Find the marginal revenue and marginal profit at 
, given the demand function
and the cost function
Should the firm produce one more item under these conditions? Justify
your answer.
| Foundations:
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Recall that the demand function,  , relates the price per unit  to the number of units sold,  .
Moreover, we have several important important functions:
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 , the total cost to produce units;
 , the total revenue (or gross receipts) from producing units;
 , the total profit from producing units.
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| In particular, we have the relations
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| and
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Finally, the marginal profit at  units is defined to be the effective profit of the next unit produced, and is precisely  . Similarly, the marginal revenue or marginal cost would be  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.
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Solution:
| Step 1:
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| Find the Important Functions: We have
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- 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}.}
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| From this,
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- 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) .}
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| Step 2:
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| Find the Marginal Revenue and Profit: The equation for marginal revenue is
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- 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}}, }
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| while the equation for marginal profit is
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- 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}}-(30+6x).}
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| 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
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- 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. }
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| On the other hand, marginal profit is
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- 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}}-(30+6x(4))\,\,=\,\,50-54\,\,=\,\,-4.}
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| Thus, it is not profitable to produce another item.
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| Final Answer:
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- 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)\,\,=\,\,-4. }
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| Thus, it is not profitable to produce another item.
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