CFA Practice Question

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CFA Practice Question

The demand curve for a monopolist is P = 16 - Q. If the monopolist's marginal cost is $10, what output should the monopolist choose? At the profit-maximizing output, what price should the monopolist charge?

A. $12
B. $13
C. $14
Correct Answer: B

Total Revenue = Q * P = 16Q - Q2, MR = ΔTR/ΔQ = 16 - 2Q = MC => 16 - 2Q = 10 => Q = 3, Price = 16 - 3 = $13

User Contributed Comments 38

User Comment
akanimo this can be solved by simulaenous equations
i) MC = 10
ii) MR = MC (at optimal price) ... MR = 10
iii)MR = P - ( Q * Slope )
iv) P = 16 - Q => Q = 16 - P

Okay lets simplify...
MR = P - ( Q * Slope ) ..where MR = 10, slope = 1
10 = P - Q .... but we can subst Q = 16 - P
10 = P - ( 16 - P )
10 = 2P - 16
26 = 2P
P = 13
kalps Marginal revenue at profit max point
marginal revenue = marginal cost
solve this for Q
Enter Q in demand function and hey ur there
bigzero The problem is confusing. For both Q=3 and Q=4,the profit are the same. You will see at Q=3, MC=MR which means producing one more unit doesn't change the profit.
BADGUY thanks akanimo, you are the best!i was almost going crazy!
sivenkova Why slope=1?
motoloco I think the slope of the MR curve is 2 (1 the the slope of the demand curve)...For mi the answer is 3.5 and 12.5
myzec Slope is 1 as in the notes it states that a Monopoly will maximise profit where it is unit elastic. Since Unit elastic = 1 the slope will be 1
raychow I agree with motoloco, the answer should be Q=3.5 P= 12.5
ljamieson well simple calculus gives:

TR = QP = 16Q - Q^2
TC = integral wrt Q of MC = 10Q + const.
profit = TR - TC = 16Q-Q^2 - 10Q - const.

max over all {Q,P}:
dprofit/dQ = 0 = 6 - 2Q => Q = 3 => P = 13.
MikeB P=(16-Q)
Let's say MR's slope is 1, MR=MC=10, sub for P
10=(16-Q) - (1*Q)

not sure why calculus is need for a simple substitution.
jassosahan Q=2,P=14 or Q=4,P=12 both the cases profit is maximised.Ask me how? remember MR=Mc=10
AT Q=3 profit is $9
Yannicklin MR = P - ( Q * Slope )

I can't find this formulae in the book :( help plzz
Mariecfa Yannicklin!
MR=P-(Q * Slope)
Slope is Change in price divided by change in quantity.
marinachow Why is profit being maximized at Q=2 P=14 or Q=4 P=12, if MC=10, profit/unit at P=12 = 12-10=2, total profit = 2X4=8
at P=13 Q=3, total profit: 3X3=9, wouldn't it be better??
qazwsxedcrfvtgb Here is the determination of the marginal revenue function.

Given demand function: P = 16 - Q
Implying Revenue function:
R = P*Q
= 16*Q - Q^2

The marginal revenue is the derivative of the revenue function with respect to the quantity, such that:
MR = dR/dQ = 16 - 2*Q

Given MC = 10, and we know the firm is maximizing profit when MC = MR, so MR = 10.

Solving the equation:
10 = MR = 16 - 2*Q
Q = 3

Substitute Q=3 into the demand function:
P = 16 - Q
We get:
P = 13
SuperKnight I am confused here. Myzec states that a monopoly maximizes profit when it is unitary elastic, elasticity of 1. This question uses this assumption, the slope having a value of 1. Yet the next question asks at which point does a Monopoly maximize profit, and the answer is when the demand is elastic, one of the choices is unitary elastic, but it is not the correct answer?
SaeedAlam I vote for the site's solution, as it was what I was taught at Uni.
BandB 4 is the correct answer, as 4*12=48 & 3*13=36

My assumption to this answer

- as it is carefully said

in this question, INITIAL Q is 3 instead of 4.
but it didn't say that you reach the profit maxmizing position at the inital Q. so the profit maxmization Q is 4 instead of 3.
Swiki MR=dR/dq
R=P*R (assuming P=16-Q)
MR=dR/dq => 16-2Q

we Know, MR=MC for MONOPOLY
so, 10=16-2Q
Substituting this value to P=16-Q
alallstar BandB - 3*13=39
jppogo the price follows the model y=ax+b so the slope is a=-1 and we take the absolute value:1
poomie83 I haven't performed calculus calcs in 15 years so pardon my ignorance but how do you turn 16Q-Q^2 to 16-2Q?

Is it a matter of square rooting all the numbers?
poomie83 Also a monopolist maximizes profit when D is elastic (see next question) and not when unitary elastic as some of you have suggested
sogah all i hear is a lot of bs haha
jqian I am totaly lost. I guess the key is the slope. I don't know where the 1 from?
dipu617 Too many convincing explanations from too many knowledgeable people!!! Really lost....
chuamj 1 came from the equation (P = 16 - Q). The slope is (-1) if you differiate the equation. 1 is just the abs of it.
dybacis Can someone explain me how is that formula for MR correct if MR=change in TR/change in quantity so at Q=4 MR=((4*12)-(3*13))/(4-3)=9 NOT 10
moneyguy I calculated exactly like Swiki, and thought I was right. Still do.
2014 Thanks qazwsxedcrfvtgb ... life becomes easy ... keep up good work
CHUCKYT MR=P-(slope*Q)
Slope is 1 because the demand curve for the monopolist(given P=16-q) is a unit elastic downward sloping straight line from point p=16 q=0, to point p=0 q=16.
When p=16-q, slope=1 and mc=mr so mr=10
then, 10=(16-Q)-(1*q)
If q=3 then p=13
sgossett86 I was in economics class and the right answer based on my U of Michigan course is Q=3....
Yass0707 I have use same logic as "gazwsxedcrfvtgb" and I get also
but I get Q=2
what do i do wrong?
Yass0707 omg im tired... 16-10=6 not 4 :)
so Q=3 :)
kay136 I give up on this question.
chesschh LOS: Explain, describe, describe explain (not calculate). I dont see why this question should be in the exam
Logaritmus On common sense let's find a profit maximizing quantity:
Our profit is P - 10, and we want to maximize total profit.
Total profit is Quantity * profit from sale = Q*(Price - 10) = Q(16-Q-10)=Q(6-Q) - a quadratic function which have max at Q = 3 so maximizing profit quantity is 3, price must be 13 and profit is 9.
pigletin it's very easy, just solve the first derivative for total revenue, and set it equal to marginal cost, then you have the quantity, and solve the price
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