022 Exam 1 Sample A, Problem 5

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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.

Recall that the demand function, , relates the price per unit to the number of units sold, .

Moreover, we have several important important functions:

  • , the total cost to produce units;
  • , the total revenue (or gross receipts) from producing units;
  • , the total profit from producing units.
In particular, we have the relations
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 , respectively.


Step 1:  
Find the Important Functions: We have
From this,
Step 2:  
Find the Marginal Revenue and Profit: The equation for marginal revenue is
while the equation for marginal profit is
At , we find the marginal revenue is
On the other hand, marginal profit is
Thus, it is profitable to produce another item.
Final Answer:  
Thus, it is profitable to produce another item.

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