Suppose you own a price-taking firm and your product sells for $5 per unit. Your

Question

Suppose you own a price-taking firm and your product sells for $5 per unit. Your total cost function is given by: C(q) 2q2+q+4. a. What are your firms' variable costs? What are the fixed costs? b. Write down your firm's profit function. c. Write down the first-order condition for a maximum and solve for the optimal q. d. Show that this q maximizes, rather than minimizes, profit. e. Write down the total profit earned by the firm. f. Should you produce q* or should you shut down and produce nothing? Explain your answer.

Explanation / Answer

a. Variable costs are that par of the cost function which are dependent on the level of output.

VC = 2q^2+q

FC is independent of the level of output ana dthe constant term

FC = 4

Optimum output at P=MC, where MC=dTC/dQ

5 = 4Q + 1

4Q = 4

Q=1

AC = C/Q = 2Q + 1 + 4/Q

Profit = (P-AC)*Q

= (5-(2Q + 1 + 4/Q))*Q

= 5Q - 2Q^2 - Q - 4

b. Profit function = -2Q^2 + 4Q - 4

c. dProfit/dQ = -4Q + 4 = 0

4Q = 4

d.Seciond order derivative d(dProfit/dQ^2) = -4, negative value means that the profit is maximum

Q = 1 profits max at Q = 1

Profits = -2*1 + 4*1+4 = 6

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