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Using Arc Formula for Elasticity

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If the price is $15 the company sells 27 units. If price is $12, the company sells 42 units. The company wants to earn a 20% return on sales. ATC=MC=$10. How do I determine the optimal price using the linear approximation method, cost plus pricing and mark-up pricing. How does the arc formula for elasticity factor in to these equations?

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This solution helps with a problem that applies elasticity.

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Suppose at optimal price P, the quantity sold is Q.
Average Total Cost = $10
Therefore, total cost (TC) for all Q units = $10Q
For a 20% profit, the selling price P = 1.2*TC = $12Q

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