Question

Given the firm demand function Q = 55 - 0.5P
(where P = Price and Q = rate of output), and the total cost function
TC = 20 + Q + 0.2Q^2
where TC = Total cost, determine
(a) The Total revenue function for the firm.(Hint: To find the total revenue function,solve the demand function for P and then multiply both sides of the equation by Q.)
(b) The marginal revenue and marginal cost functions and find the rate of output for which marginal revenue equals marginal cost.
(c) An equation for profit by subtracting the total cost function form the total revenue function. Find the level of output that maximizes total profit. Compare your answer to that obtained in part (b). Is there any corrrespondence between these answers?

Answer 1

Question #73974, Economics / Microeconomics

SOLUTION: -

$$Q = 55 - 0.5^*P$$ $$\text{So, } P = 110 - 2^*Q$$ $$\text{TC(Total Cost)} = 20 + Q + 0.2^*Q^2$$

a) TR(Total Revenue) = $Q^*(110 - 2^*Q)$

$$= 110^*Q - 2^*Q^2$$

b) MR = $110 - 4^*Q$, MC = $1 + 0.4^*Q$

To find Q at which MR = MC

$110 - 4^*Q = 1 + 0.4^*Q$

$109 = 4.4^*Q$

So, $Q = 24.77$

c) $P(\text{Profit}) = TR - TC$

$$= 109^*Q - 2.2^*Q^2 - 20$$

Now, differentiating w.r.t Q at equating to 0 to find maximum

$\frac{dP}{dQ} = 109 - 4.4^*Q = 0$

$Q = 24.77$

There is a correspondence between part b and c answers because at level of maximum profit marginal cost and marginal revenue are equal.

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