=022_Sample_Final_A,_Problem_5&feed=atom&action=history 022 Sample Final A, Problem 5 - Revision history 2022-05-24T08:16:39Z Revision history for this page on the wiki MediaWiki 1.35.0 https://wiki.math.ucr.edu/index.php?title=022_Sample_Final_A,_Problem_5&diff=902&oldid=prev MathAdmin: Created page with "300px <span class="exam"> Find producer and consumer surpluses if the supply curve is given by <math style="vertical-align: -4px"> p = 18 + 3x^2..." 2015-06-06T17:12:45Z <p>Created page with &quot;<a href="/index.php/File:022_3_A_5.png" title="File:022 3 A 5.png">right|300px</a> &lt;span class=&quot;exam&quot;&gt; Find producer and consumer surpluses if the supply curve is given by &lt;math style=&quot;vertical-align: -4px&quot;&gt; p = 18 + 3x^2...&quot;</p> <p><b>New page</b></p><div>[[File:022_3_A_5.png |right|300px]]<br /> <br /> &lt;span class=&quot;exam&quot;&gt; Find producer and consumer surpluses if the supply curve is given by &lt;math style=&quot;vertical-align: -4px&quot;&gt; p = 18 + 3x^2&lt;/math&gt;, and the demand curve is given by &lt;math style=&quot;vertical-align: -4px&quot;&gt;p = 150 - 4x&lt;/math&gt;.<br /> <br /> {| class=&quot;mw-collapsible mw-collapsed&quot; style = &quot;text-align:left;&quot;<br /> !Foundations: &amp;nbsp; <br /> |-<br /> |The supply curve and the demand curve are tied to price of a product. As the price goes up, manufacturers would be willing to produce more, so the supply curve is usually increasing. On the other hand, as the price goes up, fewer consumers might purchase the product, so the demand curve is usually decreasing.<br /> &lt;br&gt;<br /> At some sale price, these two curves intersect at an equilibrium point. When we speak of '''consumer surplus''' and '''producer surplus''', we are talking about the total area under between the horizontal line through the equilibrium level of units, and the demand curve or supply curve, respectively. We need to integrate the areas indicated on the graph to find actual values for these two surpluses.<br /> |}<br /> <br /> &amp;nbsp;'''Solution:'''<br /> <br /> {| class=&quot;mw-collapsible mw-collapsed&quot; style = &quot;text-align:left;&quot;<br /> !Step 1: &amp;nbsp;<br /> |-<br /> |'''Find the equilibrium point:''' Since we are given equations for both curves, we need to set them equal and solve for &lt;math style=&quot;vertical-align: 0px&quot;&gt;x&lt;/math&gt;. Setting them equal, we have<br /> |-<br /> |<br /> ::&lt;math&gt;18+3x^2\,=\,150-4x.&lt;/math&gt;<br /> |-<br /> |If we move everything to one side and factor, we find<br /> |-<br /> |<br /> ::&lt;math&gt;3x^2+4x-132\,=\,(3x+22)(x-6)\,=\,0.&lt;/math&gt;<br /> |-<br /> |This has only one positive root, &lt;math style=&quot;vertical-align: 0px&quot;&gt;x=6&lt;/math&gt;. at this point we have the associated height &lt;math style=&quot;vertical-align: -4px&quot;&gt;p\,=\,126.&lt;/math&gt;<br /> |}<br /> <br /> {| class=&quot;mw-collapsible mw-collapsed&quot; style = &quot;text-align:left;&quot;<br /> !Step 2: &amp;nbsp;<br /> |-<br /> |'''Consumer Surplus:''' We need to find the triangular area in the graph, and this will mean integrating between &lt;math style=&quot;vertical-align: 0px&quot;&gt;0&lt;/math&gt; and equilibrium &lt;math style=&quot;vertical-align: 0px&quot;&gt;x&lt;/math&gt;-value, &lt;math style=&quot;vertical-align: 0px&quot;&gt;6&lt;/math&gt;. Also, we will integrate from the horizontal line through the equilibrium, &lt;math style=&quot;vertical-align: -4px&quot;&gt;p\,=\,126&lt;/math&gt;, up to the demand curve, &lt;math style=&quot;vertical-align: -4px&quot;&gt;p = 150 - 4x&lt;/math&gt;. We have <br /> |-<br /> |<br /> ::&lt;math&gt;\begin{array}{rcl}<br /> \textrm{Consumer~surplus} &amp; = &amp; {\displaystyle {\displaystyle \int_{0}^{6}(150-4x)-126\, dx}}\\<br /> \\<br /> &amp; = &amp; {\displaystyle \int_{0}^{6}24-4x\, dx}\\<br /> \\<br /> &amp; = &amp; {\displaystyle 24x-2x^{2}}\biggr|_{x=0}^{6}\\<br /> \\<br /> &amp; = &amp; 144-72\,=\,72.<br /> \end{array}&lt;/math&gt;<br /> |}<br /> <br /> {| class=&quot;mw-collapsible mw-collapsed&quot; style = &quot;text-align:left;&quot;<br /> !Step 3: &amp;nbsp;<br /> |-<br /> |'''Producer Surplus:''' Now, we wish to integrate the area from the supply curve up to the horizontal line through our equilibrium point. We have<br /> |-<br /> |<br /> ::&lt;math&gt;\begin{array}{rcl}<br /> \textrm{Producer~surplus} &amp; = &amp; {\displaystyle {\displaystyle \int_{0}^{6}126-(18+3x^{2})\, dx}}\\<br /> \\<br /> &amp; = &amp; {\displaystyle \int_{0}^{6}108-3x^{2}\, dx}\\<br /> \\<br /> &amp; = &amp; 108x-x^{3}\biggr|_{x=0}^{6}\\<br /> \\<br /> &amp; = &amp; 648-216\,=\,432.<br /> \end{array}&lt;/math&gt;<br /> |}<br /> {| class=&quot;mw-collapsible mw-collapsed&quot; style = &quot;text-align:left;&quot;<br /> !Final Answer: &amp;nbsp;<br /> |-<br /> |The consumer surplus is &lt;math style=&quot;vertical-align: 0px&quot;&gt;72&lt;/math&gt;, while the producer surplus is &lt;math style=&quot;vertical-align: 0px&quot;&gt;432&lt;/math&gt;.<br /> |}<br /> [[022_Sample_Final_A|'''&lt;u&gt;Return to Sample Exam&lt;/u&gt;''']]</div> MathAdmin