Question

can you do question #2

1. Find the inflection point(s) for the function used in the inflation tax application M S = = me-(+)+aY 2. In the model of perfect competition, all firms are price-takers since they treat price as a market-determined constant. Firm Perfcomp's total revenue function is TR(9) = P Q, in which Pequals the output price. Assume that P = 12 and the total cost function TC(Q) = Q - 4.592 + 18Q - 7. (a) Determine the firm's profit function and the level of output at which Firm Perfcomp should produce in order to maximize profits. Confirm that this quantity represents maximum profits for the firm by using the second-order condition. (b) According to microeconomic theory, perfectly competitive firms will maxime profits by producing at the quantity where price equals marginal costa where the slope of the marginal revenue curve is less than that of the mare cost curve. Show that the theory holds in this example. 3. Consider month

Answer 1

(a)

TR = P x Q = 12Q

Profit (Z) = TR - TC

Z = 12Q - Q³ + 4.5Q² - 18Q + 7

Z = - 6Q - Q³ + 4.5Q² + 7

Profit is maximized when dZ/dQ = 0 (first order condition: FOC) and d²Z/dQ² < 0 (second order condition: SOC)

FOC:

dZ = dQ = - 6 - 3Q² + 9Q = 0

3Q² - 9Q + 6 = 0

Q² - 3Q + 2 = 0

Q² - Q - 2Q + 2 = 0

Q x (Q - 1) - 2 x (Q - 1) = 0

(Q - 1) (Q - 2) = 0

Q = 1 and Q = 2.

SOC:

d²Z/dQ² = d/dQ(dZ/dQ) = - 6Q + 9

When Q = 1, d²Z/dQ² = - 6 x 1 + 9 = - 6 + 9 = 3 > 0 (SOC not satisfied)

When Q = 2, d²Z/dQ² = - 6 x 2 + 9 = - 12 + 9 = - 3 < 0 (SOC satisfied)

So, profit maximizing Q = 2.

(b)

MR = dTR/dQ = 12

MC = dTC/dQ = 3Q² - 9Q + 18

Equating,

3Q² - 9Q + 18 = 12

3Q² - 9Q + 6 = 0

Q² - 3Q + 2 = 0

Q² - Q - 2Q + 2 = 0

Q x (Q - 1) - 2 x (Q - 1) = 0

(Q - 1) (Q - 2) = 0

Q = 1 and Q = 2.

So, results are same as in part (a).