Difference between revisions of "022 Exam 1 Sample A, Problem 5"

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! Foundations:    
 
! Foundations:    
 
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|Recall that the demand function, <math style="vertical-align: -25%">p(x)</math>, relates the price per unit <math style="vertical-align: -17%">p</math> to the number of units sold, <math style="vertical-align: 0%">x</math>.
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|Recall that the '''demand function''', <math style="vertical-align: -25%">p(x)</math>, relates the price per unit <math style="vertical-align: -17%">p</math> to the number of units sold, <math style="vertical-align: 0%">x</math>.
 
Moreover, we have several important important functions:
 
Moreover, we have several important important functions:
 
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*<math style="vertical-align: -20%">C(x)</math>, the total cost to produce <math style="vertical-align: 0%">x</math> units;<br>
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*<math style="vertical-align: -20%">C(x)</math>, the '''total cost''' to produce <math style="vertical-align: 0%">x</math> units;<br>
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*<math style="vertical-align: -20%">R(x)</math>, the total revenue (or gross receipts) from producing <math style="vertical-align: 0%">x</math> units;<br>
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*<math style="vertical-align: -20%">R(x)</math>, the '''total revenue''' (or gross receipts) from producing <math style="vertical-align: 0%">x</math> units;<br>
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*<math style="vertical-align: -20%">P(x)</math>, the total profit from producing <math style="vertical-align: 0%">x</math> units.<br>
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*<math style="vertical-align: -20%">P(x)</math>, the '''total profit''' from producing <math style="vertical-align: 0%">x</math> units.<br>
 
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|In particular, we have the relations
 
|In particular, we have the relations
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::<math>R(x)=x\cdot p(x).</math>
 
::<math>R(x)=x\cdot p(x).</math>
 
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|Finally, marginal profit at <math style="vertical-align: -20%">x_0</math> units is defined to be the effective cost of the next unit produced, and is precisely <math style="vertical-align: -22%">P'(x_0)</math>.  Similarly, marginal revenue or cost would be <math style="vertical-align: -22%">R'(x_0)</math> or <math style="vertical-align: -22%">C'(x_0)</math>, respectively.
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|Finally, the '''marginal profit''' at <math style="vertical-align: -20%">x_0</math> units is defined to be the effective profit of the next unit produced, and is precisely <math style="vertical-align: -22%">P'(x_0)</math>.  Similarly, the '''marginal revenue''' or '''marginal cost''' would be <math style="vertical-align: -22%">R'(x_0)</math> or <math style="vertical-align: -22%">C'(x_0)</math>, respectively.
  
 
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::<math>P'(x)\,\,=\,\,R'(x)-C'(x)\,\,=\,\,\frac{100}{\sqrt{x}}-(30+6x).</math>
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::<math>P'(x)\,\,=\,\,R'(x)-C'(x)\,\,=\,\,\frac{100}{\sqrt{x}}-(15+6x).</math>
 
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|At <math style="vertical-align: -3%">x=4</math>, we find
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|At <math style="vertical-align: -3%">x=4</math>, we find the marginal revenue is
 
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::<math>P'(4)\,\,=\,\,\frac{100}{\sqrt{4}}-(30+6x(4))\,\,=\,\,50-54\,\,=\,\,-4.</math>
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::<math>P'(4)\,\,=\,\,\frac{100}{\sqrt{4}}-(15+6(4))\,\,=\,\,50-39\,\,=\,\,11.</math>
 
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|Thus, it is <u>''not''</u> profitable to produce another item.
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|Thus, it is profitable to produce another item.
 
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::<math>R'(4)\,\,=\,\,50;\qquad P'(4)\,\,=\,\,-4. </math>
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::<math>R'(4)\,\,=\,\,50;\qquad P'(4)\,\,=\,\,11. </math>
 
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|Thus, it is <u>''not''</u> profitable to produce another item.
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|Thus, it is profitable to produce another item.
 
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[[022_Exam_1_Sample_A|'''<u>Return to Sample Exam</u>''']]
 
[[022_Exam_1_Sample_A|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 12:39, 20 April 2015

Find the marginal revenue and marginal profit at , given the demand function

and the cost function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(x)=100+15x+3x^{2}.}

Should the firm produce one more item under these conditions? Justify your answer.

Foundations:  
Recall that the demand function, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)} , relates the price per unit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} to the number of units sold, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .

Moreover, we have several important important functions:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(x)} , the total cost to produce Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} units;
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(x)} , the total revenue (or gross receipts) from producing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} units;
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x)} , the total profit from producing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} units.
In particular, we have the relations
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x)=R(x)-C(x),}
and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(x)=x\cdot p(x).}
Finally, the marginal profit at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} units is defined to be the effective profit of the next unit produced, and is precisely Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P'(x_0)} . Similarly, the marginal revenue or marginal cost would be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R'(x_0)} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C'(x_0)} , respectively.

 Solution:

Step 1:  
Find the Important Functions: We have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(x)\,\,=\,\,x\cdot p(x)\,\,=\,\,x\cdot \frac{200}{\sqrt {x}}\,\,=\,\,200 \sqrt{x}.}
From this,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x)\,\,=\,\,R(x)-C(x)\,\,=\,\,200 \sqrt{x}- \left( 100+15x+3x^{2} \right) .}
Step 2:  
Find the Marginal Revenue and Profit: The equation for marginal revenue is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R'(x)\,\,=\,\,\left(200 \sqrt{x}\right) '\,\,=\,\,200\cdot \frac{1}{2}\cdot\frac{1}{\sqrt{x}}\,\,=\,\,\frac{100}{\sqrt{x}}, }
while the equation for marginal profit is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P'(x)\,\,=\,\,R'(x)-C'(x)\,\,=\,\,\frac{100}{\sqrt{x}}-(15+6x).}
At Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=4} , we find the marginal revenue is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R'(4)\,\,=\,\,\frac{100}{\sqrt{4}}\,\,=\,\,50. }
On the other hand, marginal profit is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P'(4)\,\,=\,\,\frac{100}{\sqrt{4}}-(15+6(4))\,\,=\,\,50-39\,\,=\,\,11.}
Thus, it is profitable to produce another item.
Final Answer:  
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R'(4)\,\,=\,\,50;\qquad P'(4)\,\,=\,\,11. }
Thus, it is profitable to produce another item.


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