a) What price - quantity combination maximizes your firm's profits? b) Calculate the maximum profits. c) Is demand elastic, inelastic, or unit elastic at the profit-maximizing price-quantity...

Answer to exercise 1:
a) The demand function will be given by: Q=2600-100P+0.2*20000-500*2=
5600-100P.
) The inverse demand function will be: P=(5600-Q)/100
c) Total revenue=P*Q ={(5600-Q)/100}*Q
Marginal revenue: dTR/dQ= (5600-2Q)/100
d)
e)
TR will be maximized when MR =0.
(5600-2Q)/100=0,
Q=2800 units.
Answer to exercise 2:
a) TR= P*Q
P= (20000-Q)/100
TR=Q* (20000-Q)/100.
MR= dTR/dQ solving this we get,
MR= (20000-2Q)/100.
) At the profit maximizing level MR=MC
MC=dTC /dQ= 100+0.04*Q.
MR=MC
Equating the 2 equations.
100+0.04*Q = (20000-2Q)/100.
Solving the above we get,
Q=10000/6= 1667.
c) To find the profit we find total revenue and total cost,
TR= (10000/6)*(20000-1667)/100=$305550
TC= 10000+ (100*1667)+0.02*1667*1667=$232277.8
So, profit will be TR-TC= (305550-176733) =$73272.2
d) Since the firm is earning profit at this level of output, it need not shut
down and can continue its operations.
Answer to exercise 2:
a) TVC will be obtained by integrating MC,
SMC=80-0.1Q+0.0001Q ^2.
TVC=80Q-0.05*Q ^2+0.000033*Q ^3.
AVC= TVC/Q = {80Q-0.05*Q^2+0.000033*Q^3}/Q = 80-
0.05*Q+0.000033*Q ^1/2
) Equating price and MC we get,
75=80-0.1*Q+0.0001*Q ^2.
Solving the above expression we get Q as 54 and 946.
c) AVC at 54 and 946 will be given by,
At 54: TVC= 80-0.05*54+0.000033*54*54
Solving this we get,
AVC=$77.396
At 946: TVC= 80-0.05*946+0.000033*946*946
Solving this we get,
AVC= $62.23.
Since the AVC is lower at 946 level of output, this should be the optimal
level of production out of the two levels of production at which price is
equal to marginal cost.
Answer to exercise 3:
a) TVC will be obtained by integrating the SMC
SMC= 125-0.42*Q+0.0021*(Q ^2).
TVC= 125*Q-0.21*Q ^2+0.0007*Q ^3
AVC= 125-0.21*Q+0.0007*Q ^2.
) AVC will reach its minimum point when AVC =MC
125-0.21*Q+0.0007*Q ^2=125-0.42*Q+0.0021*(Q ^2)
0.21*Q=0.0014*Q ^2
0.21=0.0014*Q
Q=0.21/0.0014=150 units.
At this level of output, AVC=125-0.21*150+0.0007*150*150=
$109.25.
c) The firm will choose to shut down if it is not able to cover its average
cost. So if Price is
Less than AVC, the firm will choose to shut down. Given in the
question, market price is 115 which are greater than AVC at 150 level
of output. Thus the firm will not choose to shut down.
d) Optimal level of output can be obtained by equating price and SMC,
P=SMC
125-0.42*Q+0.0021*(Q ^2) =115
Q can be 172 0r 28 units.
Calculating AVC at the two levels of output,
At 172,
AVC= 125-0.21*172+0.0007*172*172=$109.59
At 28,
AVC=125-0.21*28+0.0007*28*28=$ 119.67.
Since AVC is lowest at 172 level of output, the firm will operate at this
level.
e) Profit at this level of output will be given by
TR-TC
TC= 3500+125*Q-0.21*Q^2+0.0007*Q ^3
TR=115*172=19780
TC=3500+125*172-0.21*172*172+0.0007*172*172*172
=$22349.27
Therefore firm is incu
ing a loss of =22349.27-19780= $2569.27.
Answer to exercise 4:
a) When K=2 and L=3, output will be given by,
Q=min (4, 12) = 4 units.
) Cost function of the firm will be given by,
C= 10*L+30*K
In order to minimize the cost of production of 4 units of output, the
firm will employ only 1 unit of capital and 1 unit of labor.
C= 10*1+30*1=$40
$40 is the minimum cost of producing 4 units of output.
Answer to exercise 5:
a) Lerner index is given by :P-MC/P
P-35/P=0.65
Solving the above expression, we get,
P=$100
) The factor by which firm marks up its price over MC is given by the
Lerner index which is given in the question as 0.65
c) Since the Lerner index s quite high at 0.65, the firm does enjoy
substantial market power.
Answer to exercise 6:
a) MR as function of its price will be given by the expression,
MR=P (1-1/e), where e is the elasticity of demand.
) Profit maximizing output level will be given by
MR=MC
P (1-1/e) =10, where e=-4
Solving the above...