Micro Economics Questions?

Question

1) A profit maximizing firm with the following total-cost function TC = X3 - 8X2 + 100X +512 operates under perfectly competitive conditions. Find the close-down point of this firm. [In case you have no calculus: MC = 3X2 - 16X + 100 ]

2. A monopoly firm's average variable costs are given by AVC = X2 - 3X + 14, and the market demand it confronts by X = 3 (230 - P)1/2. Find the firm's total fixed costs knowing that its maximum economic profits equal π = $400. [ In case you have no calculus: MC = 3X2 -6X +14 and MR = 230 - (27X2 / 81) ].

3. At its current output level of X = 10, a monopolistically competitive firm has MR = 4, MC = 4, ATC = 6, and P = 8. Is this market in long-run equilibrium? If not, please describe the adjustment process necessary to achieve long-run equilibrium.

Answer 1

1)

For perfect competition MR=P=AR

Short-run close-down point is if price falls below minimum of AVC

AVC=VC/X

TC=FC+VC

VC=TC-FC

FC=512

VC= X³-8X²+100X

AVC=VC/X = X²-8X+100

Minimum of AVC is at (AVC)' = 0

(X²-8X+100)' = 0

2X-8=0

2X=8

X=4

Price at this point will be MR=MC=P

MC=(TC)' = (X³-8X²+100X) ' = 3X²-16X+100

MC(4)= 3*4²-16*4+100 = 84

P=84

X=4; P=84; TR=84*4=336; TC=848 Profit=336-848= -512

So if price falls below P=84 then firm will close-down in short-run.

Long-run close-down point is at minimum of ATC

ATC=TC/Q = X²-8X+100 +512/Q

Minimum of ATC is there  (ATC) ' =0

(ATC) ' =2X-8 - 512/X²

Solving we get X=8

MC(8)= 3*8²-16*8+100 = 164

P=164

X=8; P=164; TR=164*8=1312; TC=1312; Profit=1312-1312=0

So if price falls below P=164 then firm will close-down in long-run.

2)

TR-TC=400

X=3 √(230-P)

P=230-X²/9

TR=P*X= (230-X²/9)*X = 230X - X³/9

MR= (TR)' = (230X - X³/9)' = 230 - X²/3

VC=AVC*X = (X² - 3X +14)*X = X³ - 3X² +14X

MC=(VC)' = (X³ - 3X² +14X) = 3X² - 6X +14

MR=MC

230 - X²/3 = 3X² - 6X +14

Solving we get X=9

TR(9)=230X - X³/9=1989

TC=TR - Profit= 1989-400=1589

VC=TC-FC

FC=TC(9)-VC(9) = 1589 - (X³ - 3X² +14X)=

=1589 - 9³ + 3*9² -14*9= 977

FC=977

3)

Current ATC>MC and MR=MC but P>ATC so it means that new firms will enter market until P=ATC in long-run, thus causing market price to fall.