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Compute the price elasticity of demand

Question 1:
Demand for a product produced by a firm is given by the expression: P = 60 ? 5Q. Fixed cost is 20. Variable costs of producing Q units are VC (Q) = ? Q2 + 16Q.
(I) Find expressions for total, average and marginal costs. Find expressions for total, average and marginal revenue. Find an expression for profit.
(II) Find an output that maximizes the total revenue. Compute the price elasticity of demand at this output. Explain the economic reasons for the value of the price elasticity of demand found at this level of output. How big is the maximum possible total revenue?
(III) Find an output that maximizes the profit. Compute the elasticity of demand at this output. Check if the demand is elastic or inelastic at this output and explain the economic intuition for the answer found. How big is the maximum possible profit?

Solution Preview

Hello!

(I) Total cost is the sum of average costs and fixed costs. I'll assume here that the "Q2" you wrote in the VC(Q) expression is actually Q squared, which I'll write as Q^2. Furthermore, there is a "?" in the VC(Q) equation. I'll assume that the VC(Q) equation is then VC(Q) = Q^2 + 16Q. I'll explain all the procedures so that, in case it's different, you'll be able to solve it anyway. So:

Total cost = Q^2 + 16Q + 20

Average cost is (Total Cost)/Q:

Avg Cost = Q + 16 + 20/Q

Marginal cost is the derivative of Total cost with respect to Q:

Marginal Cost = 2Q + 16

Now, revenue is price times quantity demanded (P*Q). Since we have that P = 60 - 5Q, then:

Total Revenue = (60 - 5Q)Q = 60Q - 5Q^2

Average revenue is (Total Revenue)/Q:

Avg Revenue = 60 - 5Q

Marginal Revenue is the derivative of total revenue with respect to Q:

Marg Revenue = 60 - 10Q

Finally, profit is total revenue minus total cost:

Profit = 60Q - 5Q^2 - Q^2 - 16Q - 20 = 44Q - 6Q^2 - 20

(II) The output that maximizes total revenue can be found by ...

Solution Summary

Demand for a product is considered.

$2.19