Question

A  firm’s long-run average cost curve is estimated by the equation: LAC = 1,000 – 2.3Q + .005Q² . What is the lowest price per unit sold that would prevent the firm from shutting down in the long run?

Answer 1

Assumption: Perfect Competition

Proof

In the long run the firm would operate at the minimum of LAC curve. Thus price should be equal to minimum cost, the same can be seen in the following equation;

LAC = 1,000 – 2.3Q + .005Q²

Long run average cost = Long run total cost/ Quantity produced

Long-run Total Cost(LTC)= 1000Q- 2.3Q²+.005Q³

Differentiating the above equation with respect to Q

We get,

D.LTC/D.Q = 1000-4.6Q+0.015Q²   = 0

Therefore the possible solutions can (See Note)

                        Q = 153+.207i or Q = 153.33-207.39i

D² .LTC/D.Q² = -4.6+0.030Q = 0

                        = 0.030Q= 4.6

                        = Q= 153.33

Since D² .LTC/D.Q² is greater than zero, the function i.e. 1000Q- 2.3Q²+.005Q³ will achieve its minima at Q* = 153.33

Note: Quadratic Equations i.e. ax²+bx+c can solved using the following formula

X = [-b+sqrt (b²-4ac)]/2a