Question

The demand function for a firm is; Qd = 122,000 - 500P + 4M +10,000PR where, Qd is quantity
demanded, P is price per unit. M is income, and PR the price of a related good. The estimated the
valueS of M and PR will be Rs 3200 and Rs 4, respectively.
The firm's estimated average variable coSt function IS; AVC = 500 - 0.03Q + 0.000001Q2
a. Find the profit maximizing level of output of the firm and the price to charge.
b. Should the manager continue production or shut down? Explain your answer.
c. Find the level of output at which the average variable cost is at its minimum.

Answer 1

Firms profit maximization point is achieved where :

Marginal Revenue = Marginal Costs 

Qd = 122,000 - 500P + 4M +10,000PR 

Where , PR = 4 & M = 3200

Qd = 122,000 - 500P + 12800 +40,000 

Qd = 174800 - 500P

$P =\frac{(174800 - Qd)}{500}$

Total Revenue $= P\times Q = \frac{174800Q - Q2)}{500}$

AVC = 500 - 0.03Q + 0.000001Q²

Total Variable Cost = Q(AVC) = 500Q - 0.03Q² + 0.000001Q³

$MC = \frac{d (ATC)}{ dQ}= \frac{d(FC)}{dQ} + \frac{d(VC)}{dQ}$

$\\ (Now \space \frac{d(FC)}{dQ}=0$ as fixed cost is constant )

$MC = \frac{d (500Q − 0.03Q^2 + 0.000001Q^3}{dQ} = 500 - 0.06Q + 0.000003Q^2$

$Now , MR =\frac{ d(TR)}{d(Q)}$   (Where ToTal revenue = P*Q)

$MR = \frac{(174800 - 2Q)}{500}$

Putting in equilibrium condition :

$\frac{(174800 - 2Q)}{500}$ = 500 - 0.06Q + 0.000003Q²

28Q - 75200 = 0.0015Q²

0.0015Q² -28Q + 75200 = 0

Applying quadratic formula :

$Q =\frac{28 ± \sqrt{28^2}−4(0.0015)(75200)}{ (0.0015)}= \frac{28±18.24}{0.003}$

Q* = 15,413.33 and 3,253

Putting these in demand Condition we get :

P* = 318.77 (With Q* = 15,413.33 )

P* = 343.094 (With Q * = 3253)

b.)

Manager should continue if :

Profits from shut down < Profit from Continuing   

Profit from shutdown = Zero revenue - Fixed Cost - Variable cost (also 0)

 

-FC < Total Revenue - FC - VC   

Total Revenue > VC  

Let take Q* = 3253 and P* = 343.09 from previous part .

Total Revenue = 3253*343.09 = 1116,071.77

Variable Cost = 500Q* - 0.03Q*² + 0.000001Q*³

= 1626,500 - 317,460 + 34423 = 1343,463

Now ,

VC > Total Revenue 

Hence the firm should shut down .

c.) 

AVC = 500 - 0.03Q + 0.000001Q²

AVC is minimum at the point where : $\frac{d(AVC)}{ d(Q)} = 0$

$-0.03 + 0.000002Q = 0\\ Q =\frac{ 0.03}{0.000002}$

= 15000 is the level of Q where AVC is minimum