Microecon monopoly/help/check answer?

Question

3.Consider the market for strawberry jam. Suppose there is only one producer and the market demand and supply are given by these eqns: D(p) = 100-2Pd, S(p) = 20 +6Ps, C(q) =q^2/12 -20/6q

Output is given by "y".

(i)What is the revenue function R(y)? is this PxQ where Q is y. so Py? but then P = (y-100)/2 from D(p)? or does it matter which expression I use D(p) or S(p)?

(ii)What is the marginal revenue function MR(y)? this is deriv of R(y)

(iii)The monopolist chooses the price that gives him the maximum profits. Choosing the price automatically determines the output given the demand and supply functions. Equivalently, the monopolist can choose the quantity, and this will give him the price. For analytical purposes we will choose this method since it’s easier. Set up the monopolist’s profit maximization problem.

(iv)What is the FOC? Interpret the FOC in terms of marginal revenue and marginal cost.

(v)Use FOC to solve for eq price &quantity.

(vi)Calculate profit

Answer 1

(i)

TR=P*Q

Q=100-2P

P=50-Q/2

TR=50Q-(Q^2)/2

You should take demand equation in this case since demand is exogenous variable and can not be modified any way (just usual assumption - actually marketing stuff does this job).

(ii)

MR=(TR)'=50-Q

(iii)

Profit maximization condition: MR=MC

MR=50-Q

MC=(TC)'

TC=Q^2 /12 - 10Q/3

MC=Q/6 - 10/3

MC=MR

Q/6 - 10/3=50-Q

7Q/6=50+10/3

7Q/6=(150+10)/3

7Q/6=160/3

Q=160*6/(3*7) = 20*2=45.7

P=50-Q/2 = 50-22.86=27.14

(iv)

FOC - first order condition.

Interpretation (for perfect competition):

Profit=TR-TC

TR=P*Q

Profit=PQ-TC

FOC=>  0=(PQ-TC)'=P-TC'=P-MC

FOC=> P=MC

or (most popular interpretation)

Maximum profit there marginal profit=0 (maximum point on curve)

FOC=> Profit=MAX if MR-MC=0

FOC=> (TR)'=MR

FOC=> (TC)'=MC

FOC=> MR=MC

(v)

Already solved in (iii)

Q/6 - 10/3=50-Q

7Q/6=50+10/3

7Q/6=(150+10)/3

7Q/6=160/3

Q=160*6/(3*7) = 20*2=45.7

P=50-Q/2 = 50-22.86=27.14

(vi)

TR=27.14*45.7=1240.3

TC=Q^2 /12 - 10Q/3 = 2088.5/3- 152.4/3=696.2 - 50.8= 645.4

Profit = TR-TC=1240.3-645.4=594.9

P.S. You are right, goofed sign while derived.